\(\int \frac {x^2}{(1+3 x+\sqrt {-3-2 x+4 x^2})^2} \, dx\) [37]

Optimal result

Integrand size = 25, antiderivative size = 227 \[ \int \frac {x^2}{\left (1+3 x+\sqrt {-3-2 x+4 x^2}\right )^2} \, dx=\frac {1}{100} \left (2 x+\sqrt {-3-2 x+4 x^2}\right )-\frac {13}{8 \left (1-2 \left (2 x+\sqrt {-3-2 x+4 x^2}\right )\right )}+\frac {2 \left (101-163 \left (2 x+\sqrt {-3-2 x+4 x^2}\right )\right )}{125 \left (1+2 \left (2 x+\sqrt {-3-2 x+4 x^2}\right )+5 \left (2 x+\sqrt {-3-2 x+4 x^2}\right )^2\right )}-\frac {557}{125} \arctan \left (\frac {1}{2} \left (1+5 \left (2 x+\sqrt {-3-2 x+4 x^2}\right )\right )\right )-\frac {188}{125} \log \left (7+3 x-20 x^2-\sqrt {-3-2 x+4 x^2}-10 x \sqrt {-3-2 x+4 x^2}\right )+3 \log \left (1-2 \left (2 x+\sqrt {-3-2 x+4 x^2}\right )\right ) \] Output:

1/50*x+1/100*(4*x^2-2*x-3)^(1/2)-13/(8-32*x-16*(4*x^2-2*x-3)^(1/2))+2*(101 
-326*x-163*(4*x^2-2*x-3)^(1/2))/(125+500*x+250*(4*x^2-2*x-3)^(1/2)+625*(2* 
x+(4*x^2-2*x-3)^(1/2))^2)-557/125*arctan(1/2+5*x+5/2*(4*x^2-2*x-3)^(1/2))- 
188/125*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))+3*ln 
(1-4*x-2*(4*x^2-2*x-3)^(1/2))
 

Mathematica [A] (verified)

Time = 0.45 (sec) , antiderivative size = 152, normalized size of antiderivative = 0.67 \[ \int \frac {x^2}{\left (1+3 x+\sqrt {-3-2 x+4 x^2}\right )^2} \, dx=\frac {1}{125} \left (-\frac {5 \sqrt {-3-2 x+4 x^2} \left (8+51 x+30 x^2\right )}{4+8 x+5 x^2}+\frac {-116-37 x+520 x^2+325 x^3}{4+8 x+5 x^2}+557 \arctan \left (\frac {3}{2}+x-\frac {1}{2} \sqrt {-3-2 x+4 x^2}\right )+\log \left (1-4 x+2 \sqrt {-3-2 x+4 x^2}\right )-188 \log \left (-5-5 x-4 x^2+(3+2 x) \sqrt {-3-2 x+4 x^2}\right )\right ) \] Input:

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

Output:

((-5*Sqrt[-3 - 2*x + 4*x^2]*(8 + 51*x + 30*x^2))/(4 + 8*x + 5*x^2) + (-116 
 - 37*x + 520*x^2 + 325*x^3)/(4 + 8*x + 5*x^2) + 557*ArcTan[3/2 + x - Sqrt 
[-3 - 2*x + 4*x^2]/2] + Log[1 - 4*x + 2*Sqrt[-3 - 2*x + 4*x^2]] - 188*Log[ 
-5 - 5*x - 4*x^2 + (3 + 2*x)*Sqrt[-3 - 2*x + 4*x^2]])/125
 

Rubi [A] (verified)

Time = 0.78 (sec) , antiderivative size = 221, normalized size of antiderivative = 0.97, 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.

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

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (-\frac {6 \sqrt {4 x^2-2 x-3} x}{5 \left (5 x^2+8 x+4\right )}-\frac {184 \sqrt {4 x^2-2 x-3} x}{25 \left (5 x^2+8 x+4\right )^2}-\frac {2 (470 x-51)}{125 \left (5 x^2+8 x+4\right )}+\frac {38 \sqrt {4 x^2-2 x-3}}{25 \left (5 x^2+8 x+4\right )}-\frac {8 (152 x+181)}{125 \left (5 x^2+8 x+4\right )^2}-\frac {152 \sqrt {4 x^2-2 x-3}}{25 \left (5 x^2+8 x+4\right )^2}+\frac {13}{25}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {557}{250} \arctan \left (\frac {7 x+8}{2 \sqrt {4 x^2-2 x-3}}\right )+\frac {557}{250} \arctan \left (\frac {5 x}{2}+2\right )-\frac {187}{125} \text {arctanh}\left (\frac {1-4 x}{2 \sqrt {4 x^2-2 x-3}}\right )-\frac {188}{125} \text {arctanh}\left (\frac {3 x+1}{\sqrt {4 x^2-2 x-3}}\right )-\frac {6}{25} \sqrt {4 x^2-2 x-3}-\frac {297 x+116}{125 \left (5 x^2+8 x+4\right )}+\frac {92 (x+1) \sqrt {4 x^2-2 x-3}}{25 \left (5 x^2+8 x+4\right )}-\frac {19 (5 x+4) \sqrt {4 x^2-2 x-3}}{25 \left (5 x^2+8 x+4\right )}-\frac {94}{125} \log \left (5 x^2+8 x+4\right )+\frac {13 x}{25}\)

Input:

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

Output:

(13*x)/25 - (6*Sqrt[-3 - 2*x + 4*x^2])/25 - (116 + 297*x)/(125*(4 + 8*x + 
5*x^2)) + (92*(1 + x)*Sqrt[-3 - 2*x + 4*x^2])/(25*(4 + 8*x + 5*x^2)) - (19 
*(4 + 5*x)*Sqrt[-3 - 2*x + 4*x^2])/(25*(4 + 8*x + 5*x^2)) + (557*ArcTan[2 
+ (5*x)/2])/250 + (557*ArcTan[(8 + 7*x)/(2*Sqrt[-3 - 2*x + 4*x^2])])/250 - 
 (187*ArcTanh[(1 - 4*x)/(2*Sqrt[-3 - 2*x + 4*x^2])])/125 - (188*ArcTanh[(1 
 + 3*x)/Sqrt[-3 - 2*x + 4*x^2]])/125 - (94*Log[4 + 8*x + 5*x^2])/125
 

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 [F(-1)]

Timed out.

hanged

Input:

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

Output:

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

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 274, normalized size of antiderivative = 1.21 \[ \int \frac {x^2}{\left (1+3 x+\sqrt {-3-2 x+4 x^2}\right )^2} \, dx=\frac {650 \, x^{3} + 1055 \, x^{2} + 557 \, {\left (5 \, x^{2} + 8 \, x + 4\right )} \arctan \left (\frac {5}{2} \, x + 2\right ) - 557 \, {\left (5 \, x^{2} + 8 \, x + 4\right )} \arctan \left (-x + \frac {1}{2} \, \sqrt {4 \, x^{2} - 2 \, x - 3} - \frac {3}{2}\right ) - 557 \, {\left (5 \, x^{2} + 8 \, x + 4\right )} \arctan \left (-5 \, x + \frac {5}{2} \, \sqrt {4 \, x^{2} - 2 \, x - 3} - \frac {1}{2}\right ) + 188 \, {\left (5 \, x^{2} + 8 \, x + 4\right )} \log \left (20 \, x^{2} - \sqrt {4 \, x^{2} - 2 \, x - 3} {\left (10 \, x + 1\right )} - 3 \, x - 7\right ) - 188 \, {\left (5 \, x^{2} + 8 \, x + 4\right )} \log \left (5 \, x^{2} + 8 \, x + 4\right ) - 188 \, {\left (5 \, x^{2} + 8 \, x + 4\right )} \log \left (4 \, x^{2} - \sqrt {4 \, x^{2} - 2 \, x - 3} {\left (2 \, x + 3\right )} + 5 \, x + 5\right ) - 374 \, {\left (5 \, x^{2} + 8 \, x + 4\right )} \log \left (-4 \, x + 2 \, \sqrt {4 \, x^{2} - 2 \, x - 3} + 1\right ) - 10 \, {\left (30 \, x^{2} + 51 \, x + 8\right )} \sqrt {4 \, x^{2} - 2 \, x - 3} - 50 \, x - 220}{250 \, {\left (5 \, x^{2} + 8 \, x + 4\right )}} \] Input:

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

Output:

1/250*(650*x^3 + 1055*x^2 + 557*(5*x^2 + 8*x + 4)*arctan(5/2*x + 2) - 557* 
(5*x^2 + 8*x + 4)*arctan(-x + 1/2*sqrt(4*x^2 - 2*x - 3) - 3/2) - 557*(5*x^ 
2 + 8*x + 4)*arctan(-5*x + 5/2*sqrt(4*x^2 - 2*x - 3) - 1/2) + 188*(5*x^2 + 
 8*x + 4)*log(20*x^2 - sqrt(4*x^2 - 2*x - 3)*(10*x + 1) - 3*x - 7) - 188*( 
5*x^2 + 8*x + 4)*log(5*x^2 + 8*x + 4) - 188*(5*x^2 + 8*x + 4)*log(4*x^2 - 
sqrt(4*x^2 - 2*x - 3)*(2*x + 3) + 5*x + 5) - 374*(5*x^2 + 8*x + 4)*log(-4* 
x + 2*sqrt(4*x^2 - 2*x - 3) + 1) - 10*(30*x^2 + 51*x + 8)*sqrt(4*x^2 - 2*x 
 - 3) - 50*x - 220)/(5*x^2 + 8*x + 4)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 363, normalized size of antiderivative = 1.60 \[ \int \frac {x^2}{\left (1+3 x+\sqrt {-3-2 x+4 x^2}\right )^2} \, dx=\frac {13}{25} \, x - \frac {6}{25} \, \sqrt {4 \, x^{2} - 2 \, x - 3} - \frac {2 \, {\left (431 \, {\left (2 \, x - \sqrt {4 \, x^{2} - 2 \, x - 3}\right )}^{3} - 246 \, {\left (2 \, x - \sqrt {4 \, x^{2} - 2 \, x - 3}\right )}^{2} - 1318 \, x + 659 \, \sqrt {4 \, x^{2} - 2 \, x - 3} + 654\right )}}{125 \, {\left (5 \, {\left (2 \, x - \sqrt {4 \, x^{2} - 2 \, x - 3}\right )}^{4} + 32 \, {\left (2 \, x - \sqrt {4 \, x^{2} - 2 \, x - 3}\right )}^{3} + 78 \, {\left (2 \, x - \sqrt {4 \, x^{2} - 2 \, x - 3}\right )}^{2} + 64 \, x - 32 \, \sqrt {4 \, x^{2} - 2 \, x - 3} + 13\right )}} - \frac {297 \, x + 116}{125 \, {\left (5 \, x^{2} + 8 \, x + 4\right )}} + \frac {557}{250} \, \arctan \left (\frac {5}{2} \, x + 2\right ) - \frac {557}{250} \, \arctan \left (-x + \frac {1}{2} \, \sqrt {4 \, x^{2} - 2 \, x - 3} - \frac {3}{2}\right ) - \frac {557}{250} \, \arctan \left (-5 \, x + \frac {5}{2} \, \sqrt {4 \, x^{2} - 2 \, x - 3} - \frac {1}{2}\right ) + \frac {94}{125} \, \log \left (5 \, {\left (2 \, x - \sqrt {4 \, x^{2} - 2 \, x - 3}\right )}^{2} + 4 \, x - 2 \, \sqrt {4 \, x^{2} - 2 \, x - 3} + 1\right ) - \frac {94}{125} \, \log \left ({\left (2 \, x - \sqrt {4 \, x^{2} - 2 \, x - 3}\right )}^{2} + 12 \, x - 6 \, \sqrt {4 \, x^{2} - 2 \, x - 3} + 13\right ) - \frac {94}{125} \, \log \left (5 \, x^{2} + 8 \, x + 4\right ) - \frac {187}{125} \, \log \left ({\left | -4 \, x + 2 \, \sqrt {4 \, x^{2} - 2 \, x - 3} + 1 \right |}\right ) \] Input:

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

Output:

13/25*x - 6/25*sqrt(4*x^2 - 2*x - 3) - 2/125*(431*(2*x - sqrt(4*x^2 - 2*x 
- 3))^3 - 246*(2*x - sqrt(4*x^2 - 2*x - 3))^2 - 1318*x + 659*sqrt(4*x^2 - 
2*x - 3) + 654)/(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) - 1/125*(297*x + 116)/(5*x^2 + 8*x + 4) + 557/250*arctan( 
5/2*x + 2) - 557/250*arctan(-x + 1/2*sqrt(4*x^2 - 2*x - 3) - 3/2) - 557/25 
0*arctan(-5*x + 5/2*sqrt(4*x^2 - 2*x - 3) - 1/2) + 94/125*log(5*(2*x - sqr 
t(4*x^2 - 2*x - 3))^2 + 4*x - 2*sqrt(4*x^2 - 2*x - 3) + 1) - 94/125*log((2 
*x - sqrt(4*x^2 - 2*x - 3))^2 + 12*x - 6*sqrt(4*x^2 - 2*x - 3) + 13) - 94/ 
125*log(5*x^2 + 8*x + 4) - 187/125*log(abs(-4*x + 2*sqrt(4*x^2 - 2*x - 3) 
+ 1))
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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