3.2292 \(\int \frac{(d+e x)^{5/2}}{\left (a+b x+c x^2\right )^3} \, dx\)

Optimal. Leaf size=577 \[ -\frac{3 \sqrt{d+e x} \left (-x \left (-4 c e (2 b d-a e)+b^2 e^2+8 c^2 d^2\right )-4 b \left (a e^2+c d^2\right )+4 a c d e+3 b^2 d e\right )}{4 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )}-\frac{3 \left (-8 c^2 d e \left (-d \sqrt{b^2-4 a c}-3 a e+6 b d\right )+2 c e^2 \left (-4 b d \sqrt{b^2-4 a c}+2 a e \sqrt{b^2-4 a c}-6 a b e+9 b^2 d\right )-b^2 e^3 \left (b-\sqrt{b^2-4 a c}\right )+32 c^3 d^3\right ) \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{2 c d-e \left (b-\sqrt{b^2-4 a c}\right )}}\right )}{4 \sqrt{2} \sqrt{c} \left (b^2-4 a c\right )^{5/2} \sqrt{2 c d-e \left (b-\sqrt{b^2-4 a c}\right )}}+\frac{3 \left (-8 c^2 d e \left (d \sqrt{b^2-4 a c}-3 a e+6 b d\right )+2 c e^2 \left (4 b d \sqrt{b^2-4 a c}-2 a e \sqrt{b^2-4 a c}-6 a b e+9 b^2 d\right )-b^2 e^3 \left (\sqrt{b^2-4 a c}+b\right )+32 c^3 d^3\right ) \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}\right )}{4 \sqrt{2} \sqrt{c} \left (b^2-4 a c\right )^{5/2} \sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}-\frac{(d+e x)^{3/2} (-2 a e+x (2 c d-b e)+b d)}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2} \]

[Out]

-((d + e*x)^(3/2)*(b*d - 2*a*e + (2*c*d - b*e)*x))/(2*(b^2 - 4*a*c)*(a + b*x + c
*x^2)^2) - (3*Sqrt[d + e*x]*(3*b^2*d*e + 4*a*c*d*e - 4*b*(c*d^2 + a*e^2) - (8*c^
2*d^2 + b^2*e^2 - 4*c*e*(2*b*d - a*e))*x))/(4*(b^2 - 4*a*c)^2*(a + b*x + c*x^2))
- (3*(32*c^3*d^3 - b^2*(b - Sqrt[b^2 - 4*a*c])*e^3 - 8*c^2*d*e*(6*b*d - Sqrt[b^
2 - 4*a*c]*d - 3*a*e) + 2*c*e^2*(9*b^2*d - 4*b*Sqrt[b^2 - 4*a*c]*d - 6*a*b*e + 2
*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]])/(4*Sqrt[2]*Sqrt[c]*(b^2 - 4*a*c)^(5/2)*Sqrt[2*c*d - (
b - Sqrt[b^2 - 4*a*c])*e]) + (3*(32*c^3*d^3 - b^2*(b + Sqrt[b^2 - 4*a*c])*e^3 -
8*c^2*d*e*(6*b*d + Sqrt[b^2 - 4*a*c]*d - 3*a*e) + 2*c*e^2*(9*b^2*d + 4*b*Sqrt[b^
2 - 4*a*c]*d - 6*a*b*e - 2*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]])/(4*Sqrt[2]*Sqrt[c]*(b^2 - 4
*a*c)^(5/2)*Sqrt[2*c*d - (b + Sqrt[b^2 - 4*a*c])*e])

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Rubi [A]  time = 10.664, antiderivative size = 577, 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 \[ -\frac{3 \sqrt{d+e x} \left (-x \left (-4 c e (2 b d-a e)+b^2 e^2+8 c^2 d^2\right )-4 b \left (a e^2+c d^2\right )+4 a c d e+3 b^2 d e\right )}{4 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )}-\frac{3 \left (-8 c^2 d e \left (-d \sqrt{b^2-4 a c}-3 a e+6 b d\right )+2 c e^2 \left (-4 b d \sqrt{b^2-4 a c}+2 a e \sqrt{b^2-4 a c}-6 a b e+9 b^2 d\right )-b^2 e^3 \left (b-\sqrt{b^2-4 a c}\right )+32 c^3 d^3\right ) \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{2 c d-e \left (b-\sqrt{b^2-4 a c}\right )}}\right )}{4 \sqrt{2} \sqrt{c} \left (b^2-4 a c\right )^{5/2} \sqrt{2 c d-e \left (b-\sqrt{b^2-4 a c}\right )}}+\frac{3 \left (-8 c^2 d e \left (d \sqrt{b^2-4 a c}-3 a e+6 b d\right )+2 c e^2 \left (4 b d \sqrt{b^2-4 a c}-2 a e \sqrt{b^2-4 a c}-6 a b e+9 b^2 d\right )-b^2 e^3 \left (\sqrt{b^2-4 a c}+b\right )+32 c^3 d^3\right ) \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}\right )}{4 \sqrt{2} \sqrt{c} \left (b^2-4 a c\right )^{5/2} \sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}-\frac{(d+e x)^{3/2} (-2 a e+x (2 c d-b e)+b d)}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2} \]

Antiderivative was successfully verified.

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

[Out]

-((d + e*x)^(3/2)*(b*d - 2*a*e + (2*c*d - b*e)*x))/(2*(b^2 - 4*a*c)*(a + b*x + c
*x^2)^2) - (3*Sqrt[d + e*x]*(3*b^2*d*e + 4*a*c*d*e - 4*b*(c*d^2 + a*e^2) - (8*c^
2*d^2 + b^2*e^2 - 4*c*e*(2*b*d - a*e))*x))/(4*(b^2 - 4*a*c)^2*(a + b*x + c*x^2))
- (3*(32*c^3*d^3 - b^2*(b - Sqrt[b^2 - 4*a*c])*e^3 - 8*c^2*d*e*(6*b*d - Sqrt[b^
2 - 4*a*c]*d - 3*a*e) + 2*c*e^2*(9*b^2*d - 4*b*Sqrt[b^2 - 4*a*c]*d - 6*a*b*e + 2
*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]])/(4*Sqrt[2]*Sqrt[c]*(b^2 - 4*a*c)^(5/2)*Sqrt[2*c*d - (
b - Sqrt[b^2 - 4*a*c])*e]) + (3*(32*c^3*d^3 - b^2*(b + Sqrt[b^2 - 4*a*c])*e^3 -
8*c^2*d*e*(6*b*d + Sqrt[b^2 - 4*a*c]*d - 3*a*e) + 2*c*e^2*(9*b^2*d + 4*b*Sqrt[b^
2 - 4*a*c]*d - 6*a*b*e - 2*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]])/(4*Sqrt[2]*Sqrt[c]*(b^2 - 4
*a*c)^(5/2)*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)**3,x)

[Out]

Timed out

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Mathematica [A]  time = 6.27198, size = 676, normalized size = 1.17 \[ \sqrt{d+e x} \left (\frac{a b e^2-4 a c d e-2 a c e^2 x+b^2 e^2 x+b c d^2-2 b c d e x+2 c^2 d^2 x}{2 c \left (4 a c-b^2\right ) \left (a+b x+c x^2\right )^2}+\frac{4 a b c e^2+4 a c^2 d e+12 a c^2 e^2 x+2 b^3 e^2-13 b^2 c d e+3 b^2 c e^2 x+12 b c^2 d^2-24 b c^2 d e x+24 c^3 d^2 x}{4 c \left (4 a c-b^2\right )^2 \left (a+b x+c x^2\right )}\right )-\frac{3 \left (8 c^2 d^2 e \sqrt{b^2-4 a c}-8 b c d e^2 \sqrt{b^2-4 a c}+b^2 e^3 \sqrt{b^2-4 a c}+4 a c e^3 \sqrt{b^2-4 a c}+12 a b c e^3-24 a c^2 d e^2+b^3 e^3-18 b^2 c d e^2+48 b c^2 d^2 e-32 c^3 d^3\right ) \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{-e \sqrt{b^2-4 a c}-b e+2 c d}}\right )}{4 \sqrt{2} \sqrt{c} \left (b^2-4 a c\right )^{5/2} \sqrt{-e \sqrt{b^2-4 a c}-b e+2 c d}}-\frac{3 \left (8 c^2 d^2 e \sqrt{b^2-4 a c}-8 b c d e^2 \sqrt{b^2-4 a c}+b^2 e^3 \sqrt{b^2-4 a c}+4 a c e^3 \sqrt{b^2-4 a c}-12 a b c e^3+24 a c^2 d e^2-b^3 e^3+18 b^2 c d e^2-48 b c^2 d^2 e+32 c^3 d^3\right ) \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{e \sqrt{b^2-4 a c}-b e+2 c d}}\right )}{4 \sqrt{2} \sqrt{c} \left (b^2-4 a c\right )^{5/2} \sqrt{e \sqrt{b^2-4 a c}-b e+2 c d}} \]

Antiderivative was successfully verified.

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

[Out]

Sqrt[d + e*x]*((b*c*d^2 - 4*a*c*d*e + a*b*e^2 + 2*c^2*d^2*x - 2*b*c*d*e*x + b^2*
e^2*x - 2*a*c*e^2*x)/(2*c*(-b^2 + 4*a*c)*(a + b*x + c*x^2)^2) + (12*b*c^2*d^2 -
13*b^2*c*d*e + 4*a*c^2*d*e + 2*b^3*e^2 + 4*a*b*c*e^2 + 24*c^3*d^2*x - 24*b*c^2*d
*e*x + 3*b^2*c*e^2*x + 12*a*c^2*e^2*x)/(4*c*(-b^2 + 4*a*c)^2*(a + b*x + c*x^2)))
- (3*(-32*c^3*d^3 + 48*b*c^2*d^2*e + 8*c^2*Sqrt[b^2 - 4*a*c]*d^2*e - 18*b^2*c*d
*e^2 - 24*a*c^2*d*e^2 - 8*b*c*Sqrt[b^2 - 4*a*c]*d*e^2 + b^3*e^3 + 12*a*b*c*e^3 +
b^2*Sqrt[b^2 - 4*a*c]*e^3 + 4*a*c*Sqrt[b^2 - 4*a*c]*e^3)*ArcTanh[(Sqrt[2]*Sqrt[
c]*Sqrt[d + e*x])/Sqrt[2*c*d - b*e - Sqrt[b^2 - 4*a*c]*e]])/(4*Sqrt[2]*Sqrt[c]*(
b^2 - 4*a*c)^(5/2)*Sqrt[2*c*d - b*e - Sqrt[b^2 - 4*a*c]*e]) - (3*(32*c^3*d^3 - 4
8*b*c^2*d^2*e + 8*c^2*Sqrt[b^2 - 4*a*c]*d^2*e + 18*b^2*c*d*e^2 + 24*a*c^2*d*e^2
- 8*b*c*Sqrt[b^2 - 4*a*c]*d*e^2 - b^3*e^3 - 12*a*b*c*e^3 + b^2*Sqrt[b^2 - 4*a*c]
*e^3 + 4*a*c*Sqrt[b^2 - 4*a*c]*e^3)*ArcTanh[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x])/Sqrt
[2*c*d - b*e + Sqrt[b^2 - 4*a*c]*e]])/(4*Sqrt[2]*Sqrt[c]*(b^2 - 4*a*c)^(5/2)*Sqr
t[2*c*d - b*e + Sqrt[b^2 - 4*a*c]*e])

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Maple [B]  time = 0.214, size = 21749, normalized size = 37.7 \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

result too large to display

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (e x + d\right )}^{\frac{5}{2}}}{{\left (c x^{2} + b x + a\right )}^{3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.42643, size = 6917, normalized size = 11.99 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/8*(3*sqrt(1/2)*(a^2*b^4 - 8*a^3*b^2*c + 16*a^4*c^2 + (b^4*c^2 - 8*a*b^2*c^3 +
16*a^2*c^4)*x^4 + 2*(b^5*c - 8*a*b^3*c^2 + 16*a^2*b*c^3)*x^3 + (b^6 - 6*a*b^4*c
+ 32*a^3*c^3)*x^2 + 2*(a*b^5 - 8*a^2*b^3*c + 16*a^3*b*c^2)*x)*sqrt((512*c^5*d^5
- 1280*b*c^4*d^4*e + 160*(7*b^2*c^3 + 4*a*c^4)*d^3*e^2 - 80*(5*b^3*c^2 + 12*a*b
*c^3)*d^2*e^3 + 10*(5*b^4*c + 40*a*b^2*c^2 + 16*a^2*c^3)*d*e^4 - (b^5 + 40*a*b^3
*c + 80*a^2*b*c^2)*e^5 + (b^10*c - 20*a*b^8*c^2 + 160*a^2*b^6*c^3 - 640*a^3*b^4*
c^4 + 1280*a^4*b^2*c^5 - 1024*a^5*c^6)*sqrt(e^10/(b^10*c^2 - 20*a*b^8*c^3 + 160*
a^2*b^6*c^4 - 640*a^3*b^4*c^5 + 1280*a^4*b^2*c^6 - 1024*a^5*c^7)))/(b^10*c - 20*
a*b^8*c^2 + 160*a^2*b^6*c^3 - 640*a^3*b^4*c^4 + 1280*a^4*b^2*c^5 - 1024*a^5*c^6)
)*log(27*sqrt(1/2)*(4*(b^6*c - 12*a*b^4*c^2 + 48*a^2*b^2*c^3 - 64*a^3*c^4)*d*e^6
- 2*(b^7 - 12*a*b^5*c + 48*a^2*b^3*c^2 - 64*a^3*b*c^3)*e^7 + sqrt(e^10/(b^10*c^
2 - 20*a*b^8*c^3 + 160*a^2*b^6*c^4 - 640*a^3*b^4*c^5 + 1280*a^4*b^2*c^6 - 1024*a
^5*c^7))*(16*(b^10*c^3 - 20*a*b^8*c^4 + 160*a^2*b^6*c^5 - 640*a^3*b^4*c^6 + 1280
*a^4*b^2*c^7 - 1024*a^5*c^8)*d^2 - 16*(b^11*c^2 - 20*a*b^9*c^3 + 160*a^2*b^7*c^4
- 640*a^3*b^5*c^5 + 1280*a^4*b^3*c^6 - 1024*a^5*b*c^7)*d*e + (3*b^12*c - 56*a*b
^10*c^2 + 400*a^2*b^8*c^3 - 1280*a^3*b^6*c^4 + 1280*a^4*b^4*c^5 + 2048*a^5*b^2*c
^6 - 4096*a^6*c^7)*e^2))*sqrt((512*c^5*d^5 - 1280*b*c^4*d^4*e + 160*(7*b^2*c^3 +
4*a*c^4)*d^3*e^2 - 80*(5*b^3*c^2 + 12*a*b*c^3)*d^2*e^3 + 10*(5*b^4*c + 40*a*b^2
*c^2 + 16*a^2*c^3)*d*e^4 - (b^5 + 40*a*b^3*c + 80*a^2*b*c^2)*e^5 + (b^10*c - 20*
a*b^8*c^2 + 160*a^2*b^6*c^3 - 640*a^3*b^4*c^4 + 1280*a^4*b^2*c^5 - 1024*a^5*c^6)
*sqrt(e^10/(b^10*c^2 - 20*a*b^8*c^3 + 160*a^2*b^6*c^4 - 640*a^3*b^4*c^5 + 1280*a
^4*b^2*c^6 - 1024*a^5*c^7)))/(b^10*c - 20*a*b^8*c^2 + 160*a^2*b^6*c^3 - 640*a^3*
b^4*c^4 + 1280*a^4*b^2*c^5 - 1024*a^5*c^6)) + 27*(256*c^4*d^4*e^5 - 512*b*c^3*d^
3*e^6 + 48*(7*b^2*c^2 + 4*a*c^3)*d^2*e^7 - 16*(5*b^3*c + 12*a*b*c^2)*d*e^8 + (5*
b^4 + 40*a*b^2*c + 16*a^2*c^2)*e^9)*sqrt(e*x + d)) - 3*sqrt(1/2)*(a^2*b^4 - 8*a^
3*b^2*c + 16*a^4*c^2 + (b^4*c^2 - 8*a*b^2*c^3 + 16*a^2*c^4)*x^4 + 2*(b^5*c - 8*a
*b^3*c^2 + 16*a^2*b*c^3)*x^3 + (b^6 - 6*a*b^4*c + 32*a^3*c^3)*x^2 + 2*(a*b^5 - 8
*a^2*b^3*c + 16*a^3*b*c^2)*x)*sqrt((512*c^5*d^5 - 1280*b*c^4*d^4*e + 160*(7*b^2*
c^3 + 4*a*c^4)*d^3*e^2 - 80*(5*b^3*c^2 + 12*a*b*c^3)*d^2*e^3 + 10*(5*b^4*c + 40*
a*b^2*c^2 + 16*a^2*c^3)*d*e^4 - (b^5 + 40*a*b^3*c + 80*a^2*b*c^2)*e^5 + (b^10*c
- 20*a*b^8*c^2 + 160*a^2*b^6*c^3 - 640*a^3*b^4*c^4 + 1280*a^4*b^2*c^5 - 1024*a^5
*c^6)*sqrt(e^10/(b^10*c^2 - 20*a*b^8*c^3 + 160*a^2*b^6*c^4 - 640*a^3*b^4*c^5 + 1
280*a^4*b^2*c^6 - 1024*a^5*c^7)))/(b^10*c - 20*a*b^8*c^2 + 160*a^2*b^6*c^3 - 640
*a^3*b^4*c^4 + 1280*a^4*b^2*c^5 - 1024*a^5*c^6))*log(-27*sqrt(1/2)*(4*(b^6*c - 1
2*a*b^4*c^2 + 48*a^2*b^2*c^3 - 64*a^3*c^4)*d*e^6 - 2*(b^7 - 12*a*b^5*c + 48*a^2*
b^3*c^2 - 64*a^3*b*c^3)*e^7 + sqrt(e^10/(b^10*c^2 - 20*a*b^8*c^3 + 160*a^2*b^6*c
^4 - 640*a^3*b^4*c^5 + 1280*a^4*b^2*c^6 - 1024*a^5*c^7))*(16*(b^10*c^3 - 20*a*b^
8*c^4 + 160*a^2*b^6*c^5 - 640*a^3*b^4*c^6 + 1280*a^4*b^2*c^7 - 1024*a^5*c^8)*d^2
- 16*(b^11*c^2 - 20*a*b^9*c^3 + 160*a^2*b^7*c^4 - 640*a^3*b^5*c^5 + 1280*a^4*b^
3*c^6 - 1024*a^5*b*c^7)*d*e + (3*b^12*c - 56*a*b^10*c^2 + 400*a^2*b^8*c^3 - 1280
*a^3*b^6*c^4 + 1280*a^4*b^4*c^5 + 2048*a^5*b^2*c^6 - 4096*a^6*c^7)*e^2))*sqrt((5
12*c^5*d^5 - 1280*b*c^4*d^4*e + 160*(7*b^2*c^3 + 4*a*c^4)*d^3*e^2 - 80*(5*b^3*c^
2 + 12*a*b*c^3)*d^2*e^3 + 10*(5*b^4*c + 40*a*b^2*c^2 + 16*a^2*c^3)*d*e^4 - (b^5
+ 40*a*b^3*c + 80*a^2*b*c^2)*e^5 + (b^10*c - 20*a*b^8*c^2 + 160*a^2*b^6*c^3 - 64
0*a^3*b^4*c^4 + 1280*a^4*b^2*c^5 - 1024*a^5*c^6)*sqrt(e^10/(b^10*c^2 - 20*a*b^8*
c^3 + 160*a^2*b^6*c^4 - 640*a^3*b^4*c^5 + 1280*a^4*b^2*c^6 - 1024*a^5*c^7)))/(b^
10*c - 20*a*b^8*c^2 + 160*a^2*b^6*c^3 - 640*a^3*b^4*c^4 + 1280*a^4*b^2*c^5 - 102
4*a^5*c^6)) + 27*(256*c^4*d^4*e^5 - 512*b*c^3*d^3*e^6 + 48*(7*b^2*c^2 + 4*a*c^3)
*d^2*e^7 - 16*(5*b^3*c + 12*a*b*c^2)*d*e^8 + (5*b^4 + 40*a*b^2*c + 16*a^2*c^2)*e
^9)*sqrt(e*x + d)) + 3*sqrt(1/2)*(a^2*b^4 - 8*a^3*b^2*c + 16*a^4*c^2 + (b^4*c^2
- 8*a*b^2*c^3 + 16*a^2*c^4)*x^4 + 2*(b^5*c - 8*a*b^3*c^2 + 16*a^2*b*c^3)*x^3 + (
b^6 - 6*a*b^4*c + 32*a^3*c^3)*x^2 + 2*(a*b^5 - 8*a^2*b^3*c + 16*a^3*b*c^2)*x)*sq
rt((512*c^5*d^5 - 1280*b*c^4*d^4*e + 160*(7*b^2*c^3 + 4*a*c^4)*d^3*e^2 - 80*(5*b
^3*c^2 + 12*a*b*c^3)*d^2*e^3 + 10*(5*b^4*c + 40*a*b^2*c^2 + 16*a^2*c^3)*d*e^4 -
(b^5 + 40*a*b^3*c + 80*a^2*b*c^2)*e^5 - (b^10*c - 20*a*b^8*c^2 + 160*a^2*b^6*c^3
- 640*a^3*b^4*c^4 + 1280*a^4*b^2*c^5 - 1024*a^5*c^6)*sqrt(e^10/(b^10*c^2 - 20*a
*b^8*c^3 + 160*a^2*b^6*c^4 - 640*a^3*b^4*c^5 + 1280*a^4*b^2*c^6 - 1024*a^5*c^7))
)/(b^10*c - 20*a*b^8*c^2 + 160*a^2*b^6*c^3 - 640*a^3*b^4*c^4 + 1280*a^4*b^2*c^5
- 1024*a^5*c^6))*log(27*sqrt(1/2)*(4*(b^6*c - 12*a*b^4*c^2 + 48*a^2*b^2*c^3 - 64
*a^3*c^4)*d*e^6 - 2*(b^7 - 12*a*b^5*c + 48*a^2*b^3*c^2 - 64*a^3*b*c^3)*e^7 - sqr
t(e^10/(b^10*c^2 - 20*a*b^8*c^3 + 160*a^2*b^6*c^4 - 640*a^3*b^4*c^5 + 1280*a^4*b
^2*c^6 - 1024*a^5*c^7))*(16*(b^10*c^3 - 20*a*b^8*c^4 + 160*a^2*b^6*c^5 - 640*a^3
*b^4*c^6 + 1280*a^4*b^2*c^7 - 1024*a^5*c^8)*d^2 - 16*(b^11*c^2 - 20*a*b^9*c^3 +
160*a^2*b^7*c^4 - 640*a^3*b^5*c^5 + 1280*a^4*b^3*c^6 - 1024*a^5*b*c^7)*d*e + (3*
b^12*c - 56*a*b^10*c^2 + 400*a^2*b^8*c^3 - 1280*a^3*b^6*c^4 + 1280*a^4*b^4*c^5 +
2048*a^5*b^2*c^6 - 4096*a^6*c^7)*e^2))*sqrt((512*c^5*d^5 - 1280*b*c^4*d^4*e + 1
60*(7*b^2*c^3 + 4*a*c^4)*d^3*e^2 - 80*(5*b^3*c^2 + 12*a*b*c^3)*d^2*e^3 + 10*(5*b
^4*c + 40*a*b^2*c^2 + 16*a^2*c^3)*d*e^4 - (b^5 + 40*a*b^3*c + 80*a^2*b*c^2)*e^5
- (b^10*c - 20*a*b^8*c^2 + 160*a^2*b^6*c^3 - 640*a^3*b^4*c^4 + 1280*a^4*b^2*c^5
- 1024*a^5*c^6)*sqrt(e^10/(b^10*c^2 - 20*a*b^8*c^3 + 160*a^2*b^6*c^4 - 640*a^3*b
^4*c^5 + 1280*a^4*b^2*c^6 - 1024*a^5*c^7)))/(b^10*c - 20*a*b^8*c^2 + 160*a^2*b^6
*c^3 - 640*a^3*b^4*c^4 + 1280*a^4*b^2*c^5 - 1024*a^5*c^6)) + 27*(256*c^4*d^4*e^5
- 512*b*c^3*d^3*e^6 + 48*(7*b^2*c^2 + 4*a*c^3)*d^2*e^7 - 16*(5*b^3*c + 12*a*b*c
^2)*d*e^8 + (5*b^4 + 40*a*b^2*c + 16*a^2*c^2)*e^9)*sqrt(e*x + d)) - 3*sqrt(1/2)*
(a^2*b^4 - 8*a^3*b^2*c + 16*a^4*c^2 + (b^4*c^2 - 8*a*b^2*c^3 + 16*a^2*c^4)*x^4 +
2*(b^5*c - 8*a*b^3*c^2 + 16*a^2*b*c^3)*x^3 + (b^6 - 6*a*b^4*c + 32*a^3*c^3)*x^2
+ 2*(a*b^5 - 8*a^2*b^3*c + 16*a^3*b*c^2)*x)*sqrt((512*c^5*d^5 - 1280*b*c^4*d^4*
e + 160*(7*b^2*c^3 + 4*a*c^4)*d^3*e^2 - 80*(5*b^3*c^2 + 12*a*b*c^3)*d^2*e^3 + 10
*(5*b^4*c + 40*a*b^2*c^2 + 16*a^2*c^3)*d*e^4 - (b^5 + 40*a*b^3*c + 80*a^2*b*c^2)
*e^5 - (b^10*c - 20*a*b^8*c^2 + 160*a^2*b^6*c^3 - 640*a^3*b^4*c^4 + 1280*a^4*b^2
*c^5 - 1024*a^5*c^6)*sqrt(e^10/(b^10*c^2 - 20*a*b^8*c^3 + 160*a^2*b^6*c^4 - 640*
a^3*b^4*c^5 + 1280*a^4*b^2*c^6 - 1024*a^5*c^7)))/(b^10*c - 20*a*b^8*c^2 + 160*a^
2*b^6*c^3 - 640*a^3*b^4*c^4 + 1280*a^4*b^2*c^5 - 1024*a^5*c^6))*log(-27*sqrt(1/2
)*(4*(b^6*c - 12*a*b^4*c^2 + 48*a^2*b^2*c^3 - 64*a^3*c^4)*d*e^6 - 2*(b^7 - 12*a*
b^5*c + 48*a^2*b^3*c^2 - 64*a^3*b*c^3)*e^7 - sqrt(e^10/(b^10*c^2 - 20*a*b^8*c^3
+ 160*a^2*b^6*c^4 - 640*a^3*b^4*c^5 + 1280*a^4*b^2*c^6 - 1024*a^5*c^7))*(16*(b^1
0*c^3 - 20*a*b^8*c^4 + 160*a^2*b^6*c^5 - 640*a^3*b^4*c^6 + 1280*a^4*b^2*c^7 - 10
24*a^5*c^8)*d^2 - 16*(b^11*c^2 - 20*a*b^9*c^3 + 160*a^2*b^7*c^4 - 640*a^3*b^5*c^
5 + 1280*a^4*b^3*c^6 - 1024*a^5*b*c^7)*d*e + (3*b^12*c - 56*a*b^10*c^2 + 400*a^2
*b^8*c^3 - 1280*a^3*b^6*c^4 + 1280*a^4*b^4*c^5 + 2048*a^5*b^2*c^6 - 4096*a^6*c^7
)*e^2))*sqrt((512*c^5*d^5 - 1280*b*c^4*d^4*e + 160*(7*b^2*c^3 + 4*a*c^4)*d^3*e^2
- 80*(5*b^3*c^2 + 12*a*b*c^3)*d^2*e^3 + 10*(5*b^4*c + 40*a*b^2*c^2 + 16*a^2*c^3
)*d*e^4 - (b^5 + 40*a*b^3*c + 80*a^2*b*c^2)*e^5 - (b^10*c - 20*a*b^8*c^2 + 160*a
^2*b^6*c^3 - 640*a^3*b^4*c^4 + 1280*a^4*b^2*c^5 - 1024*a^5*c^6)*sqrt(e^10/(b^10*
c^2 - 20*a*b^8*c^3 + 160*a^2*b^6*c^4 - 640*a^3*b^4*c^5 + 1280*a^4*b^2*c^6 - 1024
*a^5*c^7)))/(b^10*c - 20*a*b^8*c^2 + 160*a^2*b^6*c^3 - 640*a^3*b^4*c^4 + 1280*a^
4*b^2*c^5 - 1024*a^5*c^6)) + 27*(256*c^4*d^4*e^5 - 512*b*c^3*d^3*e^6 + 48*(7*b^2
*c^2 + 4*a*c^3)*d^2*e^7 - 16*(5*b^3*c + 12*a*b*c^2)*d*e^8 + (5*b^4 + 40*a*b^2*c
+ 16*a^2*c^2)*e^9)*sqrt(e*x + d)) - 2*(12*a^2*b*e^2 + 3*(8*c^3*d^2 - 8*b*c^2*d*e
+ (b^2*c + 4*a*c^2)*e^2)*x^3 - 2*(b^3 - 10*a*b*c)*d^2 - (5*a*b^2 + 28*a^2*c)*d*
e + (36*b*c^2*d^2 - (37*b^2*c - 4*a*c^2)*d*e + (5*b^3 + 16*a*b*c)*e^2)*x^2 + (8*
(b^2*c + 5*a*c^2)*d^2 - 9*(b^3 + 4*a*b*c)*d*e + (19*a*b^2 - 4*a^2*c)*e^2)*x)*sqr
t(e*x + d))/(a^2*b^4 - 8*a^3*b^2*c + 16*a^4*c^2 + (b^4*c^2 - 8*a*b^2*c^3 + 16*a^
2*c^4)*x^4 + 2*(b^5*c - 8*a*b^3*c^2 + 16*a^2*b*c^3)*x^3 + (b^6 - 6*a*b^4*c + 32*
a^3*c^3)*x^2 + 2*(a*b^5 - 8*a^2*b^3*c + 16*a^3*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)**3,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)^3,x, algorithm="giac")

[Out]

Timed out