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

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

[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 = 1.21633, 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 \[ -\frac{(d+e x) \sqrt [4]{a g^2+c f^2} \sqrt{\frac{\left (a+c x^2\right ) (e f-d g)^2}{(d+e x)^2 \left (a g^2+c f^2\right )}} \left (\frac{(f+g x) \sqrt{a e^2+c d^2}}{(d+e x) \sqrt{a g^2+c f^2}}+1\right ) \sqrt{\frac{\frac{(f+g x)^2 \left (a e^2+c d^2\right )}{(d+e x)^2 \left (a g^2+c f^2\right )}-\frac{2 (f+g x) (a e g+c d f)}{(d+e x) \left (a g^2+c f^2\right )}+1}{\left (\frac{(f+g x) \sqrt{a e^2+c d^2}}{(d+e x) \sqrt{a g^2+c f^2}}+1\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c d^2+a e^2} \sqrt{f+g x}}{\sqrt [4]{c f^2+a g^2} \sqrt{d+e x}}\right )|\frac{1}{2} \left (\frac{c d f+a e g}{\sqrt{c d^2+a e^2} \sqrt{c f^2+a g^2}}+1\right )\right )}{\sqrt{a+c x^2} \sqrt [4]{a e^2+c d^2} (e f-d g) \sqrt{\frac{(f+g x)^2 \left (a e^2+c d^2\right )}{(d+e x)^2 \left (a g^2+c f^2\right )}-\frac{2 (f+g x) (a e g+c d f)}{(d+e x) \left (a g^2+c f^2\right )}+1}} \]

Warning: Unable to verify antiderivative.

[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)]))

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Rubi in Sympy [A]  time = 170.923, size = 400, normalized size = 0.88 \[ \frac{\sqrt{\frac{1 - \frac{2 \left (f + g x\right ) \left (a e g + c d f\right )}{\left (d + e x\right ) \left (a g^{2} + c f^{2}\right )} + \frac{\left (f + g x\right )^{2} \left (a e^{2} + c d^{2}\right )}{\left (d + e x\right )^{2} \left (a g^{2} + c f^{2}\right )}}{\left (1 + \frac{\left (f + g x\right ) \sqrt{a e^{2} + c d^{2}}}{\left (d + e x\right ) \sqrt{a g^{2} + c f^{2}}}\right )^{2}}} \sqrt{\frac{\left (a + c x^{2}\right ) \left (d g - e f\right )^{2}}{\left (d + e x\right )^{2} \left (a g^{2} + c f^{2}\right )}} \left (1 + \frac{\left (f + g x\right ) \sqrt{a e^{2} + c d^{2}}}{\left (d + e x\right ) \sqrt{a g^{2} + c f^{2}}}\right ) \left (d + e x\right ) \sqrt [4]{a g^{2} + c f^{2}} F\left (2 \operatorname{atan}{\left (\frac{\sqrt{f + g x} \sqrt [4]{a e^{2} + c d^{2}}}{\sqrt{d + e x} \sqrt [4]{a g^{2} + c f^{2}}} \right )}\middle | \frac{1}{2} + \frac{2 a e g + 2 c d f}{4 \sqrt{a e^{2} + c d^{2}} \sqrt{a g^{2} + c f^{2}}}\right )}{\sqrt{a + c x^{2}} \sqrt [4]{a e^{2} + c d^{2}} \left (d g - e f\right ) \sqrt{1 - \frac{2 \left (f + g x\right ) \left (a e g + c d f\right )}{\left (d + e x\right ) \left (a g^{2} + c f^{2}\right )} + \frac{\left (f + g x\right )^{2} \left (a e^{2} + c d^{2}\right )}{\left (d + e x\right )^{2} \left (a g^{2} + c f^{2}\right )}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(e*x+d)**(1/2)/(g*x+f)**(1/2)/(c*x**2+a)**(1/2),x)

[Out]

sqrt((1 - 2*(f + g*x)*(a*e*g + c*d*f)/((d + e*x)*(a*g**2 + c*f**2)) + (f + g*x)*
*2*(a*e**2 + c*d**2)/((d + e*x)**2*(a*g**2 + c*f**2)))/(1 + (f + g*x)*sqrt(a*e**
2 + c*d**2)/((d + e*x)*sqrt(a*g**2 + c*f**2)))**2)*sqrt((a + c*x**2)*(d*g - e*f)
**2/((d + e*x)**2*(a*g**2 + c*f**2)))*(1 + (f + g*x)*sqrt(a*e**2 + c*d**2)/((d +
e*x)*sqrt(a*g**2 + c*f**2)))*(d + e*x)*(a*g**2 + c*f**2)**(1/4)*elliptic_f(2*at
an(sqrt(f + g*x)*(a*e**2 + c*d**2)**(1/4)/(sqrt(d + e*x)*(a*g**2 + c*f**2)**(1/4
))), 1/2 + (2*a*e*g + 2*c*d*f)/(4*sqrt(a*e**2 + c*d**2)*sqrt(a*g**2 + c*f**2)))/
(sqrt(a + c*x**2)*(a*e**2 + c*d**2)**(1/4)*(d*g - e*f)*sqrt(1 - 2*(f + g*x)*(a*e
*g + c*d*f)/((d + e*x)*(a*g**2 + c*f**2)) + (f + g*x)**2*(a*e**2 + c*d**2)/((d +
e*x)**2*(a*g**2 + c*f**2))))

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Mathematica [C]  time = 1.95133, size = 344, normalized size = 0.76 \[ \frac{\sqrt{2} \left (\sqrt{c} x+i \sqrt{a}\right ) \sqrt{d+e x} \sqrt{\frac{\frac{i \sqrt{c} d x}{\sqrt{a}}-\frac{i \sqrt{a} e}{\sqrt{c}}+d+e x}{d+e x}} \sqrt{\frac{(f+g x) \left (\sqrt{a} e+i \sqrt{c} d\right )}{(d+e x) \left (\sqrt{a} g+i \sqrt{c} f\right )}} F\left (\sin ^{-1}\left (\sqrt{\frac{(e f-d g) \left (\sqrt{c} x+i \sqrt{a}\right )}{\left (\sqrt{c} f-i \sqrt{a} g\right ) (d+e x)}}\right )|-\frac{\frac{i \sqrt{c} d f}{\sqrt{a}}-e f+d g+\frac{i \sqrt{a} e g}{\sqrt{c}}}{2 e f-2 d g}\right )}{\sqrt{a+c x^2} \sqrt{f+g x} \left (\sqrt{c} d-i \sqrt{a} e\right ) \sqrt{\frac{\left (\sqrt{c} x+i \sqrt{a}\right ) (e f-d g)}{(d+e x) \left (\sqrt{c} f-i \sqrt{a} g\right )}}} \]

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[ArcSin[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*Sqrt[a] + Sqrt[c]*x))/((Sqrt[c]*f - I*Sqr
t[a]*g)*(d + e*x))]*Sqrt[f + g*x]*Sqrt[a + c*x^2])

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Maple [A]  time = 0.221, size = 433, normalized size = 1. \[ 2\,{\frac{ \left ( c{e}^{2}f{x}^{2}-\sqrt{-ac}{x}^{2}{e}^{2}g+2\,xcdef-2\,\sqrt{-ac}xdeg+c{d}^{2}f-\sqrt{-ac}{d}^{2}g \right ) \sqrt{ex+d}\sqrt{gx+f}\sqrt{c{x}^{2}+a}}{ \left ( cd-\sqrt{-ac}e \right ) \left ( dg-ef \right ) \sqrt{ceg{x}^{4}+cdg{x}^{3}+cef{x}^{3}+aeg{x}^{2}+cdf{x}^{2}+adgx+aefx+adf}}{\it EllipticF} \left ( \sqrt{{\frac{ \left ( \sqrt{-ac}e-cd \right ) \left ( gx+f \right ) }{ \left ( g\sqrt{-ac}-cf \right ) \left ( ex+d \right ) }}},\sqrt{{\frac{ \left ( \sqrt{-ac}e+cd \right ) \left ( g\sqrt{-ac}-cf \right ) }{ \left ( g\sqrt{-ac}+cf \right ) \left ( \sqrt{-ac}e-cd \right ) }}} \right ) \sqrt{{\frac{ \left ( dg-ef \right ) \left ( cx+\sqrt{-ac} \right ) }{ \left ( g\sqrt{-ac}-cf \right ) \left ( ex+d \right ) }}}\sqrt{{\frac{ \left ( dg-ef \right ) \left ( -cx+\sqrt{-ac} \right ) }{ \left ( g\sqrt{-ac}+cf \right ) \left ( ex+d \right ) }}}\sqrt{{\frac{ \left ( \sqrt{-ac}e-cd \right ) \left ( gx+f \right ) }{ \left ( g\sqrt{-ac}-cf \right ) \left ( ex+d \right ) }}}{\frac{1}{\sqrt{-{\frac{ \left ( ex+d \right ) \left ( gx+f \right ) \left ( -cx+\sqrt{-ac} \right ) \left ( cx+\sqrt{-ac} \right ) }{c}}}}}} \]

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)/(g*(-a*c)^(1/2)-c*f
)/(e*x+d))^(1/2),(((-a*c)^(1/2)*e+c*d)*(g*(-a*c)^(1/2)-c*f)/(g*(-a*c)^(1/2)+c*f)
/((-a*c)^(1/2)*e-c*d))^(1/2))*((d*g-e*f)*(c*x+(-a*c)^(1/2))/(g*(-a*c)^(1/2)-c*f)
/(e*x+d))^(1/2)*((d*g-e*f)*(-c*x+(-a*c)^(1/2))/(g*(-a*c)^(1/2)+c*f)/(e*x+d))^(1/
2)*(((-a*c)^(1/2)*e-c*d)*(g*x+f)/(g*(-a*c)^(1/2)-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)/(c*d-(-a*c)^(1/2)*e)/(d*g-e*f)/(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. \[ \int \frac{1}{\sqrt{c x^{2} + a} \sqrt{e x + d} \sqrt{g x + f}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(c*x^2 + a)*sqrt(e*x + d)*sqrt(g*x + f)),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. \[{\rm integral}\left (\frac{1}{\sqrt{c x^{2} + a} \sqrt{e x + d} \sqrt{g x + f}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

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/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{c x^{2} + a} \sqrt{e x + d} \sqrt{g x + f}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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