Optimal. Leaf size=90 \[ \frac {b^3 (c d-b e)}{c^5 (b+c x)}+\frac {b^2 (3 c d-4 b e) \log (b+c x)}{c^5}-\frac {b x (2 c d-3 b e)}{c^4}+\frac {x^2 (c d-2 b e)}{2 c^3}+\frac {e x^3}{3 c^2} \]
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Rubi [A] time = 0.09, antiderivative size = 90, 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^3 (c d-b e)}{c^5 (b+c x)}+\frac {b^2 (3 c d-4 b e) \log (b+c x)}{c^5}+\frac {x^2 (c d-2 b e)}{2 c^3}-\frac {b x (2 c d-3 b e)}{c^4}+\frac {e x^3}{3 c^2} \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 )^2} \, dx &=\int \left (\frac {b (-2 c d+3 b e)}{c^4}+\frac {(c d-2 b e) x}{c^3}+\frac {e x^2}{c^2}+\frac {b^3 (-c d+b e)}{c^4 (b+c x)^2}-\frac {b^2 (-3 c d+4 b e)}{c^4 (b+c x)}\right ) \, dx\\ &=-\frac {b (2 c d-3 b e) x}{c^4}+\frac {(c d-2 b e) x^2}{2 c^3}+\frac {e x^3}{3 c^2}+\frac {b^3 (c d-b e)}{c^5 (b+c x)}+\frac {b^2 (3 c d-4 b e) \log (b+c x)}{c^5}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 87, normalized size = 0.97 \begin {gather*} \frac {\frac {6 b^3 (c d-b e)}{b+c x}+6 b^2 (3 c d-4 b e) \log (b+c x)+3 c^2 x^2 (c d-2 b e)+6 b c x (3 b e-2 c d)+2 c^3 e x^3}{6 c^5} \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 )^2} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.40, size = 139, normalized size = 1.54 \begin {gather*} \frac {2 \, c^{4} e x^{4} + 6 \, b^{3} c d - 6 \, b^{4} e + {\left (3 \, c^{4} d - 4 \, b c^{3} e\right )} x^{3} - 3 \, {\left (3 \, b c^{3} d - 4 \, b^{2} c^{2} e\right )} x^{2} - 6 \, {\left (2 \, b^{2} c^{2} d - 3 \, b^{3} c e\right )} x + 6 \, {\left (3 \, b^{3} c d - 4 \, b^{4} e + {\left (3 \, b^{2} c^{2} d - 4 \, b^{3} c e\right )} x\right )} \log \left (c x + b\right )}{6 \, {\left (c^{6} x + b c^{5}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 107, normalized size = 1.19 \begin {gather*} \frac {{\left (3 \, b^{2} c d - 4 \, b^{3} e\right )} \log \left ({\left | c x + b \right |}\right )}{c^{5}} + \frac {2 \, c^{4} x^{3} e + 3 \, c^{4} d x^{2} - 6 \, b c^{3} x^{2} e - 12 \, b c^{3} d x + 18 \, b^{2} c^{2} x e}{6 \, c^{6}} + \frac {b^{3} c d - b^{4} e}{{\left (c x + b\right )} c^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 109, normalized size = 1.21 \begin {gather*} \frac {e \,x^{3}}{3 c^{2}}-\frac {b e \,x^{2}}{c^{3}}+\frac {d \,x^{2}}{2 c^{2}}-\frac {b^{4} e}{\left (c x +b \right ) c^{5}}+\frac {b^{3} d}{\left (c x +b \right ) c^{4}}-\frac {4 b^{3} e \ln \left (c x +b \right )}{c^{5}}+\frac {3 b^{2} d \ln \left (c x +b \right )}{c^{4}}+\frac {3 b^{2} e x}{c^{4}}-\frac {2 b d 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 = 98, normalized size = 1.09 \begin {gather*} \frac {b^{3} c d - b^{4} e}{c^{6} x + b c^{5}} + \frac {2 \, c^{2} e x^{3} + 3 \, {\left (c^{2} d - 2 \, b c e\right )} x^{2} - 6 \, {\left (2 \, b c d - 3 \, b^{2} e\right )} x}{6 \, c^{4}} + \frac {{\left (3 \, b^{2} c d - 4 \, b^{3} e\right )} \log \left (c x + b\right )}{c^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.03, size = 115, normalized size = 1.28 \begin {gather*} x^2\,\left (\frac {d}{2\,c^2}-\frac {b\,e}{c^3}\right )-x\,\left (\frac {b^2\,e}{c^4}+\frac {2\,b\,\left (\frac {d}{c^2}-\frac {2\,b\,e}{c^3}\right )}{c}\right )-\frac {\ln \left (b+c\,x\right )\,\left (4\,b^3\,e-3\,b^2\,c\,d\right )}{c^5}+\frac {e\,x^3}{3\,c^2}-\frac {b^4\,e-b^3\,c\,d}{c\,\left (x\,c^5+b\,c^4\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.42, size = 92, normalized size = 1.02 \begin {gather*} - \frac {b^{2} \left (4 b e - 3 c d\right ) \log {\left (b + c x \right )}}{c^{5}} + x^{2} \left (- \frac {b e}{c^{3}} + \frac {d}{2 c^{2}}\right ) + x \left (\frac {3 b^{2} e}{c^{4}} - \frac {2 b d}{c^{3}}\right ) + \frac {- b^{4} e + b^{3} c d}{b c^{5} + c^{6} x} + \frac {e x^{3}}{3 c^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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