Optimal. Leaf size=105 \[ -\frac{7 a^6 b^2}{3 x^{12}}-\frac{56 a^5 b^3}{9 x^9}-\frac{35 a^4 b^4}{3 x^6}-\frac{56 a^3 b^5}{3 x^3}+28 a^2 b^6 \log (x)-\frac{8 a^7 b}{15 x^{15}}-\frac{a^8}{18 x^{18}}+\frac{8}{3} a b^7 x^3+\frac{b^8 x^6}{6} \]
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Rubi [A] time = 0.0553437, 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} \[ -\frac{7 a^6 b^2}{3 x^{12}}-\frac{56 a^5 b^3}{9 x^9}-\frac{35 a^4 b^4}{3 x^6}-\frac{56 a^3 b^5}{3 x^3}+28 a^2 b^6 \log (x)-\frac{8 a^7 b}{15 x^{15}}-\frac{a^8}{18 x^{18}}+\frac{8}{3} a b^7 x^3+\frac{b^8 x^6}{6} \]
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^{19}} \, dx &=\frac{1}{3} \operatorname{Subst}\left (\int \frac{(a+b x)^8}{x^7} \, dx,x,x^3\right )\\ &=\frac{1}{3} \operatorname{Subst}\left (\int \left (8 a b^7+\frac{a^8}{x^7}+\frac{8 a^7 b}{x^6}+\frac{28 a^6 b^2}{x^5}+\frac{56 a^5 b^3}{x^4}+\frac{70 a^4 b^4}{x^3}+\frac{56 a^3 b^5}{x^2}+\frac{28 a^2 b^6}{x}+b^8 x\right ) \, dx,x,x^3\right )\\ &=-\frac{a^8}{18 x^{18}}-\frac{8 a^7 b}{15 x^{15}}-\frac{7 a^6 b^2}{3 x^{12}}-\frac{56 a^5 b^3}{9 x^9}-\frac{35 a^4 b^4}{3 x^6}-\frac{56 a^3 b^5}{3 x^3}+\frac{8}{3} a b^7 x^3+\frac{b^8 x^6}{6}+28 a^2 b^6 \log (x)\\ \end{align*}
Mathematica [A] time = 0.0049905, size = 105, normalized size = 1. \[ -\frac{7 a^6 b^2}{3 x^{12}}-\frac{56 a^5 b^3}{9 x^9}-\frac{35 a^4 b^4}{3 x^6}-\frac{56 a^3 b^5}{3 x^3}+28 a^2 b^6 \log (x)-\frac{8 a^7 b}{15 x^{15}}-\frac{a^8}{18 x^{18}}+\frac{8}{3} a b^7 x^3+\frac{b^8 x^6}{6} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 90, normalized size = 0.9 \begin{align*} -{\frac{{a}^{8}}{18\,{x}^{18}}}-{\frac{8\,{a}^{7}b}{15\,{x}^{15}}}-{\frac{7\,{a}^{6}{b}^{2}}{3\,{x}^{12}}}-{\frac{56\,{a}^{5}{b}^{3}}{9\,{x}^{9}}}-{\frac{35\,{a}^{4}{b}^{4}}{3\,{x}^{6}}}-{\frac{56\,{a}^{3}{b}^{5}}{3\,{x}^{3}}}+{\frac{8\,a{b}^{7}{x}^{3}}{3}}+{\frac{{b}^{8}{x}^{6}}{6}}+28\,{a}^{2}{b}^{6}\ln \left ( x \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.980081, size = 127, normalized size = 1.21 \begin{align*} \frac{1}{6} \, b^{8} x^{6} + \frac{8}{3} \, a b^{7} x^{3} + \frac{28}{3} \, a^{2} b^{6} \log \left (x^{3}\right ) - \frac{1680 \, a^{3} b^{5} x^{15} + 1050 \, a^{4} b^{4} x^{12} + 560 \, a^{5} b^{3} x^{9} + 210 \, a^{6} b^{2} x^{6} + 48 \, a^{7} b x^{3} + 5 \, a^{8}}{90 \, x^{18}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.67491, size = 227, normalized size = 2.16 \begin{align*} \frac{15 \, b^{8} x^{24} + 240 \, a b^{7} x^{21} + 2520 \, a^{2} b^{6} x^{18} \log \left (x\right ) - 1680 \, a^{3} b^{5} x^{15} - 1050 \, a^{4} b^{4} x^{12} - 560 \, a^{5} b^{3} x^{9} - 210 \, a^{6} b^{2} x^{6} - 48 \, a^{7} b x^{3} - 5 \, a^{8}}{90 \, x^{18}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.23993, size = 99, normalized size = 0.94 \begin{align*} 28 a^{2} b^{6} \log{\left (x \right )} + \frac{8 a b^{7} x^{3}}{3} + \frac{b^{8} x^{6}}{6} - \frac{5 a^{8} + 48 a^{7} b x^{3} + 210 a^{6} b^{2} x^{6} + 560 a^{5} b^{3} x^{9} + 1050 a^{4} b^{4} x^{12} + 1680 a^{3} b^{5} x^{15}}{90 x^{18}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.12412, size = 140, normalized size = 1.33 \begin{align*} \frac{1}{6} \, b^{8} x^{6} + \frac{8}{3} \, a b^{7} x^{3} + 28 \, a^{2} b^{6} \log \left ({\left | x \right |}\right ) - \frac{2058 \, a^{2} b^{6} x^{18} + 1680 \, a^{3} b^{5} x^{15} + 1050 \, a^{4} b^{4} x^{12} + 560 \, a^{5} b^{3} x^{9} + 210 \, a^{6} b^{2} x^{6} + 48 \, a^{7} b x^{3} + 5 \, a^{8}}{90 \, x^{18}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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