Optimal. Leaf size=496 \[ \frac{\sqrt{a+b x+c x^2} (g+h x)^{m+1} F_1\left (m+1;-\frac{1}{2},-\frac{1}{2};m+2;\frac{2 c (g+h x)}{2 c g-\left (b-\sqrt{b^2-4 a c}\right ) h},\frac{2 c (g+h x)}{2 c g-\left (b+\sqrt{b^2-4 a c}\right ) h}\right ) \left (f h (m+1) (b g-a h)+c \left (3 f g^2-h (m+4) (e g-d h)\right )\right )}{c h^3 (m+1) (m+4) \sqrt{1-\frac{2 c (g+h x)}{2 c g-h \left (b-\sqrt{b^2-4 a c}\right )}} \sqrt{1-\frac{2 c (g+h x)}{2 c g-h \left (\sqrt{b^2-4 a c}+b\right )}}}-\frac{\sqrt{a+b x+c x^2} (g+h x)^{m+2} (b f h (2 m+5)+c (6 f g-2 e h (m+4))) F_1\left (m+2;-\frac{1}{2},-\frac{1}{2};m+3;\frac{2 c (g+h x)}{2 c g-\left (b-\sqrt{b^2-4 a c}\right ) h},\frac{2 c (g+h x)}{2 c g-\left (b+\sqrt{b^2-4 a c}\right ) h}\right )}{2 c h^3 (m+2) (m+4) \sqrt{1-\frac{2 c (g+h x)}{2 c g-h \left (b-\sqrt{b^2-4 a c}\right )}} \sqrt{1-\frac{2 c (g+h x)}{2 c g-h \left (\sqrt{b^2-4 a c}+b\right )}}}+\frac{f \left (a+b x+c x^2\right )^{3/2} (g+h x)^{m+1}}{c h (m+4)} \]
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Rubi [A] time = 0.672379, antiderivative size = 494, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {1653, 843, 759, 133} \[ \frac{\sqrt{a+b x+c x^2} (g+h x)^{m+1} F_1\left (m+1;-\frac{1}{2},-\frac{1}{2};m+2;\frac{2 c (g+h x)}{2 c g-\left (b-\sqrt{b^2-4 a c}\right ) h},\frac{2 c (g+h x)}{2 c g-\left (b+\sqrt{b^2-4 a c}\right ) h}\right ) \left (f h (m+1) (b g-a h)-c h (m+4) (e g-d h)+3 c f g^2\right )}{c h^3 (m+1) (m+4) \sqrt{1-\frac{2 c (g+h x)}{2 c g-h \left (b-\sqrt{b^2-4 a c}\right )}} \sqrt{1-\frac{2 c (g+h x)}{2 c g-h \left (\sqrt{b^2-4 a c}+b\right )}}}-\frac{\sqrt{a+b x+c x^2} (g+h x)^{m+2} (b f h (2 m+5)-2 c e h (m+4)+6 c f g) F_1\left (m+2;-\frac{1}{2},-\frac{1}{2};m+3;\frac{2 c (g+h x)}{2 c g-\left (b-\sqrt{b^2-4 a c}\right ) h},\frac{2 c (g+h x)}{2 c g-\left (b+\sqrt{b^2-4 a c}\right ) h}\right )}{2 c h^3 (m+2) (m+4) \sqrt{1-\frac{2 c (g+h x)}{2 c g-h \left (b-\sqrt{b^2-4 a c}\right )}} \sqrt{1-\frac{2 c (g+h x)}{2 c g-h \left (\sqrt{b^2-4 a c}+b\right )}}}+\frac{f \left (a+b x+c x^2\right )^{3/2} (g+h x)^{m+1}}{c h (m+4)} \]
Antiderivative was successfully verified.
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Rule 1653
Rule 843
Rule 759
Rule 133
Rubi steps
\begin{align*} \int (g+h x)^m \sqrt{a+b x+c x^2} \left (d+e x+f x^2\right ) \, dx &=\frac{f (g+h x)^{1+m} \left (a+b x+c x^2\right )^{3/2}}{c h (4+m)}+\frac{\int (g+h x)^m \left (-\frac{1}{2} h (3 b f g+2 a f h (1+m)-2 c d h (4+m))-\frac{1}{2} h (6 c f g-2 c e h (4+m)+b f h (5+2 m)) x\right ) \sqrt{a+b x+c x^2} \, dx}{c h^2 (4+m)}\\ &=\frac{f (g+h x)^{1+m} \left (a+b x+c x^2\right )^{3/2}}{c h (4+m)}+\frac{\left (3 c f g^2+f h (b g-a h) (1+m)-c h (e g-d h) (4+m)\right ) \int (g+h x)^m \sqrt{a+b x+c x^2} \, dx}{c h^2 (4+m)}-\frac{(6 c f g-2 c e h (4+m)+b f h (5+2 m)) \int (g+h x)^{1+m} \sqrt{a+b x+c x^2} \, dx}{2 c h^2 (4+m)}\\ &=\frac{f (g+h x)^{1+m} \left (a+b x+c x^2\right )^{3/2}}{c h (4+m)}+\frac{\left (\left (3 c f g^2+f h (b g-a h) (1+m)-c h (e g-d h) (4+m)\right ) \sqrt{a+b x+c x^2}\right ) \operatorname{Subst}\left (\int x^m \sqrt{1-\frac{2 c x}{2 c g-\left (b-\sqrt{b^2-4 a c}\right ) h}} \sqrt{1-\frac{2 c x}{2 c g-\left (b+\sqrt{b^2-4 a c}\right ) h}} \, dx,x,g+h x\right )}{c h^3 (4+m) \sqrt{1-\frac{g+h x}{g-\frac{\left (b-\sqrt{b^2-4 a c}\right ) h}{2 c}}} \sqrt{1-\frac{g+h x}{g-\frac{\left (b+\sqrt{b^2-4 a c}\right ) h}{2 c}}}}-\frac{\left ((6 c f g-2 c e h (4+m)+b f h (5+2 m)) \sqrt{a+b x+c x^2}\right ) \operatorname{Subst}\left (\int x^{1+m} \sqrt{1-\frac{2 c x}{2 c g-\left (b-\sqrt{b^2-4 a c}\right ) h}} \sqrt{1-\frac{2 c x}{2 c g-\left (b+\sqrt{b^2-4 a c}\right ) h}} \, dx,x,g+h x\right )}{2 c h^3 (4+m) \sqrt{1-\frac{g+h x}{g-\frac{\left (b-\sqrt{b^2-4 a c}\right ) h}{2 c}}} \sqrt{1-\frac{g+h x}{g-\frac{\left (b+\sqrt{b^2-4 a c}\right ) h}{2 c}}}}\\ &=\frac{f (g+h x)^{1+m} \left (a+b x+c x^2\right )^{3/2}}{c h (4+m)}+\frac{\left (3 c f g^2+f h (b g-a h) (1+m)-c h (e g-d h) (4+m)\right ) (g+h x)^{1+m} \sqrt{a+b x+c x^2} F_1\left (1+m;-\frac{1}{2},-\frac{1}{2};2+m;\frac{2 c (g+h x)}{2 c g-\left (b-\sqrt{b^2-4 a c}\right ) h},\frac{2 c (g+h x)}{2 c g-\left (b+\sqrt{b^2-4 a c}\right ) h}\right )}{c h^3 (1+m) (4+m) \sqrt{1-\frac{2 c (g+h x)}{2 c g-\left (b-\sqrt{b^2-4 a c}\right ) h}} \sqrt{1-\frac{2 c (g+h x)}{2 c g-\left (b+\sqrt{b^2-4 a c}\right ) h}}}-\frac{(6 c f g-2 c e h (4+m)+b f h (5+2 m)) (g+h x)^{2+m} \sqrt{a+b x+c x^2} F_1\left (2+m;-\frac{1}{2},-\frac{1}{2};3+m;\frac{2 c (g+h x)}{2 c g-\left (b-\sqrt{b^2-4 a c}\right ) h},\frac{2 c (g+h x)}{2 c g-\left (b+\sqrt{b^2-4 a c}\right ) h}\right )}{2 c h^3 (2+m) (4+m) \sqrt{1-\frac{2 c (g+h x)}{2 c g-\left (b-\sqrt{b^2-4 a c}\right ) h}} \sqrt{1-\frac{2 c (g+h x)}{2 c g-\left (b+\sqrt{b^2-4 a c}\right ) h}}}\\ \end{align*}
Mathematica [F] time = 1.46647, size = 0, normalized size = 0. \[ \int (g+h x)^m \sqrt{a+b x+c x^2} \left (d+e x+f x^2\right ) \, dx \]
Verification is Not applicable to the result.
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Maple [F] time = 1.338, size = 0, normalized size = 0. \begin{align*} \int \left ( hx+g \right ) ^{m} \left ( f{x}^{2}+ex+d \right ) \sqrt{c{x}^{2}+bx+a}\, 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 \sqrt{c x^{2} + b x + a}{\left (f x^{2} + e x + d\right )}{\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 (\sqrt{c x^{2} + b x + a}{\left (f x^{2} + e x + d\right )}{\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 \sqrt{c x^{2} + b x + a}{\left (f x^{2} + e x + d\right )}{\left (h x + g\right )}^{m}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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