3.15.32 \(\int \frac {(b+2 c x) (d+e x)^{5/2}}{(a+b x+c x^2)^3} \, dx\)

Optimal. Leaf size=398 \[ \frac {5 e \left (-2 c e \left (-d \sqrt {b^2-4 a c}-2 a e+4 b d\right )+b e^2 \left (b-\sqrt {b^2-4 a c}\right )+8 c^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-e \left (b-\sqrt {b^2-4 a c}\right )}}\right )}{4 \sqrt {2} \sqrt {c} \left (b^2-4 a c\right )^{3/2} \sqrt {2 c d-e \left (b-\sqrt {b^2-4 a c}\right )}}-\frac {5 e \left (-2 c e \left (d \sqrt {b^2-4 a c}-2 a e+4 b d\right )+b e^2 \left (\sqrt {b^2-4 a c}+b\right )+8 c^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}\right )}{4 \sqrt {2} \sqrt {c} \left (b^2-4 a c\right )^{3/2} \sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}-\frac {5 e \sqrt {d+e x} (-2 a e+x (2 c d-b e)+b d)}{4 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}-\frac {(d+e x)^{5/2}}{2 \left (a+b x+c x^2\right )^2} \]

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Rubi [A]  time = 1.70, antiderivative size = 398, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {768, 738, 826, 1166, 208} \begin {gather*} \frac {5 e \left (-2 c e \left (-d \sqrt {b^2-4 a c}-2 a e+4 b d\right )+b e^2 \left (b-\sqrt {b^2-4 a c}\right )+8 c^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-e \left (b-\sqrt {b^2-4 a c}\right )}}\right )}{4 \sqrt {2} \sqrt {c} \left (b^2-4 a c\right )^{3/2} \sqrt {2 c d-e \left (b-\sqrt {b^2-4 a c}\right )}}-\frac {5 e \left (-2 c e \left (d \sqrt {b^2-4 a c}-2 a e+4 b d\right )+b e^2 \left (\sqrt {b^2-4 a c}+b\right )+8 c^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}\right )}{4 \sqrt {2} \sqrt {c} \left (b^2-4 a c\right )^{3/2} \sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}-\frac {5 e \sqrt {d+e x} (-2 a e+x (2 c d-b e)+b d)}{4 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}-\frac {(d+e x)^{5/2}}{2 \left (a+b x+c x^2\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((b + 2*c*x)*(d + e*x)^(5/2))/(a + b*x + c*x^2)^3,x]

[Out]

-(d + e*x)^(5/2)/(2*(a + b*x + c*x^2)^2) - (5*e*Sqrt[d + e*x]*(b*d - 2*a*e + (2*c*d - b*e)*x))/(4*(b^2 - 4*a*c
)*(a + b*x + c*x^2)) + (5*e*(8*c^2*d^2 + b*(b - Sqrt[b^2 - 4*a*c])*e^2 - 2*c*e*(4*b*d - Sqrt[b^2 - 4*a*c]*d -
2*a*e))*ArcTanh[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x])/Sqrt[2*c*d - (b - Sqrt[b^2 - 4*a*c])*e]])/(4*Sqrt[2]*Sqrt[c]*(
b^2 - 4*a*c)^(3/2)*Sqrt[2*c*d - (b - Sqrt[b^2 - 4*a*c])*e]) - (5*e*(8*c^2*d^2 + b*(b + Sqrt[b^2 - 4*a*c])*e^2
- 2*c*e*(4*b*d + Sqrt[b^2 - 4*a*c]*d - 2*a*e))*ArcTanh[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x])/Sqrt[2*c*d - (b + Sqrt[
b^2 - 4*a*c])*e]])/(4*Sqrt[2]*Sqrt[c]*(b^2 - 4*a*c)^(3/2)*Sqrt[2*c*d - (b + Sqrt[b^2 - 4*a*c])*e])

Rule 208

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-(a/b), 2]*ArcTanh[x/Rt[-(a/b), 2]])/a, x] /; FreeQ[{a,
b}, x] && NegQ[a/b]

Rule 738

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((d + e*x)^(m - 1)*(
d*b - 2*a*e + (2*c*d - b*e)*x)*(a + b*x + c*x^2)^(p + 1))/((p + 1)*(b^2 - 4*a*c)), x] + Dist[1/((p + 1)*(b^2 -
4*a*c)), Int[(d + e*x)^(m - 2)*Simp[e*(2*a*e*(m - 1) + b*d*(2*p - m + 4)) - 2*c*d^2*(2*p + 3) + e*(b*e - 2*d*
c)*(m + 2*p + 2)*x, x]*(a + b*x + c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] &
& NeQ[c*d^2 - b*d*e + a*e^2, 0] && NeQ[2*c*d - b*e, 0] && LtQ[p, -1] && GtQ[m, 1] && IntQuadraticQ[a, b, c, d,
e, m, p, x]

Rule 768

Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Sim
p[(g*(d + e*x)^m*(a + b*x + c*x^2)^(p + 1))/(2*c*(p + 1)), x] - Dist[(e*g*m)/(2*c*(p + 1)), Int[(d + e*x)^(m -
1)*(a + b*x + c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && EqQ[2*c*f - b*g, 0] && LtQ[p, -1]
&& GtQ[m, 0]

Rule 826

Int[((f_.) + (g_.)*(x_))/(Sqrt[(d_.) + (e_.)*(x_)]*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)), x_Symbol] :> Dist[2,
Subst[Int[(e*f - d*g + g*x^2)/(c*d^2 - b*d*e + a*e^2 - (2*c*d - b*e)*x^2 + c*x^4), x], x, Sqrt[d + e*x]], x] /
; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0]

Rule 1166

Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Di
st[e/2 + (2*c*d - b*e)/(2*q), Int[1/(b/2 - q/2 + c*x^2), x], x] + Dist[e/2 - (2*c*d - b*e)/(2*q), Int[1/(b/2 +
q/2 + c*x^2), x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - a*e^2, 0] && PosQ[b^
2 - 4*a*c]

Rubi steps

\begin {align*} \int \frac {(b+2 c x) (d+e x)^{5/2}}{\left (a+b x+c x^2\right )^3} \, dx &=-\frac {(d+e x)^{5/2}}{2 \left (a+b x+c x^2\right )^2}+\frac {1}{4} (5 e) \int \frac {(d+e x)^{3/2}}{\left (a+b x+c x^2\right )^2} \, dx\\ &=-\frac {(d+e x)^{5/2}}{2 \left (a+b x+c x^2\right )^2}-\frac {5 e \sqrt {d+e x} (b d-2 a e+(2 c d-b e) x)}{4 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}-\frac {(5 e) \int \frac {\frac {1}{2} \left (4 c d^2-3 b d e+2 a e^2\right )+\frac {1}{2} e (2 c d-b e) x}{\sqrt {d+e x} \left (a+b x+c x^2\right )} \, dx}{4 \left (b^2-4 a c\right )}\\ &=-\frac {(d+e x)^{5/2}}{2 \left (a+b x+c x^2\right )^2}-\frac {5 e \sqrt {d+e x} (b d-2 a e+(2 c d-b e) x)}{4 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}-\frac {(5 e) \operatorname {Subst}\left (\int \frac {-\frac {1}{2} d e (2 c d-b e)+\frac {1}{2} e \left (4 c d^2-3 b d e+2 a e^2\right )+\frac {1}{2} e (2 c d-b e) x^2}{c d^2-b d e+a e^2+(-2 c d+b e) x^2+c x^4} \, dx,x,\sqrt {d+e x}\right )}{2 \left (b^2-4 a c\right )}\\ &=-\frac {(d+e x)^{5/2}}{2 \left (a+b x+c x^2\right )^2}-\frac {5 e \sqrt {d+e x} (b d-2 a e+(2 c d-b e) x)}{4 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}-\frac {\left (5 e \left (8 c^2 d^2+b \left (b-\sqrt {b^2-4 a c}\right ) e^2-2 c e \left (4 b d-\sqrt {b^2-4 a c} d-2 a e\right )\right )\right ) \operatorname {Subst}\left (\int \frac {1}{-\frac {1}{2} \sqrt {b^2-4 a c} e+\frac {1}{2} (-2 c d+b e)+c x^2} \, dx,x,\sqrt {d+e x}\right )}{8 \left (b^2-4 a c\right )^{3/2}}+\frac {\left (5 e \left (8 c^2 d^2+b \left (b+\sqrt {b^2-4 a c}\right ) e^2-2 c e \left (4 b d+\sqrt {b^2-4 a c} d-2 a e\right )\right )\right ) \operatorname {Subst}\left (\int \frac {1}{\frac {1}{2} \sqrt {b^2-4 a c} e+\frac {1}{2} (-2 c d+b e)+c x^2} \, dx,x,\sqrt {d+e x}\right )}{8 \left (b^2-4 a c\right )^{3/2}}\\ &=-\frac {(d+e x)^{5/2}}{2 \left (a+b x+c x^2\right )^2}-\frac {5 e \sqrt {d+e x} (b d-2 a e+(2 c d-b e) x)}{4 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}+\frac {5 e \left (8 c^2 d^2+b \left (b-\sqrt {b^2-4 a c}\right ) e^2-2 c e \left (4 b d-\sqrt {b^2-4 a c} d-2 a e\right )\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}}\right )}{4 \sqrt {2} \sqrt {c} \left (b^2-4 a c\right )^{3/2} \sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}}-\frac {5 e \left (8 c^2 d^2+b \left (b+\sqrt {b^2-4 a c}\right ) e^2-2 c e \left (4 b d+\sqrt {b^2-4 a c} d-2 a e\right )\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}}\right )}{4 \sqrt {2} \sqrt {c} \left (b^2-4 a c\right )^{3/2} \sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}}\\ \end {align*}

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Mathematica [B]  time = 6.59, size = 1197, normalized size = 3.01 \begin {gather*} -\frac {\left (-2 a c (2 c d-b e)+b \left (-e b^2+c d b+2 a c e\right )+c (b (2 c d-b e)-2 c (b d-2 a e)) x\right ) (d+e x)^{7/2}}{2 \left (b^2-4 a c\right ) \left (c d^2-b e d+a e^2\right ) \left (c x^2+b x+a\right )^2}-\frac {-\frac {\left (-\frac {1}{2} a c \left (b^2-4 a c\right ) (2 c d-b e) e^2-\frac {1}{2} \left (b^2-4 a c\right ) (5 c d-3 b e) \left (-e b^2+c d b+2 a c e\right ) e+c \left (-\frac {1}{2} c \left (b^2-4 a c\right ) (b d-2 a e) e^2-\frac {1}{2} \left (b^2-4 a c\right ) (5 c d-3 b e) (2 c d-b e) e\right ) x\right ) (d+e x)^{7/2}}{\left (b^2-4 a c\right ) \left (c d^2-b e d+a e^2\right ) \left (c x^2+b x+a\right )}-\frac {\frac {1}{2} \left (b^2-4 a c\right ) e^2 \left (10 c^2 d^2+3 b^2 e^2-2 c e (5 b d+a e)\right ) (d+e x)^{5/2}+\frac {2 \left (\frac {25}{4} c \left (b^2-4 a c\right ) (2 c d-b e) \left (c d^2-e (b d-a e)\right ) (d+e x)^{3/2} e^2+\frac {2 \left (\frac {75}{4} c^2 \left (b^2-4 a c\right ) e^2 \sqrt {d+e x} \left (c d^2-e (b d-a e)\right )^2+\frac {4 \left (\frac {\sqrt {2 c d-b e-\sqrt {b^2-4 a c} e} \left (-\frac {75}{32} \left (b^2-4 a c\right ) e^2 (2 c d-b e) \left (c d^2-e (b d-a e)\right )^2 c^3-\frac {\frac {75}{32} \left (b^2-4 a c\right ) e^2 (2 c d-b e) (b e-2 c d) \left (c d^2-e (b d-a e)\right )^2 c^3+2 \left (\frac {75}{32} c^3 \left (b^2-4 a c\right ) d e^2 (2 c d-b e) \left (c d^2-e (b d-a e)\right )^2-\frac {75}{32} c^3 \left (b^2-4 a c\right ) e^2 \left (4 c d^2-e (3 b d-2 a e)\right ) \left (c d^2-e (b d-a e)\right )^2\right ) c}{\sqrt {b^2-4 a c} e}\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-b e-\sqrt {b^2-4 a c} e}}\right )}{\sqrt {2} \sqrt {c} \left (-2 c d+b e+\sqrt {b^2-4 a c} e\right )}+\frac {\sqrt {2 c d-b e+\sqrt {b^2-4 a c} e} \left (\frac {\frac {75}{32} \left (b^2-4 a c\right ) e^2 (2 c d-b e) (b e-2 c d) \left (c d^2-e (b d-a e)\right )^2 c^3+2 \left (\frac {75}{32} c^3 \left (b^2-4 a c\right ) d e^2 (2 c d-b e) \left (c d^2-e (b d-a e)\right )^2-\frac {75}{32} c^3 \left (b^2-4 a c\right ) e^2 \left (4 c d^2-e (3 b d-2 a e)\right ) \left (c d^2-e (b d-a e)\right )^2\right ) c}{\sqrt {b^2-4 a c} e}-\frac {75}{32} c^3 \left (b^2-4 a c\right ) e^2 (2 c d-b e) \left (c d^2-e (b d-a e)\right )^2\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-b e+\sqrt {b^2-4 a c} e}}\right )}{\sqrt {2} \sqrt {c} \left (-2 c d+b e-\sqrt {b^2-4 a c} e\right )}\right )}{c}\right )}{3 c}\right )}{5 c}}{\left (b^2-4 a c\right ) \left (c d^2-b e d+a e^2\right )}}{2 \left (b^2-4 a c\right ) \left (c d^2-b e d+a e^2\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((b + 2*c*x)*(d + e*x)^(5/2))/(a + b*x + c*x^2)^3,x]

[Out]

-1/2*((d + e*x)^(7/2)*(-2*a*c*(2*c*d - b*e) + b*(b*c*d - b^2*e + 2*a*c*e) + c*(-2*c*(b*d - 2*a*e) + b*(2*c*d -
b*e))*x))/((b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)*(a + b*x + c*x^2)^2) - (-(((d + e*x)^(7/2)*(-1/2*(a*c*(b^2 -
4*a*c)*e^2*(2*c*d - b*e)) - ((b^2 - 4*a*c)*e*(5*c*d - 3*b*e)*(b*c*d - b^2*e + 2*a*c*e))/2 + c*(-1/2*(c*(b^2 -
4*a*c)*e^2*(b*d - 2*a*e)) - ((b^2 - 4*a*c)*e*(5*c*d - 3*b*e)*(2*c*d - b*e))/2)*x))/((b^2 - 4*a*c)*(c*d^2 - b*
d*e + a*e^2)*(a + b*x + c*x^2))) - (((b^2 - 4*a*c)*e^2*(10*c^2*d^2 + 3*b^2*e^2 - 2*c*e*(5*b*d + a*e))*(d + e*x
)^(5/2))/2 + (2*((25*c*(b^2 - 4*a*c)*e^2*(2*c*d - b*e)*(c*d^2 - e*(b*d - a*e))*(d + e*x)^(3/2))/4 + (2*((75*c^
2*(b^2 - 4*a*c)*e^2*(c*d^2 - e*(b*d - a*e))^2*Sqrt[d + e*x])/4 + (4*((Sqrt[2*c*d - b*e - Sqrt[b^2 - 4*a*c]*e]*
((-75*c^3*(b^2 - 4*a*c)*e^2*(2*c*d - b*e)*(c*d^2 - e*(b*d - a*e))^2)/32 - ((75*c^3*(b^2 - 4*a*c)*e^2*(2*c*d -
b*e)*(-2*c*d + b*e)*(c*d^2 - e*(b*d - a*e))^2)/32 + 2*c*((75*c^3*(b^2 - 4*a*c)*d*e^2*(2*c*d - b*e)*(c*d^2 - e*
(b*d - a*e))^2)/32 - (75*c^3*(b^2 - 4*a*c)*e^2*(4*c*d^2 - e*(3*b*d - 2*a*e))*(c*d^2 - e*(b*d - a*e))^2)/32))/(
Sqrt[b^2 - 4*a*c]*e))*ArcTanh[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x])/Sqrt[2*c*d - b*e - Sqrt[b^2 - 4*a*c]*e]])/(Sqrt[
2]*Sqrt[c]*(-2*c*d + b*e + Sqrt[b^2 - 4*a*c]*e)) + (Sqrt[2*c*d - b*e + Sqrt[b^2 - 4*a*c]*e]*((-75*c^3*(b^2 - 4
*a*c)*e^2*(2*c*d - b*e)*(c*d^2 - e*(b*d - a*e))^2)/32 + ((75*c^3*(b^2 - 4*a*c)*e^2*(2*c*d - b*e)*(-2*c*d + b*e
)*(c*d^2 - e*(b*d - a*e))^2)/32 + 2*c*((75*c^3*(b^2 - 4*a*c)*d*e^2*(2*c*d - b*e)*(c*d^2 - e*(b*d - a*e))^2)/32
- (75*c^3*(b^2 - 4*a*c)*e^2*(4*c*d^2 - e*(3*b*d - 2*a*e))*(c*d^2 - e*(b*d - a*e))^2)/32))/(Sqrt[b^2 - 4*a*c]*
e))*ArcTanh[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x])/Sqrt[2*c*d - b*e + Sqrt[b^2 - 4*a*c]*e]])/(Sqrt[2]*Sqrt[c]*(-2*c*d
+ b*e - Sqrt[b^2 - 4*a*c]*e))))/c))/(3*c)))/(5*c))/((b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)))/(2*(b^2 - 4*a*c)*
(c*d^2 - b*d*e + a*e^2))

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IntegrateAlgebraic [C]  time = 177.00, size = 2045, normalized size = 5.14 \begin {gather*} \text {Result too large to show} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[((b + 2*c*x)*(d + e*x)^(5/2))/(a + b*x + c*x^2)^3,x]

[Out]

(e^2*Sqrt[d + e*x]*(10*c^2*d^4 - 20*b*c*d^3*e + 10*b^2*d^2*e^2 + 20*a*c*d^2*e^2 - 20*a*b*d*e^3 + 10*a^2*e^4 -
30*c^2*d^3*(d + e*x) + 45*b*c*d^2*e*(d + e*x) - 15*b^2*d*e^2*(d + e*x) - 30*a*c*d*e^2*(d + e*x) + 15*a*b*e^3*(
d + e*x) + 30*c^2*d^2*(d + e*x)^2 - 30*b*c*d*e*(d + e*x)^2 + 3*b^2*e^2*(d + e*x)^2 + 18*a*c*e^2*(d + e*x)^2 -
10*c^2*d*(d + e*x)^3 + 5*b*c*e*(d + e*x)^3))/(4*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2 - 2*c*d*(d + e*x) + b*e*(
d + e*x) + c*(d + e*x)^2)^2) + ((48*Sqrt[2]*c^4*d^4*e - 96*Sqrt[2]*b*c^3*d^3*e^2 + 12*Sqrt[2]*c^3*Sqrt[b^2 - 4
*a*c]*d^3*e^2 + 68*Sqrt[2]*b^2*c^2*d^2*e^3 + 16*Sqrt[2]*a*c^3*d^2*e^3 - 18*Sqrt[2]*b*c^2*Sqrt[b^2 - 4*a*c]*d^2
*e^3 - 20*Sqrt[2]*b^3*c*d*e^4 - 16*Sqrt[2]*a*b*c^2*d*e^4 + 8*Sqrt[2]*b^2*c*Sqrt[b^2 - 4*a*c]*d*e^4 + 4*Sqrt[2]
*a*c^2*Sqrt[b^2 - 4*a*c]*d*e^4 - Sqrt[2]*b^4*e^5 + 28*Sqrt[2]*a*b^2*c*e^5 - 48*Sqrt[2]*a^2*c^2*e^5 - Sqrt[2]*b
^3*Sqrt[b^2 - 4*a*c]*e^5 - 2*Sqrt[2]*a*b*c*Sqrt[b^2 - 4*a*c]*e^5)*ArcTan[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x])/Sqrt[
-2*c*d + b*e - Sqrt[b^2 - 4*a*c]*e]])/(2*c^(3/2)*(b^2 - 4*a*c)^(3/2)*Sqrt[-2*c*d + b*e - Sqrt[b^2 - 4*a*c]*e]*
(-(c*d^2) + b*d*e - a*e^2)) + ((-48*Sqrt[2]*c^4*d^4*e + 96*Sqrt[2]*b*c^3*d^3*e^2 + 12*Sqrt[2]*c^3*Sqrt[b^2 - 4
*a*c]*d^3*e^2 - 68*Sqrt[2]*b^2*c^2*d^2*e^3 - 16*Sqrt[2]*a*c^3*d^2*e^3 - 18*Sqrt[2]*b*c^2*Sqrt[b^2 - 4*a*c]*d^2
*e^3 + 20*Sqrt[2]*b^3*c*d*e^4 + 16*Sqrt[2]*a*b*c^2*d*e^4 + 8*Sqrt[2]*b^2*c*Sqrt[b^2 - 4*a*c]*d*e^4 + 4*Sqrt[2]
*a*c^2*Sqrt[b^2 - 4*a*c]*d*e^4 + Sqrt[2]*b^4*e^5 - 28*Sqrt[2]*a*b^2*c*e^5 + 48*Sqrt[2]*a^2*c^2*e^5 - Sqrt[2]*b
^3*Sqrt[b^2 - 4*a*c]*e^5 - 2*Sqrt[2]*a*b*c*Sqrt[b^2 - 4*a*c]*e^5)*ArcTan[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x])/Sqrt[
-2*c*d + b*e + Sqrt[b^2 - 4*a*c]*e]])/(2*c^(3/2)*(b^2 - 4*a*c)^(3/2)*Sqrt[-2*c*d + b*e + Sqrt[b^2 - 4*a*c]*e]*
(-(c*d^2) + b*d*e - a*e^2)) + (((152*I)*Sqrt[2]*c^4*d^4*e - (304*I)*Sqrt[2]*b*c^3*d^3*e^2 - 38*Sqrt[2]*c^3*Sqr
t[-b^2 + 4*a*c]*d^3*e^2 + (227*I)*Sqrt[2]*b^2*c^2*d^2*e^3 + (4*I)*Sqrt[2]*a*c^3*d^2*e^3 + 57*Sqrt[2]*b*c^2*Sqr
t[-b^2 + 4*a*c]*d^2*e^3 - (75*I)*Sqrt[2]*b^3*c*d*e^4 - (4*I)*Sqrt[2]*a*b*c^2*d*e^4 - 27*Sqrt[2]*b^2*c*Sqrt[-b^
2 + 4*a*c]*d*e^4 - 6*Sqrt[2]*a*c^2*Sqrt[-b^2 + 4*a*c]*d*e^4 - (4*I)*Sqrt[2]*b^4*e^5 + (107*I)*Sqrt[2]*a*b^2*c*
e^5 - (212*I)*Sqrt[2]*a^2*c^2*e^5 + 4*Sqrt[2]*b^3*Sqrt[-b^2 + 4*a*c]*e^5 + 3*Sqrt[2]*a*b*c*Sqrt[-b^2 + 4*a*c]*
e^5)*ArcTan[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x])/Sqrt[-2*c*d + b*e - I*Sqrt[-b^2 + 4*a*c]*e]])/(8*c^(3/2)*(b^2 - 4*
a*c)*Sqrt[-b^2 + 4*a*c]*Sqrt[-2*c*d + b*e - I*Sqrt[-b^2 + 4*a*c]*e]*(-(c*d^2) + b*d*e - a*e^2)) + (((-152*I)*S
qrt[2]*c^4*d^4*e + (304*I)*Sqrt[2]*b*c^3*d^3*e^2 - 38*Sqrt[2]*c^3*Sqrt[-b^2 + 4*a*c]*d^3*e^2 - (227*I)*Sqrt[2]
*b^2*c^2*d^2*e^3 - (4*I)*Sqrt[2]*a*c^3*d^2*e^3 + 57*Sqrt[2]*b*c^2*Sqrt[-b^2 + 4*a*c]*d^2*e^3 + (75*I)*Sqrt[2]*
b^3*c*d*e^4 + (4*I)*Sqrt[2]*a*b*c^2*d*e^4 - 27*Sqrt[2]*b^2*c*Sqrt[-b^2 + 4*a*c]*d*e^4 - 6*Sqrt[2]*a*c^2*Sqrt[-
b^2 + 4*a*c]*d*e^4 + (4*I)*Sqrt[2]*b^4*e^5 - (107*I)*Sqrt[2]*a*b^2*c*e^5 + (212*I)*Sqrt[2]*a^2*c^2*e^5 + 4*Sqr
t[2]*b^3*Sqrt[-b^2 + 4*a*c]*e^5 + 3*Sqrt[2]*a*b*c*Sqrt[-b^2 + 4*a*c]*e^5)*ArcTan[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x
])/Sqrt[-2*c*d + b*e + I*Sqrt[-b^2 + 4*a*c]*e]])/(8*c^(3/2)*(b^2 - 4*a*c)*Sqrt[-b^2 + 4*a*c]*Sqrt[-2*c*d + b*e
+ I*Sqrt[-b^2 + 4*a*c]*e]*(-(c*d^2) + b*d*e - a*e^2))

________________________________________________________________________________________

fricas [B]  time = 0.53, size = 2773, normalized size = 6.97

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*c*x+b)*(e*x+d)^(5/2)/(c*x^2+b*x+a)^3,x, algorithm="fricas")

[Out]

1/8*(5*sqrt(1/2)*((b^2*c^2 - 4*a*c^3)*x^4 + a^2*b^2 - 4*a^3*c + 2*(b^3*c - 4*a*b*c^2)*x^3 + (b^4 - 2*a*b^2*c -
8*a^2*c^2)*x^2 + 2*(a*b^3 - 4*a^2*b*c)*x)*sqrt((32*c^3*d^3*e^2 - 48*b*c^2*d^2*e^3 + 6*(3*b^2*c + 4*a*c^2)*d*e
^4 - (b^3 + 12*a*b*c)*e^5 + sqrt(e^10/(b^6*c^2 - 12*a*b^4*c^3 + 48*a^2*b^2*c^4 - 64*a^3*c^5))*(b^6*c - 12*a*b^
4*c^2 + 48*a^2*b^2*c^3 - 64*a^3*c^4))/(b^6*c - 12*a*b^4*c^2 + 48*a^2*b^2*c^3 - 64*a^3*c^4))*log(125*sqrt(1/2)*
((b^4 - 8*a*b^2*c + 16*a^2*c^2)*e^6 + 2*sqrt(e^10/(b^6*c^2 - 12*a*b^4*c^3 + 48*a^2*b^2*c^4 - 64*a^3*c^5))*(2*(
b^6*c^2 - 12*a*b^4*c^3 + 48*a^2*b^2*c^4 - 64*a^3*c^5)*d - (b^7*c - 12*a*b^5*c^2 + 48*a^2*b^3*c^3 - 64*a^3*b*c^
4)*e))*sqrt((32*c^3*d^3*e^2 - 48*b*c^2*d^2*e^3 + 6*(3*b^2*c + 4*a*c^2)*d*e^4 - (b^3 + 12*a*b*c)*e^5 + sqrt(e^1
0/(b^6*c^2 - 12*a*b^4*c^3 + 48*a^2*b^2*c^4 - 64*a^3*c^5))*(b^6*c - 12*a*b^4*c^2 + 48*a^2*b^2*c^3 - 64*a^3*c^4)
)/(b^6*c - 12*a*b^4*c^2 + 48*a^2*b^2*c^3 - 64*a^3*c^4)) + 125*(16*c^2*d^2*e^6 - 16*b*c*d*e^7 + (3*b^2 + 4*a*c)
*e^8)*sqrt(e*x + d)) - 5*sqrt(1/2)*((b^2*c^2 - 4*a*c^3)*x^4 + a^2*b^2 - 4*a^3*c + 2*(b^3*c - 4*a*b*c^2)*x^3 +
(b^4 - 2*a*b^2*c - 8*a^2*c^2)*x^2 + 2*(a*b^3 - 4*a^2*b*c)*x)*sqrt((32*c^3*d^3*e^2 - 48*b*c^2*d^2*e^3 + 6*(3*b^
2*c + 4*a*c^2)*d*e^4 - (b^3 + 12*a*b*c)*e^5 + sqrt(e^10/(b^6*c^2 - 12*a*b^4*c^3 + 48*a^2*b^2*c^4 - 64*a^3*c^5)
)*(b^6*c - 12*a*b^4*c^2 + 48*a^2*b^2*c^3 - 64*a^3*c^4))/(b^6*c - 12*a*b^4*c^2 + 48*a^2*b^2*c^3 - 64*a^3*c^4))*
log(-125*sqrt(1/2)*((b^4 - 8*a*b^2*c + 16*a^2*c^2)*e^6 + 2*sqrt(e^10/(b^6*c^2 - 12*a*b^4*c^3 + 48*a^2*b^2*c^4
- 64*a^3*c^5))*(2*(b^6*c^2 - 12*a*b^4*c^3 + 48*a^2*b^2*c^4 - 64*a^3*c^5)*d - (b^7*c - 12*a*b^5*c^2 + 48*a^2*b^
3*c^3 - 64*a^3*b*c^4)*e))*sqrt((32*c^3*d^3*e^2 - 48*b*c^2*d^2*e^3 + 6*(3*b^2*c + 4*a*c^2)*d*e^4 - (b^3 + 12*a*
b*c)*e^5 + sqrt(e^10/(b^6*c^2 - 12*a*b^4*c^3 + 48*a^2*b^2*c^4 - 64*a^3*c^5))*(b^6*c - 12*a*b^4*c^2 + 48*a^2*b^
2*c^3 - 64*a^3*c^4))/(b^6*c - 12*a*b^4*c^2 + 48*a^2*b^2*c^3 - 64*a^3*c^4)) + 125*(16*c^2*d^2*e^6 - 16*b*c*d*e^
7 + (3*b^2 + 4*a*c)*e^8)*sqrt(e*x + d)) + 5*sqrt(1/2)*((b^2*c^2 - 4*a*c^3)*x^4 + a^2*b^2 - 4*a^3*c + 2*(b^3*c
- 4*a*b*c^2)*x^3 + (b^4 - 2*a*b^2*c - 8*a^2*c^2)*x^2 + 2*(a*b^3 - 4*a^2*b*c)*x)*sqrt((32*c^3*d^3*e^2 - 48*b*c^
2*d^2*e^3 + 6*(3*b^2*c + 4*a*c^2)*d*e^4 - (b^3 + 12*a*b*c)*e^5 - sqrt(e^10/(b^6*c^2 - 12*a*b^4*c^3 + 48*a^2*b^
2*c^4 - 64*a^3*c^5))*(b^6*c - 12*a*b^4*c^2 + 48*a^2*b^2*c^3 - 64*a^3*c^4))/(b^6*c - 12*a*b^4*c^2 + 48*a^2*b^2*
c^3 - 64*a^3*c^4))*log(125*sqrt(1/2)*((b^4 - 8*a*b^2*c + 16*a^2*c^2)*e^6 - 2*sqrt(e^10/(b^6*c^2 - 12*a*b^4*c^3
+ 48*a^2*b^2*c^4 - 64*a^3*c^5))*(2*(b^6*c^2 - 12*a*b^4*c^3 + 48*a^2*b^2*c^4 - 64*a^3*c^5)*d - (b^7*c - 12*a*b
^5*c^2 + 48*a^2*b^3*c^3 - 64*a^3*b*c^4)*e))*sqrt((32*c^3*d^3*e^2 - 48*b*c^2*d^2*e^3 + 6*(3*b^2*c + 4*a*c^2)*d*
e^4 - (b^3 + 12*a*b*c)*e^5 - sqrt(e^10/(b^6*c^2 - 12*a*b^4*c^3 + 48*a^2*b^2*c^4 - 64*a^3*c^5))*(b^6*c - 12*a*b
^4*c^2 + 48*a^2*b^2*c^3 - 64*a^3*c^4))/(b^6*c - 12*a*b^4*c^2 + 48*a^2*b^2*c^3 - 64*a^3*c^4)) + 125*(16*c^2*d^2
*e^6 - 16*b*c*d*e^7 + (3*b^2 + 4*a*c)*e^8)*sqrt(e*x + d)) - 5*sqrt(1/2)*((b^2*c^2 - 4*a*c^3)*x^4 + a^2*b^2 - 4
*a^3*c + 2*(b^3*c - 4*a*b*c^2)*x^3 + (b^4 - 2*a*b^2*c - 8*a^2*c^2)*x^2 + 2*(a*b^3 - 4*a^2*b*c)*x)*sqrt((32*c^3
*d^3*e^2 - 48*b*c^2*d^2*e^3 + 6*(3*b^2*c + 4*a*c^2)*d*e^4 - (b^3 + 12*a*b*c)*e^5 - sqrt(e^10/(b^6*c^2 - 12*a*b
^4*c^3 + 48*a^2*b^2*c^4 - 64*a^3*c^5))*(b^6*c - 12*a*b^4*c^2 + 48*a^2*b^2*c^3 - 64*a^3*c^4))/(b^6*c - 12*a*b^4
*c^2 + 48*a^2*b^2*c^3 - 64*a^3*c^4))*log(-125*sqrt(1/2)*((b^4 - 8*a*b^2*c + 16*a^2*c^2)*e^6 - 2*sqrt(e^10/(b^6
*c^2 - 12*a*b^4*c^3 + 48*a^2*b^2*c^4 - 64*a^3*c^5))*(2*(b^6*c^2 - 12*a*b^4*c^3 + 48*a^2*b^2*c^4 - 64*a^3*c^5)*
d - (b^7*c - 12*a*b^5*c^2 + 48*a^2*b^3*c^3 - 64*a^3*b*c^4)*e))*sqrt((32*c^3*d^3*e^2 - 48*b*c^2*d^2*e^3 + 6*(3*
b^2*c + 4*a*c^2)*d*e^4 - (b^3 + 12*a*b*c)*e^5 - sqrt(e^10/(b^6*c^2 - 12*a*b^4*c^3 + 48*a^2*b^2*c^4 - 64*a^3*c^
5))*(b^6*c - 12*a*b^4*c^2 + 48*a^2*b^2*c^3 - 64*a^3*c^4))/(b^6*c - 12*a*b^4*c^2 + 48*a^2*b^2*c^3 - 64*a^3*c^4)
) + 125*(16*c^2*d^2*e^6 - 16*b*c*d*e^7 + (3*b^2 + 4*a*c)*e^8)*sqrt(e*x + d)) - 2*(5*a*b*d*e - 10*a^2*e^2 + 5*(
2*c^2*d*e - b*c*e^2)*x^3 + 2*(b^2 - 4*a*c)*d^2 + 3*(5*b*c*d*e - (b^2 + 6*a*c)*e^2)*x^2 - 3*(5*a*b*e^2 - (3*b^2
- 2*a*c)*d*e)*x)*sqrt(e*x + d))/((b^2*c^2 - 4*a*c^3)*x^4 + a^2*b^2 - 4*a^3*c + 2*(b^3*c - 4*a*b*c^2)*x^3 + (b
^4 - 2*a*b^2*c - 8*a^2*c^2)*x^2 + 2*(a*b^3 - 4*a^2*b*c)*x)

________________________________________________________________________________________

giac [B]  time = 3.66, size = 1425, normalized size = 3.58

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*c*x+b)*(e*x+d)^(5/2)/(c*x^2+b*x+a)^3,x, algorithm="giac")

[Out]

-5/32*(sqrt(-4*c^2*d + 2*(b*c - sqrt(b^2 - 4*a*c)*c)*e)*(2*c*d*e^2 - b*e^3)*(b^2*e - 4*a*c*e)^2 + 4*(sqrt(b^2
- 4*a*c)*c^2*d^2*e^2 - sqrt(b^2 - 4*a*c)*b*c*d*e^3 + sqrt(b^2 - 4*a*c)*a*c*e^4)*sqrt(-4*c^2*d + 2*(b*c - sqrt(
b^2 - 4*a*c)*c)*e)*abs(b^2*e - 4*a*c*e) - (16*(b^2*c^3 - 4*a*c^4)*d^3*e^2 - 24*(b^3*c^2 - 4*a*b*c^3)*d^2*e^3 +
2*(5*b^4*c - 16*a*b^2*c^2 - 16*a^2*c^3)*d*e^4 - (b^5 - 16*a^2*b*c^2)*e^5)*sqrt(-4*c^2*d + 2*(b*c - sqrt(b^2 -
4*a*c)*c)*e))*arctan(2*sqrt(1/2)*sqrt(x*e + d)/sqrt(-(2*b^2*c*d - 8*a*c^2*d - b^3*e + 4*a*b*c*e + sqrt((2*b^2
*c*d - 8*a*c^2*d - b^3*e + 4*a*b*c*e)^2 - 4*(b^2*c*d^2 - 4*a*c^2*d^2 - b^3*d*e + 4*a*b*c*d*e + a*b^2*e^2 - 4*a
^2*c*e^2)*(b^2*c - 4*a*c^2)))/(b^2*c - 4*a*c^2)))/(((b^2*c^2 - 4*a*c^3)*sqrt(b^2 - 4*a*c)*d^2 - (b^3*c - 4*a*b
*c^2)*sqrt(b^2 - 4*a*c)*d*e + (a*b^2*c - 4*a^2*c^2)*sqrt(b^2 - 4*a*c)*e^2)*abs(b^2*e - 4*a*c*e)*abs(c)) + 5/32
*(sqrt(-4*c^2*d + 2*(b*c + sqrt(b^2 - 4*a*c)*c)*e)*(2*c*d*e^2 - b*e^3)*(b^2*e - 4*a*c*e)^2 - 4*(sqrt(b^2 - 4*a
*c)*c^2*d^2*e^2 - sqrt(b^2 - 4*a*c)*b*c*d*e^3 + sqrt(b^2 - 4*a*c)*a*c*e^4)*sqrt(-4*c^2*d + 2*(b*c + sqrt(b^2 -
4*a*c)*c)*e)*abs(b^2*e - 4*a*c*e) - (16*(b^2*c^3 - 4*a*c^4)*d^3*e^2 - 24*(b^3*c^2 - 4*a*b*c^3)*d^2*e^3 + 2*(5
*b^4*c - 16*a*b^2*c^2 - 16*a^2*c^3)*d*e^4 - (b^5 - 16*a^2*b*c^2)*e^5)*sqrt(-4*c^2*d + 2*(b*c + sqrt(b^2 - 4*a*
c)*c)*e))*arctan(2*sqrt(1/2)*sqrt(x*e + d)/sqrt(-(2*b^2*c*d - 8*a*c^2*d - b^3*e + 4*a*b*c*e - sqrt((2*b^2*c*d
- 8*a*c^2*d - b^3*e + 4*a*b*c*e)^2 - 4*(b^2*c*d^2 - 4*a*c^2*d^2 - b^3*d*e + 4*a*b*c*d*e + a*b^2*e^2 - 4*a^2*c*
e^2)*(b^2*c - 4*a*c^2)))/(b^2*c - 4*a*c^2)))/(((b^2*c^2 - 4*a*c^3)*sqrt(b^2 - 4*a*c)*d^2 - (b^3*c - 4*a*b*c^2)
*sqrt(b^2 - 4*a*c)*d*e + (a*b^2*c - 4*a^2*c^2)*sqrt(b^2 - 4*a*c)*e^2)*abs(b^2*e - 4*a*c*e)*abs(c)) - 1/4*(10*(
x*e + d)^(7/2)*c^2*d*e^2 - 30*(x*e + d)^(5/2)*c^2*d^2*e^2 + 30*(x*e + d)^(3/2)*c^2*d^3*e^2 - 10*sqrt(x*e + d)*
c^2*d^4*e^2 - 5*(x*e + d)^(7/2)*b*c*e^3 + 30*(x*e + d)^(5/2)*b*c*d*e^3 - 45*(x*e + d)^(3/2)*b*c*d^2*e^3 + 20*s
qrt(x*e + d)*b*c*d^3*e^3 - 3*(x*e + d)^(5/2)*b^2*e^4 - 18*(x*e + d)^(5/2)*a*c*e^4 + 15*(x*e + d)^(3/2)*b^2*d*e
^4 + 30*(x*e + d)^(3/2)*a*c*d*e^4 - 10*sqrt(x*e + d)*b^2*d^2*e^4 - 20*sqrt(x*e + d)*a*c*d^2*e^4 - 15*(x*e + d)
^(3/2)*a*b*e^5 + 20*sqrt(x*e + d)*a*b*d*e^5 - 10*sqrt(x*e + d)*a^2*e^6)/(((x*e + d)^2*c - 2*(x*e + d)*c*d + c*
d^2 + (x*e + d)*b*e - b*d*e + a*e^2)^2*(b^2 - 4*a*c))

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maple [B]  time = 0.15, size = 2126, normalized size = 5.34

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*c*x+b)*(e*x+d)^(5/2)/(c*x^2+b*x+a)^3,x)

[Out]

5*e^3/(4*a*c-b^2)*c/(-(4*a*c-b^2)*e^2)^(1/2)*2^(1/2)/((b*e-2*c*d+(-(4*a*c-b^2)*e^2)^(1/2))*c)^(1/2)*arctan((e*
x+d)^(1/2)*2^(1/2)/((b*e-2*c*d+(-(4*a*c-b^2)*e^2)^(1/2))*c)^(1/2)*c)*b*d+5*e^3/(4*a*c-b^2)*c/(-(4*a*c-b^2)*e^2
)^(1/2)*2^(1/2)/((-b*e+2*c*d+(-(4*a*c-b^2)*e^2)^(1/2))*c)^(1/2)*arctanh((e*x+d)^(1/2)*2^(1/2)/((-b*e+2*c*d+(-(
4*a*c-b^2)*e^2)^(1/2))*c)^(1/2)*c)*b*d-5/2*e^4/(4*a*c-b^2)*c/(-(4*a*c-b^2)*e^2)^(1/2)*2^(1/2)/((b*e-2*c*d+(-(4
*a*c-b^2)*e^2)^(1/2))*c)^(1/2)*arctan((e*x+d)^(1/2)*2^(1/2)/((b*e-2*c*d+(-(4*a*c-b^2)*e^2)^(1/2))*c)^(1/2)*c)*
a-5*e^2/(4*a*c-b^2)*c^2/(-(4*a*c-b^2)*e^2)^(1/2)*2^(1/2)/((b*e-2*c*d+(-(4*a*c-b^2)*e^2)^(1/2))*c)^(1/2)*arctan
((e*x+d)^(1/2)*2^(1/2)/((b*e-2*c*d+(-(4*a*c-b^2)*e^2)^(1/2))*c)^(1/2)*c)*d^2-5/2*e^4/(4*a*c-b^2)*c/(-(4*a*c-b^
2)*e^2)^(1/2)*2^(1/2)/((-b*e+2*c*d+(-(4*a*c-b^2)*e^2)^(1/2))*c)^(1/2)*arctanh((e*x+d)^(1/2)*2^(1/2)/((-b*e+2*c
*d+(-(4*a*c-b^2)*e^2)^(1/2))*c)^(1/2)*c)*a-5*e^2/(4*a*c-b^2)*c^2/(-(4*a*c-b^2)*e^2)^(1/2)*2^(1/2)/((-b*e+2*c*d
+(-(4*a*c-b^2)*e^2)^(1/2))*c)^(1/2)*arctanh((e*x+d)^(1/2)*2^(1/2)/((-b*e+2*c*d+(-(4*a*c-b^2)*e^2)^(1/2))*c)^(1
/2)*c)*d^2-5/8*e^3/(4*a*c-b^2)*2^(1/2)/((b*e-2*c*d+(-(4*a*c-b^2)*e^2)^(1/2))*c)^(1/2)*arctan((e*x+d)^(1/2)*2^(
1/2)/((b*e-2*c*d+(-(4*a*c-b^2)*e^2)^(1/2))*c)^(1/2)*c)*b+5/8*e^3/(4*a*c-b^2)*2^(1/2)/((-b*e+2*c*d+(-(4*a*c-b^2
)*e^2)^(1/2))*c)^(1/2)*arctanh((e*x+d)^(1/2)*2^(1/2)/((-b*e+2*c*d+(-(4*a*c-b^2)*e^2)^(1/2))*c)^(1/2)*c)*b-5/4*
e^3/(c*e^2*x^2+b*e^2*x+a*e^2)^2*c/(4*a*c-b^2)*(e*x+d)^(7/2)*b+5/2*e^2/(c*e^2*x^2+b*e^2*x+a*e^2)^2*c^2/(4*a*c-b
^2)*(e*x+d)^(7/2)*d-15/2*e^2/(c*e^2*x^2+b*e^2*x+a*e^2)^2/(4*a*c-b^2)*(e*x+d)^(5/2)*c^2*d^2-15/4*e^5/(c*e^2*x^2
+b*e^2*x+a*e^2)^2/(4*a*c-b^2)*(e*x+d)^(3/2)*a*b+15/2*e^2/(c*e^2*x^2+b*e^2*x+a*e^2)^2/(4*a*c-b^2)*(e*x+d)^(3/2)
*c^2*d^3-5/2*e^2/(c*e^2*x^2+b*e^2*x+a*e^2)^2/(4*a*c-b^2)*(e*x+d)^(1/2)*c^2*d^4-9/2*e^4/(c*e^2*x^2+b*e^2*x+a*e^
2)^2/(4*a*c-b^2)*(e*x+d)^(5/2)*a*c+15/4*e^4/(c*e^2*x^2+b*e^2*x+a*e^2)^2/(4*a*c-b^2)*(e*x+d)^(3/2)*b^2*d-5/2*e^
4/(c*e^2*x^2+b*e^2*x+a*e^2)^2/(4*a*c-b^2)*(e*x+d)^(1/2)*b^2*d^2-5/2*e^6/(c*e^2*x^2+b*e^2*x+a*e^2)^2/(4*a*c-b^2
)*(e*x+d)^(1/2)*a^2-3/4*e^4/(c*e^2*x^2+b*e^2*x+a*e^2)^2/(4*a*c-b^2)*(e*x+d)^(5/2)*b^2-45/4*e^3/(c*e^2*x^2+b*e^
2*x+a*e^2)^2/(4*a*c-b^2)*(e*x+d)^(3/2)*b*c*d^2-5/8*e^4/(4*a*c-b^2)/(-(4*a*c-b^2)*e^2)^(1/2)*2^(1/2)/((-b*e+2*c
*d+(-(4*a*c-b^2)*e^2)^(1/2))*c)^(1/2)*arctanh((e*x+d)^(1/2)*2^(1/2)/((-b*e+2*c*d+(-(4*a*c-b^2)*e^2)^(1/2))*c)^
(1/2)*c)*b^2+5*e^3/(c*e^2*x^2+b*e^2*x+a*e^2)^2/(4*a*c-b^2)*(e*x+d)^(1/2)*b*c*d^3+5*e^5/(c*e^2*x^2+b*e^2*x+a*e^
2)^2/(4*a*c-b^2)*(e*x+d)^(1/2)*a*b*d+15/2*e^3/(c*e^2*x^2+b*e^2*x+a*e^2)^2/(4*a*c-b^2)*(e*x+d)^(5/2)*b*c*d-5/4*
e^2/(4*a*c-b^2)*c*2^(1/2)/((-b*e+2*c*d+(-(4*a*c-b^2)*e^2)^(1/2))*c)^(1/2)*arctanh((e*x+d)^(1/2)*2^(1/2)/((-b*e
+2*c*d+(-(4*a*c-b^2)*e^2)^(1/2))*c)^(1/2)*c)*d+5/4*e^2/(4*a*c-b^2)*c*2^(1/2)/((b*e-2*c*d+(-(4*a*c-b^2)*e^2)^(1
/2))*c)^(1/2)*arctan((e*x+d)^(1/2)*2^(1/2)/((b*e-2*c*d+(-(4*a*c-b^2)*e^2)^(1/2))*c)^(1/2)*c)*d-5/8*e^4/(4*a*c-
b^2)/(-(4*a*c-b^2)*e^2)^(1/2)*2^(1/2)/((b*e-2*c*d+(-(4*a*c-b^2)*e^2)^(1/2))*c)^(1/2)*arctan((e*x+d)^(1/2)*2^(1
/2)/((b*e-2*c*d+(-(4*a*c-b^2)*e^2)^(1/2))*c)^(1/2)*c)*b^2+15/2*e^4/(c*e^2*x^2+b*e^2*x+a*e^2)^2/(4*a*c-b^2)*(e*
x+d)^(3/2)*a*c*d-5*e^4/(c*e^2*x^2+b*e^2*x+a*e^2)^2/(4*a*c-b^2)*(e*x+d)^(1/2)*a*c*d^2

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (2 \, c x + b\right )} {\left (e x + d\right )}^{\frac {5}{2}}}{{\left (c x^{2} + b x + a\right )}^{3}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*c*x+b)*(e*x+d)^(5/2)/(c*x^2+b*x+a)^3,x, algorithm="maxima")

[Out]

integrate((2*c*x + b)*(e*x + d)^(5/2)/(c*x^2 + b*x + a)^3, x)

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mupad [B]  time = 7.65, size = 12750, normalized size = 32.04

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((b + 2*c*x)*(d + e*x)^(5/2))/(a + b*x + c*x^2)^3,x)

[Out]

- atan(((((5*(8192*a^4*c^5*e^6 - 128*a*b^6*c^2*e^6 + 128*b^7*c^2*d*e^5 + 1536*a^2*b^4*c^3*e^6 - 6144*a^3*b^2*c
^4*e^6 + 8192*a^3*c^6*d^2*e^4 - 128*b^6*c^3*d^2*e^4 - 6144*a^2*b^2*c^5*d^2*e^4 - 1536*a*b^5*c^3*d*e^5 - 8192*a
^3*b*c^5*d*e^5 + 1536*a*b^4*c^4*d^2*e^4 + 6144*a^2*b^3*c^4*d*e^5))/(64*(b^6 - 64*a^3*c^3 + 48*a^2*b^2*c^2 - 12
*a*b^4*c)) - ((d + e*x)^(1/2)*(-(25*(b^9*e^5 + e^5*(-(4*a*c - b^2)^9)^(1/2) - 768*a^4*b*c^4*e^5 + 1536*a^4*c^5
*d*e^4 - 96*a^2*b^5*c^2*e^5 + 512*a^3*b^3*c^3*e^5 + 2048*a^3*c^6*d^3*e^2 - 32*b^6*c^3*d^3*e^2 + 48*b^7*c^2*d^2
*e^3 - 18*b^8*c*d*e^4 - 1536*a^2*b^2*c^5*d^3*e^2 + 2304*a^2*b^3*c^4*d^2*e^3 + 192*a*b^6*c^2*d*e^4 + 384*a*b^4*
c^4*d^3*e^2 - 576*a*b^5*c^3*d^2*e^3 - 576*a^2*b^4*c^3*d*e^4 - 3072*a^3*b*c^5*d^2*e^3))/(128*(b^12*c + 4096*a^6
*c^7 - 24*a*b^10*c^2 + 240*a^2*b^8*c^3 - 1280*a^3*b^6*c^4 + 3840*a^4*b^4*c^5 - 6144*a^5*b^2*c^6)))^(1/2)*(64*b
^7*c^2*e^3 - 768*a*b^5*c^3*e^3 - 4096*a^3*b*c^5*e^3 + 8192*a^3*c^6*d*e^2 - 128*b^6*c^3*d*e^2 + 3072*a^2*b^3*c^
4*e^3 + 1536*a*b^4*c^4*d*e^2 - 6144*a^2*b^2*c^5*d*e^2))/(8*(b^4 + 16*a^2*c^2 - 8*a*b^2*c)))*(-(25*(b^9*e^5 + e
^5*(-(4*a*c - b^2)^9)^(1/2) - 768*a^4*b*c^4*e^5 + 1536*a^4*c^5*d*e^4 - 96*a^2*b^5*c^2*e^5 + 512*a^3*b^3*c^3*e^
5 + 2048*a^3*c^6*d^3*e^2 - 32*b^6*c^3*d^3*e^2 + 48*b^7*c^2*d^2*e^3 - 18*b^8*c*d*e^4 - 1536*a^2*b^2*c^5*d^3*e^2
+ 2304*a^2*b^3*c^4*d^2*e^3 + 192*a*b^6*c^2*d*e^4 + 384*a*b^4*c^4*d^3*e^2 - 576*a*b^5*c^3*d^2*e^3 - 576*a^2*b^
4*c^3*d*e^4 - 3072*a^3*b*c^5*d^2*e^3))/(128*(b^12*c + 4096*a^6*c^7 - 24*a*b^10*c^2 + 240*a^2*b^8*c^3 - 1280*a^
3*b^6*c^4 + 3840*a^4*b^4*c^5 - 6144*a^5*b^2*c^6)))^(1/2) - ((d + e*x)^(1/2)*(25*b^4*c*e^8 + 200*a^2*c^3*e^8 +
800*c^5*d^4*e^4 + 50*a*b^2*c^2*e^8 + 600*a*c^4*d^2*e^6 - 1600*b*c^4*d^3*e^5 - 250*b^3*c^2*d*e^7 + 1050*b^2*c^3
*d^2*e^6 - 600*a*b*c^3*d*e^7))/(8*(b^4 + 16*a^2*c^2 - 8*a*b^2*c)))*(-(25*(b^9*e^5 + e^5*(-(4*a*c - b^2)^9)^(1/
2) - 768*a^4*b*c^4*e^5 + 1536*a^4*c^5*d*e^4 - 96*a^2*b^5*c^2*e^5 + 512*a^3*b^3*c^3*e^5 + 2048*a^3*c^6*d^3*e^2
- 32*b^6*c^3*d^3*e^2 + 48*b^7*c^2*d^2*e^3 - 18*b^8*c*d*e^4 - 1536*a^2*b^2*c^5*d^3*e^2 + 2304*a^2*b^3*c^4*d^2*e
^3 + 192*a*b^6*c^2*d*e^4 + 384*a*b^4*c^4*d^3*e^2 - 576*a*b^5*c^3*d^2*e^3 - 576*a^2*b^4*c^3*d*e^4 - 3072*a^3*b*
c^5*d^2*e^3))/(128*(b^12*c + 4096*a^6*c^7 - 24*a*b^10*c^2 + 240*a^2*b^8*c^3 - 1280*a^3*b^6*c^4 + 3840*a^4*b^4*
c^5 - 6144*a^5*b^2*c^6)))^(1/2)*1i - (((5*(8192*a^4*c^5*e^6 - 128*a*b^6*c^2*e^6 + 128*b^7*c^2*d*e^5 + 1536*a^2
*b^4*c^3*e^6 - 6144*a^3*b^2*c^4*e^6 + 8192*a^3*c^6*d^2*e^4 - 128*b^6*c^3*d^2*e^4 - 6144*a^2*b^2*c^5*d^2*e^4 -
1536*a*b^5*c^3*d*e^5 - 8192*a^3*b*c^5*d*e^5 + 1536*a*b^4*c^4*d^2*e^4 + 6144*a^2*b^3*c^4*d*e^5))/(64*(b^6 - 64*
a^3*c^3 + 48*a^2*b^2*c^2 - 12*a*b^4*c)) + ((d + e*x)^(1/2)*(-(25*(b^9*e^5 + e^5*(-(4*a*c - b^2)^9)^(1/2) - 768
*a^4*b*c^4*e^5 + 1536*a^4*c^5*d*e^4 - 96*a^2*b^5*c^2*e^5 + 512*a^3*b^3*c^3*e^5 + 2048*a^3*c^6*d^3*e^2 - 32*b^6
*c^3*d^3*e^2 + 48*b^7*c^2*d^2*e^3 - 18*b^8*c*d*e^4 - 1536*a^2*b^2*c^5*d^3*e^2 + 2304*a^2*b^3*c^4*d^2*e^3 + 192
*a*b^6*c^2*d*e^4 + 384*a*b^4*c^4*d^3*e^2 - 576*a*b^5*c^3*d^2*e^3 - 576*a^2*b^4*c^3*d*e^4 - 3072*a^3*b*c^5*d^2*
e^3))/(128*(b^12*c + 4096*a^6*c^7 - 24*a*b^10*c^2 + 240*a^2*b^8*c^3 - 1280*a^3*b^6*c^4 + 3840*a^4*b^4*c^5 - 61
44*a^5*b^2*c^6)))^(1/2)*(64*b^7*c^2*e^3 - 768*a*b^5*c^3*e^3 - 4096*a^3*b*c^5*e^3 + 8192*a^3*c^6*d*e^2 - 128*b^
6*c^3*d*e^2 + 3072*a^2*b^3*c^4*e^3 + 1536*a*b^4*c^4*d*e^2 - 6144*a^2*b^2*c^5*d*e^2))/(8*(b^4 + 16*a^2*c^2 - 8*
a*b^2*c)))*(-(25*(b^9*e^5 + e^5*(-(4*a*c - b^2)^9)^(1/2) - 768*a^4*b*c^4*e^5 + 1536*a^4*c^5*d*e^4 - 96*a^2*b^5
*c^2*e^5 + 512*a^3*b^3*c^3*e^5 + 2048*a^3*c^6*d^3*e^2 - 32*b^6*c^3*d^3*e^2 + 48*b^7*c^2*d^2*e^3 - 18*b^8*c*d*e
^4 - 1536*a^2*b^2*c^5*d^3*e^2 + 2304*a^2*b^3*c^4*d^2*e^3 + 192*a*b^6*c^2*d*e^4 + 384*a*b^4*c^4*d^3*e^2 - 576*a
*b^5*c^3*d^2*e^3 - 576*a^2*b^4*c^3*d*e^4 - 3072*a^3*b*c^5*d^2*e^3))/(128*(b^12*c + 4096*a^6*c^7 - 24*a*b^10*c^
2 + 240*a^2*b^8*c^3 - 1280*a^3*b^6*c^4 + 3840*a^4*b^4*c^5 - 6144*a^5*b^2*c^6)))^(1/2) + ((d + e*x)^(1/2)*(25*b
^4*c*e^8 + 200*a^2*c^3*e^8 + 800*c^5*d^4*e^4 + 50*a*b^2*c^2*e^8 + 600*a*c^4*d^2*e^6 - 1600*b*c^4*d^3*e^5 - 250
*b^3*c^2*d*e^7 + 1050*b^2*c^3*d^2*e^6 - 600*a*b*c^3*d*e^7))/(8*(b^4 + 16*a^2*c^2 - 8*a*b^2*c)))*(-(25*(b^9*e^5
+ e^5*(-(4*a*c - b^2)^9)^(1/2) - 768*a^4*b*c^4*e^5 + 1536*a^4*c^5*d*e^4 - 96*a^2*b^5*c^2*e^5 + 512*a^3*b^3*c^
3*e^5 + 2048*a^3*c^6*d^3*e^2 - 32*b^6*c^3*d^3*e^2 + 48*b^7*c^2*d^2*e^3 - 18*b^8*c*d*e^4 - 1536*a^2*b^2*c^5*d^3
*e^2 + 2304*a^2*b^3*c^4*d^2*e^3 + 192*a*b^6*c^2*d*e^4 + 384*a*b^4*c^4*d^3*e^2 - 576*a*b^5*c^3*d^2*e^3 - 576*a^
2*b^4*c^3*d*e^4 - 3072*a^3*b*c^5*d^2*e^3))/(128*(b^12*c + 4096*a^6*c^7 - 24*a*b^10*c^2 + 240*a^2*b^8*c^3 - 128
0*a^3*b^6*c^4 + 3840*a^4*b^4*c^5 - 6144*a^5*b^2*c^6)))^(1/2)*1i)/((5*(800*c^5*d^5*e^6 - 100*a^2*b*c^2*e^11 + 1
000*a*c^4*d^3*e^8 + 200*a^2*c^3*d*e^10 - 2000*b*c^4*d^4*e^7 + 1750*b^2*c^3*d^3*e^8 - 625*b^3*c^2*d^2*e^9 - 75*
a*b^3*c*e^11 + 75*b^4*c*d*e^10 - 1500*a*b*c^3*d^2*e^9 + 650*a*b^2*c^2*d*e^10))/(32*(b^6 - 64*a^3*c^3 + 48*a^2*
b^2*c^2 - 12*a*b^4*c)) + (((5*(8192*a^4*c^5*e^6 - 128*a*b^6*c^2*e^6 + 128*b^7*c^2*d*e^5 + 1536*a^2*b^4*c^3*e^6
- 6144*a^3*b^2*c^4*e^6 + 8192*a^3*c^6*d^2*e^4 - 128*b^6*c^3*d^2*e^4 - 6144*a^2*b^2*c^5*d^2*e^4 - 1536*a*b^5*c
^3*d*e^5 - 8192*a^3*b*c^5*d*e^5 + 1536*a*b^4*c^4*d^2*e^4 + 6144*a^2*b^3*c^4*d*e^5))/(64*(b^6 - 64*a^3*c^3 + 48
*a^2*b^2*c^2 - 12*a*b^4*c)) - ((d + e*x)^(1/2)*(-(25*(b^9*e^5 + e^5*(-(4*a*c - b^2)^9)^(1/2) - 768*a^4*b*c^4*e
^5 + 1536*a^4*c^5*d*e^4 - 96*a^2*b^5*c^2*e^5 + 512*a^3*b^3*c^3*e^5 + 2048*a^3*c^6*d^3*e^2 - 32*b^6*c^3*d^3*e^2
+ 48*b^7*c^2*d^2*e^3 - 18*b^8*c*d*e^4 - 1536*a^2*b^2*c^5*d^3*e^2 + 2304*a^2*b^3*c^4*d^2*e^3 + 192*a*b^6*c^2*d
*e^4 + 384*a*b^4*c^4*d^3*e^2 - 576*a*b^5*c^3*d^2*e^3 - 576*a^2*b^4*c^3*d*e^4 - 3072*a^3*b*c^5*d^2*e^3))/(128*(
b^12*c + 4096*a^6*c^7 - 24*a*b^10*c^2 + 240*a^2*b^8*c^3 - 1280*a^3*b^6*c^4 + 3840*a^4*b^4*c^5 - 6144*a^5*b^2*c
^6)))^(1/2)*(64*b^7*c^2*e^3 - 768*a*b^5*c^3*e^3 - 4096*a^3*b*c^5*e^3 + 8192*a^3*c^6*d*e^2 - 128*b^6*c^3*d*e^2
+ 3072*a^2*b^3*c^4*e^3 + 1536*a*b^4*c^4*d*e^2 - 6144*a^2*b^2*c^5*d*e^2))/(8*(b^4 + 16*a^2*c^2 - 8*a*b^2*c)))*(
-(25*(b^9*e^5 + e^5*(-(4*a*c - b^2)^9)^(1/2) - 768*a^4*b*c^4*e^5 + 1536*a^4*c^5*d*e^4 - 96*a^2*b^5*c^2*e^5 + 5
12*a^3*b^3*c^3*e^5 + 2048*a^3*c^6*d^3*e^2 - 32*b^6*c^3*d^3*e^2 + 48*b^7*c^2*d^2*e^3 - 18*b^8*c*d*e^4 - 1536*a^
2*b^2*c^5*d^3*e^2 + 2304*a^2*b^3*c^4*d^2*e^3 + 192*a*b^6*c^2*d*e^4 + 384*a*b^4*c^4*d^3*e^2 - 576*a*b^5*c^3*d^2
*e^3 - 576*a^2*b^4*c^3*d*e^4 - 3072*a^3*b*c^5*d^2*e^3))/(128*(b^12*c + 4096*a^6*c^7 - 24*a*b^10*c^2 + 240*a^2*
b^8*c^3 - 1280*a^3*b^6*c^4 + 3840*a^4*b^4*c^5 - 6144*a^5*b^2*c^6)))^(1/2) - ((d + e*x)^(1/2)*(25*b^4*c*e^8 + 2
00*a^2*c^3*e^8 + 800*c^5*d^4*e^4 + 50*a*b^2*c^2*e^8 + 600*a*c^4*d^2*e^6 - 1600*b*c^4*d^3*e^5 - 250*b^3*c^2*d*e
^7 + 1050*b^2*c^3*d^2*e^6 - 600*a*b*c^3*d*e^7))/(8*(b^4 + 16*a^2*c^2 - 8*a*b^2*c)))*(-(25*(b^9*e^5 + e^5*(-(4*
a*c - b^2)^9)^(1/2) - 768*a^4*b*c^4*e^5 + 1536*a^4*c^5*d*e^4 - 96*a^2*b^5*c^2*e^5 + 512*a^3*b^3*c^3*e^5 + 2048
*a^3*c^6*d^3*e^2 - 32*b^6*c^3*d^3*e^2 + 48*b^7*c^2*d^2*e^3 - 18*b^8*c*d*e^4 - 1536*a^2*b^2*c^5*d^3*e^2 + 2304*
a^2*b^3*c^4*d^2*e^3 + 192*a*b^6*c^2*d*e^4 + 384*a*b^4*c^4*d^3*e^2 - 576*a*b^5*c^3*d^2*e^3 - 576*a^2*b^4*c^3*d*
e^4 - 3072*a^3*b*c^5*d^2*e^3))/(128*(b^12*c + 4096*a^6*c^7 - 24*a*b^10*c^2 + 240*a^2*b^8*c^3 - 1280*a^3*b^6*c^
4 + 3840*a^4*b^4*c^5 - 6144*a^5*b^2*c^6)))^(1/2) + (((5*(8192*a^4*c^5*e^6 - 128*a*b^6*c^2*e^6 + 128*b^7*c^2*d*
e^5 + 1536*a^2*b^4*c^3*e^6 - 6144*a^3*b^2*c^4*e^6 + 8192*a^3*c^6*d^2*e^4 - 128*b^6*c^3*d^2*e^4 - 6144*a^2*b^2*
c^5*d^2*e^4 - 1536*a*b^5*c^3*d*e^5 - 8192*a^3*b*c^5*d*e^5 + 1536*a*b^4*c^4*d^2*e^4 + 6144*a^2*b^3*c^4*d*e^5))/
(64*(b^6 - 64*a^3*c^3 + 48*a^2*b^2*c^2 - 12*a*b^4*c)) + ((d + e*x)^(1/2)*(-(25*(b^9*e^5 + e^5*(-(4*a*c - b^2)^
9)^(1/2) - 768*a^4*b*c^4*e^5 + 1536*a^4*c^5*d*e^4 - 96*a^2*b^5*c^2*e^5 + 512*a^3*b^3*c^3*e^5 + 2048*a^3*c^6*d^
3*e^2 - 32*b^6*c^3*d^3*e^2 + 48*b^7*c^2*d^2*e^3 - 18*b^8*c*d*e^4 - 1536*a^2*b^2*c^5*d^3*e^2 + 2304*a^2*b^3*c^4
*d^2*e^3 + 192*a*b^6*c^2*d*e^4 + 384*a*b^4*c^4*d^3*e^2 - 576*a*b^5*c^3*d^2*e^3 - 576*a^2*b^4*c^3*d*e^4 - 3072*
a^3*b*c^5*d^2*e^3))/(128*(b^12*c + 4096*a^6*c^7 - 24*a*b^10*c^2 + 240*a^2*b^8*c^3 - 1280*a^3*b^6*c^4 + 3840*a^
4*b^4*c^5 - 6144*a^5*b^2*c^6)))^(1/2)*(64*b^7*c^2*e^3 - 768*a*b^5*c^3*e^3 - 4096*a^3*b*c^5*e^3 + 8192*a^3*c^6*
d*e^2 - 128*b^6*c^3*d*e^2 + 3072*a^2*b^3*c^4*e^3 + 1536*a*b^4*c^4*d*e^2 - 6144*a^2*b^2*c^5*d*e^2))/(8*(b^4 + 1
6*a^2*c^2 - 8*a*b^2*c)))*(-(25*(b^9*e^5 + e^5*(-(4*a*c - b^2)^9)^(1/2) - 768*a^4*b*c^4*e^5 + 1536*a^4*c^5*d*e^
4 - 96*a^2*b^5*c^2*e^5 + 512*a^3*b^3*c^3*e^5 + 2048*a^3*c^6*d^3*e^2 - 32*b^6*c^3*d^3*e^2 + 48*b^7*c^2*d^2*e^3
- 18*b^8*c*d*e^4 - 1536*a^2*b^2*c^5*d^3*e^2 + 2304*a^2*b^3*c^4*d^2*e^3 + 192*a*b^6*c^2*d*e^4 + 384*a*b^4*c^4*d
^3*e^2 - 576*a*b^5*c^3*d^2*e^3 - 576*a^2*b^4*c^3*d*e^4 - 3072*a^3*b*c^5*d^2*e^3))/(128*(b^12*c + 4096*a^6*c^7
- 24*a*b^10*c^2 + 240*a^2*b^8*c^3 - 1280*a^3*b^6*c^4 + 3840*a^4*b^4*c^5 - 6144*a^5*b^2*c^6)))^(1/2) + ((d + e*
x)^(1/2)*(25*b^4*c*e^8 + 200*a^2*c^3*e^8 + 800*c^5*d^4*e^4 + 50*a*b^2*c^2*e^8 + 600*a*c^4*d^2*e^6 - 1600*b*c^4
*d^3*e^5 - 250*b^3*c^2*d*e^7 + 1050*b^2*c^3*d^2*e^6 - 600*a*b*c^3*d*e^7))/(8*(b^4 + 16*a^2*c^2 - 8*a*b^2*c)))*
(-(25*(b^9*e^5 + e^5*(-(4*a*c - b^2)^9)^(1/2) - 768*a^4*b*c^4*e^5 + 1536*a^4*c^5*d*e^4 - 96*a^2*b^5*c^2*e^5 +
512*a^3*b^3*c^3*e^5 + 2048*a^3*c^6*d^3*e^2 - 32*b^6*c^3*d^3*e^2 + 48*b^7*c^2*d^2*e^3 - 18*b^8*c*d*e^4 - 1536*a
^2*b^2*c^5*d^3*e^2 + 2304*a^2*b^3*c^4*d^2*e^3 + 192*a*b^6*c^2*d*e^4 + 384*a*b^4*c^4*d^3*e^2 - 576*a*b^5*c^3*d^
2*e^3 - 576*a^2*b^4*c^3*d*e^4 - 3072*a^3*b*c^5*d^2*e^3))/(128*(b^12*c + 4096*a^6*c^7 - 24*a*b^10*c^2 + 240*a^2
*b^8*c^3 - 1280*a^3*b^6*c^4 + 3840*a^4*b^4*c^5 - 6144*a^5*b^2*c^6)))^(1/2)))*(-(25*(b^9*e^5 + e^5*(-(4*a*c - b
^2)^9)^(1/2) - 768*a^4*b*c^4*e^5 + 1536*a^4*c^5*d*e^4 - 96*a^2*b^5*c^2*e^5 + 512*a^3*b^3*c^3*e^5 + 2048*a^3*c^
6*d^3*e^2 - 32*b^6*c^3*d^3*e^2 + 48*b^7*c^2*d^2*e^3 - 18*b^8*c*d*e^4 - 1536*a^2*b^2*c^5*d^3*e^2 + 2304*a^2*b^3
*c^4*d^2*e^3 + 192*a*b^6*c^2*d*e^4 + 384*a*b^4*c^4*d^3*e^2 - 576*a*b^5*c^3*d^2*e^3 - 576*a^2*b^4*c^3*d*e^4 - 3
072*a^3*b*c^5*d^2*e^3))/(128*(b^12*c + 4096*a^6*c^7 - 24*a*b^10*c^2 + 240*a^2*b^8*c^3 - 1280*a^3*b^6*c^4 + 384
0*a^4*b^4*c^5 - 6144*a^5*b^2*c^6)))^(1/2)*2i - atan(((((5*(8192*a^4*c^5*e^6 - 128*a*b^6*c^2*e^6 + 128*b^7*c^2*
d*e^5 + 1536*a^2*b^4*c^3*e^6 - 6144*a^3*b^2*c^4*e^6 + 8192*a^3*c^6*d^2*e^4 - 128*b^6*c^3*d^2*e^4 - 6144*a^2*b^
2*c^5*d^2*e^4 - 1536*a*b^5*c^3*d*e^5 - 8192*a^3*b*c^5*d*e^5 + 1536*a*b^4*c^4*d^2*e^4 + 6144*a^2*b^3*c^4*d*e^5)
)/(64*(b^6 - 64*a^3*c^3 + 48*a^2*b^2*c^2 - 12*a*b^4*c)) - ((d + e*x)^(1/2)*((25*(e^5*(-(4*a*c - b^2)^9)^(1/2)
- b^9*e^5 + 768*a^4*b*c^4*e^5 - 1536*a^4*c^5*d*e^4 + 96*a^2*b^5*c^2*e^5 - 512*a^3*b^3*c^3*e^5 - 2048*a^3*c^6*d
^3*e^2 + 32*b^6*c^3*d^3*e^2 - 48*b^7*c^2*d^2*e^3 + 18*b^8*c*d*e^4 + 1536*a^2*b^2*c^5*d^3*e^2 - 2304*a^2*b^3*c^
4*d^2*e^3 - 192*a*b^6*c^2*d*e^4 - 384*a*b^4*c^4*d^3*e^2 + 576*a*b^5*c^3*d^2*e^3 + 576*a^2*b^4*c^3*d*e^4 + 3072
*a^3*b*c^5*d^2*e^3))/(128*(b^12*c + 4096*a^6*c^7 - 24*a*b^10*c^2 + 240*a^2*b^8*c^3 - 1280*a^3*b^6*c^4 + 3840*a
^4*b^4*c^5 - 6144*a^5*b^2*c^6)))^(1/2)*(64*b^7*c^2*e^3 - 768*a*b^5*c^3*e^3 - 4096*a^3*b*c^5*e^3 + 8192*a^3*c^6
*d*e^2 - 128*b^6*c^3*d*e^2 + 3072*a^2*b^3*c^4*e^3 + 1536*a*b^4*c^4*d*e^2 - 6144*a^2*b^2*c^5*d*e^2))/(8*(b^4 +
16*a^2*c^2 - 8*a*b^2*c)))*((25*(e^5*(-(4*a*c - b^2)^9)^(1/2) - b^9*e^5 + 768*a^4*b*c^4*e^5 - 1536*a^4*c^5*d*e^
4 + 96*a^2*b^5*c^2*e^5 - 512*a^3*b^3*c^3*e^5 - 2048*a^3*c^6*d^3*e^2 + 32*b^6*c^3*d^3*e^2 - 48*b^7*c^2*d^2*e^3
+ 18*b^8*c*d*e^4 + 1536*a^2*b^2*c^5*d^3*e^2 - 2304*a^2*b^3*c^4*d^2*e^3 - 192*a*b^6*c^2*d*e^4 - 384*a*b^4*c^4*d
^3*e^2 + 576*a*b^5*c^3*d^2*e^3 + 576*a^2*b^4*c^3*d*e^4 + 3072*a^3*b*c^5*d^2*e^3))/(128*(b^12*c + 4096*a^6*c^7
- 24*a*b^10*c^2 + 240*a^2*b^8*c^3 - 1280*a^3*b^6*c^4 + 3840*a^4*b^4*c^5 - 6144*a^5*b^2*c^6)))^(1/2) - ((d + e*
x)^(1/2)*(25*b^4*c*e^8 + 200*a^2*c^3*e^8 + 800*c^5*d^4*e^4 + 50*a*b^2*c^2*e^8 + 600*a*c^4*d^2*e^6 - 1600*b*c^4
*d^3*e^5 - 250*b^3*c^2*d*e^7 + 1050*b^2*c^3*d^2*e^6 - 600*a*b*c^3*d*e^7))/(8*(b^4 + 16*a^2*c^2 - 8*a*b^2*c)))*
((25*(e^5*(-(4*a*c - b^2)^9)^(1/2) - b^9*e^5 + 768*a^4*b*c^4*e^5 - 1536*a^4*c^5*d*e^4 + 96*a^2*b^5*c^2*e^5 - 5
12*a^3*b^3*c^3*e^5 - 2048*a^3*c^6*d^3*e^2 + 32*b^6*c^3*d^3*e^2 - 48*b^7*c^2*d^2*e^3 + 18*b^8*c*d*e^4 + 1536*a^
2*b^2*c^5*d^3*e^2 - 2304*a^2*b^3*c^4*d^2*e^3 - 192*a*b^6*c^2*d*e^4 - 384*a*b^4*c^4*d^3*e^2 + 576*a*b^5*c^3*d^2
*e^3 + 576*a^2*b^4*c^3*d*e^4 + 3072*a^3*b*c^5*d^2*e^3))/(128*(b^12*c + 4096*a^6*c^7 - 24*a*b^10*c^2 + 240*a^2*
b^8*c^3 - 1280*a^3*b^6*c^4 + 3840*a^4*b^4*c^5 - 6144*a^5*b^2*c^6)))^(1/2)*1i - (((5*(8192*a^4*c^5*e^6 - 128*a*
b^6*c^2*e^6 + 128*b^7*c^2*d*e^5 + 1536*a^2*b^4*c^3*e^6 - 6144*a^3*b^2*c^4*e^6 + 8192*a^3*c^6*d^2*e^4 - 128*b^6
*c^3*d^2*e^4 - 6144*a^2*b^2*c^5*d^2*e^4 - 1536*a*b^5*c^3*d*e^5 - 8192*a^3*b*c^5*d*e^5 + 1536*a*b^4*c^4*d^2*e^4
+ 6144*a^2*b^3*c^4*d*e^5))/(64*(b^6 - 64*a^3*c^3 + 48*a^2*b^2*c^2 - 12*a*b^4*c)) + ((d + e*x)^(1/2)*((25*(e^5
*(-(4*a*c - b^2)^9)^(1/2) - b^9*e^5 + 768*a^4*b*c^4*e^5 - 1536*a^4*c^5*d*e^4 + 96*a^2*b^5*c^2*e^5 - 512*a^3*b^
3*c^3*e^5 - 2048*a^3*c^6*d^3*e^2 + 32*b^6*c^3*d^3*e^2 - 48*b^7*c^2*d^2*e^3 + 18*b^8*c*d*e^4 + 1536*a^2*b^2*c^5
*d^3*e^2 - 2304*a^2*b^3*c^4*d^2*e^3 - 192*a*b^6*c^2*d*e^4 - 384*a*b^4*c^4*d^3*e^2 + 576*a*b^5*c^3*d^2*e^3 + 57
6*a^2*b^4*c^3*d*e^4 + 3072*a^3*b*c^5*d^2*e^3))/(128*(b^12*c + 4096*a^6*c^7 - 24*a*b^10*c^2 + 240*a^2*b^8*c^3 -
1280*a^3*b^6*c^4 + 3840*a^4*b^4*c^5 - 6144*a^5*b^2*c^6)))^(1/2)*(64*b^7*c^2*e^3 - 768*a*b^5*c^3*e^3 - 4096*a^
3*b*c^5*e^3 + 8192*a^3*c^6*d*e^2 - 128*b^6*c^3*d*e^2 + 3072*a^2*b^3*c^4*e^3 + 1536*a*b^4*c^4*d*e^2 - 6144*a^2*
b^2*c^5*d*e^2))/(8*(b^4 + 16*a^2*c^2 - 8*a*b^2*c)))*((25*(e^5*(-(4*a*c - b^2)^9)^(1/2) - b^9*e^5 + 768*a^4*b*c
^4*e^5 - 1536*a^4*c^5*d*e^4 + 96*a^2*b^5*c^2*e^5 - 512*a^3*b^3*c^3*e^5 - 2048*a^3*c^6*d^3*e^2 + 32*b^6*c^3*d^3
*e^2 - 48*b^7*c^2*d^2*e^3 + 18*b^8*c*d*e^4 + 1536*a^2*b^2*c^5*d^3*e^2 - 2304*a^2*b^3*c^4*d^2*e^3 - 192*a*b^6*c
^2*d*e^4 - 384*a*b^4*c^4*d^3*e^2 + 576*a*b^5*c^3*d^2*e^3 + 576*a^2*b^4*c^3*d*e^4 + 3072*a^3*b*c^5*d^2*e^3))/(1
28*(b^12*c + 4096*a^6*c^7 - 24*a*b^10*c^2 + 240*a^2*b^8*c^3 - 1280*a^3*b^6*c^4 + 3840*a^4*b^4*c^5 - 6144*a^5*b
^2*c^6)))^(1/2) + ((d + e*x)^(1/2)*(25*b^4*c*e^8 + 200*a^2*c^3*e^8 + 800*c^5*d^4*e^4 + 50*a*b^2*c^2*e^8 + 600*
a*c^4*d^2*e^6 - 1600*b*c^4*d^3*e^5 - 250*b^3*c^2*d*e^7 + 1050*b^2*c^3*d^2*e^6 - 600*a*b*c^3*d*e^7))/(8*(b^4 +
16*a^2*c^2 - 8*a*b^2*c)))*((25*(e^5*(-(4*a*c - b^2)^9)^(1/2) - b^9*e^5 + 768*a^4*b*c^4*e^5 - 1536*a^4*c^5*d*e^
4 + 96*a^2*b^5*c^2*e^5 - 512*a^3*b^3*c^3*e^5 - 2048*a^3*c^6*d^3*e^2 + 32*b^6*c^3*d^3*e^2 - 48*b^7*c^2*d^2*e^3
+ 18*b^8*c*d*e^4 + 1536*a^2*b^2*c^5*d^3*e^2 - 2304*a^2*b^3*c^4*d^2*e^3 - 192*a*b^6*c^2*d*e^4 - 384*a*b^4*c^4*d
^3*e^2 + 576*a*b^5*c^3*d^2*e^3 + 576*a^2*b^4*c^3*d*e^4 + 3072*a^3*b*c^5*d^2*e^3))/(128*(b^12*c + 4096*a^6*c^7
- 24*a*b^10*c^2 + 240*a^2*b^8*c^3 - 1280*a^3*b^6*c^4 + 3840*a^4*b^4*c^5 - 6144*a^5*b^2*c^6)))^(1/2)*1i)/((5*(8
00*c^5*d^5*e^6 - 100*a^2*b*c^2*e^11 + 1000*a*c^4*d^3*e^8 + 200*a^2*c^3*d*e^10 - 2000*b*c^4*d^4*e^7 + 1750*b^2*
c^3*d^3*e^8 - 625*b^3*c^2*d^2*e^9 - 75*a*b^3*c*e^11 + 75*b^4*c*d*e^10 - 1500*a*b*c^3*d^2*e^9 + 650*a*b^2*c^2*d
*e^10))/(32*(b^6 - 64*a^3*c^3 + 48*a^2*b^2*c^2 - 12*a*b^4*c)) + (((5*(8192*a^4*c^5*e^6 - 128*a*b^6*c^2*e^6 + 1
28*b^7*c^2*d*e^5 + 1536*a^2*b^4*c^3*e^6 - 6144*a^3*b^2*c^4*e^6 + 8192*a^3*c^6*d^2*e^4 - 128*b^6*c^3*d^2*e^4 -
6144*a^2*b^2*c^5*d^2*e^4 - 1536*a*b^5*c^3*d*e^5 - 8192*a^3*b*c^5*d*e^5 + 1536*a*b^4*c^4*d^2*e^4 + 6144*a^2*b^3
*c^4*d*e^5))/(64*(b^6 - 64*a^3*c^3 + 48*a^2*b^2*c^2 - 12*a*b^4*c)) - ((d + e*x)^(1/2)*((25*(e^5*(-(4*a*c - b^2
)^9)^(1/2) - b^9*e^5 + 768*a^4*b*c^4*e^5 - 1536*a^4*c^5*d*e^4 + 96*a^2*b^5*c^2*e^5 - 512*a^3*b^3*c^3*e^5 - 204
8*a^3*c^6*d^3*e^2 + 32*b^6*c^3*d^3*e^2 - 48*b^7*c^2*d^2*e^3 + 18*b^8*c*d*e^4 + 1536*a^2*b^2*c^5*d^3*e^2 - 2304
*a^2*b^3*c^4*d^2*e^3 - 192*a*b^6*c^2*d*e^4 - 384*a*b^4*c^4*d^3*e^2 + 576*a*b^5*c^3*d^2*e^3 + 576*a^2*b^4*c^3*d
*e^4 + 3072*a^3*b*c^5*d^2*e^3))/(128*(b^12*c + 4096*a^6*c^7 - 24*a*b^10*c^2 + 240*a^2*b^8*c^3 - 1280*a^3*b^6*c
^4 + 3840*a^4*b^4*c^5 - 6144*a^5*b^2*c^6)))^(1/2)*(64*b^7*c^2*e^3 - 768*a*b^5*c^3*e^3 - 4096*a^3*b*c^5*e^3 + 8
192*a^3*c^6*d*e^2 - 128*b^6*c^3*d*e^2 + 3072*a^2*b^3*c^4*e^3 + 1536*a*b^4*c^4*d*e^2 - 6144*a^2*b^2*c^5*d*e^2))
/(8*(b^4 + 16*a^2*c^2 - 8*a*b^2*c)))*((25*(e^5*(-(4*a*c - b^2)^9)^(1/2) - b^9*e^5 + 768*a^4*b*c^4*e^5 - 1536*a
^4*c^5*d*e^4 + 96*a^2*b^5*c^2*e^5 - 512*a^3*b^3*c^3*e^5 - 2048*a^3*c^6*d^3*e^2 + 32*b^6*c^3*d^3*e^2 - 48*b^7*c
^2*d^2*e^3 + 18*b^8*c*d*e^4 + 1536*a^2*b^2*c^5*d^3*e^2 - 2304*a^2*b^3*c^4*d^2*e^3 - 192*a*b^6*c^2*d*e^4 - 384*
a*b^4*c^4*d^3*e^2 + 576*a*b^5*c^3*d^2*e^3 + 576*a^2*b^4*c^3*d*e^4 + 3072*a^3*b*c^5*d^2*e^3))/(128*(b^12*c + 40
96*a^6*c^7 - 24*a*b^10*c^2 + 240*a^2*b^8*c^3 - 1280*a^3*b^6*c^4 + 3840*a^4*b^4*c^5 - 6144*a^5*b^2*c^6)))^(1/2)
- ((d + e*x)^(1/2)*(25*b^4*c*e^8 + 200*a^2*c^3*e^8 + 800*c^5*d^4*e^4 + 50*a*b^2*c^2*e^8 + 600*a*c^4*d^2*e^6 -
1600*b*c^4*d^3*e^5 - 250*b^3*c^2*d*e^7 + 1050*b^2*c^3*d^2*e^6 - 600*a*b*c^3*d*e^7))/(8*(b^4 + 16*a^2*c^2 - 8*
a*b^2*c)))*((25*(e^5*(-(4*a*c - b^2)^9)^(1/2) - b^9*e^5 + 768*a^4*b*c^4*e^5 - 1536*a^4*c^5*d*e^4 + 96*a^2*b^5*
c^2*e^5 - 512*a^3*b^3*c^3*e^5 - 2048*a^3*c^6*d^3*e^2 + 32*b^6*c^3*d^3*e^2 - 48*b^7*c^2*d^2*e^3 + 18*b^8*c*d*e^
4 + 1536*a^2*b^2*c^5*d^3*e^2 - 2304*a^2*b^3*c^4*d^2*e^3 - 192*a*b^6*c^2*d*e^4 - 384*a*b^4*c^4*d^3*e^2 + 576*a*
b^5*c^3*d^2*e^3 + 576*a^2*b^4*c^3*d*e^4 + 3072*a^3*b*c^5*d^2*e^3))/(128*(b^12*c + 4096*a^6*c^7 - 24*a*b^10*c^2
+ 240*a^2*b^8*c^3 - 1280*a^3*b^6*c^4 + 3840*a^4*b^4*c^5 - 6144*a^5*b^2*c^6)))^(1/2) + (((5*(8192*a^4*c^5*e^6
- 128*a*b^6*c^2*e^6 + 128*b^7*c^2*d*e^5 + 1536*a^2*b^4*c^3*e^6 - 6144*a^3*b^2*c^4*e^6 + 8192*a^3*c^6*d^2*e^4 -
128*b^6*c^3*d^2*e^4 - 6144*a^2*b^2*c^5*d^2*e^4 - 1536*a*b^5*c^3*d*e^5 - 8192*a^3*b*c^5*d*e^5 + 1536*a*b^4*c^4
*d^2*e^4 + 6144*a^2*b^3*c^4*d*e^5))/(64*(b^6 - 64*a^3*c^3 + 48*a^2*b^2*c^2 - 12*a*b^4*c)) + ((d + e*x)^(1/2)*(
(25*(e^5*(-(4*a*c - b^2)^9)^(1/2) - b^9*e^5 + 768*a^4*b*c^4*e^5 - 1536*a^4*c^5*d*e^4 + 96*a^2*b^5*c^2*e^5 - 51
2*a^3*b^3*c^3*e^5 - 2048*a^3*c^6*d^3*e^2 + 32*b^6*c^3*d^3*e^2 - 48*b^7*c^2*d^2*e^3 + 18*b^8*c*d*e^4 + 1536*a^2
*b^2*c^5*d^3*e^2 - 2304*a^2*b^3*c^4*d^2*e^3 - 192*a*b^6*c^2*d*e^4 - 384*a*b^4*c^4*d^3*e^2 + 576*a*b^5*c^3*d^2*
e^3 + 576*a^2*b^4*c^3*d*e^4 + 3072*a^3*b*c^5*d^2*e^3))/(128*(b^12*c + 4096*a^6*c^7 - 24*a*b^10*c^2 + 240*a^2*b
^8*c^3 - 1280*a^3*b^6*c^4 + 3840*a^4*b^4*c^5 - 6144*a^5*b^2*c^6)))^(1/2)*(64*b^7*c^2*e^3 - 768*a*b^5*c^3*e^3 -
4096*a^3*b*c^5*e^3 + 8192*a^3*c^6*d*e^2 - 128*b^6*c^3*d*e^2 + 3072*a^2*b^3*c^4*e^3 + 1536*a*b^4*c^4*d*e^2 - 6
144*a^2*b^2*c^5*d*e^2))/(8*(b^4 + 16*a^2*c^2 - 8*a*b^2*c)))*((25*(e^5*(-(4*a*c - b^2)^9)^(1/2) - b^9*e^5 + 768
*a^4*b*c^4*e^5 - 1536*a^4*c^5*d*e^4 + 96*a^2*b^5*c^2*e^5 - 512*a^3*b^3*c^3*e^5 - 2048*a^3*c^6*d^3*e^2 + 32*b^6
*c^3*d^3*e^2 - 48*b^7*c^2*d^2*e^3 + 18*b^8*c*d*e^4 + 1536*a^2*b^2*c^5*d^3*e^2 - 2304*a^2*b^3*c^4*d^2*e^3 - 192
*a*b^6*c^2*d*e^4 - 384*a*b^4*c^4*d^3*e^2 + 576*a*b^5*c^3*d^2*e^3 + 576*a^2*b^4*c^3*d*e^4 + 3072*a^3*b*c^5*d^2*
e^3))/(128*(b^12*c + 4096*a^6*c^7 - 24*a*b^10*c^2 + 240*a^2*b^8*c^3 - 1280*a^3*b^6*c^4 + 3840*a^4*b^4*c^5 - 61
44*a^5*b^2*c^6)))^(1/2) + ((d + e*x)^(1/2)*(25*b^4*c*e^8 + 200*a^2*c^3*e^8 + 800*c^5*d^4*e^4 + 50*a*b^2*c^2*e^
8 + 600*a*c^4*d^2*e^6 - 1600*b*c^4*d^3*e^5 - 250*b^3*c^2*d*e^7 + 1050*b^2*c^3*d^2*e^6 - 600*a*b*c^3*d*e^7))/(8
*(b^4 + 16*a^2*c^2 - 8*a*b^2*c)))*((25*(e^5*(-(4*a*c - b^2)^9)^(1/2) - b^9*e^5 + 768*a^4*b*c^4*e^5 - 1536*a^4*
c^5*d*e^4 + 96*a^2*b^5*c^2*e^5 - 512*a^3*b^3*c^3*e^5 - 2048*a^3*c^6*d^3*e^2 + 32*b^6*c^3*d^3*e^2 - 48*b^7*c^2*
d^2*e^3 + 18*b^8*c*d*e^4 + 1536*a^2*b^2*c^5*d^3*e^2 - 2304*a^2*b^3*c^4*d^2*e^3 - 192*a*b^6*c^2*d*e^4 - 384*a*b
^4*c^4*d^3*e^2 + 576*a*b^5*c^3*d^2*e^3 + 576*a^2*b^4*c^3*d*e^4 + 3072*a^3*b*c^5*d^2*e^3))/(128*(b^12*c + 4096*
a^6*c^7 - 24*a*b^10*c^2 + 240*a^2*b^8*c^3 - 1280*a^3*b^6*c^4 + 3840*a^4*b^4*c^5 - 6144*a^5*b^2*c^6)))^(1/2)))*
((25*(e^5*(-(4*a*c - b^2)^9)^(1/2) - b^9*e^5 + 768*a^4*b*c^4*e^5 - 1536*a^4*c^5*d*e^4 + 96*a^2*b^5*c^2*e^5 - 5
12*a^3*b^3*c^3*e^5 - 2048*a^3*c^6*d^3*e^2 + 32*b^6*c^3*d^3*e^2 - 48*b^7*c^2*d^2*e^3 + 18*b^8*c*d*e^4 + 1536*a^
2*b^2*c^5*d^3*e^2 - 2304*a^2*b^3*c^4*d^2*e^3 - 192*a*b^6*c^2*d*e^4 - 384*a*b^4*c^4*d^3*e^2 + 576*a*b^5*c^3*d^2
*e^3 + 576*a^2*b^4*c^3*d*e^4 + 3072*a^3*b*c^5*d^2*e^3))/(128*(b^12*c + 4096*a^6*c^7 - 24*a*b^10*c^2 + 240*a^2*
b^8*c^3 - 1280*a^3*b^6*c^4 + 3840*a^4*b^4*c^5 - 6144*a^5*b^2*c^6)))^(1/2)*2i - ((3*(d + e*x)^(5/2)*(b^2*e^4 +
10*c^2*d^2*e^2 + 6*a*c*e^4 - 10*b*c*d*e^3))/(4*(4*a*c - b^2)) - (15*(d + e*x)^(3/2)*(b^2*d*e^4 + 2*c^2*d^3*e^2
- a*b*e^5 + 2*a*c*d*e^4 - 3*b*c*d^2*e^3))/(4*(4*a*c - b^2)) + (5*(d + e*x)^(1/2)*(a^2*e^6 + b^2*d^2*e^4 + c^2
*d^4*e^2 - 2*a*b*d*e^5 + 2*a*c*d^2*e^4 - 2*b*c*d^3*e^3))/(2*(4*a*c - b^2)) + (5*c*(b*e^3 - 2*c*d*e^2)*(d + e*x
)^(7/2))/(4*(4*a*c - b^2)))/(c^2*(d + e*x)^4 - (d + e*x)*(4*c^2*d^3 + 2*b^2*d*e^2 - 2*a*b*e^3 + 4*a*c*d*e^2 -
6*b*c*d^2*e) - (4*c^2*d - 2*b*c*e)*(d + e*x)^3 + (d + e*x)^2*(b^2*e^2 + 6*c^2*d^2 + 2*a*c*e^2 - 6*b*c*d*e) + a
^2*e^4 + c^2*d^4 + b^2*d^2*e^2 - 2*a*b*d*e^3 - 2*b*c*d^3*e + 2*a*c*d^2*e^2)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*c*x+b)*(e*x+d)**(5/2)/(c*x**2+b*x+a)**3,x)

[Out]

Timed out

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