Optimal. Leaf size=105 \[ \frac{28}{3} a^2 b^6 x^3-\frac{70 a^4 b^4}{3 x^3}-\frac{28 a^5 b^3}{3 x^6}-\frac{28 a^6 b^2}{9 x^9}+56 a^3 b^5 \log (x)-\frac{2 a^7 b}{3 x^{12}}-\frac{a^8}{15 x^{15}}+\frac{4}{3} a b^7 x^6+\frac{b^8 x^9}{9} \]
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Rubi [A] time = 0.0598809, 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} \[ \frac{28}{3} a^2 b^6 x^3-\frac{70 a^4 b^4}{3 x^3}-\frac{28 a^5 b^3}{3 x^6}-\frac{28 a^6 b^2}{9 x^9}+56 a^3 b^5 \log (x)-\frac{2 a^7 b}{3 x^{12}}-\frac{a^8}{15 x^{15}}+\frac{4}{3} a b^7 x^6+\frac{b^8 x^9}{9} \]
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^{16}} \, dx &=\frac{1}{3} \operatorname{Subst}\left (\int \frac{(a+b x)^8}{x^6} \, dx,x,x^3\right )\\ &=\frac{1}{3} \operatorname{Subst}\left (\int \left (28 a^2 b^6+\frac{a^8}{x^6}+\frac{8 a^7 b}{x^5}+\frac{28 a^6 b^2}{x^4}+\frac{56 a^5 b^3}{x^3}+\frac{70 a^4 b^4}{x^2}+\frac{56 a^3 b^5}{x}+8 a b^7 x+b^8 x^2\right ) \, dx,x,x^3\right )\\ &=-\frac{a^8}{15 x^{15}}-\frac{2 a^7 b}{3 x^{12}}-\frac{28 a^6 b^2}{9 x^9}-\frac{28 a^5 b^3}{3 x^6}-\frac{70 a^4 b^4}{3 x^3}+\frac{28}{3} a^2 b^6 x^3+\frac{4}{3} a b^7 x^6+\frac{b^8 x^9}{9}+56 a^3 b^5 \log (x)\\ \end{align*}
Mathematica [A] time = 0.0068886, size = 105, normalized size = 1. \[ \frac{28}{3} a^2 b^6 x^3-\frac{70 a^4 b^4}{3 x^3}-\frac{28 a^5 b^3}{3 x^6}-\frac{28 a^6 b^2}{9 x^9}+56 a^3 b^5 \log (x)-\frac{2 a^7 b}{3 x^{12}}-\frac{a^8}{15 x^{15}}+\frac{4}{3} a b^7 x^6+\frac{b^8 x^9}{9} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.007, size = 90, normalized size = 0.9 \begin{align*} -{\frac{{a}^{8}}{15\,{x}^{15}}}-{\frac{2\,{a}^{7}b}{3\,{x}^{12}}}-{\frac{28\,{a}^{6}{b}^{2}}{9\,{x}^{9}}}-{\frac{28\,{a}^{5}{b}^{3}}{3\,{x}^{6}}}-{\frac{70\,{a}^{4}{b}^{4}}{3\,{x}^{3}}}+{\frac{28\,{a}^{2}{b}^{6}{x}^{3}}{3}}+{\frac{4\,a{b}^{7}{x}^{6}}{3}}+{\frac{{b}^{8}{x}^{9}}{9}}+56\,{a}^{3}{b}^{5}\ln \left ( x \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.965338, size = 127, normalized size = 1.21 \begin{align*} \frac{1}{9} \, b^{8} x^{9} + \frac{4}{3} \, a b^{7} x^{6} + \frac{28}{3} \, a^{2} b^{6} x^{3} + \frac{56}{3} \, a^{3} b^{5} \log \left (x^{3}\right ) - \frac{1050 \, a^{4} b^{4} x^{12} + 420 \, a^{5} b^{3} x^{9} + 140 \, a^{6} b^{2} x^{6} + 30 \, a^{7} b x^{3} + 3 \, a^{8}}{45 \, x^{15}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.6638, size = 223, normalized size = 2.12 \begin{align*} \frac{5 \, b^{8} x^{24} + 60 \, a b^{7} x^{21} + 420 \, a^{2} b^{6} x^{18} + 2520 \, a^{3} b^{5} x^{15} \log \left (x\right ) - 1050 \, a^{4} b^{4} x^{12} - 420 \, a^{5} b^{3} x^{9} - 140 \, a^{6} b^{2} x^{6} - 30 \, a^{7} b x^{3} - 3 \, a^{8}}{45 \, x^{15}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.05132, size = 100, normalized size = 0.95 \begin{align*} 56 a^{3} b^{5} \log{\left (x \right )} + \frac{28 a^{2} b^{6} x^{3}}{3} + \frac{4 a b^{7} x^{6}}{3} + \frac{b^{8} x^{9}}{9} - \frac{3 a^{8} + 30 a^{7} b x^{3} + 140 a^{6} b^{2} x^{6} + 420 a^{5} b^{3} x^{9} + 1050 a^{4} b^{4} x^{12}}{45 x^{15}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.1516, size = 140, normalized size = 1.33 \begin{align*} \frac{1}{9} \, b^{8} x^{9} + \frac{4}{3} \, a b^{7} x^{6} + \frac{28}{3} \, a^{2} b^{6} x^{3} + 56 \, a^{3} b^{5} \log \left ({\left | x \right |}\right ) - \frac{1918 \, a^{3} b^{5} x^{15} + 1050 \, a^{4} b^{4} x^{12} + 420 \, a^{5} b^{3} x^{9} + 140 \, a^{6} b^{2} x^{6} + 30 \, a^{7} b x^{3} + 3 \, a^{8}}{45 \, x^{15}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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