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

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 = 0.0561957, 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, Rules used = {114, 113} \[ \frac{2 \sqrt{a x} \sqrt{d f-e^2} \sqrt{\frac{e (e+f x)}{e^2-d f}} E\left (\sin ^{-1}\left (\frac{\sqrt{f} \sqrt{d+e x}}{\sqrt{d f-e^2}}\right )|1-\frac{e^2}{d f}\right )}{e \sqrt{f} \sqrt{-\frac{e x}{d}} \sqrt{e+f x}} \]

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])

Rule 114

Int[Sqrt[(e_.) + (f_.)*(x_)]/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Dist[(Sqrt[e + f*
x]*Sqrt[(b*(c + d*x))/(b*c - a*d)])/(Sqrt[c + d*x]*Sqrt[(b*(e + f*x))/(b*e - a*f)]), Int[Sqrt[(b*e)/(b*e - a*f
) + (b*f*x)/(b*e - a*f)]/(Sqrt[a + b*x]*Sqrt[(b*c)/(b*c - a*d) + (b*d*x)/(b*c - a*d)]), x], x] /; FreeQ[{a, b,
c, d, e, f}, x] &&  !(GtQ[b/(b*c - a*d), 0] && GtQ[b/(b*e - a*f), 0]) &&  !LtQ[-((b*c - a*d)/d), 0]

Rule 113

Int[Sqrt[(e_.) + (f_.)*(x_)]/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Simp[(2*Rt[-((b*e
- a*f)/d), 2]*EllipticE[ArcSin[Sqrt[a + b*x]/Rt[-((b*c - a*d)/d), 2]], (f*(b*c - a*d))/(d*(b*e - a*f))])/b, x
] /; FreeQ[{a, b, c, d, e, f}, x] && GtQ[b/(b*c - a*d), 0] && GtQ[b/(b*e - a*f), 0] &&  !LtQ[-((b*c - a*d)/d),
0] &&  !(SimplerQ[c + d*x, a + b*x] && GtQ[-(d/(b*c - a*d)), 0] && GtQ[d/(d*e - c*f), 0] &&  !LtQ[(b*c - a*d)
/b, 0])

Rubi steps

\begin{align*} \int \frac{\sqrt{a x}}{\sqrt{d+e x} \sqrt{e+f x}} \, dx &=\frac{\left (\sqrt{a x} \sqrt{\frac{e (e+f x)}{e^2-d f}}\right ) \int \frac{\sqrt{-\frac{e x}{d}}}{\sqrt{d+e x} \sqrt{\frac{e^2}{e^2-d f}+\frac{e f x}{e^2-d f}}} \, dx}{\sqrt{-\frac{e x}{d}} \sqrt{e+f x}}\\ &=\frac{2 \sqrt{-e^2+d f} \sqrt{a x} \sqrt{\frac{e (e+f x)}{e^2-d f}} E\left (\sin ^{-1}\left (\frac{\sqrt{f} \sqrt{d+e x}}{\sqrt{-e^2+d f}}\right )|1-\frac{e^2}{d f}\right )}{e \sqrt{f} \sqrt{-\frac{e x}{d}} \sqrt{e+f x}}\\ \end{align*}

Mathematica [C]  time = 0.204797, 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[Sq
rt[(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 = 0.055, size = 191, normalized size = 1.7 \begin{align*} -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}}}} \end{align*}

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. \begin{align*} \int \frac{\sqrt{a x}}{\sqrt{e x + d} \sqrt{f x + e}}\,{d x} \end{align*}

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, 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. \begin{align*}{\rm integral}\left (\frac{\sqrt{a x} \sqrt{e x + d} \sqrt{f x + e}}{e f x^{2} + d e +{\left (e^{2} + d f\right )} x}, x\right ) \end{align*}

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, algorithm="fricas")

[Out]

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

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

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

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, algorithm="giac")

[Out]

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