3.1405 \(\int \frac {1}{(c e+d e x)^{9/2} \sqrt {1-c^2-2 c d x-d^2 x^2}} \, dx\)

Optimal. Leaf size=126 \[ -\frac {10 \sqrt {-c^2-2 c d x-d^2 x^2+1}}{21 d e^3 (c e+d e x)^{3/2}}-\frac {2 \sqrt {-c^2-2 c d x-d^2 x^2+1}}{7 d e (c e+d e x)^{7/2}}+\frac {10 F\left (\left .\sin ^{-1}\left (\frac {\sqrt {c e+d x e}}{\sqrt {e}}\right )\right |-1\right )}{21 d e^{9/2}} \]

[Out]

10/21*EllipticF((d*e*x+c*e)^(1/2)/e^(1/2),I)/d/e^(9/2)-2/7*(-d^2*x^2-2*c*d*x-c^2+1)^(1/2)/d/e/(d*e*x+c*e)^(7/2
)-10/21*(-d^2*x^2-2*c*d*x-c^2+1)^(1/2)/d/e^3/(d*e*x+c*e)^(3/2)

________________________________________________________________________________________

Rubi [A]  time = 0.09, antiderivative size = 126, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 37, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.081, Rules used = {693, 689, 221} \[ -\frac {10 \sqrt {-c^2-2 c d x-d^2 x^2+1}}{21 d e^3 (c e+d e x)^{3/2}}-\frac {2 \sqrt {-c^2-2 c d x-d^2 x^2+1}}{7 d e (c e+d e x)^{7/2}}+\frac {10 F\left (\left .\sin ^{-1}\left (\frac {\sqrt {c e+d x e}}{\sqrt {e}}\right )\right |-1\right )}{21 d e^{9/2}} \]

Antiderivative was successfully verified.

[In]

Int[1/((c*e + d*e*x)^(9/2)*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])/(7*d*e*(c*e + d*e*x)^(7/2)) - (10*Sqrt[1 - c^2 - 2*c*d*x - d^2*x^2])/(2
1*d*e^3*(c*e + d*e*x)^(3/2)) + (10*EllipticF[ArcSin[Sqrt[c*e + d*e*x]/Sqrt[e]], -1])/(21*d*e^(9/2))

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 689

Int[1/(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[1/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 693

Int[((d_) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(-2*b*d*(d + e*x)^(m
+ 1)*(a + b*x + c*x^2)^(p + 1))/(d^2*(m + 1)*(b^2 - 4*a*c)), x] + Dist[(b^2*(m + 2*p + 3))/(d^2*(m + 1)*(b^2
- 4*a*c)), Int[(d + e*x)^(m + 2)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[b^2 - 4*a*
c, 0] && EqQ[2*c*d - b*e, 0] && NeQ[m + 2*p + 3, 0] && LtQ[m, -1] && (IntegerQ[2*p] || (IntegerQ[m] && Rationa
lQ[p]) || IntegerQ[(m + 2*p + 3)/2])

Rubi steps

\begin {align*} \int \frac {1}{(c e+d e x)^{9/2} \sqrt {1-c^2-2 c d x-d^2 x^2}} \, dx &=-\frac {2 \sqrt {1-c^2-2 c d x-d^2 x^2}}{7 d e (c e+d e x)^{7/2}}+\frac {5 \int \frac {1}{(c e+d e x)^{5/2} \sqrt {1-c^2-2 c d x-d^2 x^2}} \, dx}{7 e^2}\\ &=-\frac {2 \sqrt {1-c^2-2 c d x-d^2 x^2}}{7 d e (c e+d e x)^{7/2}}-\frac {10 \sqrt {1-c^2-2 c d x-d^2 x^2}}{21 d e^3 (c e+d e x)^{3/2}}+\frac {5 \int \frac {1}{\sqrt {c e+d e x} \sqrt {1-c^2-2 c d x-d^2 x^2}} \, dx}{21 e^4}\\ &=-\frac {2 \sqrt {1-c^2-2 c d x-d^2 x^2}}{7 d e (c e+d e x)^{7/2}}-\frac {10 \sqrt {1-c^2-2 c d x-d^2 x^2}}{21 d e^3 (c e+d e x)^{3/2}}+\frac {10 \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-\frac {x^4}{e^2}}} \, dx,x,\sqrt {c e+d e x}\right )}{21 d e^5}\\ &=-\frac {2 \sqrt {1-c^2-2 c d x-d^2 x^2}}{7 d e (c e+d e x)^{7/2}}-\frac {10 \sqrt {1-c^2-2 c d x-d^2 x^2}}{21 d e^3 (c e+d e x)^{3/2}}+\frac {10 F\left (\left .\sin ^{-1}\left (\frac {\sqrt {c e+d e x}}{\sqrt {e}}\right )\right |-1\right )}{21 d e^{9/2}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [C]  time = 0.03, size = 40, normalized size = 0.32 \[ -\frac {2 (c+d x) \, _2F_1\left (-\frac {7}{4},\frac {1}{2};-\frac {3}{4};(c+d x)^2\right )}{7 d (e (c+d x))^{9/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/((c*e + d*e*x)^(9/2)*Sqrt[1 - c^2 - 2*c*d*x - d^2*x^2]),x]

[Out]

(-2*(c + d*x)*Hypergeometric2F1[-7/4, 1/2, -3/4, (c + d*x)^2])/(7*d*(e*(c + d*x))^(9/2))

________________________________________________________________________________________

fricas [F]  time = 0.66, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {-d^{2} x^{2} - 2 \, c d x - c^{2} + 1} \sqrt {d e x + c e}}{d^{7} e^{5} x^{7} + 7 \, c d^{6} e^{5} x^{6} + {\left (21 \, c^{2} - 1\right )} d^{5} e^{5} x^{5} + 5 \, {\left (7 \, c^{3} - c\right )} d^{4} e^{5} x^{4} + 5 \, {\left (7 \, c^{4} - 2 \, c^{2}\right )} d^{3} e^{5} x^{3} + {\left (21 \, c^{5} - 10 \, c^{3}\right )} d^{2} e^{5} x^{2} + {\left (7 \, c^{6} - 5 \, c^{4}\right )} d e^{5} x + {\left (c^{7} - c^{5}\right )} e^{5}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(d*e*x+c*e)^(9/2)/(-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^7*e^5*x^7 + 7*c*d^6*e^5*x^6 + (21*c^2 - 1)*d
^5*e^5*x^5 + 5*(7*c^3 - c)*d^4*e^5*x^4 + 5*(7*c^4 - 2*c^2)*d^3*e^5*x^3 + (21*c^5 - 10*c^3)*d^2*e^5*x^2 + (7*c^
6 - 5*c^4)*d*e^5*x + (c^7 - c^5)*e^5), x)

________________________________________________________________________________________

giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {-d^{2} x^{2} - 2 \, c d x - c^{2} + 1} {\left (d e x + c e\right )}^{\frac {9}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(d*e*x+c*e)^(9/2)/(-d^2*x^2-2*c*d*x-c^2+1)^(1/2),x, algorithm="giac")

[Out]

integrate(1/(sqrt(-d^2*x^2 - 2*c*d*x - c^2 + 1)*(d*e*x + c*e)^(9/2)), x)

________________________________________________________________________________________

maple [B]  time = 0.12, size = 351, normalized size = 2.79 \[ -\frac {\left (10 d^{4} x^{4}+40 c \,d^{3} x^{3}+5 \sqrt {2 d x +2 c +2}\, \sqrt {-d x -c}\, \sqrt {-2 d x -2 c +2}\, d^{3} x^{3} \EllipticF \left (\frac {\sqrt {2 d x +2 c +2}}{2}, \sqrt {2}\right )+60 c^{2} d^{2} x^{2}+15 \sqrt {2 d x +2 c +2}\, \sqrt {-d x -c}\, \sqrt {-2 d x -2 c +2}\, c \,d^{2} x^{2} \EllipticF \left (\frac {\sqrt {2 d x +2 c +2}}{2}, \sqrt {2}\right )+40 c^{3} d x +15 \sqrt {2 d x +2 c +2}\, \sqrt {-d x -c}\, \sqrt {-2 d x -2 c +2}\, c^{2} d x \EllipticF \left (\frac {\sqrt {2 d x +2 c +2}}{2}, \sqrt {2}\right )+10 c^{4}+5 \sqrt {2 d x +2 c +2}\, \sqrt {-d x -c}\, \sqrt {-2 d x -2 c +2}\, c^{3} \EllipticF \left (\frac {\sqrt {2 d x +2 c +2}}{2}, \sqrt {2}\right )-4 d^{2} x^{2}-8 c d x -4 c^{2}-6\right ) \sqrt {-d^{2} x^{2}-2 c d x -c^{2}+1}\, \sqrt {\left (d x +c \right ) e}}{21 \left (d x +c \right )^{4} \left (d^{2} x^{2}+2 c d x +c^{2}-1\right ) d \,e^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(d*e*x+c*e)^(9/2)/(-d^2*x^2-2*c*d*x-c^2+1)^(1/2),x)

[Out]

-1/21*(5*(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*d^3+15*(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*d^2+15*(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^2*d+10*d^4*x^4+5*(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^3+40*x^3*c*d^3+60*c^2*d^2*x^2+40*x*c^3*d-4*d^2*x^2+10*c^4-8*c*d*x-4*c^2-6)/e^5*(-d^2*x^2-2*c*d*x-c
^2+1)^(1/2)*((d*x+c)*e)^(1/2)/(d*x+c)^4/(d^2*x^2+2*c*d*x+c^2-1)/d

________________________________________________________________________________________

maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {-d^{2} x^{2} - 2 \, c d x - c^{2} + 1} {\left (d e x + c e\right )}^{\frac {9}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(d*e*x+c*e)^(9/2)/(-d^2*x^2-2*c*d*x-c^2+1)^(1/2),x, algorithm="maxima")

[Out]

integrate(1/(sqrt(-d^2*x^2 - 2*c*d*x - c^2 + 1)*(d*e*x + c*e)^(9/2)), x)

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {1}{{\left (c\,e+d\,e\,x\right )}^{9/2}\,\sqrt {-c^2-2\,c\,d\,x-d^2\,x^2+1}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((c*e + d*e*x)^(9/2)*(1 - d^2*x^2 - 2*c*d*x - c^2)^(1/2)),x)

[Out]

int(1/((c*e + d*e*x)^(9/2)*(1 - d^2*x^2 - 2*c*d*x - c^2)^(1/2)), x)

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (e \left (c + d x\right )\right )^{\frac {9}{2}} \sqrt {- \left (c + d x - 1\right ) \left (c + d x + 1\right )}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(d*e*x+c*e)**(9/2)/(-d**2*x**2-2*c*d*x-c**2+1)**(1/2),x)

[Out]

Integral(1/((e*(c + d*x))**(9/2)*sqrt(-(c + d*x - 1)*(c + d*x + 1))), x)

________________________________________________________________________________________