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]

2/3*EllipticE(1/2*6^(1/2)*x^(1/2),1/3*(-6*e/d)^(1/2))*(e*x+d)^(1/2)*3^(1/2)/(1+e*x/d)^(1/2)

________________________________________________________________________________________

Rubi [A]  time = 0.01, antiderivative size = 51, normalized size of antiderivative = 1.00, 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 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]

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])

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 = 0.84, 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])

________________________________________________________________________________________

fricas [F]  time = 1.01, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {e x + d} \sqrt {x} \sqrt {-3 \, x + 2}}{3 \, x^{2} - 2 \, x}, x\right ) \]

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)

________________________________________________________________________________________

giac [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

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]

Timed out

________________________________________________________________________________________

maple [B]  time = 0.02, size = 212, normalized size = 4.16 \[ -\frac {2 \sqrt {e x +d}\, \sqrt {-3 x +2}\, \sqrt {\frac {e x +d}{d}}\, \sqrt {-\frac {\left (3 x -2\right ) e}{3 d +2 e}}\, \sqrt {-\frac {e x}{d}}\, \left (-3 d \EllipticE \left (\sqrt {\frac {e x +d}{d}}, \sqrt {3}\, \sqrt {\frac {d}{3 d +2 e}}\right )+3 d \EllipticF \left (\sqrt {\frac {e x +d}{d}}, \sqrt {3}\, \sqrt {\frac {d}{3 d +2 e}}\right )-2 e \EllipticE \left (\sqrt {\frac {e x +d}{d}}, \sqrt {3}\, \sqrt {\frac {d}{3 d +2 e}}\right )+2 e \EllipticF \left (\sqrt {\frac {e x +d}{d}}, \sqrt {3}\, \sqrt {\frac {d}{3 d +2 e}}\right )\right ) d}{3 \left (3 e \,x^{2}+3 d x -2 e x -2 d \right ) e \sqrt {x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {e x + d}}{\sqrt {x} \sqrt {-3 \, x + 2}}\,{d x} \]

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)

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {\sqrt {d+e\,x}}{\sqrt {x}\,\sqrt {2-3\,x}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {d + e x}}{\sqrt {x} \sqrt {2 - 3 x}}\, dx \]

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]

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

________________________________________________________________________________________