Optimal. Leaf size=137 \[ -\frac{a^2}{2 b^3 \left (a+b \sqrt [3]{x}\right )^5 \sqrt{a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}}}+\frac{6 a}{5 b^3 \left (a+b \sqrt [3]{x}\right )^4 \sqrt{a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}}}-\frac{3}{4 b^3 \left (a+b \sqrt [3]{x}\right )^3 \sqrt{a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}}} \]
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Rubi [A] time = 0.0779012, 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} \[ -\frac{a^2}{2 b^3 \left (a+b \sqrt [3]{x}\right )^5 \sqrt{a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}}}+\frac{6 a}{5 b^3 \left (a+b \sqrt [3]{x}\right )^4 \sqrt{a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}}}-\frac{3}{4 b^3 \left (a+b \sqrt [3]{x}\right )^3 \sqrt{a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}}} \]
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 )^{7/2}} \, dx &=3 \operatorname{Subst}\left (\int \frac{x^2}{\left (a^2+2 a b x+b^2 x^2\right )^{7/2}} \, dx,x,\sqrt [3]{x}\right )\\ &=\frac{\left (3 b^7 \left (a+b \sqrt [3]{x}\right )\right ) \operatorname{Subst}\left (\int \frac{x^2}{\left (a b+b^2 x\right )^7} \, dx,x,\sqrt [3]{x}\right )}{\sqrt{a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}}}\\ &=\frac{\left (3 b^7 \left (a+b \sqrt [3]{x}\right )\right ) \operatorname{Subst}\left (\int \left (\frac{a^2}{b^9 (a+b x)^7}-\frac{2 a}{b^9 (a+b x)^6}+\frac{1}{b^9 (a+b x)^5}\right ) \, dx,x,\sqrt [3]{x}\right )}{\sqrt{a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}}}\\ &=-\frac{a^2}{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}}}+\frac{6 a}{5 b^3 \left (a+b \sqrt [3]{x}\right )^4 \sqrt{a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}}}-\frac{3}{4 b^3 \left (a+b \sqrt [3]{x}\right )^3 \sqrt{a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}}}\\ \end{align*}
Mathematica [A] time = 0.0371616, size = 58, normalized size = 0.42 \[ \frac{-a^2-6 a b \sqrt [3]{x}-15 b^2 x^{2/3}}{20 b^3 \left (a+b \sqrt [3]{x}\right )^5 \sqrt{\left (a+b \sqrt [3]{x}\right )^2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.007, size = 54, normalized size = 0.4 \begin{align*} -{\frac{1}{20\,{b}^{3}}\sqrt{{a}^{2}+2\,ab\sqrt [3]{x}+{b}^{2}{x}^{{\frac{2}{3}}}} \left ( 15\,{b}^{2}{x}^{2/3}+6\,ab\sqrt [3]{x}+{a}^{2} \right ) \left ( a+b\sqrt [3]{x} \right ) ^{-7}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.10424, size = 85, normalized size = 0.62 \begin{align*} -\frac{a^{2} b^{2}}{2 \,{\left (b^{2}\right )}^{\frac{11}{2}}{\left (x^{\frac{1}{3}} + \frac{a}{b}\right )}^{6}} + \frac{6 \, a b}{5 \,{\left (b^{2}\right )}^{\frac{9}{2}}{\left (x^{\frac{1}{3}} + \frac{a}{b}\right )}^{5}} - \frac{3}{4 \,{\left (b^{2}\right )}^{\frac{7}{2}}{\left (x^{\frac{1}{3}} + \frac{a}{b}\right )}^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.93016, size = 477, normalized size = 3.48 \begin{align*} -\frac{280 \, a^{2} b^{12} x^{4} - 1400 \, a^{5} b^{9} x^{3} + 735 \, a^{8} b^{6} x^{2} - 14 \, a^{11} b^{3} x + a^{14} + 3 \,{\left (5 \, b^{14} x^{4} - 210 \, a^{3} b^{11} x^{3} + 483 \, a^{6} b^{8} x^{2} - 112 \, a^{9} b^{5} x\right )} x^{\frac{2}{3}} - 3 \,{\left (28 \, a b^{13} x^{4} - 357 \, a^{4} b^{10} x^{3} + 390 \, a^{7} b^{7} x^{2} - 35 \, a^{10} b^{4} x\right )} x^{\frac{1}{3}}}{20 \,{\left (b^{21} x^{6} + 6 \, a^{3} b^{18} x^{5} + 15 \, a^{6} b^{15} x^{4} + 20 \, a^{9} b^{12} x^{3} + 15 \, a^{12} b^{9} x^{2} + 6 \, a^{15} b^{6} x + a^{18} b^{3}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (a^{2} + 2 a b \sqrt [3]{x} + b^{2} x^{\frac{2}{3}}\right )^{\frac{7}{2}}}\, dx \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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