### 3.431 $$\int \frac{1}{\sqrt{d+e x} \sqrt{-2 x-3 x^2}} \, dx$$

Optimal. Leaf size=53 $-\frac{2 \sqrt{\frac{e x}{d}+1} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{3}{2}} \sqrt{-x}\right ),\frac{2 e}{3 d}\right )}{\sqrt{3} \sqrt{d+e x}}$

[Out]

(-2*Sqrt[1 + (e*x)/d]*EllipticF[ArcSin[Sqrt[3/2]*Sqrt[-x]], (2*e)/(3*d)])/(Sqrt[3]*Sqrt[d + e*x])

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Rubi [A]  time = 0.026315, 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, 117, 115} $-\frac{2 \sqrt{\frac{e x}{d}+1} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{2}} \sqrt{-x}\right )|\frac{2 e}{3 d}\right )}{\sqrt{3} \sqrt{d+e x}}$

Antiderivative was successfully veriﬁed.

[In]

Int[1/(Sqrt[d + e*x]*Sqrt[-2*x - 3*x^2]),x]

[Out]

(-2*Sqrt[1 + (e*x)/d]*EllipticF[ArcSin[Sqrt[3/2]*Sqrt[-x]], (2*e)/(3*d)])/(Sqrt[3]*Sqrt[d + e*x])

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 117

Int[1/(Sqrt[(b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(e_) + (f_.)*(x_)]), x_Symbol] :> Dist[(Sqrt[1 + (d*x)/c]
*Sqrt[1 + (f*x)/e])/(Sqrt[c + d*x]*Sqrt[e + f*x]), Int[1/(Sqrt[b*x]*Sqrt[1 + (d*x)/c]*Sqrt[1 + (f*x)/e]), x],
x] /; FreeQ[{b, c, d, e, f}, x] &&  !(GtQ[c, 0] && GtQ[e, 0])

Rule 115

Int[1/(Sqrt[(b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(e_) + (f_.)*(x_)]), x_Symbol] :> Simp[(2*Rt[-(b/d), 2]*E
llipticF[ArcSin[Sqrt[b*x]/(Sqrt[c]*Rt[-(b/d), 2])], (c*f)/(d*e)])/(b*Sqrt[e]), x] /; FreeQ[{b, c, d, e, f}, x]
&& GtQ[c, 0] && GtQ[e, 0] && (GtQ[-(b/d), 0] || LtQ[-(b/f), 0])

Rubi steps

\begin{align*} \int \frac{1}{\sqrt{d+e x} \sqrt{-2 x-3 x^2}} \, dx &=\int \frac{1}{\sqrt{2} \sqrt{-x} \sqrt{1+\frac{3 x}{2}} \sqrt{d+e x}} \, dx\\ &=\frac{\int \frac{1}{\sqrt{-x} \sqrt{1+\frac{3 x}{2}} \sqrt{d+e x}} \, dx}{\sqrt{2}}\\ &=\frac{\sqrt{1+\frac{e x}{d}} \int \frac{1}{\sqrt{-x} \sqrt{1+\frac{3 x}{2}} \sqrt{1+\frac{e x}{d}}} \, dx}{\sqrt{2} \sqrt{d+e x}}\\ &=-\frac{2 \sqrt{1+\frac{e x}{d}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{2}} \sqrt{-x}\right )|\frac{2 e}{3 d}\right )}{\sqrt{3} \sqrt{d+e x}}\\ \end{align*}

Mathematica [C]  time = 0.109033, size = 82, normalized size = 1.55 $\frac{i \sqrt{\frac{4}{x}+6} x^{3/2} \sqrt{\frac{d}{e x}+1} \text{EllipticF}\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{2}{3}}}{\sqrt{x}}\right ),\frac{3 d}{2 e}\right )}{\sqrt{-x (3 x+2)} \sqrt{d+e x}}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[1/(Sqrt[d + e*x]*Sqrt[-2*x - 3*x^2]),x]

[Out]

(I*Sqrt[6 + 4/x]*Sqrt[1 + d/(e*x)]*x^(3/2)*EllipticF[I*ArcSinh[Sqrt[2/3]/Sqrt[x]], (3*d)/(2*e)])/(Sqrt[-(x*(2
+ 3*x))]*Sqrt[d + e*x])

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Maple [B]  time = 0.207, size = 115, normalized size = 2.2 \begin{align*} -2\,{\frac{d\sqrt{ex+d}\sqrt{-x \left ( 2+3\,x \right ) }}{ex \left ( 3\,e{x}^{2}+3\,dx+2\,ex+2\,d \right ) }{\it EllipticF} \left ( \sqrt{{\frac{ex+d}{d}}},\sqrt{3}\sqrt{{\frac{d}{3\,d-2\,e}}} \right ) \sqrt{-{\frac{ex}{d}}}\sqrt{-{\frac{ \left ( 2+3\,x \right ) e}{3\,d-2\,e}}}\sqrt{{\frac{ex+d}{d}}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*EllipticF(((e*x+d)/d)^(1/2),3^(1/2)*(d/(3*d-2*e))^(1/2))*(-e*x/d)^(1/2)*(-(2+3*x)*e/(3*d-2*e))^(1/2)*((e*x+
d)/d)^(1/2)*d*(e*x+d)^(1/2)*(-x*(2+3*x))^(1/2)/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{1}{\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(1/(e*x+d)^(1/2)/(-3*x^2-2*x)^(1/2),x, algorithm="maxima")

[Out]

integrate(1/(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 \, e x^{3} +{\left (3 \, d + 2 \, e\right )} x^{2} + 2 \, d x}, x\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{- x \left (3 x + 2\right )} \sqrt{d + e x}}\, dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\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(1/(e*x+d)^(1/2)/(-3*x^2-2*x)^(1/2),x, algorithm="giac")

[Out]

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