Optimal. Leaf size=137 \[ -\frac{3 a^2}{8 b^3 \left (a+b \sqrt [3]{x}\right )^7 \sqrt{a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}}}+\frac{6 a}{7 b^3 \left (a+b \sqrt [3]{x}\right )^6 \sqrt{a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}}}-\frac{1}{2 b^3 \left (a+b \sqrt [3]{x}\right )^5 \sqrt{a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}}} \]
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Rubi [A] time = 0.0787639, 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} \[ -\frac{3 a^2}{8 b^3 \left (a+b \sqrt [3]{x}\right )^7 \sqrt{a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}}}+\frac{6 a}{7 b^3 \left (a+b \sqrt [3]{x}\right )^6 \sqrt{a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}}}-\frac{1}{2 b^3 \left (a+b \sqrt [3]{x}\right )^5 \sqrt{a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}}} \]
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 )^{9/2}} \, dx &=3 \operatorname{Subst}\left (\int \frac{x^2}{\left (a^2+2 a b x+b^2 x^2\right )^{9/2}} \, dx,x,\sqrt [3]{x}\right )\\ &=\frac{\left (3 b^9 \left (a+b \sqrt [3]{x}\right )\right ) \operatorname{Subst}\left (\int \frac{x^2}{\left (a b+b^2 x\right )^9} \, dx,x,\sqrt [3]{x}\right )}{\sqrt{a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}}}\\ &=\frac{\left (3 b^9 \left (a+b \sqrt [3]{x}\right )\right ) \operatorname{Subst}\left (\int \left (\frac{a^2}{b^{11} (a+b x)^9}-\frac{2 a}{b^{11} (a+b x)^8}+\frac{1}{b^{11} (a+b x)^7}\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}{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}}}+\frac{6 a}{7 b^3 \left (a+b \sqrt [3]{x}\right )^6 \sqrt{a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}}}-\frac{1}{2 b^3 \left (a+b \sqrt [3]{x}\right )^5 \sqrt{a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}}}\\ \end{align*}
Mathematica [A] time = 0.0371811, size = 58, normalized size = 0.42 \[ \frac{-a^2-8 a b \sqrt [3]{x}-28 b^2 x^{2/3}}{56 b^3 \left (a+b \sqrt [3]{x}\right )^7 \sqrt{\left (a+b \sqrt [3]{x}\right )^2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.007, size = 54, normalized size = 0.4 \begin{align*} -{\frac{1}{56\,{b}^{3}}\sqrt{{a}^{2}+2\,ab\sqrt [3]{x}+{b}^{2}{x}^{{\frac{2}{3}}}} \left ( 28\,{b}^{2}{x}^{2/3}+8\,ab\sqrt [3]{x}+{a}^{2} \right ) \left ( a+b\sqrt [3]{x} \right ) ^{-9}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.06896, size = 85, normalized size = 0.62 \begin{align*} -\frac{3 \, a^{2} b^{2}}{8 \,{\left (b^{2}\right )}^{\frac{13}{2}}{\left (x^{\frac{1}{3}} + \frac{a}{b}\right )}^{8}} + \frac{6 \, a b}{7 \,{\left (b^{2}\right )}^{\frac{11}{2}}{\left (x^{\frac{1}{3}} + \frac{a}{b}\right )}^{7}} - \frac{1}{2 \,{\left (b^{2}\right )}^{\frac{9}{2}}{\left (x^{\frac{1}{3}} + \frac{a}{b}\right )}^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.95698, size = 643, normalized size = 4.69 \begin{align*} -\frac{28 \, b^{18} x^{6} - 2856 \, a^{3} b^{15} x^{5} + 18186 \, a^{6} b^{12} x^{4} - 20608 \, a^{9} b^{9} x^{3} + 4200 \, a^{12} b^{6} x^{2} - 48 \, a^{15} b^{3} x + a^{18} - 27 \,{\left (8 \, a b^{17} x^{5} - 244 \, a^{4} b^{14} x^{4} + 840 \, a^{7} b^{11} x^{3} - 553 \, a^{10} b^{8} x^{2} + 56 \, a^{13} b^{5} x\right )} x^{\frac{2}{3}} + 27 \,{\left (35 \, a^{2} b^{16} x^{5} - 448 \, a^{5} b^{13} x^{4} + 876 \, a^{8} b^{10} x^{3} - 328 \, a^{11} b^{7} x^{2} + 14 \, a^{14} b^{4} x\right )} x^{\frac{1}{3}}}{56 \,{\left (b^{27} x^{8} + 8 \, a^{3} b^{24} x^{7} + 28 \, a^{6} b^{21} x^{6} + 56 \, a^{9} b^{18} x^{5} + 70 \, a^{12} b^{15} x^{4} + 56 \, a^{15} b^{12} x^{3} + 28 \, a^{18} b^{9} x^{2} + 8 \, a^{21} b^{6} x + a^{24} 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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