\(\int \frac {1}{(1+2 x+\sqrt {-2+8 x-5 x^2})^2} \, dx\) [54]

Optimal result

Integrand size = 21, antiderivative size = 191 \[ \int \frac {1}{\left (1+2 x+\sqrt {-2+8 x-5 x^2}\right )^2} \, dx=\frac {2 \sqrt {6} \left (13+2 \sqrt {6}-\frac {5 \sqrt {6} \sqrt {-2+8 x-5 x^2}}{4-\sqrt {6}-5 x}\right )}{23 \left (13+2 \sqrt {6}-\frac {10 \sqrt {6} \sqrt {-2+8 x-5 x^2}}{4-\sqrt {6}-5 x}-\frac {5 \left (13-2 \sqrt {6}\right ) \left (2-8 x+5 x^2\right )}{\left (4-\sqrt {6}-5 x\right )^2}\right )}+\frac {12 \arctan \left (\frac {6+\frac {\left (12-13 \sqrt {6}\right ) \sqrt {-2+8 x-5 x^2}}{4-\sqrt {6}-5 x}}{\sqrt {138}}\right )}{23 \sqrt {23}} \] Output:

2*6^(1/2)*(13+2*6^(1/2)-5*6^(1/2)*(-5*x^2+8*x-2)^(1/2)/(4-6^(1/2)-5*x))/(2 
99+46*6^(1/2)-230*6^(1/2)*(-5*x^2+8*x-2)^(1/2)/(4-6^(1/2)-5*x)-115*(13-2*6 
^(1/2))*(5*x^2-8*x+2)/(4-6^(1/2)-5*x)^2)+12/529*arctan(1/138*(6+(12-13*6^( 
1/2))*(-5*x^2+8*x-2)^(1/2)/(4-6^(1/2)-5*x))*138^(1/2))*23^(1/2)
 

Mathematica [A] (verified)

Time = 0.95 (sec) , antiderivative size = 142, normalized size of antiderivative = 0.74 \[ \int \frac {1}{\left (1+2 x+\sqrt {-2+8 x-5 x^2}\right )^2} \, dx=\frac {23 (-150+77 x)+207 (8-13 x) \sqrt {-2+8 x-5 x^2}+108 \sqrt {23} \left (3-4 x+9 x^2\right ) \arctan \left (\frac {\sqrt {23} \left (\sqrt {22+8 \sqrt {6}}-5 x\right )}{6+4 \sqrt {6}-5 \sqrt {6} x+13 \sqrt {-2+8 x-5 x^2}+2 \sqrt {6} \sqrt {-2+8 x-5 x^2}}\right )}{4761 \left (3-4 x+9 x^2\right )} \] Input:

Integrate[(1 + 2*x + Sqrt[-2 + 8*x - 5*x^2])^(-2),x]
 

Output:

(23*(-150 + 77*x) + 207*(8 - 13*x)*Sqrt[-2 + 8*x - 5*x^2] + 108*Sqrt[23]*( 
3 - 4*x + 9*x^2)*ArcTan[(Sqrt[23]*(Sqrt[22 + 8*Sqrt[6]] - 5*x))/(6 + 4*Sqr 
t[6] - 5*Sqrt[6]*x + 13*Sqrt[-2 + 8*x - 5*x^2] + 2*Sqrt[6]*Sqrt[-2 + 8*x - 
 5*x^2])])/(4761*(3 - 4*x + 9*x^2))
 

Rubi [A] (verified)

Time = 0.48 (sec) , antiderivative size = 148, normalized size of antiderivative = 0.77, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, 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.

\(\displaystyle \int \frac {1}{\left (\sqrt {-5 x^2+8 x-2}+2 x+1\right )^2} \, dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (-\frac {4 \sqrt {-5 x^2+8 x-2} x}{\left (9 x^2-4 x+3\right )^2}-\frac {1}{9 \left (9 x^2-4 x+3\right )}+\frac {2 (52 x-3)}{9 \left (9 x^2-4 x+3\right )^2}-\frac {2 \sqrt {-5 x^2+8 x-2}}{\left (9 x^2-4 x+3\right )^2}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {6 \arctan \left (\frac {8-13 x}{\sqrt {23} \sqrt {-5 x^2+8 x-2}}\right )}{23 \sqrt {23}}-\frac {6 \arctan \left (\frac {2-9 x}{\sqrt {23}}\right )}{23 \sqrt {23}}-\frac {150-77 x}{207 \left (9 x^2-4 x+3\right )}+\frac {(2-9 x) \sqrt {-5 x^2+8 x-2}}{23 \left (9 x^2-4 x+3\right )}+\frac {2 (3-2 x) \sqrt {-5 x^2+8 x-2}}{23 \left (9 x^2-4 x+3\right )}\)

Input:

Int[(1 + 2*x + Sqrt[-2 + 8*x - 5*x^2])^(-2),x]
 

Output:

-1/207*(150 - 77*x)/(3 - 4*x + 9*x^2) + ((2 - 9*x)*Sqrt[-2 + 8*x - 5*x^2]) 
/(23*(3 - 4*x + 9*x^2)) + (2*(3 - 2*x)*Sqrt[-2 + 8*x - 5*x^2])/(23*(3 - 4* 
x + 9*x^2)) - (6*ArcTan[(2 - 9*x)/Sqrt[23]])/(23*Sqrt[23]) + (6*ArcTan[(8 
- 13*x)/(Sqrt[23]*Sqrt[-2 + 8*x - 5*x^2])])/(23*Sqrt[23])
 

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 = 0.40 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.58

method	result	size
trager	\(\frac {\left (-41+150 x \right ) x}{621 x^{2}-276 x +207}-\frac {\left (-8+13 x \right ) \sqrt {-5 x^{2}+8 x -2}}{23 \left (9 x^{2}-4 x +3\right )}+\frac {6 \operatorname {RootOf}\left (\textit {\_Z}^{2}+23\right ) \ln \left (\frac {13 \operatorname {RootOf}\left (\textit {\_Z}^{2}+23\right ) x -8 \operatorname {RootOf}\left (\textit {\_Z}^{2}+23\right )+23 \sqrt {-5 x^{2}+8 x -2}}{\operatorname {RootOf}\left (\textit {\_Z}^{2}+23\right ) x -2 x +3}\right )}{529}\)	\(110\)
default	\(\text {Expression too large to display}\)	\(5693\)

Input:

int(1/(1+2*x+(-5*x^2+8*x-2)^(1/2))^2,x,method=_RETURNVERBOSE)
 

Output:

1/69*(-41+150*x)*x/(9*x^2-4*x+3)-1/23*(-8+13*x)/(9*x^2-4*x+3)*(-5*x^2+8*x- 
2)^(1/2)+6/529*RootOf(_Z^2+23)*ln((13*RootOf(_Z^2+23)*x-8*RootOf(_Z^2+23)+ 
23*(-5*x^2+8*x-2)^(1/2))/(RootOf(_Z^2+23)*x-2*x+3))
 

Fricas [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 124, normalized size of antiderivative = 0.65 \[ \int \frac {1}{\left (1+2 x+\sqrt {-2+8 x-5 x^2}\right )^2} \, dx=\frac {54 \, \sqrt {23} {\left (9 \, x^{2} - 4 \, x + 3\right )} \arctan \left (\frac {1}{23} \, \sqrt {23} {\left (9 \, x - 2\right )}\right ) + 27 \, \sqrt {23} {\left (9 \, x^{2} - 4 \, x + 3\right )} \arctan \left (\frac {\sqrt {23} {\left (142 \, x^{2} - 196 \, x + 55\right )} \sqrt {-5 \, x^{2} + 8 \, x - 2}}{23 \, {\left (65 \, x^{3} - 144 \, x^{2} + 90 \, x - 16\right )}}\right ) - 207 \, \sqrt {-5 \, x^{2} + 8 \, x - 2} {\left (13 \, x - 8\right )} + 1771 \, x - 3450}{4761 \, {\left (9 \, x^{2} - 4 \, x + 3\right )}} \] Input:

integrate(1/(1+2*x+(-5*x^2+8*x-2)^(1/2))^2,x, algorithm="fricas")
 

Output:

1/4761*(54*sqrt(23)*(9*x^2 - 4*x + 3)*arctan(1/23*sqrt(23)*(9*x - 2)) + 27 
*sqrt(23)*(9*x^2 - 4*x + 3)*arctan(1/23*sqrt(23)*(142*x^2 - 196*x + 55)*sq 
rt(-5*x^2 + 8*x - 2)/(65*x^3 - 144*x^2 + 90*x - 16)) - 207*sqrt(-5*x^2 + 8 
*x - 2)*(13*x - 8) + 1771*x - 3450)/(9*x^2 - 4*x + 3)
 

Sympy [F]

\[ \int \frac {1}{\left (1+2 x+\sqrt {-2+8 x-5 x^2}\right )^2} \, dx=\int \frac {1}{\left (2 x + \sqrt {- 5 x^{2} + 8 x - 2} + 1\right )^{2}}\, dx \] Input:

integrate(1/(1+2*x+(-5*x**2+8*x-2)**(1/2))**2,x)
 

Output:

Integral((2*x + sqrt(-5*x**2 + 8*x - 2) + 1)**(-2), x)
 

Maxima [F]

\[ \int \frac {1}{\left (1+2 x+\sqrt {-2+8 x-5 x^2}\right )^2} \, dx=\int { \frac {1}{{\left (2 \, x + \sqrt {-5 \, x^{2} + 8 \, x - 2} + 1\right )}^{2}} \,d x } \] Input:

integrate(1/(1+2*x+(-5*x^2+8*x-2)^(1/2))^2,x, algorithm="maxima")
 

Output:

integrate((2*x + sqrt(-5*x^2 + 8*x - 2) + 1)^(-2), x)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 457 vs. \(2 (148) = 296\).

Time = 0.15 (sec) , antiderivative size = 457, normalized size of antiderivative = 2.39 \[ \int \frac {1}{\left (1+2 x+\sqrt {-2+8 x-5 x^2}\right )^2} \, dx=\frac {6}{529} \, \sqrt {23} \arctan \left (\frac {1}{23} \, \sqrt {23} {\left (9 \, x - 2\right )}\right ) + \frac {6 \, {\left (5 \, \sqrt {6} + 13 \, \sqrt {5}\right )} \arctan \left (-\frac {26 \, \sqrt {6} + 12 \, \sqrt {5} - \frac {139 \, {\left (\sqrt {5} \sqrt {-5 \, x^{2} + 8 \, x - 2} - \sqrt {6}\right )}}{5 \, x - 4}}{5 \, \sqrt {138} + 13 \, \sqrt {115}}\right )}{23 \, {\left (5 \, \sqrt {138} + 13 \, \sqrt {115}\right )}} + \frac {6 \, {\left (5 \, \sqrt {6} - 13 \, \sqrt {5}\right )} \arctan \left (\frac {26 \, \sqrt {6} - 12 \, \sqrt {5} - \frac {139 \, {\left (\sqrt {5} \sqrt {-5 \, x^{2} + 8 \, x - 2} - \sqrt {6}\right )}}{5 \, x - 4}}{5 \, \sqrt {138} - 13 \, \sqrt {115}}\right )}{23 \, {\left (5 \, \sqrt {138} - 13 \, \sqrt {115}\right )}} + \frac {77 \, x - 150}{207 \, {\left (9 \, x^{2} - 4 \, x + 3\right )}} + \frac {12 \, {\left (278 \, \sqrt {30} + \frac {1183 \, \sqrt {5} {\left (\sqrt {5} \sqrt {-5 \, x^{2} + 8 \, x - 2} - \sqrt {6}\right )}^{3}}{{\left (5 \, x - 4\right )}^{3}} + \frac {494 \, \sqrt {30} {\left (\sqrt {5} \sqrt {-5 \, x^{2} + 8 \, x - 2} - \sqrt {6}\right )}^{2}}{{\left (5 \, x - 4\right )}^{2}} - \frac {2431 \, \sqrt {5} {\left (\sqrt {5} \sqrt {-5 \, x^{2} + 8 \, x - 2} - \sqrt {6}\right )}}{5 \, x - 4}\right )}}{3197 \, {\left (\frac {104 \, \sqrt {6} {\left (\sqrt {5} \sqrt {-5 \, x^{2} + 8 \, x - 2} - \sqrt {6}\right )}^{3}}{{\left (5 \, x - 4\right )}^{3}} + \frac {104 \, \sqrt {6} {\left (\sqrt {5} \sqrt {-5 \, x^{2} + 8 \, x - 2} - \sqrt {6}\right )}}{5 \, x - 4} - \frac {139 \, {\left (\sqrt {5} \sqrt {-5 \, x^{2} + 8 \, x - 2} - \sqrt {6}\right )}^{4}}{{\left (5 \, x - 4\right )}^{4}} - \frac {494 \, {\left (\sqrt {5} \sqrt {-5 \, x^{2} + 8 \, x - 2} - \sqrt {6}\right )}^{2}}{{\left (5 \, x - 4\right )}^{2}} - 139\right )}} \] Input:

integrate(1/(1+2*x+(-5*x^2+8*x-2)^(1/2))^2,x, algorithm="giac")
 

Output:

6/529*sqrt(23)*arctan(1/23*sqrt(23)*(9*x - 2)) + 6/23*(5*sqrt(6) + 13*sqrt 
(5))*arctan(-(26*sqrt(6) + 12*sqrt(5) - 139*(sqrt(5)*sqrt(-5*x^2 + 8*x - 2 
) - sqrt(6))/(5*x - 4))/(5*sqrt(138) + 13*sqrt(115)))/(5*sqrt(138) + 13*sq 
rt(115)) + 6/23*(5*sqrt(6) - 13*sqrt(5))*arctan((26*sqrt(6) - 12*sqrt(5) - 
 139*(sqrt(5)*sqrt(-5*x^2 + 8*x - 2) - sqrt(6))/(5*x - 4))/(5*sqrt(138) - 
13*sqrt(115)))/(5*sqrt(138) - 13*sqrt(115)) + 1/207*(77*x - 150)/(9*x^2 - 
4*x + 3) + 12/3197*(278*sqrt(30) + 1183*sqrt(5)*(sqrt(5)*sqrt(-5*x^2 + 8*x 
 - 2) - sqrt(6))^3/(5*x - 4)^3 + 494*sqrt(30)*(sqrt(5)*sqrt(-5*x^2 + 8*x - 
 2) - sqrt(6))^2/(5*x - 4)^2 - 2431*sqrt(5)*(sqrt(5)*sqrt(-5*x^2 + 8*x - 2 
) - sqrt(6))/(5*x - 4))/(104*sqrt(6)*(sqrt(5)*sqrt(-5*x^2 + 8*x - 2) - sqr 
t(6))^3/(5*x - 4)^3 + 104*sqrt(6)*(sqrt(5)*sqrt(-5*x^2 + 8*x - 2) - sqrt(6 
))/(5*x - 4) - 139*(sqrt(5)*sqrt(-5*x^2 + 8*x - 2) - sqrt(6))^4/(5*x - 4)^ 
4 - 494*(sqrt(5)*sqrt(-5*x^2 + 8*x - 2) - sqrt(6))^2/(5*x - 4)^2 - 139)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\left (1+2 x+\sqrt {-2+8 x-5 x^2}\right )^2} \, dx=\int \frac {1}{{\left (2\,x+\sqrt {-5\,x^2+8\,x-2}+1\right )}^2} \,d x \] Input:

int(1/(2*x + (8*x - 5*x^2 - 2)^(1/2) + 1)^2,x)
 

Output:

int(1/(2*x + (8*x - 5*x^2 - 2)^(1/2) + 1)^2, x)
 

Reduce [F]

\[ \int \frac {1}{\left (1+2 x+\sqrt {-2+8 x-5 x^2}\right )^2} \, dx=\text {too large to display} \] Input:

int(1/(1+2*x+(-5*x^2+8*x-2)^(1/2))^2,x)
 

Output:

(3337104171021340901699934769453582072500*sqrt(5)*asin((5*x - 4)/sqrt(6))* 
x**4 - 10678733347268290885439791262251462632000*sqrt(5)*asin((5*x - 4)/sq 
rt(6))*x**3 + 12481593575958338760234718984613052156600*sqrt(5)*asin((5*x 
- 4)/sqrt(6))*x**2 - 6301771037029929683012617559699011281600*sqrt(5)*asin 
((5*x - 4)/sqrt(6))*x + 2427434293291079085532841439706235255700*sqrt(5)*a 
sin((5*x - 4)/sqrt(6)) + 35779384266150332363857638619867751250000*sqrt(6) 
*asin((5*x - 4)/sqrt(6))*x**4 - 114494029651681063564344443583576804000000 
*sqrt(6)*asin((5*x - 4)/sqrt(6))*x**3 + 1338237315712704283080779036577177 
02700000*sqrt(6)*asin((5*x - 4)/sqrt(6))*x**2 - 67565612559880923930563758 
065370015200000*sqrt(6)*asin((5*x - 4)/sqrt(6))*x + 2602618914767379731948 
7556388674171650000*sqrt(6)*asin((5*x - 4)/sqrt(6)) + 30476289500894211571 
533409695999972135000*sqrt( - 5*x**2 + 8*x - 2)*sqrt(30)*x**3 - 9683914504 
2637750980836153330660404932000*sqrt( - 5*x**2 + 8*x - 2)*sqrt(30)*x**2 + 
95693382777758940941601601040099457522600*sqrt( - 5*x**2 + 8*x - 2)*sqrt(3 
0)*x - 21887937799117513267724298553180042951200*sqrt( - 5*x**2 + 8*x - 2) 
*sqrt(30) - 51954257509779098300964798000505300624125*sqrt( - 5*x**2 + 8*x 
 - 2)*x**3 + 168717983419223696552091452317731605683800*sqrt( - 5*x**2 + 8 
*x - 2)*x**2 - 191453453983614797943154461052618366322580*sqrt( - 5*x**2 + 
 8*x - 2)*x + 76676391655013209476212087819959270069896*sqrt( - 5*x**2 + 8 
*x - 2) - 503943787523333270567506417940353427767468857600000*sqrt(30)*...