Optimal. Leaf size=65 \[ \frac{5}{3} a^2 b^3 x^6+\frac{5}{2} a^3 b^2 x^4+\frac{5}{2} a^4 b x^2+a^5 \log (x)+\frac{5}{8} a b^4 x^8+\frac{b^5 x^{10}}{10} \]
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Rubi [A] time = 0.0339468, antiderivative size = 65, 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 = {266, 43} \[ \frac{5}{3} a^2 b^3 x^6+\frac{5}{2} a^3 b^2 x^4+\frac{5}{2} a^4 b x^2+a^5 \log (x)+\frac{5}{8} a b^4 x^8+\frac{b^5 x^{10}}{10} \]
Antiderivative was successfully verified.
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Rule 266
Rule 43
Rubi steps
\begin{align*} \int \frac{\left (a+b x^2\right )^5}{x} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{(a+b x)^5}{x} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (5 a^4 b+\frac{a^5}{x}+10 a^3 b^2 x+10 a^2 b^3 x^2+5 a b^4 x^3+b^5 x^4\right ) \, dx,x,x^2\right )\\ &=\frac{5}{2} a^4 b x^2+\frac{5}{2} a^3 b^2 x^4+\frac{5}{3} a^2 b^3 x^6+\frac{5}{8} a b^4 x^8+\frac{b^5 x^{10}}{10}+a^5 \log (x)\\ \end{align*}
Mathematica [A] time = 0.0038627, size = 65, normalized size = 1. \[ \frac{5}{3} a^2 b^3 x^6+\frac{5}{2} a^3 b^2 x^4+\frac{5}{2} a^4 b x^2+a^5 \log (x)+\frac{5}{8} a b^4 x^8+\frac{b^5 x^{10}}{10} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.001, size = 56, normalized size = 0.9 \begin{align*}{\frac{5\,{a}^{4}b{x}^{2}}{2}}+{\frac{5\,{a}^{3}{b}^{2}{x}^{4}}{2}}+{\frac{5\,{a}^{2}{b}^{3}{x}^{6}}{3}}+{\frac{5\,a{b}^{4}{x}^{8}}{8}}+{\frac{{b}^{5}{x}^{10}}{10}}+{a}^{5}\ln \left ( x \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.73627, size = 78, normalized size = 1.2 \begin{align*} \frac{1}{10} \, b^{5} x^{10} + \frac{5}{8} \, a b^{4} x^{8} + \frac{5}{3} \, a^{2} b^{3} x^{6} + \frac{5}{2} \, a^{3} b^{2} x^{4} + \frac{5}{2} \, a^{4} b x^{2} + \frac{1}{2} \, a^{5} \log \left (x^{2}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.27972, size = 130, normalized size = 2. \begin{align*} \frac{1}{10} \, b^{5} x^{10} + \frac{5}{8} \, a b^{4} x^{8} + \frac{5}{3} \, a^{2} b^{3} x^{6} + \frac{5}{2} \, a^{3} b^{2} x^{4} + \frac{5}{2} \, a^{4} b x^{2} + a^{5} \log \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.304247, size = 65, normalized size = 1. \begin{align*} a^{5} \log{\left (x \right )} + \frac{5 a^{4} b x^{2}}{2} + \frac{5 a^{3} b^{2} x^{4}}{2} + \frac{5 a^{2} b^{3} x^{6}}{3} + \frac{5 a b^{4} x^{8}}{8} + \frac{b^{5} x^{10}}{10} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 2.15303, size = 78, normalized size = 1.2 \begin{align*} \frac{1}{10} \, b^{5} x^{10} + \frac{5}{8} \, a b^{4} x^{8} + \frac{5}{3} \, a^{2} b^{3} x^{6} + \frac{5}{2} \, a^{3} b^{2} x^{4} + \frac{5}{2} \, a^{4} b x^{2} + \frac{1}{2} \, a^{5} \log \left (x^{2}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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