Optimal. Leaf size=64 \[ \frac{5}{2} a^2 b^3 x^4+5 a^3 b^2 x^2+5 a^4 b \log (x)-\frac{a^5}{2 x^2}+\frac{5}{6} a b^4 x^6+\frac{b^5 x^8}{8} \]
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Rubi [A] time = 0.0366834, 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} \[ \frac{5}{2} a^2 b^3 x^4+5 a^3 b^2 x^2+5 a^4 b \log (x)-\frac{a^5}{2 x^2}+\frac{5}{6} a b^4 x^6+\frac{b^5 x^8}{8} \]
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^3} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{(a+b x)^5}{x^2} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (10 a^3 b^2+\frac{a^5}{x^2}+\frac{5 a^4 b}{x}+10 a^2 b^3 x+5 a b^4 x^2+b^5 x^3\right ) \, dx,x,x^2\right )\\ &=-\frac{a^5}{2 x^2}+5 a^3 b^2 x^2+\frac{5}{2} a^2 b^3 x^4+\frac{5}{6} a b^4 x^6+\frac{b^5 x^8}{8}+5 a^4 b \log (x)\\ \end{align*}
Mathematica [A] time = 0.0043696, size = 64, normalized size = 1. \[ \frac{5}{2} a^2 b^3 x^4+5 a^3 b^2 x^2+5 a^4 b \log (x)-\frac{a^5}{2 x^2}+\frac{5}{6} a b^4 x^6+\frac{b^5 x^8}{8} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 57, normalized size = 0.9 \begin{align*} -{\frac{{a}^{5}}{2\,{x}^{2}}}+5\,{a}^{3}{b}^{2}{x}^{2}+{\frac{5\,{a}^{2}{b}^{3}{x}^{4}}{2}}+{\frac{5\,a{b}^{4}{x}^{6}}{6}}+{\frac{{b}^{5}{x}^{8}}{8}}+5\,{a}^{4}b\ln \left ( x \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 3.017, size = 78, normalized size = 1.22 \begin{align*} \frac{1}{8} \, b^{5} x^{8} + \frac{5}{6} \, a b^{4} x^{6} + \frac{5}{2} \, a^{2} b^{3} x^{4} + 5 \, a^{3} b^{2} x^{2} + \frac{5}{2} \, a^{4} b \log \left (x^{2}\right ) - \frac{a^{5}}{2 \, x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.22321, size = 142, normalized size = 2.22 \begin{align*} \frac{3 \, b^{5} x^{10} + 20 \, a b^{4} x^{8} + 60 \, a^{2} b^{3} x^{6} + 120 \, a^{3} b^{2} x^{4} + 120 \, a^{4} b x^{2} \log \left (x\right ) - 12 \, a^{5}}{24 \, x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.348286, size = 63, normalized size = 0.98 \begin{align*} - \frac{a^{5}}{2 x^{2}} + 5 a^{4} b \log{\left (x \right )} + 5 a^{3} b^{2} x^{2} + \frac{5 a^{2} b^{3} x^{4}}{2} + \frac{5 a b^{4} x^{6}}{6} + \frac{b^{5} x^{8}}{8} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 2.83745, size = 92, normalized size = 1.44 \begin{align*} \frac{1}{8} \, b^{5} x^{8} + \frac{5}{6} \, a b^{4} x^{6} + \frac{5}{2} \, a^{2} b^{3} x^{4} + 5 \, a^{3} b^{2} x^{2} + \frac{5}{2} \, a^{4} b \log \left (x^{2}\right ) - \frac{5 \, a^{4} b x^{2} + a^{5}}{2 \, x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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