\(\int \frac {x}{(1+3 x+\sqrt {-3-2 x+4 x^2})^2} \, dx\) [38]

Optimal result

Integrand size = 23, antiderivative size = 177 \[ \int \frac {x}{\left (1+3 x+\sqrt {-3-2 x+4 x^2}\right )^2} \, dx=-\frac {2 \left (7-66 \left (2 x+\sqrt {-3-2 x+4 x^2}\right )\right )}{25 \left (1+2 \left (2 x+\sqrt {-3-2 x+4 x^2}\right )+5 \left (2 x+\sqrt {-3-2 x+4 x^2}\right )^2\right )}+\frac {107}{25} \arctan \left (\frac {1}{2} \left (1+5 \left (2 x+\sqrt {-3-2 x+4 x^2}\right )\right )\right )+\frac {13}{25} \log \left (7+3 x-20 x^2-\sqrt {-3-2 x+4 x^2}-10 x \sqrt {-3-2 x+4 x^2}\right )-\log \left (1-2 \left (2 x+\sqrt {-3-2 x+4 x^2}\right )\right ) \] Output:

(-14+264*x+132*(4*x^2-2*x-3)^(1/2))/(25+100*x+50*(4*x^2-2*x-3)^(1/2)+125*( 
2*x+(4*x^2-2*x-3)^(1/2))^2)+107/25*arctan(1/2+5*x+5/2*(4*x^2-2*x-3)^(1/2)) 
+13/25*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))-ln(1- 
4*x-2*(4*x^2-2*x-3)^(1/2))
 

Mathematica [A] (verified)

Time = 0.45 (sec) , antiderivative size = 145, normalized size of antiderivative = 0.82 \[ \int \frac {x}{\left (1+3 x+\sqrt {-3-2 x+4 x^2}\right )^2} \, dx=\frac {1}{25} \left (-\frac {5 (7+4 x) \sqrt {-3-2 x+4 x^2}}{4+8 x+5 x^2}-\frac {5 \left (9+7 x+8 x^2\right )}{4+8 x+5 x^2}-107 \arctan \left (\frac {3}{2}+x-\frac {1}{2} \sqrt {-3-2 x+4 x^2}\right )-\log \left (1-4 x+2 \sqrt {-3-2 x+4 x^2}\right )+13 \log \left (-5-5 x-4 x^2+(3+2 x) \sqrt {-3-2 x+4 x^2}\right )\right ) \] Input:

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

Output:

((-5*(7 + 4*x)*Sqrt[-3 - 2*x + 4*x^2])/(4 + 8*x + 5*x^2) - (5*(9 + 7*x + 8 
*x^2))/(4 + 8*x + 5*x^2) - 107*ArcTan[3/2 + x - Sqrt[-3 - 2*x + 4*x^2]/2] 
- Log[1 - 4*x + 2*Sqrt[-3 - 2*x + 4*x^2]] + 13*Log[-5 - 5*x - 4*x^2 + (3 + 
 2*x)*Sqrt[-3 - 2*x + 4*x^2]])/25
 

Rubi [A] (verified)

Time = 0.60 (sec) , antiderivative size = 198, normalized size of antiderivative = 1.12, 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.

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

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {38 \sqrt {4 x^2-2 x-3} x}{5 \left (5 x^2+8 x+4\right )^2}+\frac {65 x-84}{25 \left (5 x^2+8 x+4\right )}-\frac {6 \sqrt {4 x^2-2 x-3}}{5 \left (5 x^2+8 x+4\right )}+\frac {2 (181 x+168)}{25 \left (5 x^2+8 x+4\right )^2}+\frac {24 \sqrt {4 x^2-2 x-3}}{5 \left (5 x^2+8 x+4\right )^2}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {107}{50} \arctan \left (\frac {7 x+8}{2 \sqrt {4 x^2-2 x-3}}\right )-\frac {107}{50} \arctan \left (\frac {5 x}{2}+2\right )+\frac {12}{25} \text {arctanh}\left (\frac {1-4 x}{2 \sqrt {4 x^2-2 x-3}}\right )+\frac {13}{25} \text {arctanh}\left (\frac {3 x+1}{\sqrt {4 x^2-2 x-3}}\right )-\frac {13-29 x}{25 \left (5 x^2+8 x+4\right )}-\frac {19 (x+1) \sqrt {4 x^2-2 x-3}}{5 \left (5 x^2+8 x+4\right )}+\frac {3 (5 x+4) \sqrt {4 x^2-2 x-3}}{5 \left (5 x^2+8 x+4\right )}+\frac {13}{50} \log \left (5 x^2+8 x+4\right )\)

Input:

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

Output:

-1/25*(13 - 29*x)/(4 + 8*x + 5*x^2) - (19*(1 + x)*Sqrt[-3 - 2*x + 4*x^2])/ 
(5*(4 + 8*x + 5*x^2)) + (3*(4 + 5*x)*Sqrt[-3 - 2*x + 4*x^2])/(5*(4 + 8*x + 
 5*x^2)) - (107*ArcTan[2 + (5*x)/2])/50 - (107*ArcTan[(8 + 7*x)/(2*Sqrt[-3 
 - 2*x + 4*x^2])])/50 + (12*ArcTanh[(1 - 4*x)/(2*Sqrt[-3 - 2*x + 4*x^2])]) 
/25 + (13*ArcTanh[(1 + 3*x)/Sqrt[-3 - 2*x + 4*x^2]])/25 + (13*Log[4 + 8*x 
+ 5*x^2])/50
 

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 = 22.62 (sec) , antiderivative size = 2977, normalized size of antiderivative = 16.82

method	result	size
trager	\(\text {Expression too large to display}\)	\(2977\)
default	\(\text {Expression too large to display}\)	\(3632\)

Input:

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

Output:

1/20*(44+13*x)*x/(5*x^2+8*x+4)-1/5*(7+4*x)/(5*x^2+8*x+4)*(4*x^2-2*x-3)^(1/ 
2)-1/25*ln((21115017504109113738234433401673393718289694623665870623712858 
766882112650845749099827200+2226806844516425181530528408044034509299660801 
71676267317924138705757073443080262745600000*x-715265447474660688407246350 
102237977028607754215189698218979887620111148370411021515776000*x^2*(4*x^2 
-2*x-3)^(1/2)-605028159231495471771644158266891273508590348610462506904321 
13741603006233677974732800*x^25-139580809940920243307336357332451289969110 
327438765763354016512373324130213627927930835279375*x^12+10136931994764898 
08011895531057954511202497631157348705978492732309116615121253760211952600 
00*x^14-410622198323495185484624399918934126721110101888939099482446393571 
49328136189742436598818500*x^10+104562755038177213998622096026308268715342 
628673794186854323379401604908597478291804305148000*x^9+683285461552861774 
8344784795331752872823477676964709943693860720413107928112197168325504000* 
x^16-379391631734948762294764848393824659434301008249986667135358802430453 
961322239783914496000*x^3+736357432826287151090559607660875830998396749401 
991978737900131156540220807730380228608000*x^2-920795352249327176489945643 
6830649496953054176984421864148509738838418314852212297783744000*x^4+12039 
63739172869121952854189471385576250876699292725473698197070995594954663612 
71877632000*x^22+492327469066405320077383732658507697211326966532195201194 
483729552545836239820026806272000*x^21+52003114909598848976945734957649...
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 264, normalized size of antiderivative = 1.49 \[ \int \frac {x}{\left (1+3 x+\sqrt {-3-2 x+4 x^2}\right )^2} \, dx=-\frac {80 \, x^{2} + 107 \, {\left (5 \, x^{2} + 8 \, x + 4\right )} \arctan \left (\frac {5}{2} \, x + 2\right ) - 107 \, {\left (5 \, x^{2} + 8 \, x + 4\right )} \arctan \left (-x + \frac {1}{2} \, \sqrt {4 \, x^{2} - 2 \, x - 3} - \frac {3}{2}\right ) - 107 \, {\left (5 \, x^{2} + 8 \, x + 4\right )} \arctan \left (-5 \, x + \frac {5}{2} \, \sqrt {4 \, x^{2} - 2 \, x - 3} - \frac {1}{2}\right ) + 13 \, {\left (5 \, x^{2} + 8 \, x + 4\right )} \log \left (20 \, x^{2} - \sqrt {4 \, x^{2} - 2 \, x - 3} {\left (10 \, x + 1\right )} - 3 \, x - 7\right ) - 13 \, {\left (5 \, x^{2} + 8 \, x + 4\right )} \log \left (5 \, x^{2} + 8 \, x + 4\right ) - 13 \, {\left (5 \, x^{2} + 8 \, x + 4\right )} \log \left (4 \, x^{2} - \sqrt {4 \, x^{2} - 2 \, x - 3} {\left (2 \, x + 3\right )} + 5 \, x + 5\right ) - 24 \, {\left (5 \, x^{2} + 8 \, x + 4\right )} \log \left (-4 \, x + 2 \, \sqrt {4 \, x^{2} - 2 \, x - 3} + 1\right ) + 10 \, \sqrt {4 \, x^{2} - 2 \, x - 3} {\left (4 \, x + 7\right )} + 70 \, x + 90}{50 \, {\left (5 \, x^{2} + 8 \, x + 4\right )}} \] Input:

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

Output:

-1/50*(80*x^2 + 107*(5*x^2 + 8*x + 4)*arctan(5/2*x + 2) - 107*(5*x^2 + 8*x 
 + 4)*arctan(-x + 1/2*sqrt(4*x^2 - 2*x - 3) - 3/2) - 107*(5*x^2 + 8*x + 4) 
*arctan(-5*x + 5/2*sqrt(4*x^2 - 2*x - 3) - 1/2) + 13*(5*x^2 + 8*x + 4)*log 
(20*x^2 - sqrt(4*x^2 - 2*x - 3)*(10*x + 1) - 3*x - 7) - 13*(5*x^2 + 8*x + 
4)*log(5*x^2 + 8*x + 4) - 13*(5*x^2 + 8*x + 4)*log(4*x^2 - sqrt(4*x^2 - 2* 
x - 3)*(2*x + 3) + 5*x + 5) - 24*(5*x^2 + 8*x + 4)*log(-4*x + 2*sqrt(4*x^2 
 - 2*x - 3) + 1) + 10*sqrt(4*x^2 - 2*x - 3)*(4*x + 7) + 70*x + 90)/(5*x^2 
+ 8*x + 4)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 346 vs. \(2 (149) = 298\).

Time = 0.15 (sec) , antiderivative size = 346, normalized size of antiderivative = 1.95 \[ \int \frac {x}{\left (1+3 x+\sqrt {-3-2 x+4 x^2}\right )^2} \, dx=-\frac {4 \, {\left (4 \, {\left (2 \, x - \sqrt {4 \, x^{2} - 2 \, x - 3}\right )}^{3} + 261 \, {\left (2 \, x - \sqrt {4 \, x^{2} - 2 \, x - 3}\right )}^{2} + 538 \, x - 269 \, \sqrt {4 \, x^{2} - 2 \, x - 3} + 11\right )}}{25 \, {\left (5 \, {\left (2 \, x - \sqrt {4 \, x^{2} - 2 \, x - 3}\right )}^{4} + 32 \, {\left (2 \, x - \sqrt {4 \, x^{2} - 2 \, x - 3}\right )}^{3} + 78 \, {\left (2 \, x - \sqrt {4 \, x^{2} - 2 \, x - 3}\right )}^{2} + 64 \, x - 32 \, \sqrt {4 \, x^{2} - 2 \, x - 3} + 13\right )}} + \frac {29 \, x - 13}{25 \, {\left (5 \, x^{2} + 8 \, x + 4\right )}} - \frac {107}{50} \, \arctan \left (\frac {5}{2} \, x + 2\right ) + \frac {107}{50} \, \arctan \left (-x + \frac {1}{2} \, \sqrt {4 \, x^{2} - 2 \, x - 3} - \frac {3}{2}\right ) + \frac {107}{50} \, \arctan \left (-5 \, x + \frac {5}{2} \, \sqrt {4 \, x^{2} - 2 \, x - 3} - \frac {1}{2}\right ) - \frac {13}{50} \, \log \left (5 \, {\left (2 \, x - \sqrt {4 \, x^{2} - 2 \, x - 3}\right )}^{2} + 4 \, x - 2 \, \sqrt {4 \, x^{2} - 2 \, x - 3} + 1\right ) + \frac {13}{50} \, \log \left ({\left (2 \, x - \sqrt {4 \, x^{2} - 2 \, x - 3}\right )}^{2} + 12 \, x - 6 \, \sqrt {4 \, x^{2} - 2 \, x - 3} + 13\right ) + \frac {13}{50} \, \log \left (5 \, x^{2} + 8 \, x + 4\right ) + \frac {12}{25} \, \log \left ({\left | -4 \, x + 2 \, \sqrt {4 \, x^{2} - 2 \, x - 3} + 1 \right |}\right ) \] Input:

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

Output:

-4/25*(4*(2*x - sqrt(4*x^2 - 2*x - 3))^3 + 261*(2*x - sqrt(4*x^2 - 2*x - 3 
))^2 + 538*x - 269*sqrt(4*x^2 - 2*x - 3) + 11)/(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) + 1/25*(29*x - 13)/(5*x^2 
+ 8*x + 4) - 107/50*arctan(5/2*x + 2) + 107/50*arctan(-x + 1/2*sqrt(4*x^2 
- 2*x - 3) - 3/2) + 107/50*arctan(-5*x + 5/2*sqrt(4*x^2 - 2*x - 3) - 1/2) 
- 13/50*log(5*(2*x - sqrt(4*x^2 - 2*x - 3))^2 + 4*x - 2*sqrt(4*x^2 - 2*x - 
 3) + 1) + 13/50*log((2*x - sqrt(4*x^2 - 2*x - 3))^2 + 12*x - 6*sqrt(4*x^2 
 - 2*x - 3) + 13) + 13/50*log(5*x^2 + 8*x + 4) + 12/25*log(abs(-4*x + 2*sq 
rt(4*x^2 - 2*x - 3) + 1))
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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