Optimal. Leaf size=167 \[ A x \left (a+c x^2\right )^p \left (\frac{c x^2}{a}+1\right )^{-p} \left (d+f x^2\right )^q \left (\frac{f x^2}{d}+1\right )^{-q} F_1\left (\frac{1}{2};-p,-q;\frac{3}{2};-\frac{c x^2}{a},-\frac{f x^2}{d}\right )+\frac{B \left (a+c x^2\right )^{p+1} \left (d+f x^2\right )^q \left (\frac{c \left (d+f x^2\right )}{c d-a f}\right )^{-q} \, _2F_1\left (p+1,-q;p+2;-\frac{f \left (c x^2+a\right )}{c d-a f}\right )}{2 c (p+1)} \]
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Rubi [A] time = 0.152508, antiderivative size = 167, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {1010, 430, 429, 444, 70, 69} \[ A x \left (a+c x^2\right )^p \left (\frac{c x^2}{a}+1\right )^{-p} \left (d+f x^2\right )^q \left (\frac{f x^2}{d}+1\right )^{-q} F_1\left (\frac{1}{2};-p,-q;\frac{3}{2};-\frac{c x^2}{a},-\frac{f x^2}{d}\right )+\frac{B \left (a+c x^2\right )^{p+1} \left (d+f x^2\right )^q \left (\frac{c \left (d+f x^2\right )}{c d-a f}\right )^{-q} \, _2F_1\left (p+1,-q;p+2;-\frac{f \left (c x^2+a\right )}{c d-a f}\right )}{2 c (p+1)} \]
Antiderivative was successfully verified.
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Rule 1010
Rule 430
Rule 429
Rule 444
Rule 70
Rule 69
Rubi steps
\begin{align*} \int (A+B x) \left (a+c x^2\right )^p \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+B \int x \left (a+c x^2\right )^p \left (d+f x^2\right )^q \, dx\\ &=\frac{1}{2} B \operatorname{Subst}\left (\int (a+c x)^p (d+f x)^q \, dx,x,x^2\right )+\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\\ &=\frac{1}{2} \left (B \left (d+f x^2\right )^q \left (\frac{c \left (d+f x^2\right )}{c d-a f}\right )^{-q}\right ) \operatorname{Subst}\left (\int (a+c x)^p \left (\frac{c d}{c d-a f}+\frac{c f x}{c d-a f}\right )^q \, dx,x,x^2\right )+\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\\ &=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{B \left (a+c x^2\right )^{1+p} \left (d+f x^2\right )^q \left (\frac{c \left (d+f x^2\right )}{c d-a f}\right )^{-q} \, _2F_1\left (1+p,-q;2+p;-\frac{f \left (a+c x^2\right )}{c d-a f}\right )}{2 c (1+p)}\\ \end{align*}
Mathematica [A] time = 0.330761, size = 236, normalized size = 1.41 \[ \frac{1}{2} x \left (a+c x^2\right )^p \left (d+f x^2\right )^q \left (\frac{6 a A d F_1\left (\frac{1}{2};-p,-q;\frac{3}{2};-\frac{c x^2}{a},-\frac{f x^2}{d}\right )}{2 x^2 \left (c d p F_1\left (\frac{3}{2};1-p,-q;\frac{5}{2};-\frac{c x^2}{a},-\frac{f x^2}{d}\right )+a f q F_1\left (\frac{3}{2};-p,1-q;\frac{5}{2};-\frac{c x^2}{a},-\frac{f x^2}{d}\right )\right )+3 a d F_1\left (\frac{1}{2};-p,-q;\frac{3}{2};-\frac{c x^2}{a},-\frac{f x^2}{d}\right )}+B x \left (\frac{c x^2}{a}+1\right )^{-p} \left (\frac{f x^2}{d}+1\right )^{-q} F_1\left (1;-p,-q;2;-\frac{c x^2}{a},-\frac{f x^2}{d}\right )\right ) \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.631, size = 0, normalized size = 0. \begin{align*} \int \left ( Bx+A \right ) \left ( c{x}^{2}+a \right ) ^{p} \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 (B x + 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 (B x + 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 (B x + 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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