Optimal. Leaf size=104 \[ -\frac{14 a^6 b^2}{9 x^{18}}-\frac{56 a^5 b^3}{15 x^{15}}-\frac{35 a^4 b^4}{6 x^{12}}-\frac{56 a^3 b^5}{9 x^9}-\frac{14 a^2 b^6}{3 x^6}-\frac{8 a^7 b}{21 x^{21}}-\frac{a^8}{24 x^{24}}-\frac{8 a b^7}{3 x^3}+b^8 \log (x) \]
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Rubi [A] time = 0.0507342, 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} \[ -\frac{14 a^6 b^2}{9 x^{18}}-\frac{56 a^5 b^3}{15 x^{15}}-\frac{35 a^4 b^4}{6 x^{12}}-\frac{56 a^3 b^5}{9 x^9}-\frac{14 a^2 b^6}{3 x^6}-\frac{8 a^7 b}{21 x^{21}}-\frac{a^8}{24 x^{24}}-\frac{8 a b^7}{3 x^3}+b^8 \log (x) \]
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^{25}} \, dx &=\frac{1}{3} \operatorname{Subst}\left (\int \frac{(a+b x)^8}{x^9} \, dx,x,x^3\right )\\ &=\frac{1}{3} \operatorname{Subst}\left (\int \left (\frac{a^8}{x^9}+\frac{8 a^7 b}{x^8}+\frac{28 a^6 b^2}{x^7}+\frac{56 a^5 b^3}{x^6}+\frac{70 a^4 b^4}{x^5}+\frac{56 a^3 b^5}{x^4}+\frac{28 a^2 b^6}{x^3}+\frac{8 a b^7}{x^2}+\frac{b^8}{x}\right ) \, dx,x,x^3\right )\\ &=-\frac{a^8}{24 x^{24}}-\frac{8 a^7 b}{21 x^{21}}-\frac{14 a^6 b^2}{9 x^{18}}-\frac{56 a^5 b^3}{15 x^{15}}-\frac{35 a^4 b^4}{6 x^{12}}-\frac{56 a^3 b^5}{9 x^9}-\frac{14 a^2 b^6}{3 x^6}-\frac{8 a b^7}{3 x^3}+b^8 \log (x)\\ \end{align*}
Mathematica [A] time = 0.0049155, size = 104, normalized size = 1. \[ -\frac{14 a^6 b^2}{9 x^{18}}-\frac{56 a^5 b^3}{15 x^{15}}-\frac{35 a^4 b^4}{6 x^{12}}-\frac{56 a^3 b^5}{9 x^9}-\frac{14 a^2 b^6}{3 x^6}-\frac{8 a^7 b}{21 x^{21}}-\frac{a^8}{24 x^{24}}-\frac{8 a b^7}{3 x^3}+b^8 \log (x) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.007, size = 89, normalized size = 0.9 \begin{align*} -{\frac{{a}^{8}}{24\,{x}^{24}}}-{\frac{8\,{a}^{7}b}{21\,{x}^{21}}}-{\frac{14\,{a}^{6}{b}^{2}}{9\,{x}^{18}}}-{\frac{56\,{a}^{5}{b}^{3}}{15\,{x}^{15}}}-{\frac{35\,{a}^{4}{b}^{4}}{6\,{x}^{12}}}-{\frac{56\,{a}^{3}{b}^{5}}{9\,{x}^{9}}}-{\frac{14\,{a}^{2}{b}^{6}}{3\,{x}^{6}}}-{\frac{8\,a{b}^{7}}{3\,{x}^{3}}}+{b}^{8}\ln \left ( x \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.951666, size = 127, normalized size = 1.22 \begin{align*} \frac{1}{3} \, b^{8} \log \left (x^{3}\right ) - \frac{6720 \, a b^{7} x^{21} + 11760 \, a^{2} b^{6} x^{18} + 15680 \, a^{3} b^{5} x^{15} + 14700 \, a^{4} b^{4} x^{12} + 9408 \, a^{5} b^{3} x^{9} + 3920 \, a^{6} b^{2} x^{6} + 960 \, a^{7} b x^{3} + 105 \, a^{8}}{2520 \, x^{24}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.69963, size = 244, normalized size = 2.35 \begin{align*} \frac{2520 \, b^{8} x^{24} \log \left (x\right ) - 6720 \, a b^{7} x^{21} - 11760 \, a^{2} b^{6} x^{18} - 15680 \, a^{3} b^{5} x^{15} - 14700 \, a^{4} b^{4} x^{12} - 9408 \, a^{5} b^{3} x^{9} - 3920 \, a^{6} b^{2} x^{6} - 960 \, a^{7} b x^{3} - 105 \, a^{8}}{2520 \, x^{24}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.26163, size = 95, normalized size = 0.91 \begin{align*} b^{8} \log{\left (x \right )} - \frac{105 a^{8} + 960 a^{7} b x^{3} + 3920 a^{6} b^{2} x^{6} + 9408 a^{5} b^{3} x^{9} + 14700 a^{4} b^{4} x^{12} + 15680 a^{3} b^{5} x^{15} + 11760 a^{2} b^{6} x^{18} + 6720 a b^{7} x^{21}}{2520 x^{24}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.14023, size = 135, normalized size = 1.3 \begin{align*} b^{8} \log \left ({\left | x \right |}\right ) - \frac{2283 \, b^{8} x^{24} + 6720 \, a b^{7} x^{21} + 11760 \, a^{2} b^{6} x^{18} + 15680 \, a^{3} b^{5} x^{15} + 14700 \, a^{4} b^{4} x^{12} + 9408 \, a^{5} b^{3} x^{9} + 3920 \, a^{6} b^{2} x^{6} + 960 \, a^{7} b x^{3} + 105 \, a^{8}}{2520 \, x^{24}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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