Optimal. Leaf size=171 \[ -\frac{54 b^7 x^{2/3}}{a^{10}}-\frac{63 b^5 x^{4/3}}{4 a^8}+\frac{9 b^4 x^{5/3}}{a^7}-\frac{5 b^3 x^2}{a^6}+\frac{18 b^2 x^{7/3}}{7 a^5}+\frac{3 b^{11}}{2 a^{12} \left (a \sqrt [3]{x}+b\right )^2}-\frac{33 b^{10}}{a^{12} \left (a \sqrt [3]{x}+b\right )}+\frac{135 b^8 \sqrt [3]{x}}{a^{11}}+\frac{28 b^6 x}{a^9}-\frac{165 b^9 \log \left (a \sqrt [3]{x}+b\right )}{a^{12}}-\frac{9 b x^{8/3}}{8 a^4}+\frac{x^3}{3 a^3} \]
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Rubi [A] time = 0.151803, antiderivative size = 171, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {263, 266, 43} \[ -\frac{54 b^7 x^{2/3}}{a^{10}}-\frac{63 b^5 x^{4/3}}{4 a^8}+\frac{9 b^4 x^{5/3}}{a^7}-\frac{5 b^3 x^2}{a^6}+\frac{18 b^2 x^{7/3}}{7 a^5}+\frac{3 b^{11}}{2 a^{12} \left (a \sqrt [3]{x}+b\right )^2}-\frac{33 b^{10}}{a^{12} \left (a \sqrt [3]{x}+b\right )}+\frac{135 b^8 \sqrt [3]{x}}{a^{11}}+\frac{28 b^6 x}{a^9}-\frac{165 b^9 \log \left (a \sqrt [3]{x}+b\right )}{a^{12}}-\frac{9 b x^{8/3}}{8 a^4}+\frac{x^3}{3 a^3} \]
Antiderivative was successfully verified.
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Rule 263
Rule 266
Rule 43
Rubi steps
\begin{align*} \int \frac{x^2}{\left (a+\frac{b}{\sqrt [3]{x}}\right )^3} \, dx &=\int \frac{x^3}{\left (b+a \sqrt [3]{x}\right )^3} \, dx\\ &=3 \operatorname{Subst}\left (\int \frac{x^{11}}{(b+a x)^3} \, dx,x,\sqrt [3]{x}\right )\\ &=3 \operatorname{Subst}\left (\int \left (\frac{45 b^8}{a^{11}}-\frac{36 b^7 x}{a^{10}}+\frac{28 b^6 x^2}{a^9}-\frac{21 b^5 x^3}{a^8}+\frac{15 b^4 x^4}{a^7}-\frac{10 b^3 x^5}{a^6}+\frac{6 b^2 x^6}{a^5}-\frac{3 b x^7}{a^4}+\frac{x^8}{a^3}-\frac{b^{11}}{a^{11} (b+a x)^3}+\frac{11 b^{10}}{a^{11} (b+a x)^2}-\frac{55 b^9}{a^{11} (b+a x)}\right ) \, dx,x,\sqrt [3]{x}\right )\\ &=\frac{3 b^{11}}{2 a^{12} \left (b+a \sqrt [3]{x}\right )^2}-\frac{33 b^{10}}{a^{12} \left (b+a \sqrt [3]{x}\right )}+\frac{135 b^8 \sqrt [3]{x}}{a^{11}}-\frac{54 b^7 x^{2/3}}{a^{10}}+\frac{28 b^6 x}{a^9}-\frac{63 b^5 x^{4/3}}{4 a^8}+\frac{9 b^4 x^{5/3}}{a^7}-\frac{5 b^3 x^2}{a^6}+\frac{18 b^2 x^{7/3}}{7 a^5}-\frac{9 b x^{8/3}}{8 a^4}+\frac{x^3}{3 a^3}-\frac{165 b^9 \log \left (b+a \sqrt [3]{x}\right )}{a^{12}}\\ \end{align*}
Mathematica [A] time = 0.155488, size = 157, normalized size = 0.92 \[ \frac{-9072 a^2 b^7 x^{2/3}-2646 a^4 b^5 x^{4/3}+1512 a^5 b^4 x^{5/3}-840 a^6 b^3 x^2+432 a^7 b^2 x^{7/3}+4704 a^3 b^6 x-189 a^8 b x^{8/3}+56 a^9 x^3+\frac{252 b^{11}}{\left (a \sqrt [3]{x}+b\right )^2}-\frac{5544 b^{10}}{a \sqrt [3]{x}+b}+22680 a b^8 \sqrt [3]{x}-27720 b^9 \log \left (a \sqrt [3]{x}+b\right )}{168 a^{12}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.008, size = 144, normalized size = 0.8 \begin{align*}{\frac{3\,{b}^{11}}{2\,{a}^{12}} \left ( b+a\sqrt [3]{x} \right ) ^{-2}}-33\,{\frac{{b}^{10}}{{a}^{12} \left ( b+a\sqrt [3]{x} \right ) }}+135\,{\frac{{b}^{8}\sqrt [3]{x}}{{a}^{11}}}-54\,{\frac{{b}^{7}{x}^{2/3}}{{a}^{10}}}+28\,{\frac{{b}^{6}x}{{a}^{9}}}-{\frac{63\,{b}^{5}}{4\,{a}^{8}}{x}^{{\frac{4}{3}}}}+9\,{\frac{{b}^{4}{x}^{5/3}}{{a}^{7}}}-5\,{\frac{{b}^{3}{x}^{2}}{{a}^{6}}}+{\frac{18\,{b}^{2}}{7\,{a}^{5}}{x}^{{\frac{7}{3}}}}-{\frac{9\,b}{8\,{a}^{4}}{x}^{{\frac{8}{3}}}}+{\frac{{x}^{3}}{3\,{a}^{3}}}-165\,{\frac{{b}^{9}\ln \left ( b+a\sqrt [3]{x} \right ) }{{a}^{12}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.0099, size = 225, normalized size = 1.32 \begin{align*} \frac{56 \, a^{10} - \frac{77 \, a^{9} b}{x^{\frac{1}{3}}} + \frac{110 \, a^{8} b^{2}}{x^{\frac{2}{3}}} - \frac{165 \, a^{7} b^{3}}{x} + \frac{264 \, a^{6} b^{4}}{x^{\frac{4}{3}}} - \frac{462 \, a^{5} b^{5}}{x^{\frac{5}{3}}} + \frac{924 \, a^{4} b^{6}}{x^{2}} - \frac{2310 \, a^{3} b^{7}}{x^{\frac{7}{3}}} + \frac{9240 \, a^{2} b^{8}}{x^{\frac{8}{3}}} + \frac{41580 \, a b^{9}}{x^{3}} + \frac{27720 \, b^{10}}{x^{\frac{10}{3}}}}{168 \,{\left (\frac{a^{13}}{x^{3}} + \frac{2 \, a^{12} b}{x^{\frac{10}{3}}} + \frac{a^{11} b^{2}}{x^{\frac{11}{3}}}\right )}} - \frac{165 \, b^{9} \log \left (a + \frac{b}{x^{\frac{1}{3}}}\right )}{a^{12}} - \frac{55 \, b^{9} \log \left (x\right )}{a^{12}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.53238, size = 543, normalized size = 3.18 \begin{align*} \frac{56 \, a^{15} x^{5} - 728 \, a^{12} b^{3} x^{4} + 3080 \, a^{9} b^{6} x^{3} + 8568 \, a^{6} b^{9} x^{2} - 1344 \, a^{3} b^{12} x - 5292 \, b^{15} - 27720 \,{\left (a^{6} b^{9} x^{2} + 2 \, a^{3} b^{12} x + b^{15}\right )} \log \left (a x^{\frac{1}{3}} + b\right ) - 63 \,{\left (3 \, a^{14} b x^{4} - 18 \, a^{11} b^{4} x^{3} + 99 \, a^{8} b^{7} x^{2} + 352 \, a^{5} b^{10} x + 220 \, a^{2} b^{13}\right )} x^{\frac{2}{3}} + 18 \,{\left (24 \, a^{13} b^{2} x^{4} - 99 \, a^{10} b^{5} x^{3} + 990 \, a^{7} b^{8} x^{2} + 2695 \, a^{4} b^{11} x + 1540 \, a b^{14}\right )} x^{\frac{1}{3}}}{168 \,{\left (a^{18} x^{2} + 2 \, a^{15} b^{3} x + a^{12} b^{6}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 6.83113, size = 624, normalized size = 3.65 \begin{align*} \begin{cases} \frac{56 a^{11} x^{\frac{11}{3}}}{168 a^{14} x^{\frac{2}{3}} + 336 a^{13} b \sqrt [3]{x} + 168 a^{12} b^{2}} - \frac{77 a^{10} b x^{\frac{10}{3}}}{168 a^{14} x^{\frac{2}{3}} + 336 a^{13} b \sqrt [3]{x} + 168 a^{12} b^{2}} + \frac{110 a^{9} b^{2} x^{3}}{168 a^{14} x^{\frac{2}{3}} + 336 a^{13} b \sqrt [3]{x} + 168 a^{12} b^{2}} - \frac{165 a^{8} b^{3} x^{\frac{8}{3}}}{168 a^{14} x^{\frac{2}{3}} + 336 a^{13} b \sqrt [3]{x} + 168 a^{12} b^{2}} + \frac{264 a^{7} b^{4} x^{\frac{7}{3}}}{168 a^{14} x^{\frac{2}{3}} + 336 a^{13} b \sqrt [3]{x} + 168 a^{12} b^{2}} - \frac{462 a^{6} b^{5} x^{2}}{168 a^{14} x^{\frac{2}{3}} + 336 a^{13} b \sqrt [3]{x} + 168 a^{12} b^{2}} + \frac{924 a^{5} b^{6} x^{\frac{5}{3}}}{168 a^{14} x^{\frac{2}{3}} + 336 a^{13} b \sqrt [3]{x} + 168 a^{12} b^{2}} - \frac{2310 a^{4} b^{7} x^{\frac{4}{3}}}{168 a^{14} x^{\frac{2}{3}} + 336 a^{13} b \sqrt [3]{x} + 168 a^{12} b^{2}} + \frac{9240 a^{3} b^{8} x}{168 a^{14} x^{\frac{2}{3}} + 336 a^{13} b \sqrt [3]{x} + 168 a^{12} b^{2}} - \frac{27720 a^{2} b^{9} x^{\frac{2}{3}} \log{\left (\sqrt [3]{x} + \frac{b}{a} \right )}}{168 a^{14} x^{\frac{2}{3}} + 336 a^{13} b \sqrt [3]{x} + 168 a^{12} b^{2}} - \frac{55440 a b^{10} \sqrt [3]{x} \log{\left (\sqrt [3]{x} + \frac{b}{a} \right )}}{168 a^{14} x^{\frac{2}{3}} + 336 a^{13} b \sqrt [3]{x} + 168 a^{12} b^{2}} - \frac{55440 a b^{10} \sqrt [3]{x}}{168 a^{14} x^{\frac{2}{3}} + 336 a^{13} b \sqrt [3]{x} + 168 a^{12} b^{2}} - \frac{27720 b^{11} \log{\left (\sqrt [3]{x} + \frac{b}{a} \right )}}{168 a^{14} x^{\frac{2}{3}} + 336 a^{13} b \sqrt [3]{x} + 168 a^{12} b^{2}} - \frac{41580 b^{11}}{168 a^{14} x^{\frac{2}{3}} + 336 a^{13} b \sqrt [3]{x} + 168 a^{12} b^{2}} & \text{for}\: a \neq 0 \\\frac{x^{4}}{4 b^{3}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.18337, size = 196, normalized size = 1.15 \begin{align*} -\frac{165 \, b^{9} \log \left ({\left | a x^{\frac{1}{3}} + b \right |}\right )}{a^{12}} - \frac{3 \,{\left (22 \, a b^{10} x^{\frac{1}{3}} + 21 \, b^{11}\right )}}{2 \,{\left (a x^{\frac{1}{3}} + b\right )}^{2} a^{12}} + \frac{56 \, a^{24} x^{3} - 189 \, a^{23} b x^{\frac{8}{3}} + 432 \, a^{22} b^{2} x^{\frac{7}{3}} - 840 \, a^{21} b^{3} x^{2} + 1512 \, a^{20} b^{4} x^{\frac{5}{3}} - 2646 \, a^{19} b^{5} x^{\frac{4}{3}} + 4704 \, a^{18} b^{6} x - 9072 \, a^{17} b^{7} x^{\frac{2}{3}} + 22680 \, a^{16} b^{8} x^{\frac{1}{3}}}{168 \, a^{27}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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