Optimal. Leaf size=105 \[ \frac{14}{3} a^2 b^6 x^6+\frac{56}{3} a^3 b^5 x^3-\frac{56 a^5 b^3}{3 x^3}-\frac{14 a^6 b^2}{3 x^6}+70 a^4 b^4 \log (x)-\frac{8 a^7 b}{9 x^9}-\frac{a^8}{12 x^{12}}+\frac{8}{9} a b^7 x^9+\frac{b^8 x^{12}}{12} \]
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Rubi [A] time = 0.0560571, 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} \[ \frac{14}{3} a^2 b^6 x^6+\frac{56}{3} a^3 b^5 x^3-\frac{56 a^5 b^3}{3 x^3}-\frac{14 a^6 b^2}{3 x^6}+70 a^4 b^4 \log (x)-\frac{8 a^7 b}{9 x^9}-\frac{a^8}{12 x^{12}}+\frac{8}{9} a b^7 x^9+\frac{b^8 x^{12}}{12} \]
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^{13}} \, dx &=\frac{1}{3} \operatorname{Subst}\left (\int \frac{(a+b x)^8}{x^5} \, dx,x,x^3\right )\\ &=\frac{1}{3} \operatorname{Subst}\left (\int \left (56 a^3 b^5+\frac{a^8}{x^5}+\frac{8 a^7 b}{x^4}+\frac{28 a^6 b^2}{x^3}+\frac{56 a^5 b^3}{x^2}+\frac{70 a^4 b^4}{x}+28 a^2 b^6 x+8 a b^7 x^2+b^8 x^3\right ) \, dx,x,x^3\right )\\ &=-\frac{a^8}{12 x^{12}}-\frac{8 a^7 b}{9 x^9}-\frac{14 a^6 b^2}{3 x^6}-\frac{56 a^5 b^3}{3 x^3}+\frac{56}{3} a^3 b^5 x^3+\frac{14}{3} a^2 b^6 x^6+\frac{8}{9} a b^7 x^9+\frac{b^8 x^{12}}{12}+70 a^4 b^4 \log (x)\\ \end{align*}
Mathematica [A] time = 0.0052424, size = 105, normalized size = 1. \[ \frac{14}{3} a^2 b^6 x^6+\frac{56}{3} a^3 b^5 x^3-\frac{56 a^5 b^3}{3 x^3}-\frac{14 a^6 b^2}{3 x^6}+70 a^4 b^4 \log (x)-\frac{8 a^7 b}{9 x^9}-\frac{a^8}{12 x^{12}}+\frac{8}{9} a b^7 x^9+\frac{b^8 x^{12}}{12} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.007, size = 90, normalized size = 0.9 \begin{align*} -{\frac{{a}^{8}}{12\,{x}^{12}}}-{\frac{8\,{a}^{7}b}{9\,{x}^{9}}}-{\frac{14\,{a}^{6}{b}^{2}}{3\,{x}^{6}}}-{\frac{56\,{a}^{5}{b}^{3}}{3\,{x}^{3}}}+{\frac{56\,{a}^{3}{b}^{5}{x}^{3}}{3}}+{\frac{14\,{a}^{2}{b}^{6}{x}^{6}}{3}}+{\frac{8\,a{b}^{7}{x}^{9}}{9}}+{\frac{{b}^{8}{x}^{12}}{12}}+70\,{a}^{4}{b}^{4}\ln \left ( x \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.981447, size = 127, normalized size = 1.21 \begin{align*} \frac{1}{12} \, b^{8} x^{12} + \frac{8}{9} \, a b^{7} x^{9} + \frac{14}{3} \, a^{2} b^{6} x^{6} + \frac{56}{3} \, a^{3} b^{5} x^{3} + \frac{70}{3} \, a^{4} b^{4} \log \left (x^{3}\right ) - \frac{672 \, a^{5} b^{3} x^{9} + 168 \, a^{6} b^{2} x^{6} + 32 \, a^{7} b x^{3} + 3 \, a^{8}}{36 \, x^{12}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.58588, size = 221, normalized size = 2.1 \begin{align*} \frac{3 \, b^{8} x^{24} + 32 \, a b^{7} x^{21} + 168 \, a^{2} b^{6} x^{18} + 672 \, a^{3} b^{5} x^{15} + 2520 \, a^{4} b^{4} x^{12} \log \left (x\right ) - 672 \, a^{5} b^{3} x^{9} - 168 \, a^{6} b^{2} x^{6} - 32 \, a^{7} b x^{3} - 3 \, a^{8}}{36 \, x^{12}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.852344, size = 102, normalized size = 0.97 \begin{align*} 70 a^{4} b^{4} \log{\left (x \right )} + \frac{56 a^{3} b^{5} x^{3}}{3} + \frac{14 a^{2} b^{6} x^{6}}{3} + \frac{8 a b^{7} x^{9}}{9} + \frac{b^{8} x^{12}}{12} - \frac{3 a^{8} + 32 a^{7} b x^{3} + 168 a^{6} b^{2} x^{6} + 672 a^{5} b^{3} x^{9}}{36 x^{12}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.12443, size = 140, normalized size = 1.33 \begin{align*} \frac{1}{12} \, b^{8} x^{12} + \frac{8}{9} \, a b^{7} x^{9} + \frac{14}{3} \, a^{2} b^{6} x^{6} + \frac{56}{3} \, a^{3} b^{5} x^{3} + 70 \, a^{4} b^{4} \log \left ({\left | x \right |}\right ) - \frac{1750 \, a^{4} b^{4} x^{12} + 672 \, a^{5} b^{3} x^{9} + 168 \, a^{6} b^{2} x^{6} + 32 \, a^{7} b x^{3} + 3 \, a^{8}}{36 \, x^{12}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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