Integrand size = 23, antiderivative size = 64 \[ \int \frac {3-x+2 x^2}{\left (2+3 x+5 x^2\right )^3} \, dx=\frac {11 (7+13 x)}{310 \left (2+3 x+5 x^2\right )^2}+\frac {553 (3+10 x)}{9610 \left (2+3 x+5 x^2\right )}+\frac {1106 \arctan \left (\frac {3+10 x}{\sqrt {31}}\right )}{961 \sqrt {31}} \] Output:
11/310*(7+13*x)/(5*x^2+3*x+2)^2+553*(3+10*x)/(48050*x^2+28830*x+19220)+110 6/29791*arctan(1/31*(3+10*x)*31^(1/2))*31^(1/2)
Time = 0.03 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.83 \[ \int \frac {3-x+2 x^2}{\left (2+3 x+5 x^2\right )^3} \, dx=\frac {\frac {31 \left (1141+4094 x+4977 x^2+5530 x^3\right )}{\left (2+3 x+5 x^2\right )^2}+2212 \sqrt {31} \arctan \left (\frac {3+10 x}{\sqrt {31}}\right )}{59582} \] Input:
Integrate[(3 - x + 2*x^2)/(2 + 3*x + 5*x^2)^3,x]
Output:
((31*(1141 + 4094*x + 4977*x^2 + 5530*x^3))/(2 + 3*x + 5*x^2)^2 + 2212*Sqr t[31]*ArcTan[(3 + 10*x)/Sqrt[31]])/59582
Time = 0.22 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.08, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {2191, 27, 1086, 1083, 217}
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.
\(\displaystyle \int \frac {2 x^2-x+3}{\left (5 x^2+3 x+2\right )^3} \, dx\) |
\(\Big \downarrow \) 2191 |
\(\displaystyle \frac {1}{62} \int \frac {553}{5 \left (5 x^2+3 x+2\right )^2}dx+\frac {11 (13 x+7)}{310 \left (5 x^2+3 x+2\right )^2}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {553}{310} \int \frac {1}{\left (5 x^2+3 x+2\right )^2}dx+\frac {11 (13 x+7)}{310 \left (5 x^2+3 x+2\right )^2}\) |
\(\Big \downarrow \) 1086 |
\(\displaystyle \frac {553}{310} \left (\frac {10}{31} \int \frac {1}{5 x^2+3 x+2}dx+\frac {10 x+3}{31 \left (5 x^2+3 x+2\right )}\right )+\frac {11 (13 x+7)}{310 \left (5 x^2+3 x+2\right )^2}\) |
\(\Big \downarrow \) 1083 |
\(\displaystyle \frac {553}{310} \left (\frac {10 x+3}{31 \left (5 x^2+3 x+2\right )}-\frac {20}{31} \int \frac {1}{-(10 x+3)^2-31}d(10 x+3)\right )+\frac {11 (13 x+7)}{310 \left (5 x^2+3 x+2\right )^2}\) |
\(\Big \downarrow \) 217 |
\(\displaystyle \frac {553}{310} \left (\frac {20 \arctan \left (\frac {10 x+3}{\sqrt {31}}\right )}{31 \sqrt {31}}+\frac {10 x+3}{31 \left (5 x^2+3 x+2\right )}\right )+\frac {11 (13 x+7)}{310 \left (5 x^2+3 x+2\right )^2}\) |
Input:
Int[(3 - x + 2*x^2)/(2 + 3*x + 5*x^2)^3,x]
Output:
(11*(7 + 13*x))/(310*(2 + 3*x + 5*x^2)^2) + (553*((3 + 10*x)/(31*(2 + 3*x + 5*x^2)) + (20*ArcTan[(3 + 10*x)/Sqrt[31]])/(31*Sqrt[31])))/310
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^( -1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & & (LtQ[a, 0] || LtQ[b, 0])
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Simp[-2 Subst[I nt[1/Simp[b^2 - 4*a*c - x^2, x], x], x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x]
Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(b + 2*c*x) *((a + b*x + c*x^2)^(p + 1)/((p + 1)*(b^2 - 4*a*c))), x] - Simp[2*c*((2*p + 3)/((p + 1)*(b^2 - 4*a*c))) Int[(a + b*x + c*x^2)^(p + 1), x], x] /; Fre eQ[{a, b, c}, x] && ILtQ[p, -1]
Int[(Pq_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = PolynomialQuotient[Pq, a + b*x + c*x^2, x], f = Coeff[PolynomialRemainder[P q, a + b*x + c*x^2, x], x, 0], g = Coeff[PolynomialRemainder[Pq, a + b*x + c*x^2, x], x, 1]}, Simp[(b*f - 2*a*g + (2*c*f - b*g)*x)*((a + b*x + c*x^2)^ (p + 1)/((p + 1)*(b^2 - 4*a*c))), x] + Simp[1/((p + 1)*(b^2 - 4*a*c)) Int [(a + b*x + c*x^2)^(p + 1)*ExpandToSum[(p + 1)*(b^2 - 4*a*c)*Q - (2*p + 3)* (2*c*f - b*g), x], x], x]] /; FreeQ[{a, b, c}, x] && PolyQ[Pq, x] && NeQ[b^ 2 - 4*a*c, 0] && LtQ[p, -1]
Time = 1.83 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.73
method | result | size |
default | \(\frac {\frac {2765}{961} x^{3}+\frac {4977}{1922} x^{2}+\frac {2047}{961} x +\frac {1141}{1922}}{\left (5 x^{2}+3 x +2\right )^{2}}+\frac {1106 \arctan \left (\frac {\left (10 x +3\right ) \sqrt {31}}{31}\right ) \sqrt {31}}{29791}\) | \(47\) |
risch | \(\frac {\frac {2765}{961} x^{3}+\frac {4977}{1922} x^{2}+\frac {2047}{961} x +\frac {1141}{1922}}{\left (5 x^{2}+3 x +2\right )^{2}}+\frac {1106 \arctan \left (\frac {\left (10 x +3\right ) \sqrt {31}}{31}\right ) \sqrt {31}}{29791}\) | \(47\) |
Input:
int((2*x^2-x+3)/(5*x^2+3*x+2)^3,x,method=_RETURNVERBOSE)
Output:
25*(553/4805*x^3+4977/48050*x^2+2047/24025*x+1141/48050)/(5*x^2+3*x+2)^2+1 106/29791*arctan(1/31*(10*x+3)*31^(1/2))*31^(1/2)
Time = 0.06 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.17 \[ \int \frac {3-x+2 x^2}{\left (2+3 x+5 x^2\right )^3} \, dx=\frac {171430 \, x^{3} + 2212 \, \sqrt {31} {\left (25 \, x^{4} + 30 \, x^{3} + 29 \, x^{2} + 12 \, x + 4\right )} \arctan \left (\frac {1}{31} \, \sqrt {31} {\left (10 \, x + 3\right )}\right ) + 154287 \, x^{2} + 126914 \, x + 35371}{59582 \, {\left (25 \, x^{4} + 30 \, x^{3} + 29 \, x^{2} + 12 \, x + 4\right )}} \] Input:
integrate((2*x^2-x+3)/(5*x^2+3*x+2)^3,x, algorithm="fricas")
Output:
1/59582*(171430*x^3 + 2212*sqrt(31)*(25*x^4 + 30*x^3 + 29*x^2 + 12*x + 4)* arctan(1/31*sqrt(31)*(10*x + 3)) + 154287*x^2 + 126914*x + 35371)/(25*x^4 + 30*x^3 + 29*x^2 + 12*x + 4)
Time = 0.10 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.98 \[ \int \frac {3-x+2 x^2}{\left (2+3 x+5 x^2\right )^3} \, dx=\frac {5530 x^{3} + 4977 x^{2} + 4094 x + 1141}{48050 x^{4} + 57660 x^{3} + 55738 x^{2} + 23064 x + 7688} + \frac {1106 \sqrt {31} \operatorname {atan}{\left (\frac {10 \sqrt {31} x}{31} + \frac {3 \sqrt {31}}{31} \right )}}{29791} \] Input:
integrate((2*x**2-x+3)/(5*x**2+3*x+2)**3,x)
Output:
(5530*x**3 + 4977*x**2 + 4094*x + 1141)/(48050*x**4 + 57660*x**3 + 55738*x **2 + 23064*x + 7688) + 1106*sqrt(31)*atan(10*sqrt(31)*x/31 + 3*sqrt(31)/3 1)/29791
Time = 0.11 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.88 \[ \int \frac {3-x+2 x^2}{\left (2+3 x+5 x^2\right )^3} \, dx=\frac {1106}{29791} \, \sqrt {31} \arctan \left (\frac {1}{31} \, \sqrt {31} {\left (10 \, x + 3\right )}\right ) + \frac {5530 \, x^{3} + 4977 \, x^{2} + 4094 \, x + 1141}{1922 \, {\left (25 \, x^{4} + 30 \, x^{3} + 29 \, x^{2} + 12 \, x + 4\right )}} \] Input:
integrate((2*x^2-x+3)/(5*x^2+3*x+2)^3,x, algorithm="maxima")
Output:
1106/29791*sqrt(31)*arctan(1/31*sqrt(31)*(10*x + 3)) + 1/1922*(5530*x^3 + 4977*x^2 + 4094*x + 1141)/(25*x^4 + 30*x^3 + 29*x^2 + 12*x + 4)
Time = 0.12 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.72 \[ \int \frac {3-x+2 x^2}{\left (2+3 x+5 x^2\right )^3} \, dx=\frac {1106}{29791} \, \sqrt {31} \arctan \left (\frac {1}{31} \, \sqrt {31} {\left (10 \, x + 3\right )}\right ) + \frac {5530 \, x^{3} + 4977 \, x^{2} + 4094 \, x + 1141}{1922 \, {\left (5 \, x^{2} + 3 \, x + 2\right )}^{2}} \] Input:
integrate((2*x^2-x+3)/(5*x^2+3*x+2)^3,x, algorithm="giac")
Output:
1106/29791*sqrt(31)*arctan(1/31*sqrt(31)*(10*x + 3)) + 1/1922*(5530*x^3 + 4977*x^2 + 4094*x + 1141)/(5*x^2 + 3*x + 2)^2
Time = 15.81 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.86 \[ \int \frac {3-x+2 x^2}{\left (2+3 x+5 x^2\right )^3} \, dx=\frac {1106\,\sqrt {31}\,\mathrm {atan}\left (\frac {10\,\sqrt {31}\,x}{31}+\frac {3\,\sqrt {31}}{31}\right )}{29791}+\frac {\frac {553\,x^3}{4805}+\frac {4977\,x^2}{48050}+\frac {2047\,x}{24025}+\frac {1141}{48050}}{x^4+\frac {6\,x^3}{5}+\frac {29\,x^2}{25}+\frac {12\,x}{25}+\frac {4}{25}} \] Input:
int((2*x^2 - x + 3)/(3*x + 5*x^2 + 2)^3,x)
Output:
(1106*31^(1/2)*atan((10*31^(1/2)*x)/31 + (3*31^(1/2))/31))/29791 + ((2047* x)/24025 + (4977*x^2)/48050 + (553*x^3)/4805 + 1141/48050)/((12*x)/25 + (2 9*x^2)/25 + (6*x^3)/5 + x^4 + 4/25)
Time = 0.19 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.92 \[ \int \frac {3-x+2 x^2}{\left (2+3 x+5 x^2\right )^3} \, dx=\frac {165900 \sqrt {31}\, \mathit {atan} \left (\frac {10 x +3}{\sqrt {31}}\right ) x^{4}+199080 \sqrt {31}\, \mathit {atan} \left (\frac {10 x +3}{\sqrt {31}}\right ) x^{3}+192444 \sqrt {31}\, \mathit {atan} \left (\frac {10 x +3}{\sqrt {31}}\right ) x^{2}+79632 \sqrt {31}\, \mathit {atan} \left (\frac {10 x +3}{\sqrt {31}}\right ) x +26544 \sqrt {31}\, \mathit {atan} \left (\frac {10 x +3}{\sqrt {31}}\right )-428575 x^{4}-34286 x^{2}+175026 x +37541}{4468650 x^{4}+5362380 x^{3}+5183634 x^{2}+2144952 x +714984} \] Input:
int((2*x^2-x+3)/(5*x^2+3*x+2)^3,x)
Output:
(165900*sqrt(31)*atan((10*x + 3)/sqrt(31))*x**4 + 199080*sqrt(31)*atan((10 *x + 3)/sqrt(31))*x**3 + 192444*sqrt(31)*atan((10*x + 3)/sqrt(31))*x**2 + 79632*sqrt(31)*atan((10*x + 3)/sqrt(31))*x + 26544*sqrt(31)*atan((10*x + 3 )/sqrt(31)) - 428575*x**4 - 34286*x**2 + 175026*x + 37541)/(178746*(25*x** 4 + 30*x**3 + 29*x**2 + 12*x + 4))