Optimal. Leaf size=137 \[ -\frac{3 a^2}{10 b^3 \left (a+b \sqrt [3]{x}\right )^9 \sqrt{a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}}}+\frac{2 a}{3 b^3 \left (a+b \sqrt [3]{x}\right )^8 \sqrt{a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}}}-\frac{3}{8 b^3 \left (a+b \sqrt [3]{x}\right )^7 \sqrt{a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}}} \]
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Rubi [A] time = 0.0780041, antiderivative size = 137, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.115, Rules used = {1341, 646, 43} \[ -\frac{3 a^2}{10 b^3 \left (a+b \sqrt [3]{x}\right )^9 \sqrt{a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}}}+\frac{2 a}{3 b^3 \left (a+b \sqrt [3]{x}\right )^8 \sqrt{a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}}}-\frac{3}{8 b^3 \left (a+b \sqrt [3]{x}\right )^7 \sqrt{a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}}} \]
Antiderivative was successfully verified.
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Rule 1341
Rule 646
Rule 43
Rubi steps
\begin{align*} \int \frac{1}{\left (a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}\right )^{11/2}} \, dx &=3 \operatorname{Subst}\left (\int \frac{x^2}{\left (a^2+2 a b x+b^2 x^2\right )^{11/2}} \, dx,x,\sqrt [3]{x}\right )\\ &=\frac{\left (3 b^{11} \left (a+b \sqrt [3]{x}\right )\right ) \operatorname{Subst}\left (\int \frac{x^2}{\left (a b+b^2 x\right )^{11}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt{a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}}}\\ &=\frac{\left (3 b^{11} \left (a+b \sqrt [3]{x}\right )\right ) \operatorname{Subst}\left (\int \left (\frac{a^2}{b^{13} (a+b x)^{11}}-\frac{2 a}{b^{13} (a+b x)^{10}}+\frac{1}{b^{13} (a+b x)^9}\right ) \, dx,x,\sqrt [3]{x}\right )}{\sqrt{a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}}}\\ &=-\frac{3 a^2}{10 b^3 \left (a+b \sqrt [3]{x}\right )^9 \sqrt{a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}}}+\frac{2 a}{3 b^3 \left (a+b \sqrt [3]{x}\right )^8 \sqrt{a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}}}-\frac{3}{8 b^3 \left (a+b \sqrt [3]{x}\right )^7 \sqrt{a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}}}\\ \end{align*}
Mathematica [A] time = 0.038088, size = 58, normalized size = 0.42 \[ \frac{-a^2-10 a b \sqrt [3]{x}-45 b^2 x^{2/3}}{120 b^3 \left (a+b \sqrt [3]{x}\right )^9 \sqrt{\left (a+b \sqrt [3]{x}\right )^2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.007, size = 54, normalized size = 0.4 \begin{align*} -{\frac{1}{120\,{b}^{3}}\sqrt{{a}^{2}+2\,ab\sqrt [3]{x}+{b}^{2}{x}^{{\frac{2}{3}}}} \left ( 45\,{b}^{2}{x}^{2/3}+10\,ab\sqrt [3]{x}+{a}^{2} \right ) \left ( a+b\sqrt [3]{x} \right ) ^{-11}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.01626, size = 85, normalized size = 0.62 \begin{align*} -\frac{3 \, a^{2} b^{2}}{10 \,{\left (b^{2}\right )}^{\frac{15}{2}}{\left (x^{\frac{1}{3}} + \frac{a}{b}\right )}^{10}} + \frac{2 \, a b}{3 \,{\left (b^{2}\right )}^{\frac{13}{2}}{\left (x^{\frac{1}{3}} + \frac{a}{b}\right )}^{9}} - \frac{3}{8 \,{\left (b^{2}\right )}^{\frac{11}{2}}{\left (x^{\frac{1}{3}} + \frac{a}{b}\right )}^{8}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.04098, size = 842, normalized size = 6.15 \begin{align*} \frac{440 \, a b^{21} x^{7} - 25630 \, a^{4} b^{18} x^{6} + 186252 \, a^{7} b^{15} x^{5} - 326150 \, a^{10} b^{12} x^{4} + 154000 \, a^{13} b^{9} x^{3} - 16005 \, a^{16} b^{6} x^{2} + 110 \, a^{19} b^{3} x - a^{22} - 27 \,{\left (88 \, a^{2} b^{20} x^{6} - 2200 \, a^{5} b^{17} x^{5} + 9625 \, a^{8} b^{14} x^{4} - 10910 \, a^{11} b^{11} x^{3} + 3245 \, a^{14} b^{8} x^{2} - 176 \, a^{17} b^{5} x\right )} x^{\frac{2}{3}} - 9 \,{\left (5 \, b^{22} x^{7} - 990 \, a^{3} b^{19} x^{6} + 12705 \, a^{6} b^{16} x^{5} - 34760 \, a^{9} b^{13} x^{4} + 25542 \, a^{12} b^{10} x^{3} - 4620 \, a^{15} b^{7} x^{2} + 110 \, a^{18} b^{4} x\right )} x^{\frac{1}{3}}}{120 \,{\left (b^{33} x^{10} + 10 \, a^{3} b^{30} x^{9} + 45 \, a^{6} b^{27} x^{8} + 120 \, a^{9} b^{24} x^{7} + 210 \, a^{12} b^{21} x^{6} + 252 \, a^{15} b^{18} x^{5} + 210 \, a^{18} b^{15} x^{4} + 120 \, a^{21} b^{12} x^{3} + 45 \, a^{24} b^{9} x^{2} + 10 \, a^{27} b^{6} x + a^{30} b^{3}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \mathit{undef} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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