3.654 \(\int \frac{1}{(d+e x) \sqrt{f+g x} \sqrt{1+c x^2}} \, dx\)

Optimal. Leaf size=110 \[ -\frac{2 \sqrt{\frac{\sqrt{-c} (f+g x)}{\sqrt{-c} f+g}} \Pi \left (\frac{2 e}{\sqrt{-c} d+e};\sin ^{-1}\left (\frac{\sqrt{1-\sqrt{-c} x}}{\sqrt{2}}\right )|\frac{2 g}{\sqrt{-c} f+g}\right )}{\left (\sqrt{-c} d+e\right ) \sqrt{f+g x}} \]

[Out]

(-2*Sqrt[(Sqrt[-c]*(f + g*x))/(Sqrt[-c]*f + g)]*EllipticPi[(2*e)/(Sqrt[-c]*d + e), ArcSin[Sqrt[1 - Sqrt[-c]*x]
/Sqrt[2]], (2*g)/(Sqrt[-c]*f + g)])/((Sqrt[-c]*d + e)*Sqrt[f + g*x])

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Rubi [A]  time = 0.304273, antiderivative size = 110, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {932, 168, 538, 537} \[ -\frac{2 \sqrt{\frac{\sqrt{-c} (f+g x)}{\sqrt{-c} f+g}} \Pi \left (\frac{2 e}{\sqrt{-c} d+e};\sin ^{-1}\left (\frac{\sqrt{1-\sqrt{-c} x}}{\sqrt{2}}\right )|\frac{2 g}{\sqrt{-c} f+g}\right )}{\left (\sqrt{-c} d+e\right ) \sqrt{f+g x}} \]

Antiderivative was successfully verified.

[In]

Int[1/((d + e*x)*Sqrt[f + g*x]*Sqrt[1 + c*x^2]),x]

[Out]

(-2*Sqrt[(Sqrt[-c]*(f + g*x))/(Sqrt[-c]*f + g)]*EllipticPi[(2*e)/(Sqrt[-c]*d + e), ArcSin[Sqrt[1 - Sqrt[-c]*x]
/Sqrt[2]], (2*g)/(Sqrt[-c]*f + g)])/((Sqrt[-c]*d + e)*Sqrt[f + g*x])

Rule 932

Int[1/(((d_.) + (e_.)*(x_))*Sqrt[(f_.) + (g_.)*(x_)]*Sqrt[(a_) + (c_.)*(x_)^2]), x_Symbol] :> With[{q = Rt[-(c
/a), 2]}, Dist[1/Sqrt[a], Int[1/((d + e*x)*Sqrt[f + g*x]*Sqrt[1 - q*x]*Sqrt[1 + q*x]), x], x]] /; FreeQ[{a, c,
d, e, f, g}, x] && NeQ[e*f - d*g, 0] && NeQ[c*d^2 + a*e^2, 0] && GtQ[a, 0]

Rule 168

Int[1/(((a_.) + (b_.)*(x_))*Sqrt[(c_.) + (d_.)*(x_)]*Sqrt[(e_.) + (f_.)*(x_)]*Sqrt[(g_.) + (h_.)*(x_)]), x_Sym
bol] :> Dist[-2, Subst[Int[1/(Simp[b*c - a*d - b*x^2, x]*Sqrt[Simp[(d*e - c*f)/d + (f*x^2)/d, x]]*Sqrt[Simp[(d
*g - c*h)/d + (h*x^2)/d, x]]), x], x, Sqrt[c + d*x]], x] /; FreeQ[{a, b, c, d, e, f, g, h}, x] && GtQ[(d*e - c
*f)/d, 0]

Rule 538

Int[1/(((a_) + (b_.)*(x_)^2)*Sqrt[(c_) + (d_.)*(x_)^2]*Sqrt[(e_) + (f_.)*(x_)^2]), x_Symbol] :> Dist[Sqrt[1 +
(d*x^2)/c]/Sqrt[c + d*x^2], Int[1/((a + b*x^2)*Sqrt[1 + (d*x^2)/c]*Sqrt[e + f*x^2]), x], x] /; FreeQ[{a, b, c,
d, e, f}, x] &&  !GtQ[c, 0]

Rule 537

Int[1/(((a_) + (b_.)*(x_)^2)*Sqrt[(c_) + (d_.)*(x_)^2]*Sqrt[(e_) + (f_.)*(x_)^2]), x_Symbol] :> Simp[(1*Ellipt
icPi[(b*c)/(a*d), ArcSin[Rt[-(d/c), 2]*x], (c*f)/(d*e)])/(a*Sqrt[c]*Sqrt[e]*Rt[-(d/c), 2]), x] /; FreeQ[{a, b,
c, d, e, f}, x] &&  !GtQ[d/c, 0] && GtQ[c, 0] && GtQ[e, 0] &&  !( !GtQ[f/e, 0] && SimplerSqrtQ[-(f/e), -(d/c)
])

Rubi steps

\begin{align*} \int \frac{1}{(d+e x) \sqrt{f+g x} \sqrt{1+c x^2}} \, dx &=\int \frac{1}{\sqrt{1-\sqrt{-c} x} \sqrt{1+\sqrt{-c} x} (d+e x) \sqrt{f+g x}} \, dx\\ &=-\left (2 \operatorname{Subst}\left (\int \frac{1}{\sqrt{2-x^2} \left (\sqrt{-c} d+e-e x^2\right ) \sqrt{f+\frac{g}{\sqrt{-c}}-\frac{g x^2}{\sqrt{-c}}}} \, dx,x,\sqrt{1-\sqrt{-c} x}\right )\right )\\ &=-\frac{\left (2 \sqrt{1+\frac{g \left (-1+\sqrt{-c} x\right )}{\sqrt{-c} f+g}}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{2-x^2} \left (\sqrt{-c} d+e-e x^2\right ) \sqrt{1-\frac{g x^2}{\sqrt{-c} \left (f+\frac{g}{\sqrt{-c}}\right )}}} \, dx,x,\sqrt{1-\sqrt{-c} x}\right )}{\sqrt{f+g x}}\\ &=-\frac{2 \sqrt{1-\frac{g \left (1-\sqrt{-c} x\right )}{\sqrt{-c} f+g}} \Pi \left (\frac{2 e}{\sqrt{-c} d+e};\sin ^{-1}\left (\frac{\sqrt{1-\sqrt{-c} x}}{\sqrt{2}}\right )|\frac{2 g}{\sqrt{-c} f+g}\right )}{\left (\sqrt{-c} d+e\right ) \sqrt{f+g x}}\\ \end{align*}

Mathematica [C]  time = 0.902458, size = 261, normalized size = 2.37 \[ -\frac{2 i (f+g x) \sqrt{\frac{g \left (x+\frac{i}{\sqrt{c}}\right )}{f+g x}} \sqrt{-\frac{-g x+\frac{i g}{\sqrt{c}}}{f+g x}} \left (\text{EllipticF}\left (i \sinh ^{-1}\left (\frac{\sqrt{-f-\frac{i g}{\sqrt{c}}}}{\sqrt{f+g x}}\right ),\frac{\sqrt{c} f-i g}{\sqrt{c} f+i g}\right )-\Pi \left (\frac{\sqrt{c} (e f-d g)}{e \left (\sqrt{c} f+i g\right )};i \sinh ^{-1}\left (\frac{\sqrt{-f-\frac{i g}{\sqrt{c}}}}{\sqrt{f+g x}}\right )|\frac{\sqrt{c} f-i g}{\sqrt{c} f+i g}\right )\right )}{\sqrt{c x^2+1} \sqrt{-f-\frac{i g}{\sqrt{c}}} (e f-d g)} \]

Antiderivative was successfully verified.

[In]

Integrate[1/((d + e*x)*Sqrt[f + g*x]*Sqrt[1 + c*x^2]),x]

[Out]

((-2*I)*Sqrt[(g*(I/Sqrt[c] + x))/(f + g*x)]*Sqrt[-(((I*g)/Sqrt[c] - g*x)/(f + g*x))]*(f + g*x)*(EllipticF[I*Ar
cSinh[Sqrt[-f - (I*g)/Sqrt[c]]/Sqrt[f + g*x]], (Sqrt[c]*f - I*g)/(Sqrt[c]*f + I*g)] - EllipticPi[(Sqrt[c]*(e*f
- d*g))/(e*(Sqrt[c]*f + I*g)), I*ArcSinh[Sqrt[-f - (I*g)/Sqrt[c]]/Sqrt[f + g*x]], (Sqrt[c]*f - I*g)/(Sqrt[c]*
f + I*g)]))/(Sqrt[-f - (I*g)/Sqrt[c]]*(e*f - d*g)*Sqrt[1 + c*x^2])

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Maple [B]  time = 0.342, size = 215, normalized size = 2. \begin{align*} 2\,{\frac{ \left ( g+f\sqrt{-c} \right ) \sqrt{c{x}^{2}+1}\sqrt{gx+f}}{\sqrt{-c} \left ( dg-ef \right ) \left ( cg{x}^{3}+cf{x}^{2}+gx+f \right ) }{\it EllipticPi} \left ( \sqrt{{\frac{ \left ( gx+f \right ) \sqrt{-c}}{g+f\sqrt{-c}}}},-{\frac{e \left ( g+f\sqrt{-c} \right ) }{\sqrt{-c} \left ( dg-ef \right ) }},\sqrt{{\frac{g+f\sqrt{-c}}{f\sqrt{-c}-g}}} \right ) \sqrt{-{\frac{ \left ( -1+x\sqrt{-c} \right ) g}{g+f\sqrt{-c}}}}\sqrt{-{\frac{ \left ( x\sqrt{-c}+1 \right ) g}{f\sqrt{-c}-g}}}\sqrt{{\frac{ \left ( gx+f \right ) \sqrt{-c}}{g+f\sqrt{-c}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(e*x+d)/(g*x+f)^(1/2)/(c*x^2+1)^(1/2),x)

[Out]

2*(g+f*(-c)^(1/2))/(-c)^(1/2)*EllipticPi(((g*x+f)*(-c)^(1/2)/(g+f*(-c)^(1/2)))^(1/2),-(g+f*(-c)^(1/2))*e/(-c)^
(1/2)/(d*g-e*f),((g+f*(-c)^(1/2))/(f*(-c)^(1/2)-g))^(1/2))*(-(-1+x*(-c)^(1/2))*g/(g+f*(-c)^(1/2)))^(1/2)*(-(x*
(-c)^(1/2)+1)*g/(f*(-c)^(1/2)-g))^(1/2)*((g*x+f)*(-c)^(1/2)/(g+f*(-c)^(1/2)))^(1/2)*(c*x^2+1)^(1/2)*(g*x+f)^(1
/2)/(d*g-e*f)/(c*g*x^3+c*f*x^2+g*x+f)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{c x^{2} + 1}{\left (e x + d\right )} \sqrt{g x + f}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(1/(sqrt(c*x^2 + 1)*(e*x + d)*sqrt(g*x + f)), x)

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Fricas [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.

[In]

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

[Out]

Timed out

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (d + e x\right ) \sqrt{f + g x} \sqrt{c x^{2} + 1}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)/(g*x+f)**(1/2)/(c*x**2+1)**(1/2),x)

[Out]

Integral(1/((d + e*x)*sqrt(f + g*x)*sqrt(c*x**2 + 1)), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{c x^{2} + 1}{\left (e x + d\right )} \sqrt{g x + f}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(1/(sqrt(c*x^2 + 1)*(e*x + d)*sqrt(g*x + f)), x)