3.874 \(\int \frac{\sqrt{d+e x}}{\sqrt{2-3 x} \sqrt{x}} \, dx\)

Optimal. Leaf size=51 \[ \frac{2 \sqrt{d+e x} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{2}} \sqrt{x}\right )|-\frac{2 e}{3 d}\right )}{\sqrt{3} \sqrt{\frac{e x}{d}+1}} \]

[Out]

(2*Sqrt[d + e*x]*EllipticE[ArcSin[Sqrt[3/2]*Sqrt[x]], (-2*e)/(3*d)])/(Sqrt[3]*Sqrt[1 + (e*x)/d])

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Rubi [A]  time = 0.0122931, antiderivative size = 51, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083, Rules used = {112, 110} \[ \frac{2 \sqrt{d+e x} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{2}} \sqrt{x}\right )|-\frac{2 e}{3 d}\right )}{\sqrt{3} \sqrt{\frac{e x}{d}+1}} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[d + e*x]/(Sqrt[2 - 3*x]*Sqrt[x]),x]

[Out]

(2*Sqrt[d + e*x]*EllipticE[ArcSin[Sqrt[3/2]*Sqrt[x]], (-2*e)/(3*d)])/(Sqrt[3]*Sqrt[1 + (e*x)/d])

Rule 112

Int[Sqrt[(e_) + (f_.)*(x_)]/(Sqrt[(b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Dist[(Sqrt[e + f*x]*Sqrt[
1 + (d*x)/c])/(Sqrt[c + d*x]*Sqrt[1 + (f*x)/e]), Int[Sqrt[1 + (f*x)/e]/(Sqrt[b*x]*Sqrt[1 + (d*x)/c]), x], x] /
; FreeQ[{b, c, d, e, f}, x] && NeQ[d*e - c*f, 0] &&  !(GtQ[c, 0] && GtQ[e, 0])

Rule 110

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

Rubi steps

\begin{align*} \int \frac{\sqrt{d+e x}}{\sqrt{2-3 x} \sqrt{x}} \, dx &=\frac{\left (\sqrt{1-\frac{3 x}{2}} \sqrt{d+e x}\right ) \int \frac{\sqrt{1+\frac{e x}{d}}}{\sqrt{1-\frac{3 x}{2}} \sqrt{x}} \, dx}{\sqrt{2-3 x} \sqrt{1+\frac{e x}{d}}}\\ &=\frac{2 \sqrt{d+e x} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{2}} \sqrt{x}\right )|-\frac{2 e}{3 d}\right )}{\sqrt{3} \sqrt{1+\frac{e x}{d}}}\\ \end{align*}

Mathematica [B]  time = 0.782408, size = 125, normalized size = 2.45 \[ \frac{2 \sqrt{x} \left (\frac{3 (d+e x)}{\sqrt{2-3 x}}-\frac{(3 d+2 e) \sqrt{\frac{d+e x}{e (3 x-2)}} E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{3 d}{e}+2}}{\sqrt{2-3 x}}\right )|\frac{2 e}{3 d+2 e}\right )}{\sqrt{\frac{x}{3 x-2}} \sqrt{\frac{3 d}{e}+2}}\right )}{3 \sqrt{d+e x}} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[d + e*x]/(Sqrt[2 - 3*x]*Sqrt[x]),x]

[Out]

(2*Sqrt[x]*((3*(d + e*x))/Sqrt[2 - 3*x] - ((3*d + 2*e)*Sqrt[(d + e*x)/(e*(-2 + 3*x))]*EllipticE[ArcSin[Sqrt[2
+ (3*d)/e]/Sqrt[2 - 3*x]], (2*e)/(3*d + 2*e)])/(Sqrt[2 + (3*d)/e]*Sqrt[x/(-2 + 3*x)])))/(3*Sqrt[d + e*x])

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Maple [B]  time = 0.015, size = 212, normalized size = 4.2 \begin{align*} -{\frac{2\,d}{3\,e \left ( 3\,e{x}^{2}+3\,dx-2\,ex-2\,d \right ) }\sqrt{ex+d}\sqrt{2-3\,x}\sqrt{{\frac{ex+d}{d}}}\sqrt{-{\frac{ \left ( -2+3\,x \right ) e}{3\,d+2\,e}}}\sqrt{-{\frac{ex}{d}}} \left ( 3\,d{\it EllipticF} \left ( \sqrt{{\frac{ex+d}{d}}},\sqrt{3}\sqrt{{\frac{d}{3\,d+2\,e}}} \right ) +2\,{\it EllipticF} \left ( \sqrt{{\frac{ex+d}{d}}},\sqrt{3}\sqrt{{\frac{d}{3\,d+2\,e}}} \right ) e-3\,{\it EllipticE} \left ( \sqrt{{\frac{ex+d}{d}}},\sqrt{3}\sqrt{{\frac{d}{3\,d+2\,e}}} \right ) d-2\,{\it EllipticE} \left ( \sqrt{{\frac{ex+d}{d}}},\sqrt{3}\sqrt{{\frac{d}{3\,d+2\,e}}} \right ) e \right ){\frac{1}{\sqrt{x}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^(1/2)/(2-3*x)^(1/2)/x^(1/2),x)

[Out]

-2/3*(e*x+d)^(1/2)*(2-3*x)^(1/2)/x^(1/2)*d*((e*x+d)/d)^(1/2)*(-(-2+3*x)*e/(3*d+2*e))^(1/2)*(-e*x/d)^(1/2)*(3*d
*EllipticF(((e*x+d)/d)^(1/2),3^(1/2)*(d/(3*d+2*e))^(1/2))+2*EllipticF(((e*x+d)/d)^(1/2),3^(1/2)*(d/(3*d+2*e))^
(1/2))*e-3*EllipticE(((e*x+d)/d)^(1/2),3^(1/2)*(d/(3*d+2*e))^(1/2))*d-2*EllipticE(((e*x+d)/d)^(1/2),3^(1/2)*(d
/(3*d+2*e))^(1/2))*e)/e/(3*e*x^2+3*d*x-2*e*x-2*d)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{e x + d}}{\sqrt{x} \sqrt{-3 \, x + 2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(1/2)/(2-3*x)^(1/2)/x^(1/2),x, algorithm="maxima")

[Out]

integrate(sqrt(e*x + d)/(sqrt(x)*sqrt(-3*x + 2)), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{\sqrt{e x + d} \sqrt{x} \sqrt{-3 \, x + 2}}{3 \, x^{2} - 2 \, x}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(1/2)/(2-3*x)^(1/2)/x^(1/2),x, algorithm="fricas")

[Out]

integral(-sqrt(e*x + d)*sqrt(x)*sqrt(-3*x + 2)/(3*x^2 - 2*x), x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)**(1/2)/(2-3*x)**(1/2)/x**(1/2),x)

[Out]

Timed out

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Giac [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(1/2)/(2-3*x)^(1/2)/x^(1/2),x, algorithm="giac")

[Out]

Exception raised: TypeError