3.903 \(\int \frac{\sqrt{f+g x}}{\sqrt{a+b x+c x^2}} \, dx\)

Optimal. Leaf size=188 \[ \frac{\sqrt{2} \sqrt{b^2-4 a c} \sqrt{f+g x} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} g}{2 c f-\left (b+\sqrt{b^2-4 a c}\right ) g}\right )}{c \sqrt{a+b x+c x^2} \sqrt{\frac{c (f+g x)}{2 c f-g \left (\sqrt{b^2-4 a c}+b\right )}}} \]

[Out]

(Sqrt[2]*Sqrt[b^2 - 4*a*c]*Sqrt[f + g*x]*Sqrt[-((c*(a + b*x + c*x^2))/(b^2 - 4*a*c))]*EllipticE[ArcSin[Sqrt[(b
+ Sqrt[b^2 - 4*a*c] + 2*c*x)/Sqrt[b^2 - 4*a*c]]/Sqrt[2]], (-2*Sqrt[b^2 - 4*a*c]*g)/(2*c*f - (b + Sqrt[b^2 - 4
*a*c])*g)])/(c*Sqrt[(c*(f + g*x))/(2*c*f - (b + Sqrt[b^2 - 4*a*c])*g)]*Sqrt[a + b*x + c*x^2])

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Rubi [A]  time = 0.064988, antiderivative size = 188, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083, Rules used = {718, 424} \[ \frac{\sqrt{2} \sqrt{b^2-4 a c} \sqrt{f+g x} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} g}{2 c f-\left (b+\sqrt{b^2-4 a c}\right ) g}\right )}{c \sqrt{a+b x+c x^2} \sqrt{\frac{c (f+g x)}{2 c f-g \left (\sqrt{b^2-4 a c}+b\right )}}} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[f + g*x]/Sqrt[a + b*x + c*x^2],x]

[Out]

(Sqrt[2]*Sqrt[b^2 - 4*a*c]*Sqrt[f + g*x]*Sqrt[-((c*(a + b*x + c*x^2))/(b^2 - 4*a*c))]*EllipticE[ArcSin[Sqrt[(b
+ Sqrt[b^2 - 4*a*c] + 2*c*x)/Sqrt[b^2 - 4*a*c]]/Sqrt[2]], (-2*Sqrt[b^2 - 4*a*c]*g)/(2*c*f - (b + Sqrt[b^2 - 4
*a*c])*g)])/(c*Sqrt[(c*(f + g*x))/(2*c*f - (b + Sqrt[b^2 - 4*a*c])*g)]*Sqrt[a + b*x + c*x^2])

Rule 718

Int[((d_.) + (e_.)*(x_))^(m_)/Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Dist[(2*Rt[b^2 - 4*a*c, 2]
*(d + e*x)^m*Sqrt[-((c*(a + b*x + c*x^2))/(b^2 - 4*a*c))])/(c*Sqrt[a + b*x + c*x^2]*((2*c*(d + e*x))/(2*c*d -
b*e - e*Rt[b^2 - 4*a*c, 2]))^m), Subst[Int[(1 + (2*e*Rt[b^2 - 4*a*c, 2]*x^2)/(2*c*d - b*e - e*Rt[b^2 - 4*a*c,
2]))^m/Sqrt[1 - x^2], x], x, Sqrt[(b + Rt[b^2 - 4*a*c, 2] + 2*c*x)/(2*Rt[b^2 - 4*a*c, 2])]], x] /; FreeQ[{a, b
, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && NeQ[2*c*d - b*e, 0] && EqQ[m^2, 1/4]

Rule 424

Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[(Sqrt[a]*EllipticE[ArcSin[Rt[-(d/c)
, 2]*x], (b*c)/(a*d)])/(Sqrt[c]*Rt[-(d/c), 2]), x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[
a, 0]

Rubi steps

\begin{align*} \int \frac{\sqrt{f+g x}}{\sqrt{a+b x+c x^2}} \, dx &=\frac{\left (\sqrt{2} \sqrt{b^2-4 a c} \sqrt{f+g x} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}}\right ) \operatorname{Subst}\left (\int \frac{\sqrt{1+\frac{2 \sqrt{b^2-4 a c} g x^2}{2 c f-b g-\sqrt{b^2-4 a c} g}}}{\sqrt{1-x^2}} \, dx,x,\frac{\sqrt{\frac{b+\sqrt{b^2-4 a c}+2 c x}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )}{c \sqrt{\frac{c (f+g x)}{2 c f-b g-\sqrt{b^2-4 a c} g}} \sqrt{a+b x+c x^2}}\\ &=\frac{\sqrt{2} \sqrt{b^2-4 a c} \sqrt{f+g x} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+\sqrt{b^2-4 a c}+2 c x}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} g}{2 c f-\left (b+\sqrt{b^2-4 a c}\right ) g}\right )}{c \sqrt{\frac{c (f+g x)}{2 c f-\left (b+\sqrt{b^2-4 a c}\right ) g}} \sqrt{a+b x+c x^2}}\\ \end{align*}

Mathematica [C]  time = 0.815616, size = 365, normalized size = 1.94 \[ \frac{i \left (g \left (\sqrt{b^2-4 a c}-b\right )+2 c f\right ) \sqrt{\frac{g \left (\sqrt{b^2-4 a c}+b+2 c x\right )}{g \left (\sqrt{b^2-4 a c}+b\right )-2 c f}} \sqrt{1-\frac{2 c (f+g x)}{g \left (\sqrt{b^2-4 a c}-b\right )+2 c f}} \left (E\left (i \sinh ^{-1}\left (\sqrt{2} \sqrt{\frac{c}{\left (b+\sqrt{b^2-4 a c}\right ) g-2 c f}} \sqrt{f+g x}\right )|\frac{2 c f-\left (b+\sqrt{b^2-4 a c}\right ) g}{2 c f+\left (\sqrt{b^2-4 a c}-b\right ) g}\right )-\text{EllipticF}\left (i \sinh ^{-1}\left (\sqrt{2} \sqrt{f+g x} \sqrt{\frac{c}{g \left (\sqrt{b^2-4 a c}+b\right )-2 c f}}\right ),\frac{2 c f-g \left (\sqrt{b^2-4 a c}+b\right )}{g \left (\sqrt{b^2-4 a c}-b\right )+2 c f}\right )\right )}{\sqrt{2} c g \sqrt{a+x (b+c x)} \sqrt{\frac{c}{g \left (\sqrt{b^2-4 a c}+b\right )-2 c f}}} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[f + g*x]/Sqrt[a + b*x + c*x^2],x]

[Out]

(I*(2*c*f + (-b + Sqrt[b^2 - 4*a*c])*g)*Sqrt[(g*(b + Sqrt[b^2 - 4*a*c] + 2*c*x))/(-2*c*f + (b + Sqrt[b^2 - 4*a
*c])*g)]*Sqrt[1 - (2*c*(f + g*x))/(2*c*f + (-b + Sqrt[b^2 - 4*a*c])*g)]*(EllipticE[I*ArcSinh[Sqrt[2]*Sqrt[c/(-
2*c*f + (b + Sqrt[b^2 - 4*a*c])*g)]*Sqrt[f + g*x]], (2*c*f - (b + Sqrt[b^2 - 4*a*c])*g)/(2*c*f + (-b + Sqrt[b^
2 - 4*a*c])*g)] - EllipticF[I*ArcSinh[Sqrt[2]*Sqrt[c/(-2*c*f + (b + Sqrt[b^2 - 4*a*c])*g)]*Sqrt[f + g*x]], (2*
c*f - (b + Sqrt[b^2 - 4*a*c])*g)/(2*c*f + (-b + Sqrt[b^2 - 4*a*c])*g)]))/(Sqrt[2]*c*g*Sqrt[c/(-2*c*f + (b + Sq
rt[b^2 - 4*a*c])*g)]*Sqrt[a + x*(b + c*x)])

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Maple [B]  time = 0.311, size = 747, normalized size = 4. \begin{align*}{\frac{\sqrt{2}}{2\,g \left ( cg{x}^{3}+bg{x}^{2}+cf{x}^{2}+agx+bfx+af \right ){c}^{2}}\sqrt{gx+f}\sqrt{c{x}^{2}+bx+a} \left ( g\sqrt{-4\,ac+{b}^{2}}+bg-2\,cf \right ) \sqrt{-{c \left ( gx+f \right ) \left ( g\sqrt{-4\,ac+{b}^{2}}+bg-2\,cf \right ) ^{-1}}}\sqrt{{g \left ( -b-2\,cx+\sqrt{-4\,ac+{b}^{2}} \right ) \left ( 2\,cf-bg+g\sqrt{-4\,ac+{b}^{2}} \right ) ^{-1}}}\sqrt{{g \left ( b+2\,cx+\sqrt{-4\,ac+{b}^{2}} \right ) \left ( g\sqrt{-4\,ac+{b}^{2}}+bg-2\,cf \right ) ^{-1}}} \left ({\it EllipticF} \left ( \sqrt{2}\sqrt{-{c \left ( gx+f \right ) \left ( g\sqrt{-4\,ac+{b}^{2}}+bg-2\,cf \right ) ^{-1}}},\sqrt{-{ \left ( g\sqrt{-4\,ac+{b}^{2}}+bg-2\,cf \right ) \left ( 2\,cf-bg+g\sqrt{-4\,ac+{b}^{2}} \right ) ^{-1}}} \right ) gb-2\,f{\it EllipticF} \left ( \sqrt{2}\sqrt{-{\frac{c \left ( gx+f \right ) }{g\sqrt{-4\,ac+{b}^{2}}+bg-2\,cf}}},\sqrt{-{\frac{g\sqrt{-4\,ac+{b}^{2}}+bg-2\,cf}{2\,cf-bg+g\sqrt{-4\,ac+{b}^{2}}}}} \right ) c-{\it EllipticF} \left ( \sqrt{2}\sqrt{-{c \left ( gx+f \right ) \left ( g\sqrt{-4\,ac+{b}^{2}}+bg-2\,cf \right ) ^{-1}}},\sqrt{-{ \left ( g\sqrt{-4\,ac+{b}^{2}}+bg-2\,cf \right ) \left ( 2\,cf-bg+g\sqrt{-4\,ac+{b}^{2}} \right ) ^{-1}}} \right ) g\sqrt{-4\,ac+{b}^{2}}-{\it EllipticE} \left ( \sqrt{2}\sqrt{-{c \left ( gx+f \right ) \left ( g\sqrt{-4\,ac+{b}^{2}}+bg-2\,cf \right ) ^{-1}}},\sqrt{-{ \left ( g\sqrt{-4\,ac+{b}^{2}}+bg-2\,cf \right ) \left ( 2\,cf-bg+g\sqrt{-4\,ac+{b}^{2}} \right ) ^{-1}}} \right ) bg+2\,{\it EllipticE} \left ( \sqrt{2}\sqrt{-{\frac{c \left ( gx+f \right ) }{g\sqrt{-4\,ac+{b}^{2}}+bg-2\,cf}}},\sqrt{-{\frac{g\sqrt{-4\,ac+{b}^{2}}+bg-2\,cf}{2\,cf-bg+g\sqrt{-4\,ac+{b}^{2}}}}} \right ) cf+\sqrt{-4\,ac+{b}^{2}}{\it EllipticE} \left ( \sqrt{2}\sqrt{-{c \left ( gx+f \right ) \left ( g\sqrt{-4\,ac+{b}^{2}}+bg-2\,cf \right ) ^{-1}}},\sqrt{-{ \left ( g\sqrt{-4\,ac+{b}^{2}}+bg-2\,cf \right ) \left ( 2\,cf-bg+g\sqrt{-4\,ac+{b}^{2}} \right ) ^{-1}}} \right ) g \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((g*x+f)^(1/2)/(c*x^2+b*x+a)^(1/2),x)

[Out]

1/2*(g*x+f)^(1/2)*(c*x^2+b*x+a)^(1/2)*(g*(-4*a*c+b^2)^(1/2)+b*g-2*c*f)*2^(1/2)*(-(g*x+f)*c/(g*(-4*a*c+b^2)^(1/
2)+b*g-2*c*f))^(1/2)*(g*(-b-2*c*x+(-4*a*c+b^2)^(1/2))/(2*c*f-b*g+g*(-4*a*c+b^2)^(1/2)))^(1/2)*(g*(b+2*c*x+(-4*
a*c+b^2)^(1/2))/(g*(-4*a*c+b^2)^(1/2)+b*g-2*c*f))^(1/2)*(EllipticF(2^(1/2)*(-(g*x+f)*c/(g*(-4*a*c+b^2)^(1/2)+b
*g-2*c*f))^(1/2),(-(g*(-4*a*c+b^2)^(1/2)+b*g-2*c*f)/(2*c*f-b*g+g*(-4*a*c+b^2)^(1/2)))^(1/2))*g*b-2*f*EllipticF
(2^(1/2)*(-(g*x+f)*c/(g*(-4*a*c+b^2)^(1/2)+b*g-2*c*f))^(1/2),(-(g*(-4*a*c+b^2)^(1/2)+b*g-2*c*f)/(2*c*f-b*g+g*(
-4*a*c+b^2)^(1/2)))^(1/2))*c-EllipticF(2^(1/2)*(-(g*x+f)*c/(g*(-4*a*c+b^2)^(1/2)+b*g-2*c*f))^(1/2),(-(g*(-4*a*
c+b^2)^(1/2)+b*g-2*c*f)/(2*c*f-b*g+g*(-4*a*c+b^2)^(1/2)))^(1/2))*g*(-4*a*c+b^2)^(1/2)-EllipticE(2^(1/2)*(-(g*x
+f)*c/(g*(-4*a*c+b^2)^(1/2)+b*g-2*c*f))^(1/2),(-(g*(-4*a*c+b^2)^(1/2)+b*g-2*c*f)/(2*c*f-b*g+g*(-4*a*c+b^2)^(1/
2)))^(1/2))*b*g+2*EllipticE(2^(1/2)*(-(g*x+f)*c/(g*(-4*a*c+b^2)^(1/2)+b*g-2*c*f))^(1/2),(-(g*(-4*a*c+b^2)^(1/2
)+b*g-2*c*f)/(2*c*f-b*g+g*(-4*a*c+b^2)^(1/2)))^(1/2))*c*f+(-4*a*c+b^2)^(1/2)*EllipticE(2^(1/2)*(-(g*x+f)*c/(g*
(-4*a*c+b^2)^(1/2)+b*g-2*c*f))^(1/2),(-(g*(-4*a*c+b^2)^(1/2)+b*g-2*c*f)/(2*c*f-b*g+g*(-4*a*c+b^2)^(1/2)))^(1/2
))*g)/g/(c*g*x^3+b*g*x^2+c*f*x^2+a*g*x+b*f*x+a*f)/c^2

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{g x + f}}{\sqrt{c x^{2} + b x + a}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((g*x+f)^(1/2)/(c*x^2+b*x+a)^(1/2),x, algorithm="maxima")

[Out]

integrate(sqrt(g*x + f)/sqrt(c*x^2 + b*x + a), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{g x + f}}{\sqrt{c x^{2} + b x + a}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((g*x+f)^(1/2)/(c*x^2+b*x+a)^(1/2),x, algorithm="fricas")

[Out]

integral(sqrt(g*x + f)/sqrt(c*x^2 + b*x + a), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{f + g x}}{\sqrt{a + b x + c x^{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((g*x+f)**(1/2)/(c*x**2+b*x+a)**(1/2),x)

[Out]

Integral(sqrt(f + g*x)/sqrt(a + b*x + c*x**2), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{g x + f}}{\sqrt{c x^{2} + b x + a}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((g*x+f)^(1/2)/(c*x^2+b*x+a)^(1/2),x, algorithm="giac")

[Out]

integrate(sqrt(g*x + f)/sqrt(c*x^2 + b*x + a), x)