Optimal. Leaf size=100 \[ \frac{7}{3} a^2 b^6 x^{12}+\frac{28}{5} a^3 b^5 x^{10}+\frac{35}{4} a^4 b^4 x^8+\frac{28}{3} a^5 b^3 x^6+7 a^6 b^2 x^4+4 a^7 b x^2+a^8 \log (x)+\frac{4}{7} a b^7 x^{14}+\frac{b^8 x^{16}}{16} \]
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Rubi [A] time = 0.058515, antiderivative size = 100, 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} \[ \frac{7}{3} a^2 b^6 x^{12}+\frac{28}{5} a^3 b^5 x^{10}+\frac{35}{4} a^4 b^4 x^8+\frac{28}{3} a^5 b^3 x^6+7 a^6 b^2 x^4+4 a^7 b x^2+a^8 \log (x)+\frac{4}{7} a b^7 x^{14}+\frac{b^8 x^{16}}{16} \]
Antiderivative was successfully verified.
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Rule 266
Rule 43
Rubi steps
\begin{align*} \int \frac{\left (a+b x^2\right )^8}{x} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{(a+b x)^8}{x} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (8 a^7 b+\frac{a^8}{x}+28 a^6 b^2 x+56 a^5 b^3 x^2+70 a^4 b^4 x^3+56 a^3 b^5 x^4+28 a^2 b^6 x^5+8 a b^7 x^6+b^8 x^7\right ) \, dx,x,x^2\right )\\ &=4 a^7 b x^2+7 a^6 b^2 x^4+\frac{28}{3} a^5 b^3 x^6+\frac{35}{4} a^4 b^4 x^8+\frac{28}{5} a^3 b^5 x^{10}+\frac{7}{3} a^2 b^6 x^{12}+\frac{4}{7} a b^7 x^{14}+\frac{b^8 x^{16}}{16}+a^8 \log (x)\\ \end{align*}
Mathematica [A] time = 0.0041288, size = 100, normalized size = 1. \[ \frac{7}{3} a^2 b^6 x^{12}+\frac{28}{5} a^3 b^5 x^{10}+\frac{35}{4} a^4 b^4 x^8+\frac{28}{3} a^5 b^3 x^6+7 a^6 b^2 x^4+4 a^7 b x^2+a^8 \log (x)+\frac{4}{7} a b^7 x^{14}+\frac{b^8 x^{16}}{16} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 89, normalized size = 0.9 \begin{align*} 4\,{a}^{7}b{x}^{2}+7\,{a}^{6}{b}^{2}{x}^{4}+{\frac{28\,{a}^{5}{b}^{3}{x}^{6}}{3}}+{\frac{35\,{a}^{4}{b}^{4}{x}^{8}}{4}}+{\frac{28\,{a}^{3}{b}^{5}{x}^{10}}{5}}+{\frac{7\,{a}^{2}{b}^{6}{x}^{12}}{3}}+{\frac{4\,a{b}^{7}{x}^{14}}{7}}+{\frac{{b}^{8}{x}^{16}}{16}}+{a}^{8}\ln \left ( x \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 2.70948, size = 123, normalized size = 1.23 \begin{align*} \frac{1}{16} \, b^{8} x^{16} + \frac{4}{7} \, a b^{7} x^{14} + \frac{7}{3} \, a^{2} b^{6} x^{12} + \frac{28}{5} \, a^{3} b^{5} x^{10} + \frac{35}{4} \, a^{4} b^{4} x^{8} + \frac{28}{3} \, a^{5} b^{3} x^{6} + 7 \, a^{6} b^{2} x^{4} + 4 \, a^{7} b x^{2} + \frac{1}{2} \, a^{8} \log \left (x^{2}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.48633, size = 205, normalized size = 2.05 \begin{align*} \frac{1}{16} \, b^{8} x^{16} + \frac{4}{7} \, a b^{7} x^{14} + \frac{7}{3} \, a^{2} b^{6} x^{12} + \frac{28}{5} \, a^{3} b^{5} x^{10} + \frac{35}{4} \, a^{4} b^{4} x^{8} + \frac{28}{3} \, a^{5} b^{3} x^{6} + 7 \, a^{6} b^{2} x^{4} + 4 \, a^{7} b x^{2} + a^{8} \log \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.324676, size = 102, normalized size = 1.02 \begin{align*} a^{8} \log{\left (x \right )} + 4 a^{7} b x^{2} + 7 a^{6} b^{2} x^{4} + \frac{28 a^{5} b^{3} x^{6}}{3} + \frac{35 a^{4} b^{4} x^{8}}{4} + \frac{28 a^{3} b^{5} x^{10}}{5} + \frac{7 a^{2} b^{6} x^{12}}{3} + \frac{4 a b^{7} x^{14}}{7} + \frac{b^{8} x^{16}}{16} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 2.4002, size = 123, normalized size = 1.23 \begin{align*} \frac{1}{16} \, b^{8} x^{16} + \frac{4}{7} \, a b^{7} x^{14} + \frac{7}{3} \, a^{2} b^{6} x^{12} + \frac{28}{5} \, a^{3} b^{5} x^{10} + \frac{35}{4} \, a^{4} b^{4} x^{8} + \frac{28}{3} \, a^{5} b^{3} x^{6} + 7 \, a^{6} b^{2} x^{4} + 4 \, a^{7} b x^{2} + \frac{1}{2} \, a^{8} \log \left (x^{2}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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