Integrand size = 25, antiderivative size = 64 \[ \int \frac {\left (3-x+2 x^2\right )^2}{\left (2+3 x+5 x^2\right )^3} \, dx=\frac {121 (61+69 x)}{7750 \left (2+3 x+5 x^2\right )^2}+\frac {11 (17557+45710 x)}{240250 \left (2+3 x+5 x^2\right )}+\frac {4330 \arctan \left (\frac {3+10 x}{\sqrt {31}}\right )}{961 \sqrt {31}} \] Output:
121/7750*(61+69*x)/(5*x^2+3*x+2)^2+11*(17557+45710*x)/(1201250*x^2+720750* x+480500)+4330/29791*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 \[ \int \frac {\left (3-x+2 x^2\right )^2}{\left (2+3 x+5 x^2\right )^3} \, dx=\frac {11 \left (11183+33524 x+44983 x^2+45710 x^3\right )}{48050 \left (2+3 x+5 x^2\right )^2}+\frac {4330 \arctan \left (\frac {3+10 x}{\sqrt {31}}\right )}{961 \sqrt {31}} \] Input:
Integrate[(3 - x + 2*x^2)^2/(2 + 3*x + 5*x^2)^3,x]
Output:
(11*(11183 + 33524*x + 44983*x^2 + 45710*x^3))/(48050*(2 + 3*x + 5*x^2)^2) + (4330*ArcTan[(3 + 10*x)/Sqrt[31]])/(961*Sqrt[31])
Time = 0.25 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.08, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, Rules used = {2191, 27, 2191, 27, 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.
\(\displaystyle \int \frac {\left (2 x^2-x+3\right )^2}{\left (5 x^2+3 x+2\right )^3} \, dx\) |
\(\Big \downarrow \) 2191 |
\(\displaystyle \frac {1}{62} \int \frac {6200 x^2-9920 x+48669}{125 \left (5 x^2+3 x+2\right )^2}dx+\frac {121 (69 x+61)}{7750 \left (5 x^2+3 x+2\right )^2}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\int \frac {6200 x^2-9920 x+48669}{\left (5 x^2+3 x+2\right )^2}dx}{7750}+\frac {121 (69 x+61)}{7750 \left (5 x^2+3 x+2\right )^2}\) |
\(\Big \downarrow \) 2191 |
\(\displaystyle \frac {\frac {1}{31} \int \frac {541250}{5 x^2+3 x+2}dx+\frac {11 (45710 x+17557)}{31 \left (5 x^2+3 x+2\right )}}{7750}+\frac {121 (69 x+61)}{7750 \left (5 x^2+3 x+2\right )^2}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\frac {541250}{31} \int \frac {1}{5 x^2+3 x+2}dx+\frac {11 (45710 x+17557)}{31 \left (5 x^2+3 x+2\right )}}{7750}+\frac {121 (69 x+61)}{7750 \left (5 x^2+3 x+2\right )^2}\) |
\(\Big \downarrow \) 1083 |
\(\displaystyle \frac {\frac {11 (45710 x+17557)}{31 \left (5 x^2+3 x+2\right )}-\frac {1082500}{31} \int \frac {1}{-(10 x+3)^2-31}d(10 x+3)}{7750}+\frac {121 (69 x+61)}{7750 \left (5 x^2+3 x+2\right )^2}\) |
\(\Big \downarrow \) 217 |
\(\displaystyle \frac {\frac {1082500 \arctan \left (\frac {10 x+3}{\sqrt {31}}\right )}{31 \sqrt {31}}+\frac {11 (45710 x+17557)}{31 \left (5 x^2+3 x+2\right )}}{7750}+\frac {121 (69 x+61)}{7750 \left (5 x^2+3 x+2\right )^2}\) |
Input:
Int[(3 - x + 2*x^2)^2/(2 + 3*x + 5*x^2)^3,x]
Output:
(121*(61 + 69*x))/(7750*(2 + 3*x + 5*x^2)^2) + ((11*(17557 + 45710*x))/(31 *(2 + 3*x + 5*x^2)) + (1082500*ArcTan[(3 + 10*x)/Sqrt[31]])/(31*Sqrt[31])) /7750
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[(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 = 2.07 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.73
method | result | size |
default | \(\frac {\frac {50281}{4805} x^{3}+\frac {494813}{48050} x^{2}+\frac {184382}{24025} x +\frac {123013}{48050}}{\left (5 x^{2}+3 x +2\right )^{2}}+\frac {4330 \arctan \left (\frac {\left (10 x +3\right ) \sqrt {31}}{31}\right ) \sqrt {31}}{29791}\) | \(47\) |
risch | \(\frac {\frac {50281}{4805} x^{3}+\frac {494813}{48050} x^{2}+\frac {184382}{24025} x +\frac {123013}{48050}}{\left (5 x^{2}+3 x +2\right )^{2}}+\frac {4330 \arctan \left (\frac {\left (10 x +3\right ) \sqrt {31}}{31}\right ) \sqrt {31}}{29791}\) | \(47\) |
Input:
int((2*x^2-x+3)^2/(5*x^2+3*x+2)^3,x,method=_RETURNVERBOSE)
Output:
25*(50281/120125*x^3+494813/1201250*x^2+184382/600625*x+123013/1201250)/(5 *x^2+3*x+2)^2+4330/29791*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 \[ \int \frac {\left (3-x+2 x^2\right )^2}{\left (2+3 x+5 x^2\right )^3} \, dx=\frac {15587110 \, x^{3} + 216500 \, \sqrt {31} {\left (25 \, x^{4} + 30 \, x^{3} + 29 \, x^{2} + 12 \, x + 4\right )} \arctan \left (\frac {1}{31} \, \sqrt {31} {\left (10 \, x + 3\right )}\right ) + 15339203 \, x^{2} + 11431684 \, x + 3813403}{1489550 \, {\left (25 \, x^{4} + 30 \, x^{3} + 29 \, x^{2} + 12 \, x + 4\right )}} \] Input:
integrate((2*x^2-x+3)^2/(5*x^2+3*x+2)^3,x, algorithm="fricas")
Output:
1/1489550*(15587110*x^3 + 216500*sqrt(31)*(25*x^4 + 30*x^3 + 29*x^2 + 12*x + 4)*arctan(1/31*sqrt(31)*(10*x + 3)) + 15339203*x^2 + 11431684*x + 38134 03)/(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 \[ \int \frac {\left (3-x+2 x^2\right )^2}{\left (2+3 x+5 x^2\right )^3} \, dx=\frac {502810 x^{3} + 494813 x^{2} + 368764 x + 123013}{1201250 x^{4} + 1441500 x^{3} + 1393450 x^{2} + 576600 x + 192200} + \frac {4330 \sqrt {31} \operatorname {atan}{\left (\frac {10 \sqrt {31} x}{31} + \frac {3 \sqrt {31}}{31} \right )}}{29791} \] Input:
integrate((2*x**2-x+3)**2/(5*x**2+3*x+2)**3,x)
Output:
(502810*x**3 + 494813*x**2 + 368764*x + 123013)/(1201250*x**4 + 1441500*x* *3 + 1393450*x**2 + 576600*x + 192200) + 4330*sqrt(31)*atan(10*sqrt(31)*x/ 31 + 3*sqrt(31)/31)/29791
Time = 0.11 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.88 \[ \int \frac {\left (3-x+2 x^2\right )^2}{\left (2+3 x+5 x^2\right )^3} \, dx=\frac {4330}{29791} \, \sqrt {31} \arctan \left (\frac {1}{31} \, \sqrt {31} {\left (10 \, x + 3\right )}\right ) + \frac {11 \, {\left (45710 \, x^{3} + 44983 \, x^{2} + 33524 \, x + 11183\right )}}{48050 \, {\left (25 \, x^{4} + 30 \, x^{3} + 29 \, x^{2} + 12 \, x + 4\right )}} \] Input:
integrate((2*x^2-x+3)^2/(5*x^2+3*x+2)^3,x, algorithm="maxima")
Output:
4330/29791*sqrt(31)*arctan(1/31*sqrt(31)*(10*x + 3)) + 11/48050*(45710*x^3 + 44983*x^2 + 33524*x + 11183)/(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 \[ \int \frac {\left (3-x+2 x^2\right )^2}{\left (2+3 x+5 x^2\right )^3} \, dx=\frac {4330}{29791} \, \sqrt {31} \arctan \left (\frac {1}{31} \, \sqrt {31} {\left (10 \, x + 3\right )}\right ) + \frac {11 \, {\left (45710 \, x^{3} + 44983 \, x^{2} + 33524 \, x + 11183\right )}}{48050 \, {\left (5 \, x^{2} + 3 \, x + 2\right )}^{2}} \] Input:
integrate((2*x^2-x+3)^2/(5*x^2+3*x+2)^3,x, algorithm="giac")
Output:
4330/29791*sqrt(31)*arctan(1/31*sqrt(31)*(10*x + 3)) + 11/48050*(45710*x^3 + 44983*x^2 + 33524*x + 11183)/(5*x^2 + 3*x + 2)^2
Time = 15.10 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.86 \[ \int \frac {\left (3-x+2 x^2\right )^2}{\left (2+3 x+5 x^2\right )^3} \, dx=\frac {4330\,\sqrt {31}\,\mathrm {atan}\left (\frac {10\,\sqrt {31}\,x}{31}+\frac {3\,\sqrt {31}}{31}\right )}{29791}+\frac {\frac {50281\,x^3}{120125}+\frac {494813\,x^2}{1201250}+\frac {184382\,x}{600625}+\frac {123013}{1201250}}{x^4+\frac {6\,x^3}{5}+\frac {29\,x^2}{25}+\frac {12\,x}{25}+\frac {4}{25}} \] Input:
int((2*x^2 - x + 3)^2/(3*x + 5*x^2 + 2)^3,x)
Output:
(4330*31^(1/2)*atan((10*31^(1/2)*x)/31 + (3*31^(1/2))/31))/29791 + ((18438 2*x)/600625 + (494813*x^2)/1201250 + (50281*x^3)/120125 + 123013/1201250)/ ((12*x)/25 + (29*x^2)/25 + (6*x^3)/5 + x^4 + 4/25)
Time = 0.20 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.92 \[ \int \frac {\left (3-x+2 x^2\right )^2}{\left (2+3 x+5 x^2\right )^3} \, dx=\frac {3247500 \sqrt {31}\, \mathit {atan} \left (\frac {10 x +3}{\sqrt {31}}\right ) x^{4}+3897000 \sqrt {31}\, \mathit {atan} \left (\frac {10 x +3}{\sqrt {31}}\right ) x^{3}+3767100 \sqrt {31}\, \mathit {atan} \left (\frac {10 x +3}{\sqrt {31}}\right ) x^{2}+1558800 \sqrt {31}\, \mathit {atan} \left (\frac {10 x +3}{\sqrt {31}}\right ) x +519600 \sqrt {31}\, \mathit {atan} \left (\frac {10 x +3}{\sqrt {31}}\right )-7793555 x^{4}+162998 x^{2}+3118104 x +1041073}{22343250 x^{4}+26811900 x^{3}+25918170 x^{2}+10724760 x +3574920} \] Input:
int((2*x^2-x+3)^2/(5*x^2+3*x+2)^3,x)
Output:
(3247500*sqrt(31)*atan((10*x + 3)/sqrt(31))*x**4 + 3897000*sqrt(31)*atan(( 10*x + 3)/sqrt(31))*x**3 + 3767100*sqrt(31)*atan((10*x + 3)/sqrt(31))*x**2 + 1558800*sqrt(31)*atan((10*x + 3)/sqrt(31))*x + 519600*sqrt(31)*atan((10 *x + 3)/sqrt(31)) - 7793555*x**4 + 162998*x**2 + 3118104*x + 1041073)/(893 730*(25*x**4 + 30*x**3 + 29*x**2 + 12*x + 4))