3.1400 \(\int \frac{1}{(c e+d e x)^{7/2} \sqrt{1-c^2-2 c d x-d^2 x^2}} \, dx\)

Optimal. Leaf size=159 \[ -\frac{6 \sqrt{-c^2-2 c d x-d^2 x^2+1}}{5 d e^3 \sqrt{c e+d e x}}-\frac{2 \sqrt{-c^2-2 c d x-d^2 x^2+1}}{5 d e (c e+d e x)^{5/2}}+\frac{6 F\left (\left .\sin ^{-1}\left (\frac{\sqrt{c e+d x e}}{\sqrt{e}}\right )\right |-1\right )}{5 d e^{7/2}}-\frac{6 E\left (\left .\sin ^{-1}\left (\frac{\sqrt{c e+d x e}}{\sqrt{e}}\right )\right |-1\right )}{5 d e^{7/2}} \]

[Out]

(-2*Sqrt[1 - c^2 - 2*c*d*x - d^2*x^2])/(5*d*e*(c*e + d*e*x)^(5/2)) - (6*Sqrt[1 -
c^2 - 2*c*d*x - d^2*x^2])/(5*d*e^3*Sqrt[c*e + d*e*x]) - (6*EllipticE[ArcSin[Sqr
t[c*e + d*e*x]/Sqrt[e]], -1])/(5*d*e^(7/2)) + (6*EllipticF[ArcSin[Sqrt[c*e + d*e
*x]/Sqrt[e]], -1])/(5*d*e^(7/2))

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Rubi [A]  time = 0.368868, antiderivative size = 159, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 37, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.162 \[ -\frac{6 \sqrt{-c^2-2 c d x-d^2 x^2+1}}{5 d e^3 \sqrt{c e+d e x}}-\frac{2 \sqrt{-c^2-2 c d x-d^2 x^2+1}}{5 d e (c e+d e x)^{5/2}}+\frac{6 F\left (\left .\sin ^{-1}\left (\frac{\sqrt{c e+d x e}}{\sqrt{e}}\right )\right |-1\right )}{5 d e^{7/2}}-\frac{6 E\left (\left .\sin ^{-1}\left (\frac{\sqrt{c e+d x e}}{\sqrt{e}}\right )\right |-1\right )}{5 d e^{7/2}} \]

Antiderivative was successfully verified.

[In]  Int[1/((c*e + d*e*x)^(7/2)*Sqrt[1 - c^2 - 2*c*d*x - d^2*x^2]),x]

[Out]

(-2*Sqrt[1 - c^2 - 2*c*d*x - d^2*x^2])/(5*d*e*(c*e + d*e*x)^(5/2)) - (6*Sqrt[1 -
c^2 - 2*c*d*x - d^2*x^2])/(5*d*e^3*Sqrt[c*e + d*e*x]) - (6*EllipticE[ArcSin[Sqr
t[c*e + d*e*x]/Sqrt[e]], -1])/(5*d*e^(7/2)) + (6*EllipticF[ArcSin[Sqrt[c*e + d*e
*x]/Sqrt[e]], -1])/(5*d*e^(7/2))

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Rubi in Sympy [A]  time = 80.134, size = 146, normalized size = 0.92 \[ - \frac{2 \sqrt{- c^{2} - 2 c d x - d^{2} x^{2} + 1}}{5 d e \left (c e + d e x\right )^{\frac{5}{2}}} - \frac{6 \sqrt{- c^{2} - 2 c d x - d^{2} x^{2} + 1}}{5 d e^{3} \sqrt{c e + d e x}} - \frac{6 E\left (\operatorname{asin}{\left (\frac{\sqrt{c e + d e x}}{\sqrt{e}} \right )}\middle | -1\right )}{5 d e^{\frac{7}{2}}} + \frac{6 F\left (\operatorname{asin}{\left (\frac{\sqrt{c e + d e x}}{\sqrt{e}} \right )}\middle | -1\right )}{5 d e^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(d*e*x+c*e)**(7/2)/(-d**2*x**2-2*c*d*x-c**2+1)**(1/2),x)

[Out]

-2*sqrt(-c**2 - 2*c*d*x - d**2*x**2 + 1)/(5*d*e*(c*e + d*e*x)**(5/2)) - 6*sqrt(-
c**2 - 2*c*d*x - d**2*x**2 + 1)/(5*d*e**3*sqrt(c*e + d*e*x)) - 6*elliptic_e(asin
(sqrt(c*e + d*e*x)/sqrt(e)), -1)/(5*d*e**(7/2)) + 6*elliptic_f(asin(sqrt(c*e + d
*e*x)/sqrt(e)), -1)/(5*d*e**(7/2))

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Mathematica [A]  time = 0.22198, size = 103, normalized size = 0.65 \[ \frac{(c+d x)^{7/2} \left (6 \left (F\left (\left .\sin ^{-1}\left (\sqrt{c+d x}\right )\right |-1\right )-E\left (\left .\sin ^{-1}\left (\sqrt{c+d x}\right )\right |-1\right )\right )-\frac{2 \left (3 c^2+6 c d x+3 d^2 x^2+1\right ) \sqrt{1-(c+d x)^2}}{(c+d x)^{5/2}}\right )}{5 d (e (c+d x))^{7/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((c*e + d*e*x)^(7/2)*Sqrt[1 - c^2 - 2*c*d*x - d^2*x^2]),x]

[Out]

((c + d*x)^(7/2)*((-2*(1 + 3*c^2 + 6*c*d*x + 3*d^2*x^2)*Sqrt[1 - (c + d*x)^2])/(
c + d*x)^(5/2) + 6*(-EllipticE[ArcSin[Sqrt[c + d*x]], -1] + EllipticF[ArcSin[Sqr
t[c + d*x]], -1])))/(5*d*(e*(c + d*x))^(7/2))

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Maple [B]  time = 0.078, size = 447, normalized size = 2.8 \[{\frac{1}{5\,{e}^{4} \left ( dx+c \right ) ^{3} \left ({d}^{2}{x}^{2}+2\,cdx+{c}^{2}-1 \right ) d} \left ( 7\,{\it EllipticE} \left ( 1/2\,\sqrt{2\,dx+2\,c+2},\sqrt{2} \right ){x}^{2}{d}^{2}\sqrt{2\,dx+2\,c+2}\sqrt{-dx-c}\sqrt{-2\,dx-2\,c+2}-10\,{\it EllipticE} \left ( 1/2\,\sqrt{-2\,dx-2\,c+2},\sqrt{2} \right ){x}^{2}{d}^{2}\sqrt{dx+c}\sqrt{2\,dx+2\,c+2}\sqrt{-2\,dx-2\,c+2}-6\,{d}^{4}{x}^{4}+14\,{\it EllipticE} \left ( 1/2\,\sqrt{2\,dx+2\,c+2},\sqrt{2} \right ) xcd\sqrt{2\,dx+2\,c+2}\sqrt{-dx-c}\sqrt{-2\,dx-2\,c+2}-20\,{\it EllipticE} \left ( 1/2\,\sqrt{-2\,dx-2\,c+2},\sqrt{2} \right ) xcd\sqrt{dx+c}\sqrt{2\,dx+2\,c+2}\sqrt{-2\,dx-2\,c+2}-24\,{x}^{3}c{d}^{3}+7\,\sqrt{-2\,dx-2\,c+2}\sqrt{2\,dx+2\,c+2}\sqrt{-dx-c}{\it EllipticE} \left ( 1/2\,\sqrt{2\,dx+2\,c+2},\sqrt{2} \right ){c}^{2}-10\,\sqrt{-2\,dx-2\,c+2}\sqrt{dx+c}\sqrt{2\,dx+2\,c+2}{\it EllipticE} \left ( 1/2\,\sqrt{-2\,dx-2\,c+2},\sqrt{2} \right ){c}^{2}-36\,{x}^{2}{c}^{2}{d}^{2}-24\,{c}^{3}dx+4\,{d}^{2}{x}^{2}-6\,{c}^{4}+8\,cdx+4\,{c}^{2}+2 \right ) \sqrt{-{d}^{2}{x}^{2}-2\,cdx-{c}^{2}+1}\sqrt{e \left ( dx+c \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(d*e*x+c*e)^(7/2)/(-d^2*x^2-2*c*d*x-c^2+1)^(1/2),x)

[Out]

1/5*(7*EllipticE(1/2*(2*d*x+2*c+2)^(1/2),2^(1/2))*x^2*d^2*(2*d*x+2*c+2)^(1/2)*(-
d*x-c)^(1/2)*(-2*d*x-2*c+2)^(1/2)-10*EllipticE(1/2*(-2*d*x-2*c+2)^(1/2),2^(1/2))
*x^2*d^2*(d*x+c)^(1/2)*(2*d*x+2*c+2)^(1/2)*(-2*d*x-2*c+2)^(1/2)-6*d^4*x^4+14*Ell
ipticE(1/2*(2*d*x+2*c+2)^(1/2),2^(1/2))*x*c*d*(2*d*x+2*c+2)^(1/2)*(-d*x-c)^(1/2)
*(-2*d*x-2*c+2)^(1/2)-20*EllipticE(1/2*(-2*d*x-2*c+2)^(1/2),2^(1/2))*x*c*d*(d*x+
c)^(1/2)*(2*d*x+2*c+2)^(1/2)*(-2*d*x-2*c+2)^(1/2)-24*x^3*c*d^3+7*(-2*d*x-2*c+2)^
(1/2)*(2*d*x+2*c+2)^(1/2)*(-d*x-c)^(1/2)*EllipticE(1/2*(2*d*x+2*c+2)^(1/2),2^(1/
2))*c^2-10*(-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))*c^2-36*x^2*c^2*d^2-24*c^3*d*x+4*d^2*x^2-6*c^4+8*c*
d*x+4*c^2+2)*(-d^2*x^2-2*c*d*x-c^2+1)^(1/2)*(e*(d*x+c))^(1/2)/e^4/(d*x+c)^3/(d^2
*x^2+2*c*d*x+c^2-1)/d

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{-d^{2} x^{2} - 2 \, c d x - c^{2} + 1}{\left (d e x + c e\right )}^{\frac{7}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(-d^2*x^2 - 2*c*d*x - c^2 + 1)*(d*e*x + c*e)^(7/2)),x, algorithm="maxima")

[Out]

integrate(1/(sqrt(-d^2*x^2 - 2*c*d*x - c^2 + 1)*(d*e*x + c*e)^(7/2)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(-d^2*x^2 - 2*c*d*x - c^2 + 1)*(d*e*x + c*e)^(7/2)),x, algorithm="fricas")

[Out]

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

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(d*e*x+c*e)**(7/2)/(-d**2*x**2-2*c*d*x-c**2+1)**(1/2),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{-d^{2} x^{2} - 2 \, c d x - c^{2} + 1}{\left (d e x + c e\right )}^{\frac{7}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(-d^2*x^2 - 2*c*d*x - c^2 + 1)*(d*e*x + c*e)^(7/2)),x, algorithm="giac")

[Out]

integrate(1/(sqrt(-d^2*x^2 - 2*c*d*x - c^2 + 1)*(d*e*x + c*e)^(7/2)), x)