\(\int \frac {1}{(c e+d e x)^{3/2} \sqrt {1-c^2-2 c d x-d^2 x^2}} \, dx\) [1410]

Optimal result

Integrand size = 37, antiderivative size = 107 \[ \int \frac {1}{(c e+d e x)^{3/2} \sqrt {1-c^2-2 c d x-d^2 x^2}} \, dx=-\frac {2 \sqrt {1-c^2-2 c d x-d^2 x^2}}{d e \sqrt {c e+d e x}}-\frac {2 E\left (\left .\arcsin \left (\frac {\sqrt {c e+d e x}}{\sqrt {e}}\right )\right |-1\right )}{d e^{3/2}}+\frac {2 \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c e+d e x}}{\sqrt {e}}\right ),-1\right )}{d e^{3/2}} \]

[Out]

-2*EllipticE((d*e*x+c*e)^(1/2)/e^(1/2),I)/d/e^(3/2)+2*EllipticF((d*e*x+c*e)^(1/2)/e^(1/2),I)/d/e^(3/2)-2*(-d^2
*x^2-2*c*d*x-c^2+1)^(1/2)/d/e/(d*e*x+c*e)^(1/2)

Rubi [A] (verified)

Time = 0.06 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.162, Rules used = {707, 704, 313, 227, 1213, 435} \[ \int \frac {1}{(c e+d e x)^{3/2} \sqrt {1-c^2-2 c d x-d^2 x^2}} \, dx=\frac {2 \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c e+d x e}}{\sqrt {e}}\right ),-1\right )}{d e^{3/2}}-\frac {2 E\left (\left .\arcsin \left (\frac {\sqrt {c e+d x e}}{\sqrt {e}}\right )\right |-1\right )}{d e^{3/2}}-\frac {2 \sqrt {-c^2-2 c d x-d^2 x^2+1}}{d e \sqrt {c e+d e x}} \]

[In]

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

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 313

Int[(x_)^2/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> With[{q = Rt[-b/a, 2]}, Dist[-q^(-1), Int[1/Sqrt[a + b*x^4]
, x], x] + Dist[1/q, Int[(1 + q*x^2)/Sqrt[a + b*x^4], x], x]] /; FreeQ[{a, b}, x] && NegQ[b/a]

Rule 435

Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[(Sqrt[a]/(Sqrt[c]*Rt[-d/c, 2]))*Ell
ipticE[ArcSin[Rt[-d/c, 2]*x], b*(c/(a*d))], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 0
]

Rule 704

Int[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[x^2/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])

Rule 1213

Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (c_.)*(x_)^4], x_Symbol] :> Dist[d/Sqrt[a], Int[Sqrt[1 + e*(x^2/d)]/Sqrt
[1 - e*(x^2/d)], x], x] /; FreeQ[{a, c, d, e}, x] && NegQ[c/a] && EqQ[c*d^2 + a*e^2, 0] && GtQ[a, 0]

Rubi steps \begin{align*} \text {integral}& = -\frac {2 \sqrt {1-c^2-2 c d x-d^2 x^2}}{d e \sqrt {c e+d e x}}-\frac {\int \frac {\sqrt {c e+d e x}}{\sqrt {1-c^2-2 c d x-d^2 x^2}} \, dx}{e^2} \\ & = -\frac {2 \sqrt {1-c^2-2 c d x-d^2 x^2}}{d e \sqrt {c e+d e x}}-\frac {2 \text {Subst}\left (\int \frac {x^2}{\sqrt {1-\frac {x^4}{e^2}}} \, dx,x,\sqrt {c e+d e x}\right )}{d e^3} \\ & = -\frac {2 \sqrt {1-c^2-2 c d x-d^2 x^2}}{d e \sqrt {c e+d e x}}+\frac {2 \text {Subst}\left (\int \frac {1}{\sqrt {1-\frac {x^4}{e^2}}} \, dx,x,\sqrt {c e+d e x}\right )}{d e^2}-\frac {2 \text {Subst}\left (\int \frac {1+\frac {x^2}{e}}{\sqrt {1-\frac {x^4}{e^2}}} \, dx,x,\sqrt {c e+d e x}\right )}{d e^2} \\ & = -\frac {2 \sqrt {1-c^2-2 c d x-d^2 x^2}}{d e \sqrt {c e+d e x}}+\frac {2 F\left (\left .\sin ^{-1}\left (\frac {\sqrt {c e+d e x}}{\sqrt {e}}\right )\right |-1\right )}{d e^{3/2}}-\frac {2 \text {Subst}\left (\int \frac {\sqrt {1+\frac {x^2}{e}}}{\sqrt {1-\frac {x^2}{e}}} \, dx,x,\sqrt {c e+d e x}\right )}{d e^2} \\ & = -\frac {2 \sqrt {1-c^2-2 c d x-d^2 x^2}}{d e \sqrt {c e+d e x}}-\frac {2 E\left (\left .\sin ^{-1}\left (\frac {\sqrt {c e+d e x}}{\sqrt {e}}\right )\right |-1\right )}{d e^{3/2}}+\frac {2 F\left (\left .\sin ^{-1}\left (\frac {\sqrt {c e+d e x}}{\sqrt {e}}\right )\right |-1\right )}{d e^{3/2}} \\ \end{align*}

Mathematica [C] (verified)

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

Time = 10.02 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.36 \[ \int \frac {1}{(c e+d e x)^{3/2} \sqrt {1-c^2-2 c d x-d^2 x^2}} \, dx=-\frac {2 (c+d x) \operatorname {Hypergeometric2F1}\left (-\frac {1}{4},\frac {1}{2},\frac {3}{4},(c+d x)^2\right )}{d (e (c+d x))^{3/2}} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 2.66 (sec) , antiderivative size = 143, normalized size of antiderivative = 1.34

method	result	size
default	\(\frac {\left (-\sqrt {-2 d x -2 c +2}\, \sqrt {d x +c}\, \sqrt {2 d x +2 c +2}\, E\left (\frac {\sqrt {-2 d x -2 c +2}}{2}, \sqrt {2}\right )-2 d^{2} x^{2}-4 c d x -2 c^{2}+2\right ) \sqrt {e \left (d x +c \right )}\, \sqrt {-d^{2} x^{2}-2 c d x -c^{2}+1}}{d \,e^{2} \left (d^{3} x^{3}+3 c \,d^{2} x^{2}+3 c^{2} d x +c^{3}-d x -c \right )}\)	\(143\)
elliptic	\(\frac {\sqrt {-e \left (d x +c \right ) \left (d^{2} x^{2}+2 c d x +c^{2}-1\right )}\, \left (-\frac {2 \left (-d^{3} e \,x^{2}-2 c \,d^{2} e x -d e \,c^{2}+d e \right )}{d^{2} e^{2} \sqrt {\left (x +\frac {c}{d}\right ) \left (-d^{3} e \,x^{2}-2 c \,d^{2} e x -d e \,c^{2}+d e \right )}}-\frac {2 c \left (-\frac {c +1}{d}+\frac {c -1}{d}\right ) \sqrt {\frac {x +\frac {c -1}{d}}{-\frac {c +1}{d}+\frac {c -1}{d}}}\, \sqrt {\frac {x +\frac {c}{d}}{-\frac {c -1}{d}+\frac {c}{d}}}\, \sqrt {\frac {x +\frac {c +1}{d}}{-\frac {c -1}{d}+\frac {c +1}{d}}}\, F\left (\sqrt {\frac {x +\frac {c -1}{d}}{-\frac {c +1}{d}+\frac {c -1}{d}}}, \sqrt {\frac {-\frac {c -1}{d}+\frac {c +1}{d}}{-\frac {c -1}{d}+\frac {c}{d}}}\right )}{e \sqrt {-d^{3} e \,x^{3}-3 c \,d^{2} e \,x^{2}-3 c^{2} d e x -e \,c^{3}+d e x +c e}}-\frac {2 d \left (-\frac {c +1}{d}+\frac {c -1}{d}\right ) \sqrt {\frac {x +\frac {c -1}{d}}{-\frac {c +1}{d}+\frac {c -1}{d}}}\, \sqrt {\frac {x +\frac {c}{d}}{-\frac {c -1}{d}+\frac {c}{d}}}\, \sqrt {\frac {x +\frac {c +1}{d}}{-\frac {c -1}{d}+\frac {c +1}{d}}}\, \left (\left (-\frac {c -1}{d}+\frac {c}{d}\right ) E\left (\sqrt {\frac {x +\frac {c -1}{d}}{-\frac {c +1}{d}+\frac {c -1}{d}}}, \sqrt {\frac {-\frac {c -1}{d}+\frac {c +1}{d}}{-\frac {c -1}{d}+\frac {c}{d}}}\right )-\frac {c F\left (\sqrt {\frac {x +\frac {c -1}{d}}{-\frac {c +1}{d}+\frac {c -1}{d}}}, \sqrt {\frac {-\frac {c -1}{d}+\frac {c +1}{d}}{-\frac {c -1}{d}+\frac {c}{d}}}\right )}{d}\right )}{e \sqrt {-d^{3} e \,x^{3}-3 c \,d^{2} e \,x^{2}-3 c^{2} d e x -e \,c^{3}+d e x +c e}}\right )}{\sqrt {e \left (d x +c \right )}\, \sqrt {-d^{2} x^{2}-2 c d x -c^{2}+1}}\)	\(657\)

[In]

int(1/(d*e*x+c*e)^(3/2)/(-d^2*x^2-2*c*d*x-c^2+1)^(1/2),x,method=_RETURNVERBOSE)

[Out]

(-(-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))-2*d^2*x^2
-4*c*d*x-2*c^2+2)*(e*(d*x+c))^(1/2)*(-d^2*x^2-2*c*d*x-c^2+1)^(1/2)/d/e^2/(d^3*x^3+3*c*d^2*x^2+3*c^2*d*x+c^3-d*
x-c)

Fricas [C] (verification not implemented)

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

Time = 0.12 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.87 \[ \int \frac {1}{(c e+d e x)^{3/2} \sqrt {1-c^2-2 c d x-d^2 x^2}} \, dx=-\frac {2 \, {\left (\sqrt {-d^{3} e} {\left (d x + c\right )} {\rm weierstrassZeta}\left (\frac {4}{d^{2}}, 0, {\rm weierstrassPInverse}\left (\frac {4}{d^{2}}, 0, \frac {d x + c}{d}\right )\right ) + \sqrt {-d^{2} x^{2} - 2 \, c d x - c^{2} + 1} \sqrt {d e x + c e} d\right )}}{d^{3} e^{2} x + c d^{2} e^{2}} \]

[In]

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

[Out]

-2*(sqrt(-d^3*e)*(d*x + c)*weierstrassZeta(4/d^2, 0, weierstrassPInverse(4/d^2, 0, (d*x + c)/d)) + sqrt(-d^2*x
^2 - 2*c*d*x - c^2 + 1)*sqrt(d*e*x + c*e)*d)/(d^3*e^2*x + c*d^2*e^2)

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

Giac [F]

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

[In]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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