Optimal. Leaf size=252 \[ A x \left (a+c x^2\right )^p \left (1+\frac {c x^2}{a}\right )^{-p} \left (d+f x^2\right )^q \left (1+\frac {f x^2}{d}\right )^{-q} F_1\left (\frac {1}{2};-p,-q;\frac {3}{2};-\frac {c x^2}{a},-\frac {f x^2}{d}\right )+\frac {1}{3} C x^3 \left (a+c x^2\right )^p \left (1+\frac {c x^2}{a}\right )^{-p} \left (d+f x^2\right )^q \left (1+\frac {f x^2}{d}\right )^{-q} F_1\left (\frac {3}{2};-p,-q;\frac {5}{2};-\frac {c x^2}{a},-\frac {f x^2}{d}\right )+\frac {B \left (a+c x^2\right )^{1+p} \left (d+f x^2\right )^q \left (\frac {c \left (d+f x^2\right )}{c d-a f}\right )^{-q} \, _2F_1\left (1+p,-q;2+p;-\frac {f \left (a+c x^2\right )}{c d-a f}\right )}{2 c (1+p)} \]
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Rubi [A]
time = 0.28, antiderivative size = 252, normalized size of antiderivative = 1.00, number of steps
used = 11, number of rules used = 8, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.276, Rules used = {6874, 441,
440, 455, 72, 71, 525, 524} \begin {gather*} 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 )+\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)} \end {gather*}
Antiderivative was successfully verified.
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Rule 71
Rule 72
Rule 440
Rule 441
Rule 455
Rule 524
Rule 525
Rule 6874
Rubi steps
\begin {align*} \int \left (a+c x^2\right )^p \left (A+B x+C x^2\right ) \left (d+f x^2\right )^q \, dx &=\int \left (A \left (a+c x^2\right )^p \left (d+f x^2\right )^q+B x \left (a+c x^2\right )^p \left (d+f x^2\right )^q+C x^2 \left (a+c x^2\right )^p \left (d+f x^2\right )^q\right ) \, 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+C \int x^2 \left (a+c x^2\right )^p \left (d+f x^2\right )^q \, dx\\ &=\frac {1}{2} B \text {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+\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\\ &=\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 ) \text {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+\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 )+\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*}
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Mathematica [A]
time = 0.41, size = 302, normalized size = 1.20 \begin {gather*} \frac {1}{6} x \left (a+c x^2\right )^p \left (d+f x^2\right )^q \left (3 B x \left (1+\frac {c x^2}{a}\right )^{-p} \left (1+\frac {f x^2}{d}\right )^{-q} F_1\left (1;-p,-q;2;-\frac {c x^2}{a},-\frac {f x^2}{d}\right )+\frac {18 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 )}{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 )+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 )}+2 C x^2 \left (1+\frac {c x^2}{a}\right )^{-p} \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 )\right ) \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [F]
time = 0.03, size = 0, normalized size = 0.00 \[\int \left (c \,x^{2}+a \right )^{p} \left (C \,x^{2}+B x +A \right ) \left (f \,x^{2}+d \right )^{q}\, dx\]
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int {\left (c\,x^2+a\right )}^p\,{\left (f\,x^2+d\right )}^q\,\left (C\,x^2+B\,x+A\right ) \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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