Optimal. Leaf size=104 \[ -\frac {b^2 x (-3 a d f+b c f+b d e)}{d^2 f^2}-\frac {(b c-a d)^3 \log (c+d x)}{d^3 (d e-c f)}+\frac {(b e-a f)^3 \log (e+f x)}{f^3 (d e-c f)}+\frac {b^3 x^2}{2 d f} \]
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Rubi [A] time = 0.12, antiderivative size = 104, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {72} \[ -\frac {b^2 x (-3 a d f+b c f+b d e)}{d^2 f^2}-\frac {(b c-a d)^3 \log (c+d x)}{d^3 (d e-c f)}+\frac {(b e-a f)^3 \log (e+f x)}{f^3 (d e-c f)}+\frac {b^3 x^2}{2 d f} \]
Antiderivative was successfully verified.
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Rule 72
Rubi steps
\begin {align*} \int \frac {(a+b x)^3}{(c+d x) (e+f x)} \, dx &=\int \left (-\frac {b^2 (b d e+b c f-3 a d f)}{d^2 f^2}+\frac {b^3 x}{d f}+\frac {(-b c+a d)^3}{d^2 (d e-c f) (c+d x)}+\frac {(-b e+a f)^3}{f^2 (-d e+c f) (e+f x)}\right ) \, dx\\ &=-\frac {b^2 (b d e+b c f-3 a d f) x}{d^2 f^2}+\frac {b^3 x^2}{2 d f}-\frac {(b c-a d)^3 \log (c+d x)}{d^3 (d e-c f)}+\frac {(b e-a f)^3 \log (e+f x)}{f^3 (d e-c f)}\\ \end {align*}
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Mathematica [A] time = 0.10, size = 99, normalized size = 0.95 \[ \frac {b^2 d f x (d e-c f) (6 a d f+b (-2 c f-2 d e+d f x))-2 f^3 (b c-a d)^3 \log (c+d x)+2 d^3 (b e-a f)^3 \log (e+f x)}{2 d^3 f^3 (d e-c f)} \]
Antiderivative was successfully verified.
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fricas [B] time = 1.09, size = 207, normalized size = 1.99 \[ -\frac {2 \, {\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} f^{3} \log \left (d x + c\right ) - {\left (b^{3} d^{3} e f^{2} - b^{3} c d^{2} f^{3}\right )} x^{2} + 2 \, {\left (b^{3} d^{3} e^{2} f - 3 \, a b^{2} d^{3} e f^{2} - {\left (b^{3} c^{2} d - 3 \, a b^{2} c d^{2}\right )} f^{3}\right )} x - 2 \, {\left (b^{3} d^{3} e^{3} - 3 \, a b^{2} d^{3} e^{2} f + 3 \, a^{2} b d^{3} e f^{2} - a^{3} d^{3} f^{3}\right )} \log \left (f x + e\right )}{2 \, {\left (d^{4} e f^{3} - c d^{3} f^{4}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.98, size = 165, normalized size = 1.59 \[ \frac {{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \log \left ({\left | d x + c \right |}\right )}{c d^{3} f - d^{4} e} + \frac {{\left (a^{3} f^{3} - 3 \, a^{2} b f^{2} e + 3 \, a b^{2} f e^{2} - b^{3} e^{3}\right )} \log \left ({\left | f x + e \right |}\right )}{c f^{4} - d f^{3} e} + \frac {b^{3} d f x^{2} - 2 \, b^{3} c f x + 6 \, a b^{2} d f x - 2 \, b^{3} d x e}{2 \, d^{2} f^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.01, size = 257, normalized size = 2.47 \[ -\frac {a^{3} \ln \left (d x +c \right )}{c f -d e}+\frac {a^{3} \ln \left (f x +e \right )}{c f -d e}+\frac {3 a^{2} b c \ln \left (d x +c \right )}{\left (c f -d e \right ) d}-\frac {3 a^{2} b e \ln \left (f x +e \right )}{\left (c f -d e \right ) f}-\frac {3 a \,b^{2} c^{2} \ln \left (d x +c \right )}{\left (c f -d e \right ) d^{2}}+\frac {3 a \,b^{2} e^{2} \ln \left (f x +e \right )}{\left (c f -d e \right ) f^{2}}+\frac {b^{3} c^{3} \ln \left (d x +c \right )}{\left (c f -d e \right ) d^{3}}-\frac {b^{3} e^{3} \ln \left (f x +e \right )}{\left (c f -d e \right ) f^{3}}+\frac {b^{3} x^{2}}{2 d f}+\frac {3 a \,b^{2} x}{d f}-\frac {b^{3} c x}{d^{2} f}-\frac {b^{3} e x}{d \,f^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.71, size = 161, normalized size = 1.55 \[ -\frac {{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \log \left (d x + c\right )}{d^{4} e - c d^{3} f} + \frac {{\left (b^{3} e^{3} - 3 \, a b^{2} e^{2} f + 3 \, a^{2} b e f^{2} - a^{3} f^{3}\right )} \log \left (f x + e\right )}{d e f^{3} - c f^{4}} + \frac {b^{3} d f x^{2} - 2 \, {\left (b^{3} d e + {\left (b^{3} c - 3 \, a b^{2} d\right )} f\right )} x}{2 \, d^{2} f^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.50, size = 109, normalized size = 1.05 \[ x\,\left (\frac {3\,a\,b^2}{d\,f}-\frac {b^3\,\left (c\,f+d\,e\right )}{d^2\,f^2}\right )+\frac {b^3\,x^2}{2\,d\,f}-\frac {\ln \left (c+d\,x\right )\,{\left (a\,d-b\,c\right )}^3}{d^3\,\left (c\,f-d\,e\right )}+\frac {\ln \left (e+f\,x\right )\,{\left (a\,f-b\,e\right )}^3}{f^3\,\left (c\,f-d\,e\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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