3.1409 \(\int \frac{\sqrt{c e+d e x}}{\sqrt{1-c^2-2 c d x-d^2 x^2}} \, dx\)

Optimal. Leaf size=63 \[ \frac{2 \sqrt{e} E\left (\left .\sin ^{-1}\left (\frac{\sqrt{c e+d x e}}{\sqrt{e}}\right )\right |-1\right )}{d}-\frac{2 \sqrt{e} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{c e+d e x}}{\sqrt{e}}\right ),-1\right )}{d} \]

[Out]

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

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Rubi [A]  time = 0.0553057, antiderivative size = 63, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 37, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.135, Rules used = {690, 307, 221, 1199, 424} \[ \frac{2 \sqrt{e} E\left (\left .\sin ^{-1}\left (\frac{\sqrt{c e+d x e}}{\sqrt{e}}\right )\right |-1\right )}{d}-\frac{2 \sqrt{e} F\left (\left .\sin ^{-1}\left (\frac{\sqrt{c e+d x e}}{\sqrt{e}}\right )\right |-1\right )}{d} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[c*e + d*e*x]/Sqrt[1 - c^2 - 2*c*d*x - d^2*x^2],x]

[Out]

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

Rule 690

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

Rule 307

Int[(x_)^2/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> With[{q = Rt[-(b/a), 2]}, -Dist[q^(-1), Int[1/Sqrt[a + b*x^
4], x], x] + Dist[1/q, Int[(1 + q*x^2)/Sqrt[a + b*x^4], x], x]] /; FreeQ[{a, b}, x] && NegQ[b/a]

Rule 221

Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> Simp[EllipticF[ArcSin[(Rt[-b, 4]*x)/Rt[a, 4]], -1]/(Rt[a, 4]*Rt[
-b, 4]), x] /; FreeQ[{a, b}, x] && NegQ[b/a] && GtQ[a, 0]

Rule 1199

Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (c_.)*(x_)^4], x_Symbol] :> Dist[d/Sqrt[a], Int[Sqrt[1 + (e*x^2)/d]/Sqrt
[1 - (e*x^2)/d], x], x] /; FreeQ[{a, c, d, e}, x] && NegQ[c/a] && EqQ[c*d^2 + a*e^2, 0] && GtQ[a, 0]

Rule 424

Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[(Sqrt[a]*EllipticE[ArcSin[Rt[-(d/c)
, 2]*x], (b*c)/(a*d)])/(Sqrt[c]*Rt[-(d/c), 2]), x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[
a, 0]

Rubi steps

\begin{align*} \int \frac{\sqrt{c e+d e x}}{\sqrt{1-c^2-2 c d x-d^2 x^2}} \, dx &=\frac{2 \operatorname{Subst}\left (\int \frac{x^2}{\sqrt{1-\frac{x^4}{e^2}}} \, dx,x,\sqrt{c e+d e x}\right )}{d e}\\ &=-\frac{2 \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-\frac{x^4}{e^2}}} \, dx,x,\sqrt{c e+d e x}\right )}{d}+\frac{2 \operatorname{Subst}\left (\int \frac{1+\frac{x^2}{e}}{\sqrt{1-\frac{x^4}{e^2}}} \, dx,x,\sqrt{c e+d e x}\right )}{d}\\ &=-\frac{2 \sqrt{e} F\left (\left .\sin ^{-1}\left (\frac{\sqrt{c e+d e x}}{\sqrt{e}}\right )\right |-1\right )}{d}+\frac{2 \operatorname{Subst}\left (\int \frac{\sqrt{1+\frac{x^2}{e}}}{\sqrt{1-\frac{x^2}{e}}} \, dx,x,\sqrt{c e+d e x}\right )}{d}\\ &=\frac{2 \sqrt{e} E\left (\left .\sin ^{-1}\left (\frac{\sqrt{c e+d e x}}{\sqrt{e}}\right )\right |-1\right )}{d}-\frac{2 \sqrt{e} F\left (\left .\sin ^{-1}\left (\frac{\sqrt{c e+d e x}}{\sqrt{e}}\right )\right |-1\right )}{d}\\ \end{align*}

Mathematica [C]  time = 0.0165099, size = 40, normalized size = 0.63 \[ \frac{2 (c+d x) \sqrt{e (c+d x)} \, _2F_1\left (\frac{1}{2},\frac{3}{4};\frac{7}{4};(c+d x)^2\right )}{3 d} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[c*e + d*e*x]/Sqrt[1 - c^2 - 2*c*d*x - d^2*x^2],x]

[Out]

(2*(c + d*x)*Sqrt[e*(c + d*x)]*Hypergeometric2F1[1/2, 3/4, 7/4, (c + d*x)^2])/(3*d)

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Maple [B]  time = 0.17, size = 204, normalized size = 3.2 \begin{align*}{\frac{1}{6\,d \left ({x}^{3}{d}^{3}+3\,{x}^{2}c{d}^{2}+3\,x{c}^{2}d+{c}^{3}-dx-c \right ) }\sqrt{e \left ( dx+c \right ) }\sqrt{-{d}^{2}{x}^{2}-2\,cdx-{c}^{2}+1}\sqrt{2\,dx+2\,c+2}\sqrt{-2\,dx-2\,c+2} \left ( \sqrt{-dx-c}{\it EllipticF} \left ({\frac{1}{2}\sqrt{2\,dx+2\,c+2}},\sqrt{2} \right ) +3\,\sqrt{-dx-c}{\it EllipticE} \left ( 1/2\,\sqrt{2\,dx+2\,c+2},\sqrt{2} \right ) +\sqrt{dx+c}{\it EllipticF} \left ({\frac{1}{2}\sqrt{-2\,dx-2\,c+2}},\sqrt{2} \right ) +3\,\sqrt{dx+c}{\it EllipticE} \left ( 1/2\,\sqrt{-2\,dx-2\,c+2},\sqrt{2} \right ) \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{d e x + c e}}{\sqrt{-d^{2} x^{2} - 2 \, c d x - c^{2} + 1}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{\sqrt{-d^{2} x^{2} - 2 \, c d x - c^{2} + 1} \sqrt{d e x + c e}}{d^{2} x^{2} + 2 \, c d x + c^{2} - 1}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(-sqrt(-d^2*x^2 - 2*c*d*x - c^2 + 1)*sqrt(d*e*x + c*e)/(d^2*x^2 + 2*c*d*x + c^2 - 1), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{d e x + c e}}{\sqrt{-d^{2} x^{2} - 2 \, c d x - c^{2} + 1}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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