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

Optimal. Leaf size=350 \[ -\frac {5 e \left (-2 c e \left (-d \sqrt {b^2-4 a c}+a e+b d\right )+b e^2 \left (b-\sqrt {b^2-4 a c}\right )+2 c^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-e \left (b-\sqrt {b^2-4 a c}\right )}}\right )}{\sqrt {2} c^{3/2} \sqrt {b^2-4 a c} \sqrt {2 c d-e \left (b-\sqrt {b^2-4 a c}\right )}}+\frac {5 e \left (-2 c e \left (d \sqrt {b^2-4 a c}+a e+b d\right )+b e^2 \left (\sqrt {b^2-4 a c}+b\right )+2 c^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}\right )}{\sqrt {2} c^{3/2} \sqrt {b^2-4 a c} \sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}-\frac {(d+e x)^{5/2}}{a+b x+c x^2}+\frac {5 e^2 \sqrt {d+e x}}{c} \]

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Rubi [A]  time = 1.29, antiderivative size = 350, 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, 703, 826, 1166, 208} \begin {gather*} -\frac {5 e \left (-2 c e \left (-d \sqrt {b^2-4 a c}+a e+b d\right )+b e^2 \left (b-\sqrt {b^2-4 a c}\right )+2 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 )}{\sqrt {2} c^{3/2} \sqrt {b^2-4 a c} \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}+a e+b d\right )+b e^2 \left (\sqrt {b^2-4 a c}+b\right )+2 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 )}{\sqrt {2} c^{3/2} \sqrt {b^2-4 a c} \sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}-\frac {(d+e x)^{5/2}}{a+b x+c x^2}+\frac {5 e^2 \sqrt {d+e x}}{c} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(5*e^2*Sqrt[d + e*x])/c - (d + e*x)^(5/2)/(a + b*x + c*x^2) - (5*e*(2*c^2*d^2 + b*(b - Sqrt[b^2 - 4*a*c])*e^2
- 2*c*e*(b*d - Sqrt[b^2 - 4*a*c]*d + a*e))*ArcTanh[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x])/Sqrt[2*c*d - (b - Sqrt[b^2
- 4*a*c])*e]])/(Sqrt[2]*c^(3/2)*Sqrt[b^2 - 4*a*c]*Sqrt[2*c*d - (b - Sqrt[b^2 - 4*a*c])*e]) + (5*e*(2*c^2*d^2 +
b*(b + Sqrt[b^2 - 4*a*c])*e^2 - 2*c*e*(b*d + Sqrt[b^2 - 4*a*c]*d + a*e))*ArcTanh[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*
x])/Sqrt[2*c*d - (b + Sqrt[b^2 - 4*a*c])*e]])/(Sqrt[2]*c^(3/2)*Sqrt[b^2 - 4*a*c]*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 703

Int[((d_.) + (e_.)*(x_))^(m_)/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(e*(d + e*x)^(m - 1))/(c*
(m - 1)), x] + Dist[1/c, Int[((d + e*x)^(m - 2)*Simp[c*d^2 - a*e^2 + e*(2*c*d - b*e)*x, x])/(a + b*x + c*x^2),
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] && GtQ[m, 1]

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

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Mathematica [A]  time = 1.84, size = 297, normalized size = 0.85 \begin {gather*} -\frac {5 e \left (-4 \sqrt {c} e \sqrt {b^2-4 a c} \sqrt {d+e x}+\sqrt {2} \left (e \left (\sqrt {b^2-4 a c}-b\right )+2 c d\right )^{3/2} \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {e \sqrt {b^2-4 a c}-b e+2 c d}}\right )-\sqrt {2} \left (2 c d-e \left (\sqrt {b^2-4 a c}+b\right )\right )^{3/2} \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 )\right )}{4 c^{3/2} \sqrt {b^2-4 a c}}-\frac {e^2 (d+e x)^{5/2}}{e (a e-b d)+c d^2}+\frac {(d+e x)^{7/2} (b e-c d+c e x)}{(a+x (b+c x)) \left (e (a e-b d)+c d^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)^2,x]

[Out]

-((e^2*(d + e*x)^(5/2))/(c*d^2 + e*(-(b*d) + a*e))) + ((d + e*x)^(7/2)*(-(c*d) + b*e + c*e*x))/((c*d^2 + e*(-(
b*d) + a*e))*(a + x*(b + c*x))) - (5*e*(-4*Sqrt[c]*Sqrt[b^2 - 4*a*c]*e*Sqrt[d + e*x] + Sqrt[2]*(2*c*d + (-b +
Sqrt[b^2 - 4*a*c])*e)^(3/2)*ArcTanh[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x])/Sqrt[2*c*d - b*e + Sqrt[b^2 - 4*a*c]*e]] -
Sqrt[2]*(2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)^(3/2)*ArcTanh[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x])/Sqrt[2*c*d - (b + S
qrt[b^2 - 4*a*c])*e]]))/(4*c^(3/2)*Sqrt[b^2 - 4*a*c])

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IntegrateAlgebraic [C]  time = 6.72, size = 925, normalized size = 2.64 \begin {gather*} \frac {\sqrt {d+e x} \left (5 c d^2-5 b e d-10 c (d+e x) d+5 a e^2+4 c (d+e x)^2+5 b e (d+e x)\right ) e^2}{c \left (c d^2-b e d-2 c (d+e x) d+a e^2+c (d+e x)^2+b e (d+e x)\right )}+\frac {\left (5 \sqrt {2} b^2 e^3-8 \sqrt {2} a c e^3-3 \sqrt {2} b \sqrt {b^2-4 a c} e^3-12 \sqrt {2} b c d e^2+6 \sqrt {2} c \sqrt {b^2-4 a c} d e^2+12 \sqrt {2} c^2 d^2 e\right ) \tan ^{-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 )}{c^{3/2} \sqrt {b^2-4 a c} \sqrt {-2 c d+b e-\sqrt {b^2-4 a c} e}}+\frac {\left (-5 \sqrt {2} b^2 e^3+8 \sqrt {2} a c e^3-3 \sqrt {2} b \sqrt {b^2-4 a c} e^3+12 \sqrt {2} b c d e^2+6 \sqrt {2} c \sqrt {b^2-4 a c} d e^2-12 \sqrt {2} c^2 d^2 e\right ) \tan ^{-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 )}{c^{3/2} \sqrt {b^2-4 a c} \sqrt {-2 c d+b e+\sqrt {b^2-4 a c} e}}+\frac {\left (5 i \sqrt {2} b^2 e^3-6 i \sqrt {2} a c e^3+\sqrt {2} b \sqrt {4 a c-b^2} e^3-14 i \sqrt {2} b c d e^2-2 \sqrt {2} c \sqrt {4 a c-b^2} d e^2+14 i \sqrt {2} c^2 d^2 e\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {-2 c d+b e-i \sqrt {4 a c-b^2} e}}\right )}{2 c^{3/2} \sqrt {4 a c-b^2} \sqrt {-2 c d+b e-i \sqrt {4 a c-b^2} e}}+\frac {\left (-5 i \sqrt {2} b^2 e^3+6 i \sqrt {2} a c e^3+\sqrt {2} b \sqrt {4 a c-b^2} e^3+14 i \sqrt {2} b c d e^2-2 \sqrt {2} c \sqrt {4 a c-b^2} d e^2-14 i \sqrt {2} c^2 d^2 e\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {-2 c d+b e+i \sqrt {4 a c-b^2} e}}\right )}{2 c^{3/2} \sqrt {4 a c-b^2} \sqrt {-2 c d+b e+i \sqrt {4 a c-b^2} e}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(e^2*Sqrt[d + e*x]*(5*c*d^2 - 5*b*d*e + 5*a*e^2 - 10*c*d*(d + e*x) + 5*b*e*(d + e*x) + 4*c*(d + e*x)^2))/(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)) + ((12*Sqrt[2]*c^2*d^2*e - 12*Sqrt[2]
*b*c*d*e^2 + 6*Sqrt[2]*c*Sqrt[b^2 - 4*a*c]*d*e^2 + 5*Sqrt[2]*b^2*e^3 - 8*Sqrt[2]*a*c*e^3 - 3*Sqrt[2]*b*Sqrt[b^
2 - 4*a*c]*e^3)*ArcTan[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x])/Sqrt[-2*c*d + b*e - Sqrt[b^2 - 4*a*c]*e]])/(c^(3/2)*Sqr
t[b^2 - 4*a*c]*Sqrt[-2*c*d + b*e - Sqrt[b^2 - 4*a*c]*e]) + ((-12*Sqrt[2]*c^2*d^2*e + 12*Sqrt[2]*b*c*d*e^2 + 6*
Sqrt[2]*c*Sqrt[b^2 - 4*a*c]*d*e^2 - 5*Sqrt[2]*b^2*e^3 + 8*Sqrt[2]*a*c*e^3 - 3*Sqrt[2]*b*Sqrt[b^2 - 4*a*c]*e^3)
*ArcTan[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x])/Sqrt[-2*c*d + b*e + Sqrt[b^2 - 4*a*c]*e]])/(c^(3/2)*Sqrt[b^2 - 4*a*c]*
Sqrt[-2*c*d + b*e + Sqrt[b^2 - 4*a*c]*e]) + (((14*I)*Sqrt[2]*c^2*d^2*e - (14*I)*Sqrt[2]*b*c*d*e^2 - 2*Sqrt[2]*
c*Sqrt[-b^2 + 4*a*c]*d*e^2 + (5*I)*Sqrt[2]*b^2*e^3 - (6*I)*Sqrt[2]*a*c*e^3 + Sqrt[2]*b*Sqrt[-b^2 + 4*a*c]*e^3)
*ArcTan[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x])/Sqrt[-2*c*d + b*e - I*Sqrt[-b^2 + 4*a*c]*e]])/(2*c^(3/2)*Sqrt[-b^2 + 4
*a*c]*Sqrt[-2*c*d + b*e - I*Sqrt[-b^2 + 4*a*c]*e]) + (((-14*I)*Sqrt[2]*c^2*d^2*e + (14*I)*Sqrt[2]*b*c*d*e^2 -
2*Sqrt[2]*c*Sqrt[-b^2 + 4*a*c]*d*e^2 - (5*I)*Sqrt[2]*b^2*e^3 + (6*I)*Sqrt[2]*a*c*e^3 + Sqrt[2]*b*Sqrt[-b^2 + 4
*a*c]*e^3)*ArcTan[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x])/Sqrt[-2*c*d + b*e + I*Sqrt[-b^2 + 4*a*c]*e]])/(2*c^(3/2)*Sqr
t[-b^2 + 4*a*c]*Sqrt[-2*c*d + b*e + I*Sqrt[-b^2 + 4*a*c]*e])

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fricas [B]  time = 0.52, size = 2928, normalized size = 8.37

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)^2,x, algorithm="fricas")

[Out]

-1/2*(5*sqrt(1/2)*(c^2*x^2 + b*c*x + a*c)*sqrt((2*c^3*d^3*e^2 - 3*b*c^2*d^2*e^3 + 3*(b^2*c - 2*a*c^2)*d*e^4 -
(b^3 - 3*a*b*c)*e^5 + (b^2*c^3 - 4*a*c^4)*sqrt((9*c^4*d^4*e^6 - 18*b*c^3*d^3*e^7 + 3*(5*b^2*c^2 - 2*a*c^3)*d^2
*e^8 - 6*(b^3*c - a*b*c^2)*d*e^9 + (b^4 - 2*a*b^2*c + a^2*c^2)*e^10)/(b^2*c^6 - 4*a*c^7)))/(b^2*c^3 - 4*a*c^4)
)*log(125*sqrt(1/2)*(3*(b^2*c^2 - 4*a*c^3)*d^2*e^4 - 3*(b^3*c - 4*a*b*c^2)*d*e^5 + (b^4 - 5*a*b^2*c + 4*a^2*c^
2)*e^6 - (2*(b^2*c^4 - 4*a*c^5)*d - (b^3*c^3 - 4*a*b*c^4)*e)*sqrt((9*c^4*d^4*e^6 - 18*b*c^3*d^3*e^7 + 3*(5*b^2
*c^2 - 2*a*c^3)*d^2*e^8 - 6*(b^3*c - a*b*c^2)*d*e^9 + (b^4 - 2*a*b^2*c + a^2*c^2)*e^10)/(b^2*c^6 - 4*a*c^7)))*
sqrt((2*c^3*d^3*e^2 - 3*b*c^2*d^2*e^3 + 3*(b^2*c - 2*a*c^2)*d*e^4 - (b^3 - 3*a*b*c)*e^5 + (b^2*c^3 - 4*a*c^4)*
sqrt((9*c^4*d^4*e^6 - 18*b*c^3*d^3*e^7 + 3*(5*b^2*c^2 - 2*a*c^3)*d^2*e^8 - 6*(b^3*c - a*b*c^2)*d*e^9 + (b^4 -
2*a*b^2*c + a^2*c^2)*e^10)/(b^2*c^6 - 4*a*c^7)))/(b^2*c^3 - 4*a*c^4)) - 250*(3*c^3*d^4*e^4 - 6*b*c^2*d^3*e^5 +
2*(2*b^2*c + a*c^2)*d^2*e^6 - (b^3 + 2*a*b*c)*d*e^7 + (a*b^2 - a^2*c)*e^8)*sqrt(e*x + d)) - 5*sqrt(1/2)*(c^2*
x^2 + b*c*x + a*c)*sqrt((2*c^3*d^3*e^2 - 3*b*c^2*d^2*e^3 + 3*(b^2*c - 2*a*c^2)*d*e^4 - (b^3 - 3*a*b*c)*e^5 + (
b^2*c^3 - 4*a*c^4)*sqrt((9*c^4*d^4*e^6 - 18*b*c^3*d^3*e^7 + 3*(5*b^2*c^2 - 2*a*c^3)*d^2*e^8 - 6*(b^3*c - a*b*c
^2)*d*e^9 + (b^4 - 2*a*b^2*c + a^2*c^2)*e^10)/(b^2*c^6 - 4*a*c^7)))/(b^2*c^3 - 4*a*c^4))*log(-125*sqrt(1/2)*(3
*(b^2*c^2 - 4*a*c^3)*d^2*e^4 - 3*(b^3*c - 4*a*b*c^2)*d*e^5 + (b^4 - 5*a*b^2*c + 4*a^2*c^2)*e^6 - (2*(b^2*c^4 -
4*a*c^5)*d - (b^3*c^3 - 4*a*b*c^4)*e)*sqrt((9*c^4*d^4*e^6 - 18*b*c^3*d^3*e^7 + 3*(5*b^2*c^2 - 2*a*c^3)*d^2*e^
8 - 6*(b^3*c - a*b*c^2)*d*e^9 + (b^4 - 2*a*b^2*c + a^2*c^2)*e^10)/(b^2*c^6 - 4*a*c^7)))*sqrt((2*c^3*d^3*e^2 -
3*b*c^2*d^2*e^3 + 3*(b^2*c - 2*a*c^2)*d*e^4 - (b^3 - 3*a*b*c)*e^5 + (b^2*c^3 - 4*a*c^4)*sqrt((9*c^4*d^4*e^6 -
18*b*c^3*d^3*e^7 + 3*(5*b^2*c^2 - 2*a*c^3)*d^2*e^8 - 6*(b^3*c - a*b*c^2)*d*e^9 + (b^4 - 2*a*b^2*c + a^2*c^2)*e
^10)/(b^2*c^6 - 4*a*c^7)))/(b^2*c^3 - 4*a*c^4)) - 250*(3*c^3*d^4*e^4 - 6*b*c^2*d^3*e^5 + 2*(2*b^2*c + a*c^2)*d
^2*e^6 - (b^3 + 2*a*b*c)*d*e^7 + (a*b^2 - a^2*c)*e^8)*sqrt(e*x + d)) + 5*sqrt(1/2)*(c^2*x^2 + b*c*x + a*c)*sqr
t((2*c^3*d^3*e^2 - 3*b*c^2*d^2*e^3 + 3*(b^2*c - 2*a*c^2)*d*e^4 - (b^3 - 3*a*b*c)*e^5 - (b^2*c^3 - 4*a*c^4)*sqr
t((9*c^4*d^4*e^6 - 18*b*c^3*d^3*e^7 + 3*(5*b^2*c^2 - 2*a*c^3)*d^2*e^8 - 6*(b^3*c - a*b*c^2)*d*e^9 + (b^4 - 2*a
*b^2*c + a^2*c^2)*e^10)/(b^2*c^6 - 4*a*c^7)))/(b^2*c^3 - 4*a*c^4))*log(125*sqrt(1/2)*(3*(b^2*c^2 - 4*a*c^3)*d^
2*e^4 - 3*(b^3*c - 4*a*b*c^2)*d*e^5 + (b^4 - 5*a*b^2*c + 4*a^2*c^2)*e^6 + (2*(b^2*c^4 - 4*a*c^5)*d - (b^3*c^3
- 4*a*b*c^4)*e)*sqrt((9*c^4*d^4*e^6 - 18*b*c^3*d^3*e^7 + 3*(5*b^2*c^2 - 2*a*c^3)*d^2*e^8 - 6*(b^3*c - a*b*c^2)
*d*e^9 + (b^4 - 2*a*b^2*c + a^2*c^2)*e^10)/(b^2*c^6 - 4*a*c^7)))*sqrt((2*c^3*d^3*e^2 - 3*b*c^2*d^2*e^3 + 3*(b^
2*c - 2*a*c^2)*d*e^4 - (b^3 - 3*a*b*c)*e^5 - (b^2*c^3 - 4*a*c^4)*sqrt((9*c^4*d^4*e^6 - 18*b*c^3*d^3*e^7 + 3*(5
*b^2*c^2 - 2*a*c^3)*d^2*e^8 - 6*(b^3*c - a*b*c^2)*d*e^9 + (b^4 - 2*a*b^2*c + a^2*c^2)*e^10)/(b^2*c^6 - 4*a*c^7
)))/(b^2*c^3 - 4*a*c^4)) - 250*(3*c^3*d^4*e^4 - 6*b*c^2*d^3*e^5 + 2*(2*b^2*c + a*c^2)*d^2*e^6 - (b^3 + 2*a*b*c
)*d*e^7 + (a*b^2 - a^2*c)*e^8)*sqrt(e*x + d)) - 5*sqrt(1/2)*(c^2*x^2 + b*c*x + a*c)*sqrt((2*c^3*d^3*e^2 - 3*b*
c^2*d^2*e^3 + 3*(b^2*c - 2*a*c^2)*d*e^4 - (b^3 - 3*a*b*c)*e^5 - (b^2*c^3 - 4*a*c^4)*sqrt((9*c^4*d^4*e^6 - 18*b
*c^3*d^3*e^7 + 3*(5*b^2*c^2 - 2*a*c^3)*d^2*e^8 - 6*(b^3*c - a*b*c^2)*d*e^9 + (b^4 - 2*a*b^2*c + a^2*c^2)*e^10)
/(b^2*c^6 - 4*a*c^7)))/(b^2*c^3 - 4*a*c^4))*log(-125*sqrt(1/2)*(3*(b^2*c^2 - 4*a*c^3)*d^2*e^4 - 3*(b^3*c - 4*a
*b*c^2)*d*e^5 + (b^4 - 5*a*b^2*c + 4*a^2*c^2)*e^6 + (2*(b^2*c^4 - 4*a*c^5)*d - (b^3*c^3 - 4*a*b*c^4)*e)*sqrt((
9*c^4*d^4*e^6 - 18*b*c^3*d^3*e^7 + 3*(5*b^2*c^2 - 2*a*c^3)*d^2*e^8 - 6*(b^3*c - a*b*c^2)*d*e^9 + (b^4 - 2*a*b^
2*c + a^2*c^2)*e^10)/(b^2*c^6 - 4*a*c^7)))*sqrt((2*c^3*d^3*e^2 - 3*b*c^2*d^2*e^3 + 3*(b^2*c - 2*a*c^2)*d*e^4 -
(b^3 - 3*a*b*c)*e^5 - (b^2*c^3 - 4*a*c^4)*sqrt((9*c^4*d^4*e^6 - 18*b*c^3*d^3*e^7 + 3*(5*b^2*c^2 - 2*a*c^3)*d^
2*e^8 - 6*(b^3*c - a*b*c^2)*d*e^9 + (b^4 - 2*a*b^2*c + a^2*c^2)*e^10)/(b^2*c^6 - 4*a*c^7)))/(b^2*c^3 - 4*a*c^4
)) - 250*(3*c^3*d^4*e^4 - 6*b*c^2*d^3*e^5 + 2*(2*b^2*c + a*c^2)*d^2*e^6 - (b^3 + 2*a*b*c)*d*e^7 + (a*b^2 - a^2
*c)*e^8)*sqrt(e*x + d)) - 2*(4*c*e^2*x^2 - c*d^2 + 5*a*e^2 - (2*c*d*e - 5*b*e^2)*x)*sqrt(e*x + d))/(c^2*x^2 +
b*c*x + a*c)

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giac [B]  time = 1.67, size = 772, normalized size = 2.21 \begin {gather*} \frac {4 \, \sqrt {x e + d} e^{2}}{c} - \frac {2 \, {\left (x e + d\right )}^{\frac {3}{2}} c d e^{2} - \sqrt {x e + d} c d^{2} e^{2} - {\left (x e + d\right )}^{\frac {3}{2}} b e^{3} + \sqrt {x e + d} b d e^{3} - \sqrt {x e + d} a e^{4}}{{\left ({\left (x e + d\right )}^{2} c - 2 \, {\left (x e + d\right )} c d + c d^{2} + {\left (x e + d\right )} b e - b d e + a e^{2}\right )} c} + \frac {5 \, {\left (4 \, c^{5} d^{3} e - 6 \, b c^{4} d^{2} e^{2} - {\left (2 \, {\left (b^{2} c - 4 \, a c^{2}\right )} d e^{3} - {\left (b^{3} - 4 \, a b c\right )} e^{4}\right )} c^{2} + 4 \, {\left (b^{2} c^{3} - a c^{4}\right )} d e^{3} + 2 \, {\left (\sqrt {b^{2} - 4 \, a c} c^{3} d^{2} e^{2} - \sqrt {b^{2} - 4 \, a c} b c^{2} d e^{3} + \sqrt {b^{2} - 4 \, a c} a c^{2} e^{4}\right )} {\left | c \right |} - {\left (b^{3} c^{2} - 2 \, a b c^{3}\right )} e^{4}\right )} \arctan \left (\frac {2 \, \sqrt {\frac {1}{2}} \sqrt {x e + d}}{\sqrt {-\frac {2 \, c^{2} d - b c e + \sqrt {-4 \, {\left (c^{2} d^{2} - b c d e + a c e^{2}\right )} c^{2} + {\left (2 \, c^{2} d - b c e\right )}^{2}}}{c^{2}}}}\right )}{{\left (2 \, \sqrt {b^{2} - 4 \, a c} c^{2} d - {\left (b^{2} c - 4 \, a c^{2} + \sqrt {b^{2} - 4 \, a c} b c\right )} e\right )} \sqrt {-4 \, c^{2} d + 2 \, {\left (b c - \sqrt {b^{2} - 4 \, a c} c\right )} e} c^{2}} - \frac {5 \, {\left (4 \, c^{5} d^{3} e - 6 \, b c^{4} d^{2} e^{2} - {\left (2 \, {\left (b^{2} c - 4 \, a c^{2}\right )} d e^{3} - {\left (b^{3} - 4 \, a b c\right )} e^{4}\right )} c^{2} + 4 \, {\left (b^{2} c^{3} - a c^{4}\right )} d e^{3} - 2 \, {\left (\sqrt {b^{2} - 4 \, a c} c^{3} d^{2} e^{2} - \sqrt {b^{2} - 4 \, a c} b c^{2} d e^{3} + \sqrt {b^{2} - 4 \, a c} a c^{2} e^{4}\right )} {\left | c \right |} - {\left (b^{3} c^{2} - 2 \, a b c^{3}\right )} e^{4}\right )} \arctan \left (\frac {2 \, \sqrt {\frac {1}{2}} \sqrt {x e + d}}{\sqrt {-\frac {2 \, c^{2} d - b c e - \sqrt {-4 \, {\left (c^{2} d^{2} - b c d e + a c e^{2}\right )} c^{2} + {\left (2 \, c^{2} d - b c e\right )}^{2}}}{c^{2}}}}\right )}{{\left (2 \, \sqrt {b^{2} - 4 \, a c} c^{2} d + {\left (b^{2} c - 4 \, a c^{2} - \sqrt {b^{2} - 4 \, a c} b c\right )} e\right )} \sqrt {-4 \, c^{2} d + 2 \, {\left (b c + \sqrt {b^{2} - 4 \, a c} c\right )} e} c^{2}} \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)^2,x, algorithm="giac")

[Out]

4*sqrt(x*e + d)*e^2/c - (2*(x*e + d)^(3/2)*c*d*e^2 - sqrt(x*e + d)*c*d^2*e^2 - (x*e + d)^(3/2)*b*e^3 + sqrt(x*
e + d)*b*d*e^3 - sqrt(x*e + d)*a*e^4)/(((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)*c) + 5*(4*c^5*d^3*e - 6*b*c^4*d^2*e^2 - (2*(b^2*c - 4*a*c^2)*d*e^3 - (b^3 - 4*a*b*c)*e^4)*c^2 + 4*(b^2*c^3
- a*c^4)*d*e^3 + 2*(sqrt(b^2 - 4*a*c)*c^3*d^2*e^2 - sqrt(b^2 - 4*a*c)*b*c^2*d*e^3 + sqrt(b^2 - 4*a*c)*a*c^2*e^
4)*abs(c) - (b^3*c^2 - 2*a*b*c^3)*e^4)*arctan(2*sqrt(1/2)*sqrt(x*e + d)/sqrt(-(2*c^2*d - b*c*e + sqrt(-4*(c^2*
d^2 - b*c*d*e + a*c*e^2)*c^2 + (2*c^2*d - b*c*e)^2))/c^2))/((2*sqrt(b^2 - 4*a*c)*c^2*d - (b^2*c - 4*a*c^2 + sq
rt(b^2 - 4*a*c)*b*c)*e)*sqrt(-4*c^2*d + 2*(b*c - sqrt(b^2 - 4*a*c)*c)*e)*c^2) - 5*(4*c^5*d^3*e - 6*b*c^4*d^2*e
^2 - (2*(b^2*c - 4*a*c^2)*d*e^3 - (b^3 - 4*a*b*c)*e^4)*c^2 + 4*(b^2*c^3 - a*c^4)*d*e^3 - 2*(sqrt(b^2 - 4*a*c)*
c^3*d^2*e^2 - sqrt(b^2 - 4*a*c)*b*c^2*d*e^3 + sqrt(b^2 - 4*a*c)*a*c^2*e^4)*abs(c) - (b^3*c^2 - 2*a*b*c^3)*e^4)
*arctan(2*sqrt(1/2)*sqrt(x*e + d)/sqrt(-(2*c^2*d - b*c*e - sqrt(-4*(c^2*d^2 - b*c*d*e + a*c*e^2)*c^2 + (2*c^2*
d - b*c*e)^2))/c^2))/((2*sqrt(b^2 - 4*a*c)*c^2*d + (b^2*c - 4*a*c^2 - sqrt(b^2 - 4*a*c)*b*c)*e)*sqrt(-4*c^2*d
+ 2*(b*c + sqrt(b^2 - 4*a*c)*c)*e)*c^2)

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maple [B]  time = 0.21, size = 1333, normalized size = 3.81

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)^2,x)

[Out]

4*e^2*(e*x+d)^(1/2)/c+e^3/c/(c*e^2*x^2+b*e^2*x+a*e^2)*(e*x+d)^(3/2)*b-2*e^2/(c*e^2*x^2+b*e^2*x+a*e^2)*(e*x+d)^
(3/2)*d+e^4/c/(c*e^2*x^2+b*e^2*x+a*e^2)*(e*x+d)^(1/2)*a-e^3/c/(c*e^2*x^2+b*e^2*x+a*e^2)*(e*x+d)^(1/2)*b*d+e^2/
(c*e^2*x^2+b*e^2*x+a*e^2)*(e*x+d)^(1/2)*d^2+5*e^4/(-(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-5/2*e^
4/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^2+5*e^3/(-(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*e^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)*d^2+5/2*e^3/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)*b-5*e^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+5*e^4/(-(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-5/2*e^4/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^2+5*e^3/(-(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^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)*d^2-5/2*e^3/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)*b+5*e^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

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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 )}^{2}}\,{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)^2,x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 1.44, size = 8776, normalized size = 25.07

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)^2,x)

[Out]

((b*e^3 - 2*c*d*e^2)*(d + e*x)^(3/2) + (d + e*x)^(1/2)*(a*e^4 + c*d^2*e^2 - b*d*e^3))/(c^2*(d + e*x)^2 - (2*c^
2*d - b*c*e)*(d + e*x) + c^2*d^2 + a*c*e^2 - b*c*d*e) - atan(((((5*(16*a^2*c^3*e^6 - 4*a*b^2*c^2*e^6 + 16*a*c^
4*d^2*e^4 + 4*b^3*c^2*d*e^5 - 4*b^2*c^3*d^2*e^4 - 16*a*b*c^3*d*e^5))/c - (2*(d + e*x)^(1/2)*(-(25*(b^5*e^5 + b
^2*e^5*(-(4*a*c - b^2)^3)^(1/2) + 12*a^2*b*c^2*e^5 + 8*a*c^4*d^3*e^2 - 24*a^2*c^3*d*e^4 - 2*b^2*c^3*d^3*e^2 +
3*b^3*c^2*d^2*e^3 + 3*c^2*d^2*e^3*(-(4*a*c - b^2)^3)^(1/2) - 7*a*b^3*c*e^5 - a*c*e^5*(-(4*a*c - b^2)^3)^(1/2)
- 3*b^4*c*d*e^4 - 3*b*c*d*e^4*(-(4*a*c - b^2)^3)^(1/2) - 12*a*b*c^3*d^2*e^3 + 18*a*b^2*c^2*d*e^4))/(8*(16*a^2*
c^5 + b^4*c^3 - 8*a*b^2*c^4)))^(1/2)*(4*b^3*c^3*e^3 - 8*b^2*c^4*d*e^2 - 16*a*b*c^4*e^3 + 32*a*c^5*d*e^2))/c)*(
-(25*(b^5*e^5 + b^2*e^5*(-(4*a*c - b^2)^3)^(1/2) + 12*a^2*b*c^2*e^5 + 8*a*c^4*d^3*e^2 - 24*a^2*c^3*d*e^4 - 2*b
^2*c^3*d^3*e^2 + 3*b^3*c^2*d^2*e^3 + 3*c^2*d^2*e^3*(-(4*a*c - b^2)^3)^(1/2) - 7*a*b^3*c*e^5 - a*c*e^5*(-(4*a*c
- b^2)^3)^(1/2) - 3*b^4*c*d*e^4 - 3*b*c*d*e^4*(-(4*a*c - b^2)^3)^(1/2) - 12*a*b*c^3*d^2*e^3 + 18*a*b^2*c^2*d*
e^4))/(8*(16*a^2*c^5 + b^4*c^3 - 8*a*b^2*c^4)))^(1/2) - (2*(d + e*x)^(1/2)*(25*b^4*e^8 + 50*a^2*c^2*e^8 + 50*c
^4*d^4*e^4 - 300*a*c^3*d^2*e^6 - 100*b*c^3*d^3*e^5 + 150*b^2*c^2*d^2*e^6 - 100*a*b^2*c*e^8 - 100*b^3*c*d*e^7 +
300*a*b*c^2*d*e^7))/c)*(-(25*(b^5*e^5 + b^2*e^5*(-(4*a*c - b^2)^3)^(1/2) + 12*a^2*b*c^2*e^5 + 8*a*c^4*d^3*e^2
- 24*a^2*c^3*d*e^4 - 2*b^2*c^3*d^3*e^2 + 3*b^3*c^2*d^2*e^3 + 3*c^2*d^2*e^3*(-(4*a*c - b^2)^3)^(1/2) - 7*a*b^3
*c*e^5 - a*c*e^5*(-(4*a*c - b^2)^3)^(1/2) - 3*b^4*c*d*e^4 - 3*b*c*d*e^4*(-(4*a*c - b^2)^3)^(1/2) - 12*a*b*c^3*
d^2*e^3 + 18*a*b^2*c^2*d*e^4))/(8*(16*a^2*c^5 + b^4*c^3 - 8*a*b^2*c^4)))^(1/2)*1i - (((5*(16*a^2*c^3*e^6 - 4*a
*b^2*c^2*e^6 + 16*a*c^4*d^2*e^4 + 4*b^3*c^2*d*e^5 - 4*b^2*c^3*d^2*e^4 - 16*a*b*c^3*d*e^5))/c + (2*(d + e*x)^(1
/2)*(-(25*(b^5*e^5 + b^2*e^5*(-(4*a*c - b^2)^3)^(1/2) + 12*a^2*b*c^2*e^5 + 8*a*c^4*d^3*e^2 - 24*a^2*c^3*d*e^4
- 2*b^2*c^3*d^3*e^2 + 3*b^3*c^2*d^2*e^3 + 3*c^2*d^2*e^3*(-(4*a*c - b^2)^3)^(1/2) - 7*a*b^3*c*e^5 - a*c*e^5*(-(
4*a*c - b^2)^3)^(1/2) - 3*b^4*c*d*e^4 - 3*b*c*d*e^4*(-(4*a*c - b^2)^3)^(1/2) - 12*a*b*c^3*d^2*e^3 + 18*a*b^2*c
^2*d*e^4))/(8*(16*a^2*c^5 + b^4*c^3 - 8*a*b^2*c^4)))^(1/2)*(4*b^3*c^3*e^3 - 8*b^2*c^4*d*e^2 - 16*a*b*c^4*e^3 +
32*a*c^5*d*e^2))/c)*(-(25*(b^5*e^5 + b^2*e^5*(-(4*a*c - b^2)^3)^(1/2) + 12*a^2*b*c^2*e^5 + 8*a*c^4*d^3*e^2 -
24*a^2*c^3*d*e^4 - 2*b^2*c^3*d^3*e^2 + 3*b^3*c^2*d^2*e^3 + 3*c^2*d^2*e^3*(-(4*a*c - b^2)^3)^(1/2) - 7*a*b^3*c*
e^5 - a*c*e^5*(-(4*a*c - b^2)^3)^(1/2) - 3*b^4*c*d*e^4 - 3*b*c*d*e^4*(-(4*a*c - b^2)^3)^(1/2) - 12*a*b*c^3*d^2
*e^3 + 18*a*b^2*c^2*d*e^4))/(8*(16*a^2*c^5 + b^4*c^3 - 8*a*b^2*c^4)))^(1/2) + (2*(d + e*x)^(1/2)*(25*b^4*e^8 +
50*a^2*c^2*e^8 + 50*c^4*d^4*e^4 - 300*a*c^3*d^2*e^6 - 100*b*c^3*d^3*e^5 + 150*b^2*c^2*d^2*e^6 - 100*a*b^2*c*e
^8 - 100*b^3*c*d*e^7 + 300*a*b*c^2*d*e^7))/c)*(-(25*(b^5*e^5 + b^2*e^5*(-(4*a*c - b^2)^3)^(1/2) + 12*a^2*b*c^2
*e^5 + 8*a*c^4*d^3*e^2 - 24*a^2*c^3*d*e^4 - 2*b^2*c^3*d^3*e^2 + 3*b^3*c^2*d^2*e^3 + 3*c^2*d^2*e^3*(-(4*a*c - b
^2)^3)^(1/2) - 7*a*b^3*c*e^5 - a*c*e^5*(-(4*a*c - b^2)^3)^(1/2) - 3*b^4*c*d*e^4 - 3*b*c*d*e^4*(-(4*a*c - b^2)^
3)^(1/2) - 12*a*b*c^3*d^2*e^3 + 18*a*b^2*c^2*d*e^4))/(8*(16*a^2*c^5 + b^4*c^3 - 8*a*b^2*c^4)))^(1/2)*1i)/((((5
*(16*a^2*c^3*e^6 - 4*a*b^2*c^2*e^6 + 16*a*c^4*d^2*e^4 + 4*b^3*c^2*d*e^5 - 4*b^2*c^3*d^2*e^4 - 16*a*b*c^3*d*e^5
))/c - (2*(d + e*x)^(1/2)*(-(25*(b^5*e^5 + b^2*e^5*(-(4*a*c - b^2)^3)^(1/2) + 12*a^2*b*c^2*e^5 + 8*a*c^4*d^3*e
^2 - 24*a^2*c^3*d*e^4 - 2*b^2*c^3*d^3*e^2 + 3*b^3*c^2*d^2*e^3 + 3*c^2*d^2*e^3*(-(4*a*c - b^2)^3)^(1/2) - 7*a*b
^3*c*e^5 - a*c*e^5*(-(4*a*c - b^2)^3)^(1/2) - 3*b^4*c*d*e^4 - 3*b*c*d*e^4*(-(4*a*c - b^2)^3)^(1/2) - 12*a*b*c^
3*d^2*e^3 + 18*a*b^2*c^2*d*e^4))/(8*(16*a^2*c^5 + b^4*c^3 - 8*a*b^2*c^4)))^(1/2)*(4*b^3*c^3*e^3 - 8*b^2*c^4*d*
e^2 - 16*a*b*c^4*e^3 + 32*a*c^5*d*e^2))/c)*(-(25*(b^5*e^5 + b^2*e^5*(-(4*a*c - b^2)^3)^(1/2) + 12*a^2*b*c^2*e^
5 + 8*a*c^4*d^3*e^2 - 24*a^2*c^3*d*e^4 - 2*b^2*c^3*d^3*e^2 + 3*b^3*c^2*d^2*e^3 + 3*c^2*d^2*e^3*(-(4*a*c - b^2)
^3)^(1/2) - 7*a*b^3*c*e^5 - a*c*e^5*(-(4*a*c - b^2)^3)^(1/2) - 3*b^4*c*d*e^4 - 3*b*c*d*e^4*(-(4*a*c - b^2)^3)^
(1/2) - 12*a*b*c^3*d^2*e^3 + 18*a*b^2*c^2*d*e^4))/(8*(16*a^2*c^5 + b^4*c^3 - 8*a*b^2*c^4)))^(1/2) - (2*(d + e*
x)^(1/2)*(25*b^4*e^8 + 50*a^2*c^2*e^8 + 50*c^4*d^4*e^4 - 300*a*c^3*d^2*e^6 - 100*b*c^3*d^3*e^5 + 150*b^2*c^2*d
^2*e^6 - 100*a*b^2*c*e^8 - 100*b^3*c*d*e^7 + 300*a*b*c^2*d*e^7))/c)*(-(25*(b^5*e^5 + b^2*e^5*(-(4*a*c - b^2)^3
)^(1/2) + 12*a^2*b*c^2*e^5 + 8*a*c^4*d^3*e^2 - 24*a^2*c^3*d*e^4 - 2*b^2*c^3*d^3*e^2 + 3*b^3*c^2*d^2*e^3 + 3*c^
2*d^2*e^3*(-(4*a*c - b^2)^3)^(1/2) - 7*a*b^3*c*e^5 - a*c*e^5*(-(4*a*c - b^2)^3)^(1/2) - 3*b^4*c*d*e^4 - 3*b*c*
d*e^4*(-(4*a*c - b^2)^3)^(1/2) - 12*a*b*c^3*d^2*e^3 + 18*a*b^2*c^2*d*e^4))/(8*(16*a^2*c^5 + b^4*c^3 - 8*a*b^2*
c^4)))^(1/2) + (((5*(16*a^2*c^3*e^6 - 4*a*b^2*c^2*e^6 + 16*a*c^4*d^2*e^4 + 4*b^3*c^2*d*e^5 - 4*b^2*c^3*d^2*e^4
- 16*a*b*c^3*d*e^5))/c + (2*(d + e*x)^(1/2)*(-(25*(b^5*e^5 + b^2*e^5*(-(4*a*c - b^2)^3)^(1/2) + 12*a^2*b*c^2*
e^5 + 8*a*c^4*d^3*e^2 - 24*a^2*c^3*d*e^4 - 2*b^2*c^3*d^3*e^2 + 3*b^3*c^2*d^2*e^3 + 3*c^2*d^2*e^3*(-(4*a*c - b^
2)^3)^(1/2) - 7*a*b^3*c*e^5 - a*c*e^5*(-(4*a*c - b^2)^3)^(1/2) - 3*b^4*c*d*e^4 - 3*b*c*d*e^4*(-(4*a*c - b^2)^3
)^(1/2) - 12*a*b*c^3*d^2*e^3 + 18*a*b^2*c^2*d*e^4))/(8*(16*a^2*c^5 + b^4*c^3 - 8*a*b^2*c^4)))^(1/2)*(4*b^3*c^3
*e^3 - 8*b^2*c^4*d*e^2 - 16*a*b*c^4*e^3 + 32*a*c^5*d*e^2))/c)*(-(25*(b^5*e^5 + b^2*e^5*(-(4*a*c - b^2)^3)^(1/2
) + 12*a^2*b*c^2*e^5 + 8*a*c^4*d^3*e^2 - 24*a^2*c^3*d*e^4 - 2*b^2*c^3*d^3*e^2 + 3*b^3*c^2*d^2*e^3 + 3*c^2*d^2*
e^3*(-(4*a*c - b^2)^3)^(1/2) - 7*a*b^3*c*e^5 - a*c*e^5*(-(4*a*c - b^2)^3)^(1/2) - 3*b^4*c*d*e^4 - 3*b*c*d*e^4*
(-(4*a*c - b^2)^3)^(1/2) - 12*a*b*c^3*d^2*e^3 + 18*a*b^2*c^2*d*e^4))/(8*(16*a^2*c^5 + b^4*c^3 - 8*a*b^2*c^4)))
^(1/2) + (2*(d + e*x)^(1/2)*(25*b^4*e^8 + 50*a^2*c^2*e^8 + 50*c^4*d^4*e^4 - 300*a*c^3*d^2*e^6 - 100*b*c^3*d^3*
e^5 + 150*b^2*c^2*d^2*e^6 - 100*a*b^2*c*e^8 - 100*b^3*c*d*e^7 + 300*a*b*c^2*d*e^7))/c)*(-(25*(b^5*e^5 + b^2*e^
5*(-(4*a*c - b^2)^3)^(1/2) + 12*a^2*b*c^2*e^5 + 8*a*c^4*d^3*e^2 - 24*a^2*c^3*d*e^4 - 2*b^2*c^3*d^3*e^2 + 3*b^3
*c^2*d^2*e^3 + 3*c^2*d^2*e^3*(-(4*a*c - b^2)^3)^(1/2) - 7*a*b^3*c*e^5 - a*c*e^5*(-(4*a*c - b^2)^3)^(1/2) - 3*b
^4*c*d*e^4 - 3*b*c*d*e^4*(-(4*a*c - b^2)^3)^(1/2) - 12*a*b*c^3*d^2*e^3 + 18*a*b^2*c^2*d*e^4))/(8*(16*a^2*c^5 +
b^4*c^3 - 8*a*b^2*c^4)))^(1/2) - (10*(50*c^3*d^5*e^6 - 25*b^3*d^2*e^9 - 25*a^2*b*e^11 + 100*a*c^2*d^3*e^8 - 1
25*b*c^2*d^4*e^7 + 100*b^2*c*d^3*e^8 + 50*a*b^2*d*e^10 + 50*a^2*c*d*e^10 - 150*a*b*c*d^2*e^9))/c))*(-(25*(b^5*
e^5 + b^2*e^5*(-(4*a*c - b^2)^3)^(1/2) + 12*a^2*b*c^2*e^5 + 8*a*c^4*d^3*e^2 - 24*a^2*c^3*d*e^4 - 2*b^2*c^3*d^3
*e^2 + 3*b^3*c^2*d^2*e^3 + 3*c^2*d^2*e^3*(-(4*a*c - b^2)^3)^(1/2) - 7*a*b^3*c*e^5 - a*c*e^5*(-(4*a*c - b^2)^3)
^(1/2) - 3*b^4*c*d*e^4 - 3*b*c*d*e^4*(-(4*a*c - b^2)^3)^(1/2) - 12*a*b*c^3*d^2*e^3 + 18*a*b^2*c^2*d*e^4))/(8*(
16*a^2*c^5 + b^4*c^3 - 8*a*b^2*c^4)))^(1/2)*2i - atan(((((5*(16*a^2*c^3*e^6 - 4*a*b^2*c^2*e^6 + 16*a*c^4*d^2*e
^4 + 4*b^3*c^2*d*e^5 - 4*b^2*c^3*d^2*e^4 - 16*a*b*c^3*d*e^5))/c - (2*(d + e*x)^(1/2)*(-(25*(b^5*e^5 - b^2*e^5*
(-(4*a*c - b^2)^3)^(1/2) + 12*a^2*b*c^2*e^5 + 8*a*c^4*d^3*e^2 - 24*a^2*c^3*d*e^4 - 2*b^2*c^3*d^3*e^2 + 3*b^3*c
^2*d^2*e^3 - 3*c^2*d^2*e^3*(-(4*a*c - b^2)^3)^(1/2) - 7*a*b^3*c*e^5 + a*c*e^5*(-(4*a*c - b^2)^3)^(1/2) - 3*b^4
*c*d*e^4 + 3*b*c*d*e^4*(-(4*a*c - b^2)^3)^(1/2) - 12*a*b*c^3*d^2*e^3 + 18*a*b^2*c^2*d*e^4))/(8*(16*a^2*c^5 + b
^4*c^3 - 8*a*b^2*c^4)))^(1/2)*(4*b^3*c^3*e^3 - 8*b^2*c^4*d*e^2 - 16*a*b*c^4*e^3 + 32*a*c^5*d*e^2))/c)*(-(25*(b
^5*e^5 - b^2*e^5*(-(4*a*c - b^2)^3)^(1/2) + 12*a^2*b*c^2*e^5 + 8*a*c^4*d^3*e^2 - 24*a^2*c^3*d*e^4 - 2*b^2*c^3*
d^3*e^2 + 3*b^3*c^2*d^2*e^3 - 3*c^2*d^2*e^3*(-(4*a*c - b^2)^3)^(1/2) - 7*a*b^3*c*e^5 + a*c*e^5*(-(4*a*c - b^2)
^3)^(1/2) - 3*b^4*c*d*e^4 + 3*b*c*d*e^4*(-(4*a*c - b^2)^3)^(1/2) - 12*a*b*c^3*d^2*e^3 + 18*a*b^2*c^2*d*e^4))/(
8*(16*a^2*c^5 + b^4*c^3 - 8*a*b^2*c^4)))^(1/2) - (2*(d + e*x)^(1/2)*(25*b^4*e^8 + 50*a^2*c^2*e^8 + 50*c^4*d^4*
e^4 - 300*a*c^3*d^2*e^6 - 100*b*c^3*d^3*e^5 + 150*b^2*c^2*d^2*e^6 - 100*a*b^2*c*e^8 - 100*b^3*c*d*e^7 + 300*a*
b*c^2*d*e^7))/c)*(-(25*(b^5*e^5 - b^2*e^5*(-(4*a*c - b^2)^3)^(1/2) + 12*a^2*b*c^2*e^5 + 8*a*c^4*d^3*e^2 - 24*a
^2*c^3*d*e^4 - 2*b^2*c^3*d^3*e^2 + 3*b^3*c^2*d^2*e^3 - 3*c^2*d^2*e^3*(-(4*a*c - b^2)^3)^(1/2) - 7*a*b^3*c*e^5
+ a*c*e^5*(-(4*a*c - b^2)^3)^(1/2) - 3*b^4*c*d*e^4 + 3*b*c*d*e^4*(-(4*a*c - b^2)^3)^(1/2) - 12*a*b*c^3*d^2*e^3
+ 18*a*b^2*c^2*d*e^4))/(8*(16*a^2*c^5 + b^4*c^3 - 8*a*b^2*c^4)))^(1/2)*1i - (((5*(16*a^2*c^3*e^6 - 4*a*b^2*c^
2*e^6 + 16*a*c^4*d^2*e^4 + 4*b^3*c^2*d*e^5 - 4*b^2*c^3*d^2*e^4 - 16*a*b*c^3*d*e^5))/c + (2*(d + e*x)^(1/2)*(-(
25*(b^5*e^5 - b^2*e^5*(-(4*a*c - b^2)^3)^(1/2) + 12*a^2*b*c^2*e^5 + 8*a*c^4*d^3*e^2 - 24*a^2*c^3*d*e^4 - 2*b^2
*c^3*d^3*e^2 + 3*b^3*c^2*d^2*e^3 - 3*c^2*d^2*e^3*(-(4*a*c - b^2)^3)^(1/2) - 7*a*b^3*c*e^5 + a*c*e^5*(-(4*a*c -
b^2)^3)^(1/2) - 3*b^4*c*d*e^4 + 3*b*c*d*e^4*(-(4*a*c - b^2)^3)^(1/2) - 12*a*b*c^3*d^2*e^3 + 18*a*b^2*c^2*d*e^
4))/(8*(16*a^2*c^5 + b^4*c^3 - 8*a*b^2*c^4)))^(1/2)*(4*b^3*c^3*e^3 - 8*b^2*c^4*d*e^2 - 16*a*b*c^4*e^3 + 32*a*c
^5*d*e^2))/c)*(-(25*(b^5*e^5 - b^2*e^5*(-(4*a*c - b^2)^3)^(1/2) + 12*a^2*b*c^2*e^5 + 8*a*c^4*d^3*e^2 - 24*a^2*
c^3*d*e^4 - 2*b^2*c^3*d^3*e^2 + 3*b^3*c^2*d^2*e^3 - 3*c^2*d^2*e^3*(-(4*a*c - b^2)^3)^(1/2) - 7*a*b^3*c*e^5 + a
*c*e^5*(-(4*a*c - b^2)^3)^(1/2) - 3*b^4*c*d*e^4 + 3*b*c*d*e^4*(-(4*a*c - b^2)^3)^(1/2) - 12*a*b*c^3*d^2*e^3 +
18*a*b^2*c^2*d*e^4))/(8*(16*a^2*c^5 + b^4*c^3 - 8*a*b^2*c^4)))^(1/2) + (2*(d + e*x)^(1/2)*(25*b^4*e^8 + 50*a^2
*c^2*e^8 + 50*c^4*d^4*e^4 - 300*a*c^3*d^2*e^6 - 100*b*c^3*d^3*e^5 + 150*b^2*c^2*d^2*e^6 - 100*a*b^2*c*e^8 - 10
0*b^3*c*d*e^7 + 300*a*b*c^2*d*e^7))/c)*(-(25*(b^5*e^5 - b^2*e^5*(-(4*a*c - b^2)^3)^(1/2) + 12*a^2*b*c^2*e^5 +
8*a*c^4*d^3*e^2 - 24*a^2*c^3*d*e^4 - 2*b^2*c^3*d^3*e^2 + 3*b^3*c^2*d^2*e^3 - 3*c^2*d^2*e^3*(-(4*a*c - b^2)^3)^
(1/2) - 7*a*b^3*c*e^5 + a*c*e^5*(-(4*a*c - b^2)^3)^(1/2) - 3*b^4*c*d*e^4 + 3*b*c*d*e^4*(-(4*a*c - b^2)^3)^(1/2
) - 12*a*b*c^3*d^2*e^3 + 18*a*b^2*c^2*d*e^4))/(8*(16*a^2*c^5 + b^4*c^3 - 8*a*b^2*c^4)))^(1/2)*1i)/((((5*(16*a^
2*c^3*e^6 - 4*a*b^2*c^2*e^6 + 16*a*c^4*d^2*e^4 + 4*b^3*c^2*d*e^5 - 4*b^2*c^3*d^2*e^4 - 16*a*b*c^3*d*e^5))/c -
(2*(d + e*x)^(1/2)*(-(25*(b^5*e^5 - b^2*e^5*(-(4*a*c - b^2)^3)^(1/2) + 12*a^2*b*c^2*e^5 + 8*a*c^4*d^3*e^2 - 24
*a^2*c^3*d*e^4 - 2*b^2*c^3*d^3*e^2 + 3*b^3*c^2*d^2*e^3 - 3*c^2*d^2*e^3*(-(4*a*c - b^2)^3)^(1/2) - 7*a*b^3*c*e^
5 + a*c*e^5*(-(4*a*c - b^2)^3)^(1/2) - 3*b^4*c*d*e^4 + 3*b*c*d*e^4*(-(4*a*c - b^2)^3)^(1/2) - 12*a*b*c^3*d^2*e
^3 + 18*a*b^2*c^2*d*e^4))/(8*(16*a^2*c^5 + b^4*c^3 - 8*a*b^2*c^4)))^(1/2)*(4*b^3*c^3*e^3 - 8*b^2*c^4*d*e^2 - 1
6*a*b*c^4*e^3 + 32*a*c^5*d*e^2))/c)*(-(25*(b^5*e^5 - b^2*e^5*(-(4*a*c - b^2)^3)^(1/2) + 12*a^2*b*c^2*e^5 + 8*a
*c^4*d^3*e^2 - 24*a^2*c^3*d*e^4 - 2*b^2*c^3*d^3*e^2 + 3*b^3*c^2*d^2*e^3 - 3*c^2*d^2*e^3*(-(4*a*c - b^2)^3)^(1/
2) - 7*a*b^3*c*e^5 + a*c*e^5*(-(4*a*c - b^2)^3)^(1/2) - 3*b^4*c*d*e^4 + 3*b*c*d*e^4*(-(4*a*c - b^2)^3)^(1/2) -
12*a*b*c^3*d^2*e^3 + 18*a*b^2*c^2*d*e^4))/(8*(16*a^2*c^5 + b^4*c^3 - 8*a*b^2*c^4)))^(1/2) - (2*(d + e*x)^(1/2
)*(25*b^4*e^8 + 50*a^2*c^2*e^8 + 50*c^4*d^4*e^4 - 300*a*c^3*d^2*e^6 - 100*b*c^3*d^3*e^5 + 150*b^2*c^2*d^2*e^6
- 100*a*b^2*c*e^8 - 100*b^3*c*d*e^7 + 300*a*b*c^2*d*e^7))/c)*(-(25*(b^5*e^5 - b^2*e^5*(-(4*a*c - b^2)^3)^(1/2)
+ 12*a^2*b*c^2*e^5 + 8*a*c^4*d^3*e^2 - 24*a^2*c^3*d*e^4 - 2*b^2*c^3*d^3*e^2 + 3*b^3*c^2*d^2*e^3 - 3*c^2*d^2*e
^3*(-(4*a*c - b^2)^3)^(1/2) - 7*a*b^3*c*e^5 + a*c*e^5*(-(4*a*c - b^2)^3)^(1/2) - 3*b^4*c*d*e^4 + 3*b*c*d*e^4*(
-(4*a*c - b^2)^3)^(1/2) - 12*a*b*c^3*d^2*e^3 + 18*a*b^2*c^2*d*e^4))/(8*(16*a^2*c^5 + b^4*c^3 - 8*a*b^2*c^4)))^
(1/2) + (((5*(16*a^2*c^3*e^6 - 4*a*b^2*c^2*e^6 + 16*a*c^4*d^2*e^4 + 4*b^3*c^2*d*e^5 - 4*b^2*c^3*d^2*e^4 - 16*a
*b*c^3*d*e^5))/c + (2*(d + e*x)^(1/2)*(-(25*(b^5*e^5 - b^2*e^5*(-(4*a*c - b^2)^3)^(1/2) + 12*a^2*b*c^2*e^5 + 8
*a*c^4*d^3*e^2 - 24*a^2*c^3*d*e^4 - 2*b^2*c^3*d^3*e^2 + 3*b^3*c^2*d^2*e^3 - 3*c^2*d^2*e^3*(-(4*a*c - b^2)^3)^(
1/2) - 7*a*b^3*c*e^5 + a*c*e^5*(-(4*a*c - b^2)^3)^(1/2) - 3*b^4*c*d*e^4 + 3*b*c*d*e^4*(-(4*a*c - b^2)^3)^(1/2)
- 12*a*b*c^3*d^2*e^3 + 18*a*b^2*c^2*d*e^4))/(8*(16*a^2*c^5 + b^4*c^3 - 8*a*b^2*c^4)))^(1/2)*(4*b^3*c^3*e^3 -
8*b^2*c^4*d*e^2 - 16*a*b*c^4*e^3 + 32*a*c^5*d*e^2))/c)*(-(25*(b^5*e^5 - b^2*e^5*(-(4*a*c - b^2)^3)^(1/2) + 12*
a^2*b*c^2*e^5 + 8*a*c^4*d^3*e^2 - 24*a^2*c^3*d*e^4 - 2*b^2*c^3*d^3*e^2 + 3*b^3*c^2*d^2*e^3 - 3*c^2*d^2*e^3*(-(
4*a*c - b^2)^3)^(1/2) - 7*a*b^3*c*e^5 + a*c*e^5*(-(4*a*c - b^2)^3)^(1/2) - 3*b^4*c*d*e^4 + 3*b*c*d*e^4*(-(4*a*
c - b^2)^3)^(1/2) - 12*a*b*c^3*d^2*e^3 + 18*a*b^2*c^2*d*e^4))/(8*(16*a^2*c^5 + b^4*c^3 - 8*a*b^2*c^4)))^(1/2)
+ (2*(d + e*x)^(1/2)*(25*b^4*e^8 + 50*a^2*c^2*e^8 + 50*c^4*d^4*e^4 - 300*a*c^3*d^2*e^6 - 100*b*c^3*d^3*e^5 + 1
50*b^2*c^2*d^2*e^6 - 100*a*b^2*c*e^8 - 100*b^3*c*d*e^7 + 300*a*b*c^2*d*e^7))/c)*(-(25*(b^5*e^5 - b^2*e^5*(-(4*
a*c - b^2)^3)^(1/2) + 12*a^2*b*c^2*e^5 + 8*a*c^4*d^3*e^2 - 24*a^2*c^3*d*e^4 - 2*b^2*c^3*d^3*e^2 + 3*b^3*c^2*d^
2*e^3 - 3*c^2*d^2*e^3*(-(4*a*c - b^2)^3)^(1/2) - 7*a*b^3*c*e^5 + a*c*e^5*(-(4*a*c - b^2)^3)^(1/2) - 3*b^4*c*d*
e^4 + 3*b*c*d*e^4*(-(4*a*c - b^2)^3)^(1/2) - 12*a*b*c^3*d^2*e^3 + 18*a*b^2*c^2*d*e^4))/(8*(16*a^2*c^5 + b^4*c^
3 - 8*a*b^2*c^4)))^(1/2) - (10*(50*c^3*d^5*e^6 - 25*b^3*d^2*e^9 - 25*a^2*b*e^11 + 100*a*c^2*d^3*e^8 - 125*b*c^
2*d^4*e^7 + 100*b^2*c*d^3*e^8 + 50*a*b^2*d*e^10 + 50*a^2*c*d*e^10 - 150*a*b*c*d^2*e^9))/c))*(-(25*(b^5*e^5 - b
^2*e^5*(-(4*a*c - b^2)^3)^(1/2) + 12*a^2*b*c^2*e^5 + 8*a*c^4*d^3*e^2 - 24*a^2*c^3*d*e^4 - 2*b^2*c^3*d^3*e^2 +
3*b^3*c^2*d^2*e^3 - 3*c^2*d^2*e^3*(-(4*a*c - b^2)^3)^(1/2) - 7*a*b^3*c*e^5 + a*c*e^5*(-(4*a*c - b^2)^3)^(1/2)
- 3*b^4*c*d*e^4 + 3*b*c*d*e^4*(-(4*a*c - b^2)^3)^(1/2) - 12*a*b*c^3*d^2*e^3 + 18*a*b^2*c^2*d*e^4))/(8*(16*a^2*
c^5 + b^4*c^3 - 8*a*b^2*c^4)))^(1/2)*2i + (4*e^2*(d + e*x)^(1/2))/c

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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)**2,x)

[Out]

Timed out

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