Optimal. Leaf size=99 \[ \frac{14}{5} a^2 b^6 x^{10}+7 a^3 b^5 x^8+\frac{35}{3} a^4 b^4 x^6+14 a^5 b^3 x^4+14 a^6 b^2 x^2+8 a^7 b \log (x)-\frac{a^8}{2 x^2}+\frac{2}{3} a b^7 x^{12}+\frac{b^8 x^{14}}{14} \]
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Rubi [A] time = 0.0599015, antiderivative size = 99, 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} \[ \frac{14}{5} a^2 b^6 x^{10}+7 a^3 b^5 x^8+\frac{35}{3} a^4 b^4 x^6+14 a^5 b^3 x^4+14 a^6 b^2 x^2+8 a^7 b \log (x)-\frac{a^8}{2 x^2}+\frac{2}{3} a b^7 x^{12}+\frac{b^8 x^{14}}{14} \]
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^3} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{(a+b x)^8}{x^2} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (28 a^6 b^2+\frac{a^8}{x^2}+\frac{8 a^7 b}{x}+56 a^5 b^3 x+70 a^4 b^4 x^2+56 a^3 b^5 x^3+28 a^2 b^6 x^4+8 a b^7 x^5+b^8 x^6\right ) \, dx,x,x^2\right )\\ &=-\frac{a^8}{2 x^2}+14 a^6 b^2 x^2+14 a^5 b^3 x^4+\frac{35}{3} a^4 b^4 x^6+7 a^3 b^5 x^8+\frac{14}{5} a^2 b^6 x^{10}+\frac{2}{3} a b^7 x^{12}+\frac{b^8 x^{14}}{14}+8 a^7 b \log (x)\\ \end{align*}
Mathematica [A] time = 0.0044414, size = 99, normalized size = 1. \[ \frac{14}{5} a^2 b^6 x^{10}+7 a^3 b^5 x^8+\frac{35}{3} a^4 b^4 x^6+14 a^5 b^3 x^4+14 a^6 b^2 x^2+8 a^7 b \log (x)-\frac{a^8}{2 x^2}+\frac{2}{3} a b^7 x^{12}+\frac{b^8 x^{14}}{14} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.007, size = 90, normalized size = 0.9 \begin{align*} -{\frac{{a}^{8}}{2\,{x}^{2}}}+14\,{a}^{6}{b}^{2}{x}^{2}+14\,{a}^{5}{b}^{3}{x}^{4}+{\frac{35\,{a}^{4}{b}^{4}{x}^{6}}{3}}+7\,{a}^{3}{b}^{5}{x}^{8}+{\frac{14\,{a}^{2}{b}^{6}{x}^{10}}{5}}+{\frac{2\,a{b}^{7}{x}^{12}}{3}}+{\frac{{b}^{8}{x}^{14}}{14}}+8\,{a}^{7}b\ln \left ( x \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.75426, size = 123, normalized size = 1.24 \begin{align*} \frac{1}{14} \, b^{8} x^{14} + \frac{2}{3} \, a b^{7} x^{12} + \frac{14}{5} \, a^{2} b^{6} x^{10} + 7 \, a^{3} b^{5} x^{8} + \frac{35}{3} \, a^{4} b^{4} x^{6} + 14 \, a^{5} b^{3} x^{4} + 14 \, a^{6} b^{2} x^{2} + 4 \, a^{7} b \log \left (x^{2}\right ) - \frac{a^{8}}{2 \, x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.50534, size = 232, normalized size = 2.34 \begin{align*} \frac{15 \, b^{8} x^{16} + 140 \, a b^{7} x^{14} + 588 \, a^{2} b^{6} x^{12} + 1470 \, a^{3} b^{5} x^{10} + 2450 \, a^{4} b^{4} x^{8} + 2940 \, a^{5} b^{3} x^{6} + 2940 \, a^{6} b^{2} x^{4} + 1680 \, a^{7} b x^{2} \log \left (x\right ) - 105 \, a^{8}}{210 \, x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.371833, size = 100, normalized size = 1.01 \begin{align*} - \frac{a^{8}}{2 x^{2}} + 8 a^{7} b \log{\left (x \right )} + 14 a^{6} b^{2} x^{2} + 14 a^{5} b^{3} x^{4} + \frac{35 a^{4} b^{4} x^{6}}{3} + 7 a^{3} b^{5} x^{8} + \frac{14 a^{2} b^{6} x^{10}}{5} + \frac{2 a b^{7} x^{12}}{3} + \frac{b^{8} x^{14}}{14} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 2.84298, size = 136, normalized size = 1.37 \begin{align*} \frac{1}{14} \, b^{8} x^{14} + \frac{2}{3} \, a b^{7} x^{12} + \frac{14}{5} \, a^{2} b^{6} x^{10} + 7 \, a^{3} b^{5} x^{8} + \frac{35}{3} \, a^{4} b^{4} x^{6} + 14 \, a^{5} b^{3} x^{4} + 14 \, a^{6} b^{2} x^{2} + 4 \, a^{7} b \log \left (x^{2}\right ) - \frac{8 \, a^{7} b x^{2} + a^{8}}{2 \, x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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