3.1952 \(\int \frac{a+b x}{(d+e x)^3 \left (a^2+2 a b x+b^2 x^2\right )^3} \, dx\)

Optimal. Leaf size=192 \[ \frac{15 b^2 e^4 \log (a+b x)}{(b d-a e)^7}-\frac{15 b^2 e^4 \log (d+e x)}{(b d-a e)^7}+\frac{10 b^2 e^3}{(a+b x) (b d-a e)^6}-\frac{3 b^2 e^2}{(a+b x)^2 (b d-a e)^5}+\frac{b^2 e}{(a+b x)^3 (b d-a e)^4}-\frac{b^2}{4 (a+b x)^4 (b d-a e)^3}+\frac{5 b e^4}{(d+e x) (b d-a e)^6}+\frac{e^4}{2 (d+e x)^2 (b d-a e)^5} \]

[Out]

-b^2/(4*(b*d - a*e)^3*(a + b*x)^4) + (b^2*e)/((b*d - a*e)^4*(a + b*x)^3) - (3*b^
2*e^2)/((b*d - a*e)^5*(a + b*x)^2) + (10*b^2*e^3)/((b*d - a*e)^6*(a + b*x)) + e^
4/(2*(b*d - a*e)^5*(d + e*x)^2) + (5*b*e^4)/((b*d - a*e)^6*(d + e*x)) + (15*b^2*
e^4*Log[a + b*x])/(b*d - a*e)^7 - (15*b^2*e^4*Log[d + e*x])/(b*d - a*e)^7

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Rubi [A]  time = 0.389442, antiderivative size = 192, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.065 \[ \frac{15 b^2 e^4 \log (a+b x)}{(b d-a e)^7}-\frac{15 b^2 e^4 \log (d+e x)}{(b d-a e)^7}+\frac{10 b^2 e^3}{(a+b x) (b d-a e)^6}-\frac{3 b^2 e^2}{(a+b x)^2 (b d-a e)^5}+\frac{b^2 e}{(a+b x)^3 (b d-a e)^4}-\frac{b^2}{4 (a+b x)^4 (b d-a e)^3}+\frac{5 b e^4}{(d+e x) (b d-a e)^6}+\frac{e^4}{2 (d+e x)^2 (b d-a e)^5} \]

Antiderivative was successfully verified.

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

[Out]

-b^2/(4*(b*d - a*e)^3*(a + b*x)^4) + (b^2*e)/((b*d - a*e)^4*(a + b*x)^3) - (3*b^
2*e^2)/((b*d - a*e)^5*(a + b*x)^2) + (10*b^2*e^3)/((b*d - a*e)^6*(a + b*x)) + e^
4/(2*(b*d - a*e)^5*(d + e*x)^2) + (5*b*e^4)/((b*d - a*e)^6*(d + e*x)) + (15*b^2*
e^4*Log[a + b*x])/(b*d - a*e)^7 - (15*b^2*e^4*Log[d + e*x])/(b*d - a*e)^7

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

[Out]

Timed out

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Mathematica [A]  time = 0.185889, size = 179, normalized size = 0.93 \[ \frac{\frac{40 b^2 e^3 (b d-a e)}{a+b x}-\frac{12 b^2 e^2 (b d-a e)^2}{(a+b x)^2}+\frac{4 b^2 e (b d-a e)^3}{(a+b x)^3}-\frac{b^2 (b d-a e)^4}{(a+b x)^4}+60 b^2 e^4 \log (a+b x)+\frac{20 b e^4 (b d-a e)}{d+e x}+\frac{2 e^4 (b d-a e)^2}{(d+e x)^2}-60 b^2 e^4 \log (d+e x)}{4 (b d-a e)^7} \]

Antiderivative was successfully verified.

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

[Out]

(-((b^2*(b*d - a*e)^4)/(a + b*x)^4) + (4*b^2*e*(b*d - a*e)^3)/(a + b*x)^3 - (12*
b^2*e^2*(b*d - a*e)^2)/(a + b*x)^2 + (40*b^2*e^3*(b*d - a*e))/(a + b*x) + (2*e^4
*(b*d - a*e)^2)/(d + e*x)^2 + (20*b*e^4*(b*d - a*e))/(d + e*x) + 60*b^2*e^4*Log[
a + b*x] - 60*b^2*e^4*Log[d + e*x])/(4*(b*d - a*e)^7)

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Maple [A]  time = 0.023, size = 189, normalized size = 1. \[{\frac{{b}^{2}}{4\, \left ( ae-bd \right ) ^{3} \left ( bx+a \right ) ^{4}}}-15\,{\frac{{b}^{2}{e}^{4}\ln \left ( bx+a \right ) }{ \left ( ae-bd \right ) ^{7}}}+10\,{\frac{{b}^{2}{e}^{3}}{ \left ( ae-bd \right ) ^{6} \left ( bx+a \right ) }}+3\,{\frac{{b}^{2}{e}^{2}}{ \left ( ae-bd \right ) ^{5} \left ( bx+a \right ) ^{2}}}+{\frac{{b}^{2}e}{ \left ( ae-bd \right ) ^{4} \left ( bx+a \right ) ^{3}}}-{\frac{{e}^{4}}{2\, \left ( ae-bd \right ) ^{5} \left ( ex+d \right ) ^{2}}}+15\,{\frac{{b}^{2}{e}^{4}\ln \left ( ex+d \right ) }{ \left ( ae-bd \right ) ^{7}}}+5\,{\frac{{e}^{4}b}{ \left ( ae-bd \right ) ^{6} \left ( ex+d \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 0.794305, size = 1620, normalized size = 8.44 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

15*b^2*e^4*log(b*x + a)/(b^7*d^7 - 7*a*b^6*d^6*e + 21*a^2*b^5*d^5*e^2 - 35*a^3*b
^4*d^4*e^3 + 35*a^4*b^3*d^3*e^4 - 21*a^5*b^2*d^2*e^5 + 7*a^6*b*d*e^6 - a^7*e^7)
- 15*b^2*e^4*log(e*x + d)/(b^7*d^7 - 7*a*b^6*d^6*e + 21*a^2*b^5*d^5*e^2 - 35*a^3
*b^4*d^4*e^3 + 35*a^4*b^3*d^3*e^4 - 21*a^5*b^2*d^2*e^5 + 7*a^6*b*d*e^6 - a^7*e^7
) + 1/4*(60*b^5*e^5*x^5 - b^5*d^5 + 7*a*b^4*d^4*e - 23*a^2*b^3*d^3*e^2 + 57*a^3*
b^2*d^2*e^3 + 22*a^4*b*d*e^4 - 2*a^5*e^5 + 30*(3*b^5*d*e^4 + 7*a*b^4*e^5)*x^4 +
20*(b^5*d^2*e^3 + 16*a*b^4*d*e^4 + 13*a^2*b^3*e^5)*x^3 - 5*(b^5*d^3*e^2 - 15*a*b
^4*d^2*e^3 - 81*a^2*b^3*d*e^4 - 25*a^3*b^2*e^5)*x^2 + 2*(b^5*d^4*e - 9*a*b^4*d^3
*e^2 + 51*a^2*b^3*d^2*e^3 + 101*a^3*b^2*d*e^4 + 6*a^4*b*e^5)*x)/(a^4*b^6*d^8 - 6
*a^5*b^5*d^7*e + 15*a^6*b^4*d^6*e^2 - 20*a^7*b^3*d^5*e^3 + 15*a^8*b^2*d^4*e^4 -
6*a^9*b*d^3*e^5 + a^10*d^2*e^6 + (b^10*d^6*e^2 - 6*a*b^9*d^5*e^3 + 15*a^2*b^8*d^
4*e^4 - 20*a^3*b^7*d^3*e^5 + 15*a^4*b^6*d^2*e^6 - 6*a^5*b^5*d*e^7 + a^6*b^4*e^8)
*x^6 + 2*(b^10*d^7*e - 4*a*b^9*d^6*e^2 + 3*a^2*b^8*d^5*e^3 + 10*a^3*b^7*d^4*e^4
- 25*a^4*b^6*d^3*e^5 + 24*a^5*b^5*d^2*e^6 - 11*a^6*b^4*d*e^7 + 2*a^7*b^3*e^8)*x^
5 + (b^10*d^8 + 2*a*b^9*d^7*e - 27*a^2*b^8*d^6*e^2 + 64*a^3*b^7*d^5*e^3 - 55*a^4
*b^6*d^4*e^4 - 6*a^5*b^5*d^3*e^5 + 43*a^6*b^4*d^2*e^6 - 28*a^7*b^3*d*e^7 + 6*a^8
*b^2*e^8)*x^4 + 4*(a*b^9*d^8 - 3*a^2*b^8*d^7*e - 2*a^3*b^7*d^6*e^2 + 19*a^4*b^6*
d^5*e^3 - 30*a^5*b^5*d^4*e^4 + 19*a^6*b^4*d^3*e^5 - 2*a^7*b^3*d^2*e^6 - 3*a^8*b^
2*d*e^7 + a^9*b*e^8)*x^3 + (6*a^2*b^8*d^8 - 28*a^3*b^7*d^7*e + 43*a^4*b^6*d^6*e^
2 - 6*a^5*b^5*d^5*e^3 - 55*a^6*b^4*d^4*e^4 + 64*a^7*b^3*d^3*e^5 - 27*a^8*b^2*d^2
*e^6 + 2*a^9*b*d*e^7 + a^10*e^8)*x^2 + 2*(2*a^3*b^7*d^8 - 11*a^4*b^6*d^7*e + 24*
a^5*b^5*d^6*e^2 - 25*a^6*b^4*d^5*e^3 + 10*a^7*b^3*d^4*e^4 + 3*a^8*b^2*d^3*e^5 -
4*a^9*b*d^2*e^6 + a^10*d*e^7)*x)

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Fricas [A]  time = 0.325932, size = 2113, normalized size = 11.01 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/4*(b^6*d^6 - 8*a*b^5*d^5*e + 30*a^2*b^4*d^4*e^2 - 80*a^3*b^3*d^3*e^3 + 35*a^4
*b^2*d^2*e^4 + 24*a^5*b*d*e^5 - 2*a^6*e^6 - 60*(b^6*d*e^5 - a*b^5*e^6)*x^5 - 30*
(3*b^6*d^2*e^4 + 4*a*b^5*d*e^5 - 7*a^2*b^4*e^6)*x^4 - 20*(b^6*d^3*e^3 + 15*a*b^5
*d^2*e^4 - 3*a^2*b^4*d*e^5 - 13*a^3*b^3*e^6)*x^3 + 5*(b^6*d^4*e^2 - 16*a*b^5*d^3
*e^3 - 66*a^2*b^4*d^2*e^4 + 56*a^3*b^3*d*e^5 + 25*a^4*b^2*e^6)*x^2 - 2*(b^6*d^5*
e - 10*a*b^5*d^4*e^2 + 60*a^2*b^4*d^3*e^3 + 50*a^3*b^3*d^2*e^4 - 95*a^4*b^2*d*e^
5 - 6*a^5*b*e^6)*x - 60*(b^6*e^6*x^6 + a^4*b^2*d^2*e^4 + 2*(b^6*d*e^5 + 2*a*b^5*
e^6)*x^5 + (b^6*d^2*e^4 + 8*a*b^5*d*e^5 + 6*a^2*b^4*e^6)*x^4 + 4*(a*b^5*d^2*e^4
+ 3*a^2*b^4*d*e^5 + a^3*b^3*e^6)*x^3 + (6*a^2*b^4*d^2*e^4 + 8*a^3*b^3*d*e^5 + a^
4*b^2*e^6)*x^2 + 2*(2*a^3*b^3*d^2*e^4 + a^4*b^2*d*e^5)*x)*log(b*x + a) + 60*(b^6
*e^6*x^6 + a^4*b^2*d^2*e^4 + 2*(b^6*d*e^5 + 2*a*b^5*e^6)*x^5 + (b^6*d^2*e^4 + 8*
a*b^5*d*e^5 + 6*a^2*b^4*e^6)*x^4 + 4*(a*b^5*d^2*e^4 + 3*a^2*b^4*d*e^5 + a^3*b^3*
e^6)*x^3 + (6*a^2*b^4*d^2*e^4 + 8*a^3*b^3*d*e^5 + a^4*b^2*e^6)*x^2 + 2*(2*a^3*b^
3*d^2*e^4 + a^4*b^2*d*e^5)*x)*log(e*x + d))/(a^4*b^7*d^9 - 7*a^5*b^6*d^8*e + 21*
a^6*b^5*d^7*e^2 - 35*a^7*b^4*d^6*e^3 + 35*a^8*b^3*d^5*e^4 - 21*a^9*b^2*d^4*e^5 +
7*a^10*b*d^3*e^6 - a^11*d^2*e^7 + (b^11*d^7*e^2 - 7*a*b^10*d^6*e^3 + 21*a^2*b^9
*d^5*e^4 - 35*a^3*b^8*d^4*e^5 + 35*a^4*b^7*d^3*e^6 - 21*a^5*b^6*d^2*e^7 + 7*a^6*
b^5*d*e^8 - a^7*b^4*e^9)*x^6 + 2*(b^11*d^8*e - 5*a*b^10*d^7*e^2 + 7*a^2*b^9*d^6*
e^3 + 7*a^3*b^8*d^5*e^4 - 35*a^4*b^7*d^4*e^5 + 49*a^5*b^6*d^3*e^6 - 35*a^6*b^5*d
^2*e^7 + 13*a^7*b^4*d*e^8 - 2*a^8*b^3*e^9)*x^5 + (b^11*d^9 + a*b^10*d^8*e - 29*a
^2*b^9*d^7*e^2 + 91*a^3*b^8*d^6*e^3 - 119*a^4*b^7*d^5*e^4 + 49*a^5*b^6*d^4*e^5 +
49*a^6*b^5*d^3*e^6 - 71*a^7*b^4*d^2*e^7 + 34*a^8*b^3*d*e^8 - 6*a^9*b^2*e^9)*x^4
+ 4*(a*b^10*d^9 - 4*a^2*b^9*d^8*e + a^3*b^8*d^7*e^2 + 21*a^4*b^7*d^6*e^3 - 49*a
^5*b^6*d^5*e^4 + 49*a^6*b^5*d^4*e^5 - 21*a^7*b^4*d^3*e^6 - a^8*b^3*d^2*e^7 + 4*a
^9*b^2*d*e^8 - a^10*b*e^9)*x^3 + (6*a^2*b^9*d^9 - 34*a^3*b^8*d^8*e + 71*a^4*b^7*
d^7*e^2 - 49*a^5*b^6*d^6*e^3 - 49*a^6*b^5*d^5*e^4 + 119*a^7*b^4*d^4*e^5 - 91*a^8
*b^3*d^3*e^6 + 29*a^9*b^2*d^2*e^7 - a^10*b*d*e^8 - a^11*e^9)*x^2 + 2*(2*a^3*b^8*
d^9 - 13*a^4*b^7*d^8*e + 35*a^5*b^6*d^7*e^2 - 49*a^6*b^5*d^6*e^3 + 35*a^7*b^4*d^
5*e^4 - 7*a^8*b^3*d^4*e^5 - 7*a^9*b^2*d^3*e^6 + 5*a^10*b*d^2*e^7 - a^11*d*e^8)*x
)

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Sympy [A]  time = 31.8535, size = 1571, normalized size = 8.18 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x+a)/(e*x+d)**3/(b**2*x**2+2*a*b*x+a**2)**3,x)

[Out]

15*b**2*e**4*log(x + (-15*a**8*b**2*e**12/(a*e - b*d)**7 + 120*a**7*b**3*d*e**11
/(a*e - b*d)**7 - 420*a**6*b**4*d**2*e**10/(a*e - b*d)**7 + 840*a**5*b**5*d**3*e
**9/(a*e - b*d)**7 - 1050*a**4*b**6*d**4*e**8/(a*e - b*d)**7 + 840*a**3*b**7*d**
5*e**7/(a*e - b*d)**7 - 420*a**2*b**8*d**6*e**6/(a*e - b*d)**7 + 120*a*b**9*d**7
*e**5/(a*e - b*d)**7 + 15*a*b**2*e**5 - 15*b**10*d**8*e**4/(a*e - b*d)**7 + 15*b
**3*d*e**4)/(30*b**3*e**5))/(a*e - b*d)**7 - 15*b**2*e**4*log(x + (15*a**8*b**2*
e**12/(a*e - b*d)**7 - 120*a**7*b**3*d*e**11/(a*e - b*d)**7 + 420*a**6*b**4*d**2
*e**10/(a*e - b*d)**7 - 840*a**5*b**5*d**3*e**9/(a*e - b*d)**7 + 1050*a**4*b**6*
d**4*e**8/(a*e - b*d)**7 - 840*a**3*b**7*d**5*e**7/(a*e - b*d)**7 + 420*a**2*b**
8*d**6*e**6/(a*e - b*d)**7 - 120*a*b**9*d**7*e**5/(a*e - b*d)**7 + 15*a*b**2*e**
5 + 15*b**10*d**8*e**4/(a*e - b*d)**7 + 15*b**3*d*e**4)/(30*b**3*e**5))/(a*e - b
*d)**7 + (-2*a**5*e**5 + 22*a**4*b*d*e**4 + 57*a**3*b**2*d**2*e**3 - 23*a**2*b**
3*d**3*e**2 + 7*a*b**4*d**4*e - b**5*d**5 + 60*b**5*e**5*x**5 + x**4*(210*a*b**4
*e**5 + 90*b**5*d*e**4) + x**3*(260*a**2*b**3*e**5 + 320*a*b**4*d*e**4 + 20*b**5
*d**2*e**3) + x**2*(125*a**3*b**2*e**5 + 405*a**2*b**3*d*e**4 + 75*a*b**4*d**2*e
**3 - 5*b**5*d**3*e**2) + x*(12*a**4*b*e**5 + 202*a**3*b**2*d*e**4 + 102*a**2*b*
*3*d**2*e**3 - 18*a*b**4*d**3*e**2 + 2*b**5*d**4*e))/(4*a**10*d**2*e**6 - 24*a**
9*b*d**3*e**5 + 60*a**8*b**2*d**4*e**4 - 80*a**7*b**3*d**5*e**3 + 60*a**6*b**4*d
**6*e**2 - 24*a**5*b**5*d**7*e + 4*a**4*b**6*d**8 + x**6*(4*a**6*b**4*e**8 - 24*
a**5*b**5*d*e**7 + 60*a**4*b**6*d**2*e**6 - 80*a**3*b**7*d**3*e**5 + 60*a**2*b**
8*d**4*e**4 - 24*a*b**9*d**5*e**3 + 4*b**10*d**6*e**2) + x**5*(16*a**7*b**3*e**8
- 88*a**6*b**4*d*e**7 + 192*a**5*b**5*d**2*e**6 - 200*a**4*b**6*d**3*e**5 + 80*
a**3*b**7*d**4*e**4 + 24*a**2*b**8*d**5*e**3 - 32*a*b**9*d**6*e**2 + 8*b**10*d**
7*e) + x**4*(24*a**8*b**2*e**8 - 112*a**7*b**3*d*e**7 + 172*a**6*b**4*d**2*e**6
- 24*a**5*b**5*d**3*e**5 - 220*a**4*b**6*d**4*e**4 + 256*a**3*b**7*d**5*e**3 - 1
08*a**2*b**8*d**6*e**2 + 8*a*b**9*d**7*e + 4*b**10*d**8) + x**3*(16*a**9*b*e**8
- 48*a**8*b**2*d*e**7 - 32*a**7*b**3*d**2*e**6 + 304*a**6*b**4*d**3*e**5 - 480*a
**5*b**5*d**4*e**4 + 304*a**4*b**6*d**5*e**3 - 32*a**3*b**7*d**6*e**2 - 48*a**2*
b**8*d**7*e + 16*a*b**9*d**8) + x**2*(4*a**10*e**8 + 8*a**9*b*d*e**7 - 108*a**8*
b**2*d**2*e**6 + 256*a**7*b**3*d**3*e**5 - 220*a**6*b**4*d**4*e**4 - 24*a**5*b**
5*d**5*e**3 + 172*a**4*b**6*d**6*e**2 - 112*a**3*b**7*d**7*e + 24*a**2*b**8*d**8
) + x*(8*a**10*d*e**7 - 32*a**9*b*d**2*e**6 + 24*a**8*b**2*d**3*e**5 + 80*a**7*b
**3*d**4*e**4 - 200*a**6*b**4*d**5*e**3 + 192*a**5*b**5*d**6*e**2 - 88*a**4*b**6
*d**7*e + 16*a**3*b**7*d**8))

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GIAC/XCAS [A]  time = 0.290865, size = 738, normalized size = 3.84 \[ \frac{15 \, b^{3} e^{4}{\rm ln}\left ({\left | b x + a \right |}\right )}{b^{8} d^{7} - 7 \, a b^{7} d^{6} e + 21 \, a^{2} b^{6} d^{5} e^{2} - 35 \, a^{3} b^{5} d^{4} e^{3} + 35 \, a^{4} b^{4} d^{3} e^{4} - 21 \, a^{5} b^{3} d^{2} e^{5} + 7 \, a^{6} b^{2} d e^{6} - a^{7} b e^{7}} - \frac{15 \, b^{2} e^{5}{\rm ln}\left ({\left | x e + d \right |}\right )}{b^{7} d^{7} e - 7 \, a b^{6} d^{6} e^{2} + 21 \, a^{2} b^{5} d^{5} e^{3} - 35 \, a^{3} b^{4} d^{4} e^{4} + 35 \, a^{4} b^{3} d^{3} e^{5} - 21 \, a^{5} b^{2} d^{2} e^{6} + 7 \, a^{6} b d e^{7} - a^{7} e^{8}} - \frac{b^{6} d^{6} - 8 \, a b^{5} d^{5} e + 30 \, a^{2} b^{4} d^{4} e^{2} - 80 \, a^{3} b^{3} d^{3} e^{3} + 35 \, a^{4} b^{2} d^{2} e^{4} + 24 \, a^{5} b d e^{5} - 2 \, a^{6} e^{6} - 60 \,{\left (b^{6} d e^{5} - a b^{5} e^{6}\right )} x^{5} - 30 \,{\left (3 \, b^{6} d^{2} e^{4} + 4 \, a b^{5} d e^{5} - 7 \, a^{2} b^{4} e^{6}\right )} x^{4} - 20 \,{\left (b^{6} d^{3} e^{3} + 15 \, a b^{5} d^{2} e^{4} - 3 \, a^{2} b^{4} d e^{5} - 13 \, a^{3} b^{3} e^{6}\right )} x^{3} + 5 \,{\left (b^{6} d^{4} e^{2} - 16 \, a b^{5} d^{3} e^{3} - 66 \, a^{2} b^{4} d^{2} e^{4} + 56 \, a^{3} b^{3} d e^{5} + 25 \, a^{4} b^{2} e^{6}\right )} x^{2} - 2 \,{\left (b^{6} d^{5} e - 10 \, a b^{5} d^{4} e^{2} + 60 \, a^{2} b^{4} d^{3} e^{3} + 50 \, a^{3} b^{3} d^{2} e^{4} - 95 \, a^{4} b^{2} d e^{5} - 6 \, a^{5} b e^{6}\right )} x}{4 \,{\left (b d - a e\right )}^{7}{\left (b x + a\right )}^{4}{\left (x e + d\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

15*b^3*e^4*ln(abs(b*x + a))/(b^8*d^7 - 7*a*b^7*d^6*e + 21*a^2*b^6*d^5*e^2 - 35*a
^3*b^5*d^4*e^3 + 35*a^4*b^4*d^3*e^4 - 21*a^5*b^3*d^2*e^5 + 7*a^6*b^2*d*e^6 - a^7
*b*e^7) - 15*b^2*e^5*ln(abs(x*e + d))/(b^7*d^7*e - 7*a*b^6*d^6*e^2 + 21*a^2*b^5*
d^5*e^3 - 35*a^3*b^4*d^4*e^4 + 35*a^4*b^3*d^3*e^5 - 21*a^5*b^2*d^2*e^6 + 7*a^6*b
*d*e^7 - a^7*e^8) - 1/4*(b^6*d^6 - 8*a*b^5*d^5*e + 30*a^2*b^4*d^4*e^2 - 80*a^3*b
^3*d^3*e^3 + 35*a^4*b^2*d^2*e^4 + 24*a^5*b*d*e^5 - 2*a^6*e^6 - 60*(b^6*d*e^5 - a
*b^5*e^6)*x^5 - 30*(3*b^6*d^2*e^4 + 4*a*b^5*d*e^5 - 7*a^2*b^4*e^6)*x^4 - 20*(b^6
*d^3*e^3 + 15*a*b^5*d^2*e^4 - 3*a^2*b^4*d*e^5 - 13*a^3*b^3*e^6)*x^3 + 5*(b^6*d^4
*e^2 - 16*a*b^5*d^3*e^3 - 66*a^2*b^4*d^2*e^4 + 56*a^3*b^3*d*e^5 + 25*a^4*b^2*e^6
)*x^2 - 2*(b^6*d^5*e - 10*a*b^5*d^4*e^2 + 60*a^2*b^4*d^3*e^3 + 50*a^3*b^3*d^2*e^
4 - 95*a^4*b^2*d*e^5 - 6*a^5*b*e^6)*x)/((b*d - a*e)^7*(b*x + a)^4*(x*e + d)^2)