Optimal. Leaf size=94 \[ \frac{b^3 (c d-b e)}{2 c^5 (b+c x)^2}-\frac{b^2 (3 c d-4 b e)}{c^5 (b+c x)}+\frac{x (c d-3 b e)}{c^4}-\frac{3 b (c d-2 b e) \log (b+c x)}{c^5}+\frac{e x^2}{2 c^3} \]
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Rubi [A] time = 0.0988481, antiderivative size = 94, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.05, Rules used = {765} \[ \frac{b^3 (c d-b e)}{2 c^5 (b+c x)^2}-\frac{b^2 (3 c d-4 b e)}{c^5 (b+c x)}+\frac{x (c d-3 b e)}{c^4}-\frac{3 b (c d-2 b e) \log (b+c x)}{c^5}+\frac{e x^2}{2 c^3} \]
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*}
Mathematica [A] time = 0.0565573, size = 86, normalized size = 0.91 \[ \frac{\frac{b^3 (c d-b e)}{(b+c x)^2}+\frac{2 b^2 (4 b e-3 c d)}{b+c x}+2 c x (c d-3 b e)+6 b (2 b e-c d) \log (b+c x)+c^2 e x^2}{2 c^5} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.007, size = 117, normalized size = 1.2 \begin{align*}{\frac{e{x}^{2}}{2\,{c}^{3}}}-3\,{\frac{bex}{{c}^{4}}}+{\frac{dx}{{c}^{3}}}+4\,{\frac{{b}^{3}e}{{c}^{5} \left ( cx+b \right ) }}-3\,{\frac{{b}^{2}d}{{c}^{4} \left ( cx+b \right ) }}+6\,{\frac{{b}^{2}\ln \left ( cx+b \right ) e}{{c}^{5}}}-3\,{\frac{b\ln \left ( cx+b \right ) d}{{c}^{4}}}-{\frac{{b}^{4}e}{2\,{c}^{5} \left ( cx+b \right ) ^{2}}}+{\frac{{b}^{3}d}{2\,{c}^{4} \left ( cx+b \right ) ^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.11124, size = 143, normalized size = 1.52 \begin{align*} -\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{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.72183, size = 350, normalized size = 3.72 \begin{align*} \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{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.51782, size = 105, normalized size = 1.12 \begin{align*} \frac{3 b \left (2 b e - c d\right ) \log{\left (b + c x \right )}}{c^{5}} + \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}} - \frac{x \left (3 b e - c d\right )}{c^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.14654, size = 140, normalized size = 1.49 \begin{align*} -\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{align*}
Verification of antiderivative is not currently implemented for this CAS.
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