3.1131 \(\int \frac{A+B x}{(a+b x)^2 (d+e x)^4} \, dx\)

Optimal. Leaf size=200 \[ -\frac{b^2 (A b-a B)}{(a+b x) (b d-a e)^4}+\frac{b^2 \log (a+b x) (3 a B e-4 A b e+b B d)}{(b d-a e)^5}-\frac{b^2 \log (d+e x) (3 a B e-4 A b e+b B d)}{(b d-a e)^5}+\frac{b (2 a B e-3 A b e+b B d)}{(d+e x) (b d-a e)^4}+\frac{a B e-2 A b e+b B d}{2 (d+e x)^2 (b d-a e)^3}+\frac{B d-A e}{3 (d+e x)^3 (b d-a e)^2} \]

[Out]

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

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Rubi [A]  time = 0.209744, antiderivative size = 200, 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, Rules used = {77} \[ -\frac{b^2 (A b-a B)}{(a+b x) (b d-a e)^4}+\frac{b^2 \log (a+b x) (3 a B e-4 A b e+b B d)}{(b d-a e)^5}-\frac{b^2 \log (d+e x) (3 a B e-4 A b e+b B d)}{(b d-a e)^5}+\frac{b (2 a B e-3 A b e+b B d)}{(d+e x) (b d-a e)^4}+\frac{a B e-2 A b e+b B d}{2 (d+e x)^2 (b d-a e)^3}+\frac{B d-A e}{3 (d+e x)^3 (b d-a e)^2} \]

Antiderivative was successfully verified.

[In]

Int[(A + B*x)/((a + b*x)^2*(d + e*x)^4),x]

[Out]

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

Rule 77

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((ILtQ[
n, 0] && ILtQ[p, 0]) || EqQ[p, 1] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1
, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))

Rubi steps

\begin{align*} \int \frac{A+B x}{(a+b x)^2 (d+e x)^4} \, dx &=\int \left (\frac{b^3 (A b-a B)}{(b d-a e)^4 (a+b x)^2}+\frac{b^3 (b B d-4 A b e+3 a B e)}{(b d-a e)^5 (a+b x)}+\frac{e (-B d+A e)}{(b d-a e)^2 (d+e x)^4}+\frac{e (-b B d+2 A b e-a B e)}{(b d-a e)^3 (d+e x)^3}+\frac{b e (-b B d+3 A b e-2 a B e)}{(b d-a e)^4 (d+e x)^2}+\frac{b^2 e (-b B d+4 A b e-3 a B e)}{(b d-a e)^5 (d+e x)}\right ) \, dx\\ &=-\frac{b^2 (A b-a B)}{(b d-a e)^4 (a+b x)}+\frac{B d-A e}{3 (b d-a e)^2 (d+e x)^3}+\frac{b B d-2 A b e+a B e}{2 (b d-a e)^3 (d+e x)^2}+\frac{b (b B d-3 A b e+2 a B e)}{(b d-a e)^4 (d+e x)}+\frac{b^2 (b B d-4 A b e+3 a B e) \log (a+b x)}{(b d-a e)^5}-\frac{b^2 (b B d-4 A b e+3 a B e) \log (d+e x)}{(b d-a e)^5}\\ \end{align*}

Mathematica [A]  time = 0.142829, size = 188, normalized size = 0.94 \[ \frac{-\frac{6 b^2 (A b-a B) (b d-a e)}{a+b x}+6 b^2 \log (a+b x) (3 a B e-4 A b e+b B d)-6 b^2 \log (d+e x) (3 a B e-4 A b e+b B d)+\frac{2 (b d-a e)^3 (B d-A e)}{(d+e x)^3}+\frac{3 (b d-a e)^2 (a B e-2 A b e+b B d)}{(d+e x)^2}+\frac{6 b (b d-a e) (2 a B e-3 A b e+b B d)}{d+e x}}{6 (b d-a e)^5} \]

Antiderivative was successfully verified.

[In]

Integrate[(A + B*x)/((a + b*x)^2*(d + e*x)^4),x]

[Out]

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

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Maple [A]  time = 0.017, size = 364, normalized size = 1.8 \begin{align*} -{\frac{Ae}{3\, \left ( ae-bd \right ) ^{2} \left ( ex+d \right ) ^{3}}}+{\frac{Bd}{3\, \left ( ae-bd \right ) ^{2} \left ( ex+d \right ) ^{3}}}-3\,{\frac{A{b}^{2}e}{ \left ( ae-bd \right ) ^{4} \left ( ex+d \right ) }}+2\,{\frac{Bbae}{ \left ( ae-bd \right ) ^{4} \left ( ex+d \right ) }}+{\frac{{b}^{2}Bd}{ \left ( ae-bd \right ) ^{4} \left ( ex+d \right ) }}+{\frac{Abe}{ \left ( ae-bd \right ) ^{3} \left ( ex+d \right ) ^{2}}}-{\frac{Bae}{2\, \left ( ae-bd \right ) ^{3} \left ( ex+d \right ) ^{2}}}-{\frac{Bbd}{2\, \left ( ae-bd \right ) ^{3} \left ( ex+d \right ) ^{2}}}-4\,{\frac{{b}^{3}\ln \left ( ex+d \right ) Ae}{ \left ( ae-bd \right ) ^{5}}}+3\,{\frac{{b}^{2}\ln \left ( ex+d \right ) Bae}{ \left ( ae-bd \right ) ^{5}}}+{\frac{{b}^{3}\ln \left ( ex+d \right ) Bd}{ \left ( ae-bd \right ) ^{5}}}+4\,{\frac{{b}^{3}\ln \left ( bx+a \right ) Ae}{ \left ( ae-bd \right ) ^{5}}}-3\,{\frac{{b}^{2}\ln \left ( bx+a \right ) Bae}{ \left ( ae-bd \right ) ^{5}}}-{\frac{{b}^{3}\ln \left ( bx+a \right ) Bd}{ \left ( ae-bd \right ) ^{5}}}-{\frac{{b}^{3}A}{ \left ( ae-bd \right ) ^{4} \left ( bx+a \right ) }}+{\frac{Ba{b}^{2}}{ \left ( ae-bd \right ) ^{4} \left ( bx+a \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)/(b*x+a)^2/(e*x+d)^4,x)

[Out]

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

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Maxima [B]  time = 1.38579, size = 1027, normalized size = 5.14 \begin{align*} \frac{{\left (B b^{3} d +{\left (3 \, B a b^{2} - 4 \, A b^{3}\right )} e\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{{\left (B b^{3} d +{\left (3 \, B a b^{2} - 4 \, A b^{3}\right )} e\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{2 \, A a^{3} e^{3} -{\left (17 \, B a b^{2} - 6 \, A b^{3}\right )} d^{3} - 2 \,{\left (4 \, B a^{2} b - 13 \, A a b^{2}\right )} d^{2} e +{\left (B a^{3} - 10 \, A a^{2} b\right )} d e^{2} - 6 \,{\left (B b^{3} d e^{2} +{\left (3 \, B a b^{2} - 4 \, A b^{3}\right )} e^{3}\right )} x^{3} - 3 \,{\left (5 \, B b^{3} d^{2} e + 4 \,{\left (4 \, B a b^{2} - 5 \, A b^{3}\right )} d e^{2} +{\left (3 \, B a^{2} b - 4 \, A a b^{2}\right )} e^{3}\right )} x^{2} -{\left (11 \, B b^{3} d^{3} +{\left (41 \, B a b^{2} - 44 \, A b^{3}\right )} d^{2} e +{\left (23 \, B a^{2} b - 32 \, A a b^{2}\right )} d e^{2} -{\left (3 \, B a^{3} - 4 \, A a^{2} b\right )} e^{3}\right )} x}{6 \,{\left (a b^{4} d^{7} - 4 \, a^{2} b^{3} d^{6} e + 6 \, a^{3} b^{2} d^{5} e^{2} - 4 \, a^{4} b d^{4} e^{3} + a^{5} d^{3} e^{4} +{\left (b^{5} d^{4} e^{3} - 4 \, a b^{4} d^{3} e^{4} + 6 \, a^{2} b^{3} d^{2} e^{5} - 4 \, a^{3} b^{2} d e^{6} + a^{4} b e^{7}\right )} x^{4} +{\left (3 \, b^{5} d^{5} e^{2} - 11 \, a b^{4} d^{4} e^{3} + 14 \, a^{2} b^{3} d^{3} e^{4} - 6 \, a^{3} b^{2} d^{2} e^{5} - a^{4} b d e^{6} + a^{5} e^{7}\right )} x^{3} + 3 \,{\left (b^{5} d^{6} e - 3 \, a b^{4} d^{5} e^{2} + 2 \, a^{2} b^{3} d^{4} e^{3} + 2 \, a^{3} b^{2} d^{3} e^{4} - 3 \, a^{4} b d^{2} e^{5} + a^{5} d e^{6}\right )} x^{2} +{\left (b^{5} d^{7} - a b^{4} d^{6} e - 6 \, a^{2} b^{3} d^{5} e^{2} + 14 \, a^{3} b^{2} d^{4} e^{3} - 11 \, a^{4} b d^{3} e^{4} + 3 \, a^{5} d^{2} e^{5}\right )} x\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)/(b*x+a)^2/(e*x+d)^4,x, algorithm="maxima")

[Out]

(B*b^3*d + (3*B*a*b^2 - 4*A*b^3)*e)*log(b*x + a)/(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) - (B*b^3*d + (3*B*a*b^2 - 4*A*b^3)*e)*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) - 1/6*(2*A*a^3*e^3 - (17*B*a*b^2 - 6*A*b^3
)*d^3 - 2*(4*B*a^2*b - 13*A*a*b^2)*d^2*e + (B*a^3 - 10*A*a^2*b)*d*e^2 - 6*(B*b^3*d*e^2 + (3*B*a*b^2 - 4*A*b^3)
*e^3)*x^3 - 3*(5*B*b^3*d^2*e + 4*(4*B*a*b^2 - 5*A*b^3)*d*e^2 + (3*B*a^2*b - 4*A*a*b^2)*e^3)*x^2 - (11*B*b^3*d^
3 + (41*B*a*b^2 - 44*A*b^3)*d^2*e + (23*B*a^2*b - 32*A*a*b^2)*d*e^2 - (3*B*a^3 - 4*A*a^2*b)*e^3)*x)/(a*b^4*d^7
- 4*a^2*b^3*d^6*e + 6*a^3*b^2*d^5*e^2 - 4*a^4*b*d^4*e^3 + a^5*d^3*e^4 + (b^5*d^4*e^3 - 4*a*b^4*d^3*e^4 + 6*a^
2*b^3*d^2*e^5 - 4*a^3*b^2*d*e^6 + a^4*b*e^7)*x^4 + (3*b^5*d^5*e^2 - 11*a*b^4*d^4*e^3 + 14*a^2*b^3*d^3*e^4 - 6*
a^3*b^2*d^2*e^5 - a^4*b*d*e^6 + a^5*e^7)*x^3 + 3*(b^5*d^6*e - 3*a*b^4*d^5*e^2 + 2*a^2*b^3*d^4*e^3 + 2*a^3*b^2*
d^3*e^4 - 3*a^4*b*d^2*e^5 + a^5*d*e^6)*x^2 + (b^5*d^7 - a*b^4*d^6*e - 6*a^2*b^3*d^5*e^2 + 14*a^3*b^2*d^4*e^3 -
11*a^4*b*d^3*e^4 + 3*a^5*d^2*e^5)*x)

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Fricas [B]  time = 1.80851, size = 2485, normalized size = 12.42 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)/(b*x+a)^2/(e*x+d)^4,x, algorithm="fricas")

[Out]

1/6*(2*A*a^4*e^4 + (17*B*a*b^3 - 6*A*b^4)*d^4 - (9*B*a^2*b^2 + 20*A*a*b^3)*d^3*e - 9*(B*a^3*b - 4*A*a^2*b^2)*d
^2*e^2 + (B*a^4 - 12*A*a^3*b)*d*e^3 + 6*(B*b^4*d^2*e^2 + 2*(B*a*b^3 - 2*A*b^4)*d*e^3 - (3*B*a^2*b^2 - 4*A*a*b^
3)*e^4)*x^3 + 3*(5*B*b^4*d^3*e + (11*B*a*b^3 - 20*A*b^4)*d^2*e^2 - (13*B*a^2*b^2 - 16*A*a*b^3)*d*e^3 - (3*B*a^
3*b - 4*A*a^2*b^2)*e^4)*x^2 + (11*B*b^4*d^4 + 2*(15*B*a*b^3 - 22*A*b^4)*d^3*e - 6*(3*B*a^2*b^2 - 2*A*a*b^3)*d^
2*e^2 - 2*(13*B*a^3*b - 18*A*a^2*b^2)*d*e^3 + (3*B*a^4 - 4*A*a^3*b)*e^4)*x + 6*(B*a*b^3*d^4 + (3*B*a^2*b^2 - 4
*A*a*b^3)*d^3*e + (B*b^4*d*e^3 + (3*B*a*b^3 - 4*A*b^4)*e^4)*x^4 + (3*B*b^4*d^2*e^2 + 2*(5*B*a*b^3 - 6*A*b^4)*d
*e^3 + (3*B*a^2*b^2 - 4*A*a*b^3)*e^4)*x^3 + 3*(B*b^4*d^3*e + 4*(B*a*b^3 - A*b^4)*d^2*e^2 + (3*B*a^2*b^2 - 4*A*
a*b^3)*d*e^3)*x^2 + (B*b^4*d^4 + 2*(3*B*a*b^3 - 2*A*b^4)*d^3*e + 3*(3*B*a^2*b^2 - 4*A*a*b^3)*d^2*e^2)*x)*log(b
*x + a) - 6*(B*a*b^3*d^4 + (3*B*a^2*b^2 - 4*A*a*b^3)*d^3*e + (B*b^4*d*e^3 + (3*B*a*b^3 - 4*A*b^4)*e^4)*x^4 + (
3*B*b^4*d^2*e^2 + 2*(5*B*a*b^3 - 6*A*b^4)*d*e^3 + (3*B*a^2*b^2 - 4*A*a*b^3)*e^4)*x^3 + 3*(B*b^4*d^3*e + 4*(B*a
*b^3 - A*b^4)*d^2*e^2 + (3*B*a^2*b^2 - 4*A*a*b^3)*d*e^3)*x^2 + (B*b^4*d^4 + 2*(3*B*a*b^3 - 2*A*b^4)*d^3*e + 3*
(3*B*a^2*b^2 - 4*A*a*b^3)*d^2*e^2)*x)*log(e*x + d))/(a*b^5*d^8 - 5*a^2*b^4*d^7*e + 10*a^3*b^3*d^6*e^2 - 10*a^4
*b^2*d^5*e^3 + 5*a^5*b*d^4*e^4 - a^6*d^3*e^5 + (b^6*d^5*e^3 - 5*a*b^5*d^4*e^4 + 10*a^2*b^4*d^3*e^5 - 10*a^3*b^
3*d^2*e^6 + 5*a^4*b^2*d*e^7 - a^5*b*e^8)*x^4 + (3*b^6*d^6*e^2 - 14*a*b^5*d^5*e^3 + 25*a^2*b^4*d^4*e^4 - 20*a^3
*b^3*d^3*e^5 + 5*a^4*b^2*d^2*e^6 + 2*a^5*b*d*e^7 - a^6*e^8)*x^3 + 3*(b^6*d^7*e - 4*a*b^5*d^6*e^2 + 5*a^2*b^4*d
^5*e^3 - 5*a^4*b^2*d^3*e^5 + 4*a^5*b*d^2*e^6 - a^6*d*e^7)*x^2 + (b^6*d^8 - 2*a*b^5*d^7*e - 5*a^2*b^4*d^6*e^2 +
20*a^3*b^3*d^5*e^3 - 25*a^4*b^2*d^4*e^4 + 14*a^5*b*d^3*e^5 - 3*a^6*d^2*e^6)*x)

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Sympy [B]  time = 5.96999, size = 1445, normalized size = 7.22 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)/(b*x+a)**2/(e*x+d)**4,x)

[Out]

b**2*(-4*A*b*e + 3*B*a*e + B*b*d)*log(x + (-4*A*a*b**3*e**2 - 4*A*b**4*d*e + 3*B*a**2*b**2*e**2 + 4*B*a*b**3*d
*e + B*b**4*d**2 - a**6*b**2*e**6*(-4*A*b*e + 3*B*a*e + B*b*d)/(a*e - b*d)**5 + 6*a**5*b**3*d*e**5*(-4*A*b*e +
3*B*a*e + B*b*d)/(a*e - b*d)**5 - 15*a**4*b**4*d**2*e**4*(-4*A*b*e + 3*B*a*e + B*b*d)/(a*e - b*d)**5 + 20*a**
3*b**5*d**3*e**3*(-4*A*b*e + 3*B*a*e + B*b*d)/(a*e - b*d)**5 - 15*a**2*b**6*d**4*e**2*(-4*A*b*e + 3*B*a*e + B*
b*d)/(a*e - b*d)**5 + 6*a*b**7*d**5*e*(-4*A*b*e + 3*B*a*e + B*b*d)/(a*e - b*d)**5 - b**8*d**6*(-4*A*b*e + 3*B*
a*e + B*b*d)/(a*e - b*d)**5)/(-8*A*b**4*e**2 + 6*B*a*b**3*e**2 + 2*B*b**4*d*e))/(a*e - b*d)**5 - b**2*(-4*A*b*
e + 3*B*a*e + B*b*d)*log(x + (-4*A*a*b**3*e**2 - 4*A*b**4*d*e + 3*B*a**2*b**2*e**2 + 4*B*a*b**3*d*e + B*b**4*d
**2 + a**6*b**2*e**6*(-4*A*b*e + 3*B*a*e + B*b*d)/(a*e - b*d)**5 - 6*a**5*b**3*d*e**5*(-4*A*b*e + 3*B*a*e + B*
b*d)/(a*e - b*d)**5 + 15*a**4*b**4*d**2*e**4*(-4*A*b*e + 3*B*a*e + B*b*d)/(a*e - b*d)**5 - 20*a**3*b**5*d**3*e
**3*(-4*A*b*e + 3*B*a*e + B*b*d)/(a*e - b*d)**5 + 15*a**2*b**6*d**4*e**2*(-4*A*b*e + 3*B*a*e + B*b*d)/(a*e - b
*d)**5 - 6*a*b**7*d**5*e*(-4*A*b*e + 3*B*a*e + B*b*d)/(a*e - b*d)**5 + b**8*d**6*(-4*A*b*e + 3*B*a*e + B*b*d)/
(a*e - b*d)**5)/(-8*A*b**4*e**2 + 6*B*a*b**3*e**2 + 2*B*b**4*d*e))/(a*e - b*d)**5 + (-2*A*a**3*e**3 + 10*A*a**
2*b*d*e**2 - 26*A*a*b**2*d**2*e - 6*A*b**3*d**3 - B*a**3*d*e**2 + 8*B*a**2*b*d**2*e + 17*B*a*b**2*d**3 + x**3*
(-24*A*b**3*e**3 + 18*B*a*b**2*e**3 + 6*B*b**3*d*e**2) + x**2*(-12*A*a*b**2*e**3 - 60*A*b**3*d*e**2 + 9*B*a**2
*b*e**3 + 48*B*a*b**2*d*e**2 + 15*B*b**3*d**2*e) + x*(4*A*a**2*b*e**3 - 32*A*a*b**2*d*e**2 - 44*A*b**3*d**2*e
- 3*B*a**3*e**3 + 23*B*a**2*b*d*e**2 + 41*B*a*b**2*d**2*e + 11*B*b**3*d**3))/(6*a**5*d**3*e**4 - 24*a**4*b*d**
4*e**3 + 36*a**3*b**2*d**5*e**2 - 24*a**2*b**3*d**6*e + 6*a*b**4*d**7 + x**4*(6*a**4*b*e**7 - 24*a**3*b**2*d*e
**6 + 36*a**2*b**3*d**2*e**5 - 24*a*b**4*d**3*e**4 + 6*b**5*d**4*e**3) + x**3*(6*a**5*e**7 - 6*a**4*b*d*e**6 -
36*a**3*b**2*d**2*e**5 + 84*a**2*b**3*d**3*e**4 - 66*a*b**4*d**4*e**3 + 18*b**5*d**5*e**2) + x**2*(18*a**5*d*
e**6 - 54*a**4*b*d**2*e**5 + 36*a**3*b**2*d**3*e**4 + 36*a**2*b**3*d**4*e**3 - 54*a*b**4*d**5*e**2 + 18*b**5*d
**6*e) + x*(18*a**5*d**2*e**5 - 66*a**4*b*d**3*e**4 + 84*a**3*b**2*d**4*e**3 - 36*a**2*b**3*d**5*e**2 - 6*a*b*
*4*d**6*e + 6*b**5*d**7))

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Giac [B]  time = 1.99375, size = 571, normalized size = 2.86 \begin{align*} -\frac{{\left (B b^{4} d + 3 \, B a b^{3} e - 4 \, A b^{4} e\right )} \log \left ({\left | -\frac{b d}{b x + a} + \frac{a e}{b x + a} - e \right |}\right )}{b^{6} d^{5} - 5 \, a b^{5} d^{4} e + 10 \, a^{2} b^{4} d^{3} e^{2} - 10 \, a^{3} b^{3} d^{2} e^{3} + 5 \, a^{4} b^{2} d e^{4} - a^{5} b e^{5}} + \frac{\frac{B a b^{6}}{b x + a} - \frac{A b^{7}}{b x + a}}{b^{8} d^{4} - 4 \, a b^{7} d^{3} e + 6 \, a^{2} b^{6} d^{2} e^{2} - 4 \, a^{3} b^{5} d e^{3} + a^{4} b^{4} e^{4}} - \frac{11 \, B b^{3} d e^{3} + 15 \, B a b^{2} e^{4} - 26 \, A b^{3} e^{4} + \frac{3 \,{\left (9 \, B b^{5} d^{2} e^{2} + 2 \, B a b^{4} d e^{3} - 20 \, A b^{5} d e^{3} - 11 \, B a^{2} b^{3} e^{4} + 20 \, A a b^{4} e^{4}\right )}}{{\left (b x + a\right )} b} + \frac{18 \,{\left (B b^{7} d^{3} e - B a b^{6} d^{2} e^{2} - 2 \, A b^{7} d^{2} e^{2} - B a^{2} b^{5} d e^{3} + 4 \, A a b^{6} d e^{3} + B a^{3} b^{4} e^{4} - 2 \, A a^{2} b^{5} e^{4}\right )}}{{\left (b x + a\right )}^{2} b^{2}}}{6 \,{\left (b d - a e\right )}^{5}{\left (\frac{b d}{b x + a} - \frac{a e}{b x + a} + e\right )}^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)/(b*x+a)^2/(e*x+d)^4,x, algorithm="giac")

[Out]

-(B*b^4*d + 3*B*a*b^3*e - 4*A*b^4*e)*log(abs(-b*d/(b*x + a) + a*e/(b*x + a) - e))/(b^6*d^5 - 5*a*b^5*d^4*e + 1
0*a^2*b^4*d^3*e^2 - 10*a^3*b^3*d^2*e^3 + 5*a^4*b^2*d*e^4 - a^5*b*e^5) + (B*a*b^6/(b*x + a) - A*b^7/(b*x + a))/
(b^8*d^4 - 4*a*b^7*d^3*e + 6*a^2*b^6*d^2*e^2 - 4*a^3*b^5*d*e^3 + a^4*b^4*e^4) - 1/6*(11*B*b^3*d*e^3 + 15*B*a*b
^2*e^4 - 26*A*b^3*e^4 + 3*(9*B*b^5*d^2*e^2 + 2*B*a*b^4*d*e^3 - 20*A*b^5*d*e^3 - 11*B*a^2*b^3*e^4 + 20*A*a*b^4*
e^4)/((b*x + a)*b) + 18*(B*b^7*d^3*e - B*a*b^6*d^2*e^2 - 2*A*b^7*d^2*e^2 - B*a^2*b^5*d*e^3 + 4*A*a*b^6*d*e^3 +
B*a^3*b^4*e^4 - 2*A*a^2*b^5*e^4)/((b*x + a)^2*b^2))/((b*d - a*e)^5*(b*d/(b*x + a) - a*e/(b*x + a) + e)^3)