Optimal. Leaf size=105 \[ \frac{28}{9} a^2 b^6 x^9+\frac{28}{3} a^3 b^5 x^6+\frac{70}{3} a^4 b^4 x^3-\frac{28 a^6 b^2}{3 x^3}+56 a^5 b^3 \log (x)-\frac{4 a^7 b}{3 x^6}-\frac{a^8}{9 x^9}+\frac{2}{3} a b^7 x^{12}+\frac{b^8 x^{15}}{15} \]
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Rubi [A] time = 0.0556131, antiderivative size = 105, 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} \[ \frac{28}{9} a^2 b^6 x^9+\frac{28}{3} a^3 b^5 x^6+\frac{70}{3} a^4 b^4 x^3-\frac{28 a^6 b^2}{3 x^3}+56 a^5 b^3 \log (x)-\frac{4 a^7 b}{3 x^6}-\frac{a^8}{9 x^9}+\frac{2}{3} a b^7 x^{12}+\frac{b^8 x^{15}}{15} \]
Antiderivative was successfully verified.
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Rule 266
Rule 43
Rubi steps
\begin{align*} \int \frac{\left (a+b x^3\right )^8}{x^{10}} \, dx &=\frac{1}{3} \operatorname{Subst}\left (\int \frac{(a+b x)^8}{x^4} \, dx,x,x^3\right )\\ &=\frac{1}{3} \operatorname{Subst}\left (\int \left (70 a^4 b^4+\frac{a^8}{x^4}+\frac{8 a^7 b}{x^3}+\frac{28 a^6 b^2}{x^2}+\frac{56 a^5 b^3}{x}+56 a^3 b^5 x+28 a^2 b^6 x^2+8 a b^7 x^3+b^8 x^4\right ) \, dx,x,x^3\right )\\ &=-\frac{a^8}{9 x^9}-\frac{4 a^7 b}{3 x^6}-\frac{28 a^6 b^2}{3 x^3}+\frac{70}{3} a^4 b^4 x^3+\frac{28}{3} a^3 b^5 x^6+\frac{28}{9} a^2 b^6 x^9+\frac{2}{3} a b^7 x^{12}+\frac{b^8 x^{15}}{15}+56 a^5 b^3 \log (x)\\ \end{align*}
Mathematica [A] time = 0.0100133, size = 105, normalized size = 1. \[ \frac{28}{9} a^2 b^6 x^9+\frac{28}{3} a^3 b^5 x^6+\frac{70}{3} a^4 b^4 x^3-\frac{28 a^6 b^2}{3 x^3}+56 a^5 b^3 \log (x)-\frac{4 a^7 b}{3 x^6}-\frac{a^8}{9 x^9}+\frac{2}{3} a b^7 x^{12}+\frac{b^8 x^{15}}{15} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 90, normalized size = 0.9 \begin{align*} -{\frac{{a}^{8}}{9\,{x}^{9}}}-{\frac{4\,{a}^{7}b}{3\,{x}^{6}}}-{\frac{28\,{a}^{6}{b}^{2}}{3\,{x}^{3}}}+{\frac{70\,{a}^{4}{b}^{4}{x}^{3}}{3}}+{\frac{28\,{a}^{3}{b}^{5}{x}^{6}}{3}}+{\frac{28\,{a}^{2}{b}^{6}{x}^{9}}{9}}+{\frac{2\,a{b}^{7}{x}^{12}}{3}}+{\frac{{b}^{8}{x}^{15}}{15}}+56\,{a}^{5}{b}^{3}\ln \left ( x \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.955577, size = 124, normalized size = 1.18 \begin{align*} \frac{1}{15} \, b^{8} x^{15} + \frac{2}{3} \, a b^{7} x^{12} + \frac{28}{9} \, a^{2} b^{6} x^{9} + \frac{28}{3} \, a^{3} b^{5} x^{6} + \frac{70}{3} \, a^{4} b^{4} x^{3} + \frac{56}{3} \, a^{5} b^{3} \log \left (x^{3}\right ) - \frac{84 \, a^{6} b^{2} x^{6} + 12 \, a^{7} b x^{3} + a^{8}}{9 \, x^{9}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.62914, size = 221, normalized size = 2.1 \begin{align*} \frac{3 \, b^{8} x^{24} + 30 \, a b^{7} x^{21} + 140 \, a^{2} b^{6} x^{18} + 420 \, a^{3} b^{5} x^{15} + 1050 \, a^{4} b^{4} x^{12} + 2520 \, a^{5} b^{3} x^{9} \log \left (x\right ) - 420 \, a^{6} b^{2} x^{6} - 60 \, a^{7} b x^{3} - 5 \, a^{8}}{45 \, x^{9}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.686961, size = 102, normalized size = 0.97 \begin{align*} 56 a^{5} b^{3} \log{\left (x \right )} + \frac{70 a^{4} b^{4} x^{3}}{3} + \frac{28 a^{3} b^{5} x^{6}}{3} + \frac{28 a^{2} b^{6} x^{9}}{9} + \frac{2 a b^{7} x^{12}}{3} + \frac{b^{8} x^{15}}{15} - \frac{a^{8} + 12 a^{7} b x^{3} + 84 a^{6} b^{2} x^{6}}{9 x^{9}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.23617, size = 138, normalized size = 1.31 \begin{align*} \frac{1}{15} \, b^{8} x^{15} + \frac{2}{3} \, a b^{7} x^{12} + \frac{28}{9} \, a^{2} b^{6} x^{9} + \frac{28}{3} \, a^{3} b^{5} x^{6} + \frac{70}{3} \, a^{4} b^{4} x^{3} + 56 \, a^{5} b^{3} \log \left ({\left | x \right |}\right ) - \frac{308 \, a^{5} b^{3} x^{9} + 84 \, a^{6} b^{2} x^{6} + 12 \, a^{7} b x^{3} + a^{8}}{9 \, x^{9}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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