\(\int \frac {\sqrt {d+e x}}{\sqrt {-2 x-3 x^2}} \, dx\) [430]

Optimal result

Integrand size = 23, antiderivative size = 53 \[ \int \frac {\sqrt {d+e x}}{\sqrt {-2 x-3 x^2}} \, dx=-\frac {2 \sqrt {d+e x} E\left (\arcsin \left (\sqrt {\frac {3}{2}} \sqrt {-x}\right )|\frac {2 e}{3 d}\right )}{\sqrt {3} \sqrt {1+\frac {e x}{d}}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {728, 12, 113, 111} \[ \int \frac {\sqrt {d+e x}}{\sqrt {-2 x-3 x^2}} \, dx=-\frac {2 \sqrt {d+e x} E\left (\arcsin \left (\sqrt {\frac {3}{2}} \sqrt {-x}\right )|\frac {2 e}{3 d}\right )}{\sqrt {3} \sqrt {\frac {e x}{d}+1}} \]

[In]

Int[Sqrt[d + e*x]/Sqrt[-2*x - 3*x^2],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 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

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])

Rule 728

Int[((d_.) + (e_.)*(x_))^(m_)/Sqrt[(b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Int[(d + e*x)^m/(Sqrt[b*x]*Sqrt[1
+ (c/b)*x]), x] /; FreeQ[{b, c, d, e}, x] && NeQ[c*d - b*e, 0] && NeQ[2*c*d - b*e, 0] && EqQ[m^2, 1/4] && LtQ[
c, 0] && RationalQ[b]

Rubi steps \begin{align*} \text {integral}& = \int \frac {\sqrt {d+e x}}{\sqrt {2} \sqrt {-x} \sqrt {1+\frac {3 x}{2}}} \, dx \\ & = \frac {\int \frac {\sqrt {d+e x}}{\sqrt {-x} \sqrt {1+\frac {3 x}{2}}} \, dx}{\sqrt {2}} \\ & = \frac {\sqrt {d+e x} \int \frac {\sqrt {1+\frac {e x}{d}}}{\sqrt {-x} \sqrt {1+\frac {3 x}{2}}} \, dx}{\sqrt {2} \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. \(117\) vs. \(2(53)=106\).

Time = 4.75 (sec) , antiderivative size = 117, normalized size of antiderivative = 2.21 \[ \int \frac {\sqrt {d+e x}}{\sqrt {-2 x-3 x^2}} \, dx=\frac {2 \sqrt {-\frac {d}{e}} (2+3 x) (d+e x)-2 d \sqrt {9+\frac {6}{x}} \sqrt {1+\frac {d}{e x}} x^{3/2} E\left (\arcsin \left (\frac {\sqrt {-\frac {d}{e}}}{\sqrt {x}}\right )|\frac {2 e}{3 d}\right )}{3 \sqrt {-\frac {d}{e}} \sqrt {-x (2+3 x)} \sqrt {d+e x}} \]

[In]

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

[Out]

(2*Sqrt[-(d/e)]*(2 + 3*x)*(d + e*x) - 2*d*Sqrt[9 + 6/x]*Sqrt[1 + d/(e*x)]*x^(3/2)*EllipticE[ArcSin[Sqrt[-(d/e)
]/Sqrt[x]], (2*e)/(3*d)])/(3*Sqrt[-(d/e)]*Sqrt[-(x*(2 + 3*x))]*Sqrt[d + e*x])

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(214\) vs. \(2(45)=90\).

Time = 2.32 (sec) , antiderivative size = 215, normalized size of antiderivative = 4.06

method	result	size
default	\(-\frac {2 \sqrt {e x +d}\, \sqrt {-x \left (2+3 x \right )}\, d \sqrt {\frac {e x +d}{d}}\, \sqrt {-\frac {\left (2+3 x \right ) e}{3 d -2 e}}\, \sqrt {-\frac {e x}{d}}\, \left (3 d F\left (\sqrt {\frac {e x +d}{d}}, \sqrt {3}\, \sqrt {\frac {d}{3 d -2 e}}\right )-2 F\left (\sqrt {\frac {e x +d}{d}}, \sqrt {3}\, \sqrt {\frac {d}{3 d -2 e}}\right ) e -3 E\left (\sqrt {\frac {e x +d}{d}}, \sqrt {3}\, \sqrt {\frac {d}{3 d -2 e}}\right ) d +2 E\left (\sqrt {\frac {e x +d}{d}}, \sqrt {3}\, \sqrt {\frac {d}{3 d -2 e}}\right ) e \right )}{3 e x \left (3 e \,x^{2}+3 d x +2 e x +2 d \right )}\)	\(215\)
elliptic	\(\frac {\sqrt {-x \left (2+3 x \right ) \left (e x +d \right )}\, \left (\frac {2 d^{2} \sqrt {\frac {\left (x +\frac {d}{e}\right ) e}{d}}\, \sqrt {\frac {\frac {2}{3}+x}{-\frac {d}{e}+\frac {2}{3}}}\, \sqrt {-\frac {e x}{d}}\, F\left (\sqrt {\frac {\left (x +\frac {d}{e}\right ) e}{d}}, \sqrt {-\frac {d}{e \left (-\frac {d}{e}+\frac {2}{3}\right )}}\right )}{e \sqrt {-3 e \,x^{3}-3 d \,x^{2}-2 e \,x^{2}-2 d x}}+\frac {2 d \sqrt {\frac {\left (x +\frac {d}{e}\right ) e}{d}}\, \sqrt {\frac {\frac {2}{3}+x}{-\frac {d}{e}+\frac {2}{3}}}\, \sqrt {-\frac {e x}{d}}\, \left (\left (-\frac {d}{e}+\frac {2}{3}\right ) E\left (\sqrt {\frac {\left (x +\frac {d}{e}\right ) e}{d}}, \sqrt {-\frac {d}{e \left (-\frac {d}{e}+\frac {2}{3}\right )}}\right )-\frac {2 F\left (\sqrt {\frac {\left (x +\frac {d}{e}\right ) e}{d}}, \sqrt {-\frac {d}{e \left (-\frac {d}{e}+\frac {2}{3}\right )}}\right )}{3}\right )}{\sqrt {-3 e \,x^{3}-3 d \,x^{2}-2 e \,x^{2}-2 d x}}\right )}{\sqrt {-x \left (2+3 x \right )}\, \sqrt {e x +d}}\)	\(285\)

[In]

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

[Out]

-2/3*(e*x+d)^(1/2)*(-x*(2+3*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*Ell
ipticF(((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/x/(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 = 0.09 (sec) , antiderivative size = 213, normalized size of antiderivative = 4.02 \[ \int \frac {\sqrt {d+e x}}{\sqrt {-2 x-3 x^2}} \, dx=-\frac {2 \, {\left (2 \, \sqrt {3} {\left (3 \, d - e\right )} \sqrt {-e} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (9 \, d^{2} - 6 \, d e + 4 \, e^{2}\right )}}{27 \, e^{2}}, -\frac {8 \, {\left (27 \, d^{3} - 27 \, d^{2} e - 18 \, d e^{2} + 8 \, e^{3}\right )}}{729 \, e^{3}}, \frac {9 \, e x + 3 \, d + 2 \, e}{9 \, e}\right ) - 9 \, \sqrt {3} \sqrt {-e} e {\rm weierstrassZeta}\left (\frac {4 \, {\left (9 \, d^{2} - 6 \, d e + 4 \, e^{2}\right )}}{27 \, e^{2}}, -\frac {8 \, {\left (27 \, d^{3} - 27 \, d^{2} e - 18 \, d e^{2} + 8 \, e^{3}\right )}}{729 \, e^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (9 \, d^{2} - 6 \, d e + 4 \, e^{2}\right )}}{27 \, e^{2}}, -\frac {8 \, {\left (27 \, d^{3} - 27 \, d^{2} e - 18 \, d e^{2} + 8 \, e^{3}\right )}}{729 \, e^{3}}, \frac {9 \, e x + 3 \, d + 2 \, e}{9 \, e}\right )\right )\right )}}{27 \, e} \]

[In]

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

\[ \int \frac {\sqrt {d+e x}}{\sqrt {-2 x-3 x^2}} \, dx=\int \frac {\sqrt {d + e x}}{\sqrt {- x \left (3 x + 2\right )}}\, dx \]

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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