\(\int \frac {x^2}{1+3 x+\sqrt {-3-2 x+4 x^2}} \, dx\) [32]

Optimal result

Integrand size = 25, antiderivative size = 209 \[ \int \frac {x^2}{1+3 x+\sqrt {-3-2 x+4 x^2}} \, dx=\frac {1}{800} \left (2 x+\sqrt {-3-2 x+4 x^2}\right )+\frac {1}{160} \left (2 x+\sqrt {-3-2 x+4 x^2}\right )^2+\frac {169}{128 \left (1-2 \left (2 x+\sqrt {-3-2 x+4 x^2}\right )\right )^2}+\frac {65}{32 \left (1-2 \left (2 x+\sqrt {-3-2 x+4 x^2}\right )\right )}-\frac {12}{125} \arctan \left (\frac {1}{2} \left (1+5 \left (2 x+\sqrt {-3-2 x+4 x^2}\right )\right )\right )+\frac {92}{125} \log \left (7+3 x-20 x^2-\sqrt {-3-2 x+4 x^2}-10 x \sqrt {-3-2 x+4 x^2}\right )-\frac {23}{16} \log \left (1-2 \left (2 x+\sqrt {-3-2 x+4 x^2}\right )\right ) \] Output:

1/400*x+1/800*(4*x^2-2*x-3)^(1/2)+1/160*(2*x+(4*x^2-2*x-3)^(1/2))^2+169/12 
8/(1-4*x-2*(4*x^2-2*x-3)^(1/2))^2+65/(32-128*x-64*(4*x^2-2*x-3)^(1/2))-12/ 
125*arctan(1/2+5*x+5/2*(4*x^2-2*x-3)^(1/2))+92/125*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))-23/16*ln(1-4*x-2*(4*x^2-2*x-3)^(1/2 
))
 

Mathematica [A] (verified)

Time = 0.39 (sec) , antiderivative size = 117, normalized size of antiderivative = 0.56 \[ \int \frac {x^2}{1+3 x+\sqrt {-3-2 x+4 x^2}} \, dx=\frac {40 x (-38+15 x)+10 (69-20 x) \sqrt {-3-2 x+4 x^2}+192 \arctan \left (\frac {3}{2}+x-\frac {1}{2} \sqrt {-3-2 x+4 x^2}\right )-69 \log \left (1-4 x+2 \sqrt {-3-2 x+4 x^2}\right )+1472 \log \left (-5-5 x-4 x^2+(3+2 x) \sqrt {-3-2 x+4 x^2}\right )}{2000} \] Input:

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

Output:

(40*x*(-38 + 15*x) + 10*(69 - 20*x)*Sqrt[-3 - 2*x + 4*x^2] + 192*ArcTan[3/ 
2 + x - Sqrt[-3 - 2*x + 4*x^2]/2] - 69*Log[1 - 4*x + 2*Sqrt[-3 - 2*x + 4*x 
^2]] + 1472*Log[-5 - 5*x - 4*x^2 + (3 + 2*x)*Sqrt[-3 - 2*x + 4*x^2]])/2000
 

Rubi [A] (verified)

Time = 0.62 (sec) , antiderivative size = 162, normalized size of antiderivative = 0.78, 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.

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

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {8 \sqrt {4 x^2-2 x-3} x}{5 \left (5 x^2+8 x+4\right )}-\frac {1}{5} \sqrt {4 x^2-2 x-3}+\frac {4 (23 x+19)}{25 \left (5 x^2+8 x+4\right )}+\frac {4 \sqrt {4 x^2-2 x-3}}{5 \left (5 x^2+8 x+4\right )}+\frac {3 x}{5}-\frac {19}{25}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {6}{125} \arctan \left (\frac {7 x+8}{2 \sqrt {4 x^2-2 x-3}}\right )+\frac {6}{125} \arctan \left (\frac {5 x}{2}+2\right )+\frac {1403 \text {arctanh}\left (\frac {1-4 x}{2 \sqrt {4 x^2-2 x-3}}\right )}{2000}+\frac {92}{125} \text {arctanh}\left (\frac {3 x+1}{\sqrt {4 x^2-2 x-3}}\right )+\frac {3 x^2}{10}+\frac {1}{40} (1-4 x) \sqrt {4 x^2-2 x-3}+\frac {8}{25} \sqrt {4 x^2-2 x-3}+\frac {46}{125} \log \left (5 x^2+8 x+4\right )-\frac {19 x}{25}\)

Input:

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

Output:

(-19*x)/25 + (3*x^2)/10 + (8*Sqrt[-3 - 2*x + 4*x^2])/25 + ((1 - 4*x)*Sqrt[ 
-3 - 2*x + 4*x^2])/40 + (6*ArcTan[2 + (5*x)/2])/125 + (6*ArcTan[(8 + 7*x)/ 
(2*Sqrt[-3 - 2*x + 4*x^2])])/125 + (1403*ArcTanh[(1 - 4*x)/(2*Sqrt[-3 - 2* 
x + 4*x^2])])/2000 + (92*ArcTanh[(1 + 3*x)/Sqrt[-3 - 2*x + 4*x^2]])/125 + 
(46*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 [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(527\) vs. \(2(170)=340\).

Time = 0.02 (sec) , antiderivative size = 528, normalized size of antiderivative = 2.53

\[-\frac {\left (8 x -2\right ) \sqrt {4 x^{2}-2 x -3}}{80}-\frac {1403 \ln \left (\frac {\left (4 x -1\right ) \sqrt {4}}{4}+\sqrt {4 x^{2}-2 x -3}\right ) \sqrt {4}}{4000}+\frac {8 \sqrt {4 x^{2}-2 x -3}}{25}-\frac {32 \sqrt {-\frac {833 \left (\frac {8}{7}+x \right )^{2}}{\left (-\frac {1}{3}-x \right )^{2}}+1989}\, \left (42 \,\operatorname {arctanh}\left (\frac {\sqrt {-\frac {833 \left (\frac {8}{7}+x \right )^{2}}{\left (-\frac {1}{3}-x \right )^{2}}+1989}}{51}\right )-19 \arctan \left (\frac {7 \sqrt {-\frac {833 \left (\frac {8}{7}+x \right )^{2}}{\left (-\frac {1}{3}-x \right )^{2}}+1989}\, \left (\frac {8}{7}+x \right )}{2 \left (\frac {49 \left (\frac {8}{7}+x \right )^{2}}{\left (-\frac {1}{3}-x \right )^{2}}-117\right ) \left (-\frac {1}{3}-x \right )}\right )\right )}{2125 \sqrt {-\frac {17 \left (\frac {49 \left (\frac {8}{7}+x \right )^{2}}{\left (-\frac {1}{3}-x \right )^{2}}-117\right )}{\left (\frac {\frac {8}{7}+x}{-\frac {1}{3}-x}+1\right )^{2}}}\, \left (\frac {\frac {8}{7}+x}{-\frac {1}{3}-x}+1\right )}-\frac {8 \sqrt {-\frac {833 \left (\frac {8}{7}+x \right )^{2}}{\left (-\frac {1}{3}-x \right )^{2}}+1989}\, \left (13 \,\operatorname {arctanh}\left (\frac {\sqrt {-\frac {833 \left (\frac {8}{7}+x \right )^{2}}{\left (-\frac {1}{3}-x \right )^{2}}+1989}}{51}\right )-16 \arctan \left (\frac {7 \sqrt {-\frac {833 \left (\frac {8}{7}+x \right )^{2}}{\left (-\frac {1}{3}-x \right )^{2}}+1989}\, \left (\frac {8}{7}+x \right )}{2 \left (\frac {49 \left (\frac {8}{7}+x \right )^{2}}{\left (-\frac {1}{3}-x \right )^{2}}-117\right ) \left (-\frac {1}{3}-x \right )}\right )\right )}{425 \sqrt {-\frac {17 \left (\frac {49 \left (\frac {8}{7}+x \right )^{2}}{\left (-\frac {1}{3}-x \right )^{2}}-117\right )}{\left (\frac {\frac {8}{7}+x}{-\frac {1}{3}-x}+1\right )^{2}}}\, \left (\frac {\frac {8}{7}+x}{-\frac {1}{3}-x}+1\right )}+\frac {6 \sqrt {-\frac {833 \left (\frac {8}{7}+x \right )^{2}}{\left (-\frac {1}{3}-x \right )^{2}}+1989}\, \left (2 \,\operatorname {arctanh}\left (\frac {\sqrt {-\frac {833 \left (\frac {8}{7}+x \right )^{2}}{\left (-\frac {1}{3}-x \right )^{2}}+1989}}{51}\right )-9 \arctan \left (\frac {7 \sqrt {-\frac {833 \left (\frac {8}{7}+x \right )^{2}}{\left (-\frac {1}{3}-x \right )^{2}}+1989}\, \left (\frac {8}{7}+x \right )}{2 \left (\frac {49 \left (\frac {8}{7}+x \right )^{2}}{\left (-\frac {1}{3}-x \right )^{2}}-117\right ) \left (-\frac {1}{3}-x \right )}\right )\right )}{85 \sqrt {-\frac {17 \left (\frac {49 \left (\frac {8}{7}+x \right )^{2}}{\left (-\frac {1}{3}-x \right )^{2}}-117\right )}{\left (\frac {\frac {8}{7}+x}{-\frac {1}{3}-x}+1\right )^{2}}}\, \left (\frac {\frac {8}{7}+x}{-\frac {1}{3}-x}+1\right )}-\frac {19 x}{25}+\frac {46 \ln \left (5 x^{2}+8 x +4\right )}{125}+\frac {6 \arctan \left (\frac {5 x}{2}+2\right )}{125}+\frac {3 x^{2}}{10}\]

Input:

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

Output:

-1/80*(8*x-2)*(4*x^2-2*x-3)^(1/2)-1403/4000*ln(1/4*(4*x-1)*4^(1/2)+(4*x^2- 
2*x-3)^(1/2))*4^(1/2)+8/25*(4*x^2-2*x-3)^(1/2)-32/2125*(-833*(8/7+x)^2/(-1 
/3-x)^2+1989)^(1/2)*(42*arctanh(1/51*(-833*(8/7+x)^2/(-1/3-x)^2+1989)^(1/2 
))-19*arctan(7/2/(49*(8/7+x)^2/(-1/3-x)^2-117)*(-833*(8/7+x)^2/(-1/3-x)^2+ 
1989)^(1/2)*(8/7+x)/(-1/3-x)))/(-17*(49*(8/7+x)^2/(-1/3-x)^2-117)/((8/7+x) 
/(-1/3-x)+1)^2)^(1/2)/((8/7+x)/(-1/3-x)+1)-8/425*(-833*(8/7+x)^2/(-1/3-x)^ 
2+1989)^(1/2)*(13*arctanh(1/51*(-833*(8/7+x)^2/(-1/3-x)^2+1989)^(1/2))-16* 
arctan(7/2/(49*(8/7+x)^2/(-1/3-x)^2-117)*(-833*(8/7+x)^2/(-1/3-x)^2+1989)^ 
(1/2)*(8/7+x)/(-1/3-x)))/(-17*(49*(8/7+x)^2/(-1/3-x)^2-117)/((8/7+x)/(-1/3 
-x)+1)^2)^(1/2)/((8/7+x)/(-1/3-x)+1)+6/85*(-833*(8/7+x)^2/(-1/3-x)^2+1989) 
^(1/2)*(2*arctanh(1/51*(-833*(8/7+x)^2/(-1/3-x)^2+1989)^(1/2))-9*arctan(7/ 
2/(49*(8/7+x)^2/(-1/3-x)^2-117)*(-833*(8/7+x)^2/(-1/3-x)^2+1989)^(1/2)*(8/ 
7+x)/(-1/3-x)))/(-17*(49*(8/7+x)^2/(-1/3-x)^2-117)/((8/7+x)/(-1/3-x)+1)^2) 
^(1/2)/((8/7+x)/(-1/3-x)+1)-19/25*x+46/125*ln(5*x^2+8*x+4)+6/125*arctan(5/ 
2*x+2)+3/10*x^2
 

Fricas [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 179, normalized size of antiderivative = 0.86 \[ \int \frac {x^2}{1+3 x+\sqrt {-3-2 x+4 x^2}} \, dx=\frac {3}{10} \, x^{2} - \frac {1}{200} \, \sqrt {4 \, x^{2} - 2 \, x - 3} {\left (20 \, x - 69\right )} - \frac {19}{25} \, x + \frac {6}{125} \, \arctan \left (\frac {5}{2} \, x + 2\right ) - \frac {6}{125} \, \arctan \left (-x + \frac {1}{2} \, \sqrt {4 \, x^{2} - 2 \, x - 3} - \frac {3}{2}\right ) - \frac {6}{125} \, \arctan \left (-5 \, x + \frac {5}{2} \, \sqrt {4 \, x^{2} - 2 \, x - 3} - \frac {1}{2}\right ) - \frac {46}{125} \, \log \left (20 \, x^{2} - \sqrt {4 \, x^{2} - 2 \, x - 3} {\left (10 \, x + 1\right )} - 3 \, x - 7\right ) + \frac {46}{125} \, \log \left (5 \, x^{2} + 8 \, x + 4\right ) + \frac {46}{125} \, \log \left (4 \, x^{2} - \sqrt {4 \, x^{2} - 2 \, x - 3} {\left (2 \, x + 3\right )} + 5 \, x + 5\right ) + \frac {1403}{2000} \, \log \left (-4 \, x + 2 \, \sqrt {4 \, x^{2} - 2 \, x - 3} + 1\right ) \] Input:

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

Output:

3/10*x^2 - 1/200*sqrt(4*x^2 - 2*x - 3)*(20*x - 69) - 19/25*x + 6/125*arcta 
n(5/2*x + 2) - 6/125*arctan(-x + 1/2*sqrt(4*x^2 - 2*x - 3) - 3/2) - 6/125* 
arctan(-5*x + 5/2*sqrt(4*x^2 - 2*x - 3) - 1/2) - 46/125*log(20*x^2 - sqrt( 
4*x^2 - 2*x - 3)*(10*x + 1) - 3*x - 7) + 46/125*log(5*x^2 + 8*x + 4) + 46/ 
125*log(4*x^2 - sqrt(4*x^2 - 2*x - 3)*(2*x + 3) + 5*x + 5) + 1403/2000*log 
(-4*x + 2*sqrt(4*x^2 - 2*x - 3) + 1)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [A] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 202, normalized size of antiderivative = 0.97 \[ \int \frac {x^2}{1+3 x+\sqrt {-3-2 x+4 x^2}} \, dx=\frac {3}{10} \, x^{2} - \frac {1}{200} \, \sqrt {4 \, x^{2} - 2 \, x - 3} {\left (20 \, x - 69\right )} - \frac {19}{25} \, x + \frac {6}{125} \, \arctan \left (\frac {5}{2} \, x + 2\right ) - \frac {6}{125} \, \arctan \left (-x + \frac {1}{2} \, \sqrt {4 \, x^{2} - 2 \, x - 3} - \frac {3}{2}\right ) - \frac {6}{125} \, \arctan \left (-5 \, x + \frac {5}{2} \, \sqrt {4 \, x^{2} - 2 \, x - 3} - \frac {1}{2}\right ) - \frac {46}{125} \, \log \left (5 \, {\left (2 \, x - \sqrt {4 \, x^{2} - 2 \, x - 3}\right )}^{2} + 4 \, x - 2 \, \sqrt {4 \, x^{2} - 2 \, x - 3} + 1\right ) + \frac {46}{125} \, \log \left ({\left (2 \, x - \sqrt {4 \, x^{2} - 2 \, x - 3}\right )}^{2} + 12 \, x - 6 \, \sqrt {4 \, x^{2} - 2 \, x - 3} + 13\right ) + \frac {46}{125} \, \log \left (5 \, x^{2} + 8 \, x + 4\right ) + \frac {1403}{2000} \, \log \left ({\left | -4 \, x + 2 \, \sqrt {4 \, x^{2} - 2 \, x - 3} + 1 \right |}\right ) \] Input:

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

Output:

3/10*x^2 - 1/200*sqrt(4*x^2 - 2*x - 3)*(20*x - 69) - 19/25*x + 6/125*arcta 
n(5/2*x + 2) - 6/125*arctan(-x + 1/2*sqrt(4*x^2 - 2*x - 3) - 3/2) - 6/125* 
arctan(-5*x + 5/2*sqrt(4*x^2 - 2*x - 3) - 1/2) - 46/125*log(5*(2*x - sqrt( 
4*x^2 - 2*x - 3))^2 + 4*x - 2*sqrt(4*x^2 - 2*x - 3) + 1) + 46/125*log((2*x 
 - sqrt(4*x^2 - 2*x - 3))^2 + 12*x - 6*sqrt(4*x^2 - 2*x - 3) + 13) + 46/12 
5*log(5*x^2 + 8*x + 4) + 1403/2000*log(abs(-4*x + 2*sqrt(4*x^2 - 2*x - 3) 
+ 1))
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 129, normalized size of antiderivative = 0.62 \[ \int \frac {x^2}{1+3 x+\sqrt {-3-2 x+4 x^2}} \, dx=-\frac {12 \mathit {atan} \left (\frac {5 \sqrt {4 x^{2}-2 x -3}}{2}+5 x +\frac {1}{2}\right )}{125}-\frac {\sqrt {4 x^{2}-2 x -3}\, x}{10}+\frac {69 \sqrt {4 x^{2}-2 x -3}}{200}+\frac {92 \,\mathrm {log}\left (\frac {80 \sqrt {4 x^{2}-2 x -3}\, x +8 \sqrt {4 x^{2}-2 x -3}+160 x^{2}-24 x -56}{\sqrt {13}}\right )}{125}-\frac {23 \,\mathrm {log}\left (\frac {2 \sqrt {4 x^{2}-2 x -3}+4 x -1}{\sqrt {13}}\right )}{16}+\frac {3 x^{2}}{10}-\frac {19 x}{25}+\frac {79}{1600} \] Input:

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

Output:

( - 768*atan((5*sqrt(4*x**2 - 2*x - 3) + 10*x + 1)/2) - 800*sqrt(4*x**2 - 
2*x - 3)*x + 2760*sqrt(4*x**2 - 2*x - 3) + 5888*log((80*sqrt(4*x**2 - 2*x 
- 3)*x + 8*sqrt(4*x**2 - 2*x - 3) + 160*x**2 - 24*x - 56)/sqrt(13)) - 1150 
0*log((2*sqrt(4*x**2 - 2*x - 3) + 4*x - 1)/sqrt(13)) + 2400*x**2 - 6080*x 
+ 395)/8000