3.236 \(\int \frac{\sqrt{a x}}{\sqrt{d+e x} \sqrt{e+f x}} \, dx\)

Optimal. Leaf size=114 \[ \frac{2 \sqrt{a x} \sqrt{d f-e^2} \sqrt{\frac{e (e+f x)}{e^2-d f}} E\left (\sin ^{-1}\left (\frac{\sqrt{f} \sqrt{d+e x}}{\sqrt{d f-e^2}}\right )|1-\frac{e^2}{d f}\right )}{e \sqrt{f} \sqrt{-\frac{e x}{d}} \sqrt{e+f x}} \]

[Out]

(2*Sqrt[-e^2 + d*f]*Sqrt[a*x]*Sqrt[(e*(e + f*x))/(e^2 - d*f)]*EllipticE[ArcSin[(
Sqrt[f]*Sqrt[d + e*x])/Sqrt[-e^2 + d*f]], 1 - e^2/(d*f)])/(e*Sqrt[f]*Sqrt[-((e*x
)/d)]*Sqrt[e + f*x])

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Rubi [A]  time = 0.230828, antiderivative size = 114, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077 \[ \frac{2 \sqrt{a x} \sqrt{d f-e^2} \sqrt{\frac{e (e+f x)}{e^2-d f}} E\left (\sin ^{-1}\left (\frac{\sqrt{f} \sqrt{d+e x}}{\sqrt{d f-e^2}}\right )|1-\frac{e^2}{d f}\right )}{e \sqrt{f} \sqrt{-\frac{e x}{d}} \sqrt{e+f x}} \]

Antiderivative was successfully verified.

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

[Out]

(2*Sqrt[-e^2 + d*f]*Sqrt[a*x]*Sqrt[(e*(e + f*x))/(e^2 - d*f)]*EllipticE[ArcSin[(
Sqrt[f]*Sqrt[d + e*x])/Sqrt[-e^2 + d*f]], 1 - e^2/(d*f)])/(e*Sqrt[f]*Sqrt[-((e*x
)/d)]*Sqrt[e + f*x])

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Rubi in Sympy [A]  time = 27.888, size = 95, normalized size = 0.83 \[ \frac{2 \sqrt{a x} \sqrt{\frac{e \left (- e - f x\right )}{d f - e^{2}}} \sqrt{d f - e^{2}} E\left (\operatorname{asin}{\left (\frac{\sqrt{f} \sqrt{d + e x}}{\sqrt{d f - e^{2}}} \right )}\middle | 1 - \frac{e^{2}}{d f}\right )}{e \sqrt{f} \sqrt{- \frac{e x}{d}} \sqrt{e + f x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*sqrt(a*x)*sqrt(e*(-e - f*x)/(d*f - e**2))*sqrt(d*f - e**2)*elliptic_e(asin(sqr
t(f)*sqrt(d + e*x)/sqrt(d*f - e**2)), 1 - e**2/(d*f))/(e*sqrt(f)*sqrt(-e*x/d)*sq
rt(e + f*x))

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Mathematica [C]  time = 0.30274, size = 106, normalized size = 0.93 \[ -\frac{2 i e \sqrt{a x} \sqrt{\frac{f x}{e}+1} \left (E\left (i \sinh ^{-1}\left (\sqrt{\frac{e x}{d}}\right )|\frac{d f}{e^2}\right )-F\left (i \sinh ^{-1}\left (\sqrt{\frac{e x}{d}}\right )|\frac{d f}{e^2}\right )\right )}{f \sqrt{\frac{e x}{d+e x}} \sqrt{d+e x} \sqrt{e+f x}} \]

Antiderivative was successfully verified.

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

[Out]

((-2*I)*e*Sqrt[a*x]*Sqrt[1 + (f*x)/e]*(EllipticE[I*ArcSinh[Sqrt[(e*x)/d]], (d*f)
/e^2] - EllipticF[I*ArcSinh[Sqrt[(e*x)/d]], (d*f)/e^2]))/(f*Sqrt[(e*x)/(d + e*x)
]*Sqrt[d + e*x]*Sqrt[e + f*x])

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Maple [A]  time = 0.084, size = 191, normalized size = 1.7 \[ -2\,{\frac{d\sqrt{fx+e}\sqrt{ex+d}\sqrt{ax}}{{e}^{2}fx \left ( ef{x}^{2}+dfx+{e}^{2}x+de \right ) } \left ({e}^{2}{\it EllipticF} \left ( \sqrt{{\frac{ex+d}{d}}},\sqrt{{\frac{df}{df-{e}^{2}}}} \right ) +{\it EllipticE} \left ( \sqrt{{\frac{ex+d}{d}}},\sqrt{{\frac{df}{df-{e}^{2}}}} \right ) df-{\it EllipticE} \left ( \sqrt{{\frac{ex+d}{d}}},\sqrt{{\frac{df}{df-{e}^{2}}}} \right ){e}^{2} \right ) \sqrt{-{\frac{ex}{d}}}\sqrt{-{\frac{ \left ( fx+e \right ) e}{df-{e}^{2}}}}\sqrt{{\frac{ex+d}{d}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*(e^2*EllipticF(((e*x+d)/d)^(1/2),(d*f/(d*f-e^2))^(1/2))+EllipticE(((e*x+d)/d)
^(1/2),(d*f/(d*f-e^2))^(1/2))*d*f-EllipticE(((e*x+d)/d)^(1/2),(d*f/(d*f-e^2))^(1
/2))*e^2)*(-e*x/d)^(1/2)*(-(f*x+e)*e/(d*f-e^2))^(1/2)*((e*x+d)/d)^(1/2)*d*(f*x+e
)^(1/2)*(e*x+d)^(1/2)*(a*x)^(1/2)/f/e^2/x/(e*f*x^2+d*f*x+e^2*x+d*e)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{a x}}{\sqrt{e x + d} \sqrt{f x + e}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(sqrt(a*x)/(sqrt(e*x + d)*sqrt(f*x + e)), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{\sqrt{a x}}{\sqrt{e x + d} \sqrt{f x + e}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral(sqrt(a*x)/(sqrt(e*x + d)*sqrt(f*x + e)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(sqrt(a*x)/(sqrt(e*x + d)*sqrt(f*x + e)), x)