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

Optimal. Leaf size=156 \[ \frac {d^3 \log (x) (4 A b e-2 A c d+b B d)}{b^3}+\frac {(b B-A c) (c d-b e)^4}{b^2 c^4 (b+c x)}-\frac {A d^4}{b^2 x}+\frac {(c d-b e)^3 \log (b+c x) \left (-b c (B d-2 A e)+2 A c^2 d-3 b^2 B e\right )}{b^3 c^4}+\frac {e^3 x (A c e-2 b B e+4 B c d)}{c^3}+\frac {B e^4 x^2}{2 c^2} \]

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Rubi [A]  time = 0.23, antiderivative size = 156, 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 {(b B-A c) (c d-b e)^4}{b^2 c^4 (b+c x)}+\frac {(c d-b e)^3 \log (b+c x) \left (-b c (B d-2 A e)+2 A c^2 d-3 b^2 B e\right )}{b^3 c^4}+\frac {d^3 \log (x) (4 A b e-2 A c d+b B d)}{b^3}-\frac {A d^4}{b^2 x}+\frac {e^3 x (A c e-2 b B e+4 B c d)}{c^3}+\frac {B e^4 x^2}{2 c^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

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Mathematica [A]  time = 0.09, size = 155, normalized size = 0.99 \begin {gather*} \frac {d^3 \log (x) (4 A b e-2 A c d+b B d)}{b^3}+\frac {(b B-A c) (c d-b e)^4}{b^2 c^4 (b+c x)}-\frac {A d^4}{b^2 x}+\frac {(b e-c d)^3 \log (b+c x) \left (b c (B d-2 A e)-2 A c^2 d+3 b^2 B e\right )}{b^3 c^4}+\frac {e^3 x (A c e-2 b B e+4 B c d)}{c^3}+\frac {B e^4 x^2}{2 c^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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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)^4}{\left (b x+c x^2\right )^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [B]  time = 0.45, size = 517, normalized size = 3.31 \begin {gather*} \frac {B b^{3} c^{3} e^{4} x^{4} - 2 \, A b^{2} c^{4} d^{4} + {\left (8 \, B b^{3} c^{3} d e^{3} - {\left (3 \, B b^{4} c^{2} - 2 \, A b^{3} c^{3}\right )} e^{4}\right )} x^{3} + 2 \, {\left (4 \, B b^{4} c^{2} d e^{3} - {\left (2 \, B b^{5} c - A b^{4} c^{2}\right )} e^{4}\right )} x^{2} + 2 \, {\left ({\left (B b^{2} c^{4} - 2 \, A b c^{5}\right )} d^{4} - 4 \, {\left (B b^{3} c^{3} - A b^{2} c^{4}\right )} d^{3} e + 6 \, {\left (B b^{4} c^{2} - A b^{3} c^{3}\right )} d^{2} e^{2} - 4 \, {\left (B b^{5} c - A b^{4} c^{2}\right )} d e^{3} + {\left (B b^{6} - A b^{5} c\right )} e^{4}\right )} x - 2 \, {\left ({\left (4 \, A b c^{5} d^{3} e - 6 \, B b^{3} c^{3} d^{2} e^{2} + {\left (B b c^{5} - 2 \, A c^{6}\right )} d^{4} + 4 \, {\left (2 \, B b^{4} c^{2} - A b^{3} c^{3}\right )} d e^{3} - {\left (3 \, B b^{5} c - 2 \, A b^{4} c^{2}\right )} e^{4}\right )} x^{2} + {\left (4 \, A b^{2} c^{4} d^{3} e - 6 \, B b^{4} c^{2} d^{2} e^{2} + {\left (B b^{2} c^{4} - 2 \, A b c^{5}\right )} d^{4} + 4 \, {\left (2 \, B b^{5} c - A b^{4} c^{2}\right )} d e^{3} - {\left (3 \, B b^{6} - 2 \, A b^{5} c\right )} e^{4}\right )} x\right )} \log \left (c x + b\right ) + 2 \, {\left ({\left (4 \, A b c^{5} d^{3} e + {\left (B b c^{5} - 2 \, A c^{6}\right )} d^{4}\right )} x^{2} + {\left (4 \, A b^{2} c^{4} d^{3} e + {\left (B b^{2} c^{4} - 2 \, A b c^{5}\right )} d^{4}\right )} x\right )} \log \relax (x)}{2 \, {\left (b^{3} c^{5} x^{2} + b^{4} c^{4} x\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(B*b^3*c^3*e^4*x^4 - 2*A*b^2*c^4*d^4 + (8*B*b^3*c^3*d*e^3 - (3*B*b^4*c^2 - 2*A*b^3*c^3)*e^4)*x^3 + 2*(4*B*
b^4*c^2*d*e^3 - (2*B*b^5*c - A*b^4*c^2)*e^4)*x^2 + 2*((B*b^2*c^4 - 2*A*b*c^5)*d^4 - 4*(B*b^3*c^3 - A*b^2*c^4)*
d^3*e + 6*(B*b^4*c^2 - A*b^3*c^3)*d^2*e^2 - 4*(B*b^5*c - A*b^4*c^2)*d*e^3 + (B*b^6 - A*b^5*c)*e^4)*x - 2*((4*A
*b*c^5*d^3*e - 6*B*b^3*c^3*d^2*e^2 + (B*b*c^5 - 2*A*c^6)*d^4 + 4*(2*B*b^4*c^2 - A*b^3*c^3)*d*e^3 - (3*B*b^5*c
- 2*A*b^4*c^2)*e^4)*x^2 + (4*A*b^2*c^4*d^3*e - 6*B*b^4*c^2*d^2*e^2 + (B*b^2*c^4 - 2*A*b*c^5)*d^4 + 4*(2*B*b^5*
c - A*b^4*c^2)*d*e^3 - (3*B*b^6 - 2*A*b^5*c)*e^4)*x)*log(c*x + b) + 2*((4*A*b*c^5*d^3*e + (B*b*c^5 - 2*A*c^6)*
d^4)*x^2 + (4*A*b^2*c^4*d^3*e + (B*b^2*c^4 - 2*A*b*c^5)*d^4)*x)*log(x))/(b^3*c^5*x^2 + b^4*c^4*x)

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giac [B]  time = 0.16, size = 315, normalized size = 2.02 \begin {gather*} \frac {{\left (B b d^{4} - 2 \, A c d^{4} + 4 \, A b d^{3} e\right )} \log \left ({\left | x \right |}\right )}{b^{3}} + \frac {B c^{2} x^{2} e^{4} + 8 \, B c^{2} d x e^{3} - 4 \, B b c x e^{4} + 2 \, A c^{2} x e^{4}}{2 \, c^{4}} - \frac {{\left (B b c^{4} d^{4} - 2 \, A c^{5} d^{4} + 4 \, A b c^{4} d^{3} e - 6 \, B b^{3} c^{2} d^{2} e^{2} + 8 \, B b^{4} c d e^{3} - 4 \, A b^{3} c^{2} d e^{3} - 3 \, B b^{5} e^{4} + 2 \, A b^{4} c e^{4}\right )} \log \left ({\left | c x + b \right |}\right )}{b^{3} c^{4}} - \frac {A b c^{4} d^{4} - {\left (B b c^{4} d^{4} - 2 \, A c^{5} d^{4} - 4 \, B b^{2} c^{3} d^{3} e + 4 \, A b c^{4} d^{3} e + 6 \, B b^{3} c^{2} d^{2} e^{2} - 6 \, A b^{2} c^{3} d^{2} e^{2} - 4 \, B b^{4} c d e^{3} + 4 \, A b^{3} c^{2} d e^{3} + B b^{5} e^{4} - A b^{4} c e^{4}\right )} x}{{\left (c x + b\right )} b^{2} c^{4} x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(B*b*d^4 - 2*A*c*d^4 + 4*A*b*d^3*e)*log(abs(x))/b^3 + 1/2*(B*c^2*x^2*e^4 + 8*B*c^2*d*x*e^3 - 4*B*b*c*x*e^4 + 2
*A*c^2*x*e^4)/c^4 - (B*b*c^4*d^4 - 2*A*c^5*d^4 + 4*A*b*c^4*d^3*e - 6*B*b^3*c^2*d^2*e^2 + 8*B*b^4*c*d*e^3 - 4*A
*b^3*c^2*d*e^3 - 3*B*b^5*e^4 + 2*A*b^4*c*e^4)*log(abs(c*x + b))/(b^3*c^4) - (A*b*c^4*d^4 - (B*b*c^4*d^4 - 2*A*
c^5*d^4 - 4*B*b^2*c^3*d^3*e + 4*A*b*c^4*d^3*e + 6*B*b^3*c^2*d^2*e^2 - 6*A*b^2*c^3*d^2*e^2 - 4*B*b^4*c*d*e^3 +
4*A*b^3*c^2*d*e^3 + B*b^5*e^4 - A*b^4*c*e^4)*x)/((c*x + b)*b^2*c^4*x)

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maple [B]  time = 0.06, size = 403, normalized size = 2.58 \begin {gather*} \frac {B \,e^{4} x^{2}}{2 c^{2}}-\frac {A \,b^{2} e^{4}}{\left (c x +b \right ) c^{3}}+\frac {4 A b d \,e^{3}}{\left (c x +b \right ) c^{2}}-\frac {2 A b \,e^{4} \ln \left (c x +b \right )}{c^{3}}+\frac {4 A \,d^{3} e}{\left (c x +b \right ) b}-\frac {A c \,d^{4}}{\left (c x +b \right ) b^{2}}+\frac {4 A \,d^{3} e \ln \relax (x )}{b^{2}}-\frac {4 A \,d^{3} e \ln \left (c x +b \right )}{b^{2}}-\frac {2 A c \,d^{4} \ln \relax (x )}{b^{3}}+\frac {2 A c \,d^{4} \ln \left (c x +b \right )}{b^{3}}-\frac {6 A \,d^{2} e^{2}}{\left (c x +b \right ) c}+\frac {4 A d \,e^{3} \ln \left (c x +b \right )}{c^{2}}+\frac {A \,e^{4} x}{c^{2}}+\frac {B \,b^{3} e^{4}}{\left (c x +b \right ) c^{4}}-\frac {4 B \,b^{2} d \,e^{3}}{\left (c x +b \right ) c^{3}}+\frac {3 B \,b^{2} e^{4} \ln \left (c x +b \right )}{c^{4}}+\frac {6 B b \,d^{2} e^{2}}{\left (c x +b \right ) c^{2}}-\frac {8 B b d \,e^{3} \ln \left (c x +b \right )}{c^{3}}-\frac {2 B b \,e^{4} x}{c^{3}}+\frac {B \,d^{4}}{\left (c x +b \right ) b}+\frac {B \,d^{4} \ln \relax (x )}{b^{2}}-\frac {B \,d^{4} \ln \left (c x +b \right )}{b^{2}}-\frac {4 B \,d^{3} e}{\left (c x +b \right ) c}+\frac {6 B \,d^{2} e^{2} \ln \left (c x +b \right )}{c^{2}}+\frac {4 B d \,e^{3} x}{c^{2}}-\frac {A \,d^{4}}{b^{2} x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

d^4/b^2*ln(x)*B+e^4/c^2*A*x-1/b^2*ln(c*x+b)*B*d^4+1/b/(c*x+b)*B*d^4-8/c^3*b*ln(c*x+b)*B*d*e^3+4/c^2*b/(c*x+b)*
A*d*e^3-4/c^3*b^2/(c*x+b)*B*d*e^3+6/c^2*b/(c*x+b)*B*d^2*e^2+4/b/(c*x+b)*A*d^3*e-4/c/(c*x+b)*B*d^3*e+6/c^2*ln(c
*x+b)*B*d^2*e^2-1/c^3*b^2/(c*x+b)*A*e^4-c/b^2/(c*x+b)*A*d^4+1/c^4*b^3/(c*x+b)*B*e^4+4*d^3/b^2*ln(x)*A*e-2*d^4/
b^3*ln(x)*A*c-2*e^4/c^3*B*b*x+4*e^3/c^2*B*d*x-4/b^2*ln(c*x+b)*A*d^3*e+2*c/b^3*ln(c*x+b)*A*d^4+3/c^4*b^2*ln(c*x
+b)*B*e^4+4/c^2*ln(c*x+b)*A*d*e^3-6/c/(c*x+b)*A*d^2*e^2-2/c^3*b*ln(c*x+b)*A*e^4-A*d^4/b^2/x+1/2*B*e^4*x^2/c^2

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maxima [B]  time = 0.50, size = 310, normalized size = 1.99 \begin {gather*} -\frac {A b c^{4} d^{4} - {\left ({\left (B b c^{4} - 2 \, A c^{5}\right )} d^{4} - 4 \, {\left (B b^{2} c^{3} - A b c^{4}\right )} d^{3} e + 6 \, {\left (B b^{3} c^{2} - A b^{2} c^{3}\right )} d^{2} e^{2} - 4 \, {\left (B b^{4} c - A b^{3} c^{2}\right )} d e^{3} + {\left (B b^{5} - A b^{4} c\right )} e^{4}\right )} x}{b^{2} c^{5} x^{2} + b^{3} c^{4} x} + \frac {{\left (4 \, A b d^{3} e + {\left (B b - 2 \, A c\right )} d^{4}\right )} \log \relax (x)}{b^{3}} + \frac {B c e^{4} x^{2} + 2 \, {\left (4 \, B c d e^{3} - {\left (2 \, B b - A c\right )} e^{4}\right )} x}{2 \, c^{3}} - \frac {{\left (4 \, A b c^{4} d^{3} e - 6 \, B b^{3} c^{2} d^{2} e^{2} + {\left (B b c^{4} - 2 \, A c^{5}\right )} d^{4} + 4 \, {\left (2 \, B b^{4} c - A b^{3} c^{2}\right )} d e^{3} - {\left (3 \, B b^{5} - 2 \, A b^{4} c\right )} e^{4}\right )} \log \left (c x + b\right )}{b^{3} c^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 1.74, size = 331, normalized size = 2.12 \begin {gather*} \ln \left (b+c\,x\right )\,\left (\frac {c^2\,\left (6\,B\,b^3\,d^2\,e^2+4\,A\,b^3\,d\,e^3\right )-c\,\left (2\,A\,b^4\,e^4+8\,B\,d\,b^4\,e^3\right )+3\,B\,b^5\,e^4}{b^3\,c^4}-\frac {B\,b\,d^4+4\,A\,b\,e\,d^3}{b^3}+\frac {2\,A\,c\,d^4}{b^3}\right )-\frac {\frac {A\,c^3\,d^4}{b}+\frac {x\,\left (-B\,b^5\,e^4+4\,B\,b^4\,c\,d\,e^3+A\,b^4\,c\,e^4-6\,B\,b^3\,c^2\,d^2\,e^2-4\,A\,b^3\,c^2\,d\,e^3+4\,B\,b^2\,c^3\,d^3\,e+6\,A\,b^2\,c^3\,d^2\,e^2-B\,b\,c^4\,d^4-4\,A\,b\,c^4\,d^3\,e+2\,A\,c^5\,d^4\right )}{b^2\,c}}{c^4\,x^2+b\,c^3\,x}+x\,\left (\frac {A\,e^4+4\,B\,d\,e^3}{c^2}-\frac {2\,B\,b\,e^4}{c^3}\right )+\frac {\ln \relax (x)\,\left (b\,\left (B\,d^4+4\,A\,e\,d^3\right )-2\,A\,c\,d^4\right )}{b^3}+\frac {B\,e^4\,x^2}{2\,c^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(b + c*x)*((c^2*(4*A*b^3*d*e^3 + 6*B*b^3*d^2*e^2) - c*(2*A*b^4*e^4 + 8*B*b^4*d*e^3) + 3*B*b^5*e^4)/(b^3*c^4
) - (B*b*d^4 + 4*A*b*d^3*e)/b^3 + (2*A*c*d^4)/b^3) - ((A*c^3*d^4)/b + (x*(2*A*c^5*d^4 - B*b^5*e^4 + A*b^4*c*e^
4 - B*b*c^4*d^4 - 4*A*b^3*c^2*d*e^3 + 4*B*b^2*c^3*d^3*e + 6*A*b^2*c^3*d^2*e^2 - 6*B*b^3*c^2*d^2*e^2 - 4*A*b*c^
4*d^3*e + 4*B*b^4*c*d*e^3))/(b^2*c))/(c^4*x^2 + b*c^3*x) + x*((A*e^4 + 4*B*d*e^3)/c^2 - (2*B*b*e^4)/c^3) + (lo
g(x)*(b*(B*d^4 + 4*A*d^3*e) - 2*A*c*d^4))/b^3 + (B*e^4*x^2)/(2*c^2)

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sympy [B]  time = 15.98, size = 644, normalized size = 4.13 \begin {gather*} \frac {B e^{4} x^{2}}{2 c^{2}} + x \left (\frac {A e^{4}}{c^{2}} - \frac {2 B b e^{4}}{c^{3}} + \frac {4 B d e^{3}}{c^{2}}\right ) + \frac {- A b c^{4} d^{4} + x \left (- A b^{4} c e^{4} + 4 A b^{3} c^{2} d e^{3} - 6 A b^{2} c^{3} d^{2} e^{2} + 4 A b c^{4} d^{3} e - 2 A c^{5} d^{4} + B b^{5} e^{4} - 4 B b^{4} c d e^{3} + 6 B b^{3} c^{2} d^{2} e^{2} - 4 B b^{2} c^{3} d^{3} e + B b c^{4} d^{4}\right )}{b^{3} c^{4} x + b^{2} c^{5} x^{2}} + \frac {d^{3} \left (4 A b e - 2 A c d + B b d\right ) \log {\left (x + \frac {- 4 A b^{2} c^{3} d^{3} e + 2 A b c^{4} d^{4} - B b^{2} c^{3} d^{4} + b c^{3} d^{3} \left (4 A b e - 2 A c d + B b d\right )}{- 2 A b^{4} c e^{4} + 4 A b^{3} c^{2} d e^{3} - 8 A b c^{4} d^{3} e + 4 A c^{5} d^{4} + 3 B b^{5} e^{4} - 8 B b^{4} c d e^{3} + 6 B b^{3} c^{2} d^{2} e^{2} - 2 B b c^{4} d^{4}} \right )}}{b^{3}} + \frac {\left (b e - c d\right )^{3} \left (- 2 A b c e - 2 A c^{2} d + 3 B b^{2} e + B b c d\right ) \log {\left (x + \frac {- 4 A b^{2} c^{3} d^{3} e + 2 A b c^{4} d^{4} - B b^{2} c^{3} d^{4} + \frac {b \left (b e - c d\right )^{3} \left (- 2 A b c e - 2 A c^{2} d + 3 B b^{2} e + B b c d\right )}{c}}{- 2 A b^{4} c e^{4} + 4 A b^{3} c^{2} d e^{3} - 8 A b c^{4} d^{3} e + 4 A c^{5} d^{4} + 3 B b^{5} e^{4} - 8 B b^{4} c d e^{3} + 6 B b^{3} c^{2} d^{2} e^{2} - 2 B b c^{4} d^{4}} \right )}}{b^{3} c^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

B*e**4*x**2/(2*c**2) + x*(A*e**4/c**2 - 2*B*b*e**4/c**3 + 4*B*d*e**3/c**2) + (-A*b*c**4*d**4 + x*(-A*b**4*c*e*
*4 + 4*A*b**3*c**2*d*e**3 - 6*A*b**2*c**3*d**2*e**2 + 4*A*b*c**4*d**3*e - 2*A*c**5*d**4 + B*b**5*e**4 - 4*B*b*
*4*c*d*e**3 + 6*B*b**3*c**2*d**2*e**2 - 4*B*b**2*c**3*d**3*e + B*b*c**4*d**4))/(b**3*c**4*x + b**2*c**5*x**2)
+ d**3*(4*A*b*e - 2*A*c*d + B*b*d)*log(x + (-4*A*b**2*c**3*d**3*e + 2*A*b*c**4*d**4 - B*b**2*c**3*d**4 + b*c**
3*d**3*(4*A*b*e - 2*A*c*d + B*b*d))/(-2*A*b**4*c*e**4 + 4*A*b**3*c**2*d*e**3 - 8*A*b*c**4*d**3*e + 4*A*c**5*d*
*4 + 3*B*b**5*e**4 - 8*B*b**4*c*d*e**3 + 6*B*b**3*c**2*d**2*e**2 - 2*B*b*c**4*d**4))/b**3 + (b*e - c*d)**3*(-2
*A*b*c*e - 2*A*c**2*d + 3*B*b**2*e + B*b*c*d)*log(x + (-4*A*b**2*c**3*d**3*e + 2*A*b*c**4*d**4 - B*b**2*c**3*d
**4 + b*(b*e - c*d)**3*(-2*A*b*c*e - 2*A*c**2*d + 3*B*b**2*e + B*b*c*d)/c)/(-2*A*b**4*c*e**4 + 4*A*b**3*c**2*d
*e**3 - 8*A*b*c**4*d**3*e + 4*A*c**5*d**4 + 3*B*b**5*e**4 - 8*B*b**4*c*d*e**3 + 6*B*b**3*c**2*d**2*e**2 - 2*B*
b*c**4*d**4))/(b**3*c**4)

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