Integrand size = 24, antiderivative size = 60 \[ \int x^2 \left (a^2+2 a b x^2+b^2 x^4\right )^p \, dx=\frac {1}{3} x^3 \left (1+\frac {b x^2}{a}\right )^{-2 p} \left (a^2+2 a b x^2+b^2 x^4\right )^p \operatorname {Hypergeometric2F1}\left (\frac {3}{2},-2 p,\frac {5}{2},-\frac {b x^2}{a}\right ) \]
[Out]
Time = 0.01 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {1127, 371} \[ \int x^2 \left (a^2+2 a b x^2+b^2 x^4\right )^p \, dx=\frac {1}{3} x^3 \left (\frac {b x^2}{a}+1\right )^{-2 p} \left (a^2+2 a b x^2+b^2 x^4\right )^p \operatorname {Hypergeometric2F1}\left (\frac {3}{2},-2 p,\frac {5}{2},-\frac {b x^2}{a}\right ) \]
[In]
[Out]
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 x^2 \left (1+\frac {b x^2}{a}\right )^{2 p} \, dx \\ & = \frac {1}{3} x^3 \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 {3}{2},-2 p;\frac {5}{2};-\frac {b x^2}{a}\right ) \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.85 \[ \int x^2 \left (a^2+2 a b x^2+b^2 x^4\right )^p \, dx=\frac {1}{3} x^3 \left (\left (a+b x^2\right )^2\right )^p \left (1+\frac {b x^2}{a}\right )^{-2 p} \operatorname {Hypergeometric2F1}\left (\frac {3}{2},-2 p,\frac {5}{2},-\frac {b x^2}{a}\right ) \]
[In]
[Out]
\[\int x^{2} \left (b^{2} x^{4}+2 a b \,x^{2}+a^{2}\right )^{p}d x\]
[In]
[Out]
\[ \int x^2 \left (a^2+2 a b x^2+b^2 x^4\right )^p \, dx=\int { {\left (b^{2} x^{4} + 2 \, a b x^{2} + a^{2}\right )}^{p} x^{2} \,d x } \]
[In]
[Out]
\[ \int x^2 \left (a^2+2 a b x^2+b^2 x^4\right )^p \, dx=\int x^{2} \left (\left (a + b x^{2}\right )^{2}\right )^{p}\, dx \]
[In]
[Out]
\[ \int x^2 \left (a^2+2 a b x^2+b^2 x^4\right )^p \, dx=\int { {\left (b^{2} x^{4} + 2 \, a b x^{2} + a^{2}\right )}^{p} x^{2} \,d x } \]
[In]
[Out]
\[ \int x^2 \left (a^2+2 a b x^2+b^2 x^4\right )^p \, dx=\int { {\left (b^{2} x^{4} + 2 \, a b x^{2} + a^{2}\right )}^{p} x^{2} \,d x } \]
[In]
[Out]
Timed out. \[ \int x^2 \left (a^2+2 a b x^2+b^2 x^4\right )^p \, dx=\int x^2\,{\left (a^2+2\,a\,b\,x^2+b^2\,x^4\right )}^p \,d x \]
[In]
[Out]