3.1398 \(\int \frac{\sqrt{c e+d e x}}{\sqrt{1-c^2-2 c d x-d^2 x^2}} \, dx\)

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

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Rubi [A]  time = 0.18825, 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 \[ \frac{2 \sqrt{e} E\left (\left .\sin ^{-1}\left (\frac{\sqrt{c e+d x e}}{\sqrt{e}}\right )\right |-1\right )}{d}-\frac{2 \sqrt{e} F\left (\left .\sin ^{-1}\left (\frac{\sqrt{c e+d x e}}{\sqrt{e}}\right )\right |-1\right )}{d} \]

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]*Elli
pticF[ArcSin[Sqrt[c*e + d*e*x]/Sqrt[e]], -1])/d

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Rubi in Sympy [A]  time = 40.3393, size = 60, normalized size = 0.95 \[ \frac{2 \sqrt{e} E\left (\operatorname{asin}{\left (\frac{\sqrt{c e + d e x}}{\sqrt{e}} \right )}\middle | -1\right )}{d} - \frac{2 \sqrt{e} F\left (\operatorname{asin}{\left (\frac{\sqrt{c e + d e x}}{\sqrt{e}} \right )}\middle | -1\right )}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((d*e*x+c*e)**(1/2)/(-d**2*x**2-2*c*d*x-c**2+1)**(1/2),x)

[Out]

2*sqrt(e)*elliptic_e(asin(sqrt(c*e + d*e*x)/sqrt(e)), -1)/d - 2*sqrt(e)*elliptic
_f(asin(sqrt(c*e + d*e*x)/sqrt(e)), -1)/d

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Mathematica [A]  time = 0.0738495, size = 52, normalized size = 0.83 \[ \frac{2 \sqrt{e (c+d x)} \left (E\left (\left .\sin ^{-1}\left (\sqrt{c+d x}\right )\right |-1\right )-F\left (\left .\sin ^{-1}\left (\sqrt{c+d x}\right )\right |-1\right )\right )}{d \sqrt{c+d x}} \]

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

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Maple [B]  time = 0.023, size = 121, normalized size = 1.9 \[{\frac{1}{d \left ({x}^{3}{d}^{3}+3\,c{d}^{2}{x}^{2}+3\,x{c}^{2}d+{c}^{3}-dx-c \right ) }\sqrt{e \left ( dx+c \right ) }\sqrt{-{d}^{2}{x}^{2}-2\,cdx-{c}^{2}+1}\sqrt{2\,dx+2\,c+2}\sqrt{-dx-c}\sqrt{-2\,dx-2\,c+2}{\it EllipticE} \left ({\frac{1}{2}\sqrt{2\,dx+2\,c+2}},\sqrt{2} \right ) } \]

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]

(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)*(-d*x-c)^(1
/2)*(-2*d*x-2*c+2)^(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. \[ \int \frac{\sqrt{d e x + c e}}{\sqrt{-d^{2} x^{2} - 2 \, c d x - c^{2} + 1}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(d*e*x + c*e)/sqrt(-d^2*x^2 - 2*c*d*x - c^2 + 1),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. \[{\rm integral}\left (\frac{\sqrt{d e x + c e}}{\sqrt{-d^{2} x^{2} - 2 \, c d x - c^{2} + 1}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral(sqrt(d*e*x + c*e)/sqrt(-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. \[ \int \frac{\sqrt{e \left (c + d x\right )}}{\sqrt{- \left (c + d x - 1\right ) \left (c + d x + 1\right )}}\, dx \]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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