Integrand size = 28, antiderivative size = 65 \[ \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^p}{(d x)^{3/2}} \, dx=-\frac {2 \left (1+\frac {b x^2}{a}\right )^{-2 p} \left (a^2+2 a b x^2+b^2 x^4\right )^p \operatorname {Hypergeometric2F1}\left (-\frac {1}{4},-2 p,\frac {3}{4},-\frac {b x^2}{a}\right )}{d \sqrt {d x}} \]
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Time = 0.02 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {1127, 371} \[ \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^p}{(d x)^{3/2}} \, dx=-\frac {2 \left (\frac {b x^2}{a}+1\right )^{-2 p} \left (a^2+2 a b x^2+b^2 x^4\right )^p \operatorname {Hypergeometric2F1}\left (-\frac {1}{4},-2 p,\frac {3}{4},-\frac {b x^2}{a}\right )}{d \sqrt {d x}} \]
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Rule 371
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}}{(d x)^{3/2}} \, dx \\ & = -\frac {2 \left (1+\frac {b x^2}{a}\right )^{-2 p} \left (a^2+2 a b x^2+b^2 x^4\right )^p \, _2F_1\left (-\frac {1}{4},-2 p;\frac {3}{4};-\frac {b x^2}{a}\right )}{d \sqrt {d x}} \\ \end{align*}
Time = 0.14 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.83 \[ \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^p}{(d x)^{3/2}} \, dx=-\frac {2 x \left (\left (a+b x^2\right )^2\right )^p \left (1+\frac {b x^2}{a}\right )^{-2 p} \operatorname {Hypergeometric2F1}\left (-\frac {1}{4},-2 p,\frac {3}{4},-\frac {b x^2}{a}\right )}{(d x)^{3/2}} \]
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\[\int \frac {\left (b^{2} x^{4}+2 a b \,x^{2}+a^{2}\right )^{p}}{\left (d x \right )^{\frac {3}{2}}}d x\]
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\[ \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^p}{(d x)^{3/2}} \, dx=\int { \frac {{\left (b^{2} x^{4} + 2 \, a b x^{2} + a^{2}\right )}^{p}}{\left (d x\right )^{\frac {3}{2}}} \,d x } \]
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\[ \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^p}{(d x)^{3/2}} \, dx=\int \frac {\left (\left (a + b x^{2}\right )^{2}\right )^{p}}{\left (d x\right )^{\frac {3}{2}}}\, dx \]
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\[ \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^p}{(d x)^{3/2}} \, dx=\int { \frac {{\left (b^{2} x^{4} + 2 \, a b x^{2} + a^{2}\right )}^{p}}{\left (d x\right )^{\frac {3}{2}}} \,d x } \]
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\[ \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^p}{(d x)^{3/2}} \, dx=\int { \frac {{\left (b^{2} x^{4} + 2 \, a b x^{2} + a^{2}\right )}^{p}}{\left (d x\right )^{\frac {3}{2}}} \,d x } \]
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Timed out. \[ \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^p}{(d x)^{3/2}} \, dx=\int \frac {{\left (a^2+2\,a\,b\,x^2+b^2\,x^4\right )}^p}{{\left (d\,x\right )}^{3/2}} \,d x \]
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