Integrand size = 25, antiderivative size = 98 \[ \int \frac {\left (2+3 x+5 x^2\right )^4}{\left (3-x+2 x^2\right )^3} \, dx=\frac {2725 x}{8}+\frac {4875 x^2}{32}+\frac {625 x^3}{24}-\frac {14641 (101+79 x)}{5888 \left (3-x+2 x^2\right )^2}+\frac {1331 (5229+76420 x)}{135424 \left (3-x+2 x^2\right )}+\frac {63799791 \arctan \left (\frac {1-4 x}{\sqrt {23}}\right )}{16928 \sqrt {23}}-\frac {13915}{64} \log \left (3-x+2 x^2\right ) \] Output:
2725/8*x+4875/32*x^2+625/24*x^3-14641/5888*(101+79*x)/(2*x^2-x+3)^2+1331*( 5229+76420*x)/(270848*x^2-135424*x+406272)+63799791/389344*arctan(1/23*(1- 4*x)*23^(1/2))*23^(1/2)-13915/64*ln(2*x^2-x+3)
Time = 0.04 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.00 \[ \int \frac {\left (2+3 x+5 x^2\right )^4}{\left (3-x+2 x^2\right )^3} \, dx=\frac {2725 x}{8}+\frac {4875 x^2}{32}+\frac {625 x^3}{24}-\frac {14641 (101+79 x)}{5888 \left (3-x+2 x^2\right )^2}+\frac {1331 (5229+76420 x)}{135424 \left (3-x+2 x^2\right )}-\frac {63799791 \arctan \left (\frac {-1+4 x}{\sqrt {23}}\right )}{16928 \sqrt {23}}-\frac {13915}{64} \log \left (3-x+2 x^2\right ) \] Input:
Integrate[(2 + 3*x + 5*x^2)^4/(3 - x + 2*x^2)^3,x]
Output:
(2725*x)/8 + (4875*x^2)/32 + (625*x^3)/24 - (14641*(101 + 79*x))/(5888*(3 - x + 2*x^2)^2) + (1331*(5229 + 76420*x))/(135424*(3 - x + 2*x^2)) - (6379 9791*ArcTan[(-1 + 4*x)/Sqrt[23]])/(16928*Sqrt[23]) - (13915*Log[3 - x + 2* x^2])/64
Time = 0.38 (sec) , antiderivative size = 106, normalized size of antiderivative = 1.08, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, Rules used = {2191, 27, 2191, 27, 2188, 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.
\(\displaystyle \int \frac {\left (5 x^2+3 x+2\right )^4}{\left (2 x^2-x+3\right )^3} \, dx\) |
\(\Big \downarrow \) 2191 |
\(\displaystyle \frac {1}{46} \int \frac {1840000 x^6+5336000 x^5+6826400 x^4+2298160 x^3-3906136 x^2-2644724 x+2173869}{128 \left (2 x^2-x+3\right )^2}dx-\frac {14641 (79 x+101)}{5888 \left (2 x^2-x+3\right )^2}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\int \frac {1840000 x^6+5336000 x^5+6826400 x^4+2298160 x^3-3906136 x^2-2644724 x+2173869}{\left (2 x^2-x+3\right )^2}dx}{5888}-\frac {14641 (79 x+101)}{5888 \left (2 x^2-x+3\right )^2}\) |
\(\Big \downarrow \) 2191 |
\(\displaystyle \frac {\frac {1}{23} \int -\frac {16 \left (-1322500 x^4-4496500 x^3-5170975 x^2+2507460 x+5460539\right )}{2 x^2-x+3}dx+\frac {1331 (76420 x+5229)}{23 \left (2 x^2-x+3\right )}}{5888}-\frac {14641 (79 x+101)}{5888 \left (2 x^2-x+3\right )^2}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\frac {1331 (76420 x+5229)}{23 \left (2 x^2-x+3\right )}-\frac {16}{23} \int \frac {-1322500 x^4-4496500 x^3-5170975 x^2+2507460 x+5460539}{2 x^2-x+3}dx}{5888}-\frac {14641 (79 x+101)}{5888 \left (2 x^2-x+3\right )^2}\) |
\(\Big \downarrow \) 2188 |
\(\displaystyle \frac {\frac {1331 (76420 x+5229)}{23 \left (2 x^2-x+3\right )}-\frac {16}{23} \int \left (-661250 x^2-2578875 x+\frac {121 (60835 x+116609)}{2 x^2-x+3}-2883050\right )dx}{5888}-\frac {14641 (79 x+101)}{5888 \left (2 x^2-x+3\right )^2}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {\frac {1331 (76420 x+5229)}{23 \left (2 x^2-x+3\right )}-\frac {16}{23} \left (-\frac {63799791 \arctan \left (\frac {1-4 x}{\sqrt {23}}\right )}{2 \sqrt {23}}-\frac {661250 x^3}{3}-\frac {2578875 x^2}{2}+\frac {7361035}{4} \log \left (2 x^2-x+3\right )-2883050 x\right )}{5888}-\frac {14641 (79 x+101)}{5888 \left (2 x^2-x+3\right )^2}\) |
Input:
Int[(2 + 3*x + 5*x^2)^4/(3 - x + 2*x^2)^3,x]
Output:
(-14641*(101 + 79*x))/(5888*(3 - x + 2*x^2)^2) + ((1331*(5229 + 76420*x))/ (23*(3 - x + 2*x^2)) - (16*(-2883050*x - (2578875*x^2)/2 - (661250*x^3)/3 - (63799791*ArcTan[(1 - 4*x)/Sqrt[23]])/(2*Sqrt[23]) + (7361035*Log[3 - x + 2*x^2])/4))/23)/5888
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[(Pq_)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[Expand Integrand[Pq*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c}, x] && PolyQ[Pq , x] && IGtQ[p, -2]
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 = 2.34 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.74
method | result | size |
default | \(\frac {625 x^{3}}{24}+\frac {4875 x^{2}}{32}+\frac {2725 x}{8}-\frac {121 \left (-\frac {210155}{4232} x^{3}+\frac {362791}{16928} x^{2}-\frac {561121}{8464} x +\frac {54263}{16928}\right )}{4 \left (2 x^{2}-x +3\right )^{2}}-\frac {13915 \ln \left (2 x^{2}-x +3\right )}{64}-\frac {63799791 \sqrt {23}\, \arctan \left (\frac {\left (4 x -1\right ) \sqrt {23}}{23}\right )}{389344}\) | \(73\) |
risch | \(\frac {625 x^{3}}{24}+\frac {4875 x^{2}}{32}+\frac {2725 x}{8}+\frac {\frac {25428755}{16928} x^{3}-\frac {43897711}{67712} x^{2}+\frac {67895641}{33856} x -\frac {6565823}{67712}}{\left (2 x^{2}-x +3\right )^{2}}-\frac {13915 \ln \left (16 x^{2}-8 x +24\right )}{64}-\frac {63799791 \sqrt {23}\, \arctan \left (\frac {\left (4 x -1\right ) \sqrt {23}}{23}\right )}{389344}\) | \(73\) |
Input:
int((5*x^2+3*x+2)^4/(2*x^2-x+3)^3,x,method=_RETURNVERBOSE)
Output:
625/24*x^3+4875/32*x^2+2725/8*x-121/4*(-210155/4232*x^3+362791/16928*x^2-5 61121/8464*x+54263/16928)/(2*x^2-x+3)^2-13915/64*ln(2*x^2-x+3)-63799791/38 9344*23^(1/2)*arctan(1/23*(4*x-1)*23^(1/2))
Time = 0.06 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.31 \[ \int \frac {\left (2+3 x+5 x^2\right )^4}{\left (3-x+2 x^2\right )^3} \, dx=\frac {486680000 \, x^{7} + 2360398000 \, x^{6} + 5100406400 \, x^{5} + 2157209100 \, x^{4} + 24531516180 \, x^{3} - 765597492 \, \sqrt {23} {\left (4 \, x^{4} - 4 \, x^{3} + 13 \, x^{2} - 6 \, x + 9\right )} \arctan \left (\frac {1}{23} \, \sqrt {23} {\left (4 \, x - 1\right )}\right ) - 6171678159 \, x^{2} - 1015822830 \, {\left (4 \, x^{4} - 4 \, x^{3} + 13 \, x^{2} - 6 \, x + 9\right )} \log \left (2 \, x^{2} - x + 3\right ) + 23692590858 \, x - 453041787}{4672128 \, {\left (4 \, x^{4} - 4 \, x^{3} + 13 \, x^{2} - 6 \, x + 9\right )}} \] Input:
integrate((5*x^2+3*x+2)^4/(2*x^2-x+3)^3,x, algorithm="fricas")
Output:
1/4672128*(486680000*x^7 + 2360398000*x^6 + 5100406400*x^5 + 2157209100*x^ 4 + 24531516180*x^3 - 765597492*sqrt(23)*(4*x^4 - 4*x^3 + 13*x^2 - 6*x + 9 )*arctan(1/23*sqrt(23)*(4*x - 1)) - 6171678159*x^2 - 1015822830*(4*x^4 - 4 *x^3 + 13*x^2 - 6*x + 9)*log(2*x^2 - x + 3) + 23692590858*x - 453041787)/( 4*x^4 - 4*x^3 + 13*x^2 - 6*x + 9)
Time = 0.12 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.97 \[ \int \frac {\left (2+3 x+5 x^2\right )^4}{\left (3-x+2 x^2\right )^3} \, dx=\frac {625 x^{3}}{24} + \frac {4875 x^{2}}{32} + \frac {2725 x}{8} + \frac {101715020 x^{3} - 43897711 x^{2} + 135791282 x - 6565823}{270848 x^{4} - 270848 x^{3} + 880256 x^{2} - 406272 x + 609408} - \frac {13915 \log {\left (x^{2} - \frac {x}{2} + \frac {3}{2} \right )}}{64} - \frac {63799791 \sqrt {23} \operatorname {atan}{\left (\frac {4 \sqrt {23} x}{23} - \frac {\sqrt {23}}{23} \right )}}{389344} \] Input:
integrate((5*x**2+3*x+2)**4/(2*x**2-x+3)**3,x)
Output:
625*x**3/24 + 4875*x**2/32 + 2725*x/8 + (101715020*x**3 - 43897711*x**2 + 135791282*x - 6565823)/(270848*x**4 - 270848*x**3 + 880256*x**2 - 406272*x + 609408) - 13915*log(x**2 - x/2 + 3/2)/64 - 63799791*sqrt(23)*atan(4*sqr t(23)*x/23 - sqrt(23)/23)/389344
Time = 0.11 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.84 \[ \int \frac {\left (2+3 x+5 x^2\right )^4}{\left (3-x+2 x^2\right )^3} \, dx=\frac {625}{24} \, x^{3} + \frac {4875}{32} \, x^{2} - \frac {63799791}{389344} \, \sqrt {23} \arctan \left (\frac {1}{23} \, \sqrt {23} {\left (4 \, x - 1\right )}\right ) + \frac {2725}{8} \, x + \frac {1331 \, {\left (76420 \, x^{3} - 32981 \, x^{2} + 102022 \, x - 4933\right )}}{67712 \, {\left (4 \, x^{4} - 4 \, x^{3} + 13 \, x^{2} - 6 \, x + 9\right )}} - \frac {13915}{64} \, \log \left (2 \, x^{2} - x + 3\right ) \] Input:
integrate((5*x^2+3*x+2)^4/(2*x^2-x+3)^3,x, algorithm="maxima")
Output:
625/24*x^3 + 4875/32*x^2 - 63799791/389344*sqrt(23)*arctan(1/23*sqrt(23)*( 4*x - 1)) + 2725/8*x + 1331/67712*(76420*x^3 - 32981*x^2 + 102022*x - 4933 )/(4*x^4 - 4*x^3 + 13*x^2 - 6*x + 9) - 13915/64*log(2*x^2 - x + 3)
Time = 0.11 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.73 \[ \int \frac {\left (2+3 x+5 x^2\right )^4}{\left (3-x+2 x^2\right )^3} \, dx=\frac {625}{24} \, x^{3} + \frac {4875}{32} \, x^{2} - \frac {63799791}{389344} \, \sqrt {23} \arctan \left (\frac {1}{23} \, \sqrt {23} {\left (4 \, x - 1\right )}\right ) + \frac {2725}{8} \, x + \frac {1331 \, {\left (76420 \, x^{3} - 32981 \, x^{2} + 102022 \, x - 4933\right )}}{67712 \, {\left (2 \, x^{2} - x + 3\right )}^{2}} - \frac {13915}{64} \, \log \left (2 \, x^{2} - x + 3\right ) \] Input:
integrate((5*x^2+3*x+2)^4/(2*x^2-x+3)^3,x, algorithm="giac")
Output:
625/24*x^3 + 4875/32*x^2 - 63799791/389344*sqrt(23)*arctan(1/23*sqrt(23)*( 4*x - 1)) + 2725/8*x + 1331/67712*(76420*x^3 - 32981*x^2 + 102022*x - 4933 )/(2*x^2 - x + 3)^2 - 13915/64*log(2*x^2 - x + 3)
Time = 15.27 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.83 \[ \int \frac {\left (2+3 x+5 x^2\right )^4}{\left (3-x+2 x^2\right )^3} \, dx=\frac {2725\,x}{8}-\frac {13915\,\ln \left (2\,x^2-x+3\right )}{64}-\frac {63799791\,\sqrt {23}\,\mathrm {atan}\left (\frac {4\,\sqrt {23}\,x}{23}-\frac {\sqrt {23}}{23}\right )}{389344}+\frac {4875\,x^2}{32}+\frac {625\,x^3}{24}+\frac {\frac {25428755\,x^3}{67712}-\frac {43897711\,x^2}{270848}+\frac {67895641\,x}{135424}-\frac {6565823}{270848}}{x^4-x^3+\frac {13\,x^2}{4}-\frac {3\,x}{2}+\frac {9}{4}} \] Input:
int((3*x + 5*x^2 + 2)^4/(2*x^2 - x + 3)^3,x)
Output:
(2725*x)/8 - (13915*log(2*x^2 - x + 3))/64 - (63799791*23^(1/2)*atan((4*23 ^(1/2)*x)/23 - 23^(1/2)/23))/389344 + (4875*x^2)/32 + (625*x^3)/24 + ((678 95641*x)/135424 - (43897711*x^2)/270848 + (25428755*x^3)/67712 - 6565823/2 70848)/((13*x^2)/4 - (3*x)/2 - x^3 + x^4 + 9/4)
Time = 0.18 (sec) , antiderivative size = 213, normalized size of antiderivative = 2.17 \[ \int \frac {\left (2+3 x+5 x^2\right )^4}{\left (3-x+2 x^2\right )^3} \, dx=\frac {-1531194984 \sqrt {23}\, \mathit {atan} \left (\frac {4 x -1}{\sqrt {23}}\right ) x^{4}+1531194984 \sqrt {23}\, \mathit {atan} \left (\frac {4 x -1}{\sqrt {23}}\right ) x^{3}-4976383698 \sqrt {23}\, \mathit {atan} \left (\frac {4 x -1}{\sqrt {23}}\right ) x^{2}+2296792476 \sqrt {23}\, \mathit {atan} \left (\frac {4 x -1}{\sqrt {23}}\right ) x -3445188714 \sqrt {23}\, \mathit {atan} \left (\frac {4 x -1}{\sqrt {23}}\right )-2031645660 \,\mathrm {log}\left (2 x^{2}-x +3\right ) x^{4}+2031645660 \,\mathrm {log}\left (2 x^{2}-x +3\right ) x^{3}-6602848395 \,\mathrm {log}\left (2 x^{2}-x +3\right ) x^{2}+3047468490 \,\mathrm {log}\left (2 x^{2}-x +3\right ) x -4571202735 \,\mathrm {log}\left (2 x^{2}-x +3\right )+243340000 x^{7}+1180199000 x^{6}+2550203200 x^{5}+13344362640 x^{4}+36777874713 x^{2}-6552341706 x +27371434809}{9344256 x^{4}-9344256 x^{3}+30368832 x^{2}-14016384 x +21024576} \] Input:
int((5*x^2+3*x+2)^4/(2*x^2-x+3)^3,x)
Output:
( - 1531194984*sqrt(23)*atan((4*x - 1)/sqrt(23))*x**4 + 1531194984*sqrt(23 )*atan((4*x - 1)/sqrt(23))*x**3 - 4976383698*sqrt(23)*atan((4*x - 1)/sqrt( 23))*x**2 + 2296792476*sqrt(23)*atan((4*x - 1)/sqrt(23))*x - 3445188714*sq rt(23)*atan((4*x - 1)/sqrt(23)) - 2031645660*log(2*x**2 - x + 3)*x**4 + 20 31645660*log(2*x**2 - x + 3)*x**3 - 6602848395*log(2*x**2 - x + 3)*x**2 + 3047468490*log(2*x**2 - x + 3)*x - 4571202735*log(2*x**2 - x + 3) + 243340 000*x**7 + 1180199000*x**6 + 2550203200*x**5 + 13344362640*x**4 + 36777874 713*x**2 - 6552341706*x + 27371434809)/(2336064*(4*x**4 - 4*x**3 + 13*x**2 - 6*x + 9))