3.11.10 \(\int \frac {(A+B x) (d+e x)^4}{(b x+c x^2)^3} \, dx\)

Optimal. Leaf size=235 \[ -\frac {d^3 (4 A b e-3 A c d+b B d)}{b^4 x}-\frac {(b B-A c) (c d-b e)^4}{2 b^3 c^3 (b+c x)^2}-\frac {A d^4}{2 b^3 x^2}+\frac {d^2 \log (x) \left (2 b^2 e (3 A e+2 B d)-3 b c d (4 A e+B d)+6 A c^2 d^2\right )}{b^5}-\frac {(c d-b e)^3 \left (-A b c e-3 A c^2 d+2 b^2 B e+2 b B c d\right )}{b^4 c^3 (b+c x)}+\frac {(c d-b e)^2 \log (b+c x) \left (-6 A c^3 d^2+b^3 B e^2+2 b^2 B c d e+3 b B c^2 d^2\right )}{b^5 c^3} \]

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Rubi [A]  time = 0.33, antiderivative size = 235, 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 \log (b+c x) \left (-6 A c^3 d^2+2 b^2 B c d e+b^3 B e^2+3 b B c^2 d^2\right )}{b^5 c^3}+\frac {d^2 \log (x) \left (2 b^2 e (3 A e+2 B d)-3 b c d (4 A e+B d)+6 A c^2 d^2\right )}{b^5}-\frac {(c d-b e)^3 \left (-A b c e-3 A c^2 d+2 b^2 B e+2 b B c d\right )}{b^4 c^3 (b+c x)}-\frac {(b B-A c) (c d-b e)^4}{2 b^3 c^3 (b+c x)^2}-\frac {d^3 (4 A b e-3 A c d+b B d)}{b^4 x}-\frac {A d^4}{2 b^3 x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

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Mathematica [A]  time = 0.16, size = 228, normalized size = 0.97 \begin {gather*} -\frac {\frac {b^2 (b B-A c) (c d-b e)^4}{c^3 (b+c x)^2}-2 d^2 \log (x) \left (2 b^2 e (3 A e+2 B d)-3 b c d (4 A e+B d)+6 A c^2 d^2\right )-\frac {2 b (b e-c d)^3 \left (b c (2 B d-A e)-3 A c^2 d+2 b^2 B e\right )}{c^3 (b+c x)}+\frac {A b^2 d^4}{x^2}-\frac {2 (c d-b e)^2 \log (b+c x) \left (-6 A c^3 d^2+b^3 B e^2+2 b^2 B c d e+3 b B c^2 d^2\right )}{c^3}+\frac {2 b d^3 (4 A b e-3 A c d+b B d)}{x}}{2 b^5} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/2*((A*b^2*d^4)/x^2 + (2*b*d^3*(b*B*d - 3*A*c*d + 4*A*b*e))/x + (b^2*(b*B - A*c)*(c*d - b*e)^4)/(c^3*(b + c*
x)^2) - (2*b*(-(c*d) + b*e)^3*(-3*A*c^2*d + 2*b^2*B*e + b*c*(2*B*d - A*e)))/(c^3*(b + c*x)) - 2*d^2*(6*A*c^2*d
^2 + 2*b^2*e*(2*B*d + 3*A*e) - 3*b*c*d*(B*d + 4*A*e))*Log[x] - (2*(c*d - b*e)^2*(3*b*B*c^2*d^2 - 6*A*c^3*d^2 +
2*b^2*B*c*d*e + b^3*B*e^2)*Log[b + c*x])/c^3)/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)^4}{\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)^4)/(b*x + c*x^2)^3,x]

[Out]

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

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fricas [B]  time = 0.46, size = 736, normalized size = 3.13 \begin {gather*} -\frac {A b^{4} c^{3} d^{4} - 2 \, {\left (6 \, A b^{3} c^{4} d^{2} e^{2} - 4 \, B b^{5} c^{2} d e^{3} - 3 \, {\left (B b^{2} c^{5} - 2 \, A b c^{6}\right )} d^{4} + 4 \, {\left (B b^{3} c^{4} - 3 \, A b^{2} c^{5}\right )} d^{3} e + {\left (2 \, B b^{6} c - A b^{5} c^{2}\right )} e^{4}\right )} x^{3} + {\left (9 \, {\left (B b^{3} c^{4} - 2 \, A b^{2} c^{5}\right )} d^{4} - 12 \, {\left (B b^{4} c^{3} - 3 \, A b^{3} c^{4}\right )} d^{3} e + 6 \, {\left (B b^{5} c^{2} - 3 \, A b^{4} c^{3}\right )} d^{2} e^{2} + 4 \, {\left (B b^{6} c + A b^{5} c^{2}\right )} d e^{3} - {\left (3 \, B b^{7} - A b^{6} c\right )} e^{4}\right )} x^{2} + 2 \, {\left (4 \, A b^{4} c^{3} d^{3} e + {\left (B b^{4} c^{3} - 2 \, A b^{3} c^{4}\right )} d^{4}\right )} x + 2 \, {\left ({\left (6 \, A b^{2} c^{5} d^{2} e^{2} - B b^{5} c^{2} e^{4} - 3 \, {\left (B b c^{6} - 2 \, A c^{7}\right )} d^{4} + 4 \, {\left (B b^{2} c^{5} - 3 \, A b c^{6}\right )} d^{3} e\right )} x^{4} + 2 \, {\left (6 \, A b^{3} c^{4} d^{2} e^{2} - B b^{6} c e^{4} - 3 \, {\left (B b^{2} c^{5} - 2 \, A b c^{6}\right )} d^{4} + 4 \, {\left (B b^{3} c^{4} - 3 \, A b^{2} c^{5}\right )} d^{3} e\right )} x^{3} + {\left (6 \, A b^{4} c^{3} d^{2} e^{2} - B b^{7} e^{4} - 3 \, {\left (B b^{3} c^{4} - 2 \, A b^{2} c^{5}\right )} d^{4} + 4 \, {\left (B b^{4} c^{3} - 3 \, A b^{3} c^{4}\right )} d^{3} e\right )} x^{2}\right )} \log \left (c x + b\right ) - 2 \, {\left ({\left (6 \, A b^{2} c^{5} d^{2} e^{2} - 3 \, {\left (B b c^{6} - 2 \, A c^{7}\right )} d^{4} + 4 \, {\left (B b^{2} c^{5} - 3 \, A b c^{6}\right )} d^{3} e\right )} x^{4} + 2 \, {\left (6 \, A b^{3} c^{4} d^{2} e^{2} - 3 \, {\left (B b^{2} c^{5} - 2 \, A b c^{6}\right )} d^{4} + 4 \, {\left (B b^{3} c^{4} - 3 \, A b^{2} c^{5}\right )} d^{3} e\right )} x^{3} + {\left (6 \, A b^{4} c^{3} d^{2} e^{2} - 3 \, {\left (B b^{3} c^{4} - 2 \, A b^{2} c^{5}\right )} d^{4} + 4 \, {\left (B b^{4} c^{3} - 3 \, A b^{3} c^{4}\right )} d^{3} e\right )} x^{2}\right )} \log \relax (x)}{2 \, {\left (b^{5} c^{5} x^{4} + 2 \, b^{6} c^{4} x^{3} + b^{7} c^{3} x^{2}\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)^3,x, algorithm="fricas")

[Out]

-1/2*(A*b^4*c^3*d^4 - 2*(6*A*b^3*c^4*d^2*e^2 - 4*B*b^5*c^2*d*e^3 - 3*(B*b^2*c^5 - 2*A*b*c^6)*d^4 + 4*(B*b^3*c^
4 - 3*A*b^2*c^5)*d^3*e + (2*B*b^6*c - A*b^5*c^2)*e^4)*x^3 + (9*(B*b^3*c^4 - 2*A*b^2*c^5)*d^4 - 12*(B*b^4*c^3 -
3*A*b^3*c^4)*d^3*e + 6*(B*b^5*c^2 - 3*A*b^4*c^3)*d^2*e^2 + 4*(B*b^6*c + A*b^5*c^2)*d*e^3 - (3*B*b^7 - A*b^6*c
)*e^4)*x^2 + 2*(4*A*b^4*c^3*d^3*e + (B*b^4*c^3 - 2*A*b^3*c^4)*d^4)*x + 2*((6*A*b^2*c^5*d^2*e^2 - B*b^5*c^2*e^4
- 3*(B*b*c^6 - 2*A*c^7)*d^4 + 4*(B*b^2*c^5 - 3*A*b*c^6)*d^3*e)*x^4 + 2*(6*A*b^3*c^4*d^2*e^2 - B*b^6*c*e^4 - 3
*(B*b^2*c^5 - 2*A*b*c^6)*d^4 + 4*(B*b^3*c^4 - 3*A*b^2*c^5)*d^3*e)*x^3 + (6*A*b^4*c^3*d^2*e^2 - B*b^7*e^4 - 3*(
B*b^3*c^4 - 2*A*b^2*c^5)*d^4 + 4*(B*b^4*c^3 - 3*A*b^3*c^4)*d^3*e)*x^2)*log(c*x + b) - 2*((6*A*b^2*c^5*d^2*e^2
- 3*(B*b*c^6 - 2*A*c^7)*d^4 + 4*(B*b^2*c^5 - 3*A*b*c^6)*d^3*e)*x^4 + 2*(6*A*b^3*c^4*d^2*e^2 - 3*(B*b^2*c^5 - 2
*A*b*c^6)*d^4 + 4*(B*b^3*c^4 - 3*A*b^2*c^5)*d^3*e)*x^3 + (6*A*b^4*c^3*d^2*e^2 - 3*(B*b^3*c^4 - 2*A*b^2*c^5)*d^
4 + 4*(B*b^4*c^3 - 3*A*b^3*c^4)*d^3*e)*x^2)*log(x))/(b^5*c^5*x^4 + 2*b^6*c^4*x^3 + b^7*c^3*x^2)

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giac [A]  time = 0.16, size = 428, normalized size = 1.82 \begin {gather*} -\frac {{\left (3 \, B b c d^{4} - 6 \, A c^{2} d^{4} - 4 \, B b^{2} d^{3} e + 12 \, A b c d^{3} e - 6 \, A b^{2} d^{2} e^{2}\right )} \log \left ({\left | x \right |}\right )}{b^{5}} + \frac {{\left (3 \, B b c^{4} d^{4} - 6 \, A c^{5} d^{4} - 4 \, B b^{2} c^{3} d^{3} e + 12 \, A b c^{4} d^{3} e - 6 \, A b^{2} c^{3} d^{2} e^{2} + B b^{5} e^{4}\right )} \log \left ({\left | c x + b \right |}\right )}{b^{5} c^{3}} - \frac {A b^{3} c^{3} d^{4} + 2 \, {\left (3 \, B b c^{5} d^{4} - 6 \, A c^{6} d^{4} - 4 \, B b^{2} c^{4} d^{3} e + 12 \, A b c^{5} d^{3} e - 6 \, A b^{2} c^{4} d^{2} e^{2} + 4 \, B b^{4} c^{2} d e^{3} - 2 \, B b^{5} c e^{4} + A b^{4} c^{2} e^{4}\right )} x^{3} + {\left (9 \, B b^{2} c^{4} d^{4} - 18 \, A b c^{5} d^{4} - 12 \, B b^{3} c^{3} d^{3} e + 36 \, A b^{2} c^{4} d^{3} e + 6 \, B b^{4} c^{2} d^{2} e^{2} - 18 \, A b^{3} c^{3} d^{2} e^{2} + 4 \, B b^{5} c d e^{3} + 4 \, A b^{4} c^{2} d e^{3} - 3 \, B b^{6} e^{4} + A b^{5} c e^{4}\right )} x^{2} + 2 \, {\left (B b^{3} c^{3} d^{4} - 2 \, A b^{2} c^{4} d^{4} + 4 \, A b^{3} c^{3} d^{3} e\right )} x}{2 \, {\left (c x + b\right )}^{2} b^{4} c^{3} x^{2}} \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)^3,x, algorithm="giac")

[Out]

-(3*B*b*c*d^4 - 6*A*c^2*d^4 - 4*B*b^2*d^3*e + 12*A*b*c*d^3*e - 6*A*b^2*d^2*e^2)*log(abs(x))/b^5 + (3*B*b*c^4*d
^4 - 6*A*c^5*d^4 - 4*B*b^2*c^3*d^3*e + 12*A*b*c^4*d^3*e - 6*A*b^2*c^3*d^2*e^2 + B*b^5*e^4)*log(abs(c*x + b))/(
b^5*c^3) - 1/2*(A*b^3*c^3*d^4 + 2*(3*B*b*c^5*d^4 - 6*A*c^6*d^4 - 4*B*b^2*c^4*d^3*e + 12*A*b*c^5*d^3*e - 6*A*b^
2*c^4*d^2*e^2 + 4*B*b^4*c^2*d*e^3 - 2*B*b^5*c*e^4 + A*b^4*c^2*e^4)*x^3 + (9*B*b^2*c^4*d^4 - 18*A*b*c^5*d^4 - 1
2*B*b^3*c^3*d^3*e + 36*A*b^2*c^4*d^3*e + 6*B*b^4*c^2*d^2*e^2 - 18*A*b^3*c^3*d^2*e^2 + 4*B*b^5*c*d*e^3 + 4*A*b^
4*c^2*d*e^3 - 3*B*b^6*e^4 + A*b^5*c*e^4)*x^2 + 2*(B*b^3*c^3*d^4 - 2*A*b^2*c^4*d^4 + 4*A*b^3*c^3*d^3*e)*x)/((c*
x + b)^2*b^4*c^3*x^2)

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maple [B]  time = 0.06, size = 536, normalized size = 2.28 \begin {gather*} \frac {A b \,e^{4}}{2 \left (c x +b \right )^{2} c^{2}}+\frac {3 A \,d^{2} e^{2}}{\left (c x +b \right )^{2} b}-\frac {2 A c \,d^{3} e}{\left (c x +b \right )^{2} b^{2}}+\frac {A \,c^{2} d^{4}}{2 \left (c x +b \right )^{2} b^{3}}-\frac {2 A d \,e^{3}}{\left (c x +b \right )^{2} c}-\frac {B \,b^{2} e^{4}}{2 \left (c x +b \right )^{2} c^{3}}+\frac {2 B b d \,e^{3}}{\left (c x +b \right )^{2} c^{2}}+\frac {2 B \,d^{3} e}{\left (c x +b \right )^{2} b}-\frac {B c \,d^{4}}{2 \left (c x +b \right )^{2} b^{2}}-\frac {3 B \,d^{2} e^{2}}{\left (c x +b \right )^{2} c}+\frac {6 A \,d^{2} e^{2}}{\left (c x +b \right ) b^{2}}-\frac {8 A c \,d^{3} e}{\left (c x +b \right ) b^{3}}+\frac {6 A \,d^{2} e^{2} \ln \relax (x )}{b^{3}}-\frac {6 A \,d^{2} e^{2} \ln \left (c x +b \right )}{b^{3}}+\frac {3 A \,c^{2} d^{4}}{\left (c x +b \right ) b^{4}}-\frac {12 A c \,d^{3} e \ln \relax (x )}{b^{4}}+\frac {12 A c \,d^{3} e \ln \left (c x +b \right )}{b^{4}}+\frac {6 A \,c^{2} d^{4} \ln \relax (x )}{b^{5}}-\frac {6 A \,c^{2} d^{4} \ln \left (c x +b \right )}{b^{5}}-\frac {A \,e^{4}}{\left (c x +b \right ) c^{2}}+\frac {2 B b \,e^{4}}{\left (c x +b \right ) c^{3}}+\frac {4 B \,d^{3} e}{\left (c x +b \right ) b^{2}}-\frac {2 B c \,d^{4}}{\left (c x +b \right ) b^{3}}+\frac {4 B \,d^{3} e \ln \relax (x )}{b^{3}}-\frac {4 B \,d^{3} e \ln \left (c x +b \right )}{b^{3}}-\frac {3 B c \,d^{4} \ln \relax (x )}{b^{4}}+\frac {3 B c \,d^{4} \ln \left (c x +b \right )}{b^{4}}-\frac {4 B d \,e^{3}}{\left (c x +b \right ) c^{2}}+\frac {B \,e^{4} \ln \left (c x +b \right )}{c^{3}}-\frac {4 A \,d^{3} e}{b^{3} x}+\frac {3 A c \,d^{4}}{b^{4} x}-\frac {B \,d^{4}}{b^{3} x}-\frac {A \,d^{4}}{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)^4/(c*x^2+b*x)^3,x)

[Out]

-8*c/b^3/(c*x+b)*A*d^3*e+2*b/c^2/(c*x+b)^2*B*d*e^3-12*d^3/b^4*ln(x)*A*e*c+12/b^4*c*ln(c*x+b)*A*d^3*e-2/b^2*c/(
c*x+b)^2*A*d^3*e-2/c/(c*x+b)^2*A*d*e^3+3/b/(c*x+b)^2*A*d^2*e^2+1/c^3*ln(c*x+b)*B*e^4-d^4/b^3/x*B-1/c^2/(c*x+b)
*A*e^4-3/c/(c*x+b)^2*B*d^2*e^2+2/b/(c*x+b)^2*B*d^3*e+6/b^2/(c*x+b)*A*d^2*e^2-4/c^2/(c*x+b)*B*d*e^3+4/b^2/(c*x+
b)*B*d^3*e-6/b^3*ln(c*x+b)*A*d^2*e^2-6/b^5*c^2*ln(c*x+b)*A*d^4+6*d^2/b^3*ln(x)*A*e^2+6*d^4/b^5*ln(x)*A*c^2+4*d
^3/b^3*ln(x)*B*e-3*d^4/b^4*ln(x)*B*c-4*d^3/b^3/x*A*e+3*d^4/b^4/x*A*c-4/b^3*ln(c*x+b)*B*d^3*e+3/b^4*c*ln(c*x+b)
*B*d^4+1/2*b/c^2/(c*x+b)^2*A*e^4+1/2/b^3*c^2/(c*x+b)^2*A*d^4-1/2*b^2/c^3/(c*x+b)^2*B*e^4-1/2/b^2*c/(c*x+b)^2*B
*d^4+3*c^2/b^4/(c*x+b)*A*d^4+2/c^3*b/(c*x+b)*B*e^4-2*c/b^3/(c*x+b)*B*d^4-1/2*A*d^4/b^3/x^2

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maxima [A]  time = 0.60, size = 430, normalized size = 1.83 \begin {gather*} -\frac {A b^{3} c^{3} d^{4} - 2 \, {\left (6 \, A b^{2} c^{4} d^{2} e^{2} - 4 \, B b^{4} c^{2} d e^{3} - 3 \, {\left (B b c^{5} - 2 \, A c^{6}\right )} d^{4} + 4 \, {\left (B b^{2} c^{4} - 3 \, A b c^{5}\right )} d^{3} e + {\left (2 \, B b^{5} c - A b^{4} c^{2}\right )} e^{4}\right )} x^{3} + {\left (9 \, {\left (B b^{2} c^{4} - 2 \, A b c^{5}\right )} d^{4} - 12 \, {\left (B b^{3} c^{3} - 3 \, A b^{2} c^{4}\right )} d^{3} e + 6 \, {\left (B b^{4} c^{2} - 3 \, 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 (3 \, B b^{6} - A b^{5} c\right )} e^{4}\right )} x^{2} + 2 \, {\left (4 \, A b^{3} c^{3} d^{3} e + {\left (B b^{3} c^{3} - 2 \, A b^{2} c^{4}\right )} d^{4}\right )} x}{2 \, {\left (b^{4} c^{5} x^{4} + 2 \, b^{5} c^{4} x^{3} + b^{6} c^{3} x^{2}\right )}} + \frac {{\left (6 \, A b^{2} d^{2} e^{2} - 3 \, {\left (B b c - 2 \, A c^{2}\right )} d^{4} + 4 \, {\left (B b^{2} - 3 \, A b c\right )} d^{3} e\right )} \log \relax (x)}{b^{5}} - \frac {{\left (6 \, A b^{2} c^{3} d^{2} e^{2} - B b^{5} e^{4} - 3 \, {\left (B b c^{4} - 2 \, A c^{5}\right )} d^{4} + 4 \, {\left (B b^{2} c^{3} - 3 \, A b c^{4}\right )} d^{3} e\right )} \log \left (c x + b\right )}{b^{5} c^{3}} \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)^3,x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 1.80, size = 403, normalized size = 1.71 \begin {gather*} \frac {\ln \relax (x)\,\left (b^2\,\left (4\,B\,d^3\,e+6\,A\,d^2\,e^2\right )-b\,\left (3\,B\,c\,d^4+12\,A\,c\,e\,d^3\right )+6\,A\,c^2\,d^4\right )}{b^5}-\frac {\frac {A\,d^4}{2\,b}+\frac {x^2\,\left (-3\,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^3-12\,B\,b^2\,c^3\,d^3\,e-18\,A\,b^2\,c^3\,d^2\,e^2+9\,B\,b\,c^4\,d^4+36\,A\,b\,c^4\,d^3\,e-18\,A\,c^5\,d^4\right )}{2\,b^3\,c^3}-\frac {x^3\,\left (2\,B\,b^5\,e^4-4\,B\,b^4\,c\,d\,e^3-A\,b^4\,c\,e^4+4\,B\,b^2\,c^3\,d^3\,e+6\,A\,b^2\,c^3\,d^2\,e^2-3\,B\,b\,c^4\,d^4-12\,A\,b\,c^4\,d^3\,e+6\,A\,c^5\,d^4\right )}{b^4\,c^2}+\frac {d^3\,x\,\left (4\,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 {\ln \left (b+c\,x\right )\,{\left (b\,e-c\,d\right )}^2\,\left (B\,b^3\,e^2+2\,B\,b^2\,c\,d\,e+3\,B\,b\,c^2\,d^2-6\,A\,c^3\,d^2\right )}{b^5\,c^3} \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)^3,x)

[Out]

(log(x)*(b^2*(6*A*d^2*e^2 + 4*B*d^3*e) - b*(3*B*c*d^4 + 12*A*c*d^3*e) + 6*A*c^2*d^4))/b^5 - ((A*d^4)/(2*b) + (
x^2*(A*b^4*c*e^4 - 3*B*b^5*e^4 - 18*A*c^5*d^4 + 9*B*b*c^4*d^4 + 4*A*b^3*c^2*d*e^3 - 12*B*b^2*c^3*d^3*e - 18*A*
b^2*c^3*d^2*e^2 + 6*B*b^3*c^2*d^2*e^2 + 36*A*b*c^4*d^3*e + 4*B*b^4*c*d*e^3))/(2*b^3*c^3) - (x^3*(6*A*c^5*d^4 +
2*B*b^5*e^4 - A*b^4*c*e^4 - 3*B*b*c^4*d^4 + 4*B*b^2*c^3*d^3*e + 6*A*b^2*c^3*d^2*e^2 - 12*A*b*c^4*d^3*e - 4*B*
b^4*c*d*e^3))/(b^4*c^2) + (d^3*x*(4*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) + (log(b +
c*x)*(b*e - c*d)^2*(B*b^3*e^2 - 6*A*c^3*d^2 + 3*B*b*c^2*d^2 + 2*B*b^2*c*d*e))/(b^5*c^3)

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sympy [B]  time = 77.98, size = 881, normalized size = 3.75 \begin {gather*} \frac {- A b^{3} c^{3} d^{4} + x^{3} \left (- 2 A b^{4} c^{2} e^{4} + 12 A b^{2} c^{4} d^{2} e^{2} - 24 A b c^{5} d^{3} e + 12 A c^{6} d^{4} + 4 B b^{5} c e^{4} - 8 B b^{4} c^{2} d e^{3} + 8 B b^{2} c^{4} d^{3} e - 6 B b c^{5} d^{4}\right ) + x^{2} \left (- A b^{5} c e^{4} - 4 A b^{4} c^{2} d e^{3} + 18 A b^{3} c^{3} d^{2} e^{2} - 36 A b^{2} c^{4} d^{3} e + 18 A b c^{5} d^{4} + 3 B b^{6} e^{4} - 4 B b^{5} c d e^{3} - 6 B b^{4} c^{2} d^{2} e^{2} + 12 B b^{3} c^{3} d^{3} e - 9 B b^{2} c^{4} d^{4}\right ) + x \left (- 8 A b^{3} c^{3} d^{3} e + 4 A b^{2} c^{4} d^{4} - 2 B b^{3} c^{3} d^{4}\right )}{2 b^{6} c^{3} x^{2} + 4 b^{5} c^{4} x^{3} + 2 b^{4} c^{5} x^{4}} + \frac {d^{2} \left (6 A b^{2} e^{2} - 12 A b c d e + 6 A c^{2} d^{2} + 4 B b^{2} d e - 3 B b c d^{2}\right ) \log {\left (x + \frac {- 6 A b^{3} c^{2} d^{2} e^{2} + 12 A b^{2} c^{3} d^{3} e - 6 A b c^{4} d^{4} - 4 B b^{3} c^{2} d^{3} e + 3 B b^{2} c^{3} d^{4} + b c^{2} d^{2} \left (6 A b^{2} e^{2} - 12 A b c d e + 6 A c^{2} d^{2} + 4 B b^{2} d e - 3 B b c d^{2}\right )}{- 12 A b^{2} c^{3} d^{2} e^{2} + 24 A b c^{4} d^{3} e - 12 A c^{5} d^{4} + B b^{5} e^{4} - 8 B b^{2} c^{3} d^{3} e + 6 B b c^{4} d^{4}} \right )}}{b^{5}} + \frac {\left (b e - c d\right )^{2} \left (- 6 A c^{3} d^{2} + B b^{3} e^{2} + 2 B b^{2} c d e + 3 B b c^{2} d^{2}\right ) \log {\left (x + \frac {- 6 A b^{3} c^{2} d^{2} e^{2} + 12 A b^{2} c^{3} d^{3} e - 6 A b c^{4} d^{4} - 4 B b^{3} c^{2} d^{3} e + 3 B b^{2} c^{3} d^{4} + \frac {b \left (b e - c d\right )^{2} \left (- 6 A c^{3} d^{2} + B b^{3} e^{2} + 2 B b^{2} c d e + 3 B b c^{2} d^{2}\right )}{c}}{- 12 A b^{2} c^{3} d^{2} e^{2} + 24 A b c^{4} d^{3} e - 12 A c^{5} d^{4} + B b^{5} e^{4} - 8 B b^{2} c^{3} d^{3} e + 6 B b c^{4} d^{4}} \right )}}{b^{5} c^{3}} \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)**3,x)

[Out]

(-A*b**3*c**3*d**4 + x**3*(-2*A*b**4*c**2*e**4 + 12*A*b**2*c**4*d**2*e**2 - 24*A*b*c**5*d**3*e + 12*A*c**6*d**
4 + 4*B*b**5*c*e**4 - 8*B*b**4*c**2*d*e**3 + 8*B*b**2*c**4*d**3*e - 6*B*b*c**5*d**4) + x**2*(-A*b**5*c*e**4 -
4*A*b**4*c**2*d*e**3 + 18*A*b**3*c**3*d**2*e**2 - 36*A*b**2*c**4*d**3*e + 18*A*b*c**5*d**4 + 3*B*b**6*e**4 - 4
*B*b**5*c*d*e**3 - 6*B*b**4*c**2*d**2*e**2 + 12*B*b**3*c**3*d**3*e - 9*B*b**2*c**4*d**4) + x*(-8*A*b**3*c**3*d
**3*e + 4*A*b**2*c**4*d**4 - 2*B*b**3*c**3*d**4))/(2*b**6*c**3*x**2 + 4*b**5*c**4*x**3 + 2*b**4*c**5*x**4) + d
**2*(6*A*b**2*e**2 - 12*A*b*c*d*e + 6*A*c**2*d**2 + 4*B*b**2*d*e - 3*B*b*c*d**2)*log(x + (-6*A*b**3*c**2*d**2*
e**2 + 12*A*b**2*c**3*d**3*e - 6*A*b*c**4*d**4 - 4*B*b**3*c**2*d**3*e + 3*B*b**2*c**3*d**4 + b*c**2*d**2*(6*A*
b**2*e**2 - 12*A*b*c*d*e + 6*A*c**2*d**2 + 4*B*b**2*d*e - 3*B*b*c*d**2))/(-12*A*b**2*c**3*d**2*e**2 + 24*A*b*c
**4*d**3*e - 12*A*c**5*d**4 + B*b**5*e**4 - 8*B*b**2*c**3*d**3*e + 6*B*b*c**4*d**4))/b**5 + (b*e - c*d)**2*(-6
*A*c**3*d**2 + B*b**3*e**2 + 2*B*b**2*c*d*e + 3*B*b*c**2*d**2)*log(x + (-6*A*b**3*c**2*d**2*e**2 + 12*A*b**2*c
**3*d**3*e - 6*A*b*c**4*d**4 - 4*B*b**3*c**2*d**3*e + 3*B*b**2*c**3*d**4 + b*(b*e - c*d)**2*(-6*A*c**3*d**2 +
B*b**3*e**2 + 2*B*b**2*c*d*e + 3*B*b*c**2*d**2)/c)/(-12*A*b**2*c**3*d**2*e**2 + 24*A*b*c**4*d**3*e - 12*A*c**5
*d**4 + B*b**5*e**4 - 8*B*b**2*c**3*d**3*e + 6*B*b*c**4*d**4))/(b**5*c**3)

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