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

Optimal. Leaf size=167 \[ -\frac{2 \sqrt{\frac{c x^2}{a}+1} \sqrt{\frac{\sqrt{c} (f+g x)}{\sqrt{-a} g+\sqrt{c} f}} \Pi \left (\frac{2 e}{\frac{\sqrt{c} d}{\sqrt{-a}}+e};\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|\frac{2 \sqrt{-a} g}{\sqrt{c} f+\sqrt{-a} g}\right )}{\sqrt{a+c x^2} \sqrt{f+g x} \left (\frac{\sqrt{c} d}{\sqrt{-a}}+e\right )} \]

[Out]

(-2*Sqrt[(Sqrt[c]*(f + g*x))/(Sqrt[c]*f + Sqrt[-a]*g)]*Sqrt[1 + (c*x^2)/a]*Ellip
ticPi[(2*e)/((Sqrt[c]*d)/Sqrt[-a] + e), ArcSin[Sqrt[1 - (Sqrt[c]*x)/Sqrt[-a]]/Sq
rt[2]], (2*Sqrt[-a]*g)/(Sqrt[c]*f + Sqrt[-a]*g)])/(((Sqrt[c]*d)/Sqrt[-a] + e)*Sq
rt[f + g*x]*Sqrt[a + c*x^2])

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Rubi [A]  time = 1.41636, antiderivative size = 167, 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 \[ -\frac{2 \sqrt{\frac{c x^2}{a}+1} \sqrt{\frac{\sqrt{c} (f+g x)}{\sqrt{-a} g+\sqrt{c} f}} \Pi \left (\frac{2 e}{\frac{\sqrt{c} d}{\sqrt{-a}}+e};\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|\frac{2 \sqrt{-a} g}{\sqrt{c} f+\sqrt{-a} g}\right )}{\sqrt{a+c x^2} \sqrt{f+g x} \left (\frac{\sqrt{c} d}{\sqrt{-a}}+e\right )} \]

Antiderivative was successfully verified.

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

[Out]

(-2*Sqrt[(Sqrt[c]*(f + g*x))/(Sqrt[c]*f + Sqrt[-a]*g)]*Sqrt[1 + (c*x^2)/a]*Ellip
ticPi[(2*e)/((Sqrt[c]*d)/Sqrt[-a] + e), ArcSin[Sqrt[1 - (Sqrt[c]*x)/Sqrt[-a]]/Sq
rt[2]], (2*Sqrt[-a]*g)/(Sqrt[c]*f + Sqrt[-a]*g)])/(((Sqrt[c]*d)/Sqrt[-a] + e)*Sq
rt[f + g*x]*Sqrt[a + c*x^2])

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Rubi in Sympy [A]  time = 18.5421, size = 201, normalized size = 1.2 \[ \frac{2 \sqrt{\frac{g \left (- \sqrt{c} x - \sqrt{- a}\right )}{\sqrt{c} f - g \sqrt{- a}}} \sqrt{\frac{g \left (- \sqrt{c} x + \sqrt{- a}\right )}{\sqrt{c} f + g \sqrt{- a}}} \Pi \left (- \frac{e \left (\sqrt{c} f + g \sqrt{- a}\right )}{\sqrt{c} \left (d g - e f\right )}; \operatorname{asin}{\left (\sqrt{\frac{c}{\sqrt{c} g \sqrt{- a} + c f}} \sqrt{f + g x} \right )}\middle | \frac{\sqrt{c} f + g \sqrt{- a}}{\sqrt{c} f - g \sqrt{- a}}\right )}{\sqrt{\frac{c}{\sqrt{c} g \sqrt{- a} + c f}} \sqrt{a + c x^{2}} \left (d g - e f\right )} \]

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+a)**(1/2),x)

[Out]

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

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Mathematica [C]  time = 0.903257, size = 311, normalized size = 1.86 \[ -\frac{2 i (f+g x) \sqrt{\frac{g \left (x+\frac{i \sqrt{a}}{\sqrt{c}}\right )}{f+g x}} \sqrt{-\frac{-g x+\frac{i \sqrt{a} g}{\sqrt{c}}}{f+g x}} \left (F\left (i \sinh ^{-1}\left (\frac{\sqrt{-f-\frac{i \sqrt{a} g}{\sqrt{c}}}}{\sqrt{f+g x}}\right )|\frac{\sqrt{c} f-i \sqrt{a} g}{\sqrt{c} f+i \sqrt{a} g}\right )-\Pi \left (\frac{\sqrt{c} (e f-d g)}{e \left (\sqrt{c} f+i \sqrt{a} g\right )};i \sinh ^{-1}\left (\frac{\sqrt{-f-\frac{i \sqrt{a} g}{\sqrt{c}}}}{\sqrt{f+g x}}\right )|\frac{\sqrt{c} f-i \sqrt{a} g}{\sqrt{c} f+i \sqrt{a} g}\right )\right )}{\sqrt{a+c x^2} \sqrt{-f-\frac{i \sqrt{a} g}{\sqrt{c}}} (e f-d g)} \]

Antiderivative was successfully verified.

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

[Out]

((-2*I)*Sqrt[(g*((I*Sqrt[a])/Sqrt[c] + x))/(f + g*x)]*Sqrt[-(((I*Sqrt[a]*g)/Sqrt
[c] - g*x)/(f + g*x))]*(f + g*x)*(EllipticF[I*ArcSinh[Sqrt[-f - (I*Sqrt[a]*g)/Sq
rt[c]]/Sqrt[f + g*x]], (Sqrt[c]*f - I*Sqrt[a]*g)/(Sqrt[c]*f + I*Sqrt[a]*g)] - El
lipticPi[(Sqrt[c]*(e*f - d*g))/(e*(Sqrt[c]*f + I*Sqrt[a]*g)), I*ArcSinh[Sqrt[-f
- (I*Sqrt[a]*g)/Sqrt[c]]/Sqrt[f + g*x]], (Sqrt[c]*f - I*Sqrt[a]*g)/(Sqrt[c]*f +
I*Sqrt[a]*g)]))/(Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]]*(e*f - d*g)*Sqrt[a + c*x^2])

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Maple [A]  time = 0.052, size = 235, normalized size = 1.4 \[ 2\,{\frac{ \left ( cf-g\sqrt{-ac} \right ) \sqrt{c{x}^{2}+a}\sqrt{gx+f}}{c \left ( dg-ef \right ) \left ( cg{x}^{3}+cf{x}^{2}+agx+fa \right ) }{\it EllipticPi} \left ( \sqrt{-{\frac{ \left ( gx+f \right ) c}{g\sqrt{-ac}-cf}}},{\frac{ \left ( g\sqrt{-ac}-cf \right ) e}{c \left ( dg-ef \right ) }},\sqrt{-{\frac{g\sqrt{-ac}-cf}{g\sqrt{-ac}+cf}}} \right ) \sqrt{{\frac{ \left ( cx+\sqrt{-ac} \right ) g}{g\sqrt{-ac}-cf}}}\sqrt{{\frac{ \left ( -cx+\sqrt{-ac} \right ) g}{g\sqrt{-ac}+cf}}}\sqrt{-{\frac{ \left ( gx+f \right ) c}{g\sqrt{-ac}-cf}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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