Integrand size = 26, antiderivative size = 268 \[ \int \frac {1}{\left (a^2+2 a b \sqrt [6]{x}+b^2 \sqrt [3]{x}\right )^{5/2}} \, dx=-\frac {60 a^2}{b^6 \sqrt {a^2+2 a b \sqrt [6]{x}+b^2 \sqrt [3]{x}}}+\frac {3 a^5}{2 b^6 \left (a+b \sqrt [6]{x}\right )^3 \sqrt {a^2+2 a b \sqrt [6]{x}+b^2 \sqrt [3]{x}}}-\frac {10 a^4}{b^6 \left (a+b \sqrt [6]{x}\right )^2 \sqrt {a^2+2 a b \sqrt [6]{x}+b^2 \sqrt [3]{x}}}+\frac {30 a^3}{b^6 \left (a+b \sqrt [6]{x}\right ) \sqrt {a^2+2 a b \sqrt [6]{x}+b^2 \sqrt [3]{x}}}+\frac {6 \left (a+b \sqrt [6]{x}\right ) \sqrt [6]{x}}{b^5 \sqrt {a^2+2 a b \sqrt [6]{x}+b^2 \sqrt [3]{x}}}-\frac {30 a \left (a+b \sqrt [6]{x}\right ) \log \left (a+b \sqrt [6]{x}\right )}{b^6 \sqrt {a^2+2 a b \sqrt [6]{x}+b^2 \sqrt [3]{x}}} \]
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Time = 0.10 (sec) , antiderivative size = 268, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {1355, 660, 45} \[ \int \frac {1}{\left (a^2+2 a b \sqrt [6]{x}+b^2 \sqrt [3]{x}\right )^{5/2}} \, dx=-\frac {60 a^2}{b^6 \sqrt {a^2+2 a b \sqrt [6]{x}+b^2 \sqrt [3]{x}}}-\frac {30 a \left (a+b \sqrt [6]{x}\right ) \log \left (a+b \sqrt [6]{x}\right )}{b^6 \sqrt {a^2+2 a b \sqrt [6]{x}+b^2 \sqrt [3]{x}}}+\frac {6 \sqrt [6]{x} \left (a+b \sqrt [6]{x}\right )}{b^5 \sqrt {a^2+2 a b \sqrt [6]{x}+b^2 \sqrt [3]{x}}}+\frac {3 a^5}{2 b^6 \left (a+b \sqrt [6]{x}\right )^3 \sqrt {a^2+2 a b \sqrt [6]{x}+b^2 \sqrt [3]{x}}}-\frac {10 a^4}{b^6 \left (a+b \sqrt [6]{x}\right )^2 \sqrt {a^2+2 a b \sqrt [6]{x}+b^2 \sqrt [3]{x}}}+\frac {30 a^3}{b^6 \left (a+b \sqrt [6]{x}\right ) \sqrt {a^2+2 a b \sqrt [6]{x}+b^2 \sqrt [3]{x}}} \]
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Rule 45
Rule 660
Rule 1355
Rubi steps \begin{align*} \text {integral}& = 6 \text {Subst}\left (\int \frac {x^5}{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx,x,\sqrt [6]{x}\right ) \\ & = \frac {\left (6 b^5 \left (a+b \sqrt [6]{x}\right )\right ) \text {Subst}\left (\int \frac {x^5}{\left (a b+b^2 x\right )^5} \, dx,x,\sqrt [6]{x}\right )}{\sqrt {a^2+2 a b \sqrt [6]{x}+b^2 \sqrt [3]{x}}} \\ & = \frac {\left (6 b^5 \left (a+b \sqrt [6]{x}\right )\right ) \text {Subst}\left (\int \left (\frac {1}{b^{10}}-\frac {a^5}{b^{10} (a+b x)^5}+\frac {5 a^4}{b^{10} (a+b x)^4}-\frac {10 a^3}{b^{10} (a+b x)^3}+\frac {10 a^2}{b^{10} (a+b x)^2}-\frac {5 a}{b^{10} (a+b x)}\right ) \, dx,x,\sqrt [6]{x}\right )}{\sqrt {a^2+2 a b \sqrt [6]{x}+b^2 \sqrt [3]{x}}} \\ & = -\frac {60 a^2}{b^6 \sqrt {a^2+2 a b \sqrt [6]{x}+b^2 \sqrt [3]{x}}}+\frac {3 a^5}{2 b^6 \left (a+b \sqrt [6]{x}\right )^3 \sqrt {a^2+2 a b \sqrt [6]{x}+b^2 \sqrt [3]{x}}}-\frac {10 a^4}{b^6 \left (a+b \sqrt [6]{x}\right )^2 \sqrt {a^2+2 a b \sqrt [6]{x}+b^2 \sqrt [3]{x}}}+\frac {30 a^3}{b^6 \left (a+b \sqrt [6]{x}\right ) \sqrt {a^2+2 a b \sqrt [6]{x}+b^2 \sqrt [3]{x}}}+\frac {6 \left (a+b \sqrt [6]{x}\right ) \sqrt [6]{x}}{b^5 \sqrt {a^2+2 a b \sqrt [6]{x}+b^2 \sqrt [3]{x}}}-\frac {30 a \left (a+b \sqrt [6]{x}\right ) \log \left (a+b \sqrt [6]{x}\right )}{b^6 \sqrt {a^2+2 a b \sqrt [6]{x}+b^2 \sqrt [3]{x}}} \\ \end{align*}
Time = 0.13 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.45 \[ \int \frac {1}{\left (a^2+2 a b \sqrt [6]{x}+b^2 \sqrt [3]{x}\right )^{5/2}} \, dx=\frac {-77 a^5-248 a^4 b \sqrt [6]{x}-252 a^3 b^2 \sqrt [3]{x}-48 a^2 b^3 \sqrt {x}+48 a b^4 x^{2/3}+12 b^5 x^{5/6}-60 a \left (a+b \sqrt [6]{x}\right )^4 \log \left (a+b \sqrt [6]{x}\right )}{2 b^6 \left (a+b \sqrt [6]{x}\right )^3 \sqrt {\left (a+b \sqrt [6]{x}\right )^2}} \]
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Time = 5.45 (sec) , antiderivative size = 163, normalized size of antiderivative = 0.61
method | result | size |
derivativedivides | \(-\frac {\left (60 \ln \left (a +b \,x^{\frac {1}{6}}\right ) a \,b^{4} x^{\frac {2}{3}}-12 b^{5} x^{\frac {5}{6}}+240 \ln \left (a +b \,x^{\frac {1}{6}}\right ) a^{2} b^{3} \sqrt {x}-48 a \,b^{4} x^{\frac {2}{3}}+360 \ln \left (a +b \,x^{\frac {1}{6}}\right ) a^{3} b^{2} x^{\frac {1}{3}}+48 a^{2} b^{3} \sqrt {x}+240 \ln \left (a +b \,x^{\frac {1}{6}}\right ) a^{4} b \,x^{\frac {1}{6}}+252 a^{3} b^{2} x^{\frac {1}{3}}+60 \ln \left (a +b \,x^{\frac {1}{6}}\right ) a^{5}+248 a^{4} b \,x^{\frac {1}{6}}+77 a^{5}\right ) \left (a +b \,x^{\frac {1}{6}}\right )}{2 b^{6} {\left (\left (a +b \,x^{\frac {1}{6}}\right )^{2}\right )}^{\frac {5}{2}}}\) | \(163\) |
default | \(\text {Expression too large to display}\) | \(5004\) |
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Timed out. \[ \int \frac {1}{\left (a^2+2 a b \sqrt [6]{x}+b^2 \sqrt [3]{x}\right )^{5/2}} \, dx=\text {Timed out} \]
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Timed out. \[ \int \frac {1}{\left (a^2+2 a b \sqrt [6]{x}+b^2 \sqrt [3]{x}\right )^{5/2}} \, dx=\text {Timed out} \]
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Time = 0.20 (sec) , antiderivative size = 119, normalized size of antiderivative = 0.44 \[ \int \frac {1}{\left (a^2+2 a b \sqrt [6]{x}+b^2 \sqrt [3]{x}\right )^{5/2}} \, dx=\frac {12 \, b^{5} x^{\frac {5}{6}} + 48 \, a b^{4} x^{\frac {2}{3}} - 48 \, a^{2} b^{3} \sqrt {x} - 252 \, a^{3} b^{2} x^{\frac {1}{3}} - 248 \, a^{4} b x^{\frac {1}{6}} - 77 \, a^{5}}{2 \, {\left (b^{10} x^{\frac {2}{3}} + 4 \, a b^{9} \sqrt {x} + 6 \, a^{2} b^{8} x^{\frac {1}{3}} + 4 \, a^{3} b^{7} x^{\frac {1}{6}} + a^{4} b^{6}\right )}} - \frac {30 \, a \log \left (b x^{\frac {1}{6}} + a\right )}{b^{6}} \]
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Time = 0.30 (sec) , antiderivative size = 105, normalized size of antiderivative = 0.39 \[ \int \frac {1}{\left (a^2+2 a b \sqrt [6]{x}+b^2 \sqrt [3]{x}\right )^{5/2}} \, dx=-\frac {30 \, a \log \left ({\left | b x^{\frac {1}{6}} + a \right |}\right )}{b^{6} \mathrm {sgn}\left (b x^{\frac {1}{6}} + a\right )} + \frac {6 \, x^{\frac {1}{6}}}{b^{5} \mathrm {sgn}\left (b x^{\frac {1}{6}} + a\right )} - \frac {120 \, a^{2} b^{3} \sqrt {x} + 300 \, a^{3} b^{2} x^{\frac {1}{3}} + 260 \, a^{4} b x^{\frac {1}{6}} + 77 \, a^{5}}{2 \, {\left (b x^{\frac {1}{6}} + a\right )}^{4} b^{6} \mathrm {sgn}\left (b x^{\frac {1}{6}} + a\right )} \]
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Timed out. \[ \int \frac {1}{\left (a^2+2 a b \sqrt [6]{x}+b^2 \sqrt [3]{x}\right )^{5/2}} \, dx=\int \frac {1}{{\left (a^2+b^2\,x^{1/3}+2\,a\,b\,x^{1/6}\right )}^{5/2}} \,d x \]
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