Optimal. Leaf size=93 \[ \frac{d x \left (a+b x^2\right )^{p+1}}{b (2 p+3)}-\frac{x \left (a+b x^2\right )^p \left (\frac{b x^2}{a}+1\right )^{-p} (a d-b c (2 p+3)) \, _2F_1\left (\frac{1}{2},-p;\frac{3}{2};-\frac{b x^2}{a}\right )}{b (2 p+3)} \]
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Rubi [A] time = 0.0390779, antiderivative size = 85, normalized size of antiderivative = 0.91, number of steps used = 3, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176, Rules used = {388, 246, 245} \[ x \left (a+b x^2\right )^p \left (\frac{b x^2}{a}+1\right )^{-p} \left (c-\frac{a d}{2 b p+3 b}\right ) \, _2F_1\left (\frac{1}{2},-p;\frac{3}{2};-\frac{b x^2}{a}\right )+\frac{d x \left (a+b x^2\right )^{p+1}}{b (2 p+3)} \]
Antiderivative was successfully verified.
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Rule 388
Rule 246
Rule 245
Rubi steps
\begin{align*} \int \left (a+b x^2\right )^p \left (c+d x^2\right ) \, dx &=\frac{d x \left (a+b x^2\right )^{1+p}}{b (3+2 p)}-\left (-c+\frac{a d}{3 b+2 b p}\right ) \int \left (a+b x^2\right )^p \, dx\\ &=\frac{d x \left (a+b x^2\right )^{1+p}}{b (3+2 p)}-\left (\left (-c+\frac{a d}{3 b+2 b p}\right ) \left (a+b x^2\right )^p \left (1+\frac{b x^2}{a}\right )^{-p}\right ) \int \left (1+\frac{b x^2}{a}\right )^p \, dx\\ &=\frac{d x \left (a+b x^2\right )^{1+p}}{b (3+2 p)}+\left (c-\frac{a d}{3 b+2 b p}\right ) x \left (a+b x^2\right )^p \left (1+\frac{b x^2}{a}\right )^{-p} \, _2F_1\left (\frac{1}{2},-p;\frac{3}{2};-\frac{b x^2}{a}\right )\\ \end{align*}
Mathematica [A] time = 0.029336, size = 90, normalized size = 0.97 \[ \frac{x \left (a+b x^2\right )^p \left (\frac{b x^2}{a}+1\right )^{-p} \left ((b c (2 p+3)-a d) \, _2F_1\left (\frac{1}{2},-p;\frac{3}{2};-\frac{b x^2}{a}\right )+d \left (a+b x^2\right ) \left (\frac{b x^2}{a}+1\right )^p\right )}{b (2 p+3)} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.03, size = 0, normalized size = 0. \begin{align*} \int \left ( b{x}^{2}+a \right ) ^{p} \left ( d{x}^{2}+c \right ) \, 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 (d x^{2} + c\right )}{\left (b x^{2} + a\right )}^{p}\,{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 (d x^{2} + c\right )}{\left (b x^{2} + a\right )}^{p}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 10.7052, size = 53, normalized size = 0.57 \begin{align*} a^{p} c x{{}_{2}F_{1}\left (\begin{matrix} \frac{1}{2}, - p \\ \frac{3}{2} \end{matrix}\middle |{\frac{b x^{2} e^{i \pi }}{a}} \right )} + \frac{a^{p} d x^{3}{{}_{2}F_{1}\left (\begin{matrix} \frac{3}{2}, - p \\ \frac{5}{2} \end{matrix}\middle |{\frac{b x^{2} e^{i \pi }}{a}} \right )}}{3} \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 (d x^{2} + c\right )}{\left (b x^{2} + a\right )}^{p}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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