Integrand size = 26, antiderivative size = 391 \[ \int \left (a^2+\frac {b^2}{x^{2/3}}+\frac {2 a b}{\sqrt [3]{x}}\right )^{7/2} \, dx=-\frac {3 b^7 \sqrt {a^2+\frac {b^2}{x^{2/3}}+\frac {2 a b}{\sqrt [3]{x}}}}{4 \left (a+\frac {b}{\sqrt [3]{x}}\right ) x^{4/3}}-\frac {7 a b^6 \sqrt {a^2+\frac {b^2}{x^{2/3}}+\frac {2 a b}{\sqrt [3]{x}}}}{\left (a+\frac {b}{\sqrt [3]{x}}\right ) x}-\frac {63 a^2 b^5 \sqrt {a^2+\frac {b^2}{x^{2/3}}+\frac {2 a b}{\sqrt [3]{x}}}}{2 \left (a+\frac {b}{\sqrt [3]{x}}\right ) x^{2/3}}-\frac {105 a^3 b^4 \sqrt {a^2+\frac {b^2}{x^{2/3}}+\frac {2 a b}{\sqrt [3]{x}}}}{\left (a+\frac {b}{\sqrt [3]{x}}\right ) \sqrt [3]{x}}+\frac {63 a^5 b^2 \sqrt {a^2+\frac {b^2}{x^{2/3}}+\frac {2 a b}{\sqrt [3]{x}}} \sqrt [3]{x}}{a+\frac {b}{\sqrt [3]{x}}}+\frac {21 a^6 b \sqrt {a^2+\frac {b^2}{x^{2/3}}+\frac {2 a b}{\sqrt [3]{x}}} x^{2/3}}{2 \left (a+\frac {b}{\sqrt [3]{x}}\right )}+\frac {a^7 \sqrt {a^2+\frac {b^2}{x^{2/3}}+\frac {2 a b}{\sqrt [3]{x}}} x}{a+\frac {b}{\sqrt [3]{x}}}+\frac {105 a^4 b^3 \sqrt {a^2+\frac {b^2}{x^{2/3}}+\frac {2 a b}{\sqrt [3]{x}}} \log \left (\sqrt [3]{x}\right )}{a+\frac {b}{\sqrt [3]{x}}} \]
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Time = 0.13 (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} \[ \int \left (a^2+\frac {b^2}{x^{2/3}}+\frac {2 a b}{\sqrt [3]{x}}\right )^{7/2} \, dx=-\frac {3 b^7 \sqrt {a^2+\frac {2 a b}{\sqrt [3]{x}}+\frac {b^2}{x^{2/3}}}}{4 x^{4/3} \left (a+\frac {b}{\sqrt [3]{x}}\right )}-\frac {7 a b^6 \sqrt {a^2+\frac {2 a b}{\sqrt [3]{x}}+\frac {b^2}{x^{2/3}}}}{x \left (a+\frac {b}{\sqrt [3]{x}}\right )}-\frac {63 a^2 b^5 \sqrt {a^2+\frac {2 a b}{\sqrt [3]{x}}+\frac {b^2}{x^{2/3}}}}{2 x^{2/3} \left (a+\frac {b}{\sqrt [3]{x}}\right )}+\frac {a^7 x \sqrt {a^2+\frac {2 a b}{\sqrt [3]{x}}+\frac {b^2}{x^{2/3}}}}{a+\frac {b}{\sqrt [3]{x}}}+\frac {21 a^6 b x^{2/3} \sqrt {a^2+\frac {2 a b}{\sqrt [3]{x}}+\frac {b^2}{x^{2/3}}}}{2 \left (a+\frac {b}{\sqrt [3]{x}}\right )}+\frac {63 a^5 b^2 \sqrt [3]{x} \sqrt {a^2+\frac {2 a b}{\sqrt [3]{x}}+\frac {b^2}{x^{2/3}}}}{a+\frac {b}{\sqrt [3]{x}}}+\frac {105 a^4 b^3 \log \left (\sqrt [3]{x}\right ) \sqrt {a^2+\frac {2 a b}{\sqrt [3]{x}}+\frac {b^2}{x^{2/3}}}}{a+\frac {b}{\sqrt [3]{x}}}-\frac {105 a^3 b^4 \sqrt {a^2+\frac {2 a b}{\sqrt [3]{x}}+\frac {b^2}{x^{2/3}}}}{\sqrt [3]{x} \left (a+\frac {b}{\sqrt [3]{x}}\right )} \]
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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 )^{7/2} 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 )^7 x^2 \, dx,x,\sqrt [3]{x}\right )}{b^6 \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 )^7}{x^5} \, dx,x,\sqrt [3]{x}\right )}{b^6 \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 (21 a^5 b^9+\frac {b^{14}}{x^5}+\frac {7 a b^{13}}{x^4}+\frac {21 a^2 b^{12}}{x^3}+\frac {35 a^3 b^{11}}{x^2}+\frac {35 a^4 b^{10}}{x}+7 a^6 b^8 x+a^7 b^7 x^2\right ) \, dx,x,\sqrt [3]{x}\right )}{b^6 \left (a b+\frac {b^2}{\sqrt [3]{x}}\right )} \\ & = -\frac {3 b^8 \sqrt {a^2+\frac {b^2}{x^{2/3}}+\frac {2 a b}{\sqrt [3]{x}}}}{4 \left (a b+\frac {b^2}{\sqrt [3]{x}}\right ) x^{4/3}}-\frac {7 a b^7 \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 ) x}-\frac {63 a^2 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 {105 a^3 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 {63 a^5 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 {21 a^6 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^7 \sqrt {a^2+\frac {b^2}{x^{2/3}}+\frac {2 a b}{\sqrt [3]{x}}} x}{a+\frac {b}{\sqrt [3]{x}}}+\frac {35 a^4 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 = 0.07 (sec) , antiderivative size = 125, normalized size of antiderivative = 0.32 \[ \int \left (a^2+\frac {b^2}{x^{2/3}}+\frac {2 a b}{\sqrt [3]{x}}\right )^{7/2} \, dx=\frac {\sqrt {\frac {\left (b+a \sqrt [3]{x}\right )^2}{x^{2/3}}} \left (-3 b^7-28 a b^6 \sqrt [3]{x}-126 a^2 b^5 x^{2/3}-420 a^3 b^4 x+252 a^5 b^2 x^{5/3}+42 a^6 b x^2+4 a^7 x^{7/3}+140 a^4 b^3 x^{4/3} \log (x)\right )}{4 \left (b+a \sqrt [3]{x}\right ) x} \]
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Time = 0.13 (sec) , antiderivative size = 113, normalized size of antiderivative = 0.29
method | result | size |
derivativedivides | \(\frac {\left (\frac {x^{\frac {2}{3}} a^{2}+2 a b \,x^{\frac {1}{3}}+b^{2}}{x^{\frac {2}{3}}}\right )^{\frac {7}{2}} x \left (4 a^{7} x^{\frac {7}{3}}+42 x^{2} a^{6} b +140 a^{4} b^{3} \ln \left (x \right ) x^{\frac {4}{3}}+252 b^{2} a^{5} x^{\frac {5}{3}}-420 b^{4} a^{3} x -126 a^{2} b^{5} x^{\frac {2}{3}}-28 a \,b^{6} x^{\frac {1}{3}}-3 b^{7}\right )}{4 \left (b +a \,x^{\frac {1}{3}}\right )^{7}}\) | \(113\) |
default | \(\frac {\left (\frac {x^{\frac {2}{3}} a^{2}+2 a b \,x^{\frac {1}{3}}+b^{2}}{x^{\frac {2}{3}}}\right )^{\frac {7}{2}} \left (42 a^{6} b \,x^{3}+252 b^{2} a^{5} x^{\frac {8}{3}}+140 a^{4} b^{3} \ln \left (x \right ) x^{\frac {7}{3}}+4 a^{7} x^{\frac {10}{3}}-28 a \,b^{6} x^{\frac {4}{3}}-420 b^{4} a^{3} x^{2}-126 a^{2} b^{5} x^{\frac {5}{3}}-3 b^{7} x \right )}{4 \left (b +a \,x^{\frac {1}{3}}\right )^{7}}\) | \(115\) |
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Timed out. \[ \int \left (a^2+\frac {b^2}{x^{2/3}}+\frac {2 a b}{\sqrt [3]{x}}\right )^{7/2} \, dx=\text {Timed out} \]
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\[ \int \left (a^2+\frac {b^2}{x^{2/3}}+\frac {2 a b}{\sqrt [3]{x}}\right )^{7/2} \, dx=\int \left (a^{2} + \frac {2 a b}{\sqrt [3]{x}} + \frac {b^{2}}{x^{\frac {2}{3}}}\right )^{\frac {7}{2}}\, dx \]
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Time = 0.21 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.20 \[ \int \left (a^2+\frac {b^2}{x^{2/3}}+\frac {2 a b}{\sqrt [3]{x}}\right )^{7/2} \, dx=35 \, a^{4} b^{3} \log \left (x\right ) + \frac {4 \, a^{7} x^{\frac {7}{3}} + 42 \, a^{6} b x^{2} + 252 \, a^{5} b^{2} x^{\frac {5}{3}} - 420 \, a^{3} b^{4} x - 126 \, a^{2} b^{5} x^{\frac {2}{3}} - 28 \, a b^{6} x^{\frac {1}{3}} - 3 \, b^{7}}{4 \, x^{\frac {4}{3}}} \]
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Time = 0.32 (sec) , antiderivative size = 173, normalized size of antiderivative = 0.44 \[ \int \left (a^2+\frac {b^2}{x^{2/3}}+\frac {2 a b}{\sqrt [3]{x}}\right )^{7/2} \, dx=a^{7} x \mathrm {sgn}\left (a x + b x^{\frac {2}{3}}\right ) \mathrm {sgn}\left (x\right ) + 35 \, a^{4} b^{3} \log \left ({\left | x \right |}\right ) \mathrm {sgn}\left (a x + b x^{\frac {2}{3}}\right ) \mathrm {sgn}\left (x\right ) + \frac {21}{2} \, a^{6} b x^{\frac {2}{3}} \mathrm {sgn}\left (a x + b x^{\frac {2}{3}}\right ) \mathrm {sgn}\left (x\right ) + 63 \, a^{5} b^{2} x^{\frac {1}{3}} \mathrm {sgn}\left (a x + b x^{\frac {2}{3}}\right ) \mathrm {sgn}\left (x\right ) - \frac {420 \, a^{3} b^{4} x \mathrm {sgn}\left (a x + b x^{\frac {2}{3}}\right ) \mathrm {sgn}\left (x\right ) + 126 \, a^{2} b^{5} x^{\frac {2}{3}} \mathrm {sgn}\left (a x + b x^{\frac {2}{3}}\right ) \mathrm {sgn}\left (x\right ) + 28 \, a b^{6} x^{\frac {1}{3}} \mathrm {sgn}\left (a x + b x^{\frac {2}{3}}\right ) \mathrm {sgn}\left (x\right ) + 3 \, b^{7} \mathrm {sgn}\left (a x + b x^{\frac {2}{3}}\right ) \mathrm {sgn}\left (x\right )}{4 \, x^{\frac {4}{3}}} \]
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Timed out. \[ \int \left (a^2+\frac {b^2}{x^{2/3}}+\frac {2 a b}{\sqrt [3]{x}}\right )^{7/2} \, dx=\int {\left (a^2+\frac {b^2}{x^{2/3}}+\frac {2\,a\,b}{x^{1/3}}\right )}^{7/2} \,d x \]
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