Optimal. Leaf size=391 \[ \frac{42 a^6 b x^{5/6} \sqrt{a^2+\frac{2 a b}{\sqrt [6]{x}}+\frac{b^2}{\sqrt [3]{x}}}}{5 \left (a+\frac{b}{\sqrt [6]{x}}\right )}+\frac{63 a^5 b^2 x^{2/3} \sqrt{a^2+\frac{2 a b}{\sqrt [6]{x}}+\frac{b^2}{\sqrt [3]{x}}}}{2 \left (a+\frac{b}{\sqrt [6]{x}}\right )}+\frac{a^7 x \sqrt{a^2+\frac{2 a b}{\sqrt [6]{x}}+\frac{b^2}{\sqrt [3]{x}}}}{a+\frac{b}{\sqrt [6]{x}}}+\frac{70 a^4 b^3 \sqrt{x} \sqrt{a^2+\frac{2 a b}{\sqrt [6]{x}}+\frac{b^2}{\sqrt [3]{x}}}}{a+\frac{b}{\sqrt [6]{x}}}+\frac{105 a^3 b^4 \sqrt [3]{x} \sqrt{a^2+\frac{2 a b}{\sqrt [6]{x}}+\frac{b^2}{\sqrt [3]{x}}}}{a+\frac{b}{\sqrt [6]{x}}}+\frac{126 a^2 b^5 \sqrt [6]{x} \sqrt{a^2+\frac{2 a b}{\sqrt [6]{x}}+\frac{b^2}{\sqrt [3]{x}}}}{a+\frac{b}{\sqrt [6]{x}}}-\frac{6 b^7 \sqrt{a^2+\frac{2 a b}{\sqrt [6]{x}}+\frac{b^2}{\sqrt [3]{x}}}}{\sqrt [6]{x} \left (a+\frac{b}{\sqrt [6]{x}}\right )}+\frac{42 a b^6 \log \left (\sqrt [6]{x}\right ) \sqrt{a^2+\frac{2 a b}{\sqrt [6]{x}}+\frac{b^2}{\sqrt [3]{x}}}}{a+\frac{b}{\sqrt [6]{x}}} \]
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Rubi [A] time = 0.179674, antiderivative size = 391, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {1341, 1355, 263, 43} \[ \frac{42 a^6 b x^{5/6} \sqrt{a^2+\frac{2 a b}{\sqrt [6]{x}}+\frac{b^2}{\sqrt [3]{x}}}}{5 \left (a+\frac{b}{\sqrt [6]{x}}\right )}+\frac{63 a^5 b^2 x^{2/3} \sqrt{a^2+\frac{2 a b}{\sqrt [6]{x}}+\frac{b^2}{\sqrt [3]{x}}}}{2 \left (a+\frac{b}{\sqrt [6]{x}}\right )}+\frac{a^7 x \sqrt{a^2+\frac{2 a b}{\sqrt [6]{x}}+\frac{b^2}{\sqrt [3]{x}}}}{a+\frac{b}{\sqrt [6]{x}}}+\frac{70 a^4 b^3 \sqrt{x} \sqrt{a^2+\frac{2 a b}{\sqrt [6]{x}}+\frac{b^2}{\sqrt [3]{x}}}}{a+\frac{b}{\sqrt [6]{x}}}+\frac{105 a^3 b^4 \sqrt [3]{x} \sqrt{a^2+\frac{2 a b}{\sqrt [6]{x}}+\frac{b^2}{\sqrt [3]{x}}}}{a+\frac{b}{\sqrt [6]{x}}}+\frac{126 a^2 b^5 \sqrt [6]{x} \sqrt{a^2+\frac{2 a b}{\sqrt [6]{x}}+\frac{b^2}{\sqrt [3]{x}}}}{a+\frac{b}{\sqrt [6]{x}}}-\frac{6 b^7 \sqrt{a^2+\frac{2 a b}{\sqrt [6]{x}}+\frac{b^2}{\sqrt [3]{x}}}}{\sqrt [6]{x} \left (a+\frac{b}{\sqrt [6]{x}}\right )}+\frac{42 a b^6 \log \left (\sqrt [6]{x}\right ) \sqrt{a^2+\frac{2 a b}{\sqrt [6]{x}}+\frac{b^2}{\sqrt [3]{x}}}}{a+\frac{b}{\sqrt [6]{x}}} \]
Antiderivative was successfully verified.
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Rule 1341
Rule 1355
Rule 263
Rule 43
Rubi steps
\begin{align*} \int \left (a^2+\frac{b^2}{\sqrt [3]{x}}+\frac{2 a b}{\sqrt [6]{x}}\right )^{7/2} \, dx &=6 \operatorname{Subst}\left (\int \left (a^2+\frac{b^2}{x^2}+\frac{2 a b}{x}\right )^{7/2} x^5 \, dx,x,\sqrt [6]{x}\right )\\ &=\frac{\left (6 \sqrt{a^2+\frac{b^2}{\sqrt [3]{x}}+\frac{2 a b}{\sqrt [6]{x}}}\right ) \operatorname{Subst}\left (\int \left (a b+\frac{b^2}{x}\right )^7 x^5 \, dx,x,\sqrt [6]{x}\right )}{b^6 \left (a b+\frac{b^2}{\sqrt [6]{x}}\right )}\\ &=\frac{\left (6 \sqrt{a^2+\frac{b^2}{\sqrt [3]{x}}+\frac{2 a b}{\sqrt [6]{x}}}\right ) \operatorname{Subst}\left (\int \frac{\left (b^2+a b x\right )^7}{x^2} \, dx,x,\sqrt [6]{x}\right )}{b^6 \left (a b+\frac{b^2}{\sqrt [6]{x}}\right )}\\ &=\frac{\left (6 \sqrt{a^2+\frac{b^2}{\sqrt [3]{x}}+\frac{2 a b}{\sqrt [6]{x}}}\right ) \operatorname{Subst}\left (\int \left (21 a^2 b^{12}+\frac{b^{14}}{x^2}+\frac{7 a b^{13}}{x}+35 a^3 b^{11} x+35 a^4 b^{10} x^2+21 a^5 b^9 x^3+7 a^6 b^8 x^4+a^7 b^7 x^5\right ) \, dx,x,\sqrt [6]{x}\right )}{b^6 \left (a b+\frac{b^2}{\sqrt [6]{x}}\right )}\\ &=-\frac{6 b^8 \sqrt{a^2+\frac{b^2}{\sqrt [3]{x}}+\frac{2 a b}{\sqrt [6]{x}}}}{\left (a b+\frac{b^2}{\sqrt [6]{x}}\right ) \sqrt [6]{x}}+\frac{126 a^2 b^6 \sqrt{a^2+\frac{b^2}{\sqrt [3]{x}}+\frac{2 a b}{\sqrt [6]{x}}} \sqrt [6]{x}}{a b+\frac{b^2}{\sqrt [6]{x}}}+\frac{105 a^3 b^5 \sqrt{a^2+\frac{b^2}{\sqrt [3]{x}}+\frac{2 a b}{\sqrt [6]{x}}} \sqrt [3]{x}}{a b+\frac{b^2}{\sqrt [6]{x}}}+\frac{70 a^4 b^4 \sqrt{a^2+\frac{b^2}{\sqrt [3]{x}}+\frac{2 a b}{\sqrt [6]{x}}} \sqrt{x}}{a b+\frac{b^2}{\sqrt [6]{x}}}+\frac{63 a^5 b^3 \sqrt{a^2+\frac{b^2}{\sqrt [3]{x}}+\frac{2 a b}{\sqrt [6]{x}}} x^{2/3}}{2 \left (a b+\frac{b^2}{\sqrt [6]{x}}\right )}+\frac{42 a^6 b^2 \sqrt{a^2+\frac{b^2}{\sqrt [3]{x}}+\frac{2 a b}{\sqrt [6]{x}}} x^{5/6}}{5 \left (a b+\frac{b^2}{\sqrt [6]{x}}\right )}+\frac{a^7 \sqrt{a^2+\frac{b^2}{\sqrt [3]{x}}+\frac{2 a b}{\sqrt [6]{x}}} x}{a+\frac{b}{\sqrt [6]{x}}}+\frac{7 a b^7 \sqrt{a^2+\frac{b^2}{\sqrt [3]{x}}+\frac{2 a b}{\sqrt [6]{x}}} \log (x)}{a b+\frac{b^2}{\sqrt [6]{x}}}\\ \end{align*}
Mathematica [A] time = 0.0701364, size = 124, normalized size = 0.32 \[ \frac{\sqrt{\frac{\left (a \sqrt [6]{x}+b\right )^2}{\sqrt [3]{x}}} \left (315 a^5 b^2 x^{5/6}+700 a^4 b^3 x^{2/3}+1050 a^3 b^4 \sqrt{x}+1260 a^2 b^5 \sqrt [3]{x}+84 a^6 b x+10 a^7 x^{7/6}+70 a b^6 \sqrt [6]{x} \log (x)-60 b^7\right )}{10 \left (a \sqrt [6]{x}+b\right )} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.024, size = 116, normalized size = 0.3 \begin{align*}{\frac{1}{10}\sqrt{{ \left ( \sqrt{x}{a}^{2}+2\,ab\sqrt [3]{x}+{b}^{2}\sqrt [6]{x} \right ){\frac{1}{\sqrt{x}}}}} \left ( 84\,{a}^{6}bx+315\,{a}^{5}{b}^{2}{x}^{5/6}+70\,a{b}^{6}\ln \left ( x \right ) \sqrt [6]{x}+1050\,{a}^{3}{b}^{4}\sqrt{x}+1260\,{a}^{2}{b}^{5}\sqrt [3]{x}+700\,{a}^{4}{b}^{3}{x}^{2/3}+10\,{a}^{7}{x}^{7/6}-60\,{b}^{7} \right ) \left ( a\sqrt [6]{x}+b \right ) ^{-1}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.993768, size = 107, normalized size = 0.27 \begin{align*} 7 \, a b^{6} \log \left (x\right ) + \frac{10 \, a^{7} x^{\frac{7}{6}} + 84 \, a^{6} b x + 315 \, a^{5} b^{2} x^{\frac{5}{6}} + 700 \, a^{4} b^{3} x^{\frac{2}{3}} + 1050 \, a^{3} b^{4} \sqrt{x} + 1260 \, a^{2} b^{5} x^{\frac{1}{3}} - 60 \, b^{7}}{10 \, x^{\frac{1}{6}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [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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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 [A] time = 1.36387, size = 232, normalized size = 0.59 \begin{align*} a^{7} x \mathrm{sgn}\left (a x + b x^{\frac{5}{6}}\right ) \mathrm{sgn}\left (x\right ) + 7 \, a b^{6} \log \left ({\left | x \right |}\right ) \mathrm{sgn}\left (a x + b x^{\frac{5}{6}}\right ) \mathrm{sgn}\left (x\right ) + \frac{42}{5} \, a^{6} b x^{\frac{5}{6}} \mathrm{sgn}\left (a x + b x^{\frac{5}{6}}\right ) \mathrm{sgn}\left (x\right ) + \frac{63}{2} \, a^{5} b^{2} x^{\frac{2}{3}} \mathrm{sgn}\left (a x + b x^{\frac{5}{6}}\right ) \mathrm{sgn}\left (x\right ) + 70 \, a^{4} b^{3} \sqrt{x} \mathrm{sgn}\left (a x + b x^{\frac{5}{6}}\right ) \mathrm{sgn}\left (x\right ) + 105 \, a^{3} b^{4} x^{\frac{1}{3}} \mathrm{sgn}\left (a x + b x^{\frac{5}{6}}\right ) \mathrm{sgn}\left (x\right ) + 126 \, a^{2} b^{5} x^{\frac{1}{6}} \mathrm{sgn}\left (a x + b x^{\frac{5}{6}}\right ) \mathrm{sgn}\left (x\right ) - \frac{6 \, b^{7} \mathrm{sgn}\left (a x + b x^{\frac{5}{6}}\right ) \mathrm{sgn}\left (x\right )}{x^{\frac{1}{6}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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