\(\int \frac {1}{(c e+d e x)^{9/2} \sqrt {1-c^2-2 c d x-d^2 x^2}} \, dx\) [1405]

Optimal result

Integrand size = 37, antiderivative size = 126 \[ \int \frac {1}{(c e+d e x)^{9/2} \sqrt {1-c^2-2 c d x-d^2 x^2}} \, dx=-\frac {2 \sqrt {1-c^2-2 c d x-d^2 x^2}}{7 d e (c e+d e x)^{7/2}}-\frac {10 \sqrt {1-c^2-2 c d x-d^2 x^2}}{21 d e^3 (c e+d e x)^{3/2}}+\frac {10 \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c e+d e x}}{\sqrt {e}}\right ),-1\right )}{21 d e^{9/2}} \]

[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] (verified)

Time = 0.06 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.081, Rules used = {707, 703, 227} \[ \int \frac {1}{(c e+d e x)^{9/2} \sqrt {1-c^2-2 c d x-d^2 x^2}} \, dx=\frac {10 \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c e+d x e}}{\sqrt {e}}\right ),-1\right )}{21 d e^{9/2}}-\frac {10 \sqrt {-c^2-2 c d x-d^2 x^2+1}}{21 d e^3 (c e+d e x)^{3/2}}-\frac {2 \sqrt {-c^2-2 c d x-d^2 x^2+1}}{7 d e (c e+d e x)^{7/2}} \]

[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 227

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 703

Int[1/(Sqrt[(d_) + (e_.)*(x_)]*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Symbol] :> Dist[(4/e)*Sqrt[-c/(b^2
- 4*a*c)], 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 707

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*} \text {integral}& = -\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 \text {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] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.

Time = 10.03 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.32 \[ \int \frac {1}{(c e+d e x)^{9/2} \sqrt {1-c^2-2 c d x-d^2 x^2}} \, dx=-\frac {2 (c+d x) \operatorname {Hypergeometric2F1}\left (-\frac {7}{4},\frac {1}{2},-\frac {3}{4},(c+d x)^2\right )}{7 d (e (c+d x))^{9/2}} \]

[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))

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(338\) vs. \(2(106)=212\).

Time = 4.35 (sec) , antiderivative size = 339, normalized size of antiderivative = 2.69

method	result	size
default	\(\frac {\left (5 \sqrt {-2 d x -2 c +2}\, \sqrt {d x +c}\, \sqrt {2 d x +2 c +2}\, F\left (\frac {\sqrt {-2 d x -2 c +2}}{2}, \sqrt {2}\right ) d^{3} x^{3}+15 \sqrt {-2 d x -2 c +2}\, \sqrt {d x +c}\, \sqrt {2 d x +2 c +2}\, F\left (\frac {\sqrt {-2 d x -2 c +2}}{2}, \sqrt {2}\right ) c \,d^{2} x^{2}+15 \sqrt {-2 d x -2 c +2}\, \sqrt {d x +c}\, \sqrt {2 d x +2 c +2}\, F\left (\frac {\sqrt {-2 d x -2 c +2}}{2}, \sqrt {2}\right ) c^{2} d x -10 d^{4} x^{4}+5 \sqrt {-2 d x -2 c +2}\, \sqrt {d x +c}\, \sqrt {2 d x +2 c +2}\, F\left (\frac {\sqrt {-2 d x -2 c +2}}{2}, \sqrt {2}\right ) c^{3}-40 c \,d^{3} x^{3}-60 c^{2} d^{2} x^{2}-40 c^{3} d x -10 c^{4}+4 d^{2} x^{2}+8 c d x +4 c^{2}+6\right ) \sqrt {-d^{2} x^{2}-2 c d x -c^{2}+1}\, \sqrt {e \left (d x +c \right )}}{21 e^{5} \left (d x +c \right )^{4} \left (d^{2} x^{2}+2 c d x +c^{2}-1\right ) d}\)	\(339\)
elliptic	\(\frac {\sqrt {-e \left (d x +c \right ) \left (d^{2} x^{2}+2 c d x +c^{2}-1\right )}\, \left (-\frac {2 \sqrt {-d^{3} e \,x^{3}-3 c \,d^{2} e \,x^{2}-3 c^{2} d e x -e \,c^{3}+d e x +c e}}{7 d^{5} e^{5} \left (x +\frac {c}{d}\right )^{4}}-\frac {10 \sqrt {-d^{3} e \,x^{3}-3 c \,d^{2} e \,x^{2}-3 c^{2} d e x -e \,c^{3}+d e x +c e}}{21 e^{5} d^{3} \left (x +\frac {c}{d}\right )^{2}}+\frac {10 \left (-\frac {c +1}{d}+\frac {c -1}{d}\right ) \sqrt {\frac {x +\frac {c -1}{d}}{-\frac {c +1}{d}+\frac {c -1}{d}}}\, \sqrt {\frac {x +\frac {c}{d}}{-\frac {c -1}{d}+\frac {c}{d}}}\, \sqrt {\frac {x +\frac {c +1}{d}}{-\frac {c -1}{d}+\frac {c +1}{d}}}\, F\left (\sqrt {\frac {x +\frac {c -1}{d}}{-\frac {c +1}{d}+\frac {c -1}{d}}}, \sqrt {\frac {-\frac {c -1}{d}+\frac {c +1}{d}}{-\frac {c -1}{d}+\frac {c}{d}}}\right )}{21 e^{4} \sqrt {-d^{3} e \,x^{3}-3 c \,d^{2} e \,x^{2}-3 c^{2} d e x -e \,c^{3}+d e x +c e}}\right )}{\sqrt {e \left (d x +c \right )}\, \sqrt {-d^{2} x^{2}-2 c d x -c^{2}+1}}\)	\(397\)

[In]

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

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

Fricas [C] (verification not implemented)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 0.10 (sec) , antiderivative size = 188, normalized size of antiderivative = 1.49 \[ \int \frac {1}{(c e+d e x)^{9/2} \sqrt {1-c^2-2 c d x-d^2 x^2}} \, dx=-\frac {2 \, {\left (5 \, {\left (d^{4} x^{4} + 4 \, c d^{3} x^{3} + 6 \, c^{2} d^{2} x^{2} + 4 \, c^{3} d x + c^{4}\right )} \sqrt {-d^{3} e} {\rm weierstrassPInverse}\left (\frac {4}{d^{2}}, 0, \frac {d x + c}{d}\right ) + {\left (5 \, d^{4} x^{2} + 10 \, c d^{3} x + {\left (5 \, c^{2} + 3\right )} d^{2}\right )} \sqrt {-d^{2} x^{2} - 2 \, c d x - c^{2} + 1} \sqrt {d e x + c e}\right )}}{21 \, {\left (d^{7} e^{5} x^{4} + 4 \, c d^{6} e^{5} x^{3} + 6 \, c^{2} d^{5} e^{5} x^{2} + 4 \, c^{3} d^{4} e^{5} x + c^{4} d^{3} e^{5}\right )}} \]

[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]

-2/21*(5*(d^4*x^4 + 4*c*d^3*x^3 + 6*c^2*d^2*x^2 + 4*c^3*d*x + c^4)*sqrt(-d^3*e)*weierstrassPInverse(4/d^2, 0,
(d*x + c)/d) + (5*d^4*x^2 + 10*c*d^3*x + (5*c^2 + 3)*d^2)*sqrt(-d^2*x^2 - 2*c*d*x - c^2 + 1)*sqrt(d*e*x + c*e)
)/(d^7*e^5*x^4 + 4*c*d^6*e^5*x^3 + 6*c^2*d^5*e^5*x^2 + 4*c^3*d^4*e^5*x + c^4*d^3*e^5)

Sympy [F]

\[ \int \frac {1}{(c e+d e x)^{9/2} \sqrt {1-c^2-2 c d x-d^2 x^2}} \, dx=\int \frac {1}{\left (e \left (c + d x\right )\right )^{\frac {9}{2}} \sqrt {- \left (c + d x - 1\right ) \left (c + d x + 1\right )}}\, dx \]

[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)

Maxima [F]

\[ \int \frac {1}{(c e+d e x)^{9/2} \sqrt {1-c^2-2 c d x-d^2 x^2}} \, dx=\int { \frac {1}{\sqrt {-d^{2} x^{2} - 2 \, c d x - c^{2} + 1} {\left (d e x + c e\right )}^{\frac {9}{2}}} \,d x } \]

[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)

Giac [F]

\[ \int \frac {1}{(c e+d e x)^{9/2} \sqrt {1-c^2-2 c d x-d^2 x^2}} \, dx=\int { \frac {1}{\sqrt {-d^{2} x^{2} - 2 \, c d x - c^{2} + 1} {\left (d e x + c e\right )}^{\frac {9}{2}}} \,d x } \]

[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)

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{(c e+d e x)^{9/2} \sqrt {1-c^2-2 c d x-d^2 x^2}} \, dx=\int \frac {1}{{\left (c\,e+d\,e\,x\right )}^{9/2}\,\sqrt {-c^2-2\,c\,d\,x-d^2\,x^2+1}} \,d x \]

[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)