\(\int \frac {1}{(1+3 x+\sqrt {-3-2 x+4 x^2})^3} \, dx\) [44]

Optimal result

Integrand size = 21, antiderivative size = 176 \[ \int \frac {1}{\left (1+3 x+\sqrt {-3-2 x+4 x^2}\right )^3} \, dx=\frac {191+567 \left (2 x+\sqrt {-3-2 x+4 x^2}\right )}{50 \left (1+2 \left (2 x+\sqrt {-3-2 x+4 x^2}\right )+5 \left (2 x+\sqrt {-3-2 x+4 x^2}\right )^2\right )^2}+\frac {1237+6825 \left (2 x+\sqrt {-3-2 x+4 x^2}\right )}{400 \left (1+2 \left (2 x+\sqrt {-3-2 x+4 x^2}\right )+5 \left (2 x+\sqrt {-3-2 x+4 x^2}\right )^2\right )}+\frac {273}{32} \arctan \left (\frac {1}{2} \left (1+5 \left (2 x+\sqrt {-3-2 x+4 x^2}\right )\right )\right ) \] Output:

1/50*(191+1134*x+567*(4*x^2-2*x-3)^(1/2))/(1+4*x+2*(4*x^2-2*x-3)^(1/2)+5*( 
2*x+(4*x^2-2*x-3)^(1/2))^2)^2+(1237+13650*x+6825*(4*x^2-2*x-3)^(1/2))/(400 
+1600*x+800*(4*x^2-2*x-3)^(1/2)+2000*(2*x+(4*x^2-2*x-3)^(1/2))^2)+273/32*a 
rctan(1/2+5*x+5/2*(4*x^2-2*x-3)^(1/2))
 

Mathematica [A] (verified)

Time = 0.41 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.57 \[ \int \frac {1}{\left (1+3 x+\sqrt {-3-2 x+4 x^2}\right )^3} \, dx=-\frac {5072+15148 x+17388 x^2+6825 x^3+5 \sqrt {-3-2 x+4 x^2} \left (576+1748 x+2064 x^2+895 x^3\right )+1365 \left (4+8 x+5 x^2\right )^2 \arctan \left (\frac {3}{2}+x-\frac {1}{2} \sqrt {-3-2 x+4 x^2}\right )}{160 \left (4+8 x+5 x^2\right )^2} \] Input:

Integrate[(1 + 3*x + Sqrt[-3 - 2*x + 4*x^2])^(-3),x]
 

Output:

-1/160*(5072 + 15148*x + 17388*x^2 + 6825*x^3 + 5*Sqrt[-3 - 2*x + 4*x^2]*( 
576 + 1748*x + 2064*x^2 + 895*x^3) + 1365*(4 + 8*x + 5*x^2)^2*ArcTan[3/2 + 
 x - Sqrt[-3 - 2*x + 4*x^2]/2])/(4 + 8*x + 5*x^2)^2
 

Rubi [A] (verified)

Time = 0.74 (sec) , antiderivative size = 277, normalized size of antiderivative = 1.57, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {7293, 2009}

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.

\(\displaystyle \int \frac {1}{\left (\sqrt {4 x^2-2 x-3}+3 x+1\right )^3} \, dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {168 \sqrt {4 x^2-2 x-3} x}{5 \left (5 x^2+8 x+4\right )^3}+\frac {21 (15 x-19)}{25 \left (5 x^2+8 x+4\right )^2}-\frac {31 \sqrt {4 x^2-2 x-3}}{5 \left (5 x^2+8 x+4\right )^2}+\frac {4 (333 x+349)}{25 \left (5 x^2+8 x+4\right )^3}+\frac {124 \sqrt {4 x^2-2 x-3}}{5 \left (5 x^2+8 x+4\right )^3}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {273}{64} \arctan \left (\frac {7 x+8}{2 \sqrt {4 x^2-2 x-3}}\right )-\frac {273}{64} \arctan \left (\frac {5 x}{2}+2\right )-\frac {42 \sqrt {4 x^2-2 x-3} (x+1)}{5 \left (5 x^2+8 x+4\right )^2}+\frac {1239 (5 x+4)}{800 \left (5 x^2+8 x+4\right )}-\frac {21 (155 x+136)}{200 \left (5 x^2+8 x+4\right )}-\frac {31 (5 x+4) \sqrt {4 x^2-2 x-3}}{40 \left (5 x^2+8 x+4\right )}-\frac {21 (965 x+701) \sqrt {4 x^2-2 x-3}}{1445 \left (5 x^2+8 x+4\right )}+\frac {31 (18355 x+13928) \sqrt {4 x^2-2 x-3}}{46240 \left (5 x^2+8 x+4\right )}+\frac {413 x+64}{100 \left (5 x^2+8 x+4\right )^2}+\frac {31 (5 x+4) \sqrt {4 x^2-2 x-3}}{20 \left (5 x^2+8 x+4\right )^2}\)

Input:

Int[(1 + 3*x + Sqrt[-3 - 2*x + 4*x^2])^(-3),x]
 

Output:

(64 + 413*x)/(100*(4 + 8*x + 5*x^2)^2) - (42*(1 + x)*Sqrt[-3 - 2*x + 4*x^2 
])/(5*(4 + 8*x + 5*x^2)^2) + (31*(4 + 5*x)*Sqrt[-3 - 2*x + 4*x^2])/(20*(4 
+ 8*x + 5*x^2)^2) + (1239*(4 + 5*x))/(800*(4 + 8*x + 5*x^2)) - (21*(136 + 
155*x))/(200*(4 + 8*x + 5*x^2)) - (31*(4 + 5*x)*Sqrt[-3 - 2*x + 4*x^2])/(4 
0*(4 + 8*x + 5*x^2)) - (21*(701 + 965*x)*Sqrt[-3 - 2*x + 4*x^2])/(1445*(4 
+ 8*x + 5*x^2)) + (31*(13928 + 18355*x)*Sqrt[-3 - 2*x + 4*x^2])/(46240*(4 
+ 8*x + 5*x^2)) - (273*ArcTan[2 + (5*x)/2])/64 - (273*ArcTan[(8 + 7*x)/(2* 
Sqrt[-3 - 2*x + 4*x^2])])/64
 

Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 7293

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v] 
]
 

Maple [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 3.

Time = 0.54 (sec) , antiderivative size = 130, normalized size of antiderivative = 0.74

method	result	size
trager	\(\frac {\left (1585 x^{3}+3707 x^{2}+3116 x +1028\right ) x}{32 \left (5 x^{2}+8 x +4\right )^{2}}-\frac {\left (895 x^{3}+2064 x^{2}+1748 x +576\right ) \sqrt {4 x^{2}-2 x -3}}{32 \left (5 x^{2}+8 x +4\right )^{2}}+\frac {273 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \ln \left (\frac {7 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x +2 \sqrt {4 x^{2}-2 x -3}+8 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right )}{\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x -2 x -2}\right )}{64}\)	\(130\)
default	\(\text {Expression too large to display}\)	\(10403\)

Input:

int(1/(1+3*x+(4*x^2-2*x-3)^(1/2))^3,x,method=_RETURNVERBOSE)
 

Output:

1/32*(1585*x^3+3707*x^2+3116*x+1028)*x/(5*x^2+8*x+4)^2-1/32*(895*x^3+2064* 
x^2+1748*x+576)/(5*x^2+8*x+4)^2*(4*x^2-2*x-3)^(1/2)+273/64*RootOf(_Z^2+1)* 
ln((7*RootOf(_Z^2+1)*x+2*(4*x^2-2*x-3)^(1/2)+8*RootOf(_Z^2+1))/(RootOf(_Z^ 
2+1)*x-2*x-2))
 

Fricas [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 162, normalized size of antiderivative = 0.92 \[ \int \frac {1}{\left (1+3 x+\sqrt {-3-2 x+4 x^2}\right )^3} \, dx=-\frac {3580 \, x^{4} + 14186 \, x^{3} + 21848 \, x^{2} + 273 \, {\left (25 \, x^{4} + 80 \, x^{3} + 104 \, x^{2} + 64 \, x + 16\right )} \arctan \left (\frac {5}{2} \, x + 2\right ) - 273 \, {\left (25 \, x^{4} + 80 \, x^{3} + 104 \, x^{2} + 64 \, x + 16\right )} \arctan \left (-\frac {70 \, x^{2} - 5 \, \sqrt {4 \, x^{2} - 2 \, x - 3} {\left (7 \, x + 8\right )} + 112 \, x + 56}{2 \, {\left (42 \, x + 31\right )}}\right ) + 2 \, {\left (895 \, x^{3} + 2064 \, x^{2} + 1748 \, x + 576\right )} \sqrt {4 \, x^{2} - 2 \, x - 3} + 15224 \, x + 4320}{64 \, {\left (25 \, x^{4} + 80 \, x^{3} + 104 \, x^{2} + 64 \, x + 16\right )}} \] Input:

integrate(1/(1+3*x+(4*x^2-2*x-3)^(1/2))^3,x, algorithm="fricas")
 

Output:

-1/64*(3580*x^4 + 14186*x^3 + 21848*x^2 + 273*(25*x^4 + 80*x^3 + 104*x^2 + 
 64*x + 16)*arctan(5/2*x + 2) - 273*(25*x^4 + 80*x^3 + 104*x^2 + 64*x + 16 
)*arctan(-1/2*(70*x^2 - 5*sqrt(4*x^2 - 2*x - 3)*(7*x + 8) + 112*x + 56)/(4 
2*x + 31)) + 2*(895*x^3 + 2064*x^2 + 1748*x + 576)*sqrt(4*x^2 - 2*x - 3) + 
 15224*x + 4320)/(25*x^4 + 80*x^3 + 104*x^2 + 64*x + 16)
 

Sympy [F]

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

integrate(1/(1+3*x+(4*x**2-2*x-3)**(1/2))**3,x)
 

Output:

Integral((3*x + sqrt(4*x**2 - 2*x - 3) + 1)**(-3), x)
 

Maxima [F]

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

integrate(1/(1+3*x+(4*x^2-2*x-3)^(1/2))^3,x, algorithm="maxima")
 

Output:

integrate((3*x + sqrt(4*x^2 - 2*x - 3) + 1)^(-3), x)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 322 vs. \(2 (147) = 294\).

Time = 0.13 (sec) , antiderivative size = 322, normalized size of antiderivative = 1.83 \[ \int \frac {1}{\left (1+3 x+\sqrt {-3-2 x+4 x^2}\right )^3} \, dx=-\frac {20475 \, {\left (2 \, x - \sqrt {4 \, x^{2} - 2 \, x - 3}\right )}^{7} + 218146 \, {\left (2 \, x - \sqrt {4 \, x^{2} - 2 \, x - 3}\right )}^{6} + 942003 \, {\left (2 \, x - \sqrt {4 \, x^{2} - 2 \, x - 3}\right )}^{5} + 2074498 \, {\left (2 \, x - \sqrt {4 \, x^{2} - 2 \, x - 3}\right )}^{4} + 2003721 \, {\left (2 \, x - \sqrt {4 \, x^{2} - 2 \, x - 3}\right )}^{3} + 1144598 \, {\left (2 \, x - \sqrt {4 \, x^{2} - 2 \, x - 3}\right )}^{2} + 873106 \, x - 436553 \, \sqrt {4 \, x^{2} - 2 \, x - 3} + 80966}{80 \, {\left (5 \, {\left (2 \, x - \sqrt {4 \, x^{2} - 2 \, x - 3}\right )}^{4} + 32 \, {\left (2 \, x - \sqrt {4 \, x^{2} - 2 \, x - 3}\right )}^{3} + 78 \, {\left (2 \, x - \sqrt {4 \, x^{2} - 2 \, x - 3}\right )}^{2} + 64 \, x - 32 \, \sqrt {4 \, x^{2} - 2 \, x - 3} + 13\right )}^{2}} - \frac {6825 \, x^{3} + 17388 \, x^{2} + 15148 \, x + 5072}{160 \, {\left (5 \, x^{2} + 8 \, x + 4\right )}^{2}} - \frac {273}{64} \, \arctan \left (\frac {5}{2} \, x + 2\right ) + \frac {273}{64} \, \arctan \left (-x + \frac {1}{2} \, \sqrt {4 \, x^{2} - 2 \, x - 3} - \frac {3}{2}\right ) + \frac {273}{64} \, \arctan \left (-5 \, x + \frac {5}{2} \, \sqrt {4 \, x^{2} - 2 \, x - 3} - \frac {1}{2}\right ) \] Input:

integrate(1/(1+3*x+(4*x^2-2*x-3)^(1/2))^3,x, algorithm="giac")
 

Output:

-1/80*(20475*(2*x - sqrt(4*x^2 - 2*x - 3))^7 + 218146*(2*x - sqrt(4*x^2 - 
2*x - 3))^6 + 942003*(2*x - sqrt(4*x^2 - 2*x - 3))^5 + 2074498*(2*x - sqrt 
(4*x^2 - 2*x - 3))^4 + 2003721*(2*x - sqrt(4*x^2 - 2*x - 3))^3 + 1144598*( 
2*x - sqrt(4*x^2 - 2*x - 3))^2 + 873106*x - 436553*sqrt(4*x^2 - 2*x - 3) + 
 80966)/(5*(2*x - sqrt(4*x^2 - 2*x - 3))^4 + 32*(2*x - sqrt(4*x^2 - 2*x - 
3))^3 + 78*(2*x - sqrt(4*x^2 - 2*x - 3))^2 + 64*x - 32*sqrt(4*x^2 - 2*x - 
3) + 13)^2 - 1/160*(6825*x^3 + 17388*x^2 + 15148*x + 5072)/(5*x^2 + 8*x + 
4)^2 - 273/64*arctan(5/2*x + 2) + 273/64*arctan(-x + 1/2*sqrt(4*x^2 - 2*x 
- 3) - 3/2) + 273/64*arctan(-5*x + 5/2*sqrt(4*x^2 - 2*x - 3) - 1/2)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\left (1+3 x+\sqrt {-3-2 x+4 x^2}\right )^3} \, dx=\int \frac {1}{{\left (3\,x+\sqrt {4\,x^2-2\,x-3}+1\right )}^3} \,d x \] Input:

int(1/(3*x + (4*x^2 - 2*x - 3)^(1/2) + 1)^3,x)
 

Output:

int(1/(3*x + (4*x^2 - 2*x - 3)^(1/2) + 1)^3, x)
 

Reduce [F]

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

int(1/(1+3*x+(4*x^2-2*x-3)^(1/2))^3,x)
 

Output:

int(1/(1+3*x+(4*x^2-2*x-3)^(1/2))^3,x)