Integrand size = 24, antiderivative size = 64 \[ \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^p}{x^3} \, dx=\frac {b \left (a+b x^2\right ) \left (a^2+2 a b x^2+b^2 x^4\right )^p \operatorname {Hypergeometric2F1}\left (2,1+2 p,2 (1+p),1+\frac {b x^2}{a}\right )}{2 a^2 (1+2 p)} \]
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Time = 0.03 (sec) , antiderivative size = 64, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {1127, 272, 67} \[ \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^p}{x^3} \, dx=\frac {b \left (a+b x^2\right ) \left (a^2+2 a b x^2+b^2 x^4\right )^p \operatorname {Hypergeometric2F1}\left (2,2 p+1,2 (p+1),\frac {b x^2}{a}+1\right )}{2 a^2 (2 p+1)} \]
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Rule 67
Rule 272
Rule 1127
Rubi steps \begin{align*} \text {integral}& = \left (\left (1+\frac {b x^2}{a}\right )^{-2 p} \left (a^2+2 a b x^2+b^2 x^4\right )^p\right ) \int \frac {\left (1+\frac {b x^2}{a}\right )^{2 p}}{x^3} \, dx \\ & = \frac {1}{2} \left (\left (1+\frac {b x^2}{a}\right )^{-2 p} \left (a^2+2 a b x^2+b^2 x^4\right )^p\right ) \text {Subst}\left (\int \frac {\left (1+\frac {b x}{a}\right )^{2 p}}{x^2} \, dx,x,x^2\right ) \\ & = \frac {b \left (a+b x^2\right ) \left (a^2+2 a b x^2+b^2 x^4\right )^p \, _2F_1\left (2,1+2 p;2 (1+p);1+\frac {b x^2}{a}\right )}{2 a^2 (1+2 p)} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.86 \[ \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^p}{x^3} \, dx=\frac {b \left (a+b x^2\right ) \left (\left (a+b x^2\right )^2\right )^p \operatorname {Hypergeometric2F1}\left (2,1+2 p,2+2 p,1+\frac {b x^2}{a}\right )}{2 a^2 (1+2 p)} \]
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\[\int \frac {\left (b^{2} x^{4}+2 a b \,x^{2}+a^{2}\right )^{p}}{x^{3}}d x\]
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\[ \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^p}{x^3} \, dx=\int { \frac {{\left (b^{2} x^{4} + 2 \, a b x^{2} + a^{2}\right )}^{p}}{x^{3}} \,d x } \]
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\[ \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^p}{x^3} \, dx=\int \frac {\left (\left (a + b x^{2}\right )^{2}\right )^{p}}{x^{3}}\, dx \]
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\[ \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^p}{x^3} \, dx=\int { \frac {{\left (b^{2} x^{4} + 2 \, a b x^{2} + a^{2}\right )}^{p}}{x^{3}} \,d x } \]
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\[ \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^p}{x^3} \, dx=\int { \frac {{\left (b^{2} x^{4} + 2 \, a b x^{2} + a^{2}\right )}^{p}}{x^{3}} \,d x } \]
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Timed out. \[ \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^p}{x^3} \, dx=\int \frac {{\left (a^2+2\,a\,b\,x^2+b^2\,x^4\right )}^p}{x^3} \,d x \]
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