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 = 1.02136, 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 \[ -\frac{2 \sqrt{\frac{\sqrt{-c} (f+g x)}{\sqrt{-c} f+g}} \Pi \left (\frac{2 e}{\sqrt{-c} d+e};\sin ^{-1}\left (\frac{\sqrt{1-\sqrt{-c} x}}{\sqrt{2}}\right )|\frac{2 g}{\sqrt{-c} f+g}\right )}{\left (\sqrt{-c} d+e\right ) \sqrt{f+g x}} \]

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])

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Rubi in Sympy [A]  time = 14.2788, size = 162, normalized size = 1.47 \[ \frac{2 \sqrt{\frac{g \left (- c x - \sqrt{- c}\right )}{c f - g \sqrt{- c}}} \sqrt{\frac{g \left (- c x + \sqrt{- c}\right )}{c f + g \sqrt{- c}}} \Pi \left (\frac{e \left (c f + g \sqrt{- c}\right )}{c \left (- d g + e f\right )}; \operatorname{asin}{\left (\sqrt{\frac{c}{c f + g \sqrt{- c}}} \sqrt{f + g x} \right )}\middle | \frac{c f + g \sqrt{- c}}{c f - g \sqrt{- c}}\right )}{\sqrt{\frac{c}{c f + g \sqrt{- c}}} \sqrt{c x^{2} + 1} \left (d g - e f\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*sqrt(g*(-c*x - sqrt(-c))/(c*f - g*sqrt(-c)))*sqrt(g*(-c*x + sqrt(-c))/(c*f + g
*sqrt(-c)))*elliptic_pi(e*(c*f + g*sqrt(-c))/(c*(-d*g + e*f)), asin(sqrt(c/(c*f
+ g*sqrt(-c)))*sqrt(f + g*x)), (c*f + g*sqrt(-c))/(c*f - g*sqrt(-c)))/(sqrt(c/(c
*f + g*sqrt(-c)))*sqrt(c*x**2 + 1)*(d*g - e*f))

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Mathematica [C]  time = 0.804204, size = 261, normalized size = 2.37 \[ -\frac{2 i (f+g x) \sqrt{\frac{g \left (x+\frac{i}{\sqrt{c}}\right )}{f+g x}} \sqrt{-\frac{-g x+\frac{i g}{\sqrt{c}}}{f+g x}} \left (F\left (i \sinh ^{-1}\left (\frac{\sqrt{-f-\frac{i g}{\sqrt{c}}}}{\sqrt{f+g x}}\right )|\frac{\sqrt{c} f-i g}{\sqrt{c} f+i g}\right )-\Pi \left (\frac{\sqrt{c} (e f-d g)}{e \left (\sqrt{c} f+i g\right )};i \sinh ^{-1}\left (\frac{\sqrt{-f-\frac{i g}{\sqrt{c}}}}{\sqrt{f+g x}}\right )|\frac{\sqrt{c} f-i g}{\sqrt{c} f+i g}\right )\right )}{\sqrt{c x^2+1} \sqrt{-f-\frac{i g}{\sqrt{c}}} (e f-d g)} \]

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*ArcSinh[Sqrt[-f - (I*g)/Sqrt[c]]/Sqrt[f + g*x]], (Sq
rt[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 = 0.128, size = 215, normalized size = 2. \[ 2\,{\frac{ \left ( g+f\sqrt{-c} \right ) \sqrt{c{x}^{2}+1}\sqrt{gx+f}}{\sqrt{-c} \left ( dg-ef \right ) \left ( cg{x}^{3}+cf{x}^{2}+gx+f \right ) }{\it EllipticPi} \left ( \sqrt{{\frac{ \left ( gx+f \right ) \sqrt{-c}}{g+f\sqrt{-c}}}},-{\frac{ \left ( g+f\sqrt{-c} \right ) e}{\sqrt{-c} \left ( dg-ef \right ) }},\sqrt{{\frac{g+f\sqrt{-c}}{f\sqrt{-c}-g}}} \right ) \sqrt{-{\frac{ \left ( -1+x\sqrt{-c} \right ) g}{g+f\sqrt{-c}}}}\sqrt{-{\frac{ \left ( x\sqrt{-c}+1 \right ) g}{f\sqrt{-c}-g}}}\sqrt{{\frac{ \left ( gx+f \right ) \sqrt{-c}}{g+f\sqrt{-c}}}}} \]

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. \[ \int \frac{1}{\sqrt{c x^{2} + 1}{\left (e x + d\right )} \sqrt{g x + f}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(c*x^2 + 1)*(e*x + d)*sqrt(g*x + f)),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. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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