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

Optimal. Leaf size=319 \[ -\frac{2 \sqrt{\frac{c x^2}{a}+1} (e f-d g) \sqrt{\frac{\sqrt{c} (f+g x)}{\sqrt{-a} g+\sqrt{c} f}} \Pi \left (\frac{2 e}{\frac{\sqrt{c} d}{\sqrt{-a}}+e};\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|\frac{2 \sqrt{-a} g}{\sqrt{c} f+\sqrt{-a} g}\right )}{e \sqrt{a+c x^2} \sqrt{f+g x} \left (\frac{\sqrt{c} d}{\sqrt{-a}}+e\right )}-\frac{2 \sqrt{-a} g \sqrt{\frac{c x^2}{a}+1} \sqrt{\frac{\sqrt{c} (f+g x)}{\sqrt{-a} g+\sqrt{c} f}} F\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a g}{\sqrt{-a} \sqrt{c} f-a g}\right )}{\sqrt{c} e \sqrt{a+c x^2} \sqrt{f+g x}} \]

[Out]

(-2*Sqrt[-a]*g*Sqrt[(Sqrt[c]*(f + g*x))/(Sqrt[c]*f + Sqrt[-a]*g)]*Sqrt[1 + (c*x^
2)/a]*EllipticF[ArcSin[Sqrt[1 - (Sqrt[c]*x)/Sqrt[-a]]/Sqrt[2]], (-2*a*g)/(Sqrt[-
a]*Sqrt[c]*f - a*g)])/(Sqrt[c]*e*Sqrt[f + g*x]*Sqrt[a + c*x^2]) - (2*(e*f - d*g)
*Sqrt[(Sqrt[c]*(f + g*x))/(Sqrt[c]*f + Sqrt[-a]*g)]*Sqrt[1 + (c*x^2)/a]*Elliptic
Pi[(2*e)/((Sqrt[c]*d)/Sqrt[-a] + e), ArcSin[Sqrt[1 - (Sqrt[c]*x)/Sqrt[-a]]/Sqrt[
2]], (2*Sqrt[-a]*g)/(Sqrt[c]*f + Sqrt[-a]*g)])/(e*((Sqrt[c]*d)/Sqrt[-a] + e)*Sqr
t[f + g*x]*Sqrt[a + c*x^2])

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Rubi [A]  time = 1.80919, antiderivative size = 319, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ -\frac{2 \sqrt{\frac{c x^2}{a}+1} (e f-d g) \sqrt{\frac{\sqrt{c} (f+g x)}{\sqrt{-a} g+\sqrt{c} f}} \Pi \left (\frac{2 e}{\frac{\sqrt{c} d}{\sqrt{-a}}+e};\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|\frac{2 \sqrt{-a} g}{\sqrt{c} f+\sqrt{-a} g}\right )}{e \sqrt{a+c x^2} \sqrt{f+g x} \left (\frac{\sqrt{c} d}{\sqrt{-a}}+e\right )}-\frac{2 \sqrt{-a} g \sqrt{\frac{c x^2}{a}+1} \sqrt{\frac{\sqrt{c} (f+g x)}{\sqrt{-a} g+\sqrt{c} f}} F\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a g}{\sqrt{-a} \sqrt{c} f-a g}\right )}{\sqrt{c} e \sqrt{a+c x^2} \sqrt{f+g x}} \]

Antiderivative was successfully verified.

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

[Out]

(-2*Sqrt[-a]*g*Sqrt[(Sqrt[c]*(f + g*x))/(Sqrt[c]*f + Sqrt[-a]*g)]*Sqrt[1 + (c*x^
2)/a]*EllipticF[ArcSin[Sqrt[1 - (Sqrt[c]*x)/Sqrt[-a]]/Sqrt[2]], (-2*a*g)/(Sqrt[-
a]*Sqrt[c]*f - a*g)])/(Sqrt[c]*e*Sqrt[f + g*x]*Sqrt[a + c*x^2]) - (2*(e*f - d*g)
*Sqrt[(Sqrt[c]*(f + g*x))/(Sqrt[c]*f + Sqrt[-a]*g)]*Sqrt[1 + (c*x^2)/a]*Elliptic
Pi[(2*e)/((Sqrt[c]*d)/Sqrt[-a] + e), ArcSin[Sqrt[1 - (Sqrt[c]*x)/Sqrt[-a]]/Sqrt[
2]], (2*Sqrt[-a]*g)/(Sqrt[c]*f + Sqrt[-a]*g)])/(e*((Sqrt[c]*d)/Sqrt[-a] + e)*Sqr
t[f + g*x]*Sqrt[a + c*x^2])

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Rubi in Sympy [A]  time = 95.722, size = 330, normalized size = 1.03 \[ - \frac{2 \sqrt{\frac{g \left (- \sqrt{c} x - \sqrt{- a}\right )}{\sqrt{c} f - g \sqrt{- a}}} \sqrt{\frac{g \left (- \sqrt{c} x + \sqrt{- a}\right )}{\sqrt{c} f + g \sqrt{- a}}} \Pi \left (- \frac{e \left (\sqrt{c} f + g \sqrt{- a}\right )}{\sqrt{c} \left (d g - e f\right )}; \operatorname{asin}{\left (\sqrt{\frac{c}{\sqrt{c} g \sqrt{- a} + c f}} \sqrt{f + g x} \right )}\middle | \frac{\sqrt{c} f + g \sqrt{- a}}{\sqrt{c} f - g \sqrt{- a}}\right )}{e \sqrt{\frac{c}{\sqrt{c} g \sqrt{- a} + c f}} \sqrt{a + c x^{2}}} - \frac{2 g \sqrt{- a} \sqrt{\frac{\sqrt{c} \sqrt{- a} \left (- f - g x\right )}{a g - \sqrt{c} f \sqrt{- a}}} \sqrt{1 + \frac{c x^{2}}{a}} F\left (\operatorname{asin}{\left (\sqrt{- \frac{\sqrt{c} x}{2 \sqrt{- a}} + \frac{1}{2}} \right )}\middle | \frac{2 a g}{a g - \sqrt{c} f \sqrt{- a}}\right )}{\sqrt{c} e \sqrt{a + c x^{2}} \sqrt{f + g x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((g*x+f)**(1/2)/(e*x+d)/(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))/(sq
rt(c)*(d*g - e*f)), asin(sqrt(c/(sqrt(c)*g*sqrt(-a) + c*f))*sqrt(f + g*x)), (sqr
t(c)*f + g*sqrt(-a))/(sqrt(c)*f - g*sqrt(-a)))/(e*sqrt(c/(sqrt(c)*g*sqrt(-a) + c
*f))*sqrt(a + c*x**2)) - 2*g*sqrt(-a)*sqrt(sqrt(c)*sqrt(-a)*(-f - g*x)/(a*g - sq
rt(c)*f*sqrt(-a)))*sqrt(1 + c*x**2/a)*elliptic_f(asin(sqrt(-sqrt(c)*x/(2*sqrt(-a
)) + 1/2)), 2*a*g/(a*g - sqrt(c)*f*sqrt(-a)))/(sqrt(c)*e*sqrt(a + c*x**2)*sqrt(f
+ g*x))

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Mathematica [C]  time = 0.90318, size = 300, normalized size = 0.94 \[ -\frac{2 i \sqrt{f+g x} \sqrt{\frac{g \left (\sqrt{a}+i \sqrt{c} x\right )}{\sqrt{a} g-i \sqrt{c} f}} \left (F\left (i \sinh ^{-1}\left (\sqrt{-\frac{\sqrt{c} (f+g x)}{\sqrt{c} f-i \sqrt{a} g}}\right )|\frac{\sqrt{c} f-i \sqrt{a} g}{\sqrt{c} f+i \sqrt{a} g}\right )-\Pi \left (\frac{e \left (f-\frac{i \sqrt{a} g}{\sqrt{c}}\right )}{e f-d g};i \sinh ^{-1}\left (\sqrt{-\frac{\sqrt{c} (f+g x)}{\sqrt{c} f-i \sqrt{a} g}}\right )|\frac{\sqrt{c} f-i \sqrt{a} g}{\sqrt{c} f+i \sqrt{a} g}\right )\right )}{e \sqrt{a+c x^2} \sqrt{\frac{\sqrt{c} (f+g x)}{g \left (\sqrt{c} x+i \sqrt{a}\right )}}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.051, size = 439, normalized size = 1.4 \[ 2\,{\frac{\sqrt{gx+f}\sqrt{c{x}^{2}+a}}{ce \left ( cg{x}^{3}+cf{x}^{2}+agx+fa \right ) }\sqrt{-{\frac{ \left ( gx+f \right ) c}{g\sqrt{-ac}-cf}}}\sqrt{{\frac{ \left ( -cx+\sqrt{-ac} \right ) g}{g\sqrt{-ac}+cf}}}\sqrt{{\frac{ \left ( cx+\sqrt{-ac} \right ) g}{g\sqrt{-ac}-cf}}} \left ( f{\it EllipticF} \left ( \sqrt{-{\frac{ \left ( gx+f \right ) c}{g\sqrt{-ac}-cf}}},\sqrt{-{\frac{g\sqrt{-ac}-cf}{g\sqrt{-ac}+cf}}} \right ) c-\sqrt{-ac}{\it EllipticF} \left ( \sqrt{-{\frac{ \left ( gx+f \right ) c}{g\sqrt{-ac}-cf}}},\sqrt{-{\frac{g\sqrt{-ac}-cf}{g\sqrt{-ac}+cf}}} \right ) g-{\it EllipticPi} \left ( \sqrt{-{\frac{ \left ( gx+f \right ) c}{g\sqrt{-ac}-cf}}},{\frac{ \left ( g\sqrt{-ac}-cf \right ) e}{c \left ( dg-ef \right ) }},\sqrt{-{\frac{g\sqrt{-ac}-cf}{g\sqrt{-ac}+cf}}} \right ) cf+{\it EllipticPi} \left ( \sqrt{-{\frac{ \left ( gx+f \right ) c}{g\sqrt{-ac}-cf}}},{\frac{ \left ( g\sqrt{-ac}-cf \right ) e}{c \left ( dg-ef \right ) }},\sqrt{-{\frac{g\sqrt{-ac}-cf}{g\sqrt{-ac}+cf}}} \right ) \sqrt{-ac}g \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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