Optimal. Leaf size=474 \[ -\frac{f \left (a+c x^2\right )^p (g+h x)^{-2 p} \left (1-\frac{g+h x}{g-\frac{\sqrt{-a} h}{\sqrt{c}}}\right )^{-p} \left (1-\frac{g+h x}{\frac{\sqrt{-a} h}{\sqrt{c}}+g}\right )^{-p} F_1\left (-2 p;-p,-p;1-2 p;\frac{g+h x}{g-\frac{\sqrt{-a} h}{\sqrt{c}}},\frac{g+h x}{g+\frac{\sqrt{-a} h}{\sqrt{c}}}\right )}{2 h^3 p}+\frac{\left (\sqrt{-a}-\sqrt{c} x\right ) \left (a+c x^2\right )^p (g+h x)^{-2 p-1} \left (-\frac{\left (\sqrt{-a}+\sqrt{c} x\right ) \left (\sqrt{-a} h+\sqrt{c} g\right )}{\left (\sqrt{-a}-\sqrt{c} x\right ) \left (\sqrt{c} g-\sqrt{-a} h\right )}\right )^{-p} \left (a h^2 (2 f g-e h)+c \left (f g^3-d g h^2\right )\right ) \, _2F_1\left (-2 p-1,-p;-2 p;\frac{2 \sqrt{-a} \sqrt{c} (g+h x)}{\left (\sqrt{c} g-\sqrt{-a} h\right ) \left (\sqrt{-a}-\sqrt{c} x\right )}\right )}{h^2 (2 p+1) \left (\sqrt{-a} h+\sqrt{c} g\right ) \left (a h^2+c g^2\right )}-\frac{\left (a+c x^2\right )^{p+1} (g+h x)^{-2 (p+1)} \left (d h^2-e g h+f g^2\right )}{2 h (p+1) \left (a h^2+c g^2\right )} \]
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Rubi [A] time = 0.519098, antiderivative size = 474, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.161, Rules used = {1656, 760, 133, 807, 727} \[ -\frac{f \left (a+c x^2\right )^p (g+h x)^{-2 p} \left (1-\frac{g+h x}{g-\frac{\sqrt{-a} h}{\sqrt{c}}}\right )^{-p} \left (1-\frac{g+h x}{\frac{\sqrt{-a} h}{\sqrt{c}}+g}\right )^{-p} F_1\left (-2 p;-p,-p;1-2 p;\frac{g+h x}{g-\frac{\sqrt{-a} h}{\sqrt{c}}},\frac{g+h x}{g+\frac{\sqrt{-a} h}{\sqrt{c}}}\right )}{2 h^3 p}+\frac{\left (\sqrt{-a}-\sqrt{c} x\right ) \left (a+c x^2\right )^p (g+h x)^{-2 p-1} \left (-\frac{\left (\sqrt{-a}+\sqrt{c} x\right ) \left (\sqrt{-a} h+\sqrt{c} g\right )}{\left (\sqrt{-a}-\sqrt{c} x\right ) \left (\sqrt{c} g-\sqrt{-a} h\right )}\right )^{-p} \left (a h^2 (2 f g-e h)+c \left (f g^3-d g h^2\right )\right ) \, _2F_1\left (-2 p-1,-p;-2 p;\frac{2 \sqrt{-a} \sqrt{c} (g+h x)}{\left (\sqrt{c} g-\sqrt{-a} h\right ) \left (\sqrt{-a}-\sqrt{c} x\right )}\right )}{h^2 (2 p+1) \left (\sqrt{-a} h+\sqrt{c} g\right ) \left (a h^2+c g^2\right )}-\frac{\left (a+c x^2\right )^{p+1} (g+h x)^{-2 (p+1)} \left (d h^2-e g h+f g^2\right )}{2 h (p+1) \left (a h^2+c g^2\right )} \]
Antiderivative was successfully verified.
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Rule 1656
Rule 760
Rule 133
Rule 807
Rule 727
Rubi steps
\begin{align*} \int (g+h x)^{-3-2 p} \left (a+c x^2\right )^p \left (d+e x+f x^2\right ) \, dx &=\frac{\int (g+h x)^{-3-2 p} \left (-f g^2+d h^2-h (2 f g-e h) x\right ) \left (a+c x^2\right )^p \, dx}{h^2}+\frac{f \int (g+h x)^{-1-2 p} \left (a+c x^2\right )^p \, dx}{h^2}\\ &=-\frac{\left (f g^2-e g h+d h^2\right ) (g+h x)^{-2 (1+p)} \left (a+c x^2\right )^{1+p}}{2 h \left (c g^2+a h^2\right ) (1+p)}-\frac{\left (a h^2 (2 f g-e h)+c \left (f g^3-d g h^2\right )\right ) \int (g+h x)^{-2-2 p} \left (a+c x^2\right )^p \, dx}{h^2 \left (c g^2+a h^2\right )}+\frac{\left (f \left (a+c x^2\right )^p \left (1-\frac{g+h x}{g-\frac{\sqrt{-a} h}{\sqrt{c}}}\right )^{-p} \left (1-\frac{g+h x}{g+\frac{\sqrt{-a} h}{\sqrt{c}}}\right )^{-p}\right ) \operatorname{Subst}\left (\int x^{-1-2 p} \left (1-\frac{x}{g-\frac{\sqrt{-a} h}{\sqrt{c}}}\right )^p \left (1-\frac{x}{g+\frac{\sqrt{-a} h}{\sqrt{c}}}\right )^p \, dx,x,g+h x\right )}{h^3}\\ &=-\frac{\left (f g^2-e g h+d h^2\right ) (g+h x)^{-2 (1+p)} \left (a+c x^2\right )^{1+p}}{2 h \left (c g^2+a h^2\right ) (1+p)}-\frac{f (g+h x)^{-2 p} \left (a+c x^2\right )^p \left (1-\frac{g+h x}{g-\frac{\sqrt{-a} h}{\sqrt{c}}}\right )^{-p} \left (1-\frac{g+h x}{g+\frac{\sqrt{-a} h}{\sqrt{c}}}\right )^{-p} F_1\left (-2 p;-p,-p;1-2 p;\frac{g+h x}{g-\frac{\sqrt{-a} h}{\sqrt{c}}},\frac{g+h x}{g+\frac{\sqrt{-a} h}{\sqrt{c}}}\right )}{2 h^3 p}+\frac{\left (a h^2 (2 f g-e h)+c \left (f g^3-d g h^2\right )\right ) \left (\sqrt{-a}-\sqrt{c} x\right ) \left (-\frac{\left (\sqrt{c} g+\sqrt{-a} h\right ) \left (\sqrt{-a}+\sqrt{c} x\right )}{\left (\sqrt{c} g-\sqrt{-a} h\right ) \left (\sqrt{-a}-\sqrt{c} x\right )}\right )^{-p} (g+h x)^{-1-2 p} \left (a+c x^2\right )^p \, _2F_1\left (-1-2 p,-p;-2 p;\frac{2 \sqrt{-a} \sqrt{c} (g+h x)}{\left (\sqrt{c} g-\sqrt{-a} h\right ) \left (\sqrt{-a}-\sqrt{c} x\right )}\right )}{h^2 \left (\sqrt{c} g+\sqrt{-a} h\right ) \left (c g^2+a h^2\right ) (1+2 p)}\\ \end{align*}
Mathematica [F] time = 2.91037, size = 0, normalized size = 0. \[ \int (g+h x)^{-3-2 p} \left (a+c x^2\right )^p \left (d+e x+f x^2\right ) \, dx \]
Verification is Not applicable to the result.
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Maple [F] time = 0.754, size = 0, normalized size = 0. \begin{align*} \int \left ( hx+g \right ) ^{-3-2\,p} \left ( c{x}^{2}+a \right ) ^{p} \left ( f{x}^{2}+ex+d \right ) \, 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 (f x^{2} + e x + d\right )}{\left (c x^{2} + a\right )}^{p}{\left (h x + g\right )}^{-2 \, p - 3}\,{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 (f x^{2} + e x + d\right )}{\left (c x^{2} + a\right )}^{p}{\left (h x + g\right )}^{-2 \, p - 3}, 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 (f x^{2} + e x + d\right )}{\left (c x^{2} + a\right )}^{p}{\left (h x + g\right )}^{-2 \, p - 3}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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