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 \[ \frac {2 \sqrt {e} E\left (\left .\sin ^{-1}\left (\frac {\sqrt {c e+d x e}}{\sqrt {e}}\right )\right |-1\right )}{d}-\frac {2 \sqrt {e} F\left (\left .\sin ^{-1}\left (\frac {\sqrt {c e+d x e}}{\sqrt {e}}\right )\right |-1\right )}{d} \]

[Out]

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

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Rubi [A]  time = 0.06, antiderivative size = 63, normalized size of antiderivative = 1.00, 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 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 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 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]

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 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]

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*}

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Mathematica [C]  time = 0.02, 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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fricas [F]  time = 0.84, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {-d^{2} x^{2} - 2 \, c d x - c^{2} + 1} \sqrt {d e x + c e}}{d^{2} x^{2} + 2 \, c d x + c^{2} - 1}, x\right ) \]

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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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {d e x + c e}}{\sqrt {-d^{2} x^{2} - 2 \, c d x - c^{2} + 1}}\,{d x} \]

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)

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maple [B]  time = 0.07, size = 121, normalized size = 1.92 \[ \frac {\sqrt {\left (d x +c \right ) e}\, \sqrt {-d^{2} x^{2}-2 c d x -c^{2}+1}\, \sqrt {2 d x +2 c +2}\, \sqrt {-d x -c}\, \sqrt {-2 d x -2 c +2}\, \EllipticE \left (\frac {\sqrt {2 d x +2 c +2}}{2}, \sqrt {2}\right )}{\left (d^{3} x^{3}+3 c \,d^{2} x^{2}+3 c^{2} d x +c^{3}-d x -c \right ) d} \]

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]

((d*x+c)*e)^(1/2)*(-d^2*x^2-2*c*d*x-c^2+1)^(1/2)*(2*d*x+2*c+2)^(1/2)*(-d*x-c)^(1/2)*(-2*d*x-2*c+2)^(1/2)*Ellip
ticE(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 = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {d e x + c e}}{\sqrt {-d^{2} x^{2} - 2 \, c d x - c^{2} + 1}}\,{d x} \]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {e \left (c + d x\right )}}{\sqrt {- \left (c + d x - 1\right ) \left (c + d x + 1\right )}}\, dx \]

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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