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

Optimal. Leaf size=454 \[ -\frac{(d+e x) \sqrt [4]{a g^2+c f^2} \sqrt{\frac{\left (a+c x^2\right ) (e f-d g)^2}{(d+e x)^2 \left (a g^2+c f^2\right )}} \left (\frac{(f+g x) \sqrt{a e^2+c d^2}}{(d+e x) \sqrt{a g^2+c f^2}}+1\right ) \sqrt{\frac{\frac{(f+g x)^2 \left (a e^2+c d^2\right )}{(d+e x)^2 \left (a g^2+c f^2\right )}-\frac{2 (f+g x) (a e g+c d f)}{(d+e x) \left (a g^2+c f^2\right )}+1}{\left (\frac{(f+g x) \sqrt{a e^2+c d^2}}{(d+e x) \sqrt{a g^2+c f^2}}+1\right )^2}} \text{EllipticF}\left (2 \tan ^{-1}\left (\frac{\sqrt{f+g x} \sqrt [4]{a e^2+c d^2}}{\sqrt{d+e x} \sqrt [4]{a g^2+c f^2}}\right ),\frac{1}{2} \left (\frac{a e g+c d f}{\sqrt{a e^2+c d^2} \sqrt{a g^2+c f^2}}+1\right )\right )}{\sqrt{a+c x^2} \sqrt [4]{a e^2+c d^2} (e f-d g) \sqrt{\frac{(f+g x)^2 \left (a e^2+c d^2\right )}{(d+e x)^2 \left (a g^2+c f^2\right )}-\frac{2 (f+g x) (a e g+c d f)}{(d+e x) \left (a g^2+c f^2\right )}+1}} \]

[Out]

-(((c*f^2 + a*g^2)^(1/4)*(d + e*x)*Sqrt[((e*f - d*g)^2*(a + c*x^2))/((c*f^2 + a*g^2)*(d + e*x)^2)]*(1 + (Sqrt[
c*d^2 + a*e^2]*(f + g*x))/(Sqrt[c*f^2 + a*g^2]*(d + e*x)))*Sqrt[(1 - (2*(c*d*f + a*e*g)*(f + g*x))/((c*f^2 + a
*g^2)*(d + e*x)) + ((c*d^2 + a*e^2)*(f + g*x)^2)/((c*f^2 + a*g^2)*(d + e*x)^2))/(1 + (Sqrt[c*d^2 + a*e^2]*(f +
g*x))/(Sqrt[c*f^2 + a*g^2]*(d + e*x)))^2]*EllipticF[2*ArcTan[((c*d^2 + a*e^2)^(1/4)*Sqrt[f + g*x])/((c*f^2 +
a*g^2)^(1/4)*Sqrt[d + e*x])], (1 + (c*d*f + a*e*g)/(Sqrt[c*d^2 + a*e^2]*Sqrt[c*f^2 + a*g^2]))/2])/((c*d^2 + a*
e^2)^(1/4)*(e*f - d*g)*Sqrt[a + c*x^2]*Sqrt[1 - (2*(c*d*f + a*e*g)*(f + g*x))/((c*f^2 + a*g^2)*(d + e*x)) + ((
c*d^2 + a*e^2)*(f + g*x)^2)/((c*f^2 + a*g^2)*(d + e*x)^2)]))

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Rubi [A]  time = 0.628851, antiderivative size = 454, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067, Rules used = {936, 1103} \[ -\frac{(d+e x) \sqrt [4]{a g^2+c f^2} \sqrt{\frac{\left (a+c x^2\right ) (e f-d g)^2}{(d+e x)^2 \left (a g^2+c f^2\right )}} \left (\frac{(f+g x) \sqrt{a e^2+c d^2}}{(d+e x) \sqrt{a g^2+c f^2}}+1\right ) \sqrt{\frac{\frac{(f+g x)^2 \left (a e^2+c d^2\right )}{(d+e x)^2 \left (a g^2+c f^2\right )}-\frac{2 (f+g x) (a e g+c d f)}{(d+e x) \left (a g^2+c f^2\right )}+1}{\left (\frac{(f+g x) \sqrt{a e^2+c d^2}}{(d+e x) \sqrt{a g^2+c f^2}}+1\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c d^2+a e^2} \sqrt{f+g x}}{\sqrt [4]{c f^2+a g^2} \sqrt{d+e x}}\right )|\frac{1}{2} \left (\frac{c d f+a e g}{\sqrt{c d^2+a e^2} \sqrt{c f^2+a g^2}}+1\right )\right )}{\sqrt{a+c x^2} \sqrt [4]{a e^2+c d^2} (e f-d g) \sqrt{\frac{(f+g x)^2 \left (a e^2+c d^2\right )}{(d+e x)^2 \left (a g^2+c f^2\right )}-\frac{2 (f+g x) (a e g+c d f)}{(d+e x) \left (a g^2+c f^2\right )}+1}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(((c*f^2 + a*g^2)^(1/4)*(d + e*x)*Sqrt[((e*f - d*g)^2*(a + c*x^2))/((c*f^2 + a*g^2)*(d + e*x)^2)]*(1 + (Sqrt[
c*d^2 + a*e^2]*(f + g*x))/(Sqrt[c*f^2 + a*g^2]*(d + e*x)))*Sqrt[(1 - (2*(c*d*f + a*e*g)*(f + g*x))/((c*f^2 + a
*g^2)*(d + e*x)) + ((c*d^2 + a*e^2)*(f + g*x)^2)/((c*f^2 + a*g^2)*(d + e*x)^2))/(1 + (Sqrt[c*d^2 + a*e^2]*(f +
g*x))/(Sqrt[c*f^2 + a*g^2]*(d + e*x)))^2]*EllipticF[2*ArcTan[((c*d^2 + a*e^2)^(1/4)*Sqrt[f + g*x])/((c*f^2 +
a*g^2)^(1/4)*Sqrt[d + e*x])], (1 + (c*d*f + a*e*g)/(Sqrt[c*d^2 + a*e^2]*Sqrt[c*f^2 + a*g^2]))/2])/((c*d^2 + a*
e^2)^(1/4)*(e*f - d*g)*Sqrt[a + c*x^2]*Sqrt[1 - (2*(c*d*f + a*e*g)*(f + g*x))/((c*f^2 + a*g^2)*(d + e*x)) + ((
c*d^2 + a*e^2)*(f + g*x)^2)/((c*f^2 + a*g^2)*(d + e*x)^2)]))

Rule 936

Int[1/(Sqrt[(d_.) + (e_.)*(x_)]*Sqrt[(f_.) + (g_.)*(x_)]*Sqrt[(a_) + (c_.)*(x_)^2]), x_Symbol] :> Dist[(-2*(d
+ e*x)*Sqrt[((e*f - d*g)^2*(a + c*x^2))/((c*f^2 + a*g^2)*(d + e*x)^2)])/((e*f - d*g)*Sqrt[a + c*x^2]), Subst[I
nt[1/Sqrt[1 - ((2*c*d*f + 2*a*e*g)*x^2)/(c*f^2 + a*g^2) + ((c*d^2 + a*e^2)*x^4)/(c*f^2 + a*g^2)], x], x, Sqrt[
f + g*x]/Sqrt[d + e*x]], x] /; FreeQ[{a, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] && NeQ[c*d^2 + a*e^2, 0]

Rule 1103

Int[1/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[c/a, 4]}, Simp[((1 + q^2*x^2)*Sqrt[(
a + b*x^2 + c*x^4)/(a*(1 + q^2*x^2)^2)]*EllipticF[2*ArcTan[q*x], 1/2 - (b*q^2)/(4*c)])/(2*q*Sqrt[a + b*x^2 + c
*x^4]), x]] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && PosQ[c/a]

Rubi steps

\begin{align*} \int \frac{1}{\sqrt{d+e x} \sqrt{f+g x} \sqrt{a+c x^2}} \, dx &=-\frac{\left (2 (d+e x) \sqrt{\frac{(e f-d g)^2 \left (a+c x^2\right )}{\left (c f^2+a g^2\right ) (d+e x)^2}}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-\frac{(2 c d f+2 a e g) x^2}{c f^2+a g^2}+\frac{\left (c d^2+a e^2\right ) x^4}{c f^2+a g^2}}} \, dx,x,\frac{\sqrt{f+g x}}{\sqrt{d+e x}}\right )}{(e f-d g) \sqrt{a+c x^2}}\\ &=-\frac{\sqrt [4]{c f^2+a g^2} (d+e x) \sqrt{\frac{(e f-d g)^2 \left (a+c x^2\right )}{\left (c f^2+a g^2\right ) (d+e x)^2}} \left (1+\frac{\sqrt{c d^2+a e^2} (f+g x)}{\sqrt{c f^2+a g^2} (d+e x)}\right ) \sqrt{\frac{1-\frac{2 (c d f+a e g) (f+g x)}{\left (c f^2+a g^2\right ) (d+e x)}+\frac{\left (c d^2+a e^2\right ) (f+g x)^2}{\left (c f^2+a g^2\right ) (d+e x)^2}}{\left (1+\frac{\sqrt{c d^2+a e^2} (f+g x)}{\sqrt{c f^2+a g^2} (d+e x)}\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c d^2+a e^2} \sqrt{f+g x}}{\sqrt [4]{c f^2+a g^2} \sqrt{d+e x}}\right )|\frac{1}{2} \left (1+\frac{c d f+a e g}{\sqrt{c d^2+a e^2} \sqrt{c f^2+a g^2}}\right )\right )}{\sqrt [4]{c d^2+a e^2} (e f-d g) \sqrt{a+c x^2} \sqrt{1-\frac{2 (c d f+a e g) (f+g x)}{\left (c f^2+a g^2\right ) (d+e x)}+\frac{\left (c d^2+a e^2\right ) (f+g x)^2}{\left (c f^2+a g^2\right ) (d+e x)^2}}}\\ \end{align*}

Mathematica [C]  time = 1.39044, size = 344, normalized size = 0.76 \[ \frac{\sqrt{2} \left (\sqrt{c} x+i \sqrt{a}\right ) \sqrt{d+e x} \sqrt{\frac{\frac{i \sqrt{c} d x}{\sqrt{a}}-\frac{i \sqrt{a} e}{\sqrt{c}}+d+e x}{d+e x}} \sqrt{\frac{(f+g x) \left (\sqrt{a} e+i \sqrt{c} d\right )}{(d+e x) \left (\sqrt{a} g+i \sqrt{c} f\right )}} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{\left (\sqrt{c} x+i \sqrt{a}\right ) (e f-d g)}{(d+e x) \left (\sqrt{c} f-i \sqrt{a} g\right )}}\right ),-\frac{\frac{i \sqrt{c} d f}{\sqrt{a}}+\frac{i \sqrt{a} e g}{\sqrt{c}}+d g-e f}{2 e f-2 d g}\right )}{\sqrt{a+c x^2} \sqrt{f+g x} \left (\sqrt{c} d-i \sqrt{a} e\right ) \sqrt{\frac{\left (\sqrt{c} x+i \sqrt{a}\right ) (e f-d g)}{(d+e x) \left (\sqrt{c} f-i \sqrt{a} g\right )}}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[2]*(I*Sqrt[a] + Sqrt[c]*x)*Sqrt[d + e*x]*Sqrt[(d - (I*Sqrt[a]*e)/Sqrt[c] + (I*Sqrt[c]*d*x)/Sqrt[a] + e*x
)/(d + e*x)]*Sqrt[((I*Sqrt[c]*d + Sqrt[a]*e)*(f + g*x))/((I*Sqrt[c]*f + Sqrt[a]*g)*(d + e*x))]*EllipticF[ArcSi
n[Sqrt[((e*f - d*g)*(I*Sqrt[a] + Sqrt[c]*x))/((Sqrt[c]*f - I*Sqrt[a]*g)*(d + e*x))]], -(((I*Sqrt[c]*d*f)/Sqrt[
a] - e*f + d*g + (I*Sqrt[a]*e*g)/Sqrt[c])/(2*e*f - 2*d*g))])/((Sqrt[c]*d - I*Sqrt[a]*e)*Sqrt[((e*f - d*g)*(I*S
qrt[a] + Sqrt[c]*x))/((Sqrt[c]*f - I*Sqrt[a]*g)*(d + e*x))]*Sqrt[f + g*x]*Sqrt[a + c*x^2])

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Maple [A]  time = 0.624, size = 433, normalized size = 1. \begin{align*} 2\,{\frac{ \left ( c{e}^{2}f{x}^{2}-\sqrt{-ac}{x}^{2}{e}^{2}g+2\,xcdef-2\,\sqrt{-ac}xdeg+c{d}^{2}f-\sqrt{-ac}{d}^{2}g \right ) \sqrt{ex+d}\sqrt{gx+f}\sqrt{c{x}^{2}+a}}{ \left ( dg-ef \right ) \left ( -\sqrt{-ac}e+cd \right ) \sqrt{ceg{x}^{4}+cdg{x}^{3}+cef{x}^{3}+aeg{x}^{2}+cdf{x}^{2}+adgx+aefx+adf}}{\it EllipticF} \left ( \sqrt{{\frac{ \left ( \sqrt{-ac}e-cd \right ) \left ( gx+f \right ) }{ \left ( \sqrt{-ac}g-cf \right ) \left ( ex+d \right ) }}},\sqrt{{\frac{ \left ( \sqrt{-ac}e+cd \right ) \left ( \sqrt{-ac}g-cf \right ) }{ \left ( \sqrt{-ac}g+cf \right ) \left ( \sqrt{-ac}e-cd \right ) }}} \right ) \sqrt{{\frac{ \left ( dg-ef \right ) \left ( cx+\sqrt{-ac} \right ) }{ \left ( \sqrt{-ac}g-cf \right ) \left ( ex+d \right ) }}}\sqrt{{\frac{ \left ( dg-ef \right ) \left ( -cx+\sqrt{-ac} \right ) }{ \left ( \sqrt{-ac}g+cf \right ) \left ( ex+d \right ) }}}\sqrt{{\frac{ \left ( \sqrt{-ac}e-cd \right ) \left ( gx+f \right ) }{ \left ( \sqrt{-ac}g-cf \right ) \left ( ex+d \right ) }}}{\frac{1}{\sqrt{-{\frac{ \left ( gx+f \right ) \left ( ex+d \right ) \left ( -cx+\sqrt{-ac} \right ) \left ( cx+\sqrt{-ac} \right ) }{c}}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{c x^{2} + a} \sqrt{e x + d} \sqrt{g x + f}}{c e g x^{4} +{\left (c e f + c d g\right )} x^{3} + a d f +{\left (c d f + a e g\right )} x^{2} +{\left (a e f + a d g\right )} x}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(sqrt(c*x^2 + a)*sqrt(e*x + d)*sqrt(g*x + f)/(c*e*g*x^4 + (c*e*f + c*d*g)*x^3 + a*d*f + (c*d*f + a*e*g
)*x^2 + (a*e*f + a*d*g)*x), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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