Integrand size = 31, antiderivative size = 64 \[ \int \frac {2+x+3 x^2-5 x^3+4 x^4}{\left (3+2 x+5 x^2\right )^3} \, dx=-\frac {1367+423 x}{7000 \left (3+2 x+5 x^2\right )^2}+\frac {34347+11015 x}{196000 \left (3+2 x+5 x^2\right )}+\frac {339 \arctan \left (\frac {1+5 x}{\sqrt {14}}\right )}{1568 \sqrt {14}} \] Output:
-1/7000*(1367+423*x)/(5*x^2+2*x+3)^2+(34347+11015*x)/(980000*x^2+392000*x+ 588000)+339/21952*arctan(1/14*(1+5*x)*14^(1/2))*14^(1/2)
Time = 0.04 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.83 \[ \int \frac {2+x+3 x^2-5 x^3+4 x^4}{\left (3+2 x+5 x^2\right )^3} \, dx=\frac {\frac {14 \left (12953+17979 x+38753 x^2+11015 x^3\right )}{\left (3+2 x+5 x^2\right )^2}+8475 \sqrt {14} \arctan \left (\frac {1+5 x}{\sqrt {14}}\right )}{548800} \] Input:
Integrate[(2 + x + 3*x^2 - 5*x^3 + 4*x^4)/(3 + 2*x + 5*x^2)^3,x]
Output:
((14*(12953 + 17979*x + 38753*x^2 + 11015*x^3))/(3 + 2*x + 5*x^2)^2 + 8475 *Sqrt[14]*ArcTan[(1 + 5*x)/Sqrt[14]])/548800
Time = 0.44 (sec) , antiderivative size = 72, normalized size of antiderivative = 1.12, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.194, 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.
\(\displaystyle \int \frac {4 x^4-5 x^3+3 x^2+x+2}{\left (5 x^2+2 x+3\right )^3} \, dx\) |
\(\Big \downarrow \) 2191 |
\(\displaystyle \frac {1}{112} \int \frac {2 \left (5600 x^2-9240 x+3267\right )}{125 \left (5 x^2+2 x+3\right )^2}dx-\frac {423 x+1367}{7000 \left (5 x^2+2 x+3\right )^2}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\int \frac {5600 x^2-9240 x+3267}{\left (5 x^2+2 x+3\right )^2}dx}{7000}-\frac {423 x+1367}{7000 \left (5 x^2+2 x+3\right )^2}\) |
\(\Big \downarrow \) 2191 |
\(\displaystyle \frac {\frac {1}{56} \int \frac {84750}{5 x^2+2 x+3}dx+\frac {11015 x+34347}{28 \left (5 x^2+2 x+3\right )}}{7000}-\frac {423 x+1367}{7000 \left (5 x^2+2 x+3\right )^2}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\frac {42375}{28} \int \frac {1}{5 x^2+2 x+3}dx+\frac {11015 x+34347}{28 \left (5 x^2+2 x+3\right )}}{7000}-\frac {423 x+1367}{7000 \left (5 x^2+2 x+3\right )^2}\) |
\(\Big \downarrow \) 1083 |
\(\displaystyle \frac {\frac {11015 x+34347}{28 \left (5 x^2+2 x+3\right )}-\frac {42375}{14} \int \frac {1}{-(10 x+2)^2-56}d(10 x+2)}{7000}-\frac {423 x+1367}{7000 \left (5 x^2+2 x+3\right )^2}\) |
\(\Big \downarrow \) 217 |
\(\displaystyle \frac {\frac {42375 \arctan \left (\frac {10 x+2}{2 \sqrt {14}}\right )}{28 \sqrt {14}}+\frac {11015 x+34347}{28 \left (5 x^2+2 x+3\right )}}{7000}-\frac {423 x+1367}{7000 \left (5 x^2+2 x+3\right )^2}\) |
Input:
Int[(2 + x + 3*x^2 - 5*x^3 + 4*x^4)/(3 + 2*x + 5*x^2)^3,x]
Output:
-1/7000*(1367 + 423*x)/(3 + 2*x + 5*x^2)^2 + ((34347 + 11015*x)/(28*(3 + 2 *x + 5*x^2)) + (42375*ArcTan[(2 + 10*x)/(2*Sqrt[14])])/(28*Sqrt[14]))/7000
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 = 0.11 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.73
method | result | size |
default | \(\frac {\frac {2203}{7840} x^{3}+\frac {38753}{39200} x^{2}+\frac {17979}{39200} x +\frac {12953}{39200}}{\left (5 x^{2}+2 x +3\right )^{2}}+\frac {339 \sqrt {14}\, \arctan \left (\frac {\left (10 x +2\right ) \sqrt {14}}{28}\right )}{21952}\) | \(47\) |
risch | \(\frac {\frac {2203}{7840} x^{3}+\frac {38753}{39200} x^{2}+\frac {17979}{39200} x +\frac {12953}{39200}}{\left (5 x^{2}+2 x +3\right )^{2}}+\frac {339 \arctan \left (\frac {\left (1+5 x \right ) \sqrt {14}}{14}\right ) \sqrt {14}}{21952}\) | \(47\) |
Input:
int((4*x^4-5*x^3+3*x^2+x+2)/(5*x^2+2*x+3)^3,x,method=_RETURNVERBOSE)
Output:
25*(2203/196000*x^3+38753/980000*x^2+17979/980000*x+12953/980000)/(5*x^2+2 *x+3)^2+339/21952*14^(1/2)*arctan(1/28*(10*x+2)*14^(1/2))
Time = 0.07 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.17 \[ \int \frac {2+x+3 x^2-5 x^3+4 x^4}{\left (3+2 x+5 x^2\right )^3} \, dx=\frac {154210 \, x^{3} + 8475 \, \sqrt {14} {\left (25 \, x^{4} + 20 \, x^{3} + 34 \, x^{2} + 12 \, x + 9\right )} \arctan \left (\frac {1}{14} \, \sqrt {14} {\left (5 \, x + 1\right )}\right ) + 542542 \, x^{2} + 251706 \, x + 181342}{548800 \, {\left (25 \, x^{4} + 20 \, x^{3} + 34 \, x^{2} + 12 \, x + 9\right )}} \] Input:
integrate((4*x^4-5*x^3+3*x^2+x+2)/(5*x^2+2*x+3)^3,x, algorithm="fricas")
Output:
1/548800*(154210*x^3 + 8475*sqrt(14)*(25*x^4 + 20*x^3 + 34*x^2 + 12*x + 9) *arctan(1/14*sqrt(14)*(5*x + 1)) + 542542*x^2 + 251706*x + 181342)/(25*x^4 + 20*x^3 + 34*x^2 + 12*x + 9)
Time = 0.08 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.95 \[ \int \frac {2+x+3 x^2-5 x^3+4 x^4}{\left (3+2 x+5 x^2\right )^3} \, dx=\frac {11015 x^{3} + 38753 x^{2} + 17979 x + 12953}{980000 x^{4} + 784000 x^{3} + 1332800 x^{2} + 470400 x + 352800} + \frac {339 \sqrt {14} \operatorname {atan}{\left (\frac {5 \sqrt {14} x}{14} + \frac {\sqrt {14}}{14} \right )}}{21952} \] Input:
integrate((4*x**4-5*x**3+3*x**2+x+2)/(5*x**2+2*x+3)**3,x)
Output:
(11015*x**3 + 38753*x**2 + 17979*x + 12953)/(980000*x**4 + 784000*x**3 + 1 332800*x**2 + 470400*x + 352800) + 339*sqrt(14)*atan(5*sqrt(14)*x/14 + sqr t(14)/14)/21952
Time = 0.11 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.88 \[ \int \frac {2+x+3 x^2-5 x^3+4 x^4}{\left (3+2 x+5 x^2\right )^3} \, dx=\frac {339}{21952} \, \sqrt {14} \arctan \left (\frac {1}{14} \, \sqrt {14} {\left (5 \, x + 1\right )}\right ) + \frac {11015 \, x^{3} + 38753 \, x^{2} + 17979 \, x + 12953}{39200 \, {\left (25 \, x^{4} + 20 \, x^{3} + 34 \, x^{2} + 12 \, x + 9\right )}} \] Input:
integrate((4*x^4-5*x^3+3*x^2+x+2)/(5*x^2+2*x+3)^3,x, algorithm="maxima")
Output:
339/21952*sqrt(14)*arctan(1/14*sqrt(14)*(5*x + 1)) + 1/39200*(11015*x^3 + 38753*x^2 + 17979*x + 12953)/(25*x^4 + 20*x^3 + 34*x^2 + 12*x + 9)
Time = 0.14 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.72 \[ \int \frac {2+x+3 x^2-5 x^3+4 x^4}{\left (3+2 x+5 x^2\right )^3} \, dx=\frac {339}{21952} \, \sqrt {14} \arctan \left (\frac {1}{14} \, \sqrt {14} {\left (5 \, x + 1\right )}\right ) + \frac {11015 \, x^{3} + 38753 \, x^{2} + 17979 \, x + 12953}{39200 \, {\left (5 \, x^{2} + 2 \, x + 3\right )}^{2}} \] Input:
integrate((4*x^4-5*x^3+3*x^2+x+2)/(5*x^2+2*x+3)^3,x, algorithm="giac")
Output:
339/21952*sqrt(14)*arctan(1/14*sqrt(14)*(5*x + 1)) + 1/39200*(11015*x^3 + 38753*x^2 + 17979*x + 12953)/(5*x^2 + 2*x + 3)^2
Time = 17.76 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.86 \[ \int \frac {2+x+3 x^2-5 x^3+4 x^4}{\left (3+2 x+5 x^2\right )^3} \, dx=\frac {339\,\sqrt {14}\,\mathrm {atan}\left (\frac {5\,\sqrt {14}\,x}{14}+\frac {\sqrt {14}}{14}\right )}{21952}+\frac {\frac {2203\,x^3}{196000}+\frac {38753\,x^2}{980000}+\frac {17979\,x}{980000}+\frac {12953}{980000}}{x^4+\frac {4\,x^3}{5}+\frac {34\,x^2}{25}+\frac {12\,x}{25}+\frac {9}{25}} \] Input:
int((x + 3*x^2 - 5*x^3 + 4*x^4 + 2)/(2*x + 5*x^2 + 3)^3,x)
Output:
(339*14^(1/2)*atan((5*14^(1/2)*x)/14 + 14^(1/2)/14))/21952 + ((17979*x)/98 0000 + (38753*x^2)/980000 + (2203*x^3)/196000 + 12953/980000)/((12*x)/25 + (34*x^2)/25 + (4*x^3)/5 + x^4 + 9/25)
Time = 0.20 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.92 \[ \int \frac {2+x+3 x^2-5 x^3+4 x^4}{\left (3+2 x+5 x^2\right )^3} \, dx=\frac {84750 \sqrt {14}\, \mathit {atan} \left (\frac {5 x +1}{\sqrt {14}}\right ) x^{4}+67800 \sqrt {14}\, \mathit {atan} \left (\frac {5 x +1}{\sqrt {14}}\right ) x^{3}+115260 \sqrt {14}\, \mathit {atan} \left (\frac {5 x +1}{\sqrt {14}}\right ) x^{2}+40680 \sqrt {14}\, \mathit {atan} \left (\frac {5 x +1}{\sqrt {14}}\right ) x +30510 \sqrt {14}\, \mathit {atan} \left (\frac {5 x +1}{\sqrt {14}}\right )-77105 x^{4}+112154 x^{2}+63672 x +44779}{5488000 x^{4}+4390400 x^{3}+7463680 x^{2}+2634240 x +1975680} \] Input:
int((4*x^4-5*x^3+3*x^2+x+2)/(5*x^2+2*x+3)^3,x)
Output:
(84750*sqrt(14)*atan((5*x + 1)/sqrt(14))*x**4 + 67800*sqrt(14)*atan((5*x + 1)/sqrt(14))*x**3 + 115260*sqrt(14)*atan((5*x + 1)/sqrt(14))*x**2 + 40680 *sqrt(14)*atan((5*x + 1)/sqrt(14))*x + 30510*sqrt(14)*atan((5*x + 1)/sqrt( 14)) - 77105*x**4 + 112154*x**2 + 63672*x + 44779)/(219520*(25*x**4 + 20*x **3 + 34*x**2 + 12*x + 9))