Optimal. Leaf size=105 \[ \frac{7}{3} a^2 b^6 x^{12}+\frac{56}{9} a^3 b^5 x^9+\frac{35}{3} a^4 b^4 x^6+\frac{56}{3} a^5 b^3 x^3+28 a^6 b^2 \log (x)-\frac{8 a^7 b}{3 x^3}-\frac{a^8}{6 x^6}+\frac{8}{15} a b^7 x^{15}+\frac{b^8 x^{18}}{18} \]
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Rubi [A] time = 0.0584267, 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} \[ \frac{7}{3} a^2 b^6 x^{12}+\frac{56}{9} a^3 b^5 x^9+\frac{35}{3} a^4 b^4 x^6+\frac{56}{3} a^5 b^3 x^3+28 a^6 b^2 \log (x)-\frac{8 a^7 b}{3 x^3}-\frac{a^8}{6 x^6}+\frac{8}{15} a b^7 x^{15}+\frac{b^8 x^{18}}{18} \]
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^7} \, dx &=\frac{1}{3} \operatorname{Subst}\left (\int \frac{(a+b x)^8}{x^3} \, dx,x,x^3\right )\\ &=\frac{1}{3} \operatorname{Subst}\left (\int \left (56 a^5 b^3+\frac{a^8}{x^3}+\frac{8 a^7 b}{x^2}+\frac{28 a^6 b^2}{x}+70 a^4 b^4 x+56 a^3 b^5 x^2+28 a^2 b^6 x^3+8 a b^7 x^4+b^8 x^5\right ) \, dx,x,x^3\right )\\ &=-\frac{a^8}{6 x^6}-\frac{8 a^7 b}{3 x^3}+\frac{56}{3} a^5 b^3 x^3+\frac{35}{3} a^4 b^4 x^6+\frac{56}{9} a^3 b^5 x^9+\frac{7}{3} a^2 b^6 x^{12}+\frac{8}{15} a b^7 x^{15}+\frac{b^8 x^{18}}{18}+28 a^6 b^2 \log (x)\\ \end{align*}
Mathematica [A] time = 0.0049907, size = 105, normalized size = 1. \[ \frac{7}{3} a^2 b^6 x^{12}+\frac{56}{9} a^3 b^5 x^9+\frac{35}{3} a^4 b^4 x^6+\frac{56}{3} a^5 b^3 x^3+28 a^6 b^2 \log (x)-\frac{8 a^7 b}{3 x^3}-\frac{a^8}{6 x^6}+\frac{8}{15} a b^7 x^{15}+\frac{b^8 x^{18}}{18} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 90, normalized size = 0.9 \begin{align*} -{\frac{{a}^{8}}{6\,{x}^{6}}}-{\frac{8\,{a}^{7}b}{3\,{x}^{3}}}+{\frac{56\,{a}^{5}{b}^{3}{x}^{3}}{3}}+{\frac{35\,{a}^{4}{b}^{4}{x}^{6}}{3}}+{\frac{56\,{a}^{3}{b}^{5}{x}^{9}}{9}}+{\frac{7\,{a}^{2}{b}^{6}{x}^{12}}{3}}+{\frac{8\,a{b}^{7}{x}^{15}}{15}}+{\frac{{b}^{8}{x}^{18}}{18}}+28\,{a}^{6}{b}^{2}\ln \left ( x \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.983206, size = 124, normalized size = 1.18 \begin{align*} \frac{1}{18} \, b^{8} x^{18} + \frac{8}{15} \, a b^{7} x^{15} + \frac{7}{3} \, a^{2} b^{6} x^{12} + \frac{56}{9} \, a^{3} b^{5} x^{9} + \frac{35}{3} \, a^{4} b^{4} x^{6} + \frac{56}{3} \, a^{5} b^{3} x^{3} + \frac{28}{3} \, a^{6} b^{2} \log \left (x^{3}\right ) - \frac{16 \, a^{7} b x^{3} + a^{8}}{6 \, x^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.58322, size = 225, normalized size = 2.14 \begin{align*} \frac{5 \, b^{8} x^{24} + 48 \, a b^{7} x^{21} + 210 \, a^{2} b^{6} x^{18} + 560 \, a^{3} b^{5} x^{15} + 1050 \, a^{4} b^{4} x^{12} + 1680 \, a^{5} b^{3} x^{9} + 2520 \, a^{6} b^{2} x^{6} \log \left (x\right ) - 240 \, a^{7} b x^{3} - 15 \, a^{8}}{90 \, x^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.591675, size = 104, normalized size = 0.99 \begin{align*} 28 a^{6} b^{2} \log{\left (x \right )} + \frac{56 a^{5} b^{3} x^{3}}{3} + \frac{35 a^{4} b^{4} x^{6}}{3} + \frac{56 a^{3} b^{5} x^{9}}{9} + \frac{7 a^{2} b^{6} x^{12}}{3} + \frac{8 a b^{7} x^{15}}{15} + \frac{b^{8} x^{18}}{18} - \frac{a^{8} + 16 a^{7} b x^{3}}{6 x^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.62545, size = 138, normalized size = 1.31 \begin{align*} \frac{1}{18} \, b^{8} x^{18} + \frac{8}{15} \, a b^{7} x^{15} + \frac{7}{3} \, a^{2} b^{6} x^{12} + \frac{56}{9} \, a^{3} b^{5} x^{9} + \frac{35}{3} \, a^{4} b^{4} x^{6} + \frac{56}{3} \, a^{5} b^{3} x^{3} + 28 \, a^{6} b^{2} \log \left ({\left | x \right |}\right ) - \frac{84 \, a^{6} b^{2} x^{6} + 16 \, a^{7} b x^{3} + a^{8}}{6 \, x^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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