3.1395 \(\int \frac{1}{(c e+d e x)^{13/2} \sqrt{1-c^2-2 c d x-d^2 x^2}} \, dx\)

Optimal. Leaf size=172 \[ -\frac{30 \sqrt{-c^2-2 c d x-d^2 x^2+1}}{77 d e^5 (c e+d e x)^{3/2}}-\frac{18 \sqrt{-c^2-2 c d x-d^2 x^2+1}}{77 d e^3 (c e+d e x)^{7/2}}-\frac{2 \sqrt{-c^2-2 c d x-d^2 x^2+1}}{11 d e (c e+d e x)^{11/2}}+\frac{30 F\left (\left .\sin ^{-1}\left (\frac{\sqrt{c e+d x e}}{\sqrt{e}}\right )\right |-1\right )}{77 d e^{13/2}} \]

[Out]

(-2*Sqrt[1 - c^2 - 2*c*d*x - d^2*x^2])/(11*d*e*(c*e + d*e*x)^(11/2)) - (18*Sqrt[
1 - c^2 - 2*c*d*x - d^2*x^2])/(77*d*e^3*(c*e + d*e*x)^(7/2)) - (30*Sqrt[1 - c^2
- 2*c*d*x - d^2*x^2])/(77*d*e^5*(c*e + d*e*x)^(3/2)) + (30*EllipticF[ArcSin[Sqrt
[c*e + d*e*x]/Sqrt[e]], -1])/(77*d*e^(13/2))

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Rubi [A]  time = 0.357219, antiderivative size = 172, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 37, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.081 \[ -\frac{30 \sqrt{-c^2-2 c d x-d^2 x^2+1}}{77 d e^5 (c e+d e x)^{3/2}}-\frac{18 \sqrt{-c^2-2 c d x-d^2 x^2+1}}{77 d e^3 (c e+d e x)^{7/2}}-\frac{2 \sqrt{-c^2-2 c d x-d^2 x^2+1}}{11 d e (c e+d e x)^{11/2}}+\frac{30 F\left (\left .\sin ^{-1}\left (\frac{\sqrt{c e+d x e}}{\sqrt{e}}\right )\right |-1\right )}{77 d e^{13/2}} \]

Antiderivative was successfully verified.

[In]  Int[1/((c*e + d*e*x)^(13/2)*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])/(11*d*e*(c*e + d*e*x)^(11/2)) - (18*Sqrt[
1 - c^2 - 2*c*d*x - d^2*x^2])/(77*d*e^3*(c*e + d*e*x)^(7/2)) - (30*Sqrt[1 - c^2
- 2*c*d*x - d^2*x^2])/(77*d*e^5*(c*e + d*e*x)^(3/2)) + (30*EllipticF[ArcSin[Sqrt
[c*e + d*e*x]/Sqrt[e]], -1])/(77*d*e^(13/2))

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Rubi in Sympy [A]  time = 83.5431, size = 156, normalized size = 0.91 \[ - \frac{2 \sqrt{- c^{2} - 2 c d x - d^{2} x^{2} + 1}}{11 d e \left (c e + d e x\right )^{\frac{11}{2}}} - \frac{18 \sqrt{- c^{2} - 2 c d x - d^{2} x^{2} + 1}}{77 d e^{3} \left (c e + d e x\right )^{\frac{7}{2}}} - \frac{30 \sqrt{- c^{2} - 2 c d x - d^{2} x^{2} + 1}}{77 d e^{5} \left (c e + d e x\right )^{\frac{3}{2}}} + \frac{30 F\left (\operatorname{asin}{\left (\frac{\sqrt{c e + d e x}}{\sqrt{e}} \right )}\middle | -1\right )}{77 d e^{\frac{13}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(d*e*x+c*e)**(13/2)/(-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)/(11*d*e*(c*e + d*e*x)**(11/2)) - 18*sqr
t(-c**2 - 2*c*d*x - d**2*x**2 + 1)/(77*d*e**3*(c*e + d*e*x)**(7/2)) - 30*sqrt(-c
**2 - 2*c*d*x - d**2*x**2 + 1)/(77*d*e**5*(c*e + d*e*x)**(3/2)) + 30*elliptic_f(
asin(sqrt(c*e + d*e*x)/sqrt(e)), -1)/(77*d*e**(13/2))

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Mathematica [A]  time = 0.284468, size = 119, normalized size = 0.69 \[ \frac{(c+d x)^{13/2} \left (-\frac{2 \left (1-(c+d x)^2\right ) \left (15 (c+d x)^4+9 (c+d x)^2+7\right )}{(c+d x)^{11/2}}-30 (c+d x) \sqrt{1-\frac{1}{(c+d x)^2}} F\left (\left .\sin ^{-1}\left (\frac{1}{\sqrt{c+d x}}\right )\right |-1\right )\right )}{77 d \sqrt{1-(c+d x)^2} (e (c+d x))^{13/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((c*e + d*e*x)^(13/2)*Sqrt[1 - c^2 - 2*c*d*x - d^2*x^2]),x]

[Out]

((c + d*x)^(13/2)*((-2*(1 - (c + d*x)^2)*(7 + 9*(c + d*x)^2 + 15*(c + d*x)^4))/(
c + d*x)^(11/2) - 30*(c + d*x)*Sqrt[1 - (c + d*x)^(-2)]*EllipticF[ArcSin[1/Sqrt[
c + d*x]], -1]))/(77*d*(e*(c + d*x))^(13/2)*Sqrt[1 - (c + d*x)^2])

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Maple [B]  time = 0.08, size = 860, normalized size = 5. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(d*e*x+c*e)^(13/2)/(-d^2*x^2-2*c*d*x-c^2+1)^(1/2),x)

[Out]

-1/231*(-42+820*(-2*d*x-2*c+2)^(1/2)*(d*x+c)^(1/2)*(2*d*x+2*c+2)^(1/2)*EllipticF
(1/2*(-2*d*x-2*c+2)^(1/2),2^(1/2))*x^4*c*d^4+1045*(-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))*x^4*c*d^4+1
640*(-2*d*x-2*c+2)^(1/2)*(d*x+c)^(1/2)*(2*d*x+2*c+2)^(1/2)*EllipticF(1/2*(-2*d*x
-2*c+2)^(1/2),2^(1/2))*x^3*c^2*d^3+2090*(-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))*x^3*c^2*d^3+1640*(-2*
d*x-2*c+2)^(1/2)*(d*x+c)^(1/2)*(2*d*x+2*c+2)^(1/2)*EllipticF(1/2*(-2*d*x-2*c+2)^
(1/2),2^(1/2))*x^2*c^3*d^2+2090*(-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))*x^2*c^3*d^2+820*(-2*d*x-2*c+2
)^(1/2)*(d*x+c)^(1/2)*(2*d*x+2*c+2)^(1/2)*EllipticF(1/2*(-2*d*x-2*c+2)^(1/2),2^(
1/2))*x*c^4*d+1045*(-2*d*x-2*c+2)^(1/2)*(2*d*x+2*c+2)^(1/2)*(-d*x-c)^(1/2)*Ellip
ticF(1/2*(2*d*x+2*c+2)^(1/2),2^(1/2))*x*c^4*d-216*x^2*c^2*d^2-144*c^3*d*x-36*d^4
*x^4-24*c*d*x+90*c^6+540*x^5*c*d^5+90*d^6*x^6-144*x^3*c*d^3-12*d^2*x^2+164*(-2*d
*x-2*c+2)^(1/2)*(d*x+c)^(1/2)*(2*d*x+2*c+2)^(1/2)*EllipticF(1/2*(-2*d*x-2*c+2)^(
1/2),2^(1/2))*c^5+209*(-2*d*x-2*c+2)^(1/2)*(2*d*x+2*c+2)^(1/2)*(-d*x-c)^(1/2)*El
lipticF(1/2*(2*d*x+2*c+2)^(1/2),2^(1/2))*c^5-12*c^2-36*c^4+209*(-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)
)*x^5*d^5+164*(-2*d*x-2*c+2)^(1/2)*(d*x+c)^(1/2)*(2*d*x+2*c+2)^(1/2)*EllipticF(1
/2*(-2*d*x-2*c+2)^(1/2),2^(1/2))*x^5*d^5+1350*x^4*c^2*d^4+540*x*c^5*d+1350*x^2*c
^4*d^2+1800*x^3*c^3*d^3)/e^7*(-d^2*x^2-2*c*d*x-c^2+1)^(1/2)*(e*(d*x+c))^(1/2)/(d
*x+c)^6/(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. \[ \int \frac{1}{\sqrt{-d^{2} x^{2} - 2 \, c d x - c^{2} + 1}{\left (d e x + c e\right )}^{\frac{13}{2}}}\,{d x} \]

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)^(13/2)),x, algorithm="maxima")

[Out]

integrate(1/(sqrt(-d^2*x^2 - 2*c*d*x - c^2 + 1)*(d*e*x + c*e)^(13/2)), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{{\left (d^{6} e^{6} x^{6} + 6 \, c d^{5} e^{6} x^{5} + 15 \, c^{2} d^{4} e^{6} x^{4} + 20 \, c^{3} d^{3} e^{6} x^{3} + 15 \, c^{4} d^{2} e^{6} x^{2} + 6 \, c^{5} d e^{6} x + c^{6} e^{6}\right )} \sqrt{-d^{2} x^{2} - 2 \, c d x - c^{2} + 1} \sqrt{d e x + c e}}, x\right ) \]

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)^(13/2)),x, algorithm="fricas")

[Out]

integral(1/((d^6*e^6*x^6 + 6*c*d^5*e^6*x^5 + 15*c^2*d^4*e^6*x^4 + 20*c^3*d^3*e^6
*x^3 + 15*c^4*d^2*e^6*x^2 + 6*c^5*d*e^6*x + c^6*e^6)*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)**(13/2)/(-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. \[ \int \frac{1}{\sqrt{-d^{2} x^{2} - 2 \, c d x - c^{2} + 1}{\left (d e x + c e\right )}^{\frac{13}{2}}}\,{d x} \]

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)^(13/2)),x, algorithm="giac")

[Out]

integrate(1/(sqrt(-d^2*x^2 - 2*c*d*x - c^2 + 1)*(d*e*x + c*e)^(13/2)), x)