\(\int \frac {\sqrt {d+e x}}{\sqrt {2-3 x} \sqrt {x}} \, dx\) [874]

Optimal result

Integrand size = 24, antiderivative size = 51 \[ \int \frac {\sqrt {d+e x}}{\sqrt {2-3 x} \sqrt {x}} \, dx=\frac {2 \sqrt {d+e x} E\left (\arcsin \left (\sqrt {\frac {3}{2}} \sqrt {x}\right )|-\frac {2 e}{3 d}\right )}{\sqrt {3} \sqrt {1+\frac {e x}{d}}} \]

[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] (verified)

Time = 0.01 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {113, 111} \[ \int \frac {\sqrt {d+e x}}{\sqrt {2-3 x} \sqrt {x}} \, dx=\frac {2 \sqrt {d+e x} E\left (\arcsin \left (\sqrt {\frac {3}{2}} \sqrt {x}\right )|-\frac {2 e}{3 d}\right )}{\sqrt {3} \sqrt {\frac {e x}{d}+1}} \]

[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 111

Int[Sqrt[(e_) + (f_.)*(x_)]/(Sqrt[(b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Simp[2*(Sqrt[e]/b)*Rt[-b/
d, 2]*EllipticE[ArcSin[Sqrt[b*x]/(Sqrt[c]*Rt[-b/d, 2])], c*(f/(d*e))], 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 113

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*} \text {integral}& = \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] (verified)

Leaf count is larger than twice the leaf count of optimal. \(125\) vs. \(2(51)=102\).

Time = 3.34 (sec) , antiderivative size = 125, normalized size of antiderivative = 2.45 \[ \int \frac {\sqrt {d+e x}}{\sqrt {2-3 x} \sqrt {x}} \, dx=\frac {2 \sqrt {x} \left (\frac {3 (d+e x)}{\sqrt {2-3 x}}-\frac {(3 d+2 e) \sqrt {\frac {d+e x}{e (-2+3 x)}} E\left (\arcsin \left (\frac {\sqrt {2+\frac {3 d}{e}}}{\sqrt {2-3 x}}\right )|\frac {2 e}{3 d+2 e}\right )}{\sqrt {2+\frac {3 d}{e}} \sqrt {\frac {x}{-2+3 x}}}\right )}{3 \sqrt {d+e x}} \]

[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])

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(211\) vs. \(2(41)=82\).

Time = 1.82 (sec) , antiderivative size = 212, normalized size of antiderivative = 4.16

method	result	size
default	\(-\frac {2 \sqrt {e x +d}\, \sqrt {2-3 x}\, d \sqrt {\frac {e x +d}{d}}\, \sqrt {-\frac {\left (-2+3 x \right ) e}{3 d +2 e}}\, \sqrt {-\frac {e x}{d}}\, \left (3 d F\left (\sqrt {\frac {e x +d}{d}}, \sqrt {3}\, \sqrt {\frac {d}{3 d +2 e}}\right )+2 F\left (\sqrt {\frac {e x +d}{d}}, \sqrt {3}\, \sqrt {\frac {d}{3 d +2 e}}\right ) e -3 E\left (\sqrt {\frac {e x +d}{d}}, \sqrt {3}\, \sqrt {\frac {d}{3 d +2 e}}\right ) d -2 E\left (\sqrt {\frac {e x +d}{d}}, \sqrt {3}\, \sqrt {\frac {d}{3 d +2 e}}\right ) e \right )}{3 \sqrt {x}\, e \left (3 e \,x^{2}+3 d x -2 e x -2 d \right )}\)	\(212\)
elliptic	\(\frac {\sqrt {-\left (-2+3 x \right ) x \left (e x +d \right )}\, \left (\frac {2 d^{2} \sqrt {\frac {\left (x +\frac {d}{e}\right ) e}{d}}\, \sqrt {\frac {-\frac {2}{3}+x}{-\frac {d}{e}-\frac {2}{3}}}\, \sqrt {-\frac {e x}{d}}\, F\left (\sqrt {\frac {\left (x +\frac {d}{e}\right ) e}{d}}, \sqrt {-\frac {d}{e \left (-\frac {d}{e}-\frac {2}{3}\right )}}\right )}{e \sqrt {-3 e \,x^{3}-3 d \,x^{2}+2 e \,x^{2}+2 d x}}+\frac {2 d \sqrt {\frac {\left (x +\frac {d}{e}\right ) e}{d}}\, \sqrt {\frac {-\frac {2}{3}+x}{-\frac {d}{e}-\frac {2}{3}}}\, \sqrt {-\frac {e x}{d}}\, \left (\left (-\frac {d}{e}-\frac {2}{3}\right ) E\left (\sqrt {\frac {\left (x +\frac {d}{e}\right ) e}{d}}, \sqrt {-\frac {d}{e \left (-\frac {d}{e}-\frac {2}{3}\right )}}\right )+\frac {2 F\left (\sqrt {\frac {\left (x +\frac {d}{e}\right ) e}{d}}, \sqrt {-\frac {d}{e \left (-\frac {d}{e}-\frac {2}{3}\right )}}\right )}{3}\right )}{\sqrt {-3 e \,x^{3}-3 d \,x^{2}+2 e \,x^{2}+2 d x}}\right )}{\sqrt {2-3 x}\, \sqrt {x}\, \sqrt {e x +d}}\)	\(285\)

[In]

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

[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)

Fricas [C] (verification not implemented)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 0.08 (sec) , antiderivative size = 211, normalized size of antiderivative = 4.14 \[ \int \frac {\sqrt {d+e x}}{\sqrt {2-3 x} \sqrt {x}} \, dx=-\frac {2 \, {\left (2 \, \sqrt {3} {\left (3 \, d + e\right )} \sqrt {-e} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (9 \, d^{2} + 6 \, d e + 4 \, e^{2}\right )}}{27 \, e^{2}}, -\frac {8 \, {\left (27 \, d^{3} + 27 \, d^{2} e - 18 \, d e^{2} - 8 \, e^{3}\right )}}{729 \, e^{3}}, \frac {9 \, e x + 3 \, d - 2 \, e}{9 \, e}\right ) - 9 \, \sqrt {3} \sqrt {-e} e {\rm weierstrassZeta}\left (\frac {4 \, {\left (9 \, d^{2} + 6 \, d e + 4 \, e^{2}\right )}}{27 \, e^{2}}, -\frac {8 \, {\left (27 \, d^{3} + 27 \, d^{2} e - 18 \, d e^{2} - 8 \, e^{3}\right )}}{729 \, e^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (9 \, d^{2} + 6 \, d e + 4 \, e^{2}\right )}}{27 \, e^{2}}, -\frac {8 \, {\left (27 \, d^{3} + 27 \, d^{2} e - 18 \, d e^{2} - 8 \, e^{3}\right )}}{729 \, e^{3}}, \frac {9 \, e x + 3 \, d - 2 \, e}{9 \, e}\right )\right )\right )}}{27 \, e} \]

[In]

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

[Out]

-2/27*(2*sqrt(3)*(3*d + e)*sqrt(-e)*weierstrassPInverse(4/27*(9*d^2 + 6*d*e + 4*e^2)/e^2, -8/729*(27*d^3 + 27*
d^2*e - 18*d*e^2 - 8*e^3)/e^3, 1/9*(9*e*x + 3*d - 2*e)/e) - 9*sqrt(3)*sqrt(-e)*e*weierstrassZeta(4/27*(9*d^2 +
6*d*e + 4*e^2)/e^2, -8/729*(27*d^3 + 27*d^2*e - 18*d*e^2 - 8*e^3)/e^3, weierstrassPInverse(4/27*(9*d^2 + 6*d*
e + 4*e^2)/e^2, -8/729*(27*d^3 + 27*d^2*e - 18*d*e^2 - 8*e^3)/e^3, 1/9*(9*e*x + 3*d - 2*e)/e)))/e

Sympy [F]

\[ \int \frac {\sqrt {d+e x}}{\sqrt {2-3 x} \sqrt {x}} \, dx=\int \frac {\sqrt {d + e x}}{\sqrt {x} \sqrt {2 - 3 x}}\, dx \]

[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)

Maxima [F]

\[ \int \frac {\sqrt {d+e x}}{\sqrt {2-3 x} \sqrt {x}} \, dx=\int { \frac {\sqrt {e x + d}}{\sqrt {x} \sqrt {-3 \, x + 2}} \,d x } \]

[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)

Giac [F(-2)]

Exception generated. \[ \int \frac {\sqrt {d+e x}}{\sqrt {2-3 x} \sqrt {x}} \, dx=\text {Exception raised: AttributeError} \]

[In]

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

[Out]

Exception raised: AttributeError >> type

Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt {d+e x}}{\sqrt {2-3 x} \sqrt {x}} \, dx=\int \frac {\sqrt {d+e\,x}}{\sqrt {x}\,\sqrt {2-3\,x}} \,d x \]

[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)