Optimal. Leaf size=166 \[ A x \left (a+c x^2\right )^p \left (\frac{c x^2}{a}+1\right )^{-p} \left (d+f x^2\right )^q \left (\frac{f x^2}{d}+1\right )^{-q} F_1\left (\frac{1}{2};-p,-q;\frac{3}{2};-\frac{c x^2}{a},-\frac{f x^2}{d}\right )+\frac{1}{3} C x^3 \left (a+c x^2\right )^p \left (\frac{c x^2}{a}+1\right )^{-p} \left (d+f x^2\right )^q \left (\frac{f x^2}{d}+1\right )^{-q} F_1\left (\frac{3}{2};-p,-q;\frac{5}{2};-\frac{c x^2}{a},-\frac{f x^2}{d}\right ) \]
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Rubi [A] time = 0.14659, antiderivative size = 166, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.192, Rules used = {531, 430, 429, 511, 510} \[ A x \left (a+c x^2\right )^p \left (\frac{c x^2}{a}+1\right )^{-p} \left (d+f x^2\right )^q \left (\frac{f x^2}{d}+1\right )^{-q} F_1\left (\frac{1}{2};-p,-q;\frac{3}{2};-\frac{c x^2}{a},-\frac{f x^2}{d}\right )+\frac{1}{3} C x^3 \left (a+c x^2\right )^p \left (\frac{c x^2}{a}+1\right )^{-p} \left (d+f x^2\right )^q \left (\frac{f x^2}{d}+1\right )^{-q} F_1\left (\frac{3}{2};-p,-q;\frac{5}{2};-\frac{c x^2}{a},-\frac{f x^2}{d}\right ) \]
Antiderivative was successfully verified.
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Rule 531
Rule 430
Rule 429
Rule 511
Rule 510
Rubi steps
\begin{align*} \int \left (a+c x^2\right )^p \left (A+C x^2\right ) \left (d+f x^2\right )^q \, dx &=A \int \left (a+c x^2\right )^p \left (d+f x^2\right )^q \, dx+C \int x^2 \left (a+c x^2\right )^p \left (d+f x^2\right )^q \, dx\\ &=\left (A \left (a+c x^2\right )^p \left (1+\frac{c x^2}{a}\right )^{-p}\right ) \int \left (1+\frac{c x^2}{a}\right )^p \left (d+f x^2\right )^q \, dx+\left (C \left (a+c x^2\right )^p \left (1+\frac{c x^2}{a}\right )^{-p}\right ) \int x^2 \left (1+\frac{c x^2}{a}\right )^p \left (d+f x^2\right )^q \, dx\\ &=\left (A \left (a+c x^2\right )^p \left (1+\frac{c x^2}{a}\right )^{-p} \left (d+f x^2\right )^q \left (1+\frac{f x^2}{d}\right )^{-q}\right ) \int \left (1+\frac{c x^2}{a}\right )^p \left (1+\frac{f x^2}{d}\right )^q \, dx+\left (C \left (a+c x^2\right )^p \left (1+\frac{c x^2}{a}\right )^{-p} \left (d+f x^2\right )^q \left (1+\frac{f x^2}{d}\right )^{-q}\right ) \int x^2 \left (1+\frac{c x^2}{a}\right )^p \left (1+\frac{f x^2}{d}\right )^q \, dx\\ &=A x \left (a+c x^2\right )^p \left (1+\frac{c x^2}{a}\right )^{-p} \left (d+f x^2\right )^q \left (1+\frac{f x^2}{d}\right )^{-q} F_1\left (\frac{1}{2};-p,-q;\frac{3}{2};-\frac{c x^2}{a},-\frac{f x^2}{d}\right )+\frac{1}{3} C x^3 \left (a+c x^2\right )^p \left (1+\frac{c x^2}{a}\right )^{-p} \left (d+f x^2\right )^q \left (1+\frac{f x^2}{d}\right )^{-q} F_1\left (\frac{3}{2};-p,-q;\frac{5}{2};-\frac{c x^2}{a},-\frac{f x^2}{d}\right )\\ \end{align*}
Mathematica [A] time = 0.42378, size = 242, normalized size = 1.46 \[ \frac{1}{3} x \left (a+c x^2\right )^p \left (d+f x^2\right )^q \left (\frac{9 a A d F_1\left (\frac{1}{2};-p,-q;\frac{3}{2};-\frac{c x^2}{a},-\frac{f x^2}{d}\right )}{2 x^2 \left (c d p F_1\left (\frac{3}{2};1-p,-q;\frac{5}{2};-\frac{c x^2}{a},-\frac{f x^2}{d}\right )+a f q F_1\left (\frac{3}{2};-p,1-q;\frac{5}{2};-\frac{c x^2}{a},-\frac{f x^2}{d}\right )\right )+3 a d F_1\left (\frac{1}{2};-p,-q;\frac{3}{2};-\frac{c x^2}{a},-\frac{f x^2}{d}\right )}+C x^2 \left (\frac{c x^2}{a}+1\right )^{-p} \left (\frac{f x^2}{d}+1\right )^{-q} F_1\left (\frac{3}{2};-p,-q;\frac{5}{2};-\frac{c x^2}{a},-\frac{f x^2}{d}\right )\right ) \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.681, size = 0, normalized size = 0. \begin{align*} \int \left ( c{x}^{2}+a \right ) ^{p} \left ( C{x}^{2}+A \right ) \left ( f{x}^{2}+d \right ) ^{q}\, 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 (C x^{2} + A\right )}{\left (c x^{2} + a\right )}^{p}{\left (f x^{2} + d\right )}^{q}\,{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 (C x^{2} + A\right )}{\left (c x^{2} + a\right )}^{p}{\left (f x^{2} + d\right )}^{q}, 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 (C x^{2} + A\right )}{\left (c x^{2} + a\right )}^{p}{\left (f x^{2} + d\right )}^{q}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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