Integrand size = 28, antiderivative size = 75 \[ \int \frac {\left (a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}\right )^p}{x^2} \, dx=\frac {3 b^3 \left (1+\frac {b \sqrt [3]{x}}{a}\right ) \left (a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}\right )^p \operatorname {Hypergeometric2F1}\left (4,1+2 p,2 (1+p),1+\frac {b \sqrt [3]{x}}{a}\right )}{a^3 (1+2 p)} \]
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Time = 0.02 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.107, Rules used = {1370, 272, 67} \[ \int \frac {\left (a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}\right )^p}{x^2} \, dx=\frac {3 b^3 \left (\frac {b \sqrt [3]{x}}{a}+1\right ) \left (a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}\right )^p \operatorname {Hypergeometric2F1}\left (4,2 p+1,2 (p+1),\frac {\sqrt [3]{x} b}{a}+1\right )}{a^3 (2 p+1)} \]
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Rule 67
Rule 272
Rule 1370
Rubi steps \begin{align*} \text {integral}& = \left (\left (1+\frac {b \sqrt [3]{x}}{a}\right )^{-2 p} \left (a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}\right )^p\right ) \int \frac {\left (1+\frac {b \sqrt [3]{x}}{a}\right )^{2 p}}{x^2} \, dx \\ & = \left (3 \left (1+\frac {b \sqrt [3]{x}}{a}\right )^{-2 p} \left (a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}\right )^p\right ) \text {Subst}\left (\int \frac {\left (1+\frac {b x}{a}\right )^{2 p}}{x^4} \, dx,x,\sqrt [3]{x}\right ) \\ & = \frac {3 b^3 \left (1+\frac {b \sqrt [3]{x}}{a}\right ) \left (a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}\right )^p \, _2F_1\left (4,1+2 p;2 (1+p);1+\frac {b \sqrt [3]{x}}{a}\right )}{a^3 (1+2 p)} \\ \end{align*}
Time = 0.10 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.81 \[ \int \frac {\left (a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}\right )^p}{x^2} \, dx=\frac {3 b^3 \left (a+b \sqrt [3]{x}\right ) \left (\left (a+b \sqrt [3]{x}\right )^2\right )^p \operatorname {Hypergeometric2F1}\left (4,1+2 p,2+2 p,1+\frac {b \sqrt [3]{x}}{a}\right )}{a^4 (1+2 p)} \]
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\[\int \frac {\left (a^{2}+2 a b \,x^{\frac {1}{3}}+b^{2} x^{\frac {2}{3}}\right )^{p}}{x^{2}}d x\]
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\[ \int \frac {\left (a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}\right )^p}{x^2} \, dx=\int { \frac {{\left (b^{2} x^{\frac {2}{3}} + 2 \, a b x^{\frac {1}{3}} + a^{2}\right )}^{p}}{x^{2}} \,d x } \]
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\[ \int \frac {\left (a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}\right )^p}{x^2} \, dx=\int \frac {\left (\left (a + b \sqrt [3]{x}\right )^{2}\right )^{p}}{x^{2}}\, dx \]
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\[ \int \frac {\left (a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}\right )^p}{x^2} \, dx=\int { \frac {{\left (b^{2} x^{\frac {2}{3}} + 2 \, a b x^{\frac {1}{3}} + a^{2}\right )}^{p}}{x^{2}} \,d x } \]
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\[ \int \frac {\left (a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}\right )^p}{x^2} \, dx=\int { \frac {{\left (b^{2} x^{\frac {2}{3}} + 2 \, a b x^{\frac {1}{3}} + a^{2}\right )}^{p}}{x^{2}} \,d x } \]
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Timed out. \[ \int \frac {\left (a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}\right )^p}{x^2} \, dx=\int \frac {{\left (a^2+b^2\,x^{2/3}+2\,a\,b\,x^{1/3}\right )}^p}{x^2} \,d x \]
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