3.2642 \(\int \frac{\sqrt{a+b x}}{\sqrt{c+\frac{b (-1+c) x}{a}} \sqrt{e+\frac{b (-1+e) x}{a}}} \, dx\)

Optimal. Leaf size=162 \[ -\frac{2 a \sqrt{c-e} \sqrt{a+b x} \sqrt{-\frac{(1-c) (a e-b (1-e) x)}{a (c-e)}} E\left (\sin ^{-1}\left (\frac{\sqrt{1-e} \sqrt{c-\frac{b (1-c) x}{a}}}{\sqrt{c-e}}\right )|\frac{c-e}{1-e}\right )}{b (1-c) \sqrt{1-e} \sqrt{\frac{(1-c) (a+b x)}{a}} \sqrt{e-\frac{b (1-e) x}{a}}} \]

[Out]

(-2*a*Sqrt[c - e]*Sqrt[a + b*x]*Sqrt[-(((1 - c)*(a*e - b*(1 - e)*x))/(a*(c - e)))]*EllipticE[ArcSin[(Sqrt[1 -
e]*Sqrt[c - (b*(1 - c)*x)/a])/Sqrt[c - e]], (c - e)/(1 - e)])/(b*(1 - c)*Sqrt[1 - e]*Sqrt[((1 - c)*(a + b*x))/
a]*Sqrt[e - (b*(1 - e)*x)/a])

________________________________________________________________________________________

Rubi [A]  time = 0.183466, antiderivative size = 162, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 40, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.05, Rules used = {114, 113} \[ -\frac{2 a \sqrt{c-e} \sqrt{a+b x} \sqrt{-\frac{(1-c) (a e-b (1-e) x)}{a (c-e)}} E\left (\sin ^{-1}\left (\frac{\sqrt{1-e} \sqrt{c-\frac{b (1-c) x}{a}}}{\sqrt{c-e}}\right )|\frac{c-e}{1-e}\right )}{b (1-c) \sqrt{1-e} \sqrt{\frac{(1-c) (a+b x)}{a}} \sqrt{e-\frac{b (1-e) x}{a}}} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[a + b*x]/(Sqrt[c + (b*(-1 + c)*x)/a]*Sqrt[e + (b*(-1 + e)*x)/a]),x]

[Out]

(-2*a*Sqrt[c - e]*Sqrt[a + b*x]*Sqrt[-(((1 - c)*(a*e - b*(1 - e)*x))/(a*(c - e)))]*EllipticE[ArcSin[(Sqrt[1 -
e]*Sqrt[c - (b*(1 - c)*x)/a])/Sqrt[c - e]], (c - e)/(1 - e)])/(b*(1 - c)*Sqrt[1 - e]*Sqrt[((1 - c)*(a + b*x))/
a]*Sqrt[e - (b*(1 - e)*x)/a])

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

Mathematica [C]  time = 0.202524, size = 103, normalized size = 0.64 \[ -\frac{2 i a \sqrt{a+b x} \left (E\left (i \sinh ^{-1}\left (\sqrt{\frac{(c-1) (a+b x)}{a}}\right )|\frac{e-1}{c-1}\right )-\text{EllipticF}\left (i \sinh ^{-1}\left (\sqrt{\frac{(c-1) (a+b x)}{a}}\right ),\frac{e-1}{c-1}\right )\right )}{b (e-1) \sqrt{\frac{(c-1) (a+b x)}{a}}} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[a + b*x]/(Sqrt[c + (b*(-1 + c)*x)/a]*Sqrt[e + (b*(-1 + e)*x)/a]),x]

[Out]

((-2*I)*a*Sqrt[a + b*x]*(EllipticE[I*ArcSinh[Sqrt[((-1 + c)*(a + b*x))/a]], (-1 + e)/(-1 + c)] - EllipticF[I*A
rcSinh[Sqrt[((-1 + c)*(a + b*x))/a]], (-1 + e)/(-1 + c)]))/(b*(-1 + e)*Sqrt[((-1 + c)*(a + b*x))/a])

________________________________________________________________________________________

Maple [A]  time = 0.028, size = 183, normalized size = 1.1 \begin{align*} 2\,{\frac{ \left ( c-e \right ){a}^{2}}{\sqrt{bx+a} \left ( c-1 \right ) ^{2}b \left ( -1+e \right ) }{\it EllipticE} \left ( \sqrt{-{\frac{ \left ( -1+e \right ) \left ( bcx+ac-bx \right ) }{a \left ( c-e \right ) }}},\sqrt{-{\frac{c-e}{-1+e}}} \right ) \sqrt{{\frac{ \left ( c-1 \right ) \left ( bxe+ae-bx \right ) }{a \left ( c-e \right ) }}}\sqrt{-{\frac{ \left ( bx+a \right ) \left ( c-1 \right ) }{a}}}\sqrt{-{\frac{ \left ( -1+e \right ) \left ( bcx+ac-bx \right ) }{a \left ( c-e \right ) }}}{\frac{1}{\sqrt{{\frac{bxe+ae-bx}{a}}}}}{\frac{1}{\sqrt{{\frac{bcx+ac-bx}{a}}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^(1/2)/(c+b*(c-1)*x/a)^(1/2)/(e+b*(-1+e)*x/a)^(1/2),x)

[Out]

2*a^2*(c-e)*EllipticE((-(-1+e)*(b*c*x+a*c-b*x)/a/(c-e))^(1/2),(-(c-e)/(-1+e))^(1/2))*((c-1)*(b*e*x+a*e-b*x)/a/
(c-e))^(1/2)*(-(b*x+a)*(c-1)/a)^(1/2)*(-(-1+e)*(b*c*x+a*c-b*x)/a/(c-e))^(1/2)/(b*x+a)^(1/2)/((b*e*x+a*e-b*x)/a
)^(1/2)/((b*c*x+a*c-b*x)/a)^(1/2)/(c-1)^2/b/(-1+e)

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{b x + a}}{\sqrt{\frac{b{\left (c - 1\right )} x}{a} + c} \sqrt{\frac{b{\left (e - 1\right )} x}{a} + e}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^(1/2)/(c+b*(-1+c)*x/a)^(1/2)/(e+b*(-1+e)*x/a)^(1/2),x, algorithm="maxima")

[Out]

integrate(sqrt(b*x + a)/(sqrt(b*(c - 1)*x/a + c)*sqrt(b*(e - 1)*x/a + e)), x)

________________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{b x + a} a^{2} \sqrt{\frac{a c +{\left (b c - b\right )} x}{a}} \sqrt{\frac{a e +{\left (b e - b\right )} x}{a}}}{a^{2} c e -{\left (b^{2} c - b^{2} -{\left (b^{2} c - b^{2}\right )} e\right )} x^{2} -{\left (a b c -{\left (2 \, a b c - a b\right )} e\right )} x}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^(1/2)/(c+b*(-1+c)*x/a)^(1/2)/(e+b*(-1+e)*x/a)^(1/2),x, algorithm="fricas")

[Out]

integral(sqrt(b*x + a)*a^2*sqrt((a*c + (b*c - b)*x)/a)*sqrt((a*e + (b*e - b)*x)/a)/(a^2*c*e - (b^2*c - b^2 - (
b^2*c - b^2)*e)*x^2 - (a*b*c - (2*a*b*c - a*b)*e)*x), x)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**(1/2)/(c+b*(-1+c)*x/a)**(1/2)/(e+b*(-1+e)*x/a)**(1/2),x)

[Out]

Integral(sqrt(a + b*x)/(sqrt(c + b*c*x/a - b*x/a)*sqrt(e + b*e*x/a - b*x/a)), x)

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{b x + a}}{\sqrt{\frac{b{\left (c - 1\right )} x}{a} + c} \sqrt{\frac{b{\left (e - 1\right )} x}{a} + e}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^(1/2)/(c+b*(-1+c)*x/a)^(1/2)/(e+b*(-1+e)*x/a)^(1/2),x, algorithm="giac")

[Out]

integrate(sqrt(b*x + a)/(sqrt(b*(c - 1)*x/a + c)*sqrt(b*(e - 1)*x/a + e)), x)