Optimal. Leaf size=87 \[ -\frac {b^3 (c d-b e) \log (b+c x)}{c^5}+\frac {b^2 x (c d-b e)}{c^4}-\frac {b x^2 (c d-b e)}{2 c^3}+\frac {x^3 (c d-b e)}{3 c^2}+\frac {e x^4}{4 c} \]
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Rubi [A] time = 0.09, antiderivative size = 87, 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 x (c d-b e)}{c^4}-\frac {b^3 (c d-b e) \log (b+c x)}{c^5}+\frac {x^3 (c d-b e)}{3 c^2}-\frac {b x^2 (c d-b e)}{2 c^3}+\frac {e x^4}{4 c} \end {gather*}
Antiderivative was successfully verified.
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Rule 765
Rubi steps
\begin {align*} \int \frac {x^4 (d+e x)}{b x+c x^2} \, dx &=\int \left (-\frac {b^2 (-c d+b e)}{c^4}+\frac {b (-c d+b e) x}{c^3}+\frac {(c d-b e) x^2}{c^2}+\frac {e x^3}{c}+\frac {b^3 (-c d+b e)}{c^4 (b+c x)}\right ) \, dx\\ &=\frac {b^2 (c d-b e) x}{c^4}-\frac {b (c d-b e) x^2}{2 c^3}+\frac {(c d-b e) x^3}{3 c^2}+\frac {e x^4}{4 c}-\frac {b^3 (c d-b e) \log (b+c x)}{c^5}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 80, normalized size = 0.92 \begin {gather*} \frac {12 b^3 (b e-c d) \log (b+c x)+c x \left (-12 b^3 e+6 b^2 c (2 d+e x)-2 b c^2 x (3 d+2 e x)+c^3 x^2 (4 d+3 e x)\right )}{12 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^4 (d+e x)}{b x+c x^2} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.39, size = 94, normalized size = 1.08 \begin {gather*} \frac {3 \, c^{4} e x^{4} + 4 \, {\left (c^{4} d - b c^{3} e\right )} x^{3} - 6 \, {\left (b c^{3} d - b^{2} c^{2} e\right )} x^{2} + 12 \, {\left (b^{2} c^{2} d - b^{3} c e\right )} x - 12 \, {\left (b^{3} c d - b^{4} e\right )} \log \left (c x + b\right )}{12 \, c^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 100, normalized size = 1.15 \begin {gather*} \frac {3 \, c^{3} x^{4} e + 4 \, c^{3} d x^{3} - 4 \, b c^{2} x^{3} e - 6 \, b c^{2} d x^{2} + 6 \, b^{2} c x^{2} e + 12 \, b^{2} c d x - 12 \, b^{3} x e}{12 \, c^{4}} - \frac {{\left (b^{3} c d - b^{4} e\right )} \log \left ({\left | c x + b \right |}\right )}{c^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 100, normalized size = 1.15 \begin {gather*} \frac {e \,x^{4}}{4 c}-\frac {b e \,x^{3}}{3 c^{2}}+\frac {d \,x^{3}}{3 c}+\frac {b^{2} e \,x^{2}}{2 c^{3}}-\frac {b d \,x^{2}}{2 c^{2}}+\frac {b^{4} e \ln \left (c x +b \right )}{c^{5}}-\frac {b^{3} d \ln \left (c x +b \right )}{c^{4}}-\frac {b^{3} e x}{c^{4}}+\frac {b^{2} d x}{c^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.89, size = 93, normalized size = 1.07 \begin {gather*} \frac {3 \, c^{3} e x^{4} + 4 \, {\left (c^{3} d - b c^{2} e\right )} x^{3} - 6 \, {\left (b c^{2} d - b^{2} c e\right )} x^{2} + 12 \, {\left (b^{2} c d - b^{3} e\right )} x}{12 \, c^{4}} - \frac {{\left (b^{3} c d - b^{4} 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 = 0.05, size = 94, normalized size = 1.08 \begin {gather*} x^3\,\left (\frac {d}{3\,c}-\frac {b\,e}{3\,c^2}\right )+\frac {\ln \left (b+c\,x\right )\,\left (b^4\,e-b^3\,c\,d\right )}{c^5}+\frac {e\,x^4}{4\,c}-\frac {b\,x^2\,\left (\frac {d}{c}-\frac {b\,e}{c^2}\right )}{2\,c}+\frac {b^2\,x\,\left (\frac {d}{c}-\frac {b\,e}{c^2}\right )}{c^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.24, size = 85, normalized size = 0.98 \begin {gather*} \frac {b^{3} \left (b e - c d\right ) \log {\left (b + c x \right )}}{c^{5}} + x^{3} \left (- \frac {b e}{3 c^{2}} + \frac {d}{3 c}\right ) + x^{2} \left (\frac {b^{2} e}{2 c^{3}} - \frac {b d}{2 c^{2}}\right ) + x \left (- \frac {b^{3} e}{c^{4}} + \frac {b^{2} d}{c^{3}}\right ) + \frac {e x^{4}}{4 c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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