Integrand size = 26, antiderivative size = 391 \[ \int \left (a^2+\frac {b^2}{\sqrt [3]{x}}+\frac {2 a b}{\sqrt [6]{x}}\right )^{7/2} \, dx=-\frac {6 b^7 \sqrt {a^2+\frac {b^2}{\sqrt [3]{x}}+\frac {2 a b}{\sqrt [6]{x}}}}{\left (a+\frac {b}{\sqrt [6]{x}}\right ) \sqrt [6]{x}}+\frac {126 a^2 b^5 \sqrt {a^2+\frac {b^2}{\sqrt [3]{x}}+\frac {2 a b}{\sqrt [6]{x}}} \sqrt [6]{x}}{a+\frac {b}{\sqrt [6]{x}}}+\frac {105 a^3 b^4 \sqrt {a^2+\frac {b^2}{\sqrt [3]{x}}+\frac {2 a b}{\sqrt [6]{x}}} \sqrt [3]{x}}{a+\frac {b}{\sqrt [6]{x}}}+\frac {70 a^4 b^3 \sqrt {a^2+\frac {b^2}{\sqrt [3]{x}}+\frac {2 a b}{\sqrt [6]{x}}} \sqrt {x}}{a+\frac {b}{\sqrt [6]{x}}}+\frac {63 a^5 b^2 \sqrt {a^2+\frac {b^2}{\sqrt [3]{x}}+\frac {2 a b}{\sqrt [6]{x}}} x^{2/3}}{2 \left (a+\frac {b}{\sqrt [6]{x}}\right )}+\frac {42 a^6 b \sqrt {a^2+\frac {b^2}{\sqrt [3]{x}}+\frac {2 a b}{\sqrt [6]{x}}} x^{5/6}}{5 \left (a+\frac {b}{\sqrt [6]{x}}\right )}+\frac {a^7 \sqrt {a^2+\frac {b^2}{\sqrt [3]{x}}+\frac {2 a b}{\sqrt [6]{x}}} x}{a+\frac {b}{\sqrt [6]{x}}}+\frac {42 a b^6 \sqrt {a^2+\frac {b^2}{\sqrt [3]{x}}+\frac {2 a b}{\sqrt [6]{x}}} \log \left (\sqrt [6]{x}\right )}{a+\frac {b}{\sqrt [6]{x}}} \]
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Time = 0.12 (sec) , antiderivative size = 391, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {1355, 1369, 269, 45} \[ \int \left (a^2+\frac {b^2}{\sqrt [3]{x}}+\frac {2 a b}{\sqrt [6]{x}}\right )^{7/2} \, dx=-\frac {6 b^7 \sqrt {a^2+\frac {2 a b}{\sqrt [6]{x}}+\frac {b^2}{\sqrt [3]{x}}}}{\sqrt [6]{x} \left (a+\frac {b}{\sqrt [6]{x}}\right )}+\frac {42 a b^6 \log \left (\sqrt [6]{x}\right ) \sqrt {a^2+\frac {2 a b}{\sqrt [6]{x}}+\frac {b^2}{\sqrt [3]{x}}}}{a+\frac {b}{\sqrt [6]{x}}}+\frac {126 a^2 b^5 \sqrt [6]{x} \sqrt {a^2+\frac {2 a b}{\sqrt [6]{x}}+\frac {b^2}{\sqrt [3]{x}}}}{a+\frac {b}{\sqrt [6]{x}}}+\frac {a^7 x \sqrt {a^2+\frac {2 a b}{\sqrt [6]{x}}+\frac {b^2}{\sqrt [3]{x}}}}{a+\frac {b}{\sqrt [6]{x}}}+\frac {42 a^6 b x^{5/6} \sqrt {a^2+\frac {2 a b}{\sqrt [6]{x}}+\frac {b^2}{\sqrt [3]{x}}}}{5 \left (a+\frac {b}{\sqrt [6]{x}}\right )}+\frac {63 a^5 b^2 x^{2/3} \sqrt {a^2+\frac {2 a b}{\sqrt [6]{x}}+\frac {b^2}{\sqrt [3]{x}}}}{2 \left (a+\frac {b}{\sqrt [6]{x}}\right )}+\frac {70 a^4 b^3 \sqrt {x} \sqrt {a^2+\frac {2 a b}{\sqrt [6]{x}}+\frac {b^2}{\sqrt [3]{x}}}}{a+\frac {b}{\sqrt [6]{x}}}+\frac {105 a^3 b^4 \sqrt [3]{x} \sqrt {a^2+\frac {2 a b}{\sqrt [6]{x}}+\frac {b^2}{\sqrt [3]{x}}}}{a+\frac {b}{\sqrt [6]{x}}} \]
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Rule 45
Rule 269
Rule 1355
Rule 1369
Rubi steps \begin{align*} \text {integral}& = 6 \text {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 ) \text {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 ) \text {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 ) \text {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*}
Time = 0.07 (sec) , antiderivative size = 124, normalized size of antiderivative = 0.32 \[ \int \left (a^2+\frac {b^2}{\sqrt [3]{x}}+\frac {2 a b}{\sqrt [6]{x}}\right )^{7/2} \, dx=\frac {\sqrt {\frac {\left (b+a \sqrt [6]{x}\right )^2}{\sqrt [3]{x}}} \left (-60 b^7+1260 a^2 b^5 \sqrt [3]{x}+1050 a^3 b^4 \sqrt {x}+700 a^4 b^3 x^{2/3}+315 a^5 b^2 x^{5/6}+84 a^6 b x+10 a^7 x^{7/6}+70 a b^6 \sqrt [6]{x} \log (x)\right )}{10 \left (b+a \sqrt [6]{x}\right )} \]
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Time = 0.06 (sec) , antiderivative size = 117, normalized size of antiderivative = 0.30
\[\frac {\left (\frac {a^{2} \sqrt {x}+2 a b \,x^{\frac {1}{3}}+x^{\frac {1}{6}} b^{2}}{\sqrt {x}}\right )^{\frac {7}{2}} x \left (10 a^{7} x^{\frac {7}{6}}+84 a^{6} b x +315 b^{2} a^{5} x^{\frac {5}{6}}+700 a^{4} b^{3} x^{\frac {2}{3}}+1050 b^{4} a^{3} \sqrt {x}+70 a \,b^{6} \ln \left (x \right ) x^{\frac {1}{6}}+1260 a^{2} b^{5} x^{\frac {1}{3}}-60 b^{7}\right )}{10 \left (a \,x^{\frac {1}{6}}+b \right )^{7}}\]
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Timed out. \[ \int \left (a^2+\frac {b^2}{\sqrt [3]{x}}+\frac {2 a b}{\sqrt [6]{x}}\right )^{7/2} \, dx=\text {Timed out} \]
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Timed out. \[ \int \left (a^2+\frac {b^2}{\sqrt [3]{x}}+\frac {2 a b}{\sqrt [6]{x}}\right )^{7/2} \, dx=\text {Timed out} \]
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Time = 0.18 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.20 \[ \int \left (a^2+\frac {b^2}{\sqrt [3]{x}}+\frac {2 a b}{\sqrt [6]{x}}\right )^{7/2} \, dx=7 \, a b^{6} \log \left (x\right ) + \frac {10 \, a^{7} x^{\frac {7}{6}} + 84 \, a^{6} b x + 315 \, a^{5} b^{2} x^{\frac {5}{6}} + 700 \, a^{4} b^{3} x^{\frac {2}{3}} + 1050 \, a^{3} b^{4} \sqrt {x} + 1260 \, a^{2} b^{5} x^{\frac {1}{3}} - 60 \, b^{7}}{10 \, x^{\frac {1}{6}}} \]
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Time = 0.43 (sec) , antiderivative size = 172, normalized size of antiderivative = 0.44 \[ \int \left (a^2+\frac {b^2}{\sqrt [3]{x}}+\frac {2 a b}{\sqrt [6]{x}}\right )^{7/2} \, dx=a^{7} x \mathrm {sgn}\left (a x + b x^{\frac {5}{6}}\right ) \mathrm {sgn}\left (x\right ) + 7 \, a b^{6} \log \left ({\left | x \right |}\right ) \mathrm {sgn}\left (a x + b x^{\frac {5}{6}}\right ) \mathrm {sgn}\left (x\right ) + \frac {42}{5} \, a^{6} b x^{\frac {5}{6}} \mathrm {sgn}\left (a x + b x^{\frac {5}{6}}\right ) \mathrm {sgn}\left (x\right ) + \frac {63}{2} \, a^{5} b^{2} x^{\frac {2}{3}} \mathrm {sgn}\left (a x + b x^{\frac {5}{6}}\right ) \mathrm {sgn}\left (x\right ) + 70 \, a^{4} b^{3} \sqrt {x} \mathrm {sgn}\left (a x + b x^{\frac {5}{6}}\right ) \mathrm {sgn}\left (x\right ) + 105 \, a^{3} b^{4} x^{\frac {1}{3}} \mathrm {sgn}\left (a x + b x^{\frac {5}{6}}\right ) \mathrm {sgn}\left (x\right ) + 126 \, a^{2} b^{5} x^{\frac {1}{6}} \mathrm {sgn}\left (a x + b x^{\frac {5}{6}}\right ) \mathrm {sgn}\left (x\right ) - \frac {6 \, b^{7} \mathrm {sgn}\left (a x + b x^{\frac {5}{6}}\right ) \mathrm {sgn}\left (x\right )}{x^{\frac {1}{6}}} \]
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Timed out. \[ \int \left (a^2+\frac {b^2}{\sqrt [3]{x}}+\frac {2 a b}{\sqrt [6]{x}}\right )^{7/2} \, dx=\int {\left (a^2+\frac {b^2}{x^{1/3}}+\frac {2\,a\,b}{x^{1/6}}\right )}^{7/2} \,d x \]
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