Optimal. Leaf size=57 \[ \frac{b^2 x^2}{2 a^3}-\frac{b^3 x}{a^4}+\frac{b^4 \log (a x+b)}{a^5}-\frac{b x^3}{3 a^2}+\frac{x^4}{4 a} \]
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Rubi [A] time = 0.0267549, antiderivative size = 57, 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} \[ \frac{b^2 x^2}{2 a^3}-\frac{b^3 x}{a^4}+\frac{b^4 \log (a x+b)}{a^5}-\frac{b x^3}{3 a^2}+\frac{x^4}{4 a} \]
Antiderivative was successfully verified.
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Rule 263
Rule 43
Rubi steps
\begin{align*} \int \frac{x^3}{a+\frac{b}{x}} \, dx &=\int \frac{x^4}{b+a x} \, dx\\ &=\int \left (-\frac{b^3}{a^4}+\frac{b^2 x}{a^3}-\frac{b x^2}{a^2}+\frac{x^3}{a}+\frac{b^4}{a^4 (b+a x)}\right ) \, dx\\ &=-\frac{b^3 x}{a^4}+\frac{b^2 x^2}{2 a^3}-\frac{b x^3}{3 a^2}+\frac{x^4}{4 a}+\frac{b^4 \log (b+a x)}{a^5}\\ \end{align*}
Mathematica [A] time = 0.0040021, size = 57, normalized size = 1. \[ \frac{b^2 x^2}{2 a^3}-\frac{b^3 x}{a^4}+\frac{b^4 \log (a x+b)}{a^5}-\frac{b x^3}{3 a^2}+\frac{x^4}{4 a} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 52, normalized size = 0.9 \begin{align*} -{\frac{{b}^{3}x}{{a}^{4}}}+{\frac{{b}^{2}{x}^{2}}{2\,{a}^{3}}}-{\frac{b{x}^{3}}{3\,{a}^{2}}}+{\frac{{x}^{4}}{4\,a}}+{\frac{{b}^{4}\ln \left ( ax+b \right ) }{{a}^{5}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.0212, size = 70, normalized size = 1.23 \begin{align*} \frac{b^{4} \log \left (a x + b\right )}{a^{5}} + \frac{3 \, a^{3} x^{4} - 4 \, a^{2} b x^{3} + 6 \, a b^{2} x^{2} - 12 \, b^{3} x}{12 \, a^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.42849, size = 117, normalized size = 2.05 \begin{align*} \frac{3 \, a^{4} x^{4} - 4 \, a^{3} b x^{3} + 6 \, a^{2} b^{2} x^{2} - 12 \, a b^{3} x + 12 \, b^{4} \log \left (a x + b\right )}{12 \, a^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.272151, size = 49, normalized size = 0.86 \begin{align*} \frac{x^{4}}{4 a} - \frac{b x^{3}}{3 a^{2}} + \frac{b^{2} x^{2}}{2 a^{3}} - \frac{b^{3} x}{a^{4}} + \frac{b^{4} \log{\left (a x + b \right )}}{a^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.10112, size = 72, normalized size = 1.26 \begin{align*} \frac{b^{4} \log \left ({\left | a x + b \right |}\right )}{a^{5}} + \frac{3 \, a^{3} x^{4} - 4 \, a^{2} b x^{3} + 6 \, a b^{2} x^{2} - 12 \, b^{3} x}{12 \, a^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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