Optimal. Leaf size=66 \[ \frac {b^2 (c d-b e) \log (b+c x)}{c^4}-\frac {b x (c d-b e)}{c^3}+\frac {x^2 (c d-b e)}{2 c^2}+\frac {e x^3}{3 c} \]
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Rubi [A] time = 0.06, antiderivative size = 66, 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} \[ \frac {b^2 (c d-b e) \log (b+c x)}{c^4}+\frac {x^2 (c d-b e)}{2 c^2}-\frac {b x (c d-b e)}{c^3}+\frac {e x^3}{3 c} \]
Antiderivative was successfully verified.
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Rule 765
Rubi steps
\begin {align*} \int \frac {x^3 (d+e x)}{b x+c x^2} \, dx &=\int \left (\frac {b (-c d+b e)}{c^3}+\frac {(c d-b e) x}{c^2}+\frac {e x^2}{c}-\frac {b^2 (-c d+b e)}{c^3 (b+c x)}\right ) \, dx\\ &=-\frac {b (c d-b e) x}{c^3}+\frac {(c d-b e) x^2}{2 c^2}+\frac {e x^3}{3 c}+\frac {b^2 (c d-b e) \log (b+c x)}{c^4}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 61, normalized size = 0.92 \[ \frac {c x \left (6 b^2 e-3 b c (2 d+e x)+c^2 x (3 d+2 e x)\right )+6 b^2 (c d-b e) \log (b+c x)}{6 c^4} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.60, size = 71, normalized size = 1.08 \[ \frac {2 \, c^{3} e x^{3} + 3 \, {\left (c^{3} d - b c^{2} e\right )} x^{2} - 6 \, {\left (b c^{2} d - b^{2} c e\right )} x + 6 \, {\left (b^{2} c d - b^{3} e\right )} \log \left (c x + b\right )}{6 \, c^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 74, normalized size = 1.12 \[ \frac {2 \, c^{2} x^{3} e + 3 \, c^{2} d x^{2} - 3 \, b c x^{2} e - 6 \, b c d x + 6 \, b^{2} x e}{6 \, c^{3}} + \frac {{\left (b^{2} c d - b^{3} e\right )} \log \left ({\left | c x + b \right |}\right )}{c^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 76, normalized size = 1.15 \[ \frac {e \,x^{3}}{3 c}-\frac {b e \,x^{2}}{2 c^{2}}+\frac {d \,x^{2}}{2 c}-\frac {b^{3} e \ln \left (c x +b \right )}{c^{4}}+\frac {b^{2} d \ln \left (c x +b \right )}{c^{3}}+\frac {b^{2} e x}{c^{3}}-\frac {b d x}{c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.95, size = 69, normalized size = 1.05 \[ \frac {2 \, c^{2} e x^{3} + 3 \, {\left (c^{2} d - b c e\right )} x^{2} - 6 \, {\left (b c d - b^{2} e\right )} x}{6 \, c^{3}} + \frac {{\left (b^{2} c d - b^{3} e\right )} \log \left (c x + b\right )}{c^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.01, size = 72, normalized size = 1.09 \[ x^2\,\left (\frac {d}{2\,c}-\frac {b\,e}{2\,c^2}\right )-\frac {\ln \left (b+c\,x\right )\,\left (b^3\,e-b^2\,c\,d\right )}{c^4}+\frac {e\,x^3}{3\,c}-\frac {b\,x\,\left (\frac {d}{c}-\frac {b\,e}{c^2}\right )}{c} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.22, size = 61, normalized size = 0.92 \[ - \frac {b^{2} \left (b e - c d\right ) \log {\left (b + c x \right )}}{c^{4}} + x^{2} \left (- \frac {b e}{2 c^{2}} + \frac {d}{2 c}\right ) + x \left (\frac {b^{2} e}{c^{3}} - \frac {b d}{c^{2}}\right ) + \frac {e x^{3}}{3 c} \]
Verification of antiderivative is not currently implemented for this CAS.
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