\(\int \frac {x^2}{(1+3 x+\sqrt {-3-2 x+4 x^2})^3} \, dx\) [42]

Optimal result

Integrand size = 25, antiderivative size = 249 \[ \int \frac {x^2}{\left (1+3 x+\sqrt {-3-2 x+4 x^2}\right )^3} \, dx=-\frac {2 \left (179-2477 \left (2 x+\sqrt {-3-2 x+4 x^2}\right )\right )}{625 \left (1+2 \left (2 x+\sqrt {-3-2 x+4 x^2}\right )+5 \left (2 x+\sqrt {-3-2 x+4 x^2}\right )^2\right )^2}+\frac {3 \left (2129+17685 \left (2 x+\sqrt {-3-2 x+4 x^2}\right )\right )}{2500 \left (1+2 \left (2 x+\sqrt {-3-2 x+4 x^2}\right )+5 \left (2 x+\sqrt {-3-2 x+4 x^2}\right )^2\right )}+\frac {12331 \arctan \left (\frac {1}{2} \left (1+5 \left (2 x+\sqrt {-3-2 x+4 x^2}\right )\right )\right )}{1000}+\frac {63}{125} \log \left (7+3 x-20 x^2-\sqrt {-3-2 x+4 x^2}-10 x \sqrt {-3-2 x+4 x^2}\right )-\log \left (1-2 \left (2 x+\sqrt {-3-2 x+4 x^2}\right )\right ) \] Output:

1/625*(-358+9908*x+4954*(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+3*(2129+35370*x+17685*(4*x^2-2*x-3)^(1/2) 
)/(2500+10000*x+5000*(4*x^2-2*x-3)^(1/2)+12500*(2*x+(4*x^2-2*x-3)^(1/2))^2 
)+12331/1000*arctan(1/2+5*x+5/2*(4*x^2-2*x-3)^(1/2))+63/125*ln(7+3*x-20*x^ 
2-(4*x^2-2*x-3)^(1/2)-10*x*(4*x^2-2*x-3)^(1/2))-ln(1-4*x-2*(4*x^2-2*x-3)^( 
1/2))
 

Mathematica [A] (verified)

Time = 0.54 (sec) , antiderivative size = 160, normalized size of antiderivative = 0.64 \[ \int \frac {x^2}{\left (1+3 x+\sqrt {-3-2 x+4 x^2}\right )^3} \, dx=\frac {-\frac {5 \sqrt {-3-2 x+4 x^2} \left (4544+13356 x+14656 x^2+5465 x^3\right )}{\left (4+8 x+5 x^2\right )^2}-\frac {28464+72596 x+66276 x^2+15455 x^3}{\left (4+8 x+5 x^2\right )^2}-12331 \arctan \left (\frac {3}{2}+x-\frac {1}{2} \sqrt {-3-2 x+4 x^2}\right )-8 \log \left (1-4 x+2 \sqrt {-3-2 x+4 x^2}\right )+504 \log \left (-5-5 x-4 x^2+(3+2 x) \sqrt {-3-2 x+4 x^2}\right )}{1000} \] Input:

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

Output:

((-5*Sqrt[-3 - 2*x + 4*x^2]*(4544 + 13356*x + 14656*x^2 + 5465*x^3))/(4 + 
8*x + 5*x^2)^2 - (28464 + 72596*x + 66276*x^2 + 15455*x^3)/(4 + 8*x + 5*x^ 
2)^2 - 12331*ArcTan[3/2 + x - Sqrt[-3 - 2*x + 4*x^2]/2] - 8*Log[1 - 4*x + 
2*Sqrt[-3 - 2*x + 4*x^2]] + 504*Log[-5 - 5*x - 4*x^2 + (3 + 2*x)*Sqrt[-3 - 
 2*x + 4*x^2]])/1000
 

Rubi [A] (verified)

Time = 1.00 (sec) , antiderivative size = 378, normalized size of antiderivative = 1.52, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, 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.

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

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {416 \sqrt {4 x^2-2 x-3} x}{25 \left (5 x^2+8 x+4\right )^2}+\frac {2432 \sqrt {4 x^2-2 x-3} x}{125 \left (5 x^2+8 x+4\right )^3}+\frac {21 (15 x-43)}{125 \left (5 x^2+8 x+4\right )}-\frac {31 \sqrt {4 x^2-2 x-3}}{25 \left (5 x^2+8 x+4\right )}+\frac {16 (2280 x+899)}{625 \left (5 x^2+8 x+4\right )^2}-\frac {104 \sqrt {4 x^2-2 x-3}}{125 \left (5 x^2+8 x+4\right )^2}+\frac {16 (173 x+919)}{625 \left (5 x^2+8 x+4\right )^3}+\frac {2896 \sqrt {4 x^2-2 x-3}}{125 \left (5 x^2+8 x+4\right )^3}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {12331 \arctan \left (\frac {7 x+8}{2 \sqrt {4 x^2-2 x-3}}\right )}{2000}-\frac {12331 \arctan \left (\frac {5 x}{2}+2\right )}{2000}+\frac {62}{125} \text {arctanh}\left (\frac {1-4 x}{2 \sqrt {4 x^2-2 x-3}}\right )+\frac {63}{125} \text {arctanh}\left (\frac {3 x+1}{\sqrt {4 x^2-2 x-3}}\right )-\frac {208 \sqrt {4 x^2-2 x-3} (x+1)}{25 \left (5 x^2+8 x+4\right )}-\frac {608 \sqrt {4 x^2-2 x-3} (x+1)}{125 \left (5 x^2+8 x+4\right )^2}+\frac {11709 (5 x+4)}{5000 \left (5 x^2+8 x+4\right )}-\frac {2 (4625 x+5524)}{625 \left (5 x^2+8 x+4\right )}-\frac {13 (5 x+4) \sqrt {4 x^2-2 x-3}}{125 \left (5 x^2+8 x+4\right )}-\frac {304 (965 x+701) \sqrt {4 x^2-2 x-3}}{36125 \left (5 x^2+8 x+4\right )}+\frac {181 (18355 x+13928) \sqrt {4 x^2-2 x-3}}{289000 \left (5 x^2+8 x+4\right )}+\frac {3903 x+2984}{625 \left (5 x^2+8 x+4\right )^2}+\frac {181 (5 x+4) \sqrt {4 x^2-2 x-3}}{125 \left (5 x^2+8 x+4\right )^2}+\frac {63}{250} \log \left (5 x^2+8 x+4\right )\)

Input:

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

Output:

(2984 + 3903*x)/(625*(4 + 8*x + 5*x^2)^2) - (608*(1 + x)*Sqrt[-3 - 2*x + 4 
*x^2])/(125*(4 + 8*x + 5*x^2)^2) + (181*(4 + 5*x)*Sqrt[-3 - 2*x + 4*x^2])/ 
(125*(4 + 8*x + 5*x^2)^2) + (11709*(4 + 5*x))/(5000*(4 + 8*x + 5*x^2)) - ( 
2*(5524 + 4625*x))/(625*(4 + 8*x + 5*x^2)) - (208*(1 + x)*Sqrt[-3 - 2*x + 
4*x^2])/(25*(4 + 8*x + 5*x^2)) - (13*(4 + 5*x)*Sqrt[-3 - 2*x + 4*x^2])/(12 
5*(4 + 8*x + 5*x^2)) - (304*(701 + 965*x)*Sqrt[-3 - 2*x + 4*x^2])/(36125*( 
4 + 8*x + 5*x^2)) + (181*(13928 + 18355*x)*Sqrt[-3 - 2*x + 4*x^2])/(289000 
*(4 + 8*x + 5*x^2)) - (12331*ArcTan[2 + (5*x)/2])/2000 - (12331*ArcTan[(8 
+ 7*x)/(2*Sqrt[-3 - 2*x + 4*x^2])])/2000 + (62*ArcTanh[(1 - 4*x)/(2*Sqrt[- 
3 - 2*x + 4*x^2])])/125 + (63*ArcTanh[(1 + 3*x)/Sqrt[-3 - 2*x + 4*x^2]])/1 
25 + (63*Log[4 + 8*x + 5*x^2])/250
 

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 [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(13827\) vs. \(2(211)=422\).

Time = 0.29 (sec) , antiderivative size = 13828, normalized size of antiderivative = 55.53

\[\text {output too large to display}\]

Input:

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

Output:

result too large to display
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 364, normalized size of antiderivative = 1.46 \[ \int \frac {x^2}{\left (1+3 x+\sqrt {-3-2 x+4 x^2}\right )^3} \, dx=-\frac {109300 \, x^{4} + 380670 \, x^{3} + 587240 \, x^{2} + 12331 \, {\left (25 \, x^{4} + 80 \, x^{3} + 104 \, x^{2} + 64 \, x + 16\right )} \arctan \left (\frac {5}{2} \, x + 2\right ) - 12331 \, {\left (25 \, x^{4} + 80 \, x^{3} + 104 \, x^{2} + 64 \, x + 16\right )} \arctan \left (-x + \frac {1}{2} \, \sqrt {4 \, x^{2} - 2 \, x - 3} - \frac {3}{2}\right ) - 12331 \, {\left (25 \, x^{4} + 80 \, x^{3} + 104 \, x^{2} + 64 \, x + 16\right )} \arctan \left (-5 \, x + \frac {5}{2} \, \sqrt {4 \, x^{2} - 2 \, x - 3} - \frac {1}{2}\right ) + 504 \, {\left (25 \, x^{4} + 80 \, x^{3} + 104 \, x^{2} + 64 \, x + 16\right )} \log \left (20 \, x^{2} - \sqrt {4 \, x^{2} - 2 \, x - 3} {\left (10 \, x + 1\right )} - 3 \, x - 7\right ) - 504 \, {\left (25 \, x^{4} + 80 \, x^{3} + 104 \, x^{2} + 64 \, x + 16\right )} \log \left (5 \, x^{2} + 8 \, x + 4\right ) - 504 \, {\left (25 \, x^{4} + 80 \, x^{3} + 104 \, x^{2} + 64 \, x + 16\right )} \log \left (4 \, x^{2} - \sqrt {4 \, x^{2} - 2 \, x - 3} {\left (2 \, x + 3\right )} + 5 \, x + 5\right ) - 992 \, {\left (25 \, x^{4} + 80 \, x^{3} + 104 \, x^{2} + 64 \, x + 16\right )} \log \left (-4 \, x + 2 \, \sqrt {4 \, x^{2} - 2 \, x - 3} + 1\right ) + 10 \, {\left (5465 \, x^{3} + 14656 \, x^{2} + 13356 \, x + 4544\right )} \sqrt {4 \, x^{2} - 2 \, x - 3} + 425000 \, x + 126880}{2000 \, {\left (25 \, x^{4} + 80 \, x^{3} + 104 \, x^{2} + 64 \, x + 16\right )}} \] Input:

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

Output:

-1/2000*(109300*x^4 + 380670*x^3 + 587240*x^2 + 12331*(25*x^4 + 80*x^3 + 1 
04*x^2 + 64*x + 16)*arctan(5/2*x + 2) - 12331*(25*x^4 + 80*x^3 + 104*x^2 + 
 64*x + 16)*arctan(-x + 1/2*sqrt(4*x^2 - 2*x - 3) - 3/2) - 12331*(25*x^4 + 
 80*x^3 + 104*x^2 + 64*x + 16)*arctan(-5*x + 5/2*sqrt(4*x^2 - 2*x - 3) - 1 
/2) + 504*(25*x^4 + 80*x^3 + 104*x^2 + 64*x + 16)*log(20*x^2 - sqrt(4*x^2 
- 2*x - 3)*(10*x + 1) - 3*x - 7) - 504*(25*x^4 + 80*x^3 + 104*x^2 + 64*x + 
 16)*log(5*x^2 + 8*x + 4) - 504*(25*x^4 + 80*x^3 + 104*x^2 + 64*x + 16)*lo 
g(4*x^2 - sqrt(4*x^2 - 2*x - 3)*(2*x + 3) + 5*x + 5) - 992*(25*x^4 + 80*x^ 
3 + 104*x^2 + 64*x + 16)*log(-4*x + 2*sqrt(4*x^2 - 2*x - 3) + 1) + 10*(546 
5*x^3 + 14656*x^2 + 13356*x + 4544)*sqrt(4*x^2 - 2*x - 3) + 425000*x + 126 
880)/(25*x^4 + 80*x^3 + 104*x^2 + 64*x + 16)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

integrate(x^2/(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. 444 vs. \(2 (211) = 422\).

Time = 0.18 (sec) , antiderivative size = 444, normalized size of antiderivative = 1.78 \[ \int \frac {x^2}{\left (1+3 x+\sqrt {-3-2 x+4 x^2}\right )^3} \, dx=-\frac {83965 \, {\left (2 \, x - \sqrt {4 \, x^{2} - 2 \, x - 3}\right )}^{7} + 1179022 \, {\left (2 \, x - \sqrt {4 \, x^{2} - 2 \, x - 3}\right )}^{6} + 5708901 \, {\left (2 \, x - \sqrt {4 \, x^{2} - 2 \, x - 3}\right )}^{5} + 13528366 \, {\left (2 \, x - \sqrt {4 \, x^{2} - 2 \, x - 3}\right )}^{4} + 13425247 \, {\left (2 \, x - \sqrt {4 \, x^{2} - 2 \, x - 3}\right )}^{3} + 7270266 \, {\left (2 \, x - \sqrt {4 \, x^{2} - 2 \, x - 3}\right )}^{2} + 5034622 \, x - 2517311 \, \sqrt {4 \, x^{2} - 2 \, x - 3} + 299722}{500 \, {\left (5 \, {\left (2 \, x - \sqrt {4 \, x^{2} - 2 \, x - 3}\right )}^{4} + 32 \, {\left (2 \, x - \sqrt {4 \, x^{2} - 2 \, x - 3}\right )}^{3} + 78 \, {\left (2 \, x - \sqrt {4 \, x^{2} - 2 \, x - 3}\right )}^{2} + 64 \, x - 32 \, \sqrt {4 \, x^{2} - 2 \, x - 3} + 13\right )}^{2}} - \frac {15455 \, x^{3} + 66276 \, x^{2} + 72596 \, x + 28464}{1000 \, {\left (5 \, x^{2} + 8 \, x + 4\right )}^{2}} - \frac {12331}{2000} \, \arctan \left (\frac {5}{2} \, x + 2\right ) + \frac {12331}{2000} \, \arctan \left (-x + \frac {1}{2} \, \sqrt {4 \, x^{2} - 2 \, x - 3} - \frac {3}{2}\right ) + \frac {12331}{2000} \, \arctan \left (-5 \, x + \frac {5}{2} \, \sqrt {4 \, x^{2} - 2 \, x - 3} - \frac {1}{2}\right ) - \frac {63}{250} \, \log \left (5 \, {\left (2 \, x - \sqrt {4 \, x^{2} - 2 \, x - 3}\right )}^{2} + 4 \, x - 2 \, \sqrt {4 \, x^{2} - 2 \, x - 3} + 1\right ) + \frac {63}{250} \, \log \left ({\left (2 \, x - \sqrt {4 \, x^{2} - 2 \, x - 3}\right )}^{2} + 12 \, x - 6 \, \sqrt {4 \, x^{2} - 2 \, x - 3} + 13\right ) + \frac {63}{250} \, \log \left (5 \, x^{2} + 8 \, x + 4\right ) + \frac {62}{125} \, \log \left ({\left | -4 \, x + 2 \, \sqrt {4 \, x^{2} - 2 \, x - 3} + 1 \right |}\right ) \] Input:

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

Output:

-1/500*(83965*(2*x - sqrt(4*x^2 - 2*x - 3))^7 + 1179022*(2*x - sqrt(4*x^2 
- 2*x - 3))^6 + 5708901*(2*x - sqrt(4*x^2 - 2*x - 3))^5 + 13528366*(2*x - 
sqrt(4*x^2 - 2*x - 3))^4 + 13425247*(2*x - sqrt(4*x^2 - 2*x - 3))^3 + 7270 
266*(2*x - sqrt(4*x^2 - 2*x - 3))^2 + 5034622*x - 2517311*sqrt(4*x^2 - 2*x 
 - 3) + 299722)/(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 - 1/1000*(15455*x^3 + 66276*x^2 + 72596*x + 28464)/(5*x 
^2 + 8*x + 4)^2 - 12331/2000*arctan(5/2*x + 2) + 12331/2000*arctan(-x + 1/ 
2*sqrt(4*x^2 - 2*x - 3) - 3/2) + 12331/2000*arctan(-5*x + 5/2*sqrt(4*x^2 - 
 2*x - 3) - 1/2) - 63/250*log(5*(2*x - sqrt(4*x^2 - 2*x - 3))^2 + 4*x - 2* 
sqrt(4*x^2 - 2*x - 3) + 1) + 63/250*log((2*x - sqrt(4*x^2 - 2*x - 3))^2 + 
12*x - 6*sqrt(4*x^2 - 2*x - 3) + 13) + 63/250*log(5*x^2 + 8*x + 4) + 62/12 
5*log(abs(-4*x + 2*sqrt(4*x^2 - 2*x - 3) + 1))
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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