3.1391 \(\int \frac{(c e+d e x)^{3/2}}{\sqrt{1-c^2-2 c d x-d^2 x^2}} \, dx\)

Optimal. Leaf size=78 \[ \frac{2 e^{3/2} F\left (\left .\sin ^{-1}\left (\frac{\sqrt{c e+d x e}}{\sqrt{e}}\right )\right |-1\right )}{3 d}-\frac{2 e \sqrt{-c^2-2 c d x-d^2 x^2+1} \sqrt{c e+d e x}}{3 d} \]

[Out]

(-2*e*Sqrt[c*e + d*e*x]*Sqrt[1 - c^2 - 2*c*d*x - d^2*x^2])/(3*d) + (2*e^(3/2)*El
lipticF[ArcSin[Sqrt[c*e + d*e*x]/Sqrt[e]], -1])/(3*d)

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Rubi [A]  time = 0.170043, antiderivative size = 78, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 37, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.081 \[ \frac{2 e^{3/2} F\left (\left .\sin ^{-1}\left (\frac{\sqrt{c e+d x e}}{\sqrt{e}}\right )\right |-1\right )}{3 d}-\frac{2 e \sqrt{-c^2-2 c d x-d^2 x^2+1} \sqrt{c e+d e x}}{3 d} \]

Antiderivative was successfully verified.

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

[Out]

(-2*e*Sqrt[c*e + d*e*x]*Sqrt[1 - c^2 - 2*c*d*x - d^2*x^2])/(3*d) + (2*e^(3/2)*El
lipticF[ArcSin[Sqrt[c*e + d*e*x]/Sqrt[e]], -1])/(3*d)

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Rubi in Sympy [A]  time = 42.9068, size = 71, normalized size = 0.91 \[ \frac{2 e^{\frac{3}{2}} F\left (\operatorname{asin}{\left (\frac{\sqrt{c e + d e x}}{\sqrt{e}} \right )}\middle | -1\right )}{3 d} - \frac{2 e \sqrt{c e + d e x} \sqrt{- c^{2} - 2 c d x - d^{2} x^{2} + 1}}{3 d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((d*e*x+c*e)**(3/2)/(-d**2*x**2-2*c*d*x-c**2+1)**(1/2),x)

[Out]

2*e**(3/2)*elliptic_f(asin(sqrt(c*e + d*e*x)/sqrt(e)), -1)/(3*d) - 2*e*sqrt(c*e
+ d*e*x)*sqrt(-c**2 - 2*c*d*x - d**2*x**2 + 1)/(3*d)

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Mathematica [A]  time = 0.262045, size = 89, normalized size = 1.14 \[ \frac{2 e \sqrt{e (c+d x)} \left (c^2+2 c d x-\sqrt{c+d x} \sqrt{1-\frac{1}{(c+d x)^2}} F\left (\left .\sin ^{-1}\left (\frac{1}{\sqrt{c+d x}}\right )\right |-1\right )+d^2 x^2-1\right )}{3 d \sqrt{1-(c+d x)^2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.023, size = 161, normalized size = 2.1 \[ -{\frac{e}{3\,d \left ({x}^{3}{d}^{3}+3\,c{d}^{2}{x}^{2}+3\,x{c}^{2}d+{c}^{3}-dx-c \right ) }\sqrt{e \left ( dx+c \right ) }\sqrt{-{d}^{2}{x}^{2}-2\,cdx-{c}^{2}+1} \left ( 2\,{x}^{3}{d}^{3}+6\,c{d}^{2}{x}^{2}+\sqrt{-2\,dx-2\,c+2}\sqrt{2\,dx+2\,c+2}\sqrt{-dx-c}{\it EllipticF} \left ({\frac{1}{2}\sqrt{2\,dx+2\,c+2}},\sqrt{2} \right ) +6\,x{c}^{2}d+2\,{c}^{3}-2\,dx-2\,c \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((d*e*x+c*e)^(3/2)/(-d^2*x^2-2*c*d*x-c^2+1)^(1/2),x)

[Out]

-1/3*(e*(d*x+c))^(1/2)*(-d^2*x^2-2*c*d*x-c^2+1)^(1/2)*e*(2*x^3*d^3+6*c*d^2*x^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))+6*x*c^2*d+2*c^3-2*d*x-2*c)/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. \[ \int \frac{{\left (d e x + c e\right )}^{\frac{3}{2}}}{\sqrt{-d^{2} x^{2} - 2 \, c d x - c^{2} + 1}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*e*x + c*e)^(3/2)/sqrt(-d^2*x^2 - 2*c*d*x - c^2 + 1),x, algorithm="maxima")

[Out]

integrate((d*e*x + c*e)^(3/2)/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. \[{\rm integral}\left (\frac{{\left (d e x + c e\right )}^{\frac{3}{2}}}{\sqrt{-d^{2} x^{2} - 2 \, c d x - c^{2} + 1}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*e*x + c*e)^(3/2)/sqrt(-d^2*x^2 - 2*c*d*x - c^2 + 1),x, algorithm="fricas")

[Out]

integral((d*e*x + c*e)^(3/2)/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. \[ \int \frac{\left (e \left (c + d x\right )\right )^{\frac{3}{2}}}{\sqrt{- \left (c + d x - 1\right ) \left (c + d x + 1\right )}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*e*x+c*e)**(3/2)/(-d**2*x**2-2*c*d*x-c**2+1)**(1/2),x)

[Out]

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

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (d e x + c e\right )}^{\frac{3}{2}}}{\sqrt{-d^{2} x^{2} - 2 \, c d x - c^{2} + 1}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*e*x + c*e)^(3/2)/sqrt(-d^2*x^2 - 2*c*d*x - c^2 + 1),x, algorithm="giac")

[Out]

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