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

Optimal. Leaf size=53 \[ -\frac {2 \sqrt {d+e x} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{2}} \sqrt {-x}\right )|\frac {2 e}{3 d}\right )}{\sqrt {3} \sqrt {\frac {e x}{d}+1}} \]

[Out]

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

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Rubi [A]  time = 0.03, antiderivative size = 53, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {714, 12, 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*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 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])

Rule 714

Int[((d_.) + (e_.)*(x_))^(m_)/Sqrt[(b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Int[(d + e*x)^m/(Sqrt[b*x]*Sqrt[1
+ (c*x)/b]), 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*} \int \frac {\sqrt {d+e x}}{\sqrt {-2 x-3 x^2}} \, dx &=\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*}

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Mathematica [B]  time = 0.21, size = 117, normalized size = 2.21 \[ \frac {2 (3 x+2) \sqrt {-\frac {d}{e}} (d+e x)-2 d \sqrt {\frac {6}{x}+9} x^{3/2} \sqrt {\frac {d}{e x}+1} E\left (\sin ^{-1}\left (\frac {\sqrt {-\frac {d}{e}}}{\sqrt {x}}\right )|\frac {2 e}{3 d}\right )}{3 \sqrt {-x (3 x+2)} \sqrt {-\frac {d}{e}} \sqrt {d+e x}} \]

Antiderivative was successfully verified.

[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])

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fricas [F]  time = 0.88, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {e x + d} \sqrt {-3 \, x^{2} - 2 \, x}}{3 \, x^{2} + 2 \, x}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[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)

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maple [B]  time = 0.12, size = 215, normalized size = 4.06 \[ -\frac {2 \sqrt {e x +d}\, \sqrt {-\left (3 x +2\right ) x}\, \sqrt {\frac {e x +d}{d}}\, \sqrt {-\frac {\left (3 x +2\right ) e}{3 d -2 e}}\, \sqrt {-\frac {e x}{d}}\, \left (-3 d \EllipticE \left (\sqrt {\frac {e x +d}{d}}, \sqrt {3}\, \sqrt {\frac {d}{3 d -2 e}}\right )+3 d \EllipticF \left (\sqrt {\frac {e x +d}{d}}, \sqrt {3}\, \sqrt {\frac {d}{3 d -2 e}}\right )+2 e \EllipticE \left (\sqrt {\frac {e x +d}{d}}, \sqrt {3}\, \sqrt {\frac {d}{3 d -2 e}}\right )-2 e \EllipticF \left (\sqrt {\frac {e x +d}{d}}, \sqrt {3}\, \sqrt {\frac {d}{3 d -2 e}}\right )\right ) d}{3 \left (3 e \,x^{2}+3 d x +2 e x +2 d \right ) e x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/3*(e*x+d)^(1/2)*(-x*(3*x+2))^(1/2)*d*((e*x+d)/d)^(1/2)*(-(3*x+2)*e/(3*d-2*e))^(1/2)*(-1/d*e*x)^(1/2)*(3*d*E
llipticF(((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)

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

Verification of antiderivative is not currently implemented for this CAS.

[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)

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

Verification of antiderivative is not currently implemented for this CAS.

[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)

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

Verification of antiderivative is not currently implemented for this CAS.

[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)

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