Optimal. Leaf size=104 \[ \frac{14}{9} a^2 b^6 x^{18}+\frac{56}{15} a^3 b^5 x^{15}+\frac{35}{6} a^4 b^4 x^{12}+\frac{56}{9} a^5 b^3 x^9+\frac{14}{3} a^6 b^2 x^6+\frac{8}{3} a^7 b x^3+a^8 \log (x)+\frac{8}{21} a b^7 x^{21}+\frac{b^8 x^{24}}{24} \]
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Rubi [A] time = 0.0551169, antiderivative size = 104, 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} \[ \frac{14}{9} a^2 b^6 x^{18}+\frac{56}{15} a^3 b^5 x^{15}+\frac{35}{6} a^4 b^4 x^{12}+\frac{56}{9} a^5 b^3 x^9+\frac{14}{3} a^6 b^2 x^6+\frac{8}{3} a^7 b x^3+a^8 \log (x)+\frac{8}{21} a b^7 x^{21}+\frac{b^8 x^{24}}{24} \]
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} \, dx &=\frac{1}{3} \operatorname{Subst}\left (\int \frac{(a+b x)^8}{x} \, dx,x,x^3\right )\\ &=\frac{1}{3} \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^3\right )\\ &=\frac{8}{3} a^7 b x^3+\frac{14}{3} a^6 b^2 x^6+\frac{56}{9} a^5 b^3 x^9+\frac{35}{6} a^4 b^4 x^{12}+\frac{56}{15} a^3 b^5 x^{15}+\frac{14}{9} a^2 b^6 x^{18}+\frac{8}{21} a b^7 x^{21}+\frac{b^8 x^{24}}{24}+a^8 \log (x)\\ \end{align*}
Mathematica [A] time = 0.0044295, size = 104, normalized size = 1. \[ \frac{14}{9} a^2 b^6 x^{18}+\frac{56}{15} a^3 b^5 x^{15}+\frac{35}{6} a^4 b^4 x^{12}+\frac{56}{9} a^5 b^3 x^9+\frac{14}{3} a^6 b^2 x^6+\frac{8}{3} a^7 b x^3+a^8 \log (x)+\frac{8}{21} a b^7 x^{21}+\frac{b^8 x^{24}}{24} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 89, normalized size = 0.9 \begin{align*}{\frac{8\,{a}^{7}b{x}^{3}}{3}}+{\frac{14\,{a}^{6}{b}^{2}{x}^{6}}{3}}+{\frac{56\,{a}^{5}{b}^{3}{x}^{9}}{9}}+{\frac{35\,{a}^{4}{b}^{4}{x}^{12}}{6}}+{\frac{56\,{a}^{3}{b}^{5}{x}^{15}}{15}}+{\frac{14\,{a}^{2}{b}^{6}{x}^{18}}{9}}+{\frac{8\,a{b}^{7}{x}^{21}}{21}}+{\frac{{b}^{8}{x}^{24}}{24}}+{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 = 0.952949, size = 123, normalized size = 1.18 \begin{align*} \frac{1}{24} \, b^{8} x^{24} + \frac{8}{21} \, a b^{7} x^{21} + \frac{14}{9} \, a^{2} b^{6} x^{18} + \frac{56}{15} \, a^{3} b^{5} x^{15} + \frac{35}{6} \, a^{4} b^{4} x^{12} + \frac{56}{9} \, a^{5} b^{3} x^{9} + \frac{14}{3} \, a^{6} b^{2} x^{6} + \frac{8}{3} \, a^{7} b x^{3} + \frac{1}{3} \, a^{8} \log \left (x^{3}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.74281, size = 217, normalized size = 2.09 \begin{align*} \frac{1}{24} \, b^{8} x^{24} + \frac{8}{21} \, a b^{7} x^{21} + \frac{14}{9} \, a^{2} b^{6} x^{18} + \frac{56}{15} \, a^{3} b^{5} x^{15} + \frac{35}{6} \, a^{4} b^{4} x^{12} + \frac{56}{9} \, a^{5} b^{3} x^{9} + \frac{14}{3} \, a^{6} b^{2} x^{6} + \frac{8}{3} \, a^{7} b x^{3} + 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 = 0.428049, size = 105, normalized size = 1.01 \begin{align*} a^{8} \log{\left (x \right )} + \frac{8 a^{7} b x^{3}}{3} + \frac{14 a^{6} b^{2} x^{6}}{3} + \frac{56 a^{5} b^{3} x^{9}}{9} + \frac{35 a^{4} b^{4} x^{12}}{6} + \frac{56 a^{3} b^{5} x^{15}}{15} + \frac{14 a^{2} b^{6} x^{18}}{9} + \frac{8 a b^{7} x^{21}}{21} + \frac{b^{8} x^{24}}{24} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.09286, size = 120, normalized size = 1.15 \begin{align*} \frac{1}{24} \, b^{8} x^{24} + \frac{8}{21} \, a b^{7} x^{21} + \frac{14}{9} \, a^{2} b^{6} x^{18} + \frac{56}{15} \, a^{3} b^{5} x^{15} + \frac{35}{6} \, a^{4} b^{4} x^{12} + \frac{56}{9} \, a^{5} b^{3} x^{9} + \frac{14}{3} \, a^{6} b^{2} x^{6} + \frac{8}{3} \, a^{7} b x^{3} + a^{8} \log \left ({\left | x \right |}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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