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

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

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Rubi [A]  time = 0.25, antiderivative size = 185, normalized size of antiderivative = 1.00, 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} \begin {gather*} -\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} \end {gather*}

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*}

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Mathematica [A]  time = 0.12, size = 177, normalized size = 0.96 \begin {gather*} -\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} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/2*((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])/b^5

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(A+B x) (d+e x)^3}{\left (b x+c x^2\right )^3} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [B]  time = 0.46, size = 627, normalized size = 3.39 \begin {gather*} -\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 \relax (x)}{2 \, {\left (b^{5} c^{4} x^{4} + 2 \, b^{6} c^{3} x^{3} + b^{7} c^{2} x^{2}\right )}} \end {gather*}

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)

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giac [B]  time = 0.18, size = 391, normalized size = 2.11 \begin {gather*} -\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 {gather*}

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)

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maple [B]  time = 0.06, size = 440, normalized size = 2.38 \begin {gather*} \frac {3 A d \,e^{2}}{2 \left (c x +b \right )^{2} b}-\frac {3 A c \,d^{2} e}{2 \left (c x +b \right )^{2} b^{2}}+\frac {A \,c^{2} d^{3}}{2 \left (c x +b \right )^{2} b^{3}}-\frac {A \,e^{3}}{2 \left (c x +b \right )^{2} c}+\frac {B b \,e^{3}}{2 \left (c x +b \right )^{2} c^{2}}+\frac {3 B \,d^{2} e}{2 \left (c x +b \right )^{2} b}-\frac {B c \,d^{3}}{2 \left (c x +b \right )^{2} b^{2}}-\frac {3 B d \,e^{2}}{2 \left (c x +b \right )^{2} c}+\frac {3 A d \,e^{2}}{\left (c x +b \right ) b^{2}}-\frac {6 A c \,d^{2} e}{\left (c x +b \right ) b^{3}}+\frac {3 A d \,e^{2} \ln \relax (x )}{b^{3}}-\frac {3 A d \,e^{2} \ln \left (c x +b \right )}{b^{3}}+\frac {3 A \,c^{2} d^{3}}{\left (c x +b \right ) b^{4}}-\frac {9 A c \,d^{2} e \ln \relax (x )}{b^{4}}+\frac {9 A c \,d^{2} e \ln \left (c x +b \right )}{b^{4}}+\frac {6 A \,c^{2} d^{3} \ln \relax (x )}{b^{5}}-\frac {6 A \,c^{2} d^{3} \ln \left (c x +b \right )}{b^{5}}+\frac {3 B \,d^{2} e}{\left (c x +b \right ) b^{2}}-\frac {2 B c \,d^{3}}{\left (c x +b \right ) b^{3}}+\frac {3 B \,d^{2} e \ln \relax (x )}{b^{3}}-\frac {3 B \,d^{2} e \ln \left (c x +b \right )}{b^{3}}-\frac {3 B c \,d^{3} \ln \relax (x )}{b^{4}}+\frac {3 B c \,d^{3} \ln \left (c x +b \right )}{b^{4}}-\frac {B \,e^{3}}{\left (c x +b \right ) c^{2}}-\frac {3 A \,d^{2} e}{b^{3} x}+\frac {3 A c \,d^{3}}{b^{4} x}-\frac {B \,d^{3}}{b^{3} x}-\frac {A \,d^{3}}{2 b^{3} x^{2}} \end {gather*}

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]

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

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maxima [A]  time = 0.55, size = 347, normalized size = 1.88 \begin {gather*} -\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 \relax (x)}{b^{5}} \end {gather*}

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

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mupad [B]  time = 1.62, size = 345, normalized size = 1.86 \begin {gather*} -\frac {\frac {A\,d^3}{2\,b}-\frac {x^3\,\left (-B\,b^4\,e^3+3\,B\,b^2\,c^2\,d^2\,e+3\,A\,b^2\,c^2\,d\,e^2-3\,B\,b\,c^3\,d^3-9\,A\,b\,c^3\,d^2\,e+6\,A\,c^4\,d^3\right )}{b^4\,c}+\frac {x^2\,\left (B\,b^4\,e^3+3\,B\,b^3\,c\,d\,e^2+A\,b^3\,c\,e^3-9\,B\,b^2\,c^2\,d^2\,e-9\,A\,b^2\,c^2\,d\,e^2+9\,B\,b\,c^3\,d^3+27\,A\,b\,c^3\,d^2\,e-18\,A\,c^4\,d^3\right )}{2\,b^3\,c^2}+\frac {d^2\,x\,\left (3\,A\,b\,e-2\,A\,c\,d+B\,b\,d\right )}{b^2}}{b^2\,x^2+2\,b\,c\,x^3+c^2\,x^4}-\frac {6\,d\,\mathrm {atanh}\left (\frac {3\,d\,\left (b\,e-c\,d\right )\,\left (b+2\,c\,x\right )\,\left (A\,b\,e-2\,A\,c\,d+B\,b\,d\right )}{b\,\left (3\,B\,b^2\,d^2\,e+3\,A\,b^2\,d\,e^2-3\,B\,b\,c\,d^3-9\,A\,b\,c\,d^2\,e+6\,A\,c^2\,d^3\right )}\right )\,\left (b\,e-c\,d\right )\,\left (A\,b\,e-2\,A\,c\,d+B\,b\,d\right )}{b^5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [B]  time = 28.93, size = 653, normalized size = 3.53 \begin {gather*} \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 {gather*}

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

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