\(\int \frac {1}{(c e+d e x)^{5/2} \sqrt {1-c^2-2 c d x-d^2 x^2}} \, dx\) [300]

Optimal result

Integrand size = 37, antiderivative size = 80 \[ \int \frac {1}{(c e+d e x)^{5/2} \sqrt {1-c^2-2 c d x-d^2 x^2}} \, dx=-\frac {2 \sqrt {1-c^2-2 c d x-d^2 x^2}}{3 d e (c e+d e x)^{3/2}}+\frac {2 \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c e+d e x}}{\sqrt {e}}\right ),-1\right )}{3 d e^{5/2}} \] Output:

-2/3*(-d^2*x^2-2*c*d*x-c^2+1)^(1/2)/d/e/(d*e*x+c*e)^(3/2)+2/3*EllipticF((d 
*e*x+c*e)^(1/2)/e^(1/2),I)/d/e^(5/2)
 

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 = 0.50 \[ \int \frac {1}{(c e+d e x)^{5/2} \sqrt {1-c^2-2 c d x-d^2 x^2}} \, dx=-\frac {2 (c+d x) \operatorname {Hypergeometric2F1}\left (-\frac {3}{4},\frac {1}{2},\frac {1}{4},(c+d x)^2\right )}{3 d (e (c+d x))^{5/2}} \] Input:

Integrate[1/((c*e + d*e*x)^(5/2)*Sqrt[1 - c^2 - 2*c*d*x - d^2*x^2]),x]
 

Output:

(-2*(c + d*x)*Hypergeometric2F1[-3/4, 1/2, 1/4, (c + d*x)^2])/(3*d*(e*(c + 
 d*x))^(5/2))
 

Rubi [A] (verified)

Time = 0.23 (sec) , antiderivative size = 80, 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 = {1117, 1113, 762}

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.

\(\displaystyle \int \frac {1}{\sqrt {-c^2-2 c d x-d^2 x^2+1} (c e+d e x)^{5/2}} \, dx\)

\(\Big \downarrow \) 1117

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

\(\Big \downarrow \) 1113

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

\(\Big \downarrow \) 762

\(\displaystyle \frac {2 \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c e+d x e}}{\sqrt {e}}\right ),-1\right )}{3 d e^{5/2}}-\frac {2 \sqrt {-c^2-2 c d x-d^2 x^2+1}}{3 d e (c e+d e x)^{3/2}}\)

Input:

Int[1/((c*e + d*e*x)^(5/2)*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])/(3*d*e*(c*e + d*e*x)^(3/2)) + (2*El 
lipticF[ArcSin[Sqrt[c*e + d*e*x]/Sqrt[e]], -1])/(3*d*e^(5/2))
 

Defintions of rubi rules used

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 1113

Int[1/(Sqrt[(d_) + (e_.)*(x_)]*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_ 
Symbol] :> Simp[(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] && 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])
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(163\) vs. \(2(66)=132\).

Time = 3.87 (sec) , antiderivative size = 164, normalized size of antiderivative = 2.05

method	result	size
default	\(-\frac {\left (\sqrt {-2 d x -2 c +2}\, \sqrt {d x +c}\, \sqrt {2 d x +2 c +2}\, \operatorname {EllipticF}\left (\frac {\sqrt {-2 d x -2 c +2}}{2}, \sqrt {2}\right ) d x +\sqrt {-2 d x -2 c +2}\, \sqrt {d x +c}\, \sqrt {2 d x +2 c +2}\, \operatorname {EllipticF}\left (\frac {\sqrt {-2 d x -2 c +2}}{2}, \sqrt {2}\right ) c -2 d^{2} x^{2}-4 c d x -2 c^{2}+2\right ) \sqrt {e \left (d x +c \right )}}{3 \left (d x +c \right )^{2} \sqrt {-d^{2} x^{2}-2 c d x -c^{2}+1}\, e^{3} d}\)	\(164\)
elliptic	\(\frac {\sqrt {-e \left (d x +c \right ) \left (d^{2} x^{2}+2 c d x +c^{2}-1\right )}\, \left (-\frac {2 \sqrt {-d^{3} e \,x^{3}-3 c \,d^{2} e \,x^{2}-3 c^{2} d e x -c^{3} e +d e x +c e}}{3 d^{3} e^{3} \left (x +\frac {c}{d}\right )^{2}}+\frac {2 \left (-\frac {c +1}{d}+\frac {c -1}{d}\right ) \sqrt {\frac {x +\frac {c -1}{d}}{-\frac {c +1}{d}+\frac {c -1}{d}}}\, \sqrt {\frac {x +\frac {c}{d}}{-\frac {c -1}{d}+\frac {c}{d}}}\, \sqrt {\frac {x +\frac {c +1}{d}}{-\frac {c -1}{d}+\frac {c +1}{d}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {x +\frac {c -1}{d}}{-\frac {c +1}{d}+\frac {c -1}{d}}}, \sqrt {\frac {-\frac {c -1}{d}+\frac {c +1}{d}}{-\frac {c -1}{d}+\frac {c}{d}}}\right )}{3 e^{2} \sqrt {-d^{3} e \,x^{3}-3 c \,d^{2} e \,x^{2}-3 c^{2} d e x -c^{3} e +d e x +c e}}\right )}{\sqrt {e \left (d x +c \right )}\, \sqrt {-d^{2} x^{2}-2 c d x -c^{2}+1}}\)	\(337\)

Input:

int(1/(d*e*x+c*e)^(5/2)/(-d^2*x^2-2*c*d*x-c^2+1)^(1/2),x,method=_RETURNVER 
BOSE)
 

Output:

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

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.41 \[ \int \frac {1}{(c e+d e x)^{5/2} \sqrt {1-c^2-2 c d x-d^2 x^2}} \, dx=-\frac {2 \, {\left (\sqrt {-d^{2} x^{2} - 2 \, c d x - c^{2} + 1} \sqrt {d e x + c e} d^{2} + \sqrt {-d^{3} e} {\left (d^{2} x^{2} + 2 \, c d x + c^{2}\right )} {\rm weierstrassPInverse}\left (\frac {4}{d^{2}}, 0, \frac {d x + c}{d}\right )\right )}}{3 \, {\left (d^{5} e^{3} x^{2} + 2 \, c d^{4} e^{3} x + c^{2} d^{3} e^{3}\right )}} \] Input:

integrate(1/(d*e*x+c*e)^(5/2)/(-d^2*x^2-2*c*d*x-c^2+1)^(1/2),x, algorithm= 
"fricas")
 

Output:

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

Sympy [F]

\[ \int \frac {1}{(c e+d e x)^{5/2} \sqrt {1-c^2-2 c d x-d^2 x^2}} \, dx=\int \frac {1}{\left (e \left (c + d x\right )\right )^{\frac {5}{2}} \sqrt {- \left (c + d x - 1\right ) \left (c + d x + 1\right )}}\, dx \] Input:

integrate(1/(d*e*x+c*e)**(5/2)/(-d**2*x**2-2*c*d*x-c**2+1)**(1/2),x)
 

Output:

Integral(1/((e*(c + d*x))**(5/2)*sqrt(-(c + d*x - 1)*(c + d*x + 1))), x)
 

Maxima [F]

\[ \int \frac {1}{(c e+d e x)^{5/2} \sqrt {1-c^2-2 c d x-d^2 x^2}} \, dx=\int { \frac {1}{\sqrt {-d^{2} x^{2} - 2 \, c d x - c^{2} + 1} {\left (d e x + c e\right )}^{\frac {5}{2}}} \,d x } \] Input:

integrate(1/(d*e*x+c*e)^(5/2)/(-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)^(5/2)), x)
 

Giac [F]

\[ \int \frac {1}{(c e+d e x)^{5/2} \sqrt {1-c^2-2 c d x-d^2 x^2}} \, dx=\int { \frac {1}{\sqrt {-d^{2} x^{2} - 2 \, c d x - c^{2} + 1} {\left (d e x + c e\right )}^{\frac {5}{2}}} \,d x } \] Input:

integrate(1/(d*e*x+c*e)^(5/2)/(-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)^(5/2)), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{(c e+d e x)^{5/2} \sqrt {1-c^2-2 c d x-d^2 x^2}} \, dx=\int \frac {1}{{\left (c\,e+d\,e\,x\right )}^{5/2}\,\sqrt {-c^2-2\,c\,d\,x-d^2\,x^2+1}} \,d x \] Input:

int(1/((c*e + d*e*x)^(5/2)*(1 - d^2*x^2 - 2*c*d*x - c^2)^(1/2)),x)
 

Output:

int(1/((c*e + d*e*x)^(5/2)*(1 - d^2*x^2 - 2*c*d*x - c^2)^(1/2)), x)
 

Reduce [F]

\[ \int \frac {1}{(c e+d e x)^{5/2} \sqrt {1-c^2-2 c d x-d^2 x^2}} \, dx=-\frac {\sqrt {e}\, \left (\int \frac {\sqrt {d x +c}\, \sqrt {-d^{2} x^{2}-2 c d x -c^{2}+1}}{d^{5} x^{5}+5 c \,d^{4} x^{4}+10 c^{2} d^{3} x^{3}+10 c^{3} d^{2} x^{2}+5 c^{4} d x -d^{3} x^{3}+c^{5}-3 c \,d^{2} x^{2}-3 c^{2} d x -c^{3}}d x \right )}{e^{3}} \] Input:

int(1/(d*e*x+c*e)^(5/2)/(-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 
**5 + 5*c**4*d*x + 10*c**3*d**2*x**2 - c**3 + 10*c**2*d**3*x**3 - 3*c**2*d 
*x + 5*c*d**4*x**4 - 3*c*d**2*x**2 + d**5*x**5 - d**3*x**3),x))/e**3