Integrand size = 26, antiderivative size = 291 \[ \int \left (a^2+\frac {b^2}{x^{2/3}}+\frac {2 a b}{\sqrt [3]{x}}\right )^{5/2} \, dx=-\frac {3 b^5 \sqrt {a^2+\frac {b^2}{x^{2/3}}+\frac {2 a b}{\sqrt [3]{x}}}}{2 \left (a+\frac {b}{\sqrt [3]{x}}\right ) x^{2/3}}-\frac {15 a b^4 \sqrt {a^2+\frac {b^2}{x^{2/3}}+\frac {2 a b}{\sqrt [3]{x}}}}{\left (a+\frac {b}{\sqrt [3]{x}}\right ) \sqrt [3]{x}}+\frac {30 a^3 b^2 \sqrt {a^2+\frac {b^2}{x^{2/3}}+\frac {2 a b}{\sqrt [3]{x}}} \sqrt [3]{x}}{a+\frac {b}{\sqrt [3]{x}}}+\frac {15 a^4 b \sqrt {a^2+\frac {b^2}{x^{2/3}}+\frac {2 a b}{\sqrt [3]{x}}} x^{2/3}}{2 \left (a+\frac {b}{\sqrt [3]{x}}\right )}+\frac {a^5 \sqrt {a^2+\frac {b^2}{x^{2/3}}+\frac {2 a b}{\sqrt [3]{x}}} x}{a+\frac {b}{\sqrt [3]{x}}}+\frac {30 a^2 b^3 \sqrt {a^2+\frac {b^2}{x^{2/3}}+\frac {2 a b}{\sqrt [3]{x}}} \log \left (\sqrt [3]{x}\right )}{a+\frac {b}{\sqrt [3]{x}}} \]
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Time = 0.09 (sec) , antiderivative size = 291, 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} \[ \int \left (a^2+\frac {b^2}{x^{2/3}}+\frac {2 a b}{\sqrt [3]{x}}\right )^{5/2} \, dx=-\frac {3 b^5 \sqrt {a^2+\frac {2 a b}{\sqrt [3]{x}}+\frac {b^2}{x^{2/3}}}}{2 x^{2/3} \left (a+\frac {b}{\sqrt [3]{x}}\right )}-\frac {15 a b^4 \sqrt {a^2+\frac {2 a b}{\sqrt [3]{x}}+\frac {b^2}{x^{2/3}}}}{\sqrt [3]{x} \left (a+\frac {b}{\sqrt [3]{x}}\right )}+\frac {30 a^2 b^3 \log \left (\sqrt [3]{x}\right ) \sqrt {a^2+\frac {2 a b}{\sqrt [3]{x}}+\frac {b^2}{x^{2/3}}}}{a+\frac {b}{\sqrt [3]{x}}}+\frac {a^5 x \sqrt {a^2+\frac {2 a b}{\sqrt [3]{x}}+\frac {b^2}{x^{2/3}}}}{a+\frac {b}{\sqrt [3]{x}}}+\frac {15 a^4 b x^{2/3} \sqrt {a^2+\frac {2 a b}{\sqrt [3]{x}}+\frac {b^2}{x^{2/3}}}}{2 \left (a+\frac {b}{\sqrt [3]{x}}\right )}+\frac {30 a^3 b^2 \sqrt [3]{x} \sqrt {a^2+\frac {2 a b}{\sqrt [3]{x}}+\frac {b^2}{x^{2/3}}}}{a+\frac {b}{\sqrt [3]{x}}} \]
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Rule 45
Rule 269
Rule 1355
Rule 1369
Rubi steps \begin{align*} \text {integral}& = 3 \text {Subst}\left (\int \left (a^2+\frac {b^2}{x^2}+\frac {2 a b}{x}\right )^{5/2} x^2 \, dx,x,\sqrt [3]{x}\right ) \\ & = \frac {\left (3 \sqrt {a^2+\frac {b^2}{x^{2/3}}+\frac {2 a b}{\sqrt [3]{x}}}\right ) \text {Subst}\left (\int \left (a b+\frac {b^2}{x}\right )^5 x^2 \, dx,x,\sqrt [3]{x}\right )}{b^4 \left (a b+\frac {b^2}{\sqrt [3]{x}}\right )} \\ & = \frac {\left (3 \sqrt {a^2+\frac {b^2}{x^{2/3}}+\frac {2 a b}{\sqrt [3]{x}}}\right ) \text {Subst}\left (\int \frac {\left (b^2+a b x\right )^5}{x^3} \, dx,x,\sqrt [3]{x}\right )}{b^4 \left (a b+\frac {b^2}{\sqrt [3]{x}}\right )} \\ & = \frac {\left (3 \sqrt {a^2+\frac {b^2}{x^{2/3}}+\frac {2 a b}{\sqrt [3]{x}}}\right ) \text {Subst}\left (\int \left (10 a^3 b^7+\frac {b^{10}}{x^3}+\frac {5 a b^9}{x^2}+\frac {10 a^2 b^8}{x}+5 a^4 b^6 x+a^5 b^5 x^2\right ) \, dx,x,\sqrt [3]{x}\right )}{b^4 \left (a b+\frac {b^2}{\sqrt [3]{x}}\right )} \\ & = -\frac {3 b^6 \sqrt {a^2+\frac {b^2}{x^{2/3}}+\frac {2 a b}{\sqrt [3]{x}}}}{2 \left (a b+\frac {b^2}{\sqrt [3]{x}}\right ) x^{2/3}}-\frac {15 a b^5 \sqrt {a^2+\frac {b^2}{x^{2/3}}+\frac {2 a b}{\sqrt [3]{x}}}}{\left (a b+\frac {b^2}{\sqrt [3]{x}}\right ) \sqrt [3]{x}}+\frac {30 a^3 b^3 \sqrt {a^2+\frac {b^2}{x^{2/3}}+\frac {2 a b}{\sqrt [3]{x}}} \sqrt [3]{x}}{a b+\frac {b^2}{\sqrt [3]{x}}}+\frac {15 a^4 b^2 \sqrt {a^2+\frac {b^2}{x^{2/3}}+\frac {2 a b}{\sqrt [3]{x}}} x^{2/3}}{2 \left (a b+\frac {b^2}{\sqrt [3]{x}}\right )}+\frac {a^5 \sqrt {a^2+\frac {b^2}{x^{2/3}}+\frac {2 a b}{\sqrt [3]{x}}} x}{a+\frac {b}{\sqrt [3]{x}}}+\frac {10 a^2 b^4 \sqrt {a^2+\frac {b^2}{x^{2/3}}+\frac {2 a b}{\sqrt [3]{x}}} \log (x)}{a b+\frac {b^2}{\sqrt [3]{x}}} \\ \end{align*}
Time = 0.06 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.34 \[ \int \left (a^2+\frac {b^2}{x^{2/3}}+\frac {2 a b}{\sqrt [3]{x}}\right )^{5/2} \, dx=\frac {\left (b+a \sqrt [3]{x}\right ) \left (-3 b^5-30 a b^4 \sqrt [3]{x}+60 a^3 b^2 x+15 a^4 b x^{4/3}+2 a^5 x^{5/3}+20 a^2 b^3 x^{2/3} \log (x)\right )}{2 \sqrt {\frac {\left (b+a \sqrt [3]{x}\right )^2}{x^{2/3}}} x} \]
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Time = 0.04 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.31
method | result | size |
derivativedivides | \(\frac {\left (\frac {x^{\frac {2}{3}} a^{2}+2 a b \,x^{\frac {1}{3}}+b^{2}}{x^{\frac {2}{3}}}\right )^{\frac {5}{2}} x \left (2 a^{5} x^{\frac {5}{3}}+15 b \,a^{4} x^{\frac {4}{3}}+20 a^{2} b^{3} \ln \left (x \right ) x^{\frac {2}{3}}+60 a^{3} b^{2} x -30 b^{4} a \,x^{\frac {1}{3}}-3 b^{5}\right )}{2 \left (b +a \,x^{\frac {1}{3}}\right )^{5}}\) | \(91\) |
default | \(\frac {\left (\frac {x^{\frac {2}{3}} a^{2}+2 a b \,x^{\frac {1}{3}}+b^{2}}{x^{\frac {2}{3}}}\right )^{\frac {5}{2}} x \left (2 a^{5} x^{\frac {5}{3}}+15 b \,a^{4} x^{\frac {4}{3}}+20 a^{2} b^{3} \ln \left (x \right ) x^{\frac {2}{3}}+60 a^{3} b^{2} x -30 b^{4} a \,x^{\frac {1}{3}}-3 b^{5}\right )}{2 \left (b +a \,x^{\frac {1}{3}}\right )^{5}}\) | \(91\) |
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Timed out. \[ \int \left (a^2+\frac {b^2}{x^{2/3}}+\frac {2 a b}{\sqrt [3]{x}}\right )^{5/2} \, dx=\text {Timed out} \]
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\[ \int \left (a^2+\frac {b^2}{x^{2/3}}+\frac {2 a b}{\sqrt [3]{x}}\right )^{5/2} \, dx=\int \left (a^{2} + \frac {2 a b}{\sqrt [3]{x}} + \frac {b^{2}}{x^{\frac {2}{3}}}\right )^{\frac {5}{2}}\, dx \]
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Time = 0.19 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.20 \[ \int \left (a^2+\frac {b^2}{x^{2/3}}+\frac {2 a b}{\sqrt [3]{x}}\right )^{5/2} \, dx=10 \, a^{2} b^{3} \log \left (x\right ) + \frac {2 \, a^{5} x^{\frac {5}{3}} + 15 \, a^{4} b x^{\frac {4}{3}} + 60 \, a^{3} b^{2} x - 30 \, a b^{4} x^{\frac {1}{3}} - 3 \, b^{5}}{2 \, x^{\frac {2}{3}}} \]
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Time = 0.33 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.44 \[ \int \left (a^2+\frac {b^2}{x^{2/3}}+\frac {2 a b}{\sqrt [3]{x}}\right )^{5/2} \, dx=a^{5} x \mathrm {sgn}\left (a x + b x^{\frac {2}{3}}\right ) \mathrm {sgn}\left (x\right ) + 10 \, a^{2} b^{3} \log \left ({\left | x \right |}\right ) \mathrm {sgn}\left (a x + b x^{\frac {2}{3}}\right ) \mathrm {sgn}\left (x\right ) + \frac {15}{2} \, a^{4} b x^{\frac {2}{3}} \mathrm {sgn}\left (a x + b x^{\frac {2}{3}}\right ) \mathrm {sgn}\left (x\right ) + 30 \, a^{3} b^{2} x^{\frac {1}{3}} \mathrm {sgn}\left (a x + b x^{\frac {2}{3}}\right ) \mathrm {sgn}\left (x\right ) - \frac {3 \, {\left (10 \, a b^{4} x^{\frac {1}{3}} \mathrm {sgn}\left (a x + b x^{\frac {2}{3}}\right ) \mathrm {sgn}\left (x\right ) + b^{5} \mathrm {sgn}\left (a x + b x^{\frac {2}{3}}\right ) \mathrm {sgn}\left (x\right )\right )}}{2 \, x^{\frac {2}{3}}} \]
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Timed out. \[ \int \left (a^2+\frac {b^2}{x^{2/3}}+\frac {2 a b}{\sqrt [3]{x}}\right )^{5/2} \, dx=\int {\left (a^2+\frac {b^2}{x^{2/3}}+\frac {2\,a\,b}{x^{1/3}}\right )}^{5/2} \,d x \]
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