\(\int \frac {(c e+d e x)^{3/2}}{\sqrt {1-c^2-2 c d x-d^2 x^2}} \, dx\) [298]

Optimal result

Integrand size = 37, antiderivative size = 78 \[ \int \frac {(c e+d e x)^{3/2}}{\sqrt {1-c^2-2 c d x-d^2 x^2}} \, dx=-\frac {2 e \sqrt {c e+d e x} \sqrt {1-c^2-2 c d x-d^2 x^2}}{3 d}+\frac {2 e^{3/2} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c e+d e x}}{\sqrt {e}}\right ),-1\right )}{3 d} \] Output:

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

Mathematica [C] (verified)

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

Time = 10.03 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.69 \[ \int \frac {(c e+d e x)^{3/2}}{\sqrt {1-c^2-2 c d x-d^2 x^2}} \, dx=-\frac {2 e \sqrt {e (c+d x)} \left (\sqrt {1-(c+d x)^2}-\operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {1}{2},\frac {5}{4},(c+d x)^2\right )\right )}{3 d} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.24 (sec) , antiderivative size = 78, 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 = {1116, 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.

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

\(\Big \downarrow \) 1116

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

\(\Big \downarrow \) 1113

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

\(\Big \downarrow \) 762

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

Input:

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

Output:

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

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 1116

Int[((d_) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_S 
ymbol] :> Simp[2*d*(d + e*x)^(m - 1)*((a + b*x + c*x^2)^(p + 1)/(b*(m + 2*p 
 + 1))), x] + Simp[d^2*(m - 1)*((b^2 - 4*a*c)/(b^2*(m + 2*p + 1)))   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] && GtQ[m, 1] && NeQ[m + 2*p 
+ 1, 0] && (IntegerQ[2*p] || (IntegerQ[m] && RationalQ[p]) || OddQ[m])
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(157\) vs. \(2(64)=128\).

Time = 2.02 (sec) , antiderivative size = 158, normalized size of antiderivative = 2.03

method	result	size
default	\(\frac {\sqrt {e \left (d x +c \right )}\, \sqrt {-d^{2} x^{2}-2 c d x -c^{2}+1}\, e \left (-2 d^{3} x^{3}-6 c \,d^{2} x^{2}+\sqrt {-2 d x -2 c +2}\, \sqrt {d x +c}\, \sqrt {2 d x +2 c +2}\, \operatorname {EllipticF}\left (\frac {\sqrt {-2 d x -2 c +2}}{2}, \sqrt {2}\right )-6 c^{2} d x -2 c^{3}+2 d x +2 c \right )}{3 d \left (d^{3} x^{3}+3 c \,d^{2} x^{2}+3 c^{2} d x +c^{3}-d x -c \right )}\)	\(158\)
elliptic	\(\frac {\sqrt {-e \left (d x +c \right ) \left (d^{2} x^{2}+2 c d x +c^{2}-1\right )}\, \sqrt {e \left (d x +c \right )}\, \left (-\frac {2 e \sqrt {-d^{3} e \,x^{3}-3 c \,d^{2} e \,x^{2}-3 c^{2} d e x -c^{3} e +d e x +c e}}{3 d}+\frac {2 \left (c^{2} e^{2}+\frac {2 e \left (-\frac {3}{2} c^{2} d e +\frac {1}{2} d e \right )}{3 d}\right ) \left (-\frac {c +1}{d}+\frac {c -1}{d}\right ) \sqrt {\frac {x +\frac {c -1}{d}}{-\frac {c +1}{d}+\frac {c -1}{d}}}\, \sqrt {\frac {x +\frac {c}{d}}{-\frac {c -1}{d}+\frac {c}{d}}}\, \sqrt {\frac {x +\frac {c +1}{d}}{-\frac {c -1}{d}+\frac {c +1}{d}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {x +\frac {c -1}{d}}{-\frac {c +1}{d}+\frac {c -1}{d}}}, \sqrt {\frac {-\frac {c -1}{d}+\frac {c +1}{d}}{-\frac {c -1}{d}+\frac {c}{d}}}\right )}{\sqrt {-d^{3} e \,x^{3}-3 c \,d^{2} e \,x^{2}-3 c^{2} d e x -c^{3} e +d e x +c e}}\right )}{\left (d x +c \right ) \sqrt {-d^{2} x^{2}-2 c d x -c^{2}+1}\, e}\)	\(359\)
risch	\(\frac {2 \left (d x +c \right ) \left (d^{2} x^{2}+2 c d x +c^{2}-1\right ) \sqrt {e \left (d x +c \right ) \left (-d^{2} x^{2}-2 c d x -c^{2}+1\right )}\, e^{2}}{3 d \sqrt {-e \left (d x +c \right ) \left (d^{2} x^{2}+2 c d x +c^{2}-1\right )}\, \sqrt {e \left (d x +c \right )}\, \sqrt {-d^{2} x^{2}-2 c d x -c^{2}+1}}+\frac {2 \left (-\frac {c -1}{d}+\frac {c +1}{d}\right ) \sqrt {\frac {x +\frac {c +1}{d}}{-\frac {c -1}{d}+\frac {c +1}{d}}}\, \sqrt {\frac {x +\frac {c}{d}}{-\frac {c +1}{d}+\frac {c}{d}}}\, \sqrt {\frac {x +\frac {c -1}{d}}{-\frac {c +1}{d}+\frac {c -1}{d}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {x +\frac {c +1}{d}}{-\frac {c -1}{d}+\frac {c +1}{d}}}, \sqrt {\frac {-\frac {c +1}{d}+\frac {c -1}{d}}{-\frac {c +1}{d}+\frac {c}{d}}}\right ) \sqrt {e \left (d x +c \right ) \left (-d^{2} x^{2}-2 c d x -c^{2}+1\right )}\, e^{2}}{3 \sqrt {-d^{3} e \,x^{3}-3 c \,d^{2} e \,x^{2}-3 c^{2} d e x -c^{3} e +d e x +c e}\, \sqrt {e \left (d x +c \right )}\, \sqrt {-d^{2} x^{2}-2 c d x -c^{2}+1}}\)	\(395\)

Input:

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

Output:

1/3*(e*(d*x+c))^(1/2)*(-d^2*x^2-2*c*d*x-c^2+1)^(1/2)*e*(-2*d^3*x^3-6*c*d^2 
*x^2+(-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))-6*c^2*d*x-2*c^3+2*d*x+2*c)/d/(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 = 0.09 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.88 \[ \int \frac {(c e+d e x)^{3/2}}{\sqrt {1-c^2-2 c d x-d^2 x^2}} \, dx=-\frac {2 \, {\left (\sqrt {-d^{2} x^{2} - 2 \, c d x - c^{2} + 1} \sqrt {d e x + c e} d^{2} e + \sqrt {-d^{3} e} e {\rm weierstrassPInverse}\left (\frac {4}{d^{2}}, 0, \frac {d x + c}{d}\right )\right )}}{3 \, d^{3}} \] Input:

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {(c e+d e x)^{3/2}}{\sqrt {1-c^2-2 c d x-d^2 x^2}} \, dx=\frac {\sqrt {e}\, e \left (-2 \sqrt {d x +c}\, \sqrt {-d^{2} x^{2}-2 c d x -c^{2}+1}\, c^{2}+3 \left (\int \frac {\sqrt {d x +c}\, \sqrt {-d^{2} x^{2}-2 c d x -c^{2}+1}\, x^{2}}{3 c^{2} d^{3} x^{3}+9 c^{3} d^{2} x^{2}+9 c^{4} d x -d^{3} x^{3}+3 c^{5}-3 c \,d^{2} x^{2}-6 c^{2} d x -4 c^{3}+d x +c}d x \right ) c^{2} d^{3}-\left (\int \frac {\sqrt {d x +c}\, \sqrt {-d^{2} x^{2}-2 c d x -c^{2}+1}\, x^{2}}{3 c^{2} d^{3} x^{3}+9 c^{3} d^{2} x^{2}+9 c^{4} d x -d^{3} x^{3}+3 c^{5}-3 c \,d^{2} x^{2}-6 c^{2} d x -4 c^{3}+d x +c}d x \right ) d^{3}+6 \left (\int \frac {\sqrt {d x +c}\, \sqrt {-d^{2} x^{2}-2 c d x -c^{2}+1}\, x}{3 c^{2} d^{3} x^{3}+9 c^{3} d^{2} x^{2}+9 c^{4} d x -d^{3} x^{3}+3 c^{5}-3 c \,d^{2} x^{2}-6 c^{2} d x -4 c^{3}+d x +c}d x \right ) c^{3} d^{2}-2 \left (\int \frac {\sqrt {d x +c}\, \sqrt {-d^{2} x^{2}-2 c d x -c^{2}+1}\, x}{3 c^{2} d^{3} x^{3}+9 c^{3} d^{2} x^{2}+9 c^{4} d x -d^{3} x^{3}+3 c^{5}-3 c \,d^{2} x^{2}-6 c^{2} d x -4 c^{3}+d x +c}d x \right ) c \,d^{2}\right )}{d \left (3 c^{2}-1\right )} \] Input:

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

Output:

(sqrt(e)*e*( - 2*sqrt(c + d*x)*sqrt( - c**2 - 2*c*d*x - d**2*x**2 + 1)*c** 
2 + 3*int((sqrt(c + d*x)*sqrt( - c**2 - 2*c*d*x - d**2*x**2 + 1)*x**2)/(3* 
c**5 + 9*c**4*d*x + 9*c**3*d**2*x**2 - 4*c**3 + 3*c**2*d**3*x**3 - 6*c**2* 
d*x - 3*c*d**2*x**2 + c - d**3*x**3 + d*x),x)*c**2*d**3 - int((sqrt(c + d* 
x)*sqrt( - c**2 - 2*c*d*x - d**2*x**2 + 1)*x**2)/(3*c**5 + 9*c**4*d*x + 9* 
c**3*d**2*x**2 - 4*c**3 + 3*c**2*d**3*x**3 - 6*c**2*d*x - 3*c*d**2*x**2 + 
c - d**3*x**3 + d*x),x)*d**3 + 6*int((sqrt(c + d*x)*sqrt( - c**2 - 2*c*d*x 
 - d**2*x**2 + 1)*x)/(3*c**5 + 9*c**4*d*x + 9*c**3*d**2*x**2 - 4*c**3 + 3* 
c**2*d**3*x**3 - 6*c**2*d*x - 3*c*d**2*x**2 + c - d**3*x**3 + d*x),x)*c**3 
*d**2 - 2*int((sqrt(c + d*x)*sqrt( - c**2 - 2*c*d*x - d**2*x**2 + 1)*x)/(3 
*c**5 + 9*c**4*d*x + 9*c**3*d**2*x**2 - 4*c**3 + 3*c**2*d**3*x**3 - 6*c**2 
*d*x - 3*c*d**2*x**2 + c - d**3*x**3 + d*x),x)*c*d**2))/(d*(3*c**2 - 1))