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 = 0.15, antiderivative size = 166, normalized size of antiderivative = 1.00, 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 429
Rule 430
Rule 510
Rule 511
Rule 531
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*}
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Mathematica [A] time = 0.40, 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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fricas [F] time = 0.89, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (C x^{2} + A\right )} {\left (c x^{2} + a\right )}^{p} {\left (f x^{2} + d\right )}^{q}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (C x^{2} + A\right )} {\left (c x^{2} + a\right )}^{p} {\left (f x^{2} + d\right )}^{q}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.08, size = 0, normalized size = 0.00 \[ \int \left (C \,x^{2}+A \right ) \left (c \,x^{2}+a \right )^{p} \left (f \,x^{2}+d \right )^{q}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (C x^{2} + A\right )} {\left (c x^{2} + a\right )}^{p} {\left (f x^{2} + d\right )}^{q}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \left (C\,x^2+A\right )\,{\left (c\,x^2+a\right )}^p\,{\left (f\,x^2+d\right )}^q \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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