Optimal. Leaf size=30 \[ \frac {a^2 x^4}{4}+\frac {2}{3} a b x^3+\frac {b^2 x^2}{2} \]
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Rubi [A] time = 0.01, antiderivative size = 30, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {263, 43} \[ \frac {a^2 x^4}{4}+\frac {2}{3} a b x^3+\frac {b^2 x^2}{2} \]
Antiderivative was successfully verified.
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Rule 43
Rule 263
Rubi steps
\begin {align*} \int \left (a+\frac {b}{x}\right )^2 x^3 \, dx &=\int x (b+a x)^2 \, dx\\ &=\int \left (b^2 x+2 a b x^2+a^2 x^3\right ) \, dx\\ &=\frac {b^2 x^2}{2}+\frac {2}{3} a b x^3+\frac {a^2 x^4}{4}\\ \end {align*}
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Mathematica [A] time = 0.00, size = 30, normalized size = 1.00 \[ \frac {a^2 x^4}{4}+\frac {2}{3} a b x^3+\frac {b^2 x^2}{2} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.02, size = 24, normalized size = 0.80 \[ \frac {1}{4} \, a^{2} x^{4} + \frac {2}{3} \, a b x^{3} + \frac {1}{2} \, b^{2} x^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.18, size = 24, normalized size = 0.80 \[ \frac {1}{4} \, a^{2} x^{4} + \frac {2}{3} \, a b x^{3} + \frac {1}{2} \, b^{2} x^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 25, normalized size = 0.83 \[ \frac {1}{4} a^{2} x^{4}+\frac {2}{3} a b \,x^{3}+\frac {1}{2} b^{2} x^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.04, size = 24, normalized size = 0.80 \[ \frac {1}{4} \, a^{2} x^{4} + \frac {2}{3} \, a b x^{3} + \frac {1}{2} \, b^{2} x^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.03, size = 24, normalized size = 0.80 \[ \frac {a^2\,x^4}{4}+\frac {2\,a\,b\,x^3}{3}+\frac {b^2\,x^2}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.07, size = 26, normalized size = 0.87 \[ \frac {a^{2} x^{4}}{4} + \frac {2 a b x^{3}}{3} + \frac {b^{2} x^{2}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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