Optimal. Leaf size=420 \[ -\frac{\left (a+c x^2\right )^p (g+h x)^{m+1} \left (1-\frac{g+h x}{g-\frac{\sqrt{-a} h}{\sqrt{c}}}\right )^{-p} \left (1-\frac{g+h x}{\frac{\sqrt{-a} h}{\sqrt{c}}+g}\right )^{-p} F_1\left (m+1;-p,-p;m+2;\frac{g+h x}{g-\frac{\sqrt{-a} h}{\sqrt{c}}},\frac{g+h x}{g+\frac{\sqrt{-a} h}{\sqrt{c}}}\right ) \left (a f h^2 (m+1)-c \left (2 f g^2 (p+1)-h (m+2 p+3) (e g-d h)\right )\right )}{c h^3 (m+1) (m+2 p+3)}-\frac{\left (a+c x^2\right )^p (g+h x)^{m+2} \left (1-\frac{g+h x}{g-\frac{\sqrt{-a} h}{\sqrt{c}}}\right )^{-p} \left (1-\frac{g+h x}{\frac{\sqrt{-a} h}{\sqrt{c}}+g}\right )^{-p} (2 f g (p+1)-e h (m+2 p+3)) F_1\left (m+2;-p,-p;m+3;\frac{g+h x}{g-\frac{\sqrt{-a} h}{\sqrt{c}}},\frac{g+h x}{g+\frac{\sqrt{-a} h}{\sqrt{c}}}\right )}{h^3 (m+2) (m+2 p+3)}+\frac{f \left (a+c x^2\right )^{p+1} (g+h x)^{m+1}}{c h (m+2 p+3)} \]
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Rubi [A] time = 0.604235, antiderivative size = 417, normalized size of antiderivative = 0.99, number of steps used = 6, number of rules used = 4, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.148, Rules used = {1654, 844, 760, 133} \[ -\frac{\left (a+c x^2\right )^p (g+h x)^{m+1} \left (1-\frac{g+h x}{g-\frac{\sqrt{-a} h}{\sqrt{c}}}\right )^{-p} \left (1-\frac{g+h x}{\frac{\sqrt{-a} h}{\sqrt{c}}+g}\right )^{-p} F_1\left (m+1;-p,-p;m+2;\frac{g+h x}{g-\frac{\sqrt{-a} h}{\sqrt{c}}},\frac{g+h x}{g+\frac{\sqrt{-a} h}{\sqrt{c}}}\right ) \left (a f h^2 (m+1)+c h (m+2 p+3) (e g-d h)-2 c f g^2 (p+1)\right )}{c h^3 (m+1) (m+2 p+3)}-\frac{\left (a+c x^2\right )^p (g+h x)^{m+2} \left (1-\frac{g+h x}{g-\frac{\sqrt{-a} h}{\sqrt{c}}}\right )^{-p} \left (1-\frac{g+h x}{\frac{\sqrt{-a} h}{\sqrt{c}}+g}\right )^{-p} (2 f g (p+1)-e h (m+2 p+3)) F_1\left (m+2;-p,-p;m+3;\frac{g+h x}{g-\frac{\sqrt{-a} h}{\sqrt{c}}},\frac{g+h x}{g+\frac{\sqrt{-a} h}{\sqrt{c}}}\right )}{h^3 (m+2) (m+2 p+3)}+\frac{f \left (a+c x^2\right )^{p+1} (g+h x)^{m+1}}{c h (m+2 p+3)} \]
Antiderivative was successfully verified.
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Rule 1654
Rule 844
Rule 760
Rule 133
Rubi steps
\begin{align*} \int (g+h x)^m \left (a+c x^2\right )^p \left (d+e x+f x^2\right ) \, dx &=\frac{f (g+h x)^{1+m} \left (a+c x^2\right )^{1+p}}{c h (3+m+2 p)}+\frac{\int (g+h x)^m \left (-h^2 (a f (1+m)-c d (3+m+2 p))-c h (2 f g (1+p)-e h (3+m+2 p)) x\right ) \left (a+c x^2\right )^p \, dx}{c h^2 (3+m+2 p)}\\ &=\frac{f (g+h x)^{1+m} \left (a+c x^2\right )^{1+p}}{c h (3+m+2 p)}+\frac{\left (e h-\frac{2 f g (1+p)}{3+m+2 p}\right ) \int (g+h x)^{1+m} \left (a+c x^2\right )^p \, dx}{h^2}-\frac{\left (a f h^2 (1+m)-2 c f g^2 (1+p)+c h (e g-d h) (3+m+2 p)\right ) \int (g+h x)^m \left (a+c x^2\right )^p \, dx}{c h^2 (3+m+2 p)}\\ &=\frac{f (g+h x)^{1+m} \left (a+c x^2\right )^{1+p}}{c h (3+m+2 p)}+\frac{\left (\left (e h-\frac{2 f g (1+p)}{3+m+2 p}\right ) \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+m} \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 (\left (a f h^2 (1+m)-2 c f g^2 (1+p)+c h (e g-d h) (3+m+2 p)\right ) \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^m \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 )}{c h^3 (3+m+2 p)}\\ &=\frac{f (g+h x)^{1+m} \left (a+c x^2\right )^{1+p}}{c h (3+m+2 p)}-\frac{\left (a f h^2 (1+m)-2 c f g^2 (1+p)+c h (e g-d h) (3+m+2 p)\right ) (g+h x)^{1+m} \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 (1+m;-p,-p;2+m;\frac{g+h x}{g-\frac{\sqrt{-a} h}{\sqrt{c}}},\frac{g+h x}{g+\frac{\sqrt{-a} h}{\sqrt{c}}}\right )}{c h^3 (1+m) (3+m+2 p)}+\frac{\left (e h-\frac{2 f g (1+p)}{3+m+2 p}\right ) (g+h x)^{2+m} \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+m;-p,-p;3+m;\frac{g+h x}{g-\frac{\sqrt{-a} h}{\sqrt{c}}},\frac{g+h x}{g+\frac{\sqrt{-a} h}{\sqrt{c}}}\right )}{h^3 (2+m)}\\ \end{align*}
Mathematica [F] time = 1.19881, size = 0, normalized size = 0. \[ \int (g+h x)^m \left (a+c x^2\right )^p \left (d+e x+f x^2\right ) \, dx \]
Verification is Not applicable to the result.
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Maple [F] time = 0.796, size = 0, normalized size = 0. \begin{align*} \int \left ( hx+g \right ) ^{m} \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 )}^{m}\,{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 )}^{m}, 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 )}^{m}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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