\(\int \frac {1}{(c e+d e x)^{3/2} \sqrt {1-c^2-2 c d x-d^2 x^2}} \, dx\) [306]

Optimal result

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

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

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 \[ \int \frac {1}{(c e+d e x)^{3/2} \sqrt {1-c^2-2 c d x-d^2 x^2}} \, dx=-\frac {2 (c+d x) \operatorname {Hypergeometric2F1}\left (-\frac {1}{4},\frac {1}{2},\frac {3}{4},(c+d x)^2\right )}{d (e (c+d x))^{3/2}} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.30 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.02, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.189, Rules used = {1117, 1114, 836, 27, 762, 1389, 327}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

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

\(\Big \downarrow \) 1117

\(\displaystyle -\frac {\int \frac {\sqrt {c e+d x e}}{\sqrt {-c^2-2 d x c-d^2 x^2+1}}dx}{e^2}-\frac {2 \sqrt {-c^2-2 c d x-d^2 x^2+1}}{d e \sqrt {c e+d e x}}\)

\(\Big \downarrow \) 1114

\(\displaystyle -\frac {2 \int \frac {c e+d x e}{\sqrt {1-\frac {(c e+d x e)^2}{e^2}}}d\sqrt {c e+d x e}}{d e^3}-\frac {2 \sqrt {-c^2-2 c d x-d^2 x^2+1}}{d e \sqrt {c e+d e x}}\)

\(\Big \downarrow \) 836

\(\displaystyle -\frac {2 \left (e \int \frac {c e+d x e+e}{e \sqrt {1-\frac {(c e+d x e)^2}{e^2}}}d\sqrt {c e+d x e}-e \int \frac {1}{\sqrt {1-\frac {(c e+d x e)^2}{e^2}}}d\sqrt {c e+d x e}\right )}{d e^3}-\frac {2 \sqrt {-c^2-2 c d x-d^2 x^2+1}}{d e \sqrt {c e+d e x}}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {2 \left (\int \frac {c e+d x e+e}{\sqrt {1-\frac {(c e+d x e)^2}{e^2}}}d\sqrt {c e+d x e}-e \int \frac {1}{\sqrt {1-\frac {(c e+d x e)^2}{e^2}}}d\sqrt {c e+d x e}\right )}{d e^3}-\frac {2 \sqrt {-c^2-2 c d x-d^2 x^2+1}}{d e \sqrt {c e+d e x}}\)

\(\Big \downarrow \) 762

\(\displaystyle -\frac {2 \left (\int \frac {c e+d x e+e}{\sqrt {1-\frac {(c e+d x e)^2}{e^2}}}d\sqrt {c e+d x e}-e^{3/2} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c e+d x e}}{\sqrt {e}}\right ),-1\right )\right )}{d e^3}-\frac {2 \sqrt {-c^2-2 c d x-d^2 x^2+1}}{d e \sqrt {c e+d e x}}\)

\(\Big \downarrow \) 1389

\(\displaystyle -\frac {2 \left (e \int \frac {\sqrt {\frac {c e+d x e}{e}+1}}{\sqrt {1-\frac {c e+d x e}{e}}}d\sqrt {c e+d x e}-e^{3/2} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c e+d x e}}{\sqrt {e}}\right ),-1\right )\right )}{d e^3}-\frac {2 \sqrt {-c^2-2 c d x-d^2 x^2+1}}{d e \sqrt {c e+d e x}}\)

\(\Big \downarrow \) 327

\(\displaystyle -\frac {2 \left (e^{3/2} E\left (\left .\arcsin \left (\frac {\sqrt {c e+d x e}}{\sqrt {e}}\right )\right |-1\right )-e^{3/2} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c e+d x e}}{\sqrt {e}}\right ),-1\right )\right )}{d e^3}-\frac {2 \sqrt {-c^2-2 c d x-d^2 x^2+1}}{d e \sqrt {c e+d e x}}\)

Input:

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

Output:

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

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 327

Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[ 
(Sqrt[a]/(Sqrt[c]*Rt[-d/c, 2]))*EllipticE[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 762

Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> Simp[(1/(Sqrt[a]*Rt[-b/a, 4]) 
)*EllipticF[ArcSin[Rt[-b/a, 4]*x], -1], x] /; FreeQ[{a, b}, x] && NegQ[b/a] 
 && GtQ[a, 0]
 

rule 836

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

rule 1114

Int[Sqrt[(d_) + (e_.)*(x_)]/Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symb 
ol] :> Simp[(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] && EqQ[2*c*d - b*e, 0] && LtQ[c/(b^2 - 4*a*c), 0]
 

rule 1117

Int[((d_) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_S 
ymbol] :> 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] + Simp[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] && EqQ[2*c*d - b*e, 0] && NeQ[m + 2*p + 3, 0] && LtQ[m, -1] & 
& (IntegerQ[2*p] || (IntegerQ[m] && RationalQ[p]) || IntegerQ[(m + 2*p + 3) 
/2])
 

rule 1389

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

Maple [A] (verified)

Time = 2.93 (sec) , antiderivative size = 143, normalized size of antiderivative = 1.34

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

Input:

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

Output:

(-(-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 [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.87 \[ \int \frac {1}{(c e+d e x)^{3/2} \sqrt {1-c^2-2 c d x-d^2 x^2}} \, dx=-\frac {2 \, {\left (\sqrt {-d^{3} e} {\left (d x + c\right )} {\rm weierstrassZeta}\left (\frac {4}{d^{2}}, 0, {\rm weierstrassPInverse}\left (\frac {4}{d^{2}}, 0, \frac {d x + c}{d}\right )\right ) + \sqrt {-d^{2} x^{2} - 2 \, c d x - c^{2} + 1} \sqrt {d e x + c e} d\right )}}{d^{3} e^{2} x + c d^{2} e^{2}} \] Input:

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")
 

Output:

-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]

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

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

Output:

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

Maxima [F]

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

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")
 

Output:

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

Giac [F]

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

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")
 

Output:

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

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

Output:

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

Reduce [F]

\[ \int \frac {1}{(c e+d e x)^{3/2} \sqrt {1-c^2-2 c d x-d^2 x^2}} \, dx=-\frac {\sqrt {e}\, \left (\int \frac {\sqrt {d x +c}\, \sqrt {-d^{2} x^{2}-2 c d x -c^{2}+1}}{d^{4} x^{4}+4 c \,d^{3} x^{3}+6 c^{2} d^{2} x^{2}+4 c^{3} d x +c^{4}-d^{2} x^{2}-2 c d x -c^{2}}d x \right )}{e^{2}} \] Input:

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

Output:

( - sqrt(e)*int((sqrt(c + d*x)*sqrt( - c**2 - 2*c*d*x - d**2*x**2 + 1))/(c 
**4 + 4*c**3*d*x + 6*c**2*d**2*x**2 - c**2 + 4*c*d**3*x**3 - 2*c*d*x + d** 
4*x**4 - d**2*x**2),x))/e**2