\(\int \frac {x^2}{(1+2 x+\sqrt {-3-2 x+4 x^2})^3} \, dx\) [12]

Optimal result

Integrand size = 25, antiderivative size = 186 \[ \int \frac {x^2}{\left (1+2 x+\sqrt {-3-2 x+4 x^2}\right )^3} \, dx=\frac {2}{81 \left (1+2 x+\sqrt {-3-2 x+4 x^2}\right )^2}+\frac {56}{243 \left (1+2 x+\sqrt {-3-2 x+4 x^2}\right )}-\frac {2197}{5184 \left (1-2 \left (2 x+\sqrt {-3-2 x+4 x^2}\right )\right )^3}-\frac {169}{384 \left (1-2 \left (2 x+\sqrt {-3-2 x+4 x^2}\right )\right )^2}-\frac {3029}{5184 \left (1-2 \left (2 x+\sqrt {-3-2 x+4 x^2}\right )\right )}-\frac {235}{729} \log \left (1+2 x+\sqrt {-3-2 x+4 x^2}\right )+\frac {15769 \log \left (1-2 \left (2 x+\sqrt {-3-2 x+4 x^2}\right )\right )}{46656} \] Output:

2/81/(1+2*x+(4*x^2-2*x-3)^(1/2))^2+56/(243+486*x+243*(4*x^2-2*x-3)^(1/2))- 
2197/5184/(1-4*x-2*(4*x^2-2*x-3)^(1/2))^3-169/384/(1-4*x-2*(4*x^2-2*x-3)^( 
1/2))^2-3029/(5184-20736*x-10368*(4*x^2-2*x-3)^(1/2))-235/729*ln(1+2*x+(4* 
x^2-2*x-3)^(1/2))+15769/46656*ln(1-4*x-2*(4*x^2-2*x-3)^(1/2))
 

Mathematica [A] (verified)

Time = 0.50 (sec) , antiderivative size = 130, normalized size of antiderivative = 0.70 \[ \int \frac {x^2}{\left (1+2 x+\sqrt {-3-2 x+4 x^2}\right )^3} \, dx=\frac {-\frac {6 \sqrt {-3-2 x+4 x^2} \left (4900+10236 x+2505 x^2-1476 x^3+1728 x^4\right )}{(2+3 x)^2}+\frac {48 \left (-108+616 x+1872 x^2+543 x^3-477 x^4+432 x^5\right )}{(2+3 x)^2}-729 \log \left (1-4 x+2 \sqrt {-3-2 x+4 x^2}\right )-15040 \log \left (-5-6 x+3 \sqrt {-3-2 x+4 x^2}\right )}{46656} \] Input:

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

Output:

((-6*Sqrt[-3 - 2*x + 4*x^2]*(4900 + 10236*x + 2505*x^2 - 1476*x^3 + 1728*x 
^4))/(2 + 3*x)^2 + (48*(-108 + 616*x + 1872*x^2 + 543*x^3 - 477*x^4 + 432* 
x^5))/(2 + 3*x)^2 - 729*Log[1 - 4*x + 2*Sqrt[-3 - 2*x + 4*x^2]] - 15040*Lo 
g[-5 - 6*x + 3*Sqrt[-3 - 2*x + 4*x^2]])/46656
 

Rubi [A] (verified)

Time = 0.56 (sec) , antiderivative size = 219, normalized size of antiderivative = 1.18, 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.

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

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {4 x^2}{27}-\frac {2}{27} \sqrt {4 x^2-2 x-3} x-\frac {17 \sqrt {4 x^2-2 x-3}}{54 (3 x+2)}+\frac {19 \sqrt {4 x^2-2 x-3}}{81 (3 x+2)^2}-\frac {2 \sqrt {4 x^2-2 x-3}}{81 (3 x+2)^3}+\frac {11}{108} \sqrt {4 x^2-2 x-3}-\frac {13 x}{54}-\frac {235}{486 (3 x+2)}+\frac {41}{243 (3 x+2)^2}-\frac {2}{243 (3 x+2)^3}+\frac {65}{324}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {8249 \text {arctanh}\left (\frac {1-4 x}{2 \sqrt {4 x^2-2 x-3}}\right )}{46656}-\frac {235 \text {arctanh}\left (\frac {11 x+7}{\sqrt {4 x^2-2 x-3}}\right )}{1458}+\frac {4 x^3}{81}-\frac {13 x^2}{108}-\frac {1}{162} \left (4 x^2-2 x-3\right )^{3/2}-\frac {1}{96} (1-4 x) \sqrt {4 x^2-2 x-3}-\frac {(11 x+7) \sqrt {4 x^2-2 x-3}}{81 (3 x+2)^2}-\frac {19 \sqrt {4 x^2-2 x-3}}{243 (3 x+2)}-\frac {17}{162} \sqrt {4 x^2-2 x-3}+\frac {65 x}{324}-\frac {41}{729 (3 x+2)}+\frac {1}{729 (3 x+2)^2}-\frac {235 \log (3 x+2)}{1458}\)

Input:

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

Output:

(65*x)/324 - (13*x^2)/108 + (4*x^3)/81 + 1/(729*(2 + 3*x)^2) - 41/(729*(2 
+ 3*x)) - (17*Sqrt[-3 - 2*x + 4*x^2])/162 - ((1 - 4*x)*Sqrt[-3 - 2*x + 4*x 
^2])/96 - (19*Sqrt[-3 - 2*x + 4*x^2])/(243*(2 + 3*x)) - ((7 + 11*x)*Sqrt[- 
3 - 2*x + 4*x^2])/(81*(2 + 3*x)^2) - (-3 - 2*x + 4*x^2)^(3/2)/162 - (8249* 
ArcTanh[(1 - 4*x)/(2*Sqrt[-3 - 2*x + 4*x^2])])/46656 - (235*ArcTanh[(7 + 1 
1*x)/Sqrt[-3 - 2*x + 4*x^2]])/1458 - (235*Log[2 + 3*x])/1458
 

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

Output:

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

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 185, normalized size of antiderivative = 0.99 \[ \int \frac {x^2}{\left (1+2 x+\sqrt {-3-2 x+4 x^2}\right )^3} \, dx=\frac {82944 \, x^{5} - 91584 \, x^{4} + 104256 \, x^{3} + 261279 \, x^{2} - 30080 \, {\left (9 \, x^{2} + 12 \, x + 4\right )} \log \left (3 \, x + 2\right ) + 30080 \, {\left (9 \, x^{2} + 12 \, x + 4\right )} \log \left (-2 \, x + \sqrt {4 \, x^{2} - 2 \, x - 3} - 1\right ) - 32996 \, {\left (9 \, x^{2} + 12 \, x + 4\right )} \log \left (-4 \, x + 2 \, \sqrt {4 \, x^{2} - 2 \, x - 3} + 1\right ) - 30080 \, {\left (9 \, x^{2} + 12 \, x + 4\right )} \log \left (-6 \, x + 3 \, \sqrt {4 \, x^{2} - 2 \, x - 3} - 5\right ) - 24 \, {\left (1728 \, x^{4} - 1476 \, x^{3} + 2505 \, x^{2} + 10236 \, x + 4900\right )} \sqrt {4 \, x^{2} - 2 \, x - 3} - 12588 \, x - 64356}{186624 \, {\left (9 \, x^{2} + 12 \, x + 4\right )}} \] Input:

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

Output:

1/186624*(82944*x^5 - 91584*x^4 + 104256*x^3 + 261279*x^2 - 30080*(9*x^2 + 
 12*x + 4)*log(3*x + 2) + 30080*(9*x^2 + 12*x + 4)*log(-2*x + sqrt(4*x^2 - 
 2*x - 3) - 1) - 32996*(9*x^2 + 12*x + 4)*log(-4*x + 2*sqrt(4*x^2 - 2*x - 
3) + 1) - 30080*(9*x^2 + 12*x + 4)*log(-6*x + 3*sqrt(4*x^2 - 2*x - 3) - 5) 
 - 24*(1728*x^4 - 1476*x^3 + 2505*x^2 + 10236*x + 4900)*sqrt(4*x^2 - 2*x - 
 3) - 12588*x - 64356)/(9*x^2 + 12*x + 4)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [A] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 236, normalized size of antiderivative = 1.27 \[ \int \frac {x^2}{\left (1+2 x+\sqrt {-3-2 x+4 x^2}\right )^3} \, dx=\frac {4}{81} \, x^{3} - \frac {13}{108} \, x^{2} - \frac {1}{2592} \, {\left (4 \, {\left (16 \, x - 35\right )} x + 251\right )} \sqrt {4 \, x^{2} - 2 \, x - 3} + \frac {65}{324} \, x - \frac {4 \, {\left (167 \, {\left (2 \, x - \sqrt {4 \, x^{2} - 2 \, x - 3}\right )}^{3} + 647 \, {\left (2 \, x - \sqrt {4 \, x^{2} - 2 \, x - 3}\right )}^{2} + 1660 \, x - 830 \, \sqrt {4 \, x^{2} - 2 \, x - 3} + 353\right )}}{243 \, {\left (3 \, {\left (2 \, x - \sqrt {4 \, x^{2} - 2 \, x - 3}\right )}^{2} + 16 \, x - 8 \, \sqrt {4 \, x^{2} - 2 \, x - 3} + 5\right )}^{2}} - \frac {41 \, x + 27}{243 \, {\left (3 \, x + 2\right )}^{2}} - \frac {235}{1458} \, \log \left ({\left | 3 \, x + 2 \right |}\right ) + \frac {235}{1458} \, \log \left ({\left | -2 \, x + \sqrt {4 \, x^{2} - 2 \, x - 3} - 1 \right |}\right ) - \frac {8249}{46656} \, \log \left ({\left | -4 \, x + 2 \, \sqrt {4 \, x^{2} - 2 \, x - 3} + 1 \right |}\right ) - \frac {235}{1458} \, \log \left ({\left | -6 \, x + 3 \, \sqrt {4 \, x^{2} - 2 \, x - 3} - 5 \right |}\right ) \] Input:

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

Output:

4/81*x^3 - 13/108*x^2 - 1/2592*(4*(16*x - 35)*x + 251)*sqrt(4*x^2 - 2*x - 
3) + 65/324*x - 4/243*(167*(2*x - sqrt(4*x^2 - 2*x - 3))^3 + 647*(2*x - sq 
rt(4*x^2 - 2*x - 3))^2 + 1660*x - 830*sqrt(4*x^2 - 2*x - 3) + 353)/(3*(2*x 
 - sqrt(4*x^2 - 2*x - 3))^2 + 16*x - 8*sqrt(4*x^2 - 2*x - 3) + 5)^2 - 1/24 
3*(41*x + 27)/(3*x + 2)^2 - 235/1458*log(abs(3*x + 2)) + 235/1458*log(abs( 
-2*x + sqrt(4*x^2 - 2*x - 3) - 1)) - 8249/46656*log(abs(-4*x + 2*sqrt(4*x^ 
2 - 2*x - 3) + 1)) - 235/1458*log(abs(-6*x + 3*sqrt(4*x^2 - 2*x - 3) - 5))
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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