Integrand size = 46, antiderivative size = 90 \[ \int \frac {\left (2-3 x+x^2\right ) \left (d+e x+f x^2+g x^3+h x^4+i x^5\right )}{4-5 x^2+x^4} \, dx=(f-3 g+7 h-15 i) x+\frac {1}{2} (g-3 h+7 i) x^2+\frac {1}{3} (h-3 i) x^3+\frac {i x^4}{4}+(d-e+f-g+h-i) \log (1+x)-(d-2 e+4 f-8 g+16 h-32 i) \log (2+x) \]
(f-3*g+7*h-15*i)*x+1/2*(g-3*h+7*i)*x^2+1/3*(h-3*i)*x^3+1/4*i*x^4+(d-e+f-g+ h-i)*ln(1+x)-(d-2*e+4*f-8*g+16*h-32*i)*ln(2+x)
Time = 0.03 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.01 \[ \int \frac {\left (2-3 x+x^2\right ) \left (d+e x+f x^2+g x^3+h x^4+i x^5\right )}{4-5 x^2+x^4} \, dx=(f-3 g+7 h-15 i) x+\frac {1}{2} (g-3 h+7 i) x^2+\frac {1}{3} (h-3 i) x^3+\frac {i x^4}{4}+(d-e+f-g+h-i) \log (1+x)+(-d+2 e-4 f+8 g-16 h+32 i) \log (2+x) \]
(f - 3*g + 7*h - 15*i)*x + ((g - 3*h + 7*i)*x^2)/2 + ((h - 3*i)*x^3)/3 + ( i*x^4)/4 + (d - e + f - g + h - i)*Log[1 + x] + (-d + 2*e - 4*f + 8*g - 16 *h + 32*i)*Log[2 + x]
Time = 0.33 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.065, Rules used = {2019, 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.
\(\displaystyle \int \frac {\left (x^2-3 x+2\right ) \left (d+e x+f x^2+g x^3+h x^4+i x^5\right )}{x^4-5 x^2+4} \, dx\) |
\(\Big \downarrow \) 2019 |
\(\displaystyle \int \frac {d+e x+f x^2+g x^3+h x^4+i x^5}{x^2+3 x+2}dx\) |
\(\Big \downarrow \) 2188 |
\(\displaystyle \int \left (\frac {d+x (e-3 f+7 g-15 h+31 i)-2 f+6 g-14 h+30 i}{x^2+3 x+2}+f+x (g-3 h+7 i)-3 g+x^2 (h-3 i)+7 h+i x^3-15 i\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \log (x+1) (d-e+f-g+h-i)-\log (x+2) (d-2 e+4 f-8 g+16 h-32 i)+x (f-3 g+7 h-15 i)+\frac {1}{2} x^2 (g-3 h+7 i)+\frac {1}{3} x^3 (h-3 i)+\frac {i x^4}{4}\) |
(f - 3*g + 7*h - 15*i)*x + ((g - 3*h + 7*i)*x^2)/2 + ((h - 3*i)*x^3)/3 + ( i*x^4)/4 + (d - e + f - g + h - i)*Log[1 + x] - (d - 2*e + 4*f - 8*g + 16* h - 32*i)*Log[2 + x]
3.1.78.3.1 Defintions of rubi rules used
Int[(u_.)*(Px_)^(p_.)*(Qx_)^(q_.), x_Symbol] :> Int[u*PolynomialQuotient[Px , Qx, x]^p*Qx^(p + q), x] /; FreeQ[q, x] && PolyQ[Px, x] && PolyQ[Qx, x] && EqQ[PolynomialRemainder[Px, Qx, x], 0] && IntegerQ[p] && LtQ[p*q, 0]
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]
Time = 0.06 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.98
method | result | size |
norman | \(\left (\frac {h}{3}-i \right ) x^{3}+\left (\frac {g}{2}-\frac {3 h}{2}+\frac {7 i}{2}\right ) x^{2}+\left (f -3 g +7 h -15 i \right ) x +\frac {i \,x^{4}}{4}+\left (-d +2 e -4 f +8 g -16 h +32 i \right ) \ln \left (x +2\right )+\left (d -e +f -g +h -i \right ) \ln \left (x +1\right )\) | \(88\) |
default | \(\frac {i \,x^{4}}{4}+\frac {h \,x^{3}}{3}-i \,x^{3}+\frac {g \,x^{2}}{2}-\frac {3 h \,x^{2}}{2}+\frac {7 i \,x^{2}}{2}+f x -3 g x +7 h x -15 i x +\left (-d +2 e -4 f +8 g -16 h +32 i \right ) \ln \left (x +2\right )+\left (d -e +f -g +h -i \right ) \ln \left (x +1\right )\) | \(95\) |
parallelrisch | \(\frac {i \,x^{4}}{4}+\frac {h \,x^{3}}{3}-i \,x^{3}+\frac {g \,x^{2}}{2}-\frac {3 h \,x^{2}}{2}+\frac {7 i \,x^{2}}{2}+f x -3 g x +7 h x -15 i x +\ln \left (x +1\right ) d -\ln \left (x +1\right ) e +\ln \left (x +1\right ) f -\ln \left (x +1\right ) g +\ln \left (x +1\right ) h -\ln \left (x +1\right ) i -\ln \left (x +2\right ) d +2 \ln \left (x +2\right ) e -4 \ln \left (x +2\right ) f +8 \ln \left (x +2\right ) g -16 \ln \left (x +2\right ) h +32 \ln \left (x +2\right ) i\) | \(134\) |
risch | \(\frac {i \,x^{4}}{4}+\frac {h \,x^{3}}{3}-i \,x^{3}+\frac {g \,x^{2}}{2}-\frac {3 h \,x^{2}}{2}+\frac {7 i \,x^{2}}{2}+f x -3 g x +7 h x -15 i x +\ln \left (-x -1\right ) d -\ln \left (-x -1\right ) e +\ln \left (-x -1\right ) f -\ln \left (-x -1\right ) g +\ln \left (-x -1\right ) h -\ln \left (-x -1\right ) i -\ln \left (x +2\right ) d +2 \ln \left (x +2\right ) e -4 \ln \left (x +2\right ) f +8 \ln \left (x +2\right ) g -16 \ln \left (x +2\right ) h +32 \ln \left (x +2\right ) i\) | \(146\) |
(1/3*h-i)*x^3+(1/2*g-3/2*h+7/2*i)*x^2+(f-3*g+7*h-15*i)*x+1/4*i*x^4+(-d+2*e -4*f+8*g-16*h+32*i)*ln(x+2)+(d-e+f-g+h-i)*ln(x+1)
Time = 0.25 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.93 \[ \int \frac {\left (2-3 x+x^2\right ) \left (d+e x+f x^2+g x^3+h x^4+i x^5\right )}{4-5 x^2+x^4} \, dx=\frac {1}{4} \, i x^{4} + \frac {1}{3} \, {\left (h - 3 \, i\right )} x^{3} + \frac {1}{2} \, {\left (g - 3 \, h + 7 \, i\right )} x^{2} + {\left (f - 3 \, g + 7 \, h - 15 \, i\right )} x - {\left (d - 2 \, e + 4 \, f - 8 \, g + 16 \, h - 32 \, i\right )} \log \left (x + 2\right ) + {\left (d - e + f - g + h - i\right )} \log \left (x + 1\right ) \]
1/4*i*x^4 + 1/3*(h - 3*i)*x^3 + 1/2*(g - 3*h + 7*i)*x^2 + (f - 3*g + 7*h - 15*i)*x - (d - 2*e + 4*f - 8*g + 16*h - 32*i)*log(x + 2) + (d - e + f - g + h - i)*log(x + 1)
Time = 1.39 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.36 \[ \int \frac {\left (2-3 x+x^2\right ) \left (d+e x+f x^2+g x^3+h x^4+i x^5\right )}{4-5 x^2+x^4} \, dx=\frac {i x^{4}}{4} + x^{3} \left (\frac {h}{3} - i\right ) + x^{2} \left (\frac {g}{2} - \frac {3 h}{2} + \frac {7 i}{2}\right ) + x \left (f - 3 g + 7 h - 15 i\right ) + \left (- d + 2 e - 4 f + 8 g - 16 h + 32 i\right ) \log {\left (x + \frac {4 d - 6 e + 10 f - 18 g + 34 h - 66 i}{2 d - 3 e + 5 f - 9 g + 17 h - 33 i} \right )} + \left (d - e + f - g + h - i\right ) \log {\left (x + 1 \right )} \]
i*x**4/4 + x**3*(h/3 - i) + x**2*(g/2 - 3*h/2 + 7*i/2) + x*(f - 3*g + 7*h - 15*i) + (-d + 2*e - 4*f + 8*g - 16*h + 32*i)*log(x + (4*d - 6*e + 10*f - 18*g + 34*h - 66*i)/(2*d - 3*e + 5*f - 9*g + 17*h - 33*i)) + (d - e + f - g + h - i)*log(x + 1)
Time = 0.18 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.93 \[ \int \frac {\left (2-3 x+x^2\right ) \left (d+e x+f x^2+g x^3+h x^4+i x^5\right )}{4-5 x^2+x^4} \, dx=\frac {1}{4} \, i x^{4} + \frac {1}{3} \, {\left (h - 3 \, i\right )} x^{3} + \frac {1}{2} \, {\left (g - 3 \, h + 7 \, i\right )} x^{2} + {\left (f - 3 \, g + 7 \, h - 15 \, i\right )} x - {\left (d - 2 \, e + 4 \, f - 8 \, g + 16 \, h - 32 \, i\right )} \log \left (x + 2\right ) + {\left (d - e + f - g + h - i\right )} \log \left (x + 1\right ) \]
1/4*i*x^4 + 1/3*(h - 3*i)*x^3 + 1/2*(g - 3*h + 7*i)*x^2 + (f - 3*g + 7*h - 15*i)*x - (d - 2*e + 4*f - 8*g + 16*h - 32*i)*log(x + 2) + (d - e + f - g + h - i)*log(x + 1)
Time = 0.30 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.06 \[ \int \frac {\left (2-3 x+x^2\right ) \left (d+e x+f x^2+g x^3+h x^4+i x^5\right )}{4-5 x^2+x^4} \, dx=\frac {1}{4} \, i x^{4} + \frac {1}{3} \, h x^{3} - i x^{3} + \frac {1}{2} \, g x^{2} - \frac {3}{2} \, h x^{2} + \frac {7}{2} \, i x^{2} + f x - 3 \, g x + 7 \, h x - 15 \, i x - {\left (d - 2 \, e + 4 \, f - 8 \, g + 16 \, h - 32 \, i\right )} \log \left ({\left | x + 2 \right |}\right ) + {\left (d - e + f - g + h - i\right )} \log \left ({\left | x + 1 \right |}\right ) \]
1/4*i*x^4 + 1/3*h*x^3 - i*x^3 + 1/2*g*x^2 - 3/2*h*x^2 + 7/2*i*x^2 + f*x - 3*g*x + 7*h*x - 15*i*x - (d - 2*e + 4*f - 8*g + 16*h - 32*i)*log(abs(x + 2 )) + (d - e + f - g + h - i)*log(abs(x + 1))
Time = 0.05 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.96 \[ \int \frac {\left (2-3 x+x^2\right ) \left (d+e x+f x^2+g x^3+h x^4+i x^5\right )}{4-5 x^2+x^4} \, dx=x^3\,\left (\frac {h}{3}-i\right )-\ln \left (x+2\right )\,\left (d-2\,e+4\,f-8\,g+16\,h-32\,i\right )+\ln \left (x+1\right )\,\left (d-e+f-g+h-i\right )+\frac {i\,x^4}{4}+x^2\,\left (\frac {g}{2}-\frac {3\,h}{2}+\frac {7\,i}{2}\right )+x\,\left (f-3\,g+7\,h-15\,i\right ) \]