3.17.24 \(\int \frac {(b+2 c x) (d+e x)^{5/2}}{(a+b x+c x^2)^3} \, dx\) [1624]

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

[Out]

-1/2*(e*x+d)^(5/2)/(c*x^2+b*x+a)^2-5/4*e*(b*d-2*a*e+(-b*e+2*c*d)*x)*(e*x+d)^(1/2)/(-4*a*c+b^2)/(c*x^2+b*x+a)+5
/8*e*arctanh(2^(1/2)*c^(1/2)*(e*x+d)^(1/2)/(2*c*d-e*(b-(-4*a*c+b^2)^(1/2)))^(1/2))*(8*c^2*d^2+b*e^2*(b-(-4*a*c
+b^2)^(1/2))-2*c*e*(4*b*d-2*a*e-d*(-4*a*c+b^2)^(1/2)))/(-4*a*c+b^2)^(3/2)*2^(1/2)/c^(1/2)/(2*c*d-e*(b-(-4*a*c+
b^2)^(1/2)))^(1/2)-5/8*e*arctanh(2^(1/2)*c^(1/2)*(e*x+d)^(1/2)/(2*c*d-e*(b+(-4*a*c+b^2)^(1/2)))^(1/2))*(8*c^2*
d^2+b*e^2*(b+(-4*a*c+b^2)^(1/2))-2*c*e*(4*b*d-2*a*e+d*(-4*a*c+b^2)^(1/2)))/(-4*a*c+b^2)^(3/2)*2^(1/2)/c^(1/2)/
(2*c*d-e*(b+(-4*a*c+b^2)^(1/2)))^(1/2)

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Rubi [A]
time = 1.10, 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 = {782, 752, 840, 1180, 214} \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]

-1/2*(d + e*x)^(5/2)/(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 214

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

Rule 752

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 782

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 840

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 1180

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) \text {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 ) \text {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 ) \text {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] Leaf count is larger than twice the leaf count of optimal. \(1197\) vs. \(2(398)=796\).
time = 16.39, size = 1197, normalized size = 3.01 \begin {gather*} -\frac {(d+e x)^{7/2} \left (-2 a c (2 c d-b e)+b \left (b c d-b^2 e+2 a c e\right )+c (-2 c (b d-2 a e)+b (2 c d-b e)) x\right )}{2 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) \left (a+b x+c x^2\right )^2}-\frac {-\frac {(d+e x)^{7/2} \left (-\frac {1}{2} a c \left (b^2-4 a c\right ) e^2 (2 c d-b e)-\frac {1}{2} \left (b^2-4 a c\right ) e (5 c d-3 b e) \left (b c d-b^2 e+2 a c e\right )+c \left (-\frac {1}{2} c \left (b^2-4 a c\right ) e^2 (b d-2 a e)-\frac {1}{2} \left (b^2-4 a c\right ) e (5 c d-3 b e) (2 c d-b e)\right ) x\right )}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) \left (a+b x+c x^2\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 ) e^2 (2 c d-b e) \left (c d^2-e (b d-a e)\right ) (d+e x)^{3/2}+\frac {2 \left (\frac {75}{4} c^2 \left (b^2-4 a c\right ) e^2 \left (c d^2-e (b d-a e)\right )^2 \sqrt {d+e x}+\frac {4 \left (\frac {\sqrt {2 c d-b e-\sqrt {b^2-4 a c} e} \left (-\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-\frac {\frac {75}{32} c^3 \left (b^2-4 a c\right ) e^2 (2 c d-b e) (-2 c d+b e) \left (c d^2-e (b d-a e)\right )^2+2 c \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 )}{\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 {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+\frac {\frac {75}{32} c^3 \left (b^2-4 a c\right ) e^2 (2 c d-b e) (-2 c d+b e) \left (c d^2-e (b d-a e)\right )^2+2 c \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 )}{\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 )}\right )}{c}\right )}{3 c}\right )}{5 c}}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )}}{2 \left (b^2-4 a c\right ) \left (c d^2-b d e+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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Maple [A]
time = 1.10, size = 601, normalized size = 1.51

method	result	size
derivativedivides	\(2 e^{4} \left (\frac {-\frac {5 \left (b e -2 c d \right ) c \left (e x +d \right )^{\frac {7}{2}}}{8 e^{2} \left (4 a c -b^{2}\right )}-\frac {3 \left (6 a c \,e^{2}+b^{2} e^{2}-10 b c d e +10 c^{2} d^{2}\right ) \left (e x +d \right )^{\frac {5}{2}}}{8 \left (4 a c -b^{2}\right ) e^{2}}-\frac {15 \left (a b \,e^{3}-2 a d \,e^{2} c -b^{2} d \,e^{2}+3 d^{2} e b c -2 c^{2} d^{3}\right ) \left (e x +d \right )^{\frac {3}{2}}}{8 e^{2} \left (4 a c -b^{2}\right )}-\frac {5 \left (e^{2} a -b d e +c \,d^{2}\right )^{2} \sqrt {e x +d}}{4 e^{2} \left (4 a c -b^{2}\right )}}{\left (\left (e x +d \right )^{2} c +b e \left (e x +d \right )-2 c d \left (e x +d \right )+e^{2} a -b d e +c \,d^{2}\right )^{2}}+\frac {5 c \left (-\frac {\left (4 a c \,e^{2}+b^{2} e^{2}-8 b c d e +8 c^{2} d^{2}-\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b e +2 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, c d \right ) \sqrt {2}\, \arctanh \left (\frac {c \sqrt {e x +d}\, \sqrt {2}}{\sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{8 c \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, \sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}+\frac {\left (-4 a c \,e^{2}-b^{2} e^{2}+8 b c d e -8 c^{2} d^{2}-\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b e +2 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, c d \right ) \sqrt {2}\, \arctan \left (\frac {c \sqrt {e x +d}\, \sqrt {2}}{\sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{8 c \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, \sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{2 e^{2} \left (4 a c -b^{2}\right )}\right )\)	\(601\)
default	\(2 e^{4} \left (\frac {-\frac {5 \left (b e -2 c d \right ) c \left (e x +d \right )^{\frac {7}{2}}}{8 e^{2} \left (4 a c -b^{2}\right )}-\frac {3 \left (6 a c \,e^{2}+b^{2} e^{2}-10 b c d e +10 c^{2} d^{2}\right ) \left (e x +d \right )^{\frac {5}{2}}}{8 \left (4 a c -b^{2}\right ) e^{2}}-\frac {15 \left (a b \,e^{3}-2 a d \,e^{2} c -b^{2} d \,e^{2}+3 d^{2} e b c -2 c^{2} d^{3}\right ) \left (e x +d \right )^{\frac {3}{2}}}{8 e^{2} \left (4 a c -b^{2}\right )}-\frac {5 \left (e^{2} a -b d e +c \,d^{2}\right )^{2} \sqrt {e x +d}}{4 e^{2} \left (4 a c -b^{2}\right )}}{\left (\left (e x +d \right )^{2} c +b e \left (e x +d \right )-2 c d \left (e x +d \right )+e^{2} a -b d e +c \,d^{2}\right )^{2}}+\frac {5 c \left (-\frac {\left (4 a c \,e^{2}+b^{2} e^{2}-8 b c d e +8 c^{2} d^{2}-\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b e +2 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, c d \right ) \sqrt {2}\, \arctanh \left (\frac {c \sqrt {e x +d}\, \sqrt {2}}{\sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{8 c \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, \sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}+\frac {\left (-4 a c \,e^{2}-b^{2} e^{2}+8 b c d e -8 c^{2} d^{2}-\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b e +2 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, c d \right ) \sqrt {2}\, \arctan \left (\frac {c \sqrt {e x +d}\, \sqrt {2}}{\sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{8 c \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, \sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{2 e^{2} \left (4 a c -b^{2}\right )}\right )\)	\(601\)

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,method=_RETURNVERBOSE)

[Out]

2*e^4*((-5/8*(b*e-2*c*d)*c/e^2/(4*a*c-b^2)*(e*x+d)^(7/2)-3/8*(6*a*c*e^2+b^2*e^2-10*b*c*d*e+10*c^2*d^2)/(4*a*c-
b^2)/e^2*(e*x+d)^(5/2)-15/8*(a*b*e^3-2*a*c*d*e^2-b^2*d*e^2+3*b*c*d^2*e-2*c^2*d^3)/e^2/(4*a*c-b^2)*(e*x+d)^(3/2
)-5/4*(a*e^2-b*d*e+c*d^2)^2/e^2/(4*a*c-b^2)*(e*x+d)^(1/2))/((e*x+d)^2*c+b*e*(e*x+d)-2*c*d*(e*x+d)+e^2*a-b*d*e+
c*d^2)^2+5/2/e^2/(4*a*c-b^2)*c*(-1/8*(4*a*c*e^2+b^2*e^2-8*b*c*d*e+8*c^2*d^2-(-e^2*(4*a*c-b^2))^(1/2)*b*e+2*(-e
^2*(4*a*c-b^2))^(1/2)*c*d)/c/(-e^2*(4*a*c-b^2))^(1/2)*2^(1/2)/((-b*e+2*c*d+(-e^2*(4*a*c-b^2))^(1/2))*c)^(1/2)*
arctanh(c*(e*x+d)^(1/2)*2^(1/2)/((-b*e+2*c*d+(-e^2*(4*a*c-b^2))^(1/2))*c)^(1/2))+1/8*(-4*a*c*e^2-b^2*e^2+8*b*c
*d*e-8*c^2*d^2-(-e^2*(4*a*c-b^2))^(1/2)*b*e+2*(-e^2*(4*a*c-b^2))^(1/2)*c*d)/c/(-e^2*(4*a*c-b^2))^(1/2)*2^(1/2)
/((b*e-2*c*d+(-e^2*(4*a*c-b^2))^(1/2))*c)^(1/2)*arctan(c*(e*x+d)^(1/2)*2^(1/2)/((b*e-2*c*d+(-e^2*(4*a*c-b^2))^
(1/2))*c)^(1/2))))

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \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)*(x*e + d)^(5/2)/(c*x^2 + b*x + a)^3, x)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 2677 vs. \(2 (353) = 706\).
time = 2.12, size = 2677, normalized size = 6.73 \begin {gather*} \text {Too large to display} \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="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 + (b^6*c - 12*a*b^4*c^2 + 48*a^2*b^2*c^3 - 64*a^3*c^4)*e^5/sqrt(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))*log(125*sqrt(1/2)*((b
^4 - 8*a*b^2*c + 16*a^2*c^2)*e^6 + 2*(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)*e^5/sqrt(b^6*c^2 - 12*a*b^4*c^3 + 48*a^2*b^2*c^4 - 64*a^3*c^5))
*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 + (b^6*c - 12*a*
b^4*c^2 + 48*a^2*b^2*c^3 - 64*a^3*c^4)*e^5/sqrt(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)) + 125*(16*c^2*d^2*e^6 - 16*b*c*d*e^7 + (3*b^2 + 4*a*c)*e^8)*sqr
t(x*e + 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 + (b^6*c - 12*a*b^4*c^2 + 48*a^2*b^2*c^3 - 64*a^3*c^4)*e^5/sqrt(b^6*c^2 - 1
2*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))*log(-125*sqr
t(1/2)*((b^4 - 8*a*b^2*c + 16*a^2*c^2)*e^6 + 2*(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)*e^5/sqrt(b^6*c^2 - 12*a*b^4*c^3 + 48*a^2*b^2*c^4 - 64
*a^3*c^5))*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 + (b^6
*c - 12*a*b^4*c^2 + 48*a^2*b^2*c^3 - 64*a^3*c^4)*e^5/sqrt(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)) + 125*(16*c^2*d^2*e^6 - 16*b*c*d*e^7 + (3*b^2 + 4*a*c
)*e^8)*sqrt(x*e + 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 - (b^6*c - 12*a*b^4*c^2 + 48*a^2*b^2*c^3 - 64*a^3*c^4)*e^5/sqrt(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))*lo
g(125*sqrt(1/2)*((b^4 - 8*a*b^2*c + 16*a^2*c^2)*e^6 - 2*(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)*e^5/sqrt(b^6*c^2 - 12*a*b^4*c^3 + 48*a^2*b^2
*c^4 - 64*a^3*c^5))*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 - (b^6*c - 12*a*b^4*c^2 + 48*a^2*b^2*c^3 - 64*a^3*c^4)*e^5/sqrt(b^6*c^2 - 12*a*b^4*c^3 + 48*a^2*b^2*c^4 - 6
4*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)) + 125*(16*c^2*d^2*e^6 - 16*b*c*d*e^7 + (3*b^
2 + 4*a*c)*e^8)*sqrt(x*e + 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 - (b^6*c - 12*a*b^4*c^2 + 48*a^2*b^2*c^3 - 64*a^3*c^4)*e
^5/sqrt(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))*log(-125*sqrt(1/2)*((b^4 - 8*a*b^2*c + 16*a^2*c^2)*e^6 - 2*(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)*e^5/sqrt(b^6*c^2 - 12*a*b^4*c^3 +
48*a^2*b^2*c^4 - 64*a^3*c^5))*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 + 1
2*a*b*c)*e^5 - (b^6*c - 12*a*b^4*c^2 + 48*a^2*b^2*c^3 - 64*a^3*c^4)*e^5/sqrt(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)) + 125*(16*c^2*d^2*e^6 - 16*b*c*d*e
^7 + (3*b^2 + 4*a*c)*e^8)*sqrt(x*e + d)) - 2*(2*(b^2 - 4*a*c)*d^2 - (5*b*c*x^3 + 15*a*b*x + 3*(b^2 + 6*a*c)*x^
2 + 10*a^2)*e^2 + (10*c^2*d*x^3 + 15*b*c*d*x^2 + 5*a*b*d + 3*(3*b^2 - 2*a*c)*d*x)*e)*sqrt(x*e + 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)

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Sympy [F(-1)] Timed out
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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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 1425 vs. \(2 (353) = 706\).
time = 3.75, size = 1425, normalized size = 3.58 \begin {gather*} -\frac {5 \, {\left (\sqrt {-4 \, c^{2} d + 2 \, {\left (b c - \sqrt {b^{2} - 4 \, a c} c\right )} e} {\left (2 \, c d e^{2} - b e^{3}\right )} {\left (b^{2} e - 4 \, a c e\right )}^{2} + 4 \, {\left (\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}\right )} \sqrt {-4 \, c^{2} d + 2 \, {\left (b c - \sqrt {b^{2} - 4 \, a c} c\right )} e} {\left | b^{2} e - 4 \, a c e \right |} - {\left (16 \, {\left (b^{2} c^{3} - 4 \, a c^{4}\right )} d^{3} e^{2} - 24 \, {\left (b^{3} c^{2} - 4 \, a b c^{3}\right )} d^{2} e^{3} + 2 \, {\left (5 \, b^{4} c - 16 \, a b^{2} c^{2} - 16 \, a^{2} c^{3}\right )} d e^{4} - {\left (b^{5} - 16 \, a^{2} b c^{2}\right )} e^{5}\right )} \sqrt {-4 \, c^{2} d + 2 \, {\left (b c - \sqrt {b^{2} - 4 \, a c} c\right )} e}\right )} \arctan \left (\frac {2 \, \sqrt {\frac {1}{2}} \sqrt {x e + d}}{\sqrt {-\frac {2 \, b^{2} c d - 8 \, a c^{2} d - b^{3} e + 4 \, a b c e + \sqrt {{\left (2 \, b^{2} c d - 8 \, a c^{2} d - b^{3} e + 4 \, a b c e\right )}^{2} - 4 \, {\left (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}\right )} {\left (b^{2} c - 4 \, a c^{2}\right )}}}{b^{2} c - 4 \, a c^{2}}}}\right )}{32 \, {\left ({\left (b^{2} c^{2} - 4 \, a c^{3}\right )} \sqrt {b^{2} - 4 \, a c} d^{2} - {\left (b^{3} c - 4 \, a b c^{2}\right )} \sqrt {b^{2} - 4 \, a c} d e + {\left (a b^{2} c - 4 \, a^{2} c^{2}\right )} \sqrt {b^{2} - 4 \, a c} e^{2}\right )} {\left | b^{2} e - 4 \, a c e \right |} {\left | c \right |}} + \frac {5 \, {\left (\sqrt {-4 \, c^{2} d + 2 \, {\left (b c + \sqrt {b^{2} - 4 \, a c} c\right )} e} {\left (2 \, c d e^{2} - b e^{3}\right )} {\left (b^{2} e - 4 \, a c e\right )}^{2} - 4 \, {\left (\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}\right )} \sqrt {-4 \, c^{2} d + 2 \, {\left (b c + \sqrt {b^{2} - 4 \, a c} c\right )} e} {\left | b^{2} e - 4 \, a c e \right |} - {\left (16 \, {\left (b^{2} c^{3} - 4 \, a c^{4}\right )} d^{3} e^{2} - 24 \, {\left (b^{3} c^{2} - 4 \, a b c^{3}\right )} d^{2} e^{3} + 2 \, {\left (5 \, b^{4} c - 16 \, a b^{2} c^{2} - 16 \, a^{2} c^{3}\right )} d e^{4} - {\left (b^{5} - 16 \, a^{2} b c^{2}\right )} e^{5}\right )} \sqrt {-4 \, c^{2} d + 2 \, {\left (b c + \sqrt {b^{2} - 4 \, a c} c\right )} e}\right )} \arctan \left (\frac {2 \, \sqrt {\frac {1}{2}} \sqrt {x e + d}}{\sqrt {-\frac {2 \, b^{2} c d - 8 \, a c^{2} d - b^{3} e + 4 \, a b c e - \sqrt {{\left (2 \, b^{2} c d - 8 \, a c^{2} d - b^{3} e + 4 \, a b c e\right )}^{2} - 4 \, {\left (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}\right )} {\left (b^{2} c - 4 \, a c^{2}\right )}}}{b^{2} c - 4 \, a c^{2}}}}\right )}{32 \, {\left ({\left (b^{2} c^{2} - 4 \, a c^{3}\right )} \sqrt {b^{2} - 4 \, a c} d^{2} - {\left (b^{3} c - 4 \, a b c^{2}\right )} \sqrt {b^{2} - 4 \, a c} d e + {\left (a b^{2} c - 4 \, a^{2} c^{2}\right )} \sqrt {b^{2} - 4 \, a c} e^{2}\right )} {\left | b^{2} e - 4 \, a c e \right |} {\left | c \right |}} - \frac {10 \, {\left (x e + d\right )}^{\frac {7}{2}} c^{2} d e^{2} - 30 \, {\left (x e + d\right )}^{\frac {5}{2}} c^{2} d^{2} e^{2} + 30 \, {\left (x e + d\right )}^{\frac {3}{2}} c^{2} d^{3} e^{2} - 10 \, \sqrt {x e + d} c^{2} d^{4} e^{2} - 5 \, {\left (x e + d\right )}^{\frac {7}{2}} b c e^{3} + 30 \, {\left (x e + d\right )}^{\frac {5}{2}} b c d e^{3} - 45 \, {\left (x e + d\right )}^{\frac {3}{2}} b c d^{2} e^{3} + 20 \, \sqrt {x e + d} b c d^{3} e^{3} - 3 \, {\left (x e + d\right )}^{\frac {5}{2}} b^{2} e^{4} - 18 \, {\left (x e + d\right )}^{\frac {5}{2}} a c e^{4} + 15 \, {\left (x e + d\right )}^{\frac {3}{2}} b^{2} d e^{4} + 30 \, {\left (x e + d\right )}^{\frac {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 \, {\left (x e + d\right )}^{\frac {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}}{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 )}^{2} {\left (b^{2} - 4 \, a c\right )}} \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="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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Mupad [B]
time = 7.65, size = 2500, normalized size = 6.28 \begin {gather*} \text {Too large to display} \end {gather*}

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...

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