3.2285 \(\int \frac{(d+e x)^{5/2}}{\left (a+b x+c x^2\right )^2} \, dx\)

Optimal. Leaf size=504 \[ \frac{\left (-2 c^2 d e \left (-d \sqrt{b^2-4 a c}-8 a e+6 b d\right )+2 c e^2 \left (-b \left (d \sqrt{b^2-4 a c}+4 a e\right )+3 a e \sqrt{b^2-4 a c}+b^2 d\right )+b^2 e^3 \left (b-\sqrt{b^2-4 a c}\right )+8 c^3 d^3\right ) \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{2 c d-e \left (b-\sqrt{b^2-4 a c}\right )}}\right )}{\sqrt{2} c^{3/2} \left (b^2-4 a c\right )^{3/2} \sqrt{2 c d-e \left (b-\sqrt{b^2-4 a c}\right )}}-\frac{\left (-2 c^2 d e \left (d \sqrt{b^2-4 a c}-8 a e+6 b d\right )+2 c e^2 \left (b d \sqrt{b^2-4 a c}-3 a e \sqrt{b^2-4 a c}-4 a b e+b^2 d\right )+b^2 e^3 \left (\sqrt{b^2-4 a c}+b\right )+8 c^3 d^3\right ) \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}\right )}{\sqrt{2} c^{3/2} \left (b^2-4 a c\right )^{3/2} \sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}-\frac{(d+e x)^{3/2} (-2 a e+x (2 c d-b e)+b d)}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}+\frac{e \sqrt{d+e x} (2 c d-b e)}{c \left (b^2-4 a c\right )} \]

[Out]

(e*(2*c*d - b*e)*Sqrt[d + e*x])/(c*(b^2 - 4*a*c)) - ((d + e*x)^(3/2)*(b*d - 2*a*
e + (2*c*d - b*e)*x))/((b^2 - 4*a*c)*(a + b*x + c*x^2)) + ((8*c^3*d^3 + b^2*(b -
Sqrt[b^2 - 4*a*c])*e^3 - 2*c^2*d*e*(6*b*d - Sqrt[b^2 - 4*a*c]*d - 8*a*e) + 2*c*
e^2*(b^2*d + 3*a*Sqrt[b^2 - 4*a*c]*e - b*(Sqrt[b^2 - 4*a*c]*d + 4*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)*(b^2 - 4*a*c)^(3/2)*Sqrt[2*c*d - (b - Sqrt[b^2 - 4*a*c])*e]) - ((8*c
^3*d^3 + b^2*(b + Sqrt[b^2 - 4*a*c])*e^3 - 2*c^2*d*e*(6*b*d + Sqrt[b^2 - 4*a*c]*
d - 8*a*e) + 2*c*e^2*(b^2*d + b*Sqrt[b^2 - 4*a*c]*d - 4*a*b*e - 3*a*Sqrt[b^2 - 4
*a*c]*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)*(b^2 - 4*a*c)^(3/2)*Sqrt[2*c*d - (b + Sqrt[b^2 - 4*
a*c])*e])

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Rubi [A]  time = 10.0162, antiderivative size = 504, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227 \[ \frac{\left (-2 c^2 d e \left (-d \sqrt{b^2-4 a c}-8 a e+6 b d\right )+2 c e^2 \left (-b \left (d \sqrt{b^2-4 a c}+4 a e\right )+3 a e \sqrt{b^2-4 a c}+b^2 d\right )+b^2 e^3 \left (b-\sqrt{b^2-4 a c}\right )+8 c^3 d^3\right ) \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{2 c d-e \left (b-\sqrt{b^2-4 a c}\right )}}\right )}{\sqrt{2} c^{3/2} \left (b^2-4 a c\right )^{3/2} \sqrt{2 c d-e \left (b-\sqrt{b^2-4 a c}\right )}}-\frac{\left (-2 c^2 d e \left (d \sqrt{b^2-4 a c}-8 a e+6 b d\right )+2 c e^2 \left (b d \sqrt{b^2-4 a c}-3 a e \sqrt{b^2-4 a c}-4 a b e+b^2 d\right )+b^2 e^3 \left (\sqrt{b^2-4 a c}+b\right )+8 c^3 d^3\right ) \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}\right )}{\sqrt{2} c^{3/2} \left (b^2-4 a c\right )^{3/2} \sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}-\frac{(d+e x)^{3/2} (-2 a e+x (2 c d-b e)+b d)}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}+\frac{e \sqrt{d+e x} (2 c d-b e)}{c \left (b^2-4 a c\right )} \]

Antiderivative was successfully verified.

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

[Out]

(e*(2*c*d - b*e)*Sqrt[d + e*x])/(c*(b^2 - 4*a*c)) - ((d + e*x)^(3/2)*(b*d - 2*a*
e + (2*c*d - b*e)*x))/((b^2 - 4*a*c)*(a + b*x + c*x^2)) + ((8*c^3*d^3 + b^2*(b -
Sqrt[b^2 - 4*a*c])*e^3 - 2*c^2*d*e*(6*b*d - Sqrt[b^2 - 4*a*c]*d - 8*a*e) + 2*c*
e^2*(b^2*d + 3*a*Sqrt[b^2 - 4*a*c]*e - b*(Sqrt[b^2 - 4*a*c]*d + 4*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)*(b^2 - 4*a*c)^(3/2)*Sqrt[2*c*d - (b - Sqrt[b^2 - 4*a*c])*e]) - ((8*c
^3*d^3 + b^2*(b + Sqrt[b^2 - 4*a*c])*e^3 - 2*c^2*d*e*(6*b*d + Sqrt[b^2 - 4*a*c]*
d - 8*a*e) + 2*c*e^2*(b^2*d + b*Sqrt[b^2 - 4*a*c]*d - 4*a*b*e - 3*a*Sqrt[b^2 - 4
*a*c]*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)*(b^2 - 4*a*c)^(3/2)*Sqrt[2*c*d - (b + Sqrt[b^2 - 4*
a*c])*e])

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Rubi in Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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Mathematica [A]  time = 3.06696, size = 491, normalized size = 0.97 \[ \frac{-\frac{2 \sqrt{c} \sqrt{d+e x} \left (a b e^2-2 a c e (2 d+e x)+b^2 e^2 x+b c d (d-2 e x)+2 c^2 d^2 x\right )}{\left (b^2-4 a c\right ) (a+x (b+c x))}+\frac{\left (2 c^2 d e \left (d \sqrt{b^2-4 a c}+8 a e-6 b d\right )+2 c e^2 \left (-b \left (d \sqrt{b^2-4 a c}+4 a e\right )+3 a e \sqrt{b^2-4 a c}+b^2 d\right )+b^2 e^3 \left (b-\sqrt{b^2-4 a c}\right )+8 c^3 d^3\right ) \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{e \sqrt{b^2-4 a c}-b e+2 c d}}\right )}{\left (b^2-4 a c\right )^{3/2} \sqrt{\frac{1}{2} e \left (\sqrt{b^2-4 a c}-b\right )+c d}}-\frac{\left (-2 c^2 d e \left (d \sqrt{b^2-4 a c}-8 a e+6 b d\right )+2 c e^2 \left (b d \sqrt{b^2-4 a c}-3 a e \sqrt{b^2-4 a c}-4 a b e+b^2 d\right )+b^2 e^3 \left (\sqrt{b^2-4 a c}+b\right )+8 c^3 d^3\right ) \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}\right )}{\left (b^2-4 a c\right )^{3/2} \sqrt{c d-\frac{1}{2} e \left (\sqrt{b^2-4 a c}+b\right )}}}{2 c^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

((-2*Sqrt[c]*Sqrt[d + e*x]*(a*b*e^2 + 2*c^2*d^2*x + b^2*e^2*x + b*c*d*(d - 2*e*x
) - 2*a*c*e*(2*d + e*x)))/((b^2 - 4*a*c)*(a + x*(b + c*x))) + ((8*c^3*d^3 + b^2*
(b - Sqrt[b^2 - 4*a*c])*e^3 + 2*c^2*d*e*(-6*b*d + Sqrt[b^2 - 4*a*c]*d + 8*a*e) +
2*c*e^2*(b^2*d + 3*a*Sqrt[b^2 - 4*a*c]*e - b*(Sqrt[b^2 - 4*a*c]*d + 4*a*e)))*Ar
cTanh[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x])/Sqrt[2*c*d - b*e + Sqrt[b^2 - 4*a*c]*e]])/
((b^2 - 4*a*c)^(3/2)*Sqrt[c*d + ((-b + Sqrt[b^2 - 4*a*c])*e)/2]) - ((8*c^3*d^3 +
b^2*(b + Sqrt[b^2 - 4*a*c])*e^3 - 2*c^2*d*e*(6*b*d + Sqrt[b^2 - 4*a*c]*d - 8*a*
e) + 2*c*e^2*(b^2*d + b*Sqrt[b^2 - 4*a*c]*d - 4*a*b*e - 3*a*Sqrt[b^2 - 4*a*c]*e)
)*ArcTanh[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x])/Sqrt[2*c*d - (b + Sqrt[b^2 - 4*a*c])*e
]])/((b^2 - 4*a*c)^(3/2)*Sqrt[c*d - ((b + Sqrt[b^2 - 4*a*c])*e)/2]))/(2*c^(3/2))

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Maple [B]  time = 0.171, size = 9812, normalized size = 19.5 \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

result too large to display

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (e x + d\right )}^{\frac{5}{2}}}{{\left (c x^{2} + b x + a\right )}^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.601489, size = 7096, normalized size = 14.08 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*(sqrt(1/2)*(a*b^2*c - 4*a^2*c^2 + (b^2*c^2 - 4*a*c^3)*x^2 + (b^3*c - 4*a*b*c
^2)*x)*sqrt((32*c^5*d^5 - 80*b*c^4*d^4*e + 10*(5*b^2*c^3 + 12*a*c^4)*d^3*e^2 + 5
*(b^3*c^2 - 36*a*b*c^3)*d^2*e^3 - 5*(b^4*c - 6*a*b^2*c^2 - 24*a^2*c^3)*d*e^4 - (
b^5 - 15*a*b^3*c + 60*a^2*b*c^2)*e^5 + (b^6*c^3 - 12*a*b^4*c^4 + 48*a^2*b^2*c^5
- 64*a^3*c^6)*sqrt((25*c^4*d^4*e^6 - 50*b*c^3*d^3*e^7 + 15*(b^2*c^2 + 6*a*c^3)*d
^2*e^8 + 10*(b^3*c - 9*a*b*c^2)*d*e^9 + (b^4 - 18*a*b^2*c + 81*a^2*c^2)*e^10)/(b
^6*c^6 - 12*a*b^4*c^7 + 48*a^2*b^2*c^8 - 64*a^3*c^9)))/(b^6*c^3 - 12*a*b^4*c^4 +
48*a^2*b^2*c^5 - 64*a^3*c^6))*log(sqrt(1/2)*(10*(b^4*c^3 - 8*a*b^2*c^4 + 16*a^2
*c^5)*d^3*e^4 - 15*(b^5*c^2 - 8*a*b^3*c^3 + 16*a^2*b*c^4)*d^2*e^5 + 3*(b^6*c - 2
*a*b^4*c^2 - 32*a^2*b^2*c^3 + 96*a^3*c^4)*d*e^6 + (b^7 - 17*a*b^5*c + 88*a^2*b^3
*c^2 - 144*a^3*b*c^3)*e^7 - (8*(b^6*c^5 - 12*a*b^4*c^6 + 48*a^2*b^2*c^7 - 64*a^3
*c^8)*d^2 - 8*(b^7*c^4 - 12*a*b^5*c^5 + 48*a^2*b^3*c^6 - 64*a^3*b*c^7)*d*e - (b^
8*c^3 - 24*a*b^6*c^4 + 192*a^2*b^4*c^5 - 640*a^3*b^2*c^6 + 768*a^4*c^7)*e^2)*sqr
t((25*c^4*d^4*e^6 - 50*b*c^3*d^3*e^7 + 15*(b^2*c^2 + 6*a*c^3)*d^2*e^8 + 10*(b^3*
c - 9*a*b*c^2)*d*e^9 + (b^4 - 18*a*b^2*c + 81*a^2*c^2)*e^10)/(b^6*c^6 - 12*a*b^4
*c^7 + 48*a^2*b^2*c^8 - 64*a^3*c^9)))*sqrt((32*c^5*d^5 - 80*b*c^4*d^4*e + 10*(5*
b^2*c^3 + 12*a*c^4)*d^3*e^2 + 5*(b^3*c^2 - 36*a*b*c^3)*d^2*e^3 - 5*(b^4*c - 6*a*
b^2*c^2 - 24*a^2*c^3)*d*e^4 - (b^5 - 15*a*b^3*c + 60*a^2*b*c^2)*e^5 + (b^6*c^3 -
12*a*b^4*c^4 + 48*a^2*b^2*c^5 - 64*a^3*c^6)*sqrt((25*c^4*d^4*e^6 - 50*b*c^3*d^3
*e^7 + 15*(b^2*c^2 + 6*a*c^3)*d^2*e^8 + 10*(b^3*c - 9*a*b*c^2)*d*e^9 + (b^4 - 18
*a*b^2*c + 81*a^2*c^2)*e^10)/(b^6*c^6 - 12*a*b^4*c^7 + 48*a^2*b^2*c^8 - 64*a^3*c
^9)))/(b^6*c^3 - 12*a*b^4*c^4 + 48*a^2*b^2*c^5 - 64*a^3*c^6)) + 2*(80*c^5*d^6*e^
3 - 240*b*c^4*d^5*e^4 + (199*b^2*c^3 + 404*a*c^4)*d^4*e^5 + 2*(b^3*c^2 - 404*a*b
*c^3)*d^3*e^6 - 6*(6*b^4*c - 47*a*b^2*c^2 - 108*a^2*c^3)*d^2*e^7 - (5*b^5 - 122*
a*b^3*c + 648*a^2*b*c^2)*d*e^8 + (5*a*b^4 - 81*a^2*b^2*c + 324*a^3*c^2)*e^9)*sqr
t(e*x + d)) - sqrt(1/2)*(a*b^2*c - 4*a^2*c^2 + (b^2*c^2 - 4*a*c^3)*x^2 + (b^3*c
- 4*a*b*c^2)*x)*sqrt((32*c^5*d^5 - 80*b*c^4*d^4*e + 10*(5*b^2*c^3 + 12*a*c^4)*d^
3*e^2 + 5*(b^3*c^2 - 36*a*b*c^3)*d^2*e^3 - 5*(b^4*c - 6*a*b^2*c^2 - 24*a^2*c^3)*
d*e^4 - (b^5 - 15*a*b^3*c + 60*a^2*b*c^2)*e^5 + (b^6*c^3 - 12*a*b^4*c^4 + 48*a^2
*b^2*c^5 - 64*a^3*c^6)*sqrt((25*c^4*d^4*e^6 - 50*b*c^3*d^3*e^7 + 15*(b^2*c^2 + 6
*a*c^3)*d^2*e^8 + 10*(b^3*c - 9*a*b*c^2)*d*e^9 + (b^4 - 18*a*b^2*c + 81*a^2*c^2)
*e^10)/(b^6*c^6 - 12*a*b^4*c^7 + 48*a^2*b^2*c^8 - 64*a^3*c^9)))/(b^6*c^3 - 12*a*
b^4*c^4 + 48*a^2*b^2*c^5 - 64*a^3*c^6))*log(-sqrt(1/2)*(10*(b^4*c^3 - 8*a*b^2*c^
4 + 16*a^2*c^5)*d^3*e^4 - 15*(b^5*c^2 - 8*a*b^3*c^3 + 16*a^2*b*c^4)*d^2*e^5 + 3*
(b^6*c - 2*a*b^4*c^2 - 32*a^2*b^2*c^3 + 96*a^3*c^4)*d*e^6 + (b^7 - 17*a*b^5*c +
88*a^2*b^3*c^2 - 144*a^3*b*c^3)*e^7 - (8*(b^6*c^5 - 12*a*b^4*c^6 + 48*a^2*b^2*c^
7 - 64*a^3*c^8)*d^2 - 8*(b^7*c^4 - 12*a*b^5*c^5 + 48*a^2*b^3*c^6 - 64*a^3*b*c^7)
*d*e - (b^8*c^3 - 24*a*b^6*c^4 + 192*a^2*b^4*c^5 - 640*a^3*b^2*c^6 + 768*a^4*c^7
)*e^2)*sqrt((25*c^4*d^4*e^6 - 50*b*c^3*d^3*e^7 + 15*(b^2*c^2 + 6*a*c^3)*d^2*e^8
+ 10*(b^3*c - 9*a*b*c^2)*d*e^9 + (b^4 - 18*a*b^2*c + 81*a^2*c^2)*e^10)/(b^6*c^6
- 12*a*b^4*c^7 + 48*a^2*b^2*c^8 - 64*a^3*c^9)))*sqrt((32*c^5*d^5 - 80*b*c^4*d^4*
e + 10*(5*b^2*c^3 + 12*a*c^4)*d^3*e^2 + 5*(b^3*c^2 - 36*a*b*c^3)*d^2*e^3 - 5*(b^
4*c - 6*a*b^2*c^2 - 24*a^2*c^3)*d*e^4 - (b^5 - 15*a*b^3*c + 60*a^2*b*c^2)*e^5 +
(b^6*c^3 - 12*a*b^4*c^4 + 48*a^2*b^2*c^5 - 64*a^3*c^6)*sqrt((25*c^4*d^4*e^6 - 50
*b*c^3*d^3*e^7 + 15*(b^2*c^2 + 6*a*c^3)*d^2*e^8 + 10*(b^3*c - 9*a*b*c^2)*d*e^9 +
(b^4 - 18*a*b^2*c + 81*a^2*c^2)*e^10)/(b^6*c^6 - 12*a*b^4*c^7 + 48*a^2*b^2*c^8
- 64*a^3*c^9)))/(b^6*c^3 - 12*a*b^4*c^4 + 48*a^2*b^2*c^5 - 64*a^3*c^6)) + 2*(80*
c^5*d^6*e^3 - 240*b*c^4*d^5*e^4 + (199*b^2*c^3 + 404*a*c^4)*d^4*e^5 + 2*(b^3*c^2
- 404*a*b*c^3)*d^3*e^6 - 6*(6*b^4*c - 47*a*b^2*c^2 - 108*a^2*c^3)*d^2*e^7 - (5*
b^5 - 122*a*b^3*c + 648*a^2*b*c^2)*d*e^8 + (5*a*b^4 - 81*a^2*b^2*c + 324*a^3*c^2
)*e^9)*sqrt(e*x + d)) + sqrt(1/2)*(a*b^2*c - 4*a^2*c^2 + (b^2*c^2 - 4*a*c^3)*x^2
+ (b^3*c - 4*a*b*c^2)*x)*sqrt((32*c^5*d^5 - 80*b*c^4*d^4*e + 10*(5*b^2*c^3 + 12
*a*c^4)*d^3*e^2 + 5*(b^3*c^2 - 36*a*b*c^3)*d^2*e^3 - 5*(b^4*c - 6*a*b^2*c^2 - 24
*a^2*c^3)*d*e^4 - (b^5 - 15*a*b^3*c + 60*a^2*b*c^2)*e^5 - (b^6*c^3 - 12*a*b^4*c^
4 + 48*a^2*b^2*c^5 - 64*a^3*c^6)*sqrt((25*c^4*d^4*e^6 - 50*b*c^3*d^3*e^7 + 15*(b
^2*c^2 + 6*a*c^3)*d^2*e^8 + 10*(b^3*c - 9*a*b*c^2)*d*e^9 + (b^4 - 18*a*b^2*c + 8
1*a^2*c^2)*e^10)/(b^6*c^6 - 12*a*b^4*c^7 + 48*a^2*b^2*c^8 - 64*a^3*c^9)))/(b^6*c
^3 - 12*a*b^4*c^4 + 48*a^2*b^2*c^5 - 64*a^3*c^6))*log(sqrt(1/2)*(10*(b^4*c^3 - 8
*a*b^2*c^4 + 16*a^2*c^5)*d^3*e^4 - 15*(b^5*c^2 - 8*a*b^3*c^3 + 16*a^2*b*c^4)*d^2
*e^5 + 3*(b^6*c - 2*a*b^4*c^2 - 32*a^2*b^2*c^3 + 96*a^3*c^4)*d*e^6 + (b^7 - 17*a
*b^5*c + 88*a^2*b^3*c^2 - 144*a^3*b*c^3)*e^7 + (8*(b^6*c^5 - 12*a*b^4*c^6 + 48*a
^2*b^2*c^7 - 64*a^3*c^8)*d^2 - 8*(b^7*c^4 - 12*a*b^5*c^5 + 48*a^2*b^3*c^6 - 64*a
^3*b*c^7)*d*e - (b^8*c^3 - 24*a*b^6*c^4 + 192*a^2*b^4*c^5 - 640*a^3*b^2*c^6 + 76
8*a^4*c^7)*e^2)*sqrt((25*c^4*d^4*e^6 - 50*b*c^3*d^3*e^7 + 15*(b^2*c^2 + 6*a*c^3)
*d^2*e^8 + 10*(b^3*c - 9*a*b*c^2)*d*e^9 + (b^4 - 18*a*b^2*c + 81*a^2*c^2)*e^10)/
(b^6*c^6 - 12*a*b^4*c^7 + 48*a^2*b^2*c^8 - 64*a^3*c^9)))*sqrt((32*c^5*d^5 - 80*b
*c^4*d^4*e + 10*(5*b^2*c^3 + 12*a*c^4)*d^3*e^2 + 5*(b^3*c^2 - 36*a*b*c^3)*d^2*e^
3 - 5*(b^4*c - 6*a*b^2*c^2 - 24*a^2*c^3)*d*e^4 - (b^5 - 15*a*b^3*c + 60*a^2*b*c^
2)*e^5 - (b^6*c^3 - 12*a*b^4*c^4 + 48*a^2*b^2*c^5 - 64*a^3*c^6)*sqrt((25*c^4*d^4
*e^6 - 50*b*c^3*d^3*e^7 + 15*(b^2*c^2 + 6*a*c^3)*d^2*e^8 + 10*(b^3*c - 9*a*b*c^2
)*d*e^9 + (b^4 - 18*a*b^2*c + 81*a^2*c^2)*e^10)/(b^6*c^6 - 12*a*b^4*c^7 + 48*a^2
*b^2*c^8 - 64*a^3*c^9)))/(b^6*c^3 - 12*a*b^4*c^4 + 48*a^2*b^2*c^5 - 64*a^3*c^6))
+ 2*(80*c^5*d^6*e^3 - 240*b*c^4*d^5*e^4 + (199*b^2*c^3 + 404*a*c^4)*d^4*e^5 + 2
*(b^3*c^2 - 404*a*b*c^3)*d^3*e^6 - 6*(6*b^4*c - 47*a*b^2*c^2 - 108*a^2*c^3)*d^2*
e^7 - (5*b^5 - 122*a*b^3*c + 648*a^2*b*c^2)*d*e^8 + (5*a*b^4 - 81*a^2*b^2*c + 32
4*a^3*c^2)*e^9)*sqrt(e*x + d)) - sqrt(1/2)*(a*b^2*c - 4*a^2*c^2 + (b^2*c^2 - 4*a
*c^3)*x^2 + (b^3*c - 4*a*b*c^2)*x)*sqrt((32*c^5*d^5 - 80*b*c^4*d^4*e + 10*(5*b^2
*c^3 + 12*a*c^4)*d^3*e^2 + 5*(b^3*c^2 - 36*a*b*c^3)*d^2*e^3 - 5*(b^4*c - 6*a*b^2
*c^2 - 24*a^2*c^3)*d*e^4 - (b^5 - 15*a*b^3*c + 60*a^2*b*c^2)*e^5 - (b^6*c^3 - 12
*a*b^4*c^4 + 48*a^2*b^2*c^5 - 64*a^3*c^6)*sqrt((25*c^4*d^4*e^6 - 50*b*c^3*d^3*e^
7 + 15*(b^2*c^2 + 6*a*c^3)*d^2*e^8 + 10*(b^3*c - 9*a*b*c^2)*d*e^9 + (b^4 - 18*a*
b^2*c + 81*a^2*c^2)*e^10)/(b^6*c^6 - 12*a*b^4*c^7 + 48*a^2*b^2*c^8 - 64*a^3*c^9)
))/(b^6*c^3 - 12*a*b^4*c^4 + 48*a^2*b^2*c^5 - 64*a^3*c^6))*log(-sqrt(1/2)*(10*(b
^4*c^3 - 8*a*b^2*c^4 + 16*a^2*c^5)*d^3*e^4 - 15*(b^5*c^2 - 8*a*b^3*c^3 + 16*a^2*
b*c^4)*d^2*e^5 + 3*(b^6*c - 2*a*b^4*c^2 - 32*a^2*b^2*c^3 + 96*a^3*c^4)*d*e^6 + (
b^7 - 17*a*b^5*c + 88*a^2*b^3*c^2 - 144*a^3*b*c^3)*e^7 + (8*(b^6*c^5 - 12*a*b^4*
c^6 + 48*a^2*b^2*c^7 - 64*a^3*c^8)*d^2 - 8*(b^7*c^4 - 12*a*b^5*c^5 + 48*a^2*b^3*
c^6 - 64*a^3*b*c^7)*d*e - (b^8*c^3 - 24*a*b^6*c^4 + 192*a^2*b^4*c^5 - 640*a^3*b^
2*c^6 + 768*a^4*c^7)*e^2)*sqrt((25*c^4*d^4*e^6 - 50*b*c^3*d^3*e^7 + 15*(b^2*c^2
+ 6*a*c^3)*d^2*e^8 + 10*(b^3*c - 9*a*b*c^2)*d*e^9 + (b^4 - 18*a*b^2*c + 81*a^2*c
^2)*e^10)/(b^6*c^6 - 12*a*b^4*c^7 + 48*a^2*b^2*c^8 - 64*a^3*c^9)))*sqrt((32*c^5*
d^5 - 80*b*c^4*d^4*e + 10*(5*b^2*c^3 + 12*a*c^4)*d^3*e^2 + 5*(b^3*c^2 - 36*a*b*c
^3)*d^2*e^3 - 5*(b^4*c - 6*a*b^2*c^2 - 24*a^2*c^3)*d*e^4 - (b^5 - 15*a*b^3*c + 6
0*a^2*b*c^2)*e^5 - (b^6*c^3 - 12*a*b^4*c^4 + 48*a^2*b^2*c^5 - 64*a^3*c^6)*sqrt((
25*c^4*d^4*e^6 - 50*b*c^3*d^3*e^7 + 15*(b^2*c^2 + 6*a*c^3)*d^2*e^8 + 10*(b^3*c -
9*a*b*c^2)*d*e^9 + (b^4 - 18*a*b^2*c + 81*a^2*c^2)*e^10)/(b^6*c^6 - 12*a*b^4*c^
7 + 48*a^2*b^2*c^8 - 64*a^3*c^9)))/(b^6*c^3 - 12*a*b^4*c^4 + 48*a^2*b^2*c^5 - 64
*a^3*c^6)) + 2*(80*c^5*d^6*e^3 - 240*b*c^4*d^5*e^4 + (199*b^2*c^3 + 404*a*c^4)*d
^4*e^5 + 2*(b^3*c^2 - 404*a*b*c^3)*d^3*e^6 - 6*(6*b^4*c - 47*a*b^2*c^2 - 108*a^2
*c^3)*d^2*e^7 - (5*b^5 - 122*a*b^3*c + 648*a^2*b*c^2)*d*e^8 + (5*a*b^4 - 81*a^2*
b^2*c + 324*a^3*c^2)*e^9)*sqrt(e*x + d)) - 2*(b*c*d^2 - 4*a*c*d*e + a*b*e^2 + (2
*c^2*d^2 - 2*b*c*d*e + (b^2 - 2*a*c)*e^2)*x)*sqrt(e*x + d))/(a*b^2*c - 4*a^2*c^2
+ (b^2*c^2 - 4*a*c^3)*x^2 + (b^3*c - 4*a*b*c^2)*x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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GIAC/XCAS [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out