Optimal. Leaf size=94 \[ \frac {b^3 (c d-b e)}{2 c^5 (b+c x)^2}-\frac {b^2 (3 c d-4 b e)}{c^5 (b+c x)}-\frac {3 b (c d-2 b e) \log (b+c x)}{c^5}+\frac {x (c d-3 b e)}{c^4}+\frac {e x^2}{2 c^3} \]
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Rubi [A] time = 0.10, antiderivative size = 94, 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)}{2 c^5 (b+c x)^2}-\frac {b^2 (3 c d-4 b e)}{c^5 (b+c x)}+\frac {x (c d-3 b e)}{c^4}-\frac {3 b (c d-2 b e) \log (b+c x)}{c^5}+\frac {e x^2}{2 c^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 765
Rubi steps
\begin {align*} \int \frac {x^6 (d+e x)}{\left (b x+c x^2\right )^3} \, dx &=\int \left (\frac {c d-3 b e}{c^4}+\frac {e x}{c^3}+\frac {b^3 (-c d+b e)}{c^4 (b+c x)^3}-\frac {b^2 (-3 c d+4 b e)}{c^4 (b+c x)^2}+\frac {3 b (-c d+2 b e)}{c^4 (b+c x)}\right ) \, dx\\ &=\frac {(c d-3 b e) x}{c^4}+\frac {e x^2}{2 c^3}+\frac {b^3 (c d-b e)}{2 c^5 (b+c x)^2}-\frac {b^2 (3 c d-4 b e)}{c^5 (b+c x)}-\frac {3 b (c d-2 b e) \log (b+c x)}{c^5}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 86, normalized size = 0.91 \begin {gather*} \frac {\frac {b^3 (c d-b e)}{(b+c x)^2}+\frac {2 b^2 (4 b e-3 c d)}{b+c x}+2 c x (c d-3 b e)+6 b (2 b e-c d) \log (b+c x)+c^2 e x^2}{2 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^6 (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 = 167, normalized size = 1.78 \begin {gather*} \frac {c^{4} e x^{4} - 5 \, b^{3} c d + 7 \, b^{4} e + 2 \, {\left (c^{4} d - 2 \, b c^{3} e\right )} x^{3} + {\left (4 \, b c^{3} d - 11 \, b^{2} c^{2} e\right )} x^{2} - 2 \, {\left (2 \, b^{2} c^{2} d - b^{3} c e\right )} x - 6 \, {\left (b^{3} c d - 2 \, b^{4} e + {\left (b c^{3} d - 2 \, b^{2} c^{2} e\right )} x^{2} + 2 \, {\left (b^{2} c^{2} d - 2 \, b^{3} c e\right )} x\right )} \log \left (c x + b\right )}{2 \, {\left (c^{7} x^{2} + 2 \, b c^{6} x + b^{2} c^{5}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 104, normalized size = 1.11 \begin {gather*} -\frac {3 \, {\left (b c d - 2 \, b^{2} e\right )} \log \left ({\left | c x + b \right |}\right )}{c^{5}} + \frac {c^{3} x^{2} e + 2 \, c^{3} d x - 6 \, b c^{2} x e}{2 \, c^{6}} - \frac {5 \, b^{3} c d - 7 \, b^{4} e + 2 \, {\left (3 \, b^{2} c^{2} d - 4 \, b^{3} c e\right )} x}{2 \, {\left (c x + b\right )}^{2} c^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.07, size = 117, normalized size = 1.24 \begin {gather*} -\frac {b^{4} e}{2 \left (c x +b \right )^{2} c^{5}}+\frac {b^{3} d}{2 \left (c x +b \right )^{2} c^{4}}+\frac {e \,x^{2}}{2 c^{3}}+\frac {4 b^{3} e}{\left (c x +b \right ) c^{5}}-\frac {3 b^{2} d}{\left (c x +b \right ) c^{4}}+\frac {6 b^{2} e \ln \left (c x +b \right )}{c^{5}}-\frac {3 b d \ln \left (c x +b \right )}{c^{4}}-\frac {3 b e x}{c^{4}}+\frac {d x}{c^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.81, size = 106, normalized size = 1.13 \begin {gather*} -\frac {5 \, b^{3} c d - 7 \, b^{4} e + 2 \, {\left (3 \, b^{2} c^{2} d - 4 \, b^{3} c e\right )} x}{2 \, {\left (c^{7} x^{2} + 2 \, b c^{6} x + b^{2} c^{5}\right )}} + \frac {c e x^{2} + 2 \, {\left (c d - 3 \, b e\right )} x}{2 \, c^{4}} - \frac {3 \, {\left (b c d - 2 \, b^{2} 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 = 1.04, size = 108, normalized size = 1.15 \begin {gather*} x\,\left (\frac {d}{c^3}-\frac {3\,b\,e}{c^4}\right )+\frac {x\,\left (4\,b^3\,e-3\,b^2\,c\,d\right )+\frac {7\,b^4\,e-5\,b^3\,c\,d}{2\,c}}{b^2\,c^4+2\,b\,c^5\,x+c^6\,x^2}+\frac {e\,x^2}{2\,c^3}+\frac {\ln \left (b+c\,x\right )\,\left (6\,b^2\,e-3\,b\,c\,d\right )}{c^5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.68, size = 107, normalized size = 1.14 \begin {gather*} \frac {3 b \left (2 b e - c d\right ) \log {\left (b + c x \right )}}{c^{5}} + x \left (- \frac {3 b e}{c^{4}} + \frac {d}{c^{3}}\right ) + \frac {7 b^{4} e - 5 b^{3} c d + x \left (8 b^{3} c e - 6 b^{2} c^{2} d\right )}{2 b^{2} c^{5} + 4 b c^{6} x + 2 c^{7} x^{2}} + \frac {e x^{2}}{2 c^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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