\(\int \frac {x}{(1+2 x+\sqrt {-3-2 x+4 x^2})^3} \, dx\) [13]

Optimal result

Integrand size = 23, antiderivative size = 194 \[ \int \frac {x}{\left (1+2 x+\sqrt {-3-2 x+4 x^2}\right )^3} \, dx=\frac {137+233 \left (2 x+\sqrt {-3-2 x+4 x^2}\right )}{288 \left (1-2 x-\sqrt {-3-2 x+4 x^2}-2 \left (2 x+\sqrt {-3-2 x+4 x^2}\right )^2\right )^2}+\frac {179+1148 \left (2 x+\sqrt {-3-2 x+4 x^2}\right )}{864 \left (1-2 x-\sqrt {-3-2 x+4 x^2}-2 \left (2 x+\sqrt {-3-2 x+4 x^2}\right )^2\right )}+\frac {65}{162} \log \left (1+2 x+\sqrt {-3-2 x+4 x^2}\right )-\frac {65}{162} \log \left (1-2 \left (2 x+\sqrt {-3-2 x+4 x^2}\right )\right ) \] Output:

1/288*(137+466*x+233*(4*x^2-2*x-3)^(1/2))/(1-2*x-(4*x^2-2*x-3)^(1/2)-2*(2* 
x+(4*x^2-2*x-3)^(1/2))^2)^2+(179+2296*x+1148*(4*x^2-2*x-3)^(1/2))/(864-172 
8*x-864*(4*x^2-2*x-3)^(1/2)-1728*(2*x+(4*x^2-2*x-3)^(1/2))^2)+65/162*ln(1+ 
2*x+(4*x^2-2*x-3)^(1/2))-65/162*ln(1-4*x-2*(4*x^2-2*x-3)^(1/2))
 

Mathematica [A] (verified)

Time = 0.40 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.48 \[ \int \frac {x}{\left (1+2 x+\sqrt {-3-2 x+4 x^2}\right )^3} \, dx=\frac {79-348 x-1260 x^2-621 x^3+324 x^4+9 \sqrt {-3-2 x+4 x^2} \left (43+93 x+30 x^2-18 x^3\right )+195 (2+3 x)^2 \log \left (-5-6 x+3 \sqrt {-3-2 x+4 x^2}\right )}{486 (2+3 x)^2} \] Input:

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

Output:

(79 - 348*x - 1260*x^2 - 621*x^3 + 324*x^4 + 9*Sqrt[-3 - 2*x + 4*x^2]*(43 
+ 93*x + 30*x^2 - 18*x^3) + 195*(2 + 3*x)^2*Log[-5 - 6*x + 3*Sqrt[-3 - 2*x 
 + 4*x^2]])/(486*(2 + 3*x)^2)
 

Rubi [A] (verified)

Time = 0.50 (sec) , antiderivative size = 194, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, 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.

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

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {11 \sqrt {4 x^2-2 x-3}}{36 (3 x+2)}-\frac {\sqrt {4 x^2-2 x-3}}{3 (3 x+2)^2}+\frac {\sqrt {4 x^2-2 x-3}}{27 (3 x+2)^3}-\frac {2}{27} \sqrt {4 x^2-2 x-3}+\frac {4 x}{27}+\frac {65}{108 (3 x+2)}-\frac {20}{81 (3 x+2)^2}+\frac {1}{81 (3 x+2)^3}-\frac {13}{54}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {65}{324} \text {arctanh}\left (\frac {1-4 x}{2 \sqrt {4 x^2-2 x-3}}\right )+\frac {65}{324} \text {arctanh}\left (\frac {11 x+7}{\sqrt {4 x^2-2 x-3}}\right )+\frac {2 x^2}{27}+\frac {1}{108} (1-4 x) \sqrt {4 x^2-2 x-3}+\frac {(11 x+7) \sqrt {4 x^2-2 x-3}}{54 (3 x+2)^2}+\frac {\sqrt {4 x^2-2 x-3}}{9 (3 x+2)}+\frac {11}{108} \sqrt {4 x^2-2 x-3}-\frac {13 x}{54}+\frac {20}{243 (3 x+2)}-\frac {1}{486 (3 x+2)^2}+\frac {65}{324} \log (3 x+2)\)

Input:

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

Output:

(-13*x)/54 + (2*x^2)/27 - 1/(486*(2 + 3*x)^2) + 20/(243*(2 + 3*x)) + (11*S 
qrt[-3 - 2*x + 4*x^2])/108 + ((1 - 4*x)*Sqrt[-3 - 2*x + 4*x^2])/108 + Sqrt 
[-3 - 2*x + 4*x^2]/(9*(2 + 3*x)) + ((7 + 11*x)*Sqrt[-3 - 2*x + 4*x^2])/(54 
*(2 + 3*x)^2) + (65*ArcTanh[(1 - 4*x)/(2*Sqrt[-3 - 2*x + 4*x^2])])/324 + ( 
65*ArcTanh[(7 + 11*x)/Sqrt[-3 - 2*x + 4*x^2]])/324 + (65*Log[2 + 3*x])/324
 

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 [A] (verified)

Time = 0.08 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.44

method	result	size
trager	\(\frac {\left (144 x^{3}-276 x^{2}-639 x -260\right ) x}{216 \left (2+3 x \right )^{2}}-\frac {\left (18 x^{3}-30 x^{2}-93 x -43\right ) \sqrt {4 x^{2}-2 x -3}}{54 \left (2+3 x \right )^{2}}+\frac {65 \ln \left (-3 \sqrt {4 x^{2}-2 x -3}+5+6 x \right )}{162}\)	\(85\)
default	\(\frac {20}{243 \left (2+3 x \right )}-\frac {1}{486 \left (2+3 x \right )^{2}}+\frac {65 \ln \left (2+3 x \right )}{324}-\frac {13 x}{54}+\frac {2 x^{2}}{27}-\frac {\left (4 \left (x +\frac {2}{3}\right )^{2}-\frac {22 x}{3}-\frac {43}{9}\right )^{\frac {3}{2}}}{162 \left (x +\frac {2}{3}\right )^{2}}+\frac {7 \left (4 \left (x +\frac {2}{3}\right )^{2}-\frac {22 x}{3}-\frac {43}{9}\right )^{\frac {3}{2}}}{54 \left (x +\frac {2}{3}\right )}+\frac {65 \sqrt {36 \left (x +\frac {2}{3}\right )^{2}-66 x -43}}{324}-\frac {65 \,\operatorname {arctanh}\left (\frac {-21-33 x}{\sqrt {36 \left (x +\frac {2}{3}\right )^{2}-66 x -43}}\right )}{324}-\frac {7 \left (8 x -2\right ) \sqrt {4 \left (x +\frac {2}{3}\right )^{2}-\frac {22 x}{3}-\frac {43}{9}}}{108}-\frac {169 \ln \left (\frac {\left (4 x -1\right ) \sqrt {4}}{4}+\sqrt {4 \left (x +\frac {2}{3}\right )^{2}-\frac {22 x}{3}-\frac {43}{9}}\right ) \sqrt {4}}{1296}-\frac {\left (8 x -2\right ) \sqrt {4 x^{2}-2 x -3}}{216}+\frac {13 \ln \left (\frac {\left (4 x -1\right ) \sqrt {4}}{4}+\sqrt {4 x^{2}-2 x -3}\right ) \sqrt {4}}{432}\)	\(218\)

Input:

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

Output:

1/216*(144*x^3-276*x^2-639*x-260)*x/(2+3*x)^2-1/54*(18*x^3-30*x^2-93*x-43) 
/(2+3*x)^2*(4*x^2-2*x-3)^(1/2)+65/162*ln(-3*(4*x^2-2*x-3)^(1/2)+5+6*x)
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 153, normalized size of antiderivative = 0.79 \[ \int \frac {x}{\left (1+2 x+\sqrt {-3-2 x+4 x^2}\right )^3} \, dx=\frac {5184 \, x^{4} - 9936 \, x^{3} - 13671 \, x^{2} + 1560 \, {\left (9 \, x^{2} + 12 \, x + 4\right )} \log \left (48 \, x^{2} - \sqrt {4 \, x^{2} - 2 \, x - 3} {\left (24 \, x + 7\right )} + 2 \, x - 23\right ) + 1560 \, {\left (9 \, x^{2} + 12 \, x + 4\right )} \log \left (3 \, x + 2\right ) - 1560 \, {\left (9 \, x^{2} + 12 \, x + 4\right )} \log \left (-2 \, x + \sqrt {4 \, x^{2} - 2 \, x - 3} - 1\right ) - 144 \, {\left (18 \, x^{3} - 30 \, x^{2} - 93 \, x - 43\right )} \sqrt {4 \, x^{2} - 2 \, x - 3} + 3084 \, x + 4148}{7776 \, {\left (9 \, x^{2} + 12 \, x + 4\right )}} \] Input:

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

Output:

1/7776*(5184*x^4 - 9936*x^3 - 13671*x^2 + 1560*(9*x^2 + 12*x + 4)*log(48*x 
^2 - sqrt(4*x^2 - 2*x - 3)*(24*x + 7) + 2*x - 23) + 1560*(9*x^2 + 12*x + 4 
)*log(3*x + 2) - 1560*(9*x^2 + 12*x + 4)*log(-2*x + sqrt(4*x^2 - 2*x - 3) 
- 1) - 144*(18*x^3 - 30*x^2 - 93*x - 43)*sqrt(4*x^2 - 2*x - 3) + 3084*x + 
4148)/(9*x^2 + 12*x + 4)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [A] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 224, normalized size of antiderivative = 1.15 \[ \int \frac {x}{\left (1+2 x+\sqrt {-3-2 x+4 x^2}\right )^3} \, dx=\frac {2}{27} \, x^{2} - \frac {1}{27} \, \sqrt {4 \, x^{2} - 2 \, x - 3} {\left (x - 3\right )} - \frac {13}{54} \, x + \frac {969 \, {\left (2 \, x - \sqrt {4 \, x^{2} - 2 \, x - 3}\right )}^{3} + 3752 \, {\left (2 \, x - \sqrt {4 \, x^{2} - 2 \, x - 3}\right )}^{2} + 9626 \, x - 4813 \, \sqrt {4 \, x^{2} - 2 \, x - 3} + 2048}{243 \, {\left (3 \, {\left (2 \, x - \sqrt {4 \, x^{2} - 2 \, x - 3}\right )}^{2} + 16 \, x - 8 \, \sqrt {4 \, x^{2} - 2 \, x - 3} + 5\right )}^{2}} + \frac {120 \, x + 79}{486 \, {\left (3 \, x + 2\right )}^{2}} + \frac {65}{324} \, \log \left ({\left | 3 \, x + 2 \right |}\right ) - \frac {65}{324} \, \log \left ({\left | -2 \, x + \sqrt {4 \, x^{2} - 2 \, x - 3} - 1 \right |}\right ) + \frac {65}{324} \, \log \left ({\left | -4 \, x + 2 \, \sqrt {4 \, x^{2} - 2 \, x - 3} + 1 \right |}\right ) + \frac {65}{324} \, \log \left ({\left | -6 \, x + 3 \, \sqrt {4 \, x^{2} - 2 \, x - 3} - 5 \right |}\right ) \] Input:

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

Output:

2/27*x^2 - 1/27*sqrt(4*x^2 - 2*x - 3)*(x - 3) - 13/54*x + 1/243*(969*(2*x 
- sqrt(4*x^2 - 2*x - 3))^3 + 3752*(2*x - sqrt(4*x^2 - 2*x - 3))^2 + 9626*x 
 - 4813*sqrt(4*x^2 - 2*x - 3) + 2048)/(3*(2*x - sqrt(4*x^2 - 2*x - 3))^2 + 
 16*x - 8*sqrt(4*x^2 - 2*x - 3) + 5)^2 + 1/486*(120*x + 79)/(3*x + 2)^2 + 
65/324*log(abs(3*x + 2)) - 65/324*log(abs(-2*x + sqrt(4*x^2 - 2*x - 3) - 1 
)) + 65/324*log(abs(-4*x + 2*sqrt(4*x^2 - 2*x - 3) + 1)) + 65/324*log(abs( 
-6*x + 3*sqrt(4*x^2 - 2*x - 3) - 5))
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 156, normalized size of antiderivative = 0.80 \[ \int \frac {x}{\left (1+2 x+\sqrt {-3-2 x+4 x^2}\right )^3} \, dx=\frac {-54 \sqrt {4 x^{2}-2 x -3}\, x^{3}+90 \sqrt {4 x^{2}-2 x -3}\, x^{2}+279 \sqrt {4 x^{2}-2 x -3}\, x +129 \sqrt {4 x^{2}-2 x -3}+585 \,\mathrm {log}\left (-3 \sqrt {4 x^{2}-2 x -3}+6 x +5\right ) x^{2}+780 \,\mathrm {log}\left (-3 \sqrt {4 x^{2}-2 x -3}+6 x +5\right ) x +260 \,\mathrm {log}\left (-3 \sqrt {4 x^{2}-2 x -3}+6 x +5\right )+108 x^{4}-207 x^{3}-333 x^{2}+65}{1458 x^{2}+1944 x +648} \] Input:

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

Output:

( - 54*sqrt(4*x**2 - 2*x - 3)*x**3 + 90*sqrt(4*x**2 - 2*x - 3)*x**2 + 279* 
sqrt(4*x**2 - 2*x - 3)*x + 129*sqrt(4*x**2 - 2*x - 3) + 585*log( - 3*sqrt( 
4*x**2 - 2*x - 3) + 6*x + 5)*x**2 + 780*log( - 3*sqrt(4*x**2 - 2*x - 3) + 
6*x + 5)*x + 260*log( - 3*sqrt(4*x**2 - 2*x - 3) + 6*x + 5) + 108*x**4 - 2 
07*x**3 - 333*x**2 + 65)/(162*(9*x**2 + 12*x + 4))