Optimal. Leaf size=85 \[ \frac{\left (a+b x^2\right )^{p+1} \left (c+d x^2\right )^q \left (\frac{b \left (c+d x^2\right )}{b c-a d}\right )^{-q} \, _2F_1\left (p+1,-q;p+2;-\frac{d \left (b x^2+a\right )}{b c-a d}\right )}{2 b (p+1)} \]
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Rubi [A] time = 0.062654, antiderivative size = 85, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15, Rules used = {444, 70, 69} \[ \frac{\left (a+b x^2\right )^{p+1} \left (c+d x^2\right )^q \left (\frac{b \left (c+d x^2\right )}{b c-a d}\right )^{-q} \, _2F_1\left (p+1,-q;p+2;-\frac{d \left (b x^2+a\right )}{b c-a d}\right )}{2 b (p+1)} \]
Antiderivative was successfully verified.
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Rule 444
Rule 70
Rule 69
Rubi steps
\begin{align*} \int x \left (a+b x^2\right )^p \left (c+d x^2\right )^q \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int (a+b x)^p (c+d x)^q \, dx,x,x^2\right )\\ &=\frac{1}{2} \left (\left (c+d x^2\right )^q \left (\frac{b \left (c+d x^2\right )}{b c-a d}\right )^{-q}\right ) \operatorname{Subst}\left (\int (a+b x)^p \left (\frac{b c}{b c-a d}+\frac{b d x}{b c-a d}\right )^q \, dx,x,x^2\right )\\ &=\frac{\left (a+b x^2\right )^{1+p} \left (c+d x^2\right )^q \left (\frac{b \left (c+d x^2\right )}{b c-a d}\right )^{-q} \, _2F_1\left (1+p,-q;2+p;-\frac{d \left (a+b x^2\right )}{b c-a d}\right )}{2 b (1+p)}\\ \end{align*}
Mathematica [A] time = 0.0216457, size = 84, normalized size = 0.99 \[ \frac{\left (a+b x^2\right )^{p+1} \left (c+d x^2\right )^q \left (\frac{b \left (c+d x^2\right )}{b c-a d}\right )^{-q} \, _2F_1\left (p+1,-q;p+2;\frac{d \left (b x^2+a\right )}{a d-b c}\right )}{2 b (p+1)} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.057, size = 0, normalized size = 0. \begin{align*} \int x \left ( b{x}^{2}+a \right ) ^{p} \left ( d{x}^{2}+c \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^{2} + a\right )}^{p}{\left (d x^{2} + c\right )}^{q} x\,{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^{2} + a\right )}^{p}{\left (d x^{2} + c\right )}^{q} x, 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^{2} + a\right )}^{p}{\left (d x^{2} + c\right )}^{q} x\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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