Optimal. Leaf size=71 \[ -\frac {b^2 (c d-b e)}{2 c^4 (b+c x)^2}+\frac {b (2 c d-3 b e)}{c^4 (b+c x)}+\frac {(c d-3 b e) \log (b+c x)}{c^4}+\frac {e x}{c^3} \]
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Rubi [A] time = 0.06, antiderivative size = 71, 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} \begin {gather*} -\frac {b^2 (c d-b e)}{2 c^4 (b+c x)^2}+\frac {b (2 c d-3 b e)}{c^4 (b+c x)}+\frac {(c d-3 b e) \log (b+c x)}{c^4}+\frac {e x}{c^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 765
Rubi steps
\begin {align*} \int \frac {x^5 (d+e x)}{\left (b x+c x^2\right )^3} \, dx &=\int \left (\frac {e}{c^3}-\frac {b^2 (-c d+b e)}{c^3 (b+c x)^3}+\frac {b (-2 c d+3 b e)}{c^3 (b+c x)^2}+\frac {c d-3 b e}{c^3 (b+c x)}\right ) \, dx\\ &=\frac {e x}{c^3}-\frac {b^2 (c d-b e)}{2 c^4 (b+c x)^2}+\frac {b (2 c d-3 b e)}{c^4 (b+c x)}+\frac {(c d-3 b e) \log (b+c x)}{c^4}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 75, normalized size = 1.06 \begin {gather*} \frac {2 b c d-3 b^2 e}{c^4 (b+c x)}+\frac {b^3 e-b^2 c d}{2 c^4 (b+c x)^2}+\frac {(c d-3 b e) \log (b+c x)}{c^4}+\frac {e x}{c^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 {x^5 (d+e x)}{\left (b x+c x^2\right )^3} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.39, size = 131, normalized size = 1.85 \begin {gather*} \frac {2 \, c^{3} e x^{3} + 4 \, b c^{2} e x^{2} + 3 \, b^{2} c d - 5 \, b^{3} e + 4 \, {\left (b c^{2} d - b^{2} c e\right )} x + 2 \, {\left (b^{2} c d - 3 \, b^{3} e + {\left (c^{3} d - 3 \, b c^{2} e\right )} x^{2} + 2 \, {\left (b c^{2} d - 3 \, b^{2} c e\right )} x\right )} \log \left (c x + b\right )}{2 \, {\left (c^{6} x^{2} + 2 \, b c^{5} x + b^{2} c^{4}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.19, size = 74, normalized size = 1.04 \begin {gather*} \frac {x e}{c^{3}} + \frac {{\left (c d - 3 \, b e\right )} \log \left ({\left | c x + b \right |}\right )}{c^{4}} + \frac {3 \, b^{2} c d - 5 \, b^{3} e + 2 \, {\left (2 \, b c^{2} d - 3 \, b^{2} c e\right )} x}{2 \, {\left (c x + b\right )}^{2} c^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 94, normalized size = 1.32 \begin {gather*} \frac {b^{3} e}{2 \left (c x +b \right )^{2} c^{4}}-\frac {b^{2} d}{2 \left (c x +b \right )^{2} c^{3}}-\frac {3 b^{2} e}{\left (c x +b \right ) c^{4}}+\frac {2 b d}{\left (c x +b \right ) c^{3}}-\frac {3 b e \ln \left (c x +b \right )}{c^{4}}+\frac {d \ln \left (c x +b \right )}{c^{3}}+\frac {e x}{c^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.93, size = 83, normalized size = 1.17 \begin {gather*} \frac {3 \, b^{2} c d - 5 \, b^{3} e + 2 \, {\left (2 \, b c^{2} d - 3 \, b^{2} c e\right )} x}{2 \, {\left (c^{6} x^{2} + 2 \, b c^{5} x + b^{2} c^{4}\right )}} + \frac {e x}{c^{3}} + \frac {{\left (c d - 3 \, b e\right )} \log \left (c x + b\right )}{c^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.08, size = 87, normalized size = 1.23 \begin {gather*} \frac {e\,x}{c^3}-\frac {\ln \left (b+c\,x\right )\,\left (3\,b\,e-c\,d\right )}{c^4}-\frac {x\,\left (3\,b^2\,e-2\,b\,c\,d\right )+\frac {5\,b^3\,e-3\,b^2\,c\,d}{2\,c}}{b^2\,c^3+2\,b\,c^4\,x+c^5\,x^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.55, size = 83, normalized size = 1.17 \begin {gather*} \frac {- 5 b^{3} e + 3 b^{2} c d + x \left (- 6 b^{2} c e + 4 b c^{2} d\right )}{2 b^{2} c^{4} + 4 b c^{5} x + 2 c^{6} x^{2}} + \frac {e x}{c^{3}} - \frac {\left (3 b e - c d\right ) \log {\left (b + c x \right )}}{c^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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