\(\int \frac {(2-3 x+x^2) (d+e x+f x^2+g x^3+h x^4+i x^5)}{4-5 x^2+x^4} \, dx\) [78]

Optimal result

Integrand size = 46, antiderivative size = 90 \[ \int \frac {\left (2-3 x+x^2\right ) \left (d+e x+f x^2+g x^3+h x^4+i x^5\right )}{4-5 x^2+x^4} \, dx=(f-3 g+7 h-15 i) x+\frac {1}{2} (g-3 h+7 i) x^2+\frac {1}{3} (h-3 i) x^3+\frac {i x^4}{4}+(d-e+f-g+h-i) \log (1+x)-(d-2 e+4 f-8 g+16 h-32 i) \log (2+x) \]

[Out]

(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)

Rubi [A] (verified)

Time = 0.07 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {1600, 1671, 646, 31} \[ \int \frac {\left (2-3 x+x^2\right ) \left (d+e x+f x^2+g x^3+h x^4+i x^5\right )}{4-5 x^2+x^4} \, dx=\log (x+1) (d-e+f-g+h-i)-\log (x+2) (d-2 e+4 f-8 g+16 h-32 i)+x (f-3 g+7 h-15 i)+\frac {1}{2} x^2 (g-3 h+7 i)+\frac {1}{3} x^3 (h-3 i)+\frac {i x^4}{4} \]

[In]

Int[((2 - 3*x + x^2)*(d + e*x + f*x^2 + g*x^3 + h*x^4 + i*x^5))/(4 - 5*x^2 + x^4),x]

[Out]

(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)*L
og[1 + x] - (d - 2*e + 4*f - 8*g + 16*h - 32*i)*Log[2 + x]

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 646

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Dist[
(c*d - e*(b/2 - q/2))/q, Int[1/(b/2 - q/2 + c*x), x], x] - Dist[(c*d - e*(b/2 + q/2))/q, Int[1/(b/2 + q/2 + c*
x), x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[2*c*d - b*e, 0] && NeQ[b^2 - 4*a*c, 0] && NiceSqrtQ[b^2 - 4*a*
c]

Rule 1600

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]

Rule 1671

Int[(Pq_)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[Pq*(a + b*x + c*x^2)^p, x
], x] /; FreeQ[{a, b, c}, x] && PolyQ[Pq, x] && IGtQ[p, -2]

Rubi steps \begin{align*} \text {integral}& = \int \frac {d+e x+f x^2+g x^3+h x^4+i x^5}{2+3 x+x^2} \, dx \\ & = \int \left (f-3 g+7 h-15 i+(g-3 h+7 i) x+(h-3 i) x^2+i x^3+\frac {d-2 f+6 g-14 h+30 i+(e-3 f+7 g-15 h+31 i) x}{2+3 x+x^2}\right ) \, 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}+\int \frac {d-2 f+6 g-14 h+30 i+(e-3 f+7 g-15 h+31 i) x}{2+3 x+x^2} \, 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-2 e+4 f-8 g+16 h-32 i) \int \frac {1}{2+x} \, dx+(d-e+f-g+h-i) \int \frac {1}{1+x} \, 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) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.01 \[ \int \frac {\left (2-3 x+x^2\right ) \left (d+e x+f x^2+g x^3+h x^4+i x^5\right )}{4-5 x^2+x^4} \, dx=(f-3 g+7 h-15 i) x+\frac {1}{2} (g-3 h+7 i) x^2+\frac {1}{3} (h-3 i) x^3+\frac {i x^4}{4}+(d-e+f-g+h-i) \log (1+x)+(-d+2 e-4 f+8 g-16 h+32 i) \log (2+x) \]

[In]

Integrate[((2 - 3*x + x^2)*(d + e*x + f*x^2 + g*x^3 + h*x^4 + i*x^5))/(4 - 5*x^2 + x^4),x]

[Out]

(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)*L
og[1 + x] + (-d + 2*e - 4*f + 8*g - 16*h + 32*i)*Log[2 + x]

Maple [A] (verified)

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	\(\frac {i \,x^{4}}{4}+\frac {h \,x^{3}}{3}-i \,x^{3}+\frac {g \,x^{2}}{2}-\frac {3 h \,x^{2}}{2}+\frac {7 i \,x^{2}}{2}+f x -3 g x +7 h x -15 i x +\ln \left (-x -1\right ) d -\ln \left (-x -1\right ) e +\ln \left (-x -1\right ) f -\ln \left (-x -1\right ) g +\ln \left (-x -1\right ) h -\ln \left (-x -1\right ) i -\ln \left (x +2\right ) d +2 \ln \left (x +2\right ) e -4 \ln \left (x +2\right ) f +8 \ln \left (x +2\right ) g -16 \ln \left (x +2\right ) h +32 \ln \left (x +2\right ) i\)	\(146\)

[In]

int((x^2-3*x+2)*(i*x^5+h*x^4+g*x^3+f*x^2+e*x+d)/(x^4-5*x^2+4),x,method=_RETURNVERBOSE)

[Out]

(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)

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.93 \[ \int \frac {\left (2-3 x+x^2\right ) \left (d+e x+f x^2+g x^3+h x^4+i x^5\right )}{4-5 x^2+x^4} \, dx=\frac {1}{4} \, i x^{4} + \frac {1}{3} \, {\left (h - 3 \, i\right )} x^{3} + \frac {1}{2} \, {\left (g - 3 \, h + 7 \, i\right )} x^{2} + {\left (f - 3 \, g + 7 \, h - 15 \, i\right )} x - {\left (d - 2 \, e + 4 \, f - 8 \, g + 16 \, h - 32 \, i\right )} \log \left (x + 2\right ) + {\left (d - e + f - g + h - i\right )} \log \left (x + 1\right ) \]

[In]

integrate((x^2-3*x+2)*(i*x^5+h*x^4+g*x^3+f*x^2+e*x+d)/(x^4-5*x^2+4),x, algorithm="fricas")

[Out]

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)

Sympy [A] (verification not implemented)

Time = 1.39 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.36 \[ \int \frac {\left (2-3 x+x^2\right ) \left (d+e x+f x^2+g x^3+h x^4+i x^5\right )}{4-5 x^2+x^4} \, dx=\frac {i x^{4}}{4} + x^{3} \left (\frac {h}{3} - i\right ) + x^{2} \left (\frac {g}{2} - \frac {3 h}{2} + \frac {7 i}{2}\right ) + x \left (f - 3 g + 7 h - 15 i\right ) + \left (- d + 2 e - 4 f + 8 g - 16 h + 32 i\right ) \log {\left (x + \frac {4 d - 6 e + 10 f - 18 g + 34 h - 66 i}{2 d - 3 e + 5 f - 9 g + 17 h - 33 i} \right )} + \left (d - e + f - g + h - i\right ) \log {\left (x + 1 \right )} \]

[In]

integrate((x**2-3*x+2)*(i*x**5+h*x**4+g*x**3+f*x**2+e*x+d)/(x**4-5*x**2+4),x)

[Out]

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)

Maxima [A] (verification not implemented)

none

Time = 0.18 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.93 \[ \int \frac {\left (2-3 x+x^2\right ) \left (d+e x+f x^2+g x^3+h x^4+i x^5\right )}{4-5 x^2+x^4} \, dx=\frac {1}{4} \, i x^{4} + \frac {1}{3} \, {\left (h - 3 \, i\right )} x^{3} + \frac {1}{2} \, {\left (g - 3 \, h + 7 \, i\right )} x^{2} + {\left (f - 3 \, g + 7 \, h - 15 \, i\right )} x - {\left (d - 2 \, e + 4 \, f - 8 \, g + 16 \, h - 32 \, i\right )} \log \left (x + 2\right ) + {\left (d - e + f - g + h - i\right )} \log \left (x + 1\right ) \]

[In]

integrate((x^2-3*x+2)*(i*x^5+h*x^4+g*x^3+f*x^2+e*x+d)/(x^4-5*x^2+4),x, algorithm="maxima")

[Out]

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)

Giac [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.06 \[ \int \frac {\left (2-3 x+x^2\right ) \left (d+e x+f x^2+g x^3+h x^4+i x^5\right )}{4-5 x^2+x^4} \, dx=\frac {1}{4} \, i x^{4} + \frac {1}{3} \, h x^{3} - i x^{3} + \frac {1}{2} \, g x^{2} - \frac {3}{2} \, h x^{2} + \frac {7}{2} \, i x^{2} + f x - 3 \, g x + 7 \, h x - 15 \, i x - {\left (d - 2 \, e + 4 \, f - 8 \, g + 16 \, h - 32 \, i\right )} \log \left ({\left | x + 2 \right |}\right ) + {\left (d - e + f - g + h - i\right )} \log \left ({\left | x + 1 \right |}\right ) \]

[In]

integrate((x^2-3*x+2)*(i*x^5+h*x^4+g*x^3+f*x^2+e*x+d)/(x^4-5*x^2+4),x, algorithm="giac")

[Out]

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))

Mupad [B] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.96 \[ \int \frac {\left (2-3 x+x^2\right ) \left (d+e x+f x^2+g x^3+h x^4+i x^5\right )}{4-5 x^2+x^4} \, dx=x^3\,\left (\frac {h}{3}-i\right )-\ln \left (x+2\right )\,\left (d-2\,e+4\,f-8\,g+16\,h-32\,i\right )+\ln \left (x+1\right )\,\left (d-e+f-g+h-i\right )+\frac {i\,x^4}{4}+x^2\,\left (\frac {g}{2}-\frac {3\,h}{2}+\frac {7\,i}{2}\right )+x\,\left (f-3\,g+7\,h-15\,i\right ) \]

[In]

int(((x^2 - 3*x + 2)*(d + e*x + f*x^2 + g*x^3 + h*x^4 + i*x^5))/(x^4 - 5*x^2 + 4),x)

[Out]

x^3*(h/3 - i) - log(x + 2)*(d - 2*e + 4*f - 8*g + 16*h - 32*i) + log(x + 1)*(d - e + f - g + h - i) + (i*x^4)/
4 + x^2*(g/2 - (3*h)/2 + (7*i)/2) + x*(f - 3*g + 7*h - 15*i)