Optimal. Leaf size=64 \[ 5 a^2 b^3 x^2+10 a^3 b^2 \log (x)-\frac{5 a^4 b}{2 x^2}-\frac{a^5}{4 x^4}+\frac{5}{4} a b^4 x^4+\frac{b^5 x^6}{6} \]
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Rubi [A] time = 0.0361221, antiderivative size = 64, 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} \[ 5 a^2 b^3 x^2+10 a^3 b^2 \log (x)-\frac{5 a^4 b}{2 x^2}-\frac{a^5}{4 x^4}+\frac{5}{4} a b^4 x^4+\frac{b^5 x^6}{6} \]
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^5} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{(a+b x)^5}{x^3} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (10 a^2 b^3+\frac{a^5}{x^3}+\frac{5 a^4 b}{x^2}+\frac{10 a^3 b^2}{x}+5 a b^4 x+b^5 x^2\right ) \, dx,x,x^2\right )\\ &=-\frac{a^5}{4 x^4}-\frac{5 a^4 b}{2 x^2}+5 a^2 b^3 x^2+\frac{5}{4} a b^4 x^4+\frac{b^5 x^6}{6}+10 a^3 b^2 \log (x)\\ \end{align*}
Mathematica [A] time = 0.006366, size = 64, normalized size = 1. \[ 5 a^2 b^3 x^2+10 a^3 b^2 \log (x)-\frac{5 a^4 b}{2 x^2}-\frac{a^5}{4 x^4}+\frac{5}{4} a b^4 x^4+\frac{b^5 x^6}{6} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 57, normalized size = 0.9 \begin{align*} -{\frac{{a}^{5}}{4\,{x}^{4}}}-{\frac{5\,{a}^{4}b}{2\,{x}^{2}}}+5\,{a}^{2}{b}^{3}{x}^{2}+{\frac{5\,a{b}^{4}{x}^{4}}{4}}+{\frac{{b}^{5}{x}^{6}}{6}}+10\,{a}^{3}{b}^{2}\ln \left ( x \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 2.05834, size = 80, normalized size = 1.25 \begin{align*} \frac{1}{6} \, b^{5} x^{6} + \frac{5}{4} \, a b^{4} x^{4} + 5 \, a^{2} b^{3} x^{2} + 5 \, a^{3} b^{2} \log \left (x^{2}\right ) - \frac{10 \, a^{4} b x^{2} + a^{5}}{4 \, x^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.37146, size = 139, normalized size = 2.17 \begin{align*} \frac{2 \, b^{5} x^{10} + 15 \, a b^{4} x^{8} + 60 \, a^{2} b^{3} x^{6} + 120 \, a^{3} b^{2} x^{4} \log \left (x\right ) - 30 \, a^{4} b x^{2} - 3 \, a^{5}}{12 \, x^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.486586, size = 61, normalized size = 0.95 \begin{align*} 10 a^{3} b^{2} \log{\left (x \right )} + 5 a^{2} b^{3} x^{2} + \frac{5 a b^{4} x^{4}}{4} + \frac{b^{5} x^{6}}{6} - \frac{a^{5} + 10 a^{4} b x^{2}}{4 x^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 2.1646, size = 95, normalized size = 1.48 \begin{align*} \frac{1}{6} \, b^{5} x^{6} + \frac{5}{4} \, a b^{4} x^{4} + 5 \, a^{2} b^{3} x^{2} + 5 \, a^{3} b^{2} \log \left (x^{2}\right ) - \frac{30 \, a^{3} b^{2} x^{4} + 10 \, a^{4} b x^{2} + a^{5}}{4 \, x^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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