\(\int \frac {1}{(1+2 x+\sqrt {2+3 x+5 x^2})^3} \, dx\) [29]

Optimal result

Integrand size = 21, antiderivative size = 335 \[ \int \frac {1}{\left (1+2 x+\sqrt {2+3 x+5 x^2}\right )^3} \, dx=\frac {157+42 \sqrt {2}-\frac {\left (161+71 \sqrt {2}\right ) \left (\sqrt {2}-\sqrt {2+3 x+5 x^2}\right )}{x}}{3 \left (1-5 \sqrt {2}-\frac {\left (3-4 \sqrt {2}\right ) \left (\sqrt {2}-\sqrt {2+3 x+5 x^2}\right )}{x}+\frac {\left (1-\sqrt {2}\right ) \left (\sqrt {2}-\sqrt {2+3 x+5 x^2}\right )^2}{x^2}\right )^2}-\frac {\left (1+\sqrt {2}\right ) \left (327-526 \sqrt {2}-\frac {2 \left (130-121 \sqrt {2}\right ) \left (\sqrt {2}-\sqrt {2+3 x+5 x^2}\right )}{x}\right )}{3 \left (1-5 \sqrt {2}-\frac {\left (3-4 \sqrt {2}\right ) \left (\sqrt {2}-\sqrt {2+3 x+5 x^2}\right )}{x}+\frac {\left (1-\sqrt {2}\right ) \left (\sqrt {2}-\sqrt {2+3 x+5 x^2}\right )^2}{x^2}\right )}+\frac {496 \arctan \left (\frac {3-4 \sqrt {2}-\frac {2 \left (1-\sqrt {2}\right ) \left (\sqrt {2}-\sqrt {2+3 x+5 x^2}\right )}{x}}{\sqrt {3}}\right )}{3 \sqrt {3}} \] Output:

1/3*(157+42*2^(1/2)-(161+71*2^(1/2))*(2^(1/2)-(5*x^2+3*x+2)^(1/2))/x)/(1-5 
*2^(1/2)-(3-4*2^(1/2))*(2^(1/2)-(5*x^2+3*x+2)^(1/2))/x+(1-2^(1/2))*(2^(1/2 
)-(5*x^2+3*x+2)^(1/2))^2/x^2)^2-(1+2^(1/2))*(327-526*2^(1/2)-2*(130-121*2^ 
(1/2))*(2^(1/2)-(5*x^2+3*x+2)^(1/2))/x)/(3-15*2^(1/2)-3*(3-4*2^(1/2))*(2^( 
1/2)-(5*x^2+3*x+2)^(1/2))/x+3*(1-2^(1/2))*(2^(1/2)-(5*x^2+3*x+2)^(1/2))^2/ 
x^2)+496/9*arctan(1/3*(3-4*2^(1/2)-2*(1-2^(1/2))*(2^(1/2)-(5*x^2+3*x+2)^(1 
/2))/x)*3^(1/2))*3^(1/2)
 

Mathematica [A] (verified)

Time = 2.41 (sec) , antiderivative size = 233, normalized size of antiderivative = 0.70 \[ \int \frac {1}{\left (1+2 x+\sqrt {2+3 x+5 x^2}\right )^3} \, dx=\frac {1}{9} \left (\frac {459-609 x+729 x^2-372 x^3}{\left (1-x+x^2\right )^2}+\frac {6 \sqrt {2+3 x+5 x^2} \left (-22+37 x-35 x^2+26 x^3\right )}{\left (1-x+x^2\right )^2}-96 \sqrt {15 \left (9+4 \sqrt {5}\right )} \arctan \left (\sqrt {3+\frac {4 \sqrt {5}}{3}} \left (-16+7 \sqrt {5}+2 \left (-20+9 \sqrt {5}\right ) x-18 \sqrt {2+3 x+5 x^2}+8 \sqrt {5} \sqrt {2+3 x+5 x^2}\right )\right )+16 \sqrt {2163-72 \sqrt {5}} \arctan \left (\frac {3+10 x+4 \sqrt {2+3 x+5 x^2}-2 \sqrt {5} \left (1+2 x+\sqrt {2+3 x+5 x^2}\right )}{\sqrt {3}}\right )\right ) \] Input:

Integrate[(1 + 2*x + Sqrt[2 + 3*x + 5*x^2])^(-3),x]
 

Output:

((459 - 609*x + 729*x^2 - 372*x^3)/(1 - x + x^2)^2 + (6*Sqrt[2 + 3*x + 5*x 
^2]*(-22 + 37*x - 35*x^2 + 26*x^3))/(1 - x + x^2)^2 - 96*Sqrt[15*(9 + 4*Sq 
rt[5])]*ArcTan[Sqrt[3 + (4*Sqrt[5])/3]*(-16 + 7*Sqrt[5] + 2*(-20 + 9*Sqrt[ 
5])*x - 18*Sqrt[2 + 3*x + 5*x^2] + 8*Sqrt[5]*Sqrt[2 + 3*x + 5*x^2])] + 16* 
Sqrt[2163 - 72*Sqrt[5]]*ArcTan[(3 + 10*x + 4*Sqrt[2 + 3*x + 5*x^2] - 2*Sqr 
t[5]*(1 + 2*x + Sqrt[2 + 3*x + 5*x^2]))/Sqrt[3]])/9
 

Rubi [A] (verified)

Time = 0.78 (sec) , antiderivative size = 309, normalized size of antiderivative = 0.92, 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.

\(\displaystyle \int \frac {1}{\left (\sqrt {5 x^2+3 x+2}+2 x+1\right )^3} \, dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {-38 x-83}{\left (x^2-x+1\right )^2}+\frac {17 \sqrt {5 x^2+3 x+2}}{\left (x^2-x+1\right )^2}+\frac {32 x \sqrt {5 x^2+3 x+2}}{\left (x^2-x+1\right )^3}-\frac {12 \sqrt {5 x^2+3 x+2}}{\left (x^2-x+1\right )^3}-\frac {4 (18 x-19)}{\left (x^2-x+1\right )^3}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {4154}{343} \sqrt {3} \arctan \left (\frac {5-4 x}{\sqrt {3} \sqrt {5 x^2+3 x+2}}\right )-\frac {122450 \arctan \left (\frac {5-4 x}{\sqrt {3} \sqrt {5 x^2+3 x+2}}\right )}{1029 \sqrt {3}}+\frac {248 \arctan \left (\frac {1-2 x}{\sqrt {3}}\right )}{3 \sqrt {3}}+\frac {\sqrt {5 x^2+3 x+2} (101-338 x)}{49 \left (x^2-x+1\right )}-\frac {8 (44-237 x) \sqrt {5 x^2+3 x+2}}{147 \left (x^2-x+1\right )}-\frac {17 (1-2 x) \sqrt {5 x^2+3 x+2}}{3 \left (x^2-x+1\right )}+\frac {2 (1-2 x) \sqrt {5 x^2+3 x+2}}{\left (x^2-x+1\right )^2}-\frac {16 (2-x) \sqrt {5 x^2+3 x+2}}{3 \left (x^2-x+1\right )^2}+\frac {53-68 x}{x^2-x+1}-\frac {40 (1-2 x)}{3 \left (x^2-x+1\right )}+\frac {2 (20 x+17)}{3 \left (x^2-x+1\right )^2}\)

Input:

Int[(1 + 2*x + Sqrt[2 + 3*x + 5*x^2])^(-3),x]
 

Output:

(2*(17 + 20*x))/(3*(1 - x + x^2)^2) + (53 - 68*x)/(1 - x + x^2) - (40*(1 - 
 2*x))/(3*(1 - x + x^2)) + (2*(1 - 2*x)*Sqrt[2 + 3*x + 5*x^2])/(1 - x + x^ 
2)^2 - (16*(2 - x)*Sqrt[2 + 3*x + 5*x^2])/(3*(1 - x + x^2)^2) + ((101 - 33 
8*x)*Sqrt[2 + 3*x + 5*x^2])/(49*(1 - x + x^2)) - (8*(44 - 237*x)*Sqrt[2 + 
3*x + 5*x^2])/(147*(1 - x + x^2)) - (17*(1 - 2*x)*Sqrt[2 + 3*x + 5*x^2])/( 
3*(1 - x + x^2)) + (248*ArcTan[(1 - 2*x)/Sqrt[3]])/(3*Sqrt[3]) - (122450*A 
rcTan[(5 - 4*x)/(Sqrt[3]*Sqrt[2 + 3*x + 5*x^2])])/(1029*Sqrt[3]) + (4154*S 
qrt[3]*ArcTan[(5 - 4*x)/(Sqrt[3]*Sqrt[2 + 3*x + 5*x^2])])/343
 

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 = 0.26 (sec) , antiderivative size = 125, normalized size of antiderivative = 0.37

method	result	size
trager	\(-\frac {\left (153 x^{3}-182 x^{2}+216 x -103\right ) x}{3 \left (x^{2}-x +1\right )^{2}}+\frac {2 \left (26 x^{3}-35 x^{2}+37 x -22\right ) \sqrt {5 x^{2}+3 x +2}}{3 \left (x^{2}-x +1\right )^{2}}+\frac {248 \operatorname {RootOf}\left (\textit {\_Z}^{2}+3\right ) \ln \left (-\frac {4 \operatorname {RootOf}\left (\textit {\_Z}^{2}+3\right ) x -5 \operatorname {RootOf}\left (\textit {\_Z}^{2}+3\right )-3 \sqrt {5 x^{2}+3 x +2}}{\operatorname {RootOf}\left (\textit {\_Z}^{2}+3\right ) x +x -2}\right )}{9}\)	\(125\)
default	\(\text {Expression too large to display}\)	\(8267\)

Input:

int(1/(1+2*x+(5*x^2+3*x+2)^(1/2))^3,x,method=_RETURNVERBOSE)
 

Output:

-1/3*(153*x^3-182*x^2+216*x-103)*x/(x^2-x+1)^2+2/3*(26*x^3-35*x^2+37*x-22) 
/(x^2-x+1)^2*(5*x^2+3*x+2)^(1/2)+248/9*RootOf(_Z^2+3)*ln(-(4*RootOf(_Z^2+3 
)*x-5*RootOf(_Z^2+3)-3*(5*x^2+3*x+2)^(1/2))/(RootOf(_Z^2+3)*x+x-2))
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 166, normalized size of antiderivative = 0.50 \[ \int \frac {1}{\left (1+2 x+\sqrt {2+3 x+5 x^2}\right )^3} \, dx=-\frac {372 \, x^{3} + 248 \, \sqrt {3} {\left (x^{4} - 2 \, x^{3} + 3 \, x^{2} - 2 \, x + 1\right )} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x - 1\right )}\right ) - 124 \, \sqrt {3} {\left (x^{4} - 2 \, x^{3} + 3 \, x^{2} - 2 \, x + 1\right )} \arctan \left (\frac {\sqrt {3} \sqrt {5 \, x^{2} + 3 \, x + 2} {\left (x^{2} - 49 \, x + 19\right )}}{6 \, {\left (20 \, x^{3} - 13 \, x^{2} - 7 \, x - 10\right )}}\right ) - 729 \, x^{2} - 6 \, {\left (26 \, x^{3} - 35 \, x^{2} + 37 \, x - 22\right )} \sqrt {5 \, x^{2} + 3 \, x + 2} + 609 \, x - 459}{9 \, {\left (x^{4} - 2 \, x^{3} + 3 \, x^{2} - 2 \, x + 1\right )}} \] Input:

integrate(1/(1+2*x+(5*x^2+3*x+2)^(1/2))^3,x, algorithm="fricas")
 

Output:

-1/9*(372*x^3 + 248*sqrt(3)*(x^4 - 2*x^3 + 3*x^2 - 2*x + 1)*arctan(1/3*sqr 
t(3)*(2*x - 1)) - 124*sqrt(3)*(x^4 - 2*x^3 + 3*x^2 - 2*x + 1)*arctan(1/6*s 
qrt(3)*sqrt(5*x^2 + 3*x + 2)*(x^2 - 49*x + 19)/(20*x^3 - 13*x^2 - 7*x - 10 
)) - 729*x^2 - 6*(26*x^3 - 35*x^2 + 37*x - 22)*sqrt(5*x^2 + 3*x + 2) + 609 
*x - 459)/(x^4 - 2*x^3 + 3*x^2 - 2*x + 1)
 

Sympy [F]

\[ \int \frac {1}{\left (1+2 x+\sqrt {2+3 x+5 x^2}\right )^3} \, dx=\int \frac {1}{\left (2 x + \sqrt {5 x^{2} + 3 x + 2} + 1\right )^{3}}\, dx \] Input:

integrate(1/(1+2*x+(5*x**2+3*x+2)**(1/2))**3,x)
 

Output:

Integral((2*x + sqrt(5*x**2 + 3*x + 2) + 1)**(-3), x)
 

Maxima [F]

\[ \int \frac {1}{\left (1+2 x+\sqrt {2+3 x+5 x^2}\right )^3} \, dx=\int { \frac {1}{{\left (2 \, x + \sqrt {5 \, x^{2} + 3 \, x + 2} + 1\right )}^{3}} \,d x } \] Input:

integrate(1/(1+2*x+(5*x^2+3*x+2)^(1/2))^3,x, algorithm="maxima")
 

Output:

integrate((2*x + sqrt(5*x^2 + 3*x + 2) + 1)^(-3), x)
 

Giac [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 445, normalized size of antiderivative = 1.33 \[ \int \frac {1}{\left (1+2 x+\sqrt {2+3 x+5 x^2}\right )^3} \, dx=-\frac {248}{9} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x - 1\right )}\right ) + \frac {248 \, {\left (\sqrt {5} + 2\right )} \arctan \left (-\frac {2 \, \sqrt {5} x - \sqrt {5} - 2 \, \sqrt {5 \, x^{2} + 3 \, x + 2} - 4}{\sqrt {15} + 2 \, \sqrt {3}}\right )}{3 \, {\left (\sqrt {15} + 2 \, \sqrt {3}\right )}} - \frac {248 \, {\left (\sqrt {5} - 2\right )} \arctan \left (-\frac {2 \, \sqrt {5} x - \sqrt {5} - 2 \, \sqrt {5 \, x^{2} + 3 \, x + 2} + 4}{\sqrt {15} - 2 \, \sqrt {3}}\right )}{3 \, {\left (\sqrt {15} - 2 \, \sqrt {3}\right )}} - \frac {2 \, {\left (248 \, {\left (\sqrt {5} x - \sqrt {5 \, x^{2} + 3 \, x + 2}\right )}^{7} - 875 \, \sqrt {5} {\left (\sqrt {5} x - \sqrt {5 \, x^{2} + 3 \, x + 2}\right )}^{6} + 2557 \, {\left (\sqrt {5} x - \sqrt {5 \, x^{2} + 3 \, x + 2}\right )}^{5} - 947 \, \sqrt {5} {\left (\sqrt {5} x - \sqrt {5 \, x^{2} + 3 \, x + 2}\right )}^{4} - 70665 \, {\left (\sqrt {5} x - \sqrt {5 \, x^{2} + 3 \, x + 2}\right )}^{3} - 63830 \, \sqrt {5} {\left (\sqrt {5} x - \sqrt {5 \, x^{2} + 3 \, x + 2}\right )}^{2} - 108264 \, \sqrt {5} x - 13162 \, \sqrt {5} + 108264 \, \sqrt {5 \, x^{2} + 3 \, x + 2}\right )}}{3 \, {\left ({\left (\sqrt {5} x - \sqrt {5 \, x^{2} + 3 \, x + 2}\right )}^{4} - 2 \, \sqrt {5} {\left (\sqrt {5} x - \sqrt {5 \, x^{2} + 3 \, x + 2}\right )}^{3} + 13 \, {\left (\sqrt {5} x - \sqrt {5 \, x^{2} + 3 \, x + 2}\right )}^{2} + 16 \, \sqrt {5} {\left (\sqrt {5} x - \sqrt {5 \, x^{2} + 3 \, x + 2}\right )} + 19\right )}^{2}} - \frac {124 \, x^{3} - 243 \, x^{2} + 203 \, x - 153}{3 \, {\left (x^{2} - x + 1\right )}^{2}} \] Input:

integrate(1/(1+2*x+(5*x^2+3*x+2)^(1/2))^3,x, algorithm="giac")
 

Output:

-248/9*sqrt(3)*arctan(1/3*sqrt(3)*(2*x - 1)) + 248/3*(sqrt(5) + 2)*arctan( 
-(2*sqrt(5)*x - sqrt(5) - 2*sqrt(5*x^2 + 3*x + 2) - 4)/(sqrt(15) + 2*sqrt( 
3)))/(sqrt(15) + 2*sqrt(3)) - 248/3*(sqrt(5) - 2)*arctan(-(2*sqrt(5)*x - s 
qrt(5) - 2*sqrt(5*x^2 + 3*x + 2) + 4)/(sqrt(15) - 2*sqrt(3)))/(sqrt(15) - 
2*sqrt(3)) - 2/3*(248*(sqrt(5)*x - sqrt(5*x^2 + 3*x + 2))^7 - 875*sqrt(5)* 
(sqrt(5)*x - sqrt(5*x^2 + 3*x + 2))^6 + 2557*(sqrt(5)*x - sqrt(5*x^2 + 3*x 
 + 2))^5 - 947*sqrt(5)*(sqrt(5)*x - sqrt(5*x^2 + 3*x + 2))^4 - 70665*(sqrt 
(5)*x - sqrt(5*x^2 + 3*x + 2))^3 - 63830*sqrt(5)*(sqrt(5)*x - sqrt(5*x^2 + 
 3*x + 2))^2 - 108264*sqrt(5)*x - 13162*sqrt(5) + 108264*sqrt(5*x^2 + 3*x 
+ 2))/((sqrt(5)*x - sqrt(5*x^2 + 3*x + 2))^4 - 2*sqrt(5)*(sqrt(5)*x - sqrt 
(5*x^2 + 3*x + 2))^3 + 13*(sqrt(5)*x - sqrt(5*x^2 + 3*x + 2))^2 + 16*sqrt( 
5)*(sqrt(5)*x - sqrt(5*x^2 + 3*x + 2)) + 19)^2 - 1/3*(124*x^3 - 243*x^2 + 
203*x - 153)/(x^2 - x + 1)^2
 

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\left (1+2 x+\sqrt {2+3 x+5 x^2}\right )^3} \, dx=\int \frac {1}{{\left (2\,x+\sqrt {5\,x^2+3\,x+2}+1\right )}^3} \,d x \] Input:

int(1/(2*x + (3*x + 5*x^2 + 2)^(1/2) + 1)^3,x)
 

Output:

int(1/(2*x + (3*x + 5*x^2 + 2)^(1/2) + 1)^3, x)
 

Reduce [F]

\[ \int \frac {1}{\left (1+2 x+\sqrt {2+3 x+5 x^2}\right )^3} \, dx=\int \frac {1}{17 \sqrt {5 x^{2}+3 x +2}\, x^{2}+15 \sqrt {5 x^{2}+3 x +2}\, x +5 \sqrt {5 x^{2}+3 x +2}+38 x^{3}+45 x^{2}+27 x +7}d x \] Input:

int(1/(1+2*x+(5*x^2+3*x+2)^(1/2))^3,x)
 

Output:

int(1/(17*sqrt(5*x**2 + 3*x + 2)*x**2 + 15*sqrt(5*x**2 + 3*x + 2)*x + 5*sq 
rt(5*x**2 + 3*x + 2) + 38*x**3 + 45*x**2 + 27*x + 7),x)