### 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*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 = 0.0259557, antiderivative size = 53, normalized size of antiderivative = 1., 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 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]

Rule 12

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

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 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*}

Mathematica [B]  time = 0.211225, 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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Maple [B]  time = 0.245, size = 215, normalized size = 4.1 \begin{align*} -{\frac{2\,d}{3\,ex \left ( 3\,e{x}^{2}+3\,dx+2\,ex+2\,d \right ) }\sqrt{ex+d}\sqrt{-x \left ( 2+3\,x \right ) }\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 ) } \end{align*}

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*(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)

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

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

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

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)