Optimal. Leaf size=171 \[ -\frac{54 a^7 x^{2/3}}{b^{10}}-\frac{63 a^5 x^{4/3}}{4 b^8}+\frac{9 a^4 x^{5/3}}{b^7}-\frac{5 a^3 x^2}{b^6}+\frac{18 a^2 x^{7/3}}{7 b^5}+\frac{3 a^{11}}{2 b^{12} \left (a+b \sqrt [3]{x}\right )^2}-\frac{33 a^{10}}{b^{12} \left (a+b \sqrt [3]{x}\right )}+\frac{135 a^8 \sqrt [3]{x}}{b^{11}}+\frac{28 a^6 x}{b^9}-\frac{165 a^9 \log \left (a+b \sqrt [3]{x}\right )}{b^{12}}-\frac{9 a x^{8/3}}{8 b^4}+\frac{x^3}{3 b^3} \]
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Rubi [A] time = 0.148063, antiderivative size = 171, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {266, 43} \[ -\frac{54 a^7 x^{2/3}}{b^{10}}-\frac{63 a^5 x^{4/3}}{4 b^8}+\frac{9 a^4 x^{5/3}}{b^7}-\frac{5 a^3 x^2}{b^6}+\frac{18 a^2 x^{7/3}}{7 b^5}+\frac{3 a^{11}}{2 b^{12} \left (a+b \sqrt [3]{x}\right )^2}-\frac{33 a^{10}}{b^{12} \left (a+b \sqrt [3]{x}\right )}+\frac{135 a^8 \sqrt [3]{x}}{b^{11}}+\frac{28 a^6 x}{b^9}-\frac{165 a^9 \log \left (a+b \sqrt [3]{x}\right )}{b^{12}}-\frac{9 a x^{8/3}}{8 b^4}+\frac{x^3}{3 b^3} \]
Antiderivative was successfully verified.
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Rule 266
Rule 43
Rubi steps
\begin{align*} \int \frac{x^3}{\left (a+b \sqrt [3]{x}\right )^3} \, dx &=3 \operatorname{Subst}\left (\int \frac{x^{11}}{(a+b x)^3} \, dx,x,\sqrt [3]{x}\right )\\ &=3 \operatorname{Subst}\left (\int \left (\frac{45 a^8}{b^{11}}-\frac{36 a^7 x}{b^{10}}+\frac{28 a^6 x^2}{b^9}-\frac{21 a^5 x^3}{b^8}+\frac{15 a^4 x^4}{b^7}-\frac{10 a^3 x^5}{b^6}+\frac{6 a^2 x^6}{b^5}-\frac{3 a x^7}{b^4}+\frac{x^8}{b^3}-\frac{a^{11}}{b^{11} (a+b x)^3}+\frac{11 a^{10}}{b^{11} (a+b x)^2}-\frac{55 a^9}{b^{11} (a+b x)}\right ) \, dx,x,\sqrt [3]{x}\right )\\ &=\frac{3 a^{11}}{2 b^{12} \left (a+b \sqrt [3]{x}\right )^2}-\frac{33 a^{10}}{b^{12} \left (a+b \sqrt [3]{x}\right )}+\frac{135 a^8 \sqrt [3]{x}}{b^{11}}-\frac{54 a^7 x^{2/3}}{b^{10}}+\frac{28 a^6 x}{b^9}-\frac{63 a^5 x^{4/3}}{4 b^8}+\frac{9 a^4 x^{5/3}}{b^7}-\frac{5 a^3 x^2}{b^6}+\frac{18 a^2 x^{7/3}}{7 b^5}-\frac{9 a x^{8/3}}{8 b^4}+\frac{x^3}{3 b^3}-\frac{165 a^9 \log \left (a+b \sqrt [3]{x}\right )}{b^{12}}\\ \end{align*}
Mathematica [A] time = 0.151713, size = 157, normalized size = 0.92 \[ \frac{-9072 a^7 b^2 x^{2/3}-2646 a^5 b^4 x^{4/3}+1512 a^4 b^5 x^{5/3}-840 a^3 b^6 x^2+432 a^2 b^7 x^{7/3}+4704 a^6 b^3 x+\frac{252 a^{11}}{\left (a+b \sqrt [3]{x}\right )^2}-\frac{5544 a^{10}}{a+b \sqrt [3]{x}}+22680 a^8 b \sqrt [3]{x}-27720 a^9 \log \left (a+b \sqrt [3]{x}\right )-189 a b^8 x^{8/3}+56 b^9 x^3}{168 b^{12}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.007, size = 144, normalized size = 0.8 \begin{align*}{\frac{3\,{a}^{11}}{2\,{b}^{12}} \left ( a+b\sqrt [3]{x} \right ) ^{-2}}-33\,{\frac{{a}^{10}}{{b}^{12} \left ( a+b\sqrt [3]{x} \right ) }}+135\,{\frac{{a}^{8}\sqrt [3]{x}}{{b}^{11}}}-54\,{\frac{{a}^{7}{x}^{2/3}}{{b}^{10}}}+28\,{\frac{{a}^{6}x}{{b}^{9}}}-{\frac{63\,{a}^{5}}{4\,{b}^{8}}{x}^{{\frac{4}{3}}}}+9\,{\frac{{a}^{4}{x}^{5/3}}{{b}^{7}}}-5\,{\frac{{x}^{2}{a}^{3}}{{b}^{6}}}+{\frac{18\,{a}^{2}}{7\,{b}^{5}}{x}^{{\frac{7}{3}}}}-{\frac{9\,a}{8\,{b}^{4}}{x}^{{\frac{8}{3}}}}+{\frac{{x}^{3}}{3\,{b}^{3}}}-165\,{\frac{{a}^{9}\ln \left ( a+b\sqrt [3]{x} \right ) }{{b}^{12}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.969329, size = 266, normalized size = 1.56 \begin{align*} -\frac{165 \, a^{9} \log \left (b x^{\frac{1}{3}} + a\right )}{b^{12}} + \frac{{\left (b x^{\frac{1}{3}} + a\right )}^{9}}{3 \, b^{12}} - \frac{33 \,{\left (b x^{\frac{1}{3}} + a\right )}^{8} a}{8 \, b^{12}} + \frac{165 \,{\left (b x^{\frac{1}{3}} + a\right )}^{7} a^{2}}{7 \, b^{12}} - \frac{165 \,{\left (b x^{\frac{1}{3}} + a\right )}^{6} a^{3}}{2 \, b^{12}} + \frac{198 \,{\left (b x^{\frac{1}{3}} + a\right )}^{5} a^{4}}{b^{12}} - \frac{693 \,{\left (b x^{\frac{1}{3}} + a\right )}^{4} a^{5}}{2 \, b^{12}} + \frac{462 \,{\left (b x^{\frac{1}{3}} + a\right )}^{3} a^{6}}{b^{12}} - \frac{495 \,{\left (b x^{\frac{1}{3}} + a\right )}^{2} a^{7}}{b^{12}} + \frac{495 \,{\left (b x^{\frac{1}{3}} + a\right )} a^{8}}{b^{12}} - \frac{33 \, a^{10}}{{\left (b x^{\frac{1}{3}} + a\right )} b^{12}} + \frac{3 \, a^{11}}{2 \,{\left (b x^{\frac{1}{3}} + a\right )}^{2} b^{12}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.5674, size = 543, normalized size = 3.18 \begin{align*} \frac{56 \, b^{15} x^{5} - 728 \, a^{3} b^{12} x^{4} + 3080 \, a^{6} b^{9} x^{3} + 8568 \, a^{9} b^{6} x^{2} - 1344 \, a^{12} b^{3} x - 5292 \, a^{15} - 27720 \,{\left (a^{9} b^{6} x^{2} + 2 \, a^{12} b^{3} x + a^{15}\right )} \log \left (b x^{\frac{1}{3}} + a\right ) - 63 \,{\left (3 \, a b^{14} x^{4} - 18 \, a^{4} b^{11} x^{3} + 99 \, a^{7} b^{8} x^{2} + 352 \, a^{10} b^{5} x + 220 \, a^{13} b^{2}\right )} x^{\frac{2}{3}} + 18 \,{\left (24 \, a^{2} b^{13} x^{4} - 99 \, a^{5} b^{10} x^{3} + 990 \, a^{8} b^{7} x^{2} + 2695 \, a^{11} b^{4} x + 1540 \, a^{14} b\right )} x^{\frac{1}{3}}}{168 \,{\left (b^{18} x^{2} + 2 \, a^{3} b^{15} x + a^{6} b^{12}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 7.25401, size = 624, normalized size = 3.65 \begin{align*} \begin{cases} - \frac{27720 a^{11} \log{\left (\frac{a}{b} + \sqrt [3]{x} \right )}}{168 a^{2} b^{12} + 336 a b^{13} \sqrt [3]{x} + 168 b^{14} x^{\frac{2}{3}}} - \frac{41580 a^{11}}{168 a^{2} b^{12} + 336 a b^{13} \sqrt [3]{x} + 168 b^{14} x^{\frac{2}{3}}} - \frac{55440 a^{10} b \sqrt [3]{x} \log{\left (\frac{a}{b} + \sqrt [3]{x} \right )}}{168 a^{2} b^{12} + 336 a b^{13} \sqrt [3]{x} + 168 b^{14} x^{\frac{2}{3}}} - \frac{55440 a^{10} b \sqrt [3]{x}}{168 a^{2} b^{12} + 336 a b^{13} \sqrt [3]{x} + 168 b^{14} x^{\frac{2}{3}}} - \frac{27720 a^{9} b^{2} x^{\frac{2}{3}} \log{\left (\frac{a}{b} + \sqrt [3]{x} \right )}}{168 a^{2} b^{12} + 336 a b^{13} \sqrt [3]{x} + 168 b^{14} x^{\frac{2}{3}}} + \frac{9240 a^{8} b^{3} x}{168 a^{2} b^{12} + 336 a b^{13} \sqrt [3]{x} + 168 b^{14} x^{\frac{2}{3}}} - \frac{2310 a^{7} b^{4} x^{\frac{4}{3}}}{168 a^{2} b^{12} + 336 a b^{13} \sqrt [3]{x} + 168 b^{14} x^{\frac{2}{3}}} + \frac{924 a^{6} b^{5} x^{\frac{5}{3}}}{168 a^{2} b^{12} + 336 a b^{13} \sqrt [3]{x} + 168 b^{14} x^{\frac{2}{3}}} - \frac{462 a^{5} b^{6} x^{2}}{168 a^{2} b^{12} + 336 a b^{13} \sqrt [3]{x} + 168 b^{14} x^{\frac{2}{3}}} + \frac{264 a^{4} b^{7} x^{\frac{7}{3}}}{168 a^{2} b^{12} + 336 a b^{13} \sqrt [3]{x} + 168 b^{14} x^{\frac{2}{3}}} - \frac{165 a^{3} b^{8} x^{\frac{8}{3}}}{168 a^{2} b^{12} + 336 a b^{13} \sqrt [3]{x} + 168 b^{14} x^{\frac{2}{3}}} + \frac{110 a^{2} b^{9} x^{3}}{168 a^{2} b^{12} + 336 a b^{13} \sqrt [3]{x} + 168 b^{14} x^{\frac{2}{3}}} - \frac{77 a b^{10} x^{\frac{10}{3}}}{168 a^{2} b^{12} + 336 a b^{13} \sqrt [3]{x} + 168 b^{14} x^{\frac{2}{3}}} + \frac{56 b^{11} x^{\frac{11}{3}}}{168 a^{2} b^{12} + 336 a b^{13} \sqrt [3]{x} + 168 b^{14} x^{\frac{2}{3}}} & \text{for}\: b \neq 0 \\\frac{x^{4}}{4 a^{3}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.1914, size = 196, normalized size = 1.15 \begin{align*} -\frac{165 \, a^{9} \log \left ({\left | b x^{\frac{1}{3}} + a \right |}\right )}{b^{12}} - \frac{3 \,{\left (22 \, a^{10} b x^{\frac{1}{3}} + 21 \, a^{11}\right )}}{2 \,{\left (b x^{\frac{1}{3}} + a\right )}^{2} b^{12}} + \frac{56 \, b^{24} x^{3} - 189 \, a b^{23} x^{\frac{8}{3}} + 432 \, a^{2} b^{22} x^{\frac{7}{3}} - 840 \, a^{3} b^{21} x^{2} + 1512 \, a^{4} b^{20} x^{\frac{5}{3}} - 2646 \, a^{5} b^{19} x^{\frac{4}{3}} + 4704 \, a^{6} b^{18} x - 9072 \, a^{7} b^{17} x^{\frac{2}{3}} + 22680 \, a^{8} b^{16} x^{\frac{1}{3}}}{168 \, b^{27}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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