Integrand size = 23, antiderivative size = 322 \[ \int \frac {x}{\left (1+2 x+\sqrt {2+3 x+5 x^2}\right )^3} \, dx=-\frac {2 \left (269+231 \sqrt {2}\right )-\frac {5 \left (7+13 \sqrt {2}\right ) \left (\sqrt {2}-\sqrt {2+3 x+5 x^2}\right )}{x}}{3 \left (1-5 \sqrt {2}-\frac {\left (3-4 \sqrt {2}\right ) \left (\sqrt {2}-\sqrt {2+3 x+5 x^2}\right )}{x}+\frac {\left (1-\sqrt {2}\right ) \left (\sqrt {2}-\sqrt {2+3 x+5 x^2}\right )^2}{x^2}\right )^2}+\frac {811+179 \sqrt {2}-\frac {310 \left (\sqrt {2}-\sqrt {2+3 x+5 x^2}\right )}{x}}{3 \left (1-5 \sqrt {2}-\frac {\left (3-4 \sqrt {2}\right ) \left (\sqrt {2}-\sqrt {2+3 x+5 x^2}\right )}{x}+\frac {\left (1-\sqrt {2}\right ) \left (\sqrt {2}-\sqrt {2+3 x+5 x^2}\right )^2}{x^2}\right )}+\frac {620 \arctan \left (\frac {3-4 \sqrt {2}-\frac {2 \left (1-\sqrt {2}\right ) \left (\sqrt {2}-\sqrt {2+3 x+5 x^2}\right )}{x}}{\sqrt {3}}\right )}{3 \sqrt {3}} \] Output:
-1/3*(538+462*2^(1/2)-5*(7+13*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+(811+179*2^(1/2)-310*(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)+620/9*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)
Result contains complex when optimal does not.
Time = 8.25 (sec) , antiderivative size = 1077, normalized size of antiderivative = 3.34 \[ \int \frac {x}{\left (1+2 x+\sqrt {2+3 x+5 x^2}\right )^3} \, dx =\text {Too large to display} \] Input:
Integrate[x/(1 + 2*x + Sqrt[2 + 3*x + 5*x^2])^3,x]
Output:
(2*(-20 + 37*x))/(3*(1 - x + x^2)^2) + (202 - 41*x)/(3 - 3*x + 3*x^2) + (2 *Sqrt[2 + 3*x + 5*x^2]*(-29 + 41*x - 52*x^2 + 25*x^3))/(3*(1 - x + x^2)^2) - (310*ArcTan[(-1 + 2*x)/Sqrt[3]])/(3*Sqrt[3]) + (((155*I)/3)*(2*I + Sqrt [3])*ArcTan[(3*(9*(7 + (8*I)*Sqrt[3]) + 3*(71 + (4*I)*Sqrt[3])*x + (163 + (68*I)*Sqrt[3])*x^2 + 96*(1 - (2*I)*Sqrt[3])*x^3 + (-16 + (80*I)*Sqrt[3])* x^4))/(72*I - 42*Sqrt[3] + 6*(-80*I + 17*Sqrt[3])*x^4 + 21*Sqrt[3 - (12*I) *Sqrt[3]]*Sqrt[2 + 3*x + 5*x^2] + x^2*(348*I + 206*Sqrt[3] - 49*Sqrt[3 - ( 12*I)*Sqrt[3]]*Sqrt[2 + 3*x + 5*x^2]) + x*(348*I - 104*Sqrt[3] + 49*Sqrt[3 - (12*I)*Sqrt[3]]*Sqrt[2 + 3*x + 5*x^2]) + x^3*(312*I + 340*Sqrt[3] + 70* Sqrt[3 - (12*I)*Sqrt[3]]*Sqrt[2 + 3*x + 5*x^2]))])/Sqrt[3 - (12*I)*Sqrt[3] ] + (155*(2 + I*Sqrt[3])*ArcTan[(3*(-63 + (72*I)*Sqrt[3] + (3*I)*(71*I + 4 *Sqrt[3])*x + (-163 + (68*I)*Sqrt[3])*x^2 + (-96 - (192*I)*Sqrt[3])*x^3 + 16*(1 + (5*I)*Sqrt[3])*x^4))/(-72*I - 42*Sqrt[3] - 4*(87*I + 26*Sqrt[3])*x + 6*(80*I + 17*Sqrt[3])*x^4 + 21*Sqrt[3 + (12*I)*Sqrt[3]]*Sqrt[2 + 3*x + 5*x^2] + 49*Sqrt[3 + (12*I)*Sqrt[3]]*x*Sqrt[2 + 3*x + 5*x^2] + x^2*(-348*I + 206*Sqrt[3] - 49*Sqrt[3 + (12*I)*Sqrt[3]]*Sqrt[2 + 3*x + 5*x^2]) + x^3* (-312*I + 340*Sqrt[3] + 70*Sqrt[3 + (12*I)*Sqrt[3]]*Sqrt[2 + 3*x + 5*x^2]) )])/(3*Sqrt[3 + (12*I)*Sqrt[3]]) + (155*(-2*I + Sqrt[3])*Log[16*(1 - x + x ^2)^2])/(6*Sqrt[3 + (12*I)*Sqrt[3]]) + (155*(2*I + Sqrt[3])*Log[16*(1 - x + x^2)^2])/(6*Sqrt[3 - (12*I)*Sqrt[3]]) - (155*(2*I + Sqrt[3])*Log[(1 -...
Time = 0.88 (sec) , antiderivative size = 328, normalized size of antiderivative = 1.02, 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.
| \(\displaystyle \int \frac {x}{\left (\sqrt {5 x^2+3 x+2}+2 x+1\right )^3} \, dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {-121 x-34}{\left (x^2-x+1\right )^2}+\frac {17 x \sqrt {5 x^2+3 x+2}}{\left (x^2-x+1\right )^2}+\frac {32 \sqrt {5 x^2+3 x+2}}{\left (x^2-x+1\right )^2}+\frac {20 x \sqrt {5 x^2+3 x+2}}{\left (x^2-x+1\right )^3}-\frac {32 \sqrt {5 x^2+3 x+2}}{\left (x^2-x+1\right )^3}-\frac {38}{x^2-x+1}+\frac {4 (x+18)}{\left (x^2-x+1\right )^3}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {310 \arctan \left (\frac {5-4 x}{\sqrt {3} \sqrt {5 x^2+3 x+2}}\right )}{3 \sqrt {3}}+42 \sqrt {3} \arctan \left (\frac {1-2 x}{\sqrt {3}}\right )-\frac {68 \arctan \left (\frac {1-2 x}{\sqrt {3}}\right )}{3 \sqrt {3}}+\frac {8 \sqrt {5 x^2+3 x+2} (101-338 x)}{147 \left (x^2-x+1\right )}-\frac {5 (44-237 x) \sqrt {5 x^2+3 x+2}}{147 \left (x^2-x+1\right )}-\frac {32 (1-2 x) \sqrt {5 x^2+3 x+2}}{3 \left (x^2-x+1\right )}-\frac {17 (2-x) \sqrt {5 x^2+3 x+2}}{3 \left (x^2-x+1\right )}+\frac {16 (1-2 x) \sqrt {5 x^2+3 x+2}}{3 \left (x^2-x+1\right )^2}-\frac {10 (2-x) \sqrt {5 x^2+3 x+2}}{3 \left (x^2-x+1\right )^2}+\frac {92-63 x}{x^2-x+1}-\frac {74 (1-2 x)}{3 \left (x^2-x+1\right )}-\frac {2 (20-37 x)}{3 \left (x^2-x+1\right )^2}\) |
Input:
Int[x/(1 + 2*x + Sqrt[2 + 3*x + 5*x^2])^3,x]
Output:
(-2*(20 - 37*x))/(3*(1 - x + x^2)^2) + (92 - 63*x)/(1 - x + x^2) - (74*(1 - 2*x))/(3*(1 - x + x^2)) + (16*(1 - 2*x)*Sqrt[2 + 3*x + 5*x^2])/(3*(1 - x + x^2)^2) - (10*(2 - x)*Sqrt[2 + 3*x + 5*x^2])/(3*(1 - x + x^2)^2) + (8*( 101 - 338*x)*Sqrt[2 + 3*x + 5*x^2])/(147*(1 - x + x^2)) - (5*(44 - 237*x)* Sqrt[2 + 3*x + 5*x^2])/(147*(1 - x + x^2)) - (32*(1 - 2*x)*Sqrt[2 + 3*x + 5*x^2])/(3*(1 - x + x^2)) - (17*(2 - x)*Sqrt[2 + 3*x + 5*x^2])/(3*(1 - x + x^2)) - (68*ArcTan[(1 - 2*x)/Sqrt[3]])/(3*Sqrt[3]) + 42*Sqrt[3]*ArcTan[(1 - 2*x)/Sqrt[3]] - (310*ArcTan[(5 - 4*x)/(Sqrt[3]*Sqrt[2 + 3*x + 5*x^2])]) /(3*Sqrt[3])
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.25 (sec) , antiderivative size = 126, normalized size of antiderivative = 0.39
| method | result | size |
| trager | \(-\frac {\left (162 x^{3}-283 x^{2}+243 x -155\right ) x}{3 \left (x^{2}-x +1\right )^{2}}+\frac {2 \left (25 x^{3}-52 x^{2}+41 x -29\right ) \sqrt {5 x^{2}+3 x +2}}{3 \left (x^{2}-x +1\right )^{2}}-\frac {310 \operatorname {RootOf}\left (\textit {\_Z}^{2}+3\right ) \ln \left (\frac {4 \operatorname {RootOf}\left (\textit {\_Z}^{2}+3\right ) x -5 \operatorname {RootOf}\left (\textit {\_Z}^{2}+3\right )+3 \sqrt {5 x^{2}+3 x +2}}{\operatorname {RootOf}\left (\textit {\_Z}^{2}+3\right ) x -x +2}\right )}{9}\) | \(126\) |
| default | \(\text {Expression too large to display}\) | \(9881\) |
Input:
int(x/(1+2*x+(5*x^2+3*x+2)^(1/2))^3,x,method=_RETURNVERBOSE)
Output:
-1/3*(162*x^3-283*x^2+243*x-155)*x/(x^2-x+1)^2+2/3*(25*x^3-52*x^2+41*x-29) /(x^2-x+1)^2*(5*x^2+3*x+2)^(1/2)-310/9*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))
Time = 0.07 (sec) , antiderivative size = 166, normalized size of antiderivative = 0.52 \[ \int \frac {x}{\left (1+2 x+\sqrt {2+3 x+5 x^2}\right )^3} \, dx=-\frac {123 \, x^{3} + 310 \, \sqrt {3} {\left (x^{4} - 2 \, x^{3} + 3 \, x^{2} - 2 \, x + 1\right )} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x - 1\right )}\right ) - 155 \, \sqrt {3} {\left (x^{4} - 2 \, x^{3} + 3 \, x^{2} - 2 \, x + 1\right )} \arctan \left (\frac {\sqrt {3} \sqrt {5 \, x^{2} + 3 \, x + 2} {\left (x^{2} - 49 \, x + 19\right )}}{6 \, {\left (20 \, x^{3} - 13 \, x^{2} - 7 \, x - 10\right )}}\right ) - 729 \, x^{2} - 6 \, {\left (25 \, x^{3} - 52 \, x^{2} + 41 \, x - 29\right )} \sqrt {5 \, x^{2} + 3 \, x + 2} + 507 \, x - 486}{9 \, {\left (x^{4} - 2 \, x^{3} + 3 \, x^{2} - 2 \, x + 1\right )}} \] Input:
integrate(x/(1+2*x+(5*x^2+3*x+2)^(1/2))^3,x, algorithm="fricas")
Output:
-1/9*(123*x^3 + 310*sqrt(3)*(x^4 - 2*x^3 + 3*x^2 - 2*x + 1)*arctan(1/3*sqr t(3)*(2*x - 1)) - 155*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*(25*x^3 - 52*x^2 + 41*x - 29)*sqrt(5*x^2 + 3*x + 2) + 507 *x - 486)/(x^4 - 2*x^3 + 3*x^2 - 2*x + 1)
\[ \int \frac {x}{\left (1+2 x+\sqrt {2+3 x+5 x^2}\right )^3} \, dx=\int \frac {x}{\left (2 x + \sqrt {5 x^{2} + 3 x + 2} + 1\right )^{3}}\, dx \] Input:
integrate(x/(1+2*x+(5*x**2+3*x+2)**(1/2))**3,x)
Output:
Integral(x/(2*x + sqrt(5*x**2 + 3*x + 2) + 1)**3, x)
\[ \int \frac {x}{\left (1+2 x+\sqrt {2+3 x+5 x^2}\right )^3} \, dx=\int { \frac {x}{{\left (2 \, x + \sqrt {5 \, x^{2} + 3 \, x + 2} + 1\right )}^{3}} \,d x } \] Input:
integrate(x/(1+2*x+(5*x^2+3*x+2)^(1/2))^3,x, algorithm="maxima")
Output:
integrate(x/(2*x + sqrt(5*x^2 + 3*x + 2) + 1)^3, x)
Time = 0.18 (sec) , antiderivative size = 445, normalized size of antiderivative = 1.38 \[ \int \frac {x}{\left (1+2 x+\sqrt {2+3 x+5 x^2}\right )^3} \, dx=-\frac {310}{9} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x - 1\right )}\right ) + \frac {310 \, {\left (\sqrt {5} + 2\right )} \arctan \left (-\frac {2 \, \sqrt {5} x - \sqrt {5} - 2 \, \sqrt {5 \, x^{2} + 3 \, x + 2} - 4}{\sqrt {15} + 2 \, \sqrt {3}}\right )}{3 \, {\left (\sqrt {15} + 2 \, \sqrt {3}\right )}} - \frac {310 \, {\left (\sqrt {5} - 2\right )} \arctan \left (-\frac {2 \, \sqrt {5} x - \sqrt {5} - 2 \, \sqrt {5 \, x^{2} + 3 \, x + 2} + 4}{\sqrt {15} - 2 \, \sqrt {3}}\right )}{3 \, {\left (\sqrt {15} - 2 \, \sqrt {3}\right )}} - \frac {2 \, {\left (55 \, {\left (\sqrt {5} x - \sqrt {5 \, x^{2} + 3 \, x + 2}\right )}^{7} - 898 \, \sqrt {5} {\left (\sqrt {5} x - \sqrt {5 \, x^{2} + 3 \, x + 2}\right )}^{6} + 1742 \, {\left (\sqrt {5} x - \sqrt {5 \, x^{2} + 3 \, x + 2}\right )}^{5} - 2470 \, \sqrt {5} {\left (\sqrt {5} x - \sqrt {5 \, x^{2} + 3 \, x + 2}\right )}^{4} - 76551 \, {\left (\sqrt {5} x - \sqrt {5 \, x^{2} + 3 \, x + 2}\right )}^{3} - 66727 \, \sqrt {5} {\left (\sqrt {5} x - \sqrt {5 \, x^{2} + 3 \, x + 2}\right )}^{2} - 112695 \, \sqrt {5} x - 13715 \, \sqrt {5} + 112695 \, \sqrt {5 \, x^{2} + 3 \, x + 2}\right )}}{3 \, {\left ({\left (\sqrt {5} x - \sqrt {5 \, x^{2} + 3 \, x + 2}\right )}^{4} - 2 \, \sqrt {5} {\left (\sqrt {5} x - \sqrt {5 \, x^{2} + 3 \, x + 2}\right )}^{3} + 13 \, {\left (\sqrt {5} x - \sqrt {5 \, x^{2} + 3 \, x + 2}\right )}^{2} + 16 \, \sqrt {5} {\left (\sqrt {5} x - \sqrt {5 \, x^{2} + 3 \, x + 2}\right )} + 19\right )}^{2}} - \frac {41 \, x^{3} - 243 \, x^{2} + 169 \, x - 162}{3 \, {\left (x^{2} - x + 1\right )}^{2}} \] Input:
integrate(x/(1+2*x+(5*x^2+3*x+2)^(1/2))^3,x, algorithm="giac")
Output:
-310/9*sqrt(3)*arctan(1/3*sqrt(3)*(2*x - 1)) + 310/3*(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)) - 310/3*(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*(55*(sqrt(5)*x - sqrt(5*x^2 + 3*x + 2))^7 - 898*sqrt(5)*( sqrt(5)*x - sqrt(5*x^2 + 3*x + 2))^6 + 1742*(sqrt(5)*x - sqrt(5*x^2 + 3*x + 2))^5 - 2470*sqrt(5)*(sqrt(5)*x - sqrt(5*x^2 + 3*x + 2))^4 - 76551*(sqrt (5)*x - sqrt(5*x^2 + 3*x + 2))^3 - 66727*sqrt(5)*(sqrt(5)*x - sqrt(5*x^2 + 3*x + 2))^2 - 112695*sqrt(5)*x - 13715*sqrt(5) + 112695*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*(41*x^3 - 243*x^2 + 1 69*x - 162)/(x^2 - x + 1)^2
Timed out. \[ \int \frac {x}{\left (1+2 x+\sqrt {2+3 x+5 x^2}\right )^3} \, dx=\int \frac {x\,\sqrt {5\,x^2+3\,x+2}\,\left (17\,x^2+15\,x+5\right )}{{\left (x^2-x+1\right )}^3} \,d x-\frac {310\,\sqrt {3}\,\mathrm {atan}\left (\frac {2\,\sqrt {3}\,x}{3}-\frac {\sqrt {3}}{3}\right )}{9}-\frac {\frac {41\,x^3}{3}-81\,x^2+\frac {169\,x}{3}-54}{x^4-2\,x^3+3\,x^2-2\,x+1} \] Input:
int(x/(2*x + (3*x + 5*x^2 + 2)^(1/2) + 1)^3,x)
Output:
int((x*(3*x + 5*x^2 + 2)^(1/2)*(15*x + 17*x^2 + 5))/(x^2 - x + 1)^3, x) - (310*3^(1/2)*atan((2*3^(1/2)*x)/3 - 3^(1/2)/3))/9 - ((169*x)/3 - 81*x^2 + (41*x^3)/3 - 54)/(3*x^2 - 2*x - 2*x^3 + x^4 + 1)
\[ \int \frac {x}{\left (1+2 x+\sqrt {2+3 x+5 x^2}\right )^3} \, dx=\int \frac {x}{17 \sqrt {5 x^{2}+3 x +2}\, x^{2}+15 \sqrt {5 x^{2}+3 x +2}\, x +5 \sqrt {5 x^{2}+3 x +2}+38 x^{3}+45 x^{2}+27 x +7}d x \] Input:
int(x/(1+2*x+(5*x^2+3*x+2)^(1/2))^3,x)
Output:
int(x/(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)