Optimal. Leaf size=87 \[ \frac {(c d-b e)^2 (b e+2 c d) \log (b+c x)}{b^3 c^2}-\frac {d^2 \log (x) (2 c d-3 b e)}{b^3}-\frac {(c d-b e)^3}{b^2 c^2 (b+c x)}-\frac {d^3}{b^2 x} \]
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Rubi [A] time = 0.08, antiderivative size = 87, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {698} \begin {gather*} -\frac {(c d-b e)^3}{b^2 c^2 (b+c x)}+\frac {(c d-b e)^2 (b e+2 c d) \log (b+c x)}{b^3 c^2}-\frac {d^2 \log (x) (2 c d-3 b e)}{b^3}-\frac {d^3}{b^2 x} \end {gather*}
Antiderivative was successfully verified.
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Rule 698
Rubi steps
\begin {align*} \int \frac {(d+e x)^3}{\left (b x+c x^2\right )^2} \, dx &=\int \left (\frac {d^3}{b^2 x^2}+\frac {d^2 (-2 c d+3 b e)}{b^3 x}-\frac {(-c d+b e)^3}{b^2 c (b+c x)^2}+\frac {(-c d+b e)^2 (2 c d+b e)}{b^3 c (b+c x)}\right ) \, dx\\ &=-\frac {d^3}{b^2 x}-\frac {(c d-b e)^3}{b^2 c^2 (b+c x)}-\frac {d^2 (2 c d-3 b e) \log (x)}{b^3}+\frac {(c d-b e)^2 (2 c d+b e) \log (b+c x)}{b^3 c^2}\\ \end {align*}
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Mathematica [A] time = 0.07, size = 79, normalized size = 0.91 \begin {gather*} \frac {\frac {b (b e-c d)^3}{c^2 (b+c x)}+\frac {(c d-b e)^2 (b e+2 c d) \log (b+c x)}{c^2}+d^2 \log (x) (3 b e-2 c d)-\frac {b d^3}{x}}{b^3} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(d+e x)^3}{\left (b x+c x^2\right )^2} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [B] time = 0.42, size = 198, normalized size = 2.28 \begin {gather*} -\frac {b^{2} c^{2} d^{3} + {\left (2 \, b c^{3} d^{3} - 3 \, b^{2} c^{2} d^{2} e + 3 \, b^{3} c d e^{2} - b^{4} e^{3}\right )} x - {\left ({\left (2 \, c^{4} d^{3} - 3 \, b c^{3} d^{2} e + b^{3} c e^{3}\right )} x^{2} + {\left (2 \, b c^{3} d^{3} - 3 \, b^{2} c^{2} d^{2} e + b^{4} e^{3}\right )} x\right )} \log \left (c x + b\right ) + {\left ({\left (2 \, c^{4} d^{3} - 3 \, b c^{3} d^{2} e\right )} x^{2} + {\left (2 \, b c^{3} d^{3} - 3 \, b^{2} c^{2} d^{2} e\right )} x\right )} \log \relax (x)}{b^{3} c^{3} x^{2} + b^{4} c^{2} x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.20, size = 129, normalized size = 1.48 \begin {gather*} -\frac {{\left (2 \, c d^{3} - 3 \, b d^{2} e\right )} \log \left ({\left | x \right |}\right )}{b^{3}} + \frac {{\left (2 \, c^{3} d^{3} - 3 \, b c^{2} d^{2} e + b^{3} e^{3}\right )} \log \left ({\left | c x + b \right |}\right )}{b^{3} c^{2}} - \frac {b c^{2} d^{3} + {\left (2 \, c^{3} d^{3} - 3 \, b c^{2} d^{2} e + 3 \, b^{2} c d e^{2} - b^{3} e^{3}\right )} x}{{\left (c x + b\right )} b^{2} c^{2} x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 141, normalized size = 1.62 \begin {gather*} \frac {b \,e^{3}}{\left (c x +b \right ) c^{2}}+\frac {3 d^{2} e}{\left (c x +b \right ) b}-\frac {c \,d^{3}}{\left (c x +b \right ) b^{2}}+\frac {3 d^{2} e \ln \relax (x )}{b^{2}}-\frac {3 d^{2} e \ln \left (c x +b \right )}{b^{2}}-\frac {2 c \,d^{3} \ln \relax (x )}{b^{3}}+\frac {2 c \,d^{3} \ln \left (c x +b \right )}{b^{3}}-\frac {3 d \,e^{2}}{\left (c x +b \right ) c}+\frac {e^{3} \ln \left (c x +b \right )}{c^{2}}-\frac {d^{3}}{b^{2} x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.37, size = 132, normalized size = 1.52 \begin {gather*} -\frac {b c^{2} d^{3} + {\left (2 \, c^{3} d^{3} - 3 \, b c^{2} d^{2} e + 3 \, b^{2} c d e^{2} - b^{3} e^{3}\right )} x}{b^{2} c^{3} x^{2} + b^{3} c^{2} x} - \frac {{\left (2 \, c d^{3} - 3 \, b d^{2} e\right )} \log \relax (x)}{b^{3}} + \frac {{\left (2 \, c^{3} d^{3} - 3 \, b c^{2} d^{2} e + b^{3} e^{3}\right )} \log \left (c x + b\right )}{b^{3} c^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.30, size = 118, normalized size = 1.36 \begin {gather*} \ln \left (b+c\,x\right )\,\left (\frac {e^3}{c^2}+\frac {2\,c\,d^3}{b^3}-\frac {3\,d^2\,e}{b^2}\right )-\frac {\frac {d^3}{b}-\frac {x\,\left (b^3\,e^3-3\,b^2\,c\,d\,e^2+3\,b\,c^2\,d^2\,e-2\,c^3\,d^3\right )}{b^2\,c^2}}{c\,x^2+b\,x}+\frac {d^2\,\ln \relax (x)\,\left (3\,b\,e-2\,c\,d\right )}{b^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 1.55, size = 250, normalized size = 2.87 \begin {gather*} \frac {- b c^{2} d^{3} + x \left (b^{3} e^{3} - 3 b^{2} c d e^{2} + 3 b c^{2} d^{2} e - 2 c^{3} d^{3}\right )}{b^{3} c^{2} x + b^{2} c^{3} x^{2}} + \frac {d^{2} \left (3 b e - 2 c d\right ) \log {\left (x + \frac {- 3 b^{2} c d^{2} e + 2 b c^{2} d^{3} + b c d^{2} \left (3 b e - 2 c d\right )}{b^{3} e^{3} - 6 b c^{2} d^{2} e + 4 c^{3} d^{3}} \right )}}{b^{3}} + \frac {\left (b e - c d\right )^{2} \left (b e + 2 c d\right ) \log {\left (x + \frac {- 3 b^{2} c d^{2} e + 2 b c^{2} d^{3} + \frac {b \left (b e - c d\right )^{2} \left (b e + 2 c d\right )}{c}}{b^{3} e^{3} - 6 b c^{2} d^{2} e + 4 c^{3} d^{3}} \right )}}{b^{3} c^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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