Optimal. Leaf size=69 \[ -\frac {b^2 (c d-b e)}{c^4 (b+c x)}-\frac {b (2 c d-3 b e) \log (b+c x)}{c^4}+\frac {x (c d-2 b e)}{c^3}+\frac {e x^2}{2 c^2} \]
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Rubi [A] time = 0.07, antiderivative size = 69, 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)}{c^4 (b+c x)}+\frac {x (c d-2 b e)}{c^3}-\frac {b (2 c d-3 b e) \log (b+c x)}{c^4}+\frac {e x^2}{2 c^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 765
Rubi steps
\begin {align*} \int \frac {x^4 (d+e x)}{\left (b x+c x^2\right )^2} \, dx &=\int \left (\frac {c d-2 b e}{c^3}+\frac {e x}{c^2}-\frac {b^2 (-c d+b e)}{c^3 (b+c x)^2}+\frac {b (-2 c d+3 b e)}{c^3 (b+c x)}\right ) \, dx\\ &=\frac {(c d-2 b e) x}{c^3}+\frac {e x^2}{2 c^2}-\frac {b^2 (c d-b e)}{c^4 (b+c x)}-\frac {b (2 c d-3 b e) \log (b+c x)}{c^4}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 66, normalized size = 0.96 \begin {gather*} \frac {\frac {2 b^2 (b e-c d)}{b+c x}+2 c x (c d-2 b e)+2 b (3 b e-2 c d) \log (b+c x)+c^2 e x^2}{2 c^4} \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^4 (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 = 0.39, size = 111, normalized size = 1.61 \begin {gather*} \frac {c^{3} e x^{3} - 2 \, b^{2} c d + 2 \, b^{3} e + {\left (2 \, c^{3} d - 3 \, b c^{2} e\right )} x^{2} + 2 \, {\left (b c^{2} d - 2 \, b^{2} c e\right )} x - 2 \, {\left (2 \, b^{2} c d - 3 \, b^{3} e + {\left (2 \, b c^{2} d - 3 \, b^{2} c e\right )} x\right )} \log \left (c x + b\right )}{2 \, {\left (c^{5} x + b c^{4}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 81, normalized size = 1.17 \begin {gather*} -\frac {{\left (2 \, b c d - 3 \, b^{2} e\right )} \log \left ({\left | c x + b \right |}\right )}{c^{4}} + \frac {c^{2} x^{2} e + 2 \, c^{2} d x - 4 \, b c x e}{2 \, c^{4}} - \frac {b^{2} c d - b^{3} e}{{\left (c x + b\right )} c^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 84, normalized size = 1.22 \begin {gather*} \frac {e \,x^{2}}{2 c^{2}}+\frac {b^{3} e}{\left (c x +b \right ) c^{4}}-\frac {b^{2} d}{\left (c x +b \right ) c^{3}}+\frac {3 b^{2} e \ln \left (c x +b \right )}{c^{4}}-\frac {2 b d \ln \left (c x +b \right )}{c^{3}}-\frac {2 b e x}{c^{3}}+\frac {d x}{c^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.91, size = 75, normalized size = 1.09 \begin {gather*} -\frac {b^{2} c d - b^{3} e}{c^{5} x + b c^{4}} + \frac {c e x^{2} + 2 \, {\left (c d - 2 \, b e\right )} x}{2 \, c^{3}} - \frac {{\left (2 \, b c d - 3 \, b^{2} 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 = 0.06, size = 77, normalized size = 1.12 \begin {gather*} x\,\left (\frac {d}{c^2}-\frac {2\,b\,e}{c^3}\right )+\frac {e\,x^2}{2\,c^2}+\frac {b^3\,e-b^2\,c\,d}{c\,\left (x\,c^4+b\,c^3\right )}+\frac {\ln \left (b+c\,x\right )\,\left (3\,b^2\,e-2\,b\,c\,d\right )}{c^4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.37, size = 68, normalized size = 0.99 \begin {gather*} \frac {b \left (3 b e - 2 c d\right ) \log {\left (b + c x \right )}}{c^{4}} + x \left (- \frac {2 b e}{c^{3}} + \frac {d}{c^{2}}\right ) + \frac {b^{3} e - b^{2} c d}{b c^{4} + c^{5} x} + \frac {e x^{2}}{2 c^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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