Optimal. Leaf size=57 \[ -\frac {d \log (b+c x)}{b^3}+\frac {d \log (x)}{b^3}+\frac {d}{b^2 (b+c x)}+\frac {c d-b e}{2 b c (b+c x)^2} \]
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Rubi [A] time = 0.04, antiderivative size = 57, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {765} \[ \frac {d}{b^2 (b+c x)}-\frac {d \log (b+c x)}{b^3}+\frac {d \log (x)}{b^3}+\frac {c d-b e}{2 b c (b+c x)^2} \]
Antiderivative was successfully verified.
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Rule 765
Rubi steps
\begin {align*} \int \frac {x^2 (d+e x)}{\left (b x+c x^2\right )^3} \, dx &=\int \left (\frac {d}{b^3 x}+\frac {-c d+b e}{b (b+c x)^3}-\frac {c d}{b^2 (b+c x)^2}-\frac {c d}{b^3 (b+c x)}\right ) \, dx\\ &=\frac {c d-b e}{2 b c (b+c x)^2}+\frac {d}{b^2 (b+c x)}+\frac {d \log (x)}{b^3}-\frac {d \log (b+c x)}{b^3}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 53, normalized size = 0.93 \[ \frac {\frac {b \left (b^2 (-e)+3 b c d+2 c^2 d x\right )}{c (b+c x)^2}-2 d \log (b+c x)+2 d \log (x)}{2 b^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.92, size = 109, normalized size = 1.91 \[ \frac {2 \, b c^{2} d x + 3 \, b^{2} c d - b^{3} e - 2 \, {\left (c^{3} d x^{2} + 2 \, b c^{2} d x + b^{2} c d\right )} \log \left (c x + b\right ) + 2 \, {\left (c^{3} d x^{2} + 2 \, b c^{2} d x + b^{2} c d\right )} \log \relax (x)}{2 \, {\left (b^{3} c^{3} x^{2} + 2 \, b^{4} c^{2} x + b^{5} c\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 60, normalized size = 1.05 \[ -\frac {d \log \left ({\left | c x + b \right |}\right )}{b^{3}} + \frac {d \log \left ({\left | x \right |}\right )}{b^{3}} + \frac {2 \, b c^{2} d x + 3 \, b^{2} c d - b^{3} e}{2 \, {\left (c x + b\right )}^{2} b^{3} c} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 59, normalized size = 1.04 \[ \frac {d}{2 \left (c x +b \right )^{2} b}-\frac {e}{2 \left (c x +b \right )^{2} c}+\frac {d}{\left (c x +b \right ) b^{2}}+\frac {d \ln \relax (x )}{b^{3}}-\frac {d \ln \left (c x +b \right )}{b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.92, size = 68, normalized size = 1.19 \[ \frac {2 \, c^{2} d x + 3 \, b c d - b^{2} e}{2 \, {\left (b^{2} c^{3} x^{2} + 2 \, b^{3} c^{2} x + b^{4} c\right )}} - \frac {d \log \left (c x + b\right )}{b^{3}} + \frac {d \log \relax (x)}{b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.07, size = 62, normalized size = 1.09 \[ -\frac {\frac {b\,e-3\,c\,d}{2\,b\,c}-\frac {c\,d\,x}{b^2}}{b^2+2\,b\,c\,x+c^2\,x^2}-\frac {2\,d\,\mathrm {atanh}\left (\frac {2\,c\,x}{b}+1\right )}{b^3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.41, size = 63, normalized size = 1.11 \[ \frac {- b^{2} e + 3 b c d + 2 c^{2} d x}{2 b^{4} c + 4 b^{3} c^{2} x + 2 b^{2} c^{3} x^{2}} + \frac {d \left (\log {\relax (x )} - \log {\left (\frac {b}{c} + x \right )}\right )}{b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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