3.2641 \(\int \frac{\sqrt{c+d x}}{\sqrt{a+b x} \sqrt{e+\frac{b (-1+e) x}{a}}} \, dx\)

Optimal. Leaf size=96 \[ \frac{2 \sqrt{a} \sqrt{c+d x} E\left (\sin ^{-1}\left (\frac{\sqrt{1-e} \sqrt{a+b x}}{\sqrt{a}}\right )|-\frac{a d}{(b c-a d) (1-e)}\right )}{b \sqrt{1-e} \sqrt{\frac{b (c+d x)}{b c-a d}}} \]

[Out]

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

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Rubi [A]  time = 0.0569543, antiderivative size = 96, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 34, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.059, Rules used = {114, 113} \[ \frac{2 \sqrt{a} \sqrt{c+d x} E\left (\sin ^{-1}\left (\frac{\sqrt{1-e} \sqrt{a+b x}}{\sqrt{a}}\right )|-\frac{a d}{(b c-a d) (1-e)}\right )}{b \sqrt{1-e} \sqrt{\frac{b (c+d x)}{b c-a d}}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

Mathematica [B]  time = 1.12612, size = 200, normalized size = 2.08 \[ \frac{2 \sqrt{\frac{\frac{a}{a+b x}+e-1}{e-1}} \left (b \sqrt{a+b x} (c+d x) \sqrt{a-\frac{b c}{d}} \sqrt{\frac{a e+b (e-1) x}{(e-1) (a+b x)}}-(a+b x) (b c-a d) \sqrt{\frac{b (c+d x)}{d (a+b x)}} E\left (\sin ^{-1}\left (\frac{\sqrt{a-\frac{b c}{d}}}{\sqrt{a+b x}}\right )|\frac{a d}{(b c-a d) (e-1)}\right )\right )}{b^2 \sqrt{c+d x} \sqrt{a-\frac{b c}{d}} \sqrt{\frac{b (e-1) x}{a}+e}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B]  time = 0.026, size = 822, normalized size = 8.6 \begin{align*} -2\,{\frac{\sqrt{bx+a}\sqrt{dx+c}}{ \left ( d{x}^{2}b+adx+bcx+ac \right ) \left ( -1+e \right ) ^{2}{b}^{2}d}\sqrt{{\frac{d \left ( bxe+ae-bx \right ) }{ade-bce+bc}}}\sqrt{-{\frac{ \left ( bx+a \right ) \left ( -1+e \right ) }{a}}}\sqrt{-{\frac{ \left ( dx+c \right ) b \left ( -1+e \right ) }{ade-bce+bc}}} \left ({\it EllipticF} \left ( \sqrt{{\frac{d \left ( bxe+ae-bx \right ) }{ade-bce+bc}}},\sqrt{{\frac{ade-bce+bc}{ad}}} \right ){a}^{2}{d}^{2}{e}^{2}-2\,{\it EllipticF} \left ( \sqrt{{\frac{d \left ( bxe+ae-bx \right ) }{ade-bce+bc}}},\sqrt{{\frac{ade-bce+bc}{ad}}} \right ) abcd{e}^{2}+{\it EllipticF} \left ( \sqrt{{\frac{d \left ( bxe+ae-bx \right ) }{ade-bce+bc}}},\sqrt{{\frac{ade-bce+bc}{ad}}} \right ){b}^{2}{c}^{2}{e}^{2}-{\it EllipticF} \left ( \sqrt{{\frac{d \left ( bxe+ae-bx \right ) }{ade-bce+bc}}},\sqrt{{\frac{ade-bce+bc}{ad}}} \right ){a}^{2}{d}^{2}e+3\,{\it EllipticF} \left ( \sqrt{{\frac{d \left ( bxe+ae-bx \right ) }{ade-bce+bc}}},\sqrt{{\frac{ade-bce+bc}{ad}}} \right ) abcde-2\,{\it EllipticF} \left ( \sqrt{{\frac{d \left ( bxe+ae-bx \right ) }{ade-bce+bc}}},\sqrt{{\frac{ade-bce+bc}{ad}}} \right ){b}^{2}{c}^{2}e+{\it EllipticE} \left ( \sqrt{{\frac{d \left ( bxe+ae-bx \right ) }{ade-bce+bc}}},\sqrt{{\frac{ade-bce+bc}{ad}}} \right ){a}^{2}{d}^{2}e-{\it EllipticE} \left ( \sqrt{{\frac{d \left ( bxe+ae-bx \right ) }{ade-bce+bc}}},\sqrt{{\frac{ade-bce+bc}{ad}}} \right ) abcde-{\it EllipticF} \left ( \sqrt{{\frac{d \left ( bxe+ae-bx \right ) }{ade-bce+bc}}},\sqrt{{\frac{ade-bce+bc}{ad}}} \right ) abcd+{\it EllipticF} \left ( \sqrt{{\frac{d \left ( bxe+ae-bx \right ) }{ade-bce+bc}}},\sqrt{{\frac{ade-bce+bc}{ad}}} \right ){b}^{2}{c}^{2}+{\it EllipticE} \left ( \sqrt{{\frac{d \left ( bxe+ae-bx \right ) }{ade-bce+bc}}},\sqrt{{\frac{ade-bce+bc}{ad}}} \right ) abcd \right ){\frac{1}{\sqrt{{\frac{bxe+ae-bx}{a}}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*(d*x+c)^(1/2)*(b*x+a)^(1/2)*(d*(b*e*x+a*e-b*x)/(a*d*e-b*c*e+b*c))^(1/2)*(-(b*x+a)*(-1+e)/a)^(1/2)*(-(d*x+c)
*b*(-1+e)/(a*d*e-b*c*e+b*c))^(1/2)*(EllipticF((d*(b*e*x+a*e-b*x)/(a*d*e-b*c*e+b*c))^(1/2),((a*d*e-b*c*e+b*c)/d
/a)^(1/2))*a^2*d^2*e^2-2*EllipticF((d*(b*e*x+a*e-b*x)/(a*d*e-b*c*e+b*c))^(1/2),((a*d*e-b*c*e+b*c)/d/a)^(1/2))*
a*b*c*d*e^2+EllipticF((d*(b*e*x+a*e-b*x)/(a*d*e-b*c*e+b*c))^(1/2),((a*d*e-b*c*e+b*c)/d/a)^(1/2))*b^2*c^2*e^2-E
llipticF((d*(b*e*x+a*e-b*x)/(a*d*e-b*c*e+b*c))^(1/2),((a*d*e-b*c*e+b*c)/d/a)^(1/2))*a^2*d^2*e+3*EllipticF((d*(
b*e*x+a*e-b*x)/(a*d*e-b*c*e+b*c))^(1/2),((a*d*e-b*c*e+b*c)/d/a)^(1/2))*a*b*c*d*e-2*EllipticF((d*(b*e*x+a*e-b*x
)/(a*d*e-b*c*e+b*c))^(1/2),((a*d*e-b*c*e+b*c)/d/a)^(1/2))*b^2*c^2*e+EllipticE((d*(b*e*x+a*e-b*x)/(a*d*e-b*c*e+
b*c))^(1/2),((a*d*e-b*c*e+b*c)/d/a)^(1/2))*a^2*d^2*e-EllipticE((d*(b*e*x+a*e-b*x)/(a*d*e-b*c*e+b*c))^(1/2),((a
*d*e-b*c*e+b*c)/d/a)^(1/2))*a*b*c*d*e-EllipticF((d*(b*e*x+a*e-b*x)/(a*d*e-b*c*e+b*c))^(1/2),((a*d*e-b*c*e+b*c)
/d/a)^(1/2))*a*b*c*d+EllipticF((d*(b*e*x+a*e-b*x)/(a*d*e-b*c*e+b*c))^(1/2),((a*d*e-b*c*e+b*c)/d/a)^(1/2))*b^2*
c^2+EllipticE((d*(b*e*x+a*e-b*x)/(a*d*e-b*c*e+b*c))^(1/2),((a*d*e-b*c*e+b*c)/d/a)^(1/2))*a*b*c*d)/((b*e*x+a*e-
b*x)/a)^(1/2)/(b*d*x^2+a*d*x+b*c*x+a*c)/(-1+e)^2/b^2/d

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{d x + c}}{\sqrt{b x + a} \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((d*x+c)^(1/2)/(b*x+a)^(1/2)/(e+b*(-1+e)*x/a)^(1/2),x, algorithm="maxima")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{c + d x}}{\sqrt{a + b x} \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((d*x+c)**(1/2)/(b*x+a)**(1/2)/(e+b*(-1+e)*x/a)**(1/2),x)

[Out]

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

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{d x + c}}{\sqrt{b x + a} \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((d*x+c)^(1/2)/(b*x+a)^(1/2)/(e+b*(-1+e)*x/a)^(1/2),x, algorithm="giac")

[Out]

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