Optimal. Leaf size=105 \[ \frac{28}{15} a^2 b^6 x^{15}+\frac{14}{3} a^3 b^5 x^{12}+\frac{70}{9} a^4 b^4 x^9+\frac{28}{3} a^5 b^3 x^6+\frac{28}{3} a^6 b^2 x^3+8 a^7 b \log (x)-\frac{a^8}{3 x^3}+\frac{4}{9} a b^7 x^{18}+\frac{b^8 x^{21}}{21} \]
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Rubi [A] time = 0.0582046, 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} \[ \frac{28}{15} a^2 b^6 x^{15}+\frac{14}{3} a^3 b^5 x^{12}+\frac{70}{9} a^4 b^4 x^9+\frac{28}{3} a^5 b^3 x^6+\frac{28}{3} a^6 b^2 x^3+8 a^7 b \log (x)-\frac{a^8}{3 x^3}+\frac{4}{9} a b^7 x^{18}+\frac{b^8 x^{21}}{21} \]
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^4} \, dx &=\frac{1}{3} \operatorname{Subst}\left (\int \frac{(a+b x)^8}{x^2} \, dx,x,x^3\right )\\ &=\frac{1}{3} \operatorname{Subst}\left (\int \left (28 a^6 b^2+\frac{a^8}{x^2}+\frac{8 a^7 b}{x}+56 a^5 b^3 x+70 a^4 b^4 x^2+56 a^3 b^5 x^3+28 a^2 b^6 x^4+8 a b^7 x^5+b^8 x^6\right ) \, dx,x,x^3\right )\\ &=-\frac{a^8}{3 x^3}+\frac{28}{3} a^6 b^2 x^3+\frac{28}{3} a^5 b^3 x^6+\frac{70}{9} a^4 b^4 x^9+\frac{14}{3} a^3 b^5 x^{12}+\frac{28}{15} a^2 b^6 x^{15}+\frac{4}{9} a b^7 x^{18}+\frac{b^8 x^{21}}{21}+8 a^7 b \log (x)\\ \end{align*}
Mathematica [A] time = 0.0088502, size = 105, normalized size = 1. \[ \frac{28}{15} a^2 b^6 x^{15}+\frac{14}{3} a^3 b^5 x^{12}+\frac{70}{9} a^4 b^4 x^9+\frac{28}{3} a^5 b^3 x^6+\frac{28}{3} a^6 b^2 x^3+8 a^7 b \log (x)-\frac{a^8}{3 x^3}+\frac{4}{9} a b^7 x^{18}+\frac{b^8 x^{21}}{21} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 90, normalized size = 0.9 \begin{align*} -{\frac{{a}^{8}}{3\,{x}^{3}}}+{\frac{28\,{a}^{6}{b}^{2}{x}^{3}}{3}}+{\frac{28\,{a}^{5}{b}^{3}{x}^{6}}{3}}+{\frac{70\,{a}^{4}{b}^{4}{x}^{9}}{9}}+{\frac{14\,{a}^{3}{b}^{5}{x}^{12}}{3}}+{\frac{28\,{a}^{2}{b}^{6}{x}^{15}}{15}}+{\frac{4\,a{b}^{7}{x}^{18}}{9}}+{\frac{{b}^{8}{x}^{21}}{21}}+8\,{a}^{7}b\ln \left ( x \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.955478, size = 123, normalized size = 1.17 \begin{align*} \frac{1}{21} \, b^{8} x^{21} + \frac{4}{9} \, a b^{7} x^{18} + \frac{28}{15} \, a^{2} b^{6} x^{15} + \frac{14}{3} \, a^{3} b^{5} x^{12} + \frac{70}{9} \, a^{4} b^{4} x^{9} + \frac{28}{3} \, a^{5} b^{3} x^{6} + \frac{28}{3} \, a^{6} b^{2} x^{3} + \frac{8}{3} \, a^{7} b \log \left (x^{3}\right ) - \frac{a^{8}}{3 \, x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.66007, size = 234, normalized size = 2.23 \begin{align*} \frac{15 \, b^{8} x^{24} + 140 \, a b^{7} x^{21} + 588 \, a^{2} b^{6} x^{18} + 1470 \, a^{3} b^{5} x^{15} + 2450 \, a^{4} b^{4} x^{12} + 2940 \, a^{5} b^{3} x^{9} + 2940 \, a^{6} b^{2} x^{6} + 2520 \, a^{7} b x^{3} \log \left (x\right ) - 105 \, a^{8}}{315 \, x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.493005, size = 105, normalized size = 1. \begin{align*} - \frac{a^{8}}{3 x^{3}} + 8 a^{7} b \log{\left (x \right )} + \frac{28 a^{6} b^{2} x^{3}}{3} + \frac{28 a^{5} b^{3} x^{6}}{3} + \frac{70 a^{4} b^{4} x^{9}}{9} + \frac{14 a^{3} b^{5} x^{12}}{3} + \frac{28 a^{2} b^{6} x^{15}}{15} + \frac{4 a b^{7} x^{18}}{9} + \frac{b^{8} x^{21}}{21} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.13569, size = 135, normalized size = 1.29 \begin{align*} \frac{1}{21} \, b^{8} x^{21} + \frac{4}{9} \, a b^{7} x^{18} + \frac{28}{15} \, a^{2} b^{6} x^{15} + \frac{14}{3} \, a^{3} b^{5} x^{12} + \frac{70}{9} \, a^{4} b^{4} x^{9} + \frac{28}{3} \, a^{5} b^{3} x^{6} + \frac{28}{3} \, a^{6} b^{2} x^{3} + 8 \, a^{7} b \log \left ({\left | x \right |}\right ) - \frac{8 \, a^{7} b x^{3} + a^{8}}{3 \, x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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