3.1156 \(\int \frac{(A+B x) (d+e x)^3}{(b x+c x^2)^3} \, dx\)

Optimal. Leaf size=185 \[ -\frac{(c d-b e)^2 \left (-3 A c^2 d+b^2 B e+2 b B c d\right )}{b^4 c^2 (b+c x)}-\frac{(b B-A c) (c d-b e)^3}{2 b^3 c^2 (b+c x)^2}-\frac{d^2 (3 A b e-3 A c d+b B d)}{b^4 x}-\frac{3 d \log (x) (c d-b e) (A b e-2 A c d+b B d)}{b^5}+\frac{3 d (c d-b e) \log (b+c x) (A b e-2 A c d+b B d)}{b^5}-\frac{A d^3}{2 b^3 x^2} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.246013, antiderivative size = 185, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.042, Rules used = {771} \[ -\frac{(c d-b e)^2 \left (-3 A c^2 d+b^2 B e+2 b B c d\right )}{b^4 c^2 (b+c x)}-\frac{(b B-A c) (c d-b e)^3}{2 b^3 c^2 (b+c x)^2}-\frac{d^2 (3 A b e-3 A c d+b B d)}{b^4 x}-\frac{3 d \log (x) (c d-b e) (A b e-2 A c d+b B d)}{b^5}+\frac{3 d (c d-b e) \log (b+c x) (A b e-2 A c d+b B d)}{b^5}-\frac{A d^3}{2 b^3 x^2} \]

Antiderivative was successfully verified.

[In]

Int[((A + B*x)*(d + e*x)^3)/(b*x + c*x^2)^3,x]

[Out]

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

Rule 771

Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> In
t[ExpandIntegrand[(d + e*x)^m*(f + g*x)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && N
eQ[b^2 - 4*a*c, 0] && IntegerQ[p] && (GtQ[p, 0] || (EqQ[a, 0] && IntegerQ[m]))

Rubi steps

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

Mathematica [A]  time = 0.139838, size = 177, normalized size = 0.96 \[ -\frac{\frac{2 b (c d-b e)^2 \left (-3 A c^2 d+b^2 B e+2 b B c d\right )}{c^2 (b+c x)}-\frac{b^2 (b B-A c) (b e-c d)^3}{c^2 (b+c x)^2}+\frac{A b^2 d^3}{x^2}+\frac{2 b d^2 (3 A b e-3 A c d+b B d)}{x}-6 d \log (x) (b e-c d) (A b e-2 A c d+b B d)+6 d (b e-c d) \log (b+c x) (A b e-2 A c d+b B d)}{2 b^5} \]

Antiderivative was successfully verified.

[In]

Integrate[((A + B*x)*(d + e*x)^3)/(b*x + c*x^2)^3,x]

[Out]

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

________________________________________________________________________________________

Maple [B]  time = 0.014, size = 440, normalized size = 2.4 \begin{align*} -{\frac{A{d}^{3}}{2\,{b}^{3}{x}^{2}}}-6\,{\frac{Ac{d}^{2}e}{{b}^{3} \left ( cx+b \right ) }}-9\,{\frac{{d}^{2}\ln \left ( x \right ) Ace}{{b}^{4}}}-{\frac{3\,Ac{d}^{2}e}{2\,{b}^{2} \left ( cx+b \right ) ^{2}}}+9\,{\frac{{d}^{2}\ln \left ( cx+b \right ) Ace}{{b}^{4}}}-{\frac{B{d}^{3}}{{b}^{3}x}}-{\frac{A{e}^{3}}{2\,c \left ( cx+b \right ) ^{2}}}-{\frac{B{e}^{3}}{{c}^{2} \left ( cx+b \right ) }}-{\frac{Bc{d}^{3}}{2\,{b}^{2} \left ( cx+b \right ) ^{2}}}-3\,{\frac{d\ln \left ( cx+b \right ) A{e}^{2}}{{b}^{3}}}-6\,{\frac{{d}^{3}\ln \left ( cx+b \right ) A{c}^{2}}{{b}^{5}}}-3\,{\frac{{d}^{2}\ln \left ( cx+b \right ) Be}{{b}^{3}}}+3\,{\frac{{d}^{3}\ln \left ( cx+b \right ) Bc}{{b}^{4}}}+3\,{\frac{d\ln \left ( x \right ) A{e}^{2}}{{b}^{3}}}+6\,{\frac{{d}^{3}\ln \left ( x \right ) A{c}^{2}}{{b}^{5}}}+3\,{\frac{{d}^{2}\ln \left ( x \right ) Be}{{b}^{3}}}-3\,{\frac{{d}^{3}\ln \left ( x \right ) Bc}{{b}^{4}}}+3\,{\frac{A{d}^{3}{c}^{2}}{{b}^{4} \left ( cx+b \right ) }}-2\,{\frac{Bc{d}^{3}}{{b}^{3} \left ( cx+b \right ) }}-3\,{\frac{A{d}^{2}e}{{b}^{3}x}}+3\,{\frac{A{d}^{3}c}{{b}^{4}x}}+{\frac{3\,Ad{e}^{2}}{2\,b \left ( cx+b \right ) ^{2}}}+3\,{\frac{B{d}^{2}e}{{b}^{2} \left ( cx+b \right ) }}+3\,{\frac{Ad{e}^{2}}{{b}^{2} \left ( cx+b \right ) }}+{\frac{A{d}^{3}{c}^{2}}{2\,{b}^{3} \left ( cx+b \right ) ^{2}}}+{\frac{B{e}^{3}b}{2\,{c}^{2} \left ( cx+b \right ) ^{2}}}+{\frac{3\,B{d}^{2}e}{2\,b \left ( cx+b \right ) ^{2}}}-{\frac{3\,Bd{e}^{2}}{2\,c \left ( cx+b \right ) ^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)*(e*x+d)^3/(c*x^2+b*x)^3,x)

[Out]

-1/2*A*d^3/b^3/x^2-6/b^3*c/(c*x+b)*A*d^2*e-9*d^2/b^4*ln(x)*A*c*e-3/2*c/b^2/(c*x+b)^2*A*d^2*e+9*d^2/b^4*ln(c*x+
b)*A*c*e-d^3/b^3/x*B-1/2/c/(c*x+b)^2*A*e^3-1/c^2/(c*x+b)*B*e^3-1/2*c/b^2/(c*x+b)^2*B*d^3-3*d/b^3*ln(c*x+b)*A*e
^2-6*d^3/b^5*ln(c*x+b)*A*c^2-3*d^2/b^3*ln(c*x+b)*B*e+3*d^3/b^4*ln(c*x+b)*B*c+3*d/b^3*ln(x)*A*e^2+6*d^3/b^5*ln(
x)*A*c^2+3*d^2/b^3*ln(x)*B*e-3*d^3/b^4*ln(x)*B*c+3/b^4*c^2/(c*x+b)*A*d^3-2/b^3*c/(c*x+b)*B*d^3-3*d^2/b^3/x*A*e
+3*d^3/b^4/x*A*c+3/2/b/(c*x+b)^2*A*d*e^2+3/b^2/(c*x+b)*B*d^2*e+3/b^2/(c*x+b)*A*d*e^2+1/2*c^2/b^3/(c*x+b)^2*A*d
^3+1/2/c^2*b/(c*x+b)^2*B*e^3+3/2/b/(c*x+b)^2*B*d^2*e-3/2/c/(c*x+b)^2*B*d*e^2

________________________________________________________________________________________

Maxima [A]  time = 1.09174, size = 468, normalized size = 2.53 \begin{align*} -\frac{A b^{3} c^{2} d^{3} - 2 \,{\left (3 \, A b^{2} c^{3} d e^{2} - B b^{4} c e^{3} - 3 \,{\left (B b c^{4} - 2 \, A c^{5}\right )} d^{3} + 3 \,{\left (B b^{2} c^{3} - 3 \, A b c^{4}\right )} d^{2} e\right )} x^{3} +{\left (9 \,{\left (B b^{2} c^{3} - 2 \, A b c^{4}\right )} d^{3} - 9 \,{\left (B b^{3} c^{2} - 3 \, A b^{2} c^{3}\right )} d^{2} e + 3 \,{\left (B b^{4} c - 3 \, A b^{3} c^{2}\right )} d e^{2} +{\left (B b^{5} + A b^{4} c\right )} e^{3}\right )} x^{2} + 2 \,{\left (3 \, A b^{3} c^{2} d^{2} e +{\left (B b^{3} c^{2} - 2 \, A b^{2} c^{3}\right )} d^{3}\right )} x}{2 \,{\left (b^{4} c^{4} x^{4} + 2 \, b^{5} c^{3} x^{3} + b^{6} c^{2} x^{2}\right )}} - \frac{3 \,{\left (A b^{2} d e^{2} -{\left (B b c - 2 \, A c^{2}\right )} d^{3} +{\left (B b^{2} - 3 \, A b c\right )} d^{2} e\right )} \log \left (c x + b\right )}{b^{5}} + \frac{3 \,{\left (A b^{2} d e^{2} -{\left (B b c - 2 \, A c^{2}\right )} d^{3} +{\left (B b^{2} - 3 \, A b c\right )} d^{2} e\right )} \log \left (x\right )}{b^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)^3/(c*x^2+b*x)^3,x, algorithm="maxima")

[Out]

-1/2*(A*b^3*c^2*d^3 - 2*(3*A*b^2*c^3*d*e^2 - B*b^4*c*e^3 - 3*(B*b*c^4 - 2*A*c^5)*d^3 + 3*(B*b^2*c^3 - 3*A*b*c^
4)*d^2*e)*x^3 + (9*(B*b^2*c^3 - 2*A*b*c^4)*d^3 - 9*(B*b^3*c^2 - 3*A*b^2*c^3)*d^2*e + 3*(B*b^4*c - 3*A*b^3*c^2)
*d*e^2 + (B*b^5 + A*b^4*c)*e^3)*x^2 + 2*(3*A*b^3*c^2*d^2*e + (B*b^3*c^2 - 2*A*b^2*c^3)*d^3)*x)/(b^4*c^4*x^4 +
2*b^5*c^3*x^3 + b^6*c^2*x^2) - 3*(A*b^2*d*e^2 - (B*b*c - 2*A*c^2)*d^3 + (B*b^2 - 3*A*b*c)*d^2*e)*log(c*x + b)/
b^5 + 3*(A*b^2*d*e^2 - (B*b*c - 2*A*c^2)*d^3 + (B*b^2 - 3*A*b*c)*d^2*e)*log(x)/b^5

________________________________________________________________________________________

Fricas [B]  time = 2.0316, size = 1242, normalized size = 6.71 \begin{align*} -\frac{A b^{4} c^{2} d^{3} - 2 \,{\left (3 \, A b^{3} c^{3} d e^{2} - B b^{5} c e^{3} - 3 \,{\left (B b^{2} c^{4} - 2 \, A b c^{5}\right )} d^{3} + 3 \,{\left (B b^{3} c^{3} - 3 \, A b^{2} c^{4}\right )} d^{2} e\right )} x^{3} +{\left (9 \,{\left (B b^{3} c^{3} - 2 \, A b^{2} c^{4}\right )} d^{3} - 9 \,{\left (B b^{4} c^{2} - 3 \, A b^{3} c^{3}\right )} d^{2} e + 3 \,{\left (B b^{5} c - 3 \, A b^{4} c^{2}\right )} d e^{2} +{\left (B b^{6} + A b^{5} c\right )} e^{3}\right )} x^{2} + 2 \,{\left (3 \, A b^{4} c^{2} d^{2} e +{\left (B b^{4} c^{2} - 2 \, A b^{3} c^{3}\right )} d^{3}\right )} x + 6 \,{\left ({\left (A b^{2} c^{4} d e^{2} -{\left (B b c^{5} - 2 \, A c^{6}\right )} d^{3} +{\left (B b^{2} c^{4} - 3 \, A b c^{5}\right )} d^{2} e\right )} x^{4} + 2 \,{\left (A b^{3} c^{3} d e^{2} -{\left (B b^{2} c^{4} - 2 \, A b c^{5}\right )} d^{3} +{\left (B b^{3} c^{3} - 3 \, A b^{2} c^{4}\right )} d^{2} e\right )} x^{3} +{\left (A b^{4} c^{2} d e^{2} -{\left (B b^{3} c^{3} - 2 \, A b^{2} c^{4}\right )} d^{3} +{\left (B b^{4} c^{2} - 3 \, A b^{3} c^{3}\right )} d^{2} e\right )} x^{2}\right )} \log \left (c x + b\right ) - 6 \,{\left ({\left (A b^{2} c^{4} d e^{2} -{\left (B b c^{5} - 2 \, A c^{6}\right )} d^{3} +{\left (B b^{2} c^{4} - 3 \, A b c^{5}\right )} d^{2} e\right )} x^{4} + 2 \,{\left (A b^{3} c^{3} d e^{2} -{\left (B b^{2} c^{4} - 2 \, A b c^{5}\right )} d^{3} +{\left (B b^{3} c^{3} - 3 \, A b^{2} c^{4}\right )} d^{2} e\right )} x^{3} +{\left (A b^{4} c^{2} d e^{2} -{\left (B b^{3} c^{3} - 2 \, A b^{2} c^{4}\right )} d^{3} +{\left (B b^{4} c^{2} - 3 \, A b^{3} c^{3}\right )} d^{2} e\right )} x^{2}\right )} \log \left (x\right )}{2 \,{\left (b^{5} c^{4} x^{4} + 2 \, b^{6} c^{3} x^{3} + b^{7} c^{2} x^{2}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)^3/(c*x^2+b*x)^3,x, algorithm="fricas")

[Out]

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

________________________________________________________________________________________

Sympy [B]  time = 74.1382, size = 653, normalized size = 3.53 \begin{align*} - \frac{A b^{3} c^{2} d^{3} + x^{3} \left (- 6 A b^{2} c^{3} d e^{2} + 18 A b c^{4} d^{2} e - 12 A c^{5} d^{3} + 2 B b^{4} c e^{3} - 6 B b^{2} c^{3} d^{2} e + 6 B b c^{4} d^{3}\right ) + x^{2} \left (A b^{4} c e^{3} - 9 A b^{3} c^{2} d e^{2} + 27 A b^{2} c^{3} d^{2} e - 18 A b c^{4} d^{3} + B b^{5} e^{3} + 3 B b^{4} c d e^{2} - 9 B b^{3} c^{2} d^{2} e + 9 B b^{2} c^{3} d^{3}\right ) + x \left (6 A b^{3} c^{2} d^{2} e - 4 A b^{2} c^{3} d^{3} + 2 B b^{3} c^{2} d^{3}\right )}{2 b^{6} c^{2} x^{2} + 4 b^{5} c^{3} x^{3} + 2 b^{4} c^{4} x^{4}} + \frac{3 d \left (b e - c d\right ) \left (A b e - 2 A c d + B b d\right ) \log{\left (x + \frac{3 A b^{3} d e^{2} - 9 A b^{2} c d^{2} e + 6 A b c^{2} d^{3} + 3 B b^{3} d^{2} e - 3 B b^{2} c d^{3} - 3 b d \left (b e - c d\right ) \left (A b e - 2 A c d + B b d\right )}{6 A b^{2} c d e^{2} - 18 A b c^{2} d^{2} e + 12 A c^{3} d^{3} + 6 B b^{2} c d^{2} e - 6 B b c^{2} d^{3}} \right )}}{b^{5}} - \frac{3 d \left (b e - c d\right ) \left (A b e - 2 A c d + B b d\right ) \log{\left (x + \frac{3 A b^{3} d e^{2} - 9 A b^{2} c d^{2} e + 6 A b c^{2} d^{3} + 3 B b^{3} d^{2} e - 3 B b^{2} c d^{3} + 3 b d \left (b e - c d\right ) \left (A b e - 2 A c d + B b d\right )}{6 A b^{2} c d e^{2} - 18 A b c^{2} d^{2} e + 12 A c^{3} d^{3} + 6 B b^{2} c d^{2} e - 6 B b c^{2} d^{3}} \right )}}{b^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)**3/(c*x**2+b*x)**3,x)

[Out]

-(A*b**3*c**2*d**3 + x**3*(-6*A*b**2*c**3*d*e**2 + 18*A*b*c**4*d**2*e - 12*A*c**5*d**3 + 2*B*b**4*c*e**3 - 6*B
*b**2*c**3*d**2*e + 6*B*b*c**4*d**3) + x**2*(A*b**4*c*e**3 - 9*A*b**3*c**2*d*e**2 + 27*A*b**2*c**3*d**2*e - 18
*A*b*c**4*d**3 + B*b**5*e**3 + 3*B*b**4*c*d*e**2 - 9*B*b**3*c**2*d**2*e + 9*B*b**2*c**3*d**3) + x*(6*A*b**3*c*
*2*d**2*e - 4*A*b**2*c**3*d**3 + 2*B*b**3*c**2*d**3))/(2*b**6*c**2*x**2 + 4*b**5*c**3*x**3 + 2*b**4*c**4*x**4)
+ 3*d*(b*e - c*d)*(A*b*e - 2*A*c*d + B*b*d)*log(x + (3*A*b**3*d*e**2 - 9*A*b**2*c*d**2*e + 6*A*b*c**2*d**3 +
3*B*b**3*d**2*e - 3*B*b**2*c*d**3 - 3*b*d*(b*e - c*d)*(A*b*e - 2*A*c*d + B*b*d))/(6*A*b**2*c*d*e**2 - 18*A*b*c
**2*d**2*e + 12*A*c**3*d**3 + 6*B*b**2*c*d**2*e - 6*B*b*c**2*d**3))/b**5 - 3*d*(b*e - c*d)*(A*b*e - 2*A*c*d +
B*b*d)*log(x + (3*A*b**3*d*e**2 - 9*A*b**2*c*d**2*e + 6*A*b*c**2*d**3 + 3*B*b**3*d**2*e - 3*B*b**2*c*d**3 + 3*
b*d*(b*e - c*d)*(A*b*e - 2*A*c*d + B*b*d))/(6*A*b**2*c*d*e**2 - 18*A*b*c**2*d**2*e + 12*A*c**3*d**3 + 6*B*b**2
*c*d**2*e - 6*B*b*c**2*d**3))/b**5

________________________________________________________________________________________

Giac [B]  time = 1.2529, size = 528, normalized size = 2.85 \begin{align*} -\frac{3 \,{\left (B b c d^{3} - 2 \, A c^{2} d^{3} - B b^{2} d^{2} e + 3 \, A b c d^{2} e - A b^{2} d e^{2}\right )} \log \left ({\left | x \right |}\right )}{b^{5}} + \frac{3 \,{\left (B b c^{2} d^{3} - 2 \, A c^{3} d^{3} - B b^{2} c d^{2} e + 3 \, A b c^{2} d^{2} e - A b^{2} c d e^{2}\right )} \log \left ({\left | c x + b \right |}\right )}{b^{5} c} - \frac{6 \, B b c^{4} d^{3} x^{3} - 12 \, A c^{5} d^{3} x^{3} - 6 \, B b^{2} c^{3} d^{2} x^{3} e + 18 \, A b c^{4} d^{2} x^{3} e + 9 \, B b^{2} c^{3} d^{3} x^{2} - 18 \, A b c^{4} d^{3} x^{2} - 6 \, A b^{2} c^{3} d x^{3} e^{2} - 9 \, B b^{3} c^{2} d^{2} x^{2} e + 27 \, A b^{2} c^{3} d^{2} x^{2} e + 2 \, B b^{3} c^{2} d^{3} x - 4 \, A b^{2} c^{3} d^{3} x + 2 \, B b^{4} c x^{3} e^{3} + 3 \, B b^{4} c d x^{2} e^{2} - 9 \, A b^{3} c^{2} d x^{2} e^{2} + 6 \, A b^{3} c^{2} d^{2} x e + A b^{3} c^{2} d^{3} + B b^{5} x^{2} e^{3} + A b^{4} c x^{2} e^{3}}{2 \,{\left (c x^{2} + b x\right )}^{2} b^{4} c^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)^3/(c*x^2+b*x)^3,x, algorithm="giac")

[Out]

-3*(B*b*c*d^3 - 2*A*c^2*d^3 - B*b^2*d^2*e + 3*A*b*c*d^2*e - A*b^2*d*e^2)*log(abs(x))/b^5 + 3*(B*b*c^2*d^3 - 2*
A*c^3*d^3 - B*b^2*c*d^2*e + 3*A*b*c^2*d^2*e - A*b^2*c*d*e^2)*log(abs(c*x + b))/(b^5*c) - 1/2*(6*B*b*c^4*d^3*x^
3 - 12*A*c^5*d^3*x^3 - 6*B*b^2*c^3*d^2*x^3*e + 18*A*b*c^4*d^2*x^3*e + 9*B*b^2*c^3*d^3*x^2 - 18*A*b*c^4*d^3*x^2
- 6*A*b^2*c^3*d*x^3*e^2 - 9*B*b^3*c^2*d^2*x^2*e + 27*A*b^2*c^3*d^2*x^2*e + 2*B*b^3*c^2*d^3*x - 4*A*b^2*c^3*d^
3*x + 2*B*b^4*c*x^3*e^3 + 3*B*b^4*c*d*x^2*e^2 - 9*A*b^3*c^2*d*x^2*e^2 + 6*A*b^3*c^2*d^2*x*e + A*b^3*c^2*d^3 +
B*b^5*x^2*e^3 + A*b^4*c*x^2*e^3)/((c*x^2 + b*x)^2*b^4*c^2)