3.1124 \(\int \frac{A+B x}{(a+b x)^3 (d+e x)^3} \, dx\)

Optimal. Leaf size=199 \[ -\frac{b (A b-a B)}{2 (a+b x)^2 (b d-a e)^3}-\frac{b (2 a B e-3 A b e+b B d)}{(a+b x) (b d-a e)^4}-\frac{e (a B e-3 A b e+2 b B d)}{(d+e x) (b d-a e)^4}-\frac{e (B d-A e)}{2 (d+e x)^2 (b d-a e)^3}-\frac{3 b e \log (a+b x) (a B e-2 A b e+b B d)}{(b d-a e)^5}+\frac{3 b e \log (d+e x) (a B e-2 A b e+b B d)}{(b d-a e)^5} \]

[Out]

-(b*(A*b - a*B))/(2*(b*d - a*e)^3*(a + b*x)^2) - (b*(b*B*d - 3*A*b*e + 2*a*B*e))
/((b*d - a*e)^4*(a + b*x)) - (e*(B*d - A*e))/(2*(b*d - a*e)^3*(d + e*x)^2) - (e*
(2*b*B*d - 3*A*b*e + a*B*e))/((b*d - a*e)^4*(d + e*x)) - (3*b*e*(b*B*d - 2*A*b*e
+ a*B*e)*Log[a + b*x])/(b*d - a*e)^5 + (3*b*e*(b*B*d - 2*A*b*e + a*B*e)*Log[d +
e*x])/(b*d - a*e)^5

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Rubi [A]  time = 0.491061, antiderivative size = 199, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.05 \[ -\frac{b (A b-a B)}{2 (a+b x)^2 (b d-a e)^3}-\frac{b (2 a B e-3 A b e+b B d)}{(a+b x) (b d-a e)^4}-\frac{e (a B e-3 A b e+2 b B d)}{(d+e x) (b d-a e)^4}-\frac{e (B d-A e)}{2 (d+e x)^2 (b d-a e)^3}-\frac{3 b e \log (a+b x) (a B e-2 A b e+b B d)}{(b d-a e)^5}+\frac{3 b e \log (d+e x) (a B e-2 A b e+b B d)}{(b d-a e)^5} \]

Antiderivative was successfully verified.

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

[Out]

-(b*(A*b - a*B))/(2*(b*d - a*e)^3*(a + b*x)^2) - (b*(b*B*d - 3*A*b*e + 2*a*B*e))
/((b*d - a*e)^4*(a + b*x)) - (e*(B*d - A*e))/(2*(b*d - a*e)^3*(d + e*x)^2) - (e*
(2*b*B*d - 3*A*b*e + a*B*e))/((b*d - a*e)^4*(d + e*x)) - (3*b*e*(b*B*d - 2*A*b*e
+ a*B*e)*Log[a + b*x])/(b*d - a*e)^5 + (3*b*e*(b*B*d - 2*A*b*e + a*B*e)*Log[d +
e*x])/(b*d - a*e)^5

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Rubi in Sympy [A]  time = 116.923, size = 192, normalized size = 0.96 \[ - \frac{3 b e \left (2 A b e - B a e - B b d\right ) \log{\left (a + b x \right )}}{\left (a e - b d\right )^{5}} + \frac{3 b e \left (2 A b e - B a e - B b d\right ) \log{\left (d + e x \right )}}{\left (a e - b d\right )^{5}} + \frac{b \left (3 A b e - 2 B a e - B b d\right )}{\left (a + b x\right ) \left (a e - b d\right )^{4}} + \frac{b \left (A b - B a\right )}{2 \left (a + b x\right )^{2} \left (a e - b d\right )^{3}} + \frac{e \left (3 A b e - B a e - 2 B b d\right )}{\left (d + e x\right ) \left (a e - b d\right )^{4}} - \frac{e \left (A e - B d\right )}{2 \left (d + e x\right )^{2} \left (a e - b d\right )^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x+A)/(b*x+a)**3/(e*x+d)**3,x)

[Out]

-3*b*e*(2*A*b*e - B*a*e - B*b*d)*log(a + b*x)/(a*e - b*d)**5 + 3*b*e*(2*A*b*e -
B*a*e - B*b*d)*log(d + e*x)/(a*e - b*d)**5 + b*(3*A*b*e - 2*B*a*e - B*b*d)/((a +
b*x)*(a*e - b*d)**4) + b*(A*b - B*a)/(2*(a + b*x)**2*(a*e - b*d)**3) + e*(3*A*b
*e - B*a*e - 2*B*b*d)/((d + e*x)*(a*e - b*d)**4) - e*(A*e - B*d)/(2*(d + e*x)**2
*(a*e - b*d)**3)

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Mathematica [A]  time = 0.251137, size = 185, normalized size = 0.93 \[ \frac{-\frac{b (A b-a B) (b d-a e)^2}{(a+b x)^2}+\frac{e (b d-a e)^2 (A e-B d)}{(d+e x)^2}-\frac{2 b (b d-a e) (2 a B e-3 A b e+b B d)}{a+b x}+\frac{2 e (b d-a e) (-a B e+3 A b e-2 b B d)}{d+e x}-6 b e \log (a+b x) (a B e-2 A b e+b B d)+6 b e \log (d+e x) (a B e-2 A b e+b B d)}{2 (b d-a e)^5} \]

Antiderivative was successfully verified.

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

[Out]

(-((b*(A*b - a*B)*(b*d - a*e)^2)/(a + b*x)^2) - (2*b*(b*d - a*e)*(b*B*d - 3*A*b*
e + 2*a*B*e))/(a + b*x) + (e*(b*d - a*e)^2*(-(B*d) + A*e))/(d + e*x)^2 + (2*e*(b
*d - a*e)*(-2*b*B*d + 3*A*b*e - a*B*e))/(d + e*x) - 6*b*e*(b*B*d - 2*A*b*e + a*B
*e)*Log[a + b*x] + 6*b*e*(b*B*d - 2*A*b*e + a*B*e)*Log[d + e*x])/(2*(b*d - a*e)^
5)

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Maple [A]  time = 0.024, size = 380, normalized size = 1.9 \[ -{\frac{{e}^{2}A}{2\, \left ( ae-bd \right ) ^{3} \left ( ex+d \right ) ^{2}}}+{\frac{eBd}{2\, \left ( ae-bd \right ) ^{3} \left ( ex+d \right ) ^{2}}}+3\,{\frac{{e}^{2}Ab}{ \left ( ae-bd \right ) ^{4} \left ( ex+d \right ) }}-{\frac{{e}^{2}Ba}{ \left ( ae-bd \right ) ^{4} \left ( ex+d \right ) }}-2\,{\frac{bBde}{ \left ( ae-bd \right ) ^{4} \left ( ex+d \right ) }}+6\,{\frac{{b}^{2}{e}^{2}\ln \left ( ex+d \right ) A}{ \left ( ae-bd \right ) ^{5}}}-3\,{\frac{{e}^{2}b\ln \left ( ex+d \right ) Ba}{ \left ( ae-bd \right ) ^{5}}}-3\,{\frac{e{b}^{2}\ln \left ( ex+d \right ) Bd}{ \left ( ae-bd \right ) ^{5}}}+3\,{\frac{{b}^{2}Ae}{ \left ( ae-bd \right ) ^{4} \left ( bx+a \right ) }}-2\,{\frac{Bbae}{ \left ( ae-bd \right ) ^{4} \left ( bx+a \right ) }}-{\frac{{b}^{2}Bd}{ \left ( ae-bd \right ) ^{4} \left ( bx+a \right ) }}+{\frac{{b}^{2}A}{2\, \left ( ae-bd \right ) ^{3} \left ( bx+a \right ) ^{2}}}-{\frac{Bba}{2\, \left ( ae-bd \right ) ^{3} \left ( bx+a \right ) ^{2}}}-6\,{\frac{{b}^{2}{e}^{2}\ln \left ( bx+a \right ) A}{ \left ( ae-bd \right ) ^{5}}}+3\,{\frac{{e}^{2}b\ln \left ( bx+a \right ) Ba}{ \left ( ae-bd \right ) ^{5}}}+3\,{\frac{e{b}^{2}\ln \left ( bx+a \right ) Bd}{ \left ( ae-bd \right ) ^{5}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2*e^2/(a*e-b*d)^3/(e*x+d)^2*A+1/2*e/(a*e-b*d)^3/(e*x+d)^2*B*d+3*e^2/(a*e-b*d)
^4/(e*x+d)*A*b-e^2/(a*e-b*d)^4/(e*x+d)*B*a-2*e/(a*e-b*d)^4/(e*x+d)*B*b*d+6*e^2*b
^2/(a*e-b*d)^5*ln(e*x+d)*A-3*e^2*b/(a*e-b*d)^5*ln(e*x+d)*B*a-3*e*b^2/(a*e-b*d)^5
*ln(e*x+d)*B*d+3*b^2/(a*e-b*d)^4/(b*x+a)*A*e-2*b/(a*e-b*d)^4/(b*x+a)*B*a*e-b^2/(
a*e-b*d)^4/(b*x+a)*B*d+1/2*b^2/(a*e-b*d)^3/(b*x+a)^2*A-1/2*b/(a*e-b*d)^3/(b*x+a)
^2*B*a-6*e^2*b^2/(a*e-b*d)^5*ln(b*x+a)*A+3*e^2*b/(a*e-b*d)^5*ln(b*x+a)*B*a+3*e*b
^2/(a*e-b*d)^5*ln(b*x+a)*B*d

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Maxima [A]  time = 1.40423, size = 1006, normalized size = 5.06 \[ -\frac{3 \,{\left (B b^{2} d e +{\left (B a b - 2 \, A b^{2}\right )} e^{2}\right )} \log \left (b x + a\right )}{b^{5} d^{5} - 5 \, a b^{4} d^{4} e + 10 \, a^{2} b^{3} d^{3} e^{2} - 10 \, a^{3} b^{2} d^{2} e^{3} + 5 \, a^{4} b d e^{4} - a^{5} e^{5}} + \frac{3 \,{\left (B b^{2} d e +{\left (B a b - 2 \, A b^{2}\right )} e^{2}\right )} \log \left (e x + d\right )}{b^{5} d^{5} - 5 \, a b^{4} d^{4} e + 10 \, a^{2} b^{3} d^{3} e^{2} - 10 \, a^{3} b^{2} d^{2} e^{3} + 5 \, a^{4} b d e^{4} - a^{5} e^{5}} - \frac{A a^{3} e^{3} +{\left (B a b^{2} + A b^{3}\right )} d^{3} +{\left (10 \, B a^{2} b - 7 \, A a b^{2}\right )} d^{2} e +{\left (B a^{3} - 7 \, A a^{2} b\right )} d e^{2} + 6 \,{\left (B b^{3} d e^{2} +{\left (B a b^{2} - 2 \, A b^{3}\right )} e^{3}\right )} x^{3} + 9 \,{\left (B b^{3} d^{2} e + 2 \,{\left (B a b^{2} - A b^{3}\right )} d e^{2} +{\left (B a^{2} b - 2 \, A a b^{2}\right )} e^{3}\right )} x^{2} + 2 \,{\left (B b^{3} d^{3} + 2 \,{\left (4 \, B a b^{2} - A b^{3}\right )} d^{2} e + 2 \,{\left (4 \, B a^{2} b - 7 \, A a b^{2}\right )} d e^{2} +{\left (B a^{3} - 2 \, A a^{2} b\right )} e^{3}\right )} x}{2 \,{\left (a^{2} b^{4} d^{6} - 4 \, a^{3} b^{3} d^{5} e + 6 \, a^{4} b^{2} d^{4} e^{2} - 4 \, a^{5} b d^{3} e^{3} + a^{6} d^{2} e^{4} +{\left (b^{6} d^{4} e^{2} - 4 \, a b^{5} d^{3} e^{3} + 6 \, a^{2} b^{4} d^{2} e^{4} - 4 \, a^{3} b^{3} d e^{5} + a^{4} b^{2} e^{6}\right )} x^{4} + 2 \,{\left (b^{6} d^{5} e - 3 \, a b^{5} d^{4} e^{2} + 2 \, a^{2} b^{4} d^{3} e^{3} + 2 \, a^{3} b^{3} d^{2} e^{4} - 3 \, a^{4} b^{2} d e^{5} + a^{5} b e^{6}\right )} x^{3} +{\left (b^{6} d^{6} - 9 \, a^{2} b^{4} d^{4} e^{2} + 16 \, a^{3} b^{3} d^{3} e^{3} - 9 \, a^{4} b^{2} d^{2} e^{4} + a^{6} e^{6}\right )} x^{2} + 2 \,{\left (a b^{5} d^{6} - 3 \, a^{2} b^{4} d^{5} e + 2 \, a^{3} b^{3} d^{4} e^{2} + 2 \, a^{4} b^{2} d^{3} e^{3} - 3 \, a^{5} b d^{2} e^{4} + a^{6} d e^{5}\right )} x\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-3*(B*b^2*d*e + (B*a*b - 2*A*b^2)*e^2)*log(b*x + a)/(b^5*d^5 - 5*a*b^4*d^4*e + 1
0*a^2*b^3*d^3*e^2 - 10*a^3*b^2*d^2*e^3 + 5*a^4*b*d*e^4 - a^5*e^5) + 3*(B*b^2*d*e
+ (B*a*b - 2*A*b^2)*e^2)*log(e*x + d)/(b^5*d^5 - 5*a*b^4*d^4*e + 10*a^2*b^3*d^3
*e^2 - 10*a^3*b^2*d^2*e^3 + 5*a^4*b*d*e^4 - a^5*e^5) - 1/2*(A*a^3*e^3 + (B*a*b^2
+ A*b^3)*d^3 + (10*B*a^2*b - 7*A*a*b^2)*d^2*e + (B*a^3 - 7*A*a^2*b)*d*e^2 + 6*(
B*b^3*d*e^2 + (B*a*b^2 - 2*A*b^3)*e^3)*x^3 + 9*(B*b^3*d^2*e + 2*(B*a*b^2 - A*b^3
)*d*e^2 + (B*a^2*b - 2*A*a*b^2)*e^3)*x^2 + 2*(B*b^3*d^3 + 2*(4*B*a*b^2 - A*b^3)*
d^2*e + 2*(4*B*a^2*b - 7*A*a*b^2)*d*e^2 + (B*a^3 - 2*A*a^2*b)*e^3)*x)/(a^2*b^4*d
^6 - 4*a^3*b^3*d^5*e + 6*a^4*b^2*d^4*e^2 - 4*a^5*b*d^3*e^3 + a^6*d^2*e^4 + (b^6*
d^4*e^2 - 4*a*b^5*d^3*e^3 + 6*a^2*b^4*d^2*e^4 - 4*a^3*b^3*d*e^5 + a^4*b^2*e^6)*x
^4 + 2*(b^6*d^5*e - 3*a*b^5*d^4*e^2 + 2*a^2*b^4*d^3*e^3 + 2*a^3*b^3*d^2*e^4 - 3*
a^4*b^2*d*e^5 + a^5*b*e^6)*x^3 + (b^6*d^6 - 9*a^2*b^4*d^4*e^2 + 16*a^3*b^3*d^3*e
^3 - 9*a^4*b^2*d^2*e^4 + a^6*e^6)*x^2 + 2*(a*b^5*d^6 - 3*a^2*b^4*d^5*e + 2*a^3*b
^3*d^4*e^2 + 2*a^4*b^2*d^3*e^3 - 3*a^5*b*d^2*e^4 + a^6*d*e^5)*x)

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Fricas [A]  time = 0.236162, size = 1640, normalized size = 8.24 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*(9*B*a^3*b*d^2*e^2 + A*a^4*e^4 - (B*a*b^3 + A*b^4)*d^4 - (9*B*a^2*b^2 - 8*A*
a*b^3)*d^3*e + (B*a^4 - 8*A*a^3*b)*d*e^3 - 6*(B*b^4*d^2*e^2 - 2*A*b^4*d*e^3 - (B
*a^2*b^2 - 2*A*a*b^3)*e^4)*x^3 - 9*(B*b^4*d^3*e - B*a^2*b^2*d*e^3 + (B*a*b^3 - 2
*A*b^4)*d^2*e^2 - (B*a^3*b - 2*A*a^2*b^2)*e^4)*x^2 - 2*(B*b^4*d^4 - 12*A*a*b^3*d
^2*e^2 + (7*B*a*b^3 - 2*A*b^4)*d^3*e - (7*B*a^3*b - 12*A*a^2*b^2)*d*e^3 - (B*a^4
- 2*A*a^3*b)*e^4)*x - 6*(B*a^2*b^2*d^3*e + (B*a^3*b - 2*A*a^2*b^2)*d^2*e^2 + (B
*b^4*d*e^3 + (B*a*b^3 - 2*A*b^4)*e^4)*x^4 + 2*(B*b^4*d^2*e^2 + 2*(B*a*b^3 - A*b^
4)*d*e^3 + (B*a^2*b^2 - 2*A*a*b^3)*e^4)*x^3 + (B*b^4*d^3*e + (5*B*a*b^3 - 2*A*b^
4)*d^2*e^2 + (5*B*a^2*b^2 - 8*A*a*b^3)*d*e^3 + (B*a^3*b - 2*A*a^2*b^2)*e^4)*x^2
+ 2*(B*a*b^3*d^3*e + 2*(B*a^2*b^2 - A*a*b^3)*d^2*e^2 + (B*a^3*b - 2*A*a^2*b^2)*d
*e^3)*x)*log(b*x + a) + 6*(B*a^2*b^2*d^3*e + (B*a^3*b - 2*A*a^2*b^2)*d^2*e^2 + (
B*b^4*d*e^3 + (B*a*b^3 - 2*A*b^4)*e^4)*x^4 + 2*(B*b^4*d^2*e^2 + 2*(B*a*b^3 - A*b
^4)*d*e^3 + (B*a^2*b^2 - 2*A*a*b^3)*e^4)*x^3 + (B*b^4*d^3*e + (5*B*a*b^3 - 2*A*b
^4)*d^2*e^2 + (5*B*a^2*b^2 - 8*A*a*b^3)*d*e^3 + (B*a^3*b - 2*A*a^2*b^2)*e^4)*x^2
+ 2*(B*a*b^3*d^3*e + 2*(B*a^2*b^2 - A*a*b^3)*d^2*e^2 + (B*a^3*b - 2*A*a^2*b^2)*
d*e^3)*x)*log(e*x + d))/(a^2*b^5*d^7 - 5*a^3*b^4*d^6*e + 10*a^4*b^3*d^5*e^2 - 10
*a^5*b^2*d^4*e^3 + 5*a^6*b*d^3*e^4 - a^7*d^2*e^5 + (b^7*d^5*e^2 - 5*a*b^6*d^4*e^
3 + 10*a^2*b^5*d^3*e^4 - 10*a^3*b^4*d^2*e^5 + 5*a^4*b^3*d*e^6 - a^5*b^2*e^7)*x^4
+ 2*(b^7*d^6*e - 4*a*b^6*d^5*e^2 + 5*a^2*b^5*d^4*e^3 - 5*a^4*b^3*d^2*e^5 + 4*a^
5*b^2*d*e^6 - a^6*b*e^7)*x^3 + (b^7*d^7 - a*b^6*d^6*e - 9*a^2*b^5*d^5*e^2 + 25*a
^3*b^4*d^4*e^3 - 25*a^4*b^3*d^3*e^4 + 9*a^5*b^2*d^2*e^5 + a^6*b*d*e^6 - a^7*e^7)
*x^2 + 2*(a*b^6*d^7 - 4*a^2*b^5*d^6*e + 5*a^3*b^4*d^5*e^2 - 5*a^5*b^2*d^3*e^4 +
4*a^6*b*d^2*e^5 - a^7*d*e^6)*x)

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Sympy [A]  time = 30.2583, size = 1431, normalized size = 7.19 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-3*b*e*(-2*A*b*e + B*a*e + B*b*d)*log(x + (-6*A*a*b**2*e**3 - 6*A*b**3*d*e**2 +
3*B*a**2*b*e**3 + 6*B*a*b**2*d*e**2 + 3*B*b**3*d**2*e - 3*a**6*b*e**7*(-2*A*b*e
+ B*a*e + B*b*d)/(a*e - b*d)**5 + 18*a**5*b**2*d*e**6*(-2*A*b*e + B*a*e + B*b*d)
/(a*e - b*d)**5 - 45*a**4*b**3*d**2*e**5*(-2*A*b*e + B*a*e + B*b*d)/(a*e - b*d)*
*5 + 60*a**3*b**4*d**3*e**4*(-2*A*b*e + B*a*e + B*b*d)/(a*e - b*d)**5 - 45*a**2*
b**5*d**4*e**3*(-2*A*b*e + B*a*e + B*b*d)/(a*e - b*d)**5 + 18*a*b**6*d**5*e**2*(
-2*A*b*e + B*a*e + B*b*d)/(a*e - b*d)**5 - 3*b**7*d**6*e*(-2*A*b*e + B*a*e + B*b
*d)/(a*e - b*d)**5)/(-12*A*b**3*e**3 + 6*B*a*b**2*e**3 + 6*B*b**3*d*e**2))/(a*e
- b*d)**5 + 3*b*e*(-2*A*b*e + B*a*e + B*b*d)*log(x + (-6*A*a*b**2*e**3 - 6*A*b**
3*d*e**2 + 3*B*a**2*b*e**3 + 6*B*a*b**2*d*e**2 + 3*B*b**3*d**2*e + 3*a**6*b*e**7
*(-2*A*b*e + B*a*e + B*b*d)/(a*e - b*d)**5 - 18*a**5*b**2*d*e**6*(-2*A*b*e + B*a
*e + B*b*d)/(a*e - b*d)**5 + 45*a**4*b**3*d**2*e**5*(-2*A*b*e + B*a*e + B*b*d)/(
a*e - b*d)**5 - 60*a**3*b**4*d**3*e**4*(-2*A*b*e + B*a*e + B*b*d)/(a*e - b*d)**5
+ 45*a**2*b**5*d**4*e**3*(-2*A*b*e + B*a*e + B*b*d)/(a*e - b*d)**5 - 18*a*b**6*
d**5*e**2*(-2*A*b*e + B*a*e + B*b*d)/(a*e - b*d)**5 + 3*b**7*d**6*e*(-2*A*b*e +
B*a*e + B*b*d)/(a*e - b*d)**5)/(-12*A*b**3*e**3 + 6*B*a*b**2*e**3 + 6*B*b**3*d*e
**2))/(a*e - b*d)**5 - (A*a**3*e**3 - 7*A*a**2*b*d*e**2 - 7*A*a*b**2*d**2*e + A*
b**3*d**3 + B*a**3*d*e**2 + 10*B*a**2*b*d**2*e + B*a*b**2*d**3 + x**3*(-12*A*b**
3*e**3 + 6*B*a*b**2*e**3 + 6*B*b**3*d*e**2) + x**2*(-18*A*a*b**2*e**3 - 18*A*b**
3*d*e**2 + 9*B*a**2*b*e**3 + 18*B*a*b**2*d*e**2 + 9*B*b**3*d**2*e) + x*(-4*A*a**
2*b*e**3 - 28*A*a*b**2*d*e**2 - 4*A*b**3*d**2*e + 2*B*a**3*e**3 + 16*B*a**2*b*d*
e**2 + 16*B*a*b**2*d**2*e + 2*B*b**3*d**3))/(2*a**6*d**2*e**4 - 8*a**5*b*d**3*e*
*3 + 12*a**4*b**2*d**4*e**2 - 8*a**3*b**3*d**5*e + 2*a**2*b**4*d**6 + x**4*(2*a*
*4*b**2*e**6 - 8*a**3*b**3*d*e**5 + 12*a**2*b**4*d**2*e**4 - 8*a*b**5*d**3*e**3
+ 2*b**6*d**4*e**2) + x**3*(4*a**5*b*e**6 - 12*a**4*b**2*d*e**5 + 8*a**3*b**3*d*
*2*e**4 + 8*a**2*b**4*d**3*e**3 - 12*a*b**5*d**4*e**2 + 4*b**6*d**5*e) + x**2*(2
*a**6*e**6 - 18*a**4*b**2*d**2*e**4 + 32*a**3*b**3*d**3*e**3 - 18*a**2*b**4*d**4
*e**2 + 2*b**6*d**6) + x*(4*a**6*d*e**5 - 12*a**5*b*d**2*e**4 + 8*a**4*b**2*d**3
*e**3 + 8*a**3*b**3*d**4*e**2 - 12*a**2*b**4*d**5*e + 4*a*b**5*d**6))

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GIAC/XCAS [A]  time = 0.256223, size = 617, normalized size = 3.1 \[ -\frac{3 \,{\left (B b^{2} d e + B a b e^{2} - 2 \, A b^{2} e^{2}\right )}{\rm ln}\left (\frac{{\left | 2 \, b x e + b d + a e -{\left | b d - a e \right |} \right |}}{{\left | 2 \, b x e + b d + a e +{\left | b d - a e \right |} \right |}}\right )}{{\left (b^{4} d^{4} - 4 \, a b^{3} d^{3} e + 6 \, a^{2} b^{2} d^{2} e^{2} - 4 \, a^{3} b d e^{3} + a^{4} e^{4}\right )}{\left | b d - a e \right |}} - \frac{6 \, B b^{3} d x^{3} e^{2} + 9 \, B b^{3} d^{2} x^{2} e + 2 \, B b^{3} d^{3} x + 6 \, B a b^{2} x^{3} e^{3} - 12 \, A b^{3} x^{3} e^{3} + 18 \, B a b^{2} d x^{2} e^{2} - 18 \, A b^{3} d x^{2} e^{2} + 16 \, B a b^{2} d^{2} x e - 4 \, A b^{3} d^{2} x e + B a b^{2} d^{3} + A b^{3} d^{3} + 9 \, B a^{2} b x^{2} e^{3} - 18 \, A a b^{2} x^{2} e^{3} + 16 \, B a^{2} b d x e^{2} - 28 \, A a b^{2} d x e^{2} + 10 \, B a^{2} b d^{2} e - 7 \, A a b^{2} d^{2} e + 2 \, B a^{3} x e^{3} - 4 \, A a^{2} b x e^{3} + B a^{3} d e^{2} - 7 \, A a^{2} b d e^{2} + A a^{3} e^{3}}{2 \,{\left (b^{4} d^{4} - 4 \, a b^{3} d^{3} e + 6 \, a^{2} b^{2} d^{2} e^{2} - 4 \, a^{3} b d e^{3} + a^{4} e^{4}\right )}{\left (b x^{2} e + b d x + a x e + a d\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-3*(B*b^2*d*e + B*a*b*e^2 - 2*A*b^2*e^2)*ln(abs(2*b*x*e + b*d + a*e - abs(b*d -
a*e))/abs(2*b*x*e + b*d + a*e + abs(b*d - a*e)))/((b^4*d^4 - 4*a*b^3*d^3*e + 6*a
^2*b^2*d^2*e^2 - 4*a^3*b*d*e^3 + a^4*e^4)*abs(b*d - a*e)) - 1/2*(6*B*b^3*d*x^3*e
^2 + 9*B*b^3*d^2*x^2*e + 2*B*b^3*d^3*x + 6*B*a*b^2*x^3*e^3 - 12*A*b^3*x^3*e^3 +
18*B*a*b^2*d*x^2*e^2 - 18*A*b^3*d*x^2*e^2 + 16*B*a*b^2*d^2*x*e - 4*A*b^3*d^2*x*e
+ B*a*b^2*d^3 + A*b^3*d^3 + 9*B*a^2*b*x^2*e^3 - 18*A*a*b^2*x^2*e^3 + 16*B*a^2*b
*d*x*e^2 - 28*A*a*b^2*d*x*e^2 + 10*B*a^2*b*d^2*e - 7*A*a*b^2*d^2*e + 2*B*a^3*x*e
^3 - 4*A*a^2*b*x*e^3 + B*a^3*d*e^2 - 7*A*a^2*b*d*e^2 + A*a^3*e^3)/((b^4*d^4 - 4*
a*b^3*d^3*e + 6*a^2*b^2*d^2*e^2 - 4*a^3*b*d*e^3 + a^4*e^4)*(b*x^2*e + b*d*x + a*
x*e + a*d)^2)