Optimal. Leaf size=30 \[ \frac{(a x+b)^5}{5 a^2}-\frac{b (a x+b)^4}{4 a^2} \]
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Rubi [A] time = 0.0099667, antiderivative size = 30, normalized size of antiderivative = 1., 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} \[ \frac{(a x+b)^5}{5 a^2}-\frac{b (a x+b)^4}{4 a^2} \]
Antiderivative was successfully verified.
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Rule 263
Rule 43
Rubi steps
\begin{align*} \int \left (a+\frac{b}{x}\right )^3 x^4 \, dx &=\int x (b+a x)^3 \, dx\\ &=\int \left (-\frac{b (b+a x)^3}{a}+\frac{(b+a x)^4}{a}\right ) \, dx\\ &=-\frac{b (b+a x)^4}{4 a^2}+\frac{(b+a x)^5}{5 a^2}\\ \end{align*}
Mathematica [A] time = 0.0015151, size = 40, normalized size = 1.33 \[ \frac{3}{4} a^2 b x^4+\frac{a^3 x^5}{5}+a b^2 x^3+\frac{b^3 x^2}{2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.001, size = 35, normalized size = 1.2 \begin{align*}{\frac{{a}^{3}{x}^{5}}{5}}+{\frac{3\,{a}^{2}b{x}^{4}}{4}}+{x}^{3}a{b}^{2}+{\frac{{b}^{3}{x}^{2}}{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.980427, size = 46, normalized size = 1.53 \begin{align*} \frac{1}{5} \, a^{3} x^{5} + \frac{3}{4} \, a^{2} b x^{4} + a b^{2} x^{3} + \frac{1}{2} \, b^{3} x^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.40899, size = 74, normalized size = 2.47 \begin{align*} \frac{1}{5} \, a^{3} x^{5} + \frac{3}{4} \, a^{2} b x^{4} + a b^{2} x^{3} + \frac{1}{2} \, b^{3} x^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.061702, size = 36, normalized size = 1.2 \begin{align*} \frac{a^{3} x^{5}}{5} + \frac{3 a^{2} b x^{4}}{4} + a b^{2} x^{3} + \frac{b^{3} x^{2}}{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.13982, size = 46, normalized size = 1.53 \begin{align*} \frac{1}{5} \, a^{3} x^{5} + \frac{3}{4} \, a^{2} b x^{4} + a b^{2} x^{3} + \frac{1}{2} \, b^{3} x^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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