Optimal. Leaf size=403 \[ -\frac{\sqrt{a+c x^2} (g+h x)^{m+1} F_1\left (m+1;-\frac{1}{2},-\frac{1}{2};m+2;\frac{g+h x}{g-\frac{\sqrt{-a} h}{\sqrt{c}}},\frac{g+h x}{g+\frac{\sqrt{-a} h}{\sqrt{c}}}\right ) \left (a f h^2 (m+1)-c \left (3 f g^2-h (m+4) (e g-d h)\right )\right )}{c h^3 (m+1) (m+4) \sqrt{1-\frac{g+h x}{g-\frac{\sqrt{-a} h}{\sqrt{c}}}} \sqrt{1-\frac{g+h x}{\frac{\sqrt{-a} h}{\sqrt{c}}+g}}}-\frac{\sqrt{a+c x^2} (g+h x)^{m+2} (3 f g-e h (m+4)) F_1\left (m+2;-\frac{1}{2},-\frac{1}{2};m+3;\frac{g+h x}{g-\frac{\sqrt{-a} h}{\sqrt{c}}},\frac{g+h x}{g+\frac{\sqrt{-a} h}{\sqrt{c}}}\right )}{h^3 (m+2) (m+4) \sqrt{1-\frac{g+h x}{g-\frac{\sqrt{-a} h}{\sqrt{c}}}} \sqrt{1-\frac{g+h x}{\frac{\sqrt{-a} h}{\sqrt{c}}+g}}}+\frac{f \left (a+c x^2\right )^{3/2} (g+h x)^{m+1}}{c h (m+4)} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.453214, antiderivative size = 401, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.138, Rules used = {1654, 844, 760, 133} \[ \frac{\sqrt{a+c x^2} (g+h x)^{m+1} F_1\left (m+1;-\frac{1}{2},-\frac{1}{2};m+2;\frac{g+h x}{g-\frac{\sqrt{-a} h}{\sqrt{c}}},\frac{g+h x}{g+\frac{\sqrt{-a} h}{\sqrt{c}}}\right ) \left (-a f h^2 (m+1)-c h (m+4) (e g-d h)+3 c f g^2\right )}{c h^3 (m+1) (m+4) \sqrt{1-\frac{g+h x}{g-\frac{\sqrt{-a} h}{\sqrt{c}}}} \sqrt{1-\frac{g+h x}{\frac{\sqrt{-a} h}{\sqrt{c}}+g}}}-\frac{\sqrt{a+c x^2} (g+h x)^{m+2} (3 f g-e h (m+4)) F_1\left (m+2;-\frac{1}{2},-\frac{1}{2};m+3;\frac{g+h x}{g-\frac{\sqrt{-a} h}{\sqrt{c}}},\frac{g+h x}{g+\frac{\sqrt{-a} h}{\sqrt{c}}}\right )}{h^3 (m+2) (m+4) \sqrt{1-\frac{g+h x}{g-\frac{\sqrt{-a} h}{\sqrt{c}}}} \sqrt{1-\frac{g+h x}{\frac{\sqrt{-a} h}{\sqrt{c}}+g}}}+\frac{f \left (a+c x^2\right )^{3/2} (g+h x)^{m+1}}{c h (m+4)} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 1654
Rule 844
Rule 760
Rule 133
Rubi steps
\begin{align*} \int (g+h x)^m \sqrt{a+c x^2} \left (d+e x+f x^2\right ) \, dx &=\frac{f (g+h x)^{1+m} \left (a+c x^2\right )^{3/2}}{c h (4+m)}+\frac{\int (g+h x)^m \left (-h^2 (a f (1+m)-c d (4+m))-c h (3 f g-e h (4+m)) x\right ) \sqrt{a+c x^2} \, dx}{c h^2 (4+m)}\\ &=\frac{f (g+h x)^{1+m} \left (a+c x^2\right )^{3/2}}{c h (4+m)}-\frac{(3 f g-e h (4+m)) \int (g+h x)^{1+m} \sqrt{a+c x^2} \, dx}{h^2 (4+m)}+\frac{\left (3 c f g^2-a f h^2 (1+m)-c h (e g-d h) (4+m)\right ) \int (g+h x)^m \sqrt{a+c x^2} \, dx}{c h^2 (4+m)}\\ &=\frac{f (g+h x)^{1+m} \left (a+c x^2\right )^{3/2}}{c h (4+m)}-\frac{\left ((3 f g-e h (4+m)) \sqrt{a+c x^2}\right ) \operatorname{Subst}\left (\int x^{1+m} \sqrt{1-\frac{x}{g-\frac{\sqrt{-a} h}{\sqrt{c}}}} \sqrt{1-\frac{x}{g+\frac{\sqrt{-a} h}{\sqrt{c}}}} \, dx,x,g+h x\right )}{h^3 (4+m) \sqrt{1-\frac{g+h x}{g-\frac{\sqrt{-a} h}{\sqrt{c}}}} \sqrt{1-\frac{g+h x}{g+\frac{\sqrt{-a} h}{\sqrt{c}}}}}+\frac{\left (\left (3 c f g^2-a f h^2 (1+m)-c h (e g-d h) (4+m)\right ) \sqrt{a+c x^2}\right ) \operatorname{Subst}\left (\int x^m \sqrt{1-\frac{x}{g-\frac{\sqrt{-a} h}{\sqrt{c}}}} \sqrt{1-\frac{x}{g+\frac{\sqrt{-a} h}{\sqrt{c}}}} \, dx,x,g+h x\right )}{c h^3 (4+m) \sqrt{1-\frac{g+h x}{g-\frac{\sqrt{-a} h}{\sqrt{c}}}} \sqrt{1-\frac{g+h x}{g+\frac{\sqrt{-a} h}{\sqrt{c}}}}}\\ &=\frac{f (g+h x)^{1+m} \left (a+c x^2\right )^{3/2}}{c h (4+m)}+\frac{\left (3 c f g^2-a f h^2 (1+m)-c h (e g-d h) (4+m)\right ) (g+h x)^{1+m} \sqrt{a+c x^2} F_1\left (1+m;-\frac{1}{2},-\frac{1}{2};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) (4+m) \sqrt{1-\frac{g+h x}{g-\frac{\sqrt{-a} h}{\sqrt{c}}}} \sqrt{1-\frac{g+h x}{g+\frac{\sqrt{-a} h}{\sqrt{c}}}}}-\frac{(3 f g-e h (4+m)) (g+h x)^{2+m} \sqrt{a+c x^2} F_1\left (2+m;-\frac{1}{2},-\frac{1}{2};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) (4+m) \sqrt{1-\frac{g+h x}{g-\frac{\sqrt{-a} h}{\sqrt{c}}}} \sqrt{1-\frac{g+h x}{g+\frac{\sqrt{-a} h}{\sqrt{c}}}}}\\ \end{align*}
Mathematica [F] time = 0.725941, size = 0, normalized size = 0. \[ \int (g+h x)^m \sqrt{a+c x^2} \left (d+e x+f x^2\right ) \, dx \]
Verification is Not applicable to the result.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.742, size = 0, normalized size = 0. \begin{align*} \int \left ( hx+g \right ) ^{m} \left ( f{x}^{2}+ex+d \right ) \sqrt{c{x}^{2}+a}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{c x^{2} + 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.
[In]
[Out]
________________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\sqrt{c x^{2} + 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.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{a + c x^{2}} \left (g + h x\right )^{m} \left (d + e x + f x^{2}\right )\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{c x^{2} + 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.
[In]
[Out]