3.1142 \(\int \frac{A+B x}{(a+b x)^3 (d+e x)^4} \, dx\)

Optimal. Leaf size=248 \[ -\frac{b^2 (3 a B e-4 A b e+b B d)}{(a+b x) (b d-a e)^5}-\frac{b^2 (A b-a B)}{2 (a+b x)^2 (b d-a e)^4}-\frac{2 b^2 e \log (a+b x) (3 a B e-5 A b e+2 b B d)}{(b d-a e)^6}+\frac{2 b^2 e \log (d+e x) (3 a B e-5 A b e+2 b B d)}{(b d-a e)^6}-\frac{3 b e (a B e-2 A b e+b B d)}{(d+e x) (b d-a e)^5}-\frac{e (a B e-3 A b e+2 b B d)}{2 (d+e x)^2 (b d-a e)^4}-\frac{e (B d-A e)}{3 (d+e x)^3 (b d-a e)^3} \]

[Out]

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

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Rubi [A]  time = 0.310272, antiderivative size = 248, 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} \[ -\frac{b^2 (3 a B e-4 A b e+b B d)}{(a+b x) (b d-a e)^5}-\frac{b^2 (A b-a B)}{2 (a+b x)^2 (b d-a e)^4}-\frac{2 b^2 e \log (a+b x) (3 a B e-5 A b e+2 b B d)}{(b d-a e)^6}+\frac{2 b^2 e \log (d+e x) (3 a B e-5 A b e+2 b B d)}{(b d-a e)^6}-\frac{3 b e (a B e-2 A b e+b B d)}{(d+e x) (b d-a e)^5}-\frac{e (a B e-3 A b e+2 b B d)}{2 (d+e x)^2 (b d-a e)^4}-\frac{e (B d-A e)}{3 (d+e x)^3 (b d-a e)^3} \]

Antiderivative was successfully verified.

[In]

Int[(A + B*x)/((a + b*x)^3*(d + e*x)^4),x]

[Out]

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

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)^3 (d+e x)^4} \, dx &=\int \left (\frac{b^3 (A b-a B)}{(b d-a e)^4 (a+b x)^3}+\frac{b^3 (b B d-4 A b e+3 a B e)}{(b d-a e)^5 (a+b x)^2}+\frac{2 b^3 e (-2 b B d+5 A b e-3 a B e)}{(b d-a e)^6 (a+b x)}-\frac{e^2 (-B d+A e)}{(b d-a e)^3 (d+e x)^4}-\frac{e^2 (-2 b B d+3 A b e-a B e)}{(b d-a e)^4 (d+e x)^3}-\frac{3 b e^2 (-b B d+2 A b e-a B e)}{(b d-a e)^5 (d+e x)^2}-\frac{2 b^2 e^2 (-2 b B d+5 A b e-3 a B e)}{(b d-a e)^6 (d+e x)}\right ) \, dx\\ &=-\frac{b^2 (A b-a B)}{2 (b d-a e)^4 (a+b x)^2}-\frac{b^2 (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)}{3 (b d-a e)^3 (d+e x)^3}-\frac{e (2 b B d-3 A b e+a B e)}{2 (b d-a e)^4 (d+e x)^2}-\frac{3 b e (b B d-2 A b e+a B e)}{(b d-a e)^5 (d+e x)}-\frac{2 b^2 e (2 b B d-5 A b e+3 a B e) \log (a+b x)}{(b d-a e)^6}+\frac{2 b^2 e (2 b B d-5 A b e+3 a B e) \log (d+e x)}{(b d-a e)^6}\\ \end{align*}

Mathematica [A]  time = 0.196913, size = 233, normalized size = 0.94 \[ \frac{-\frac{3 b^2 (A b-a B) (b d-a e)^2}{(a+b x)^2}-\frac{6 b^2 (b d-a e) (3 a B e-4 A b e+b B d)}{a+b x}+12 b^2 e \log (a+b x) (-3 a B e+5 A b e-2 b B d)+12 b^2 e \log (d+e x) (3 a B e-5 A b e+2 b B d)+\frac{2 e (b d-a e)^3 (A e-B d)}{(d+e x)^3}+\frac{3 e (b d-a e)^2 (-a B e+3 A b e-2 b B d)}{(d+e x)^2}+\frac{18 b e (a e-b d) (a B e-2 A b e+b B d)}{d+e x}}{6 (b d-a e)^6} \]

Antiderivative was successfully verified.

[In]

Integrate[(A + B*x)/((a + b*x)^3*(d + e*x)^4),x]

[Out]

((-3*b^2*(A*b - a*B)*(b*d - a*e)^2)/(a + b*x)^2 - (6*b^2*(b*d - a*e)*(b*B*d - 4*A*b*e + 3*a*B*e))/(a + b*x) +
(2*e*(b*d - a*e)^3*(-(B*d) + A*e))/(d + e*x)^3 + (3*e*(b*d - a*e)^2*(-2*b*B*d + 3*A*b*e - a*B*e))/(d + e*x)^2
+ (18*b*e*(-(b*d) + a*e)*(b*B*d - 2*A*b*e + a*B*e))/(d + e*x) + 12*b^2*e*(-2*b*B*d + 5*A*b*e - 3*a*B*e)*Log[a
+ b*x] + 12*b^2*e*(2*b*B*d - 5*A*b*e + 3*a*B*e)*Log[d + e*x])/(6*(b*d - a*e)^6)

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Maple [A]  time = 0.018, size = 463, normalized size = 1.9 \begin{align*} -{\frac{{e}^{2}A}{3\, \left ( ae-bd \right ) ^{3} \left ( ex+d \right ) ^{3}}}+{\frac{eBd}{3\, \left ( ae-bd \right ) ^{3} \left ( ex+d \right ) ^{3}}}+{\frac{3\,{e}^{2}Ab}{2\, \left ( ae-bd \right ) ^{4} \left ( ex+d \right ) ^{2}}}-{\frac{{e}^{2}Ba}{2\, \left ( ae-bd \right ) ^{4} \left ( ex+d \right ) ^{2}}}-{\frac{bBde}{ \left ( ae-bd \right ) ^{4} \left ( ex+d \right ) ^{2}}}-6\,{\frac{{e}^{2}{b}^{2}A}{ \left ( ae-bd \right ) ^{5} \left ( ex+d \right ) }}+3\,{\frac{{e}^{2}bBa}{ \left ( ae-bd \right ) ^{5} \left ( ex+d \right ) }}+3\,{\frac{{b}^{2}Bde}{ \left ( ae-bd \right ) ^{5} \left ( ex+d \right ) }}-10\,{\frac{{e}^{2}{b}^{3}\ln \left ( ex+d \right ) A}{ \left ( ae-bd \right ) ^{6}}}+6\,{\frac{{e}^{2}{b}^{2}\ln \left ( ex+d \right ) Ba}{ \left ( ae-bd \right ) ^{6}}}+4\,{\frac{{b}^{3}e\ln \left ( ex+d \right ) Bd}{ \left ( ae-bd \right ) ^{6}}}-4\,{\frac{{b}^{3}Ae}{ \left ( ae-bd \right ) ^{5} \left ( bx+a \right ) }}+3\,{\frac{Ba{b}^{2}e}{ \left ( ae-bd \right ) ^{5} \left ( bx+a \right ) }}+{\frac{{b}^{3}Bd}{ \left ( ae-bd \right ) ^{5} \left ( bx+a \right ) }}-{\frac{{b}^{3}A}{2\, \left ( ae-bd \right ) ^{4} \left ( bx+a \right ) ^{2}}}+{\frac{Ba{b}^{2}}{2\, \left ( ae-bd \right ) ^{4} \left ( bx+a \right ) ^{2}}}+10\,{\frac{{e}^{2}{b}^{3}\ln \left ( bx+a \right ) A}{ \left ( ae-bd \right ) ^{6}}}-6\,{\frac{{e}^{2}{b}^{2}\ln \left ( bx+a \right ) Ba}{ \left ( ae-bd \right ) ^{6}}}-4\,{\frac{{b}^{3}e\ln \left ( bx+a \right ) Bd}{ \left ( ae-bd \right ) ^{6}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)/(b*x+a)^3/(e*x+d)^4,x)

[Out]

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

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Maxima [B]  time = 3.52765, size = 1520, normalized size = 6.13 \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)^3/(e*x+d)^4,x, algorithm="maxima")

[Out]

-2*(2*B*b^3*d*e + (3*B*a*b^2 - 5*A*b^3)*e^2)*log(b*x + a)/(b^6*d^6 - 6*a*b^5*d^5*e + 15*a^2*b^4*d^4*e^2 - 20*a
^3*b^3*d^3*e^3 + 15*a^4*b^2*d^2*e^4 - 6*a^5*b*d*e^5 + a^6*e^6) + 2*(2*B*b^3*d*e + (3*B*a*b^2 - 5*A*b^3)*e^2)*l
og(e*x + d)/(b^6*d^6 - 6*a*b^5*d^5*e + 15*a^2*b^4*d^4*e^2 - 20*a^3*b^3*d^3*e^3 + 15*a^4*b^2*d^2*e^4 - 6*a^5*b*
d*e^5 + a^6*e^6) + 1/6*(2*A*a^4*e^4 - 3*(B*a*b^3 + A*b^4)*d^4 - (47*B*a^2*b^2 - 27*A*a*b^3)*d^3*e - (11*B*a^3*
b - 47*A*a^2*b^2)*d^2*e^2 + (B*a^4 - 13*A*a^3*b)*d*e^3 - 12*(2*B*b^4*d*e^3 + (3*B*a*b^3 - 5*A*b^4)*e^4)*x^4 -
6*(10*B*b^4*d^2*e^2 + (21*B*a*b^3 - 25*A*b^4)*d*e^3 + 3*(3*B*a^2*b^2 - 5*A*a*b^3)*e^4)*x^3 - 2*(22*B*b^4*d^3*e
+ (79*B*a*b^3 - 55*A*b^4)*d^2*e^2 + (73*B*a^2*b^2 - 115*A*a*b^3)*d*e^3 + 2*(3*B*a^3*b - 5*A*a^2*b^2)*e^4)*x^2
- (6*B*b^4*d^4 + (79*B*a*b^3 - 15*A*b^4)*d^3*e + (127*B*a^2*b^2 - 175*A*a*b^3)*d^2*e^2 + (31*B*a^3*b - 55*A*a
^2*b^2)*d*e^3 - (3*B*a^4 - 5*A*a^3*b)*e^4)*x)/(a^2*b^5*d^8 - 5*a^3*b^4*d^7*e + 10*a^4*b^3*d^6*e^2 - 10*a^5*b^2
*d^5*e^3 + 5*a^6*b*d^4*e^4 - a^7*d^3*e^5 + (b^7*d^5*e^3 - 5*a*b^6*d^4*e^4 + 10*a^2*b^5*d^3*e^5 - 10*a^3*b^4*d^
2*e^6 + 5*a^4*b^3*d*e^7 - a^5*b^2*e^8)*x^5 + (3*b^7*d^6*e^2 - 13*a*b^6*d^5*e^3 + 20*a^2*b^5*d^4*e^4 - 10*a^3*b
^4*d^3*e^5 - 5*a^4*b^3*d^2*e^6 + 7*a^5*b^2*d*e^7 - 2*a^6*b*e^8)*x^4 + (3*b^7*d^7*e - 9*a*b^6*d^6*e^2 + a^2*b^5
*d^5*e^3 + 25*a^3*b^4*d^4*e^4 - 35*a^4*b^3*d^3*e^5 + 17*a^5*b^2*d^2*e^6 - a^6*b*d*e^7 - a^7*e^8)*x^3 + (b^7*d^
8 + a*b^6*d^7*e - 17*a^2*b^5*d^6*e^2 + 35*a^3*b^4*d^5*e^3 - 25*a^4*b^3*d^4*e^4 - a^5*b^2*d^3*e^5 + 9*a^6*b*d^2
*e^6 - 3*a^7*d*e^7)*x^2 + (2*a*b^6*d^8 - 7*a^2*b^5*d^7*e + 5*a^3*b^4*d^6*e^2 + 10*a^4*b^3*d^5*e^3 - 20*a^5*b^2
*d^4*e^4 + 13*a^6*b*d^3*e^5 - 3*a^7*d^2*e^6)*x)

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Fricas [B]  time = 2.01714, size = 3785, normalized size = 15.26 \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)^3/(e*x+d)^4,x, algorithm="fricas")

[Out]

-1/6*(2*A*a^5*e^5 + 3*(B*a*b^4 + A*b^5)*d^5 + 2*(22*B*a^2*b^3 - 15*A*a*b^4)*d^4*e - 4*(9*B*a^3*b^2 + 5*A*a^2*b
^3)*d^3*e^2 - 12*(B*a^4*b - 5*A*a^3*b^2)*d^2*e^3 + (B*a^5 - 15*A*a^4*b)*d*e^4 + 12*(2*B*b^5*d^2*e^3 + (B*a*b^4
- 5*A*b^5)*d*e^4 - (3*B*a^2*b^3 - 5*A*a*b^4)*e^5)*x^4 + 6*(10*B*b^5*d^3*e^2 + (11*B*a*b^4 - 25*A*b^5)*d^2*e^3
- 2*(6*B*a^2*b^3 - 5*A*a*b^4)*d*e^4 - 3*(3*B*a^3*b^2 - 5*A*a^2*b^3)*e^5)*x^3 + 2*(22*B*b^5*d^4*e + (57*B*a*b^
4 - 55*A*b^5)*d^3*e^2 - 6*(B*a^2*b^3 + 10*A*a*b^4)*d^2*e^3 - (67*B*a^3*b^2 - 105*A*a^2*b^3)*d*e^4 - 2*(3*B*a^4
*b - 5*A*a^3*b^2)*e^5)*x^2 + (6*B*b^5*d^5 + (73*B*a*b^4 - 15*A*b^5)*d^4*e + 16*(3*B*a^2*b^3 - 10*A*a*b^4)*d^3*
e^2 - 24*(4*B*a^3*b^2 - 5*A*a^2*b^3)*d^2*e^3 - 2*(17*B*a^4*b - 30*A*a^3*b^2)*d*e^4 + (3*B*a^5 - 5*A*a^4*b)*e^5
)*x + 12*(2*B*a^2*b^3*d^4*e + (3*B*a^3*b^2 - 5*A*a^2*b^3)*d^3*e^2 + (2*B*b^5*d*e^4 + (3*B*a*b^4 - 5*A*b^5)*e^5
)*x^5 + (6*B*b^5*d^2*e^3 + (13*B*a*b^4 - 15*A*b^5)*d*e^4 + 2*(3*B*a^2*b^3 - 5*A*a*b^4)*e^5)*x^4 + (6*B*b^5*d^3
*e^2 + 3*(7*B*a*b^4 - 5*A*b^5)*d^2*e^3 + 10*(2*B*a^2*b^3 - 3*A*a*b^4)*d*e^4 + (3*B*a^3*b^2 - 5*A*a^2*b^3)*e^5)
*x^3 + (2*B*b^5*d^4*e + 5*(3*B*a*b^4 - A*b^5)*d^3*e^2 + 6*(4*B*a^2*b^3 - 5*A*a*b^4)*d^2*e^3 + 3*(3*B*a^3*b^2 -
5*A*a^2*b^3)*d*e^4)*x^2 + (4*B*a*b^4*d^4*e + 2*(6*B*a^2*b^3 - 5*A*a*b^4)*d^3*e^2 + 3*(3*B*a^3*b^2 - 5*A*a^2*b
^3)*d^2*e^3)*x)*log(b*x + a) - 12*(2*B*a^2*b^3*d^4*e + (3*B*a^3*b^2 - 5*A*a^2*b^3)*d^3*e^2 + (2*B*b^5*d*e^4 +
(3*B*a*b^4 - 5*A*b^5)*e^5)*x^5 + (6*B*b^5*d^2*e^3 + (13*B*a*b^4 - 15*A*b^5)*d*e^4 + 2*(3*B*a^2*b^3 - 5*A*a*b^4
)*e^5)*x^4 + (6*B*b^5*d^3*e^2 + 3*(7*B*a*b^4 - 5*A*b^5)*d^2*e^3 + 10*(2*B*a^2*b^3 - 3*A*a*b^4)*d*e^4 + (3*B*a^
3*b^2 - 5*A*a^2*b^3)*e^5)*x^3 + (2*B*b^5*d^4*e + 5*(3*B*a*b^4 - A*b^5)*d^3*e^2 + 6*(4*B*a^2*b^3 - 5*A*a*b^4)*d
^2*e^3 + 3*(3*B*a^3*b^2 - 5*A*a^2*b^3)*d*e^4)*x^2 + (4*B*a*b^4*d^4*e + 2*(6*B*a^2*b^3 - 5*A*a*b^4)*d^3*e^2 + 3
*(3*B*a^3*b^2 - 5*A*a^2*b^3)*d^2*e^3)*x)*log(e*x + d))/(a^2*b^6*d^9 - 6*a^3*b^5*d^8*e + 15*a^4*b^4*d^7*e^2 - 2
0*a^5*b^3*d^6*e^3 + 15*a^6*b^2*d^5*e^4 - 6*a^7*b*d^4*e^5 + a^8*d^3*e^6 + (b^8*d^6*e^3 - 6*a*b^7*d^5*e^4 + 15*a
^2*b^6*d^4*e^5 - 20*a^3*b^5*d^3*e^6 + 15*a^4*b^4*d^2*e^7 - 6*a^5*b^3*d*e^8 + a^6*b^2*e^9)*x^5 + (3*b^8*d^7*e^2
- 16*a*b^7*d^6*e^3 + 33*a^2*b^6*d^5*e^4 - 30*a^3*b^5*d^4*e^5 + 5*a^4*b^4*d^3*e^6 + 12*a^5*b^3*d^2*e^7 - 9*a^6
*b^2*d*e^8 + 2*a^7*b*e^9)*x^4 + (3*b^8*d^8*e - 12*a*b^7*d^7*e^2 + 10*a^2*b^6*d^6*e^3 + 24*a^3*b^5*d^5*e^4 - 60
*a^4*b^4*d^4*e^5 + 52*a^5*b^3*d^3*e^6 - 18*a^6*b^2*d^2*e^7 + a^8*e^9)*x^3 + (b^8*d^9 - 18*a^2*b^6*d^7*e^2 + 52
*a^3*b^5*d^6*e^3 - 60*a^4*b^4*d^5*e^4 + 24*a^5*b^3*d^4*e^5 + 10*a^6*b^2*d^3*e^6 - 12*a^7*b*d^2*e^7 + 3*a^8*d*e
^8)*x^2 + (2*a*b^7*d^9 - 9*a^2*b^6*d^8*e + 12*a^3*b^5*d^7*e^2 + 5*a^4*b^4*d^6*e^3 - 30*a^5*b^3*d^5*e^4 + 33*a^
6*b^2*d^4*e^5 - 16*a^7*b*d^3*e^6 + 3*a^8*d^2*e^7)*x)

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Sympy [B]  time = 9.75186, size = 1975, normalized size = 7.96 \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)**3/(e*x+d)**4,x)

[Out]

2*b**2*e*(-5*A*b*e + 3*B*a*e + 2*B*b*d)*log(x + (-10*A*a*b**3*e**3 - 10*A*b**4*d*e**2 + 6*B*a**2*b**2*e**3 + 1
0*B*a*b**3*d*e**2 + 4*B*b**4*d**2*e - 2*a**7*b**2*e**8*(-5*A*b*e + 3*B*a*e + 2*B*b*d)/(a*e - b*d)**6 + 14*a**6
*b**3*d*e**7*(-5*A*b*e + 3*B*a*e + 2*B*b*d)/(a*e - b*d)**6 - 42*a**5*b**4*d**2*e**6*(-5*A*b*e + 3*B*a*e + 2*B*
b*d)/(a*e - b*d)**6 + 70*a**4*b**5*d**3*e**5*(-5*A*b*e + 3*B*a*e + 2*B*b*d)/(a*e - b*d)**6 - 70*a**3*b**6*d**4
*e**4*(-5*A*b*e + 3*B*a*e + 2*B*b*d)/(a*e - b*d)**6 + 42*a**2*b**7*d**5*e**3*(-5*A*b*e + 3*B*a*e + 2*B*b*d)/(a
*e - b*d)**6 - 14*a*b**8*d**6*e**2*(-5*A*b*e + 3*B*a*e + 2*B*b*d)/(a*e - b*d)**6 + 2*b**9*d**7*e*(-5*A*b*e + 3
*B*a*e + 2*B*b*d)/(a*e - b*d)**6)/(-20*A*b**4*e**3 + 12*B*a*b**3*e**3 + 8*B*b**4*d*e**2))/(a*e - b*d)**6 - 2*b
**2*e*(-5*A*b*e + 3*B*a*e + 2*B*b*d)*log(x + (-10*A*a*b**3*e**3 - 10*A*b**4*d*e**2 + 6*B*a**2*b**2*e**3 + 10*B
*a*b**3*d*e**2 + 4*B*b**4*d**2*e + 2*a**7*b**2*e**8*(-5*A*b*e + 3*B*a*e + 2*B*b*d)/(a*e - b*d)**6 - 14*a**6*b*
*3*d*e**7*(-5*A*b*e + 3*B*a*e + 2*B*b*d)/(a*e - b*d)**6 + 42*a**5*b**4*d**2*e**6*(-5*A*b*e + 3*B*a*e + 2*B*b*d
)/(a*e - b*d)**6 - 70*a**4*b**5*d**3*e**5*(-5*A*b*e + 3*B*a*e + 2*B*b*d)/(a*e - b*d)**6 + 70*a**3*b**6*d**4*e*
*4*(-5*A*b*e + 3*B*a*e + 2*B*b*d)/(a*e - b*d)**6 - 42*a**2*b**7*d**5*e**3*(-5*A*b*e + 3*B*a*e + 2*B*b*d)/(a*e
- b*d)**6 + 14*a*b**8*d**6*e**2*(-5*A*b*e + 3*B*a*e + 2*B*b*d)/(a*e - b*d)**6 - 2*b**9*d**7*e*(-5*A*b*e + 3*B*
a*e + 2*B*b*d)/(a*e - b*d)**6)/(-20*A*b**4*e**3 + 12*B*a*b**3*e**3 + 8*B*b**4*d*e**2))/(a*e - b*d)**6 + (-2*A*
a**4*e**4 + 13*A*a**3*b*d*e**3 - 47*A*a**2*b**2*d**2*e**2 - 27*A*a*b**3*d**3*e + 3*A*b**4*d**4 - B*a**4*d*e**3
+ 11*B*a**3*b*d**2*e**2 + 47*B*a**2*b**2*d**3*e + 3*B*a*b**3*d**4 + x**4*(-60*A*b**4*e**4 + 36*B*a*b**3*e**4
+ 24*B*b**4*d*e**3) + x**3*(-90*A*a*b**3*e**4 - 150*A*b**4*d*e**3 + 54*B*a**2*b**2*e**4 + 126*B*a*b**3*d*e**3
+ 60*B*b**4*d**2*e**2) + x**2*(-20*A*a**2*b**2*e**4 - 230*A*a*b**3*d*e**3 - 110*A*b**4*d**2*e**2 + 12*B*a**3*b
*e**4 + 146*B*a**2*b**2*d*e**3 + 158*B*a*b**3*d**2*e**2 + 44*B*b**4*d**3*e) + x*(5*A*a**3*b*e**4 - 55*A*a**2*b
**2*d*e**3 - 175*A*a*b**3*d**2*e**2 - 15*A*b**4*d**3*e - 3*B*a**4*e**4 + 31*B*a**3*b*d*e**3 + 127*B*a**2*b**2*
d**2*e**2 + 79*B*a*b**3*d**3*e + 6*B*b**4*d**4))/(6*a**7*d**3*e**5 - 30*a**6*b*d**4*e**4 + 60*a**5*b**2*d**5*e
**3 - 60*a**4*b**3*d**6*e**2 + 30*a**3*b**4*d**7*e - 6*a**2*b**5*d**8 + x**5*(6*a**5*b**2*e**8 - 30*a**4*b**3*
d*e**7 + 60*a**3*b**4*d**2*e**6 - 60*a**2*b**5*d**3*e**5 + 30*a*b**6*d**4*e**4 - 6*b**7*d**5*e**3) + x**4*(12*
a**6*b*e**8 - 42*a**5*b**2*d*e**7 + 30*a**4*b**3*d**2*e**6 + 60*a**3*b**4*d**3*e**5 - 120*a**2*b**5*d**4*e**4
+ 78*a*b**6*d**5*e**3 - 18*b**7*d**6*e**2) + x**3*(6*a**7*e**8 + 6*a**6*b*d*e**7 - 102*a**5*b**2*d**2*e**6 + 2
10*a**4*b**3*d**3*e**5 - 150*a**3*b**4*d**4*e**4 - 6*a**2*b**5*d**5*e**3 + 54*a*b**6*d**6*e**2 - 18*b**7*d**7*
e) + x**2*(18*a**7*d*e**7 - 54*a**6*b*d**2*e**6 + 6*a**5*b**2*d**3*e**5 + 150*a**4*b**3*d**4*e**4 - 210*a**3*b
**4*d**5*e**3 + 102*a**2*b**5*d**6*e**2 - 6*a*b**6*d**7*e - 6*b**7*d**8) + x*(18*a**7*d**2*e**6 - 78*a**6*b*d*
*3*e**5 + 120*a**5*b**2*d**4*e**4 - 60*a**4*b**3*d**5*e**3 - 30*a**3*b**4*d**6*e**2 + 42*a**2*b**5*d**7*e - 12
*a*b**6*d**8))

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Giac [B]  time = 2.12749, size = 1026, normalized size = 4.14 \begin{align*} -\frac{2 \,{\left (2 \, B b^{4} d e + 3 \, B a b^{3} e^{2} - 5 \, A b^{4} e^{2}\right )} \log \left ({\left | b x + a \right |}\right )}{b^{7} d^{6} - 6 \, a b^{6} d^{5} e + 15 \, a^{2} b^{5} d^{4} e^{2} - 20 \, a^{3} b^{4} d^{3} e^{3} + 15 \, a^{4} b^{3} d^{2} e^{4} - 6 \, a^{5} b^{2} d e^{5} + a^{6} b e^{6}} + \frac{2 \,{\left (2 \, B b^{3} d e^{2} + 3 \, B a b^{2} e^{3} - 5 \, A b^{3} e^{3}\right )} \log \left ({\left | x e + d \right |}\right )}{b^{6} d^{6} e - 6 \, a b^{5} d^{5} e^{2} + 15 \, a^{2} b^{4} d^{4} e^{3} - 20 \, a^{3} b^{3} d^{3} e^{4} + 15 \, a^{4} b^{2} d^{2} e^{5} - 6 \, a^{5} b d e^{6} + a^{6} e^{7}} - \frac{3 \, B a b^{4} d^{5} + 3 \, A b^{5} d^{5} + 44 \, B a^{2} b^{3} d^{4} e - 30 \, A a b^{4} d^{4} e - 36 \, B a^{3} b^{2} d^{3} e^{2} - 20 \, A a^{2} b^{3} d^{3} e^{2} - 12 \, B a^{4} b d^{2} e^{3} + 60 \, A a^{3} b^{2} d^{2} e^{3} + B a^{5} d e^{4} - 15 \, A a^{4} b d e^{4} + 2 \, A a^{5} e^{5} + 12 \,{\left (2 \, B b^{5} d^{2} e^{3} + B a b^{4} d e^{4} - 5 \, A b^{5} d e^{4} - 3 \, B a^{2} b^{3} e^{5} + 5 \, A a b^{4} e^{5}\right )} x^{4} + 6 \,{\left (10 \, B b^{5} d^{3} e^{2} + 11 \, B a b^{4} d^{2} e^{3} - 25 \, A b^{5} d^{2} e^{3} - 12 \, B a^{2} b^{3} d e^{4} + 10 \, A a b^{4} d e^{4} - 9 \, B a^{3} b^{2} e^{5} + 15 \, A a^{2} b^{3} e^{5}\right )} x^{3} + 2 \,{\left (22 \, B b^{5} d^{4} e + 57 \, B a b^{4} d^{3} e^{2} - 55 \, A b^{5} d^{3} e^{2} - 6 \, B a^{2} b^{3} d^{2} e^{3} - 60 \, A a b^{4} d^{2} e^{3} - 67 \, B a^{3} b^{2} d e^{4} + 105 \, A a^{2} b^{3} d e^{4} - 6 \, B a^{4} b e^{5} + 10 \, A a^{3} b^{2} e^{5}\right )} x^{2} +{\left (6 \, B b^{5} d^{5} + 73 \, B a b^{4} d^{4} e - 15 \, A b^{5} d^{4} e + 48 \, B a^{2} b^{3} d^{3} e^{2} - 160 \, A a b^{4} d^{3} e^{2} - 96 \, B a^{3} b^{2} d^{2} e^{3} + 120 \, A a^{2} b^{3} d^{2} e^{3} - 34 \, B a^{4} b d e^{4} + 60 \, A a^{3} b^{2} d e^{4} + 3 \, B a^{5} e^{5} - 5 \, A a^{4} b e^{5}\right )} x}{6 \,{\left (b d - a e\right )}^{6}{\left (b x + a\right )}^{2}{\left (x e + d\right )}^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)/(b*x+a)^3/(e*x+d)^4,x, algorithm="giac")

[Out]

-2*(2*B*b^4*d*e + 3*B*a*b^3*e^2 - 5*A*b^4*e^2)*log(abs(b*x + a))/(b^7*d^6 - 6*a*b^6*d^5*e + 15*a^2*b^5*d^4*e^2
- 20*a^3*b^4*d^3*e^3 + 15*a^4*b^3*d^2*e^4 - 6*a^5*b^2*d*e^5 + a^6*b*e^6) + 2*(2*B*b^3*d*e^2 + 3*B*a*b^2*e^3 -
5*A*b^3*e^3)*log(abs(x*e + d))/(b^6*d^6*e - 6*a*b^5*d^5*e^2 + 15*a^2*b^4*d^4*e^3 - 20*a^3*b^3*d^3*e^4 + 15*a^
4*b^2*d^2*e^5 - 6*a^5*b*d*e^6 + a^6*e^7) - 1/6*(3*B*a*b^4*d^5 + 3*A*b^5*d^5 + 44*B*a^2*b^3*d^4*e - 30*A*a*b^4*
d^4*e - 36*B*a^3*b^2*d^3*e^2 - 20*A*a^2*b^3*d^3*e^2 - 12*B*a^4*b*d^2*e^3 + 60*A*a^3*b^2*d^2*e^3 + B*a^5*d*e^4
- 15*A*a^4*b*d*e^4 + 2*A*a^5*e^5 + 12*(2*B*b^5*d^2*e^3 + B*a*b^4*d*e^4 - 5*A*b^5*d*e^4 - 3*B*a^2*b^3*e^5 + 5*A
*a*b^4*e^5)*x^4 + 6*(10*B*b^5*d^3*e^2 + 11*B*a*b^4*d^2*e^3 - 25*A*b^5*d^2*e^3 - 12*B*a^2*b^3*d*e^4 + 10*A*a*b^
4*d*e^4 - 9*B*a^3*b^2*e^5 + 15*A*a^2*b^3*e^5)*x^3 + 2*(22*B*b^5*d^4*e + 57*B*a*b^4*d^3*e^2 - 55*A*b^5*d^3*e^2
- 6*B*a^2*b^3*d^2*e^3 - 60*A*a*b^4*d^2*e^3 - 67*B*a^3*b^2*d*e^4 + 105*A*a^2*b^3*d*e^4 - 6*B*a^4*b*e^5 + 10*A*a
^3*b^2*e^5)*x^2 + (6*B*b^5*d^5 + 73*B*a*b^4*d^4*e - 15*A*b^5*d^4*e + 48*B*a^2*b^3*d^3*e^2 - 160*A*a*b^4*d^3*e^
2 - 96*B*a^3*b^2*d^2*e^3 + 120*A*a^2*b^3*d^2*e^3 - 34*B*a^4*b*d*e^4 + 60*A*a^3*b^2*d*e^4 + 3*B*a^5*e^5 - 5*A*a
^4*b*e^5)*x)/((b*d - a*e)^6*(b*x + a)^2*(x*e + d)^3)