Optimal. Leaf size=52 \[ \frac{x^{3-2 p} \left (a+b x^2\right )^{p+1} \, _2F_1\left (1,\frac{5}{2};\frac{1}{2} (5-2 p);-\frac{b x^2}{a}\right )}{a (3-2 p)} \]
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Rubi [A] time = 0.0196595, antiderivative size = 69, normalized size of antiderivative = 1.33, number of steps used = 2, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118, Rules used = {365, 364} \[ \frac{x^{3-2 p} \left (a+b x^2\right )^p \left (\frac{b x^2}{a}+1\right )^{-p} \, _2F_1\left (\frac{1}{2} (3-2 p),-p;\frac{1}{2} (5-2 p);-\frac{b x^2}{a}\right )}{3-2 p} \]
Antiderivative was successfully verified.
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Rule 365
Rule 364
Rubi steps
\begin{align*} \int x^{2-2 p} \left (a+b x^2\right )^p \, dx &=\left (\left (a+b x^2\right )^p \left (1+\frac{b x^2}{a}\right )^{-p}\right ) \int x^{2-2 p} \left (1+\frac{b x^2}{a}\right )^p \, dx\\ &=\frac{x^{3-2 p} \left (a+b x^2\right )^p \left (1+\frac{b x^2}{a}\right )^{-p} \, _2F_1\left (\frac{1}{2} (3-2 p),-p;\frac{1}{2} (5-2 p);-\frac{b x^2}{a}\right )}{3-2 p}\\ \end{align*}
Mathematica [A] time = 0.0169598, size = 65, normalized size = 1.25 \[ \frac{x^{3-2 p} \left (a+b x^2\right )^p \left (\frac{b x^2}{a}+1\right )^{-p} \, _2F_1\left (\frac{3}{2}-p,-p;\frac{5}{2}-p;-\frac{b x^2}{a}\right )}{3-2 p} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.055, size = 0, normalized size = 0. \begin{align*} \int{x}^{2-2\,p} \left ( b{x}^{2}+a \right ) ^{p}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b x^{2} + a\right )}^{p} x^{-2 \, p + 2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (b x^{2} + a\right )}^{p} x^{-2 \, p + 2}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b x^{2} + a\right )}^{p} x^{-2 \, p + 2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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