Optimal. Leaf size=128 \[ \frac {d^2 \log (x) (3 A b e-2 A c d+b B d)}{b^3}+\frac {(b B-A c) (c d-b e)^3}{b^2 c^3 (b+c x)}-\frac {A d^3}{b^2 x}+\frac {(c d-b e)^2 \log (b+c x) \left (-b c (B d-A e)+2 A c^2 d-2 b^2 B e\right )}{b^3 c^3}+\frac {B e^3 x}{c^2} \]
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Rubi [A] time = 0.16, antiderivative size = 128, 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} \[ \frac {(b B-A c) (c d-b e)^3}{b^2 c^3 (b+c x)}+\frac {(c d-b e)^2 \log (b+c x) \left (-b c (B d-A e)+2 A c^2 d-2 b^2 B e\right )}{b^3 c^3}+\frac {d^2 \log (x) (3 A b e-2 A c d+b B d)}{b^3}-\frac {A d^3}{b^2 x}+\frac {B e^3 x}{c^2} \]
Antiderivative was successfully verified.
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Rule 771
Rubi steps
\begin {align*} \int \frac {(A+B x) (d+e x)^3}{\left (b x+c x^2\right )^2} \, dx &=\int \left (\frac {B e^3}{c^2}+\frac {A d^3}{b^2 x^2}+\frac {d^2 (b B d-2 A c d+3 A b e)}{b^3 x}+\frac {(b B-A c) (-c d+b e)^3}{b^2 c^2 (b+c x)^2}+\frac {(c d-b e)^2 \left (2 A c^2 d-2 b^2 B e-b c (B d-A e)\right )}{b^3 c^2 (b+c x)}\right ) \, dx\\ &=-\frac {A d^3}{b^2 x}+\frac {B e^3 x}{c^2}+\frac {(b B-A c) (c d-b e)^3}{b^2 c^3 (b+c x)}+\frac {d^2 (b B d-2 A c d+3 A b e) \log (x)}{b^3}+\frac {(c d-b e)^2 \left (2 A c^2 d-2 b^2 B e-b c (B d-A e)\right ) \log (b+c x)}{b^3 c^3}\\ \end {align*}
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Mathematica [A] time = 0.07, size = 128, normalized size = 1.00 \[ \frac {d^2 \log (x) (3 A b e-2 A c d+b B d)}{b^3}-\frac {(b B-A c) (b e-c d)^3}{b^2 c^3 (b+c x)}-\frac {A d^3}{b^2 x}+\frac {(c d-b e)^2 \log (b+c x) \left (A b c e+2 A c^2 d-2 b^2 B e-b B c d\right )}{b^3 c^3}+\frac {B e^3 x}{c^2} \]
Antiderivative was successfully verified.
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fricas [B] time = 1.24, size = 362, normalized size = 2.83 \[ \frac {B b^{3} c^{2} e^{3} x^{3} + B b^{4} c e^{3} x^{2} - A b^{2} c^{3} d^{3} + {\left ({\left (B b^{2} c^{3} - 2 \, A b c^{4}\right )} d^{3} - 3 \, {\left (B b^{3} c^{2} - A b^{2} c^{3}\right )} d^{2} e + 3 \, {\left (B b^{4} c - A b^{3} c^{2}\right )} d e^{2} - {\left (B b^{5} - A b^{4} c\right )} e^{3}\right )} x - {\left ({\left (3 \, A b c^{4} d^{2} e - 3 \, B b^{3} c^{2} d e^{2} + {\left (B b c^{4} - 2 \, A c^{5}\right )} d^{3} + {\left (2 \, B b^{4} c - A b^{3} c^{2}\right )} e^{3}\right )} x^{2} + {\left (3 \, A b^{2} c^{3} d^{2} e - 3 \, B b^{4} c d e^{2} + {\left (B b^{2} c^{3} - 2 \, A b c^{4}\right )} d^{3} + {\left (2 \, B b^{5} - A b^{4} c\right )} e^{3}\right )} x\right )} \log \left (c x + b\right ) + {\left ({\left (3 \, A b c^{4} d^{2} e + {\left (B b c^{4} - 2 \, A c^{5}\right )} d^{3}\right )} x^{2} + {\left (3 \, A b^{2} c^{3} d^{2} e + {\left (B b^{2} c^{3} - 2 \, A b c^{4}\right )} d^{3}\right )} x\right )} \log \relax (x)}{b^{3} c^{4} x^{2} + b^{4} c^{3} x} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.19, size = 229, normalized size = 1.79 \[ \frac {B x e^{3}}{c^{2}} + \frac {{\left (B b d^{3} - 2 \, A c d^{3} + 3 \, A b d^{2} e\right )} \log \left ({\left | x \right |}\right )}{b^{3}} - \frac {{\left (B b c^{3} d^{3} - 2 \, A c^{4} d^{3} + 3 \, A b c^{3} d^{2} e - 3 \, B b^{3} c d e^{2} + 2 \, B b^{4} e^{3} - A b^{3} c e^{3}\right )} \log \left ({\left | c x + b \right |}\right )}{b^{3} c^{3}} - \frac {A b c^{2} d^{3} - \frac {{\left (B b c^{3} d^{3} - 2 \, A c^{4} d^{3} - 3 \, B b^{2} c^{2} d^{2} e + 3 \, A b c^{3} d^{2} e + 3 \, B b^{3} c d e^{2} - 3 \, A b^{2} c^{2} d e^{2} - B b^{4} e^{3} + A b^{3} c e^{3}\right )} x}{c}}{{\left (c x + b\right )} b^{2} c^{2} x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.06, size = 286, normalized size = 2.23 \[ \frac {A b \,e^{3}}{\left (c x +b \right ) c^{2}}+\frac {3 A \,d^{2} e}{\left (c x +b \right ) b}-\frac {A c \,d^{3}}{\left (c x +b \right ) b^{2}}+\frac {3 A \,d^{2} e \ln \relax (x )}{b^{2}}-\frac {3 A \,d^{2} e \ln \left (c x +b \right )}{b^{2}}-\frac {2 A c \,d^{3} \ln \relax (x )}{b^{3}}+\frac {2 A c \,d^{3} \ln \left (c x +b \right )}{b^{3}}-\frac {3 A d \,e^{2}}{\left (c x +b \right ) c}+\frac {A \,e^{3} \ln \left (c x +b \right )}{c^{2}}-\frac {B \,b^{2} e^{3}}{\left (c x +b \right ) c^{3}}+\frac {3 B b d \,e^{2}}{\left (c x +b \right ) c^{2}}-\frac {2 B b \,e^{3} \ln \left (c x +b \right )}{c^{3}}+\frac {B \,d^{3}}{\left (c x +b \right ) b}+\frac {B \,d^{3} \ln \relax (x )}{b^{2}}-\frac {B \,d^{3} \ln \left (c x +b \right )}{b^{2}}-\frac {3 B \,d^{2} e}{\left (c x +b \right ) c}+\frac {3 B d \,e^{2} \ln \left (c x +b \right )}{c^{2}}+\frac {B \,e^{3} x}{c^{2}}-\frac {A \,d^{3}}{b^{2} x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.51, size = 225, normalized size = 1.76 \[ \frac {B e^{3} x}{c^{2}} - \frac {A b c^{3} d^{3} - {\left ({\left (B b c^{3} - 2 \, A c^{4}\right )} d^{3} - 3 \, {\left (B b^{2} c^{2} - A b c^{3}\right )} d^{2} e + 3 \, {\left (B b^{3} c - A b^{2} c^{2}\right )} d e^{2} - {\left (B b^{4} - A b^{3} c\right )} e^{3}\right )} x}{b^{2} c^{4} x^{2} + b^{3} c^{3} x} + \frac {{\left (3 \, A b d^{2} e + {\left (B b - 2 \, A c\right )} d^{3}\right )} \log \relax (x)}{b^{3}} - \frac {{\left (3 \, A b c^{3} d^{2} e - 3 \, B b^{3} c d e^{2} + {\left (B b c^{3} - 2 \, A c^{4}\right )} d^{3} + {\left (2 \, B b^{4} - A b^{3} c\right )} e^{3}\right )} \log \left (c x + b\right )}{b^{3} c^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.59, size = 212, normalized size = 1.66 \[ \frac {\ln \relax (x)\,\left (b\,\left (B\,d^3+3\,A\,e\,d^2\right )-2\,A\,c\,d^3\right )}{b^3}-\frac {\frac {x\,\left (B\,b^4\,e^3-3\,B\,b^3\,c\,d\,e^2-A\,b^3\,c\,e^3+3\,B\,b^2\,c^2\,d^2\,e+3\,A\,b^2\,c^2\,d\,e^2-B\,b\,c^3\,d^3-3\,A\,b\,c^3\,d^2\,e+2\,A\,c^4\,d^3\right )}{b^2\,c}+\frac {A\,c^2\,d^3}{b}}{c^3\,x^2+b\,c^2\,x}+\frac {B\,e^3\,x}{c^2}+\frac {\ln \left (b+c\,x\right )\,{\left (b\,e-c\,d\right )}^2\,\left (2\,A\,c^2\,d-2\,B\,b^2\,e+A\,b\,c\,e-B\,b\,c\,d\right )}{b^3\,c^3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 9.38, size = 502, normalized size = 3.92 \[ \frac {B e^{3} x}{c^{2}} + \frac {- A b c^{3} d^{3} + x \left (A b^{3} c e^{3} - 3 A b^{2} c^{2} d e^{2} + 3 A b c^{3} d^{2} e - 2 A c^{4} d^{3} - B b^{4} e^{3} + 3 B b^{3} c d e^{2} - 3 B b^{2} c^{2} d^{2} e + B b c^{3} d^{3}\right )}{b^{3} c^{3} x + b^{2} c^{4} x^{2}} + \frac {d^{2} \left (3 A b e - 2 A c d + B b d\right ) \log {\left (x + \frac {3 A b^{2} c^{2} d^{2} e - 2 A b c^{3} d^{3} + B b^{2} c^{2} d^{3} - b c^{2} d^{2} \left (3 A b e - 2 A c d + B b d\right )}{- A b^{3} c e^{3} + 6 A b c^{3} d^{2} e - 4 A c^{4} d^{3} + 2 B b^{4} e^{3} - 3 B b^{3} c d e^{2} + 2 B b c^{3} d^{3}} \right )}}{b^{3}} - \frac {\left (b e - c d\right )^{2} \left (- A b c e - 2 A c^{2} d + 2 B b^{2} e + B b c d\right ) \log {\left (x + \frac {3 A b^{2} c^{2} d^{2} e - 2 A b c^{3} d^{3} + B b^{2} c^{2} d^{3} + \frac {b \left (b e - c d\right )^{2} \left (- A b c e - 2 A c^{2} d + 2 B b^{2} e + B b c d\right )}{c}}{- A b^{3} c e^{3} + 6 A b c^{3} d^{2} e - 4 A c^{4} d^{3} + 2 B b^{4} e^{3} - 3 B b^{3} c d e^{2} + 2 B b c^{3} d^{3}} \right )}}{b^{3} c^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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