\(\int \frac {x}{(1+3 x+\sqrt {-3-2 x+4 x^2})^3} \, dx\) [43]

Optimal result

Integrand size = 23, antiderivative size = 176 \[ \int \frac {x}{\left (1+3 x+\sqrt {-3-2 x+4 x^2}\right )^3} \, dx=-\frac {2 \left (97+664 \left (2 x+\sqrt {-3-2 x+4 x^2}\right )\right )}{125 \left (1+2 \left (2 x+\sqrt {-3-2 x+4 x^2}\right )+5 \left (2 x+\sqrt {-3-2 x+4 x^2}\right )^2\right )^2}-\frac {947+4915 \left (2 x+\sqrt {-3-2 x+4 x^2}\right )}{250 \left (1+2 \left (2 x+\sqrt {-3-2 x+4 x^2}\right )+5 \left (2 x+\sqrt {-3-2 x+4 x^2}\right )^2\right )}-\frac {39}{4} \arctan \left (\frac {1}{2} \left (1+5 \left (2 x+\sqrt {-3-2 x+4 x^2}\right )\right )\right ) \] Output:

1/125*(-194-2656*x-1328*(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-(947+9830*x+4915*(4*x^2-2*x-3)^(1/2))/(25 
0+1000*x+500*(4*x^2-2*x-3)^(1/2)+1250*(2*x+(4*x^2-2*x-3)^(1/2))^2)-39/4*ar 
ctan(1/2+5*x+5/2*(4*x^2-2*x-3)^(1/2))
 

Mathematica [A] (verified)

Time = 0.41 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.57 \[ \int \frac {x}{\left (1+3 x+\sqrt {-3-2 x+4 x^2}\right )^3} \, dx=\frac {3428+9812 x+10482 x^2+3615 x^3+25 \sqrt {-3-2 x+4 x^2} \left (84+256 x+296 x^2+121 x^3\right )+975 \left (4+8 x+5 x^2\right )^2 \arctan \left (\frac {3}{2}+x-\frac {1}{2} \sqrt {-3-2 x+4 x^2}\right )}{100 \left (4+8 x+5 x^2\right )^2} \] Input:

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

Output:

(3428 + 9812*x + 10482*x^2 + 3615*x^3 + 25*Sqrt[-3 - 2*x + 4*x^2]*(84 + 25 
6*x + 296*x^2 + 121*x^3) + 975*(4 + 8*x + 5*x^2)^2*ArcTan[3/2 + x - Sqrt[- 
3 - 2*x + 4*x^2]/2])/(100*(4 + 8*x + 5*x^2)^2)
 

Rubi [A] (verified)

Time = 0.86 (sec) , antiderivative size = 310, normalized size of antiderivative = 1.76, 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.

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

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (-\frac {31 \sqrt {4 x^2-2 x-3} x}{5 \left (5 x^2+8 x+4\right )^2}-\frac {724 \sqrt {4 x^2-2 x-3} x}{25 \left (5 x^2+8 x+4\right )^3}+\frac {63}{25 \left (5 x^2+8 x+4\right )}-\frac {3 (1505 x-24)}{125 \left (5 x^2+8 x+4\right )^2}+\frac {168 \sqrt {4 x^2-2 x-3}}{25 \left (5 x^2+8 x+4\right )^2}-\frac {4 (919 x+1332)}{125 \left (5 x^2+8 x+4\right )^3}-\frac {672 \sqrt {4 x^2-2 x-3}}{25 \left (5 x^2+8 x+4\right )^3}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {39}{8} \arctan \left (\frac {7 x+8}{2 \sqrt {4 x^2-2 x-3}}\right )+\frac {39}{8} \arctan \left (\frac {5 x}{2}+2\right )+\frac {31 \sqrt {4 x^2-2 x-3} (x+1)}{10 \left (5 x^2+8 x+4\right )}+\frac {181 \sqrt {4 x^2-2 x-3} (x+1)}{25 \left (5 x^2+8 x+4\right )^2}-\frac {1119 (5 x+4)}{500 \left (5 x^2+8 x+4\right )}+\frac {3 (1535 x+1529)}{250 \left (5 x^2+8 x+4\right )}+\frac {21 (5 x+4) \sqrt {4 x^2-2 x-3}}{25 \left (5 x^2+8 x+4\right )}+\frac {181 (965 x+701) \sqrt {4 x^2-2 x-3}}{14450 \left (5 x^2+8 x+4\right )}-\frac {21 (18355 x+13928) \sqrt {4 x^2-2 x-3}}{28900 \left (5 x^2+8 x+4\right )}-\frac {746 x+413}{125 \left (5 x^2+8 x+4\right )^2}-\frac {42 (5 x+4) \sqrt {4 x^2-2 x-3}}{25 \left (5 x^2+8 x+4\right )^2}\)

Input:

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

Output:

-1/125*(413 + 746*x)/(4 + 8*x + 5*x^2)^2 + (181*(1 + x)*Sqrt[-3 - 2*x + 4* 
x^2])/(25*(4 + 8*x + 5*x^2)^2) - (42*(4 + 5*x)*Sqrt[-3 - 2*x + 4*x^2])/(25 
*(4 + 8*x + 5*x^2)^2) - (1119*(4 + 5*x))/(500*(4 + 8*x + 5*x^2)) + (3*(152 
9 + 1535*x))/(250*(4 + 8*x + 5*x^2)) + (31*(1 + x)*Sqrt[-3 - 2*x + 4*x^2]) 
/(10*(4 + 8*x + 5*x^2)) + (21*(4 + 5*x)*Sqrt[-3 - 2*x + 4*x^2])/(25*(4 + 8 
*x + 5*x^2)) + (181*(701 + 965*x)*Sqrt[-3 - 2*x + 4*x^2])/(14450*(4 + 8*x 
+ 5*x^2)) - (21*(13928 + 18355*x)*Sqrt[-3 - 2*x + 4*x^2])/(28900*(4 + 8*x 
+ 5*x^2)) + (39*ArcTan[2 + (5*x)/2])/8 + (39*ArcTan[(8 + 7*x)/(2*Sqrt[-3 - 
 2*x + 4*x^2])])/8
 

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 [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 3.

Time = 0.46 (sec) , antiderivative size = 131, normalized size of antiderivative = 0.74

method	result	size
trager	\(-\frac {\left (857 x^{3}+2164 x^{2}+1888 x +624\right ) x}{16 \left (5 x^{2}+8 x +4\right )^{2}}+\frac {\left (121 x^{3}+296 x^{2}+256 x +84\right ) \sqrt {4 x^{2}-2 x -3}}{4 \left (5 x^{2}+8 x +4\right )^{2}}+\frac {39 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \ln \left (-\frac {7 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x +8 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right )-2 \sqrt {4 x^{2}-2 x -3}}{\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x +2 x +2}\right )}{8}\)	\(131\)
default	\(\text {Expression too large to display}\)	\(12661\)

Input:

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

Output:

-1/16*(857*x^3+2164*x^2+1888*x+624)*x/(5*x^2+8*x+4)^2+1/4*(121*x^3+296*x^2 
+256*x+84)/(5*x^2+8*x+4)^2*(4*x^2-2*x-3)^(1/2)+39/8*RootOf(_Z^2+1)*ln(-(7* 
RootOf(_Z^2+1)*x+8*RootOf(_Z^2+1)-2*(4*x^2-2*x-3)^(1/2))/(RootOf(_Z^2+1)*x 
+2*x+2))
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 162, normalized size of antiderivative = 0.92 \[ \int \frac {x}{\left (1+3 x+\sqrt {-3-2 x+4 x^2}\right )^3} \, dx=\frac {484 \, x^{4} + 1838 \, x^{3} + 2852 \, x^{2} + 39 \, {\left (25 \, x^{4} + 80 \, x^{3} + 104 \, x^{2} + 64 \, x + 16\right )} \arctan \left (\frac {5}{2} \, x + 2\right ) - 39 \, {\left (25 \, x^{4} + 80 \, x^{3} + 104 \, x^{2} + 64 \, x + 16\right )} \arctan \left (-\frac {70 \, x^{2} - 5 \, \sqrt {4 \, x^{2} - 2 \, x - 3} {\left (7 \, x + 8\right )} + 112 \, x + 56}{2 \, {\left (42 \, x + 31\right )}}\right ) + 2 \, {\left (121 \, x^{3} + 296 \, x^{2} + 256 \, x + 84\right )} \sqrt {4 \, x^{2} - 2 \, x - 3} + 2024 \, x + 584}{8 \, {\left (25 \, x^{4} + 80 \, x^{3} + 104 \, x^{2} + 64 \, x + 16\right )}} \] Input:

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

Output:

1/8*(484*x^4 + 1838*x^3 + 2852*x^2 + 39*(25*x^4 + 80*x^3 + 104*x^2 + 64*x 
+ 16)*arctan(5/2*x + 2) - 39*(25*x^4 + 80*x^3 + 104*x^2 + 64*x + 16)*arcta 
n(-1/2*(70*x^2 - 5*sqrt(4*x^2 - 2*x - 3)*(7*x + 8) + 112*x + 56)/(42*x + 3 
1)) + 2*(121*x^3 + 296*x^2 + 256*x + 84)*sqrt(4*x^2 - 2*x - 3) + 2024*x + 
584)/(25*x^4 + 80*x^3 + 104*x^2 + 64*x + 16)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

integrate(x/(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. 322 vs. \(2 (147) = 294\).

Time = 0.15 (sec) , antiderivative size = 322, normalized size of antiderivative = 1.83 \[ \int \frac {x}{\left (1+3 x+\sqrt {-3-2 x+4 x^2}\right )^3} \, dx=\frac {12145 \, {\left (2 \, x - \sqrt {4 \, x^{2} - 2 \, x - 3}\right )}^{7} + 142134 \, {\left (2 \, x - \sqrt {4 \, x^{2} - 2 \, x - 3}\right )}^{6} + 636777 \, {\left (2 \, x - \sqrt {4 \, x^{2} - 2 \, x - 3}\right )}^{5} + 1429702 \, {\left (2 \, x - \sqrt {4 \, x^{2} - 2 \, x - 3}\right )}^{4} + 1397579 \, {\left (2 \, x - \sqrt {4 \, x^{2} - 2 \, x - 3}\right )}^{3} + 793042 \, {\left (2 \, x - \sqrt {4 \, x^{2} - 2 \, x - 3}\right )}^{2} + 578294 \, x - 289147 \, \sqrt {4 \, x^{2} - 2 \, x - 3} + 45874}{50 \, {\left (5 \, {\left (2 \, x - \sqrt {4 \, x^{2} - 2 \, x - 3}\right )}^{4} + 32 \, {\left (2 \, x - \sqrt {4 \, x^{2} - 2 \, x - 3}\right )}^{3} + 78 \, {\left (2 \, x - \sqrt {4 \, x^{2} - 2 \, x - 3}\right )}^{2} + 64 \, x - 32 \, \sqrt {4 \, x^{2} - 2 \, x - 3} + 13\right )}^{2}} + \frac {3615 \, x^{3} + 10482 \, x^{2} + 9812 \, x + 3428}{100 \, {\left (5 \, x^{2} + 8 \, x + 4\right )}^{2}} + \frac {39}{8} \, \arctan \left (\frac {5}{2} \, x + 2\right ) - \frac {39}{8} \, \arctan \left (-x + \frac {1}{2} \, \sqrt {4 \, x^{2} - 2 \, x - 3} - \frac {3}{2}\right ) - \frac {39}{8} \, \arctan \left (-5 \, x + \frac {5}{2} \, \sqrt {4 \, x^{2} - 2 \, x - 3} - \frac {1}{2}\right ) \] Input:

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

Output:

1/50*(12145*(2*x - sqrt(4*x^2 - 2*x - 3))^7 + 142134*(2*x - sqrt(4*x^2 - 2 
*x - 3))^6 + 636777*(2*x - sqrt(4*x^2 - 2*x - 3))^5 + 1429702*(2*x - sqrt( 
4*x^2 - 2*x - 3))^4 + 1397579*(2*x - sqrt(4*x^2 - 2*x - 3))^3 + 793042*(2* 
x - sqrt(4*x^2 - 2*x - 3))^2 + 578294*x - 289147*sqrt(4*x^2 - 2*x - 3) + 4 
5874)/(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 + 1/100*(3615*x^3 + 10482*x^2 + 9812*x + 3428)/(5*x^2 + 8*x + 4)^ 
2 + 39/8*arctan(5/2*x + 2) - 39/8*arctan(-x + 1/2*sqrt(4*x^2 - 2*x - 3) - 
3/2) - 39/8*arctan(-5*x + 5/2*sqrt(4*x^2 - 2*x - 3) - 1/2)
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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