\(\int \frac {1}{(2+e x)^{3/2} (12-3 e^2 x^2)^{3/2}} \, dx\) [237]

Optimal result

Integrand size = 24, antiderivative size = 108 \[ \int \frac {1}{(2+e x)^{3/2} \left (12-3 e^2 x^2\right )^{3/2}} \, dx=\frac {5}{256 \sqrt {3} e \sqrt {2-e x}}-\frac {1}{24 \sqrt {3} e \sqrt {2-e x} (2+e x)^2}-\frac {5}{192 \sqrt {3} e \sqrt {2-e x} (2+e x)}-\frac {5 \text {arctanh}\left (\frac {1}{2} \sqrt {2-e x}\right )}{512 \sqrt {3} e} \] Output:

5/768*3^(1/2)/e/(-e*x+2)^(1/2)-1/72*3^(1/2)/e/(-e*x+2)^(1/2)/(e*x+2)^2-5/5 
76*3^(1/2)/e/(-e*x+2)^(1/2)/(e*x+2)-5/1536*3^(1/2)*arctanh(1/2*(-e*x+2)^(1 
/2))/e
 

Mathematica [A] (verified)

Time = 0.44 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.81 \[ \int \frac {1}{(2+e x)^{3/2} \left (12-3 e^2 x^2\right )^{3/2}} \, dx=\frac {-\frac {2 \sqrt {4-e^2 x^2} \left (-12+40 e x+15 e^2 x^2\right )}{(-2+e x) (2+e x)^{5/2}}-15 \text {arctanh}\left (\frac {2 \sqrt {2+e x}}{\sqrt {4-e^2 x^2}}\right )}{1536 \sqrt {3} e} \] Input:

Integrate[1/((2 + e*x)^(3/2)*(12 - 3*e^2*x^2)^(3/2)),x]
 

Output:

((-2*Sqrt[4 - e^2*x^2]*(-12 + 40*e*x + 15*e^2*x^2))/((-2 + e*x)*(2 + e*x)^ 
(5/2)) - 15*ArcTanh[(2*Sqrt[2 + e*x])/Sqrt[4 - e^2*x^2]])/(1536*Sqrt[3]*e)
 

Rubi [A] (verified)

Time = 0.34 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {456, 52, 27, 52, 61, 73, 219}

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.

\(\displaystyle \int \frac {1}{(e x+2)^{3/2} \left (12-3 e^2 x^2\right )^{3/2}} \, dx\)

\(\Big \downarrow \) 456

\(\displaystyle \int \frac {1}{(6-3 e x)^{3/2} (e x+2)^3}dx\)

\(\Big \downarrow \) 52

\(\displaystyle \frac {5}{16} \int \frac {1}{3 \sqrt {3} (2-e x)^{3/2} (e x+2)^2}dx-\frac {1}{24 \sqrt {3} e \sqrt {2-e x} (e x+2)^2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {5 \int \frac {1}{(2-e x)^{3/2} (e x+2)^2}dx}{48 \sqrt {3}}-\frac {1}{24 \sqrt {3} e \sqrt {2-e x} (e x+2)^2}\)

\(\Big \downarrow \) 52

\(\displaystyle \frac {5 \left (\frac {3}{8} \int \frac {1}{(2-e x)^{3/2} (e x+2)}dx-\frac {1}{4 e \sqrt {2-e x} (e x+2)}\right )}{48 \sqrt {3}}-\frac {1}{24 \sqrt {3} e \sqrt {2-e x} (e x+2)^2}\)

\(\Big \downarrow \) 61

\(\displaystyle \frac {5 \left (\frac {3}{8} \left (\frac {1}{4} \int \frac {1}{\sqrt {2-e x} (e x+2)}dx+\frac {1}{2 e \sqrt {2-e x}}\right )-\frac {1}{4 e \sqrt {2-e x} (e x+2)}\right )}{48 \sqrt {3}}-\frac {1}{24 \sqrt {3} e \sqrt {2-e x} (e x+2)^2}\)

\(\Big \downarrow \) 73

\(\displaystyle \frac {5 \left (\frac {3}{8} \left (\frac {1}{2 e \sqrt {2-e x}}-\frac {\int \frac {1}{e x+2}d\sqrt {2-e x}}{2 e}\right )-\frac {1}{4 e \sqrt {2-e x} (e x+2)}\right )}{48 \sqrt {3}}-\frac {1}{24 \sqrt {3} e \sqrt {2-e x} (e x+2)^2}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {5 \left (\frac {3}{8} \left (\frac {1}{2 e \sqrt {2-e x}}-\frac {\text {arctanh}\left (\frac {1}{2} \sqrt {2-e x}\right )}{4 e}\right )-\frac {1}{4 e \sqrt {2-e x} (e x+2)}\right )}{48 \sqrt {3}}-\frac {1}{24 \sqrt {3} e \sqrt {2-e x} (e x+2)^2}\)

Input:

Int[1/((2 + e*x)^(3/2)*(12 - 3*e^2*x^2)^(3/2)),x]
 

Output:

-1/24*1/(Sqrt[3]*e*Sqrt[2 - e*x]*(2 + e*x)^2) + (5*(-1/4*1/(e*Sqrt[2 - e*x 
]*(2 + e*x)) + (3*(1/(2*e*Sqrt[2 - e*x]) - ArcTanh[Sqrt[2 - e*x]/2]/(4*e)) 
)/8))/(48*Sqrt[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 52

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ 
(a + b*x)^(m + 1)*((c + d*x)^(n + 1)/((b*c - a*d)*(m + 1))), x] - Simp[d*(( 
m + n + 2)/((b*c - a*d)*(m + 1)))   Int[(a + b*x)^(m + 1)*(c + d*x)^n, x], 
x] /; FreeQ[{a, b, c, d, n}, x] && ILtQ[m, -1] && FractionQ[n] && LtQ[n, 0]
 

rule 61

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ 
(a + b*x)^(m + 1)*((c + d*x)^(n + 1)/((b*c - a*d)*(m + 1))), x] - Simp[d*(( 
m + n + 2)/((b*c - a*d)*(m + 1)))   Int[(a + b*x)^(m + 1)*(c + d*x)^n, x], 
x] /; FreeQ[{a, b, c, d, n}, x] && LtQ[m, -1] &&  !(LtQ[n, -1] && (EqQ[a, 0 
] || (NeQ[c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && IntLinearQ[a, b, c, d 
, m, n, x]
 

rule 73

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ 
{p = Denominator[m]}, Simp[p/b   Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + 
 d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt 
Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL 
inearQ[a, b, c, d, m, n, x]
 

rule 219

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* 
ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt 
Q[a, 0] || LtQ[b, 0])
 

rule 456

Int[((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ 
(c + d*x)^(n + p)*(a/c + (b/d)*x)^p, x] /; FreeQ[{a, b, c, d, n, p}, x] && 
EqQ[b*c^2 + a*d^2, 0] && (IntegerQ[p] || (GtQ[a, 0] && GtQ[c, 0] &&  !Integ 
erQ[n]))
 

Maple [A] (verified)

Time = 0.17 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.25

method	result	size
default	\(\frac {\sqrt {-3 e^{2} x^{2}+12}\, \left (5 \sqrt {3}\, \operatorname {arctanh}\left (\frac {\sqrt {-3 e x +6}\, \sqrt {3}}{6}\right ) \sqrt {-3 e x +6}\, e^{2} x^{2}+20 \sqrt {3}\, \operatorname {arctanh}\left (\frac {\sqrt {-3 e x +6}\, \sqrt {3}}{6}\right ) \sqrt {-3 e x +6}\, e x -30 e^{2} x^{2}+20 \sqrt {3}\, \operatorname {arctanh}\left (\frac {\sqrt {-3 e x +6}\, \sqrt {3}}{6}\right ) \sqrt {-3 e x +6}-80 e x +24\right )}{4608 \left (e x +2\right )^{\frac {5}{2}} \left (e x -2\right ) e}\)	\(135\)

Input:

int(1/(e*x+2)^(3/2)/(-3*e^2*x^2+12)^(3/2),x,method=_RETURNVERBOSE)
 

Output:

1/4608/(e*x+2)^(5/2)*(-3*e^2*x^2+12)^(1/2)*(5*3^(1/2)*arctanh(1/6*(-3*e*x+ 
6)^(1/2)*3^(1/2))*(-3*e*x+6)^(1/2)*e^2*x^2+20*3^(1/2)*arctanh(1/6*(-3*e*x+ 
6)^(1/2)*3^(1/2))*(-3*e*x+6)^(1/2)*e*x-30*e^2*x^2+20*3^(1/2)*arctanh(1/6*( 
-3*e*x+6)^(1/2)*3^(1/2))*(-3*e*x+6)^(1/2)-80*e*x+24)/(e*x-2)/e
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 147, normalized size of antiderivative = 1.36 \[ \int \frac {1}{(2+e x)^{3/2} \left (12-3 e^2 x^2\right )^{3/2}} \, dx=\frac {15 \, \sqrt {3} {\left (e^{4} x^{4} + 4 \, e^{3} x^{3} - 16 \, e x - 16\right )} \log \left (-\frac {3 \, e^{2} x^{2} - 12 \, e x + 4 \, \sqrt {3} \sqrt {-3 \, e^{2} x^{2} + 12} \sqrt {e x + 2} - 36}{e^{2} x^{2} + 4 \, e x + 4}\right ) - 4 \, {\left (15 \, e^{2} x^{2} + 40 \, e x - 12\right )} \sqrt {-3 \, e^{2} x^{2} + 12} \sqrt {e x + 2}}{9216 \, {\left (e^{5} x^{4} + 4 \, e^{4} x^{3} - 16 \, e^{2} x - 16 \, e\right )}} \] Input:

integrate(1/(e*x+2)^(3/2)/(-3*e^2*x^2+12)^(3/2),x, algorithm="fricas")
                                                                                    
                                                                                    
 

Output:

1/9216*(15*sqrt(3)*(e^4*x^4 + 4*e^3*x^3 - 16*e*x - 16)*log(-(3*e^2*x^2 - 1 
2*e*x + 4*sqrt(3)*sqrt(-3*e^2*x^2 + 12)*sqrt(e*x + 2) - 36)/(e^2*x^2 + 4*e 
*x + 4)) - 4*(15*e^2*x^2 + 40*e*x - 12)*sqrt(-3*e^2*x^2 + 12)*sqrt(e*x + 2 
))/(e^5*x^4 + 4*e^4*x^3 - 16*e^2*x - 16*e)
 

Sympy [F]

\[ \int \frac {1}{(2+e x)^{3/2} \left (12-3 e^2 x^2\right )^{3/2}} \, dx=\frac {\sqrt {3} \int \frac {1}{- e^{3} x^{3} \sqrt {e x + 2} \sqrt {- e^{2} x^{2} + 4} - 2 e^{2} x^{2} \sqrt {e x + 2} \sqrt {- e^{2} x^{2} + 4} + 4 e x \sqrt {e x + 2} \sqrt {- e^{2} x^{2} + 4} + 8 \sqrt {e x + 2} \sqrt {- e^{2} x^{2} + 4}}\, dx}{9} \] Input:

integrate(1/(e*x+2)**(3/2)/(-3*e**2*x**2+12)**(3/2),x)
 

Output:

sqrt(3)*Integral(1/(-e**3*x**3*sqrt(e*x + 2)*sqrt(-e**2*x**2 + 4) - 2*e**2 
*x**2*sqrt(e*x + 2)*sqrt(-e**2*x**2 + 4) + 4*e*x*sqrt(e*x + 2)*sqrt(-e**2* 
x**2 + 4) + 8*sqrt(e*x + 2)*sqrt(-e**2*x**2 + 4)), x)/9
 

Maxima [F]

\[ \int \frac {1}{(2+e x)^{3/2} \left (12-3 e^2 x^2\right )^{3/2}} \, dx=\int { \frac {1}{{\left (-3 \, e^{2} x^{2} + 12\right )}^{\frac {3}{2}} {\left (e x + 2\right )}^{\frac {3}{2}}} \,d x } \] Input:

integrate(1/(e*x+2)^(3/2)/(-3*e^2*x^2+12)^(3/2),x, algorithm="maxima")
 

Output:

integrate(1/((-3*e^2*x^2 + 12)^(3/2)*(e*x + 2)^(3/2)), x)
 

Giac [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.82 \[ \int \frac {1}{(2+e x)^{3/2} \left (12-3 e^2 x^2\right )^{3/2}} \, dx=-\frac {15 \, \sqrt {3} \log \left (\sqrt {-e x + 2} + 2\right ) - 15 \, \sqrt {3} \log \left (-\sqrt {-e x + 2} + 2\right ) - \frac {32 \, \sqrt {3}}{\sqrt {-e x + 2}} - \frac {4 \, {\left (7 \, \sqrt {3} {\left (-e x + 2\right )}^{\frac {3}{2}} - 36 \, \sqrt {3} \sqrt {-e x + 2}\right )}}{{\left (e x + 2\right )}^{2}}}{9216 \, e} \] Input:

integrate(1/(e*x+2)^(3/2)/(-3*e^2*x^2+12)^(3/2),x, algorithm="giac")
 

Output:

-1/9216*(15*sqrt(3)*log(sqrt(-e*x + 2) + 2) - 15*sqrt(3)*log(-sqrt(-e*x + 
2) + 2) - 32*sqrt(3)/sqrt(-e*x + 2) - 4*(7*sqrt(3)*(-e*x + 2)^(3/2) - 36*s 
qrt(3)*sqrt(-e*x + 2))/(e*x + 2)^2)/e
 

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{(2+e x)^{3/2} \left (12-3 e^2 x^2\right )^{3/2}} \, dx=\int \frac {1}{{\left (12-3\,e^2\,x^2\right )}^{3/2}\,{\left (e\,x+2\right )}^{3/2}} \,d x \] Input:

int(1/((12 - 3*e^2*x^2)^(3/2)*(e*x + 2)^(3/2)),x)
 

Output:

int(1/((12 - 3*e^2*x^2)^(3/2)*(e*x + 2)^(3/2)), x)
 

Reduce [B] (verification not implemented)

Time = 0.23 (sec) , antiderivative size = 175, normalized size of antiderivative = 1.62 \[ \int \frac {1}{(2+e x)^{3/2} \left (12-3 e^2 x^2\right )^{3/2}} \, dx=\frac {\sqrt {3}\, \left (15 \sqrt {-e x +2}\, \mathrm {log}\left (\sqrt {-e x +2}-2\right ) e^{2} x^{2}+60 \sqrt {-e x +2}\, \mathrm {log}\left (\sqrt {-e x +2}-2\right ) e x +60 \sqrt {-e x +2}\, \mathrm {log}\left (\sqrt {-e x +2}-2\right )-15 \sqrt {-e x +2}\, \mathrm {log}\left (\sqrt {-e x +2}+2\right ) e^{2} x^{2}-60 \sqrt {-e x +2}\, \mathrm {log}\left (\sqrt {-e x +2}+2\right ) e x -60 \sqrt {-e x +2}\, \mathrm {log}\left (\sqrt {-e x +2}+2\right )+60 e^{2} x^{2}+160 e x -48\right )}{9216 \sqrt {-e x +2}\, e \left (e^{2} x^{2}+4 e x +4\right )} \] Input:

int(1/(e*x+2)^(3/2)/(-3*e^2*x^2+12)^(3/2),x)
 

Output:

(sqrt(3)*(15*sqrt( - e*x + 2)*log(sqrt( - e*x + 2) - 2)*e**2*x**2 + 60*sqr 
t( - e*x + 2)*log(sqrt( - e*x + 2) - 2)*e*x + 60*sqrt( - e*x + 2)*log(sqrt 
( - e*x + 2) - 2) - 15*sqrt( - e*x + 2)*log(sqrt( - e*x + 2) + 2)*e**2*x** 
2 - 60*sqrt( - e*x + 2)*log(sqrt( - e*x + 2) + 2)*e*x - 60*sqrt( - e*x + 2 
)*log(sqrt( - e*x + 2) + 2) + 60*e**2*x**2 + 160*e*x - 48))/(9216*sqrt( - 
e*x + 2)*e*(e**2*x**2 + 4*e*x + 4))