Optimal. Leaf size=122 \[ -\frac{b^4 \left (a+b \sqrt [3]{x}\right )^{11}}{5005 a^5 x^{11/3}}+\frac{b^3 \left (a+b \sqrt [3]{x}\right )^{11}}{455 a^4 x^4}-\frac{6 b^2 \left (a+b \sqrt [3]{x}\right )^{11}}{455 a^3 x^{13/3}}+\frac{2 b \left (a+b \sqrt [3]{x}\right )^{11}}{35 a^2 x^{14/3}}-\frac{\left (a+b \sqrt [3]{x}\right )^{11}}{5 a x^5} \]
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Rubi [A] time = 0.0465489, antiderivative size = 122, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {266, 45, 37} \[ -\frac{b^4 \left (a+b \sqrt [3]{x}\right )^{11}}{5005 a^5 x^{11/3}}+\frac{b^3 \left (a+b \sqrt [3]{x}\right )^{11}}{455 a^4 x^4}-\frac{6 b^2 \left (a+b \sqrt [3]{x}\right )^{11}}{455 a^3 x^{13/3}}+\frac{2 b \left (a+b \sqrt [3]{x}\right )^{11}}{35 a^2 x^{14/3}}-\frac{\left (a+b \sqrt [3]{x}\right )^{11}}{5 a x^5} \]
Antiderivative was successfully verified.
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Rule 266
Rule 45
Rule 37
Rubi steps
\begin{align*} \int \frac{\left (a+b \sqrt [3]{x}\right )^{10}}{x^6} \, dx &=3 \operatorname{Subst}\left (\int \frac{(a+b x)^{10}}{x^{16}} \, dx,x,\sqrt [3]{x}\right )\\ &=-\frac{\left (a+b \sqrt [3]{x}\right )^{11}}{5 a x^5}-\frac{(4 b) \operatorname{Subst}\left (\int \frac{(a+b x)^{10}}{x^{15}} \, dx,x,\sqrt [3]{x}\right )}{5 a}\\ &=-\frac{\left (a+b \sqrt [3]{x}\right )^{11}}{5 a x^5}+\frac{2 b \left (a+b \sqrt [3]{x}\right )^{11}}{35 a^2 x^{14/3}}+\frac{\left (6 b^2\right ) \operatorname{Subst}\left (\int \frac{(a+b x)^{10}}{x^{14}} \, dx,x,\sqrt [3]{x}\right )}{35 a^2}\\ &=-\frac{\left (a+b \sqrt [3]{x}\right )^{11}}{5 a x^5}+\frac{2 b \left (a+b \sqrt [3]{x}\right )^{11}}{35 a^2 x^{14/3}}-\frac{6 b^2 \left (a+b \sqrt [3]{x}\right )^{11}}{455 a^3 x^{13/3}}-\frac{\left (12 b^3\right ) \operatorname{Subst}\left (\int \frac{(a+b x)^{10}}{x^{13}} \, dx,x,\sqrt [3]{x}\right )}{455 a^3}\\ &=-\frac{\left (a+b \sqrt [3]{x}\right )^{11}}{5 a x^5}+\frac{2 b \left (a+b \sqrt [3]{x}\right )^{11}}{35 a^2 x^{14/3}}-\frac{6 b^2 \left (a+b \sqrt [3]{x}\right )^{11}}{455 a^3 x^{13/3}}+\frac{b^3 \left (a+b \sqrt [3]{x}\right )^{11}}{455 a^4 x^4}+\frac{b^4 \operatorname{Subst}\left (\int \frac{(a+b x)^{10}}{x^{12}} \, dx,x,\sqrt [3]{x}\right )}{455 a^4}\\ &=-\frac{\left (a+b \sqrt [3]{x}\right )^{11}}{5 a x^5}+\frac{2 b \left (a+b \sqrt [3]{x}\right )^{11}}{35 a^2 x^{14/3}}-\frac{6 b^2 \left (a+b \sqrt [3]{x}\right )^{11}}{455 a^3 x^{13/3}}+\frac{b^3 \left (a+b \sqrt [3]{x}\right )^{11}}{455 a^4 x^4}-\frac{b^4 \left (a+b \sqrt [3]{x}\right )^{11}}{5005 a^5 x^{11/3}}\\ \end{align*}
Mathematica [A] time = 0.0132769, size = 67, normalized size = 0.55 \[ -\frac{\left (a+b \sqrt [3]{x}\right )^{11} \left (66 a^2 b^2 x^{2/3}-286 a^3 b \sqrt [3]{x}+1001 a^4-11 a b^3 x+b^4 x^{4/3}\right )}{5005 a^5 x^5} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.007, size = 113, normalized size = 0.9 \begin{align*} -{\frac{3\,{b}^{10}}{5}{x}^{-{\frac{5}{3}}}}-30\,{\frac{{a}^{7}{b}^{3}}{{x}^{4}}}-{\frac{{a}^{10}}{5\,{x}^{5}}}-{\frac{630\,{a}^{6}{b}^{4}}{11}{x}^{-{\frac{11}{3}}}}-{\frac{15\,{a}^{9}b}{7}{x}^{-{\frac{14}{3}}}}-{\frac{378\,{a}^{5}{b}^{5}}{5}{x}^{-{\frac{10}{3}}}}-45\,{\frac{{a}^{3}{b}^{7}}{{x}^{8/3}}}-{\frac{135\,{a}^{8}{b}^{2}}{13}{x}^{-{\frac{13}{3}}}}-5\,{\frac{a{b}^{9}}{{x}^{2}}}-{\frac{135\,{a}^{2}{b}^{8}}{7}{x}^{-{\frac{7}{3}}}}-70\,{\frac{{a}^{4}{b}^{6}}{{x}^{3}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.980982, size = 151, normalized size = 1.24 \begin{align*} -\frac{3003 \, b^{10} x^{\frac{10}{3}} + 25025 \, a b^{9} x^{3} + 96525 \, a^{2} b^{8} x^{\frac{8}{3}} + 225225 \, a^{3} b^{7} x^{\frac{7}{3}} + 350350 \, a^{4} b^{6} x^{2} + 378378 \, a^{5} b^{5} x^{\frac{5}{3}} + 286650 \, a^{6} b^{4} x^{\frac{4}{3}} + 150150 \, a^{7} b^{3} x + 51975 \, a^{8} b^{2} x^{\frac{2}{3}} + 10725 \, a^{9} b x^{\frac{1}{3}} + 1001 \, a^{10}}{5005 \, x^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.48243, size = 298, normalized size = 2.44 \begin{align*} -\frac{25025 \, a b^{9} x^{3} + 350350 \, a^{4} b^{6} x^{2} + 150150 \, a^{7} b^{3} x + 1001 \, a^{10} + 297 \,{\left (325 \, a^{2} b^{8} x^{2} + 1274 \, a^{5} b^{5} x + 175 \, a^{8} b^{2}\right )} x^{\frac{2}{3}} + 39 \,{\left (77 \, b^{10} x^{3} + 5775 \, a^{3} b^{7} x^{2} + 7350 \, a^{6} b^{4} x + 275 \, a^{9} b\right )} x^{\frac{1}{3}}}{5005 \, x^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 6.70453, size = 143, normalized size = 1.17 \begin{align*} - \frac{a^{10}}{5 x^{5}} - \frac{15 a^{9} b}{7 x^{\frac{14}{3}}} - \frac{135 a^{8} b^{2}}{13 x^{\frac{13}{3}}} - \frac{30 a^{7} b^{3}}{x^{4}} - \frac{630 a^{6} b^{4}}{11 x^{\frac{11}{3}}} - \frac{378 a^{5} b^{5}}{5 x^{\frac{10}{3}}} - \frac{70 a^{4} b^{6}}{x^{3}} - \frac{45 a^{3} b^{7}}{x^{\frac{8}{3}}} - \frac{135 a^{2} b^{8}}{7 x^{\frac{7}{3}}} - \frac{5 a b^{9}}{x^{2}} - \frac{3 b^{10}}{5 x^{\frac{5}{3}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.20654, size = 151, normalized size = 1.24 \begin{align*} -\frac{3003 \, b^{10} x^{\frac{10}{3}} + 25025 \, a b^{9} x^{3} + 96525 \, a^{2} b^{8} x^{\frac{8}{3}} + 225225 \, a^{3} b^{7} x^{\frac{7}{3}} + 350350 \, a^{4} b^{6} x^{2} + 378378 \, a^{5} b^{5} x^{\frac{5}{3}} + 286650 \, a^{6} b^{4} x^{\frac{4}{3}} + 150150 \, a^{7} b^{3} x + 51975 \, a^{8} b^{2} x^{\frac{2}{3}} + 10725 \, a^{9} b x^{\frac{1}{3}} + 1001 \, a^{10}}{5005 \, x^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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