Optimal. Leaf size=31 \[ \frac {a^2 \log (a+b x)}{b^3}-\frac {a x}{b^2}+\frac {x^2}{2 b} \]
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Rubi [A] time = 0.02, antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {1584, 43} \[ \frac {a^2 \log (a+b x)}{b^3}-\frac {a x}{b^2}+\frac {x^2}{2 b} \]
Antiderivative was successfully verified.
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Rule 43
Rule 1584
Rubi steps
\begin {align*} \int \frac {x^4}{a x^2+b x^3} \, dx &=\int \frac {x^2}{a+b x} \, dx\\ &=\int \left (-\frac {a}{b^2}+\frac {x}{b}+\frac {a^2}{b^2 (a+b x)}\right ) \, dx\\ &=-\frac {a x}{b^2}+\frac {x^2}{2 b}+\frac {a^2 \log (a+b x)}{b^3}\\ \end {align*}
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Mathematica [A] time = 0.00, size = 31, normalized size = 1.00 \[ \frac {a^2 \log (a+b x)}{b^3}-\frac {a x}{b^2}+\frac {x^2}{2 b} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.47, size = 29, normalized size = 0.94 \[ \frac {b^{2} x^{2} - 2 \, a b x + 2 \, a^{2} \log \left (b x + a\right )}{2 \, b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.19, size = 30, normalized size = 0.97 \[ \frac {a^{2} \log \left ({\left | b x + a \right |}\right )}{b^{3}} + \frac {b x^{2} - 2 \, a x}{2 \, b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 30, normalized size = 0.97 \[ \frac {x^{2}}{2 b}+\frac {a^{2} \ln \left (b x +a \right )}{b^{3}}-\frac {a x}{b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.37, size = 29, normalized size = 0.94 \[ \frac {a^{2} \log \left (b x + a\right )}{b^{3}} + \frac {b x^{2} - 2 \, a x}{2 \, b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.04, size = 29, normalized size = 0.94 \[ \frac {2\,a^2\,\ln \left (a+b\,x\right )+b^2\,x^2-2\,a\,b\,x}{2\,b^3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.12, size = 26, normalized size = 0.84 \[ \frac {a^{2} \log {\left (a + b x \right )}}{b^{3}} - \frac {a x}{b^{2}} + \frac {x^{2}}{2 b} \]
Verification of antiderivative is not currently implemented for this CAS.
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