Integrand size = 58, antiderivative size = 22 \[ \int \frac {e^3 (1-34 x)-3 x^2+6 x^3-105 x^4+102 x^5-867 x^6}{1-2 x+35 x^2-34 x^3+289 x^4} \, dx=-x^3+\frac {e^3}{1-x+17 x^2} \] Output:
exp(3)/(17*x^2-x+1)-x^3
Time = 0.01 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.95 \[ \int \frac {e^3 (1-34 x)-3 x^2+6 x^3-105 x^4+102 x^5-867 x^6}{1-2 x+35 x^2-34 x^3+289 x^4} \, dx=-x^3-\frac {e^3}{-1+x-17 x^2} \] Input:
Integrate[(E^3*(1 - 34*x) - 3*x^2 + 6*x^3 - 105*x^4 + 102*x^5 - 867*x^6)/( 1 - 2*x + 35*x^2 - 34*x^3 + 289*x^4),x]
Output:
-x^3 - E^3/(-1 + x - 17*x^2)
Time = 0.40 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.73, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.121, Rules used = {2459, 1380, 27, 2345, 27, 2006, 17}
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.
\(\displaystyle \int \frac {-867 x^6+102 x^5-105 x^4+6 x^3-3 x^2+e^3 (1-34 x)}{289 x^4-34 x^3+35 x^2-2 x+1} \, dx\) |
\(\Big \downarrow \) 2459 |
\(\displaystyle \int \frac {-867 \left (x-\frac {1}{34}\right )^6-51 \left (x-\frac {1}{34}\right )^5-\frac {405}{4} \left (x-\frac {1}{34}\right )^4-\frac {201}{34} \left (x-\frac {1}{34}\right )^3-\frac {13869 \left (x-\frac {1}{34}\right )^2}{4624}-\frac {\left (13467+2672672 e^3\right ) \left (x-\frac {1}{34}\right )}{78608}-\frac {13467}{5345344}}{289 \left (x-\frac {1}{34}\right )^4+\frac {67}{2} \left (x-\frac {1}{34}\right )^2+\frac {4489}{4624}}d\left (x-\frac {1}{34}\right )\) |
\(\Big \downarrow \) 1380 |
\(\displaystyle 289 \int -\frac {4634413248 \left (x-\frac {1}{34}\right )^6+272612544 \left (x-\frac {1}{34}\right )^5+541216080 \left (x-\frac {1}{34}\right )^4+31600416 \left (x-\frac {1}{34}\right )^3+16032564 \left (x-\frac {1}{34}\right )^2+68 \left (13467+2672672 e^3\right ) \left (x-\frac {1}{34}\right )+13467}{334084 \left (1156 \left (x-\frac {1}{34}\right )^2+67\right )^2}d\left (x-\frac {1}{34}\right )\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {\int \frac {4634413248 \left (x-\frac {1}{34}\right )^6+272612544 \left (x-\frac {1}{34}\right )^5+541216080 \left (x-\frac {1}{34}\right )^4+31600416 \left (x-\frac {1}{34}\right )^3+16032564 \left (x-\frac {1}{34}\right )^2+68 \left (13467+2672672 e^3\right ) \left (x-\frac {1}{34}\right )+13467}{\left (1156 \left (x-\frac {1}{34}\right )^2+67\right )^2}d\left (x-\frac {1}{34}\right )}{1156}\) |
\(\Big \downarrow \) 2345 |
\(\displaystyle \frac {\frac {1}{134} \int -\frac {402 \left (1336336 \left (x-\frac {1}{34}\right )^4+78608 \left (x-\frac {1}{34}\right )^3+78608 \left (x-\frac {1}{34}\right )^2+4556 \left (x-\frac {1}{34}\right )+67\right )}{1156 \left (x-\frac {1}{34}\right )^2+67}d\left (x-\frac {1}{34}\right )+\frac {78608 e^3}{1156 \left (x-\frac {1}{34}\right )^2+67}}{1156}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\frac {78608 e^3}{1156 \left (x-\frac {1}{34}\right )^2+67}-3 \int \frac {1336336 \left (x-\frac {1}{34}\right )^4+78608 \left (x-\frac {1}{34}\right )^3+78608 \left (x-\frac {1}{34}\right )^2+4556 \left (x-\frac {1}{34}\right )+67}{1156 \left (x-\frac {1}{34}\right )^2+67}d\left (x-\frac {1}{34}\right )}{1156}\) |
\(\Big \downarrow \) 2006 |
\(\displaystyle \frac {\frac {78608 e^3}{1156 \left (x-\frac {1}{34}\right )^2+67}-3 \int \left (34 \left (x-\frac {1}{34}\right )+1\right )^2d\left (x-\frac {1}{34}\right )}{1156}\) |
\(\Big \downarrow \) 17 |
\(\displaystyle \frac {\frac {78608 e^3}{1156 \left (x-\frac {1}{34}\right )^2+67}-\frac {1}{34} \left (34 \left (x-\frac {1}{34}\right )+1\right )^3}{1156}\) |
Input:
Int[(E^3*(1 - 34*x) - 3*x^2 + 6*x^3 - 105*x^4 + 102*x^5 - 867*x^6)/(1 - 2* x + 35*x^2 - 34*x^3 + 289*x^4),x]
Output:
(-1/34*(1 + 34*(-1/34 + x))^3 + (78608*E^3)/(67 + 1156*(-1/34 + x)^2))/115 6
Int[(c_.)*((a_.) + (b_.)*(x_))^(m_.), x_Symbol] :> Simp[c*((a + b*x)^(m + 1 )/(b*(m + 1))), x] /; FreeQ[{a, b, c, m}, x] && NeQ[m, -1]
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[(u_)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> S imp[1/c^p Int[u*(b/2 + c*x^n)^(2*p), x], x] /; FreeQ[{a, b, c, n, p}, x] && EqQ[n2, 2*n] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]
Int[(u_.)*(Px_), x_Symbol] :> With[{a = Rt[Coeff[Px, x, 0], Expon[Px, x]], b = Rt[Coeff[Px, x, Expon[Px, x]], Expon[Px, x]]}, Int[u*(a + b*x)^Expon[Px , x], x] /; EqQ[Px, (a + b*x)^Expon[Px, x]]] /; PolyQ[Px, x] && GtQ[Expon[P x, x], 1] && NeQ[Coeff[Px, x, 0], 0] && !MatchQ[Px, (a_.)*(v_)^Expon[Px, x ] /; FreeQ[a, x] && LinearQ[v, x]]
Int[(Pq_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = PolynomialQuot ient[Pq, a + b*x^2, x], f = Coeff[PolynomialRemainder[Pq, a + b*x^2, x], x, 0], g = Coeff[PolynomialRemainder[Pq, a + b*x^2, x], x, 1]}, Simp[(a*g - b *f*x)*((a + b*x^2)^(p + 1)/(2*a*b*(p + 1))), x] + Simp[1/(2*a*(p + 1)) In t[(a + b*x^2)^(p + 1)*ExpandToSum[2*a*(p + 1)*Q + f*(2*p + 3), x], x], x]] /; FreeQ[{a, b}, x] && PolyQ[Pq, x] && LtQ[p, -1]
Int[(Pn_)^(p_.)*(Qx_), x_Symbol] :> With[{S = Coeff[Pn, x, Expon[Pn, x] - 1 ]/(Expon[Pn, x]*Coeff[Pn, x, Expon[Pn, x]])}, Subst[Int[ExpandToSum[Pn /. x -> x - S, x]^p*ExpandToSum[Qx /. x -> x - S, x], x], x, x + S] /; Binomial Q[Pn /. x -> x - S, x] || (IntegerQ[Expon[Pn, x]/2] && TrinomialQ[Pn /. x - > x - S, x])] /; FreeQ[p, x] && PolyQ[Pn, x] && GtQ[Expon[Pn, x], 2] && NeQ [Coeff[Pn, x, Expon[Pn, x] - 1], 0] && PolyQ[Qx, x] && !(MonomialQ[Qx, x] && IGtQ[p, 0])
Time = 0.08 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.95
method | result | size |
risch | \(\frac {{\mathrm e}^{3}}{17 x^{2}-x +1}-x^{3}\) | \(21\) |
default | \(\frac {{\mathrm e}^{3}}{17 x^{2}-x +1}-x^{3}\) | \(22\) |
gosper | \(\frac {-17 x^{5}+x^{4}-x^{3}+{\mathrm e}^{3}}{17 x^{2}-x +1}\) | \(30\) |
norman | \(\frac {-17 x^{5}+x^{4}-x^{3}+{\mathrm e}^{3}}{17 x^{2}-x +1}\) | \(30\) |
parallelrisch | \(\frac {-17 x^{5}+x^{4}-x^{3}+{\mathrm e}^{3}}{17 x^{2}-x +1}\) | \(30\) |
Input:
int(((-34*x+1)*exp(3)-867*x^6+102*x^5-105*x^4+6*x^3-3*x^2)/(289*x^4-34*x^3 +35*x^2-2*x+1),x,method=_RETURNVERBOSE)
Output:
-x^3+1/17*exp(3)/(x^2-1/17*x+1/17)
Time = 0.08 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.45 \[ \int \frac {e^3 (1-34 x)-3 x^2+6 x^3-105 x^4+102 x^5-867 x^6}{1-2 x+35 x^2-34 x^3+289 x^4} \, dx=-\frac {17 \, x^{5} - x^{4} + x^{3} - e^{3}}{17 \, x^{2} - x + 1} \] Input:
integrate(((-34*x+1)*exp(3)-867*x^6+102*x^5-105*x^4+6*x^3-3*x^2)/(289*x^4- 34*x^3+35*x^2-2*x+1),x, algorithm="fricas")
Output:
-(17*x^5 - x^4 + x^3 - e^3)/(17*x^2 - x + 1)
Time = 0.09 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.64 \[ \int \frac {e^3 (1-34 x)-3 x^2+6 x^3-105 x^4+102 x^5-867 x^6}{1-2 x+35 x^2-34 x^3+289 x^4} \, dx=- x^{3} + \frac {e^{3}}{17 x^{2} - x + 1} \] Input:
integrate(((-34*x+1)*exp(3)-867*x**6+102*x**5-105*x**4+6*x**3-3*x**2)/(289 *x**4-34*x**3+35*x**2-2*x+1),x)
Output:
-x**3 + exp(3)/(17*x**2 - x + 1)
Time = 0.03 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.95 \[ \int \frac {e^3 (1-34 x)-3 x^2+6 x^3-105 x^4+102 x^5-867 x^6}{1-2 x+35 x^2-34 x^3+289 x^4} \, dx=-x^{3} + \frac {e^{3}}{17 \, x^{2} - x + 1} \] Input:
integrate(((-34*x+1)*exp(3)-867*x^6+102*x^5-105*x^4+6*x^3-3*x^2)/(289*x^4- 34*x^3+35*x^2-2*x+1),x, algorithm="maxima")
Output:
-x^3 + e^3/(17*x^2 - x + 1)
Time = 0.11 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.95 \[ \int \frac {e^3 (1-34 x)-3 x^2+6 x^3-105 x^4+102 x^5-867 x^6}{1-2 x+35 x^2-34 x^3+289 x^4} \, dx=-x^{3} + \frac {e^{3}}{17 \, x^{2} - x + 1} \] Input:
integrate(((-34*x+1)*exp(3)-867*x^6+102*x^5-105*x^4+6*x^3-3*x^2)/(289*x^4- 34*x^3+35*x^2-2*x+1),x, algorithm="giac")
Output:
-x^3 + e^3/(17*x^2 - x + 1)
Time = 0.04 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.95 \[ \int \frac {e^3 (1-34 x)-3 x^2+6 x^3-105 x^4+102 x^5-867 x^6}{1-2 x+35 x^2-34 x^3+289 x^4} \, dx=\frac {{\mathrm {e}}^3}{17\,x^2-x+1}-x^3 \] Input:
int(-(3*x^2 - 6*x^3 + 105*x^4 - 102*x^5 + 867*x^6 + exp(3)*(34*x - 1))/(35 *x^2 - 2*x - 34*x^3 + 289*x^4 + 1),x)
Output:
exp(3)/(17*x^2 - x + 1) - x^3
Time = 0.15 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.36 \[ \int \frac {e^3 (1-34 x)-3 x^2+6 x^3-105 x^4+102 x^5-867 x^6}{1-2 x+35 x^2-34 x^3+289 x^4} \, dx=\frac {-17 x^{5}+x^{4}+e^{3}-x^{3}}{17 x^{2}-x +1} \] Input:
int(((-34*x+1)*exp(3)-867*x^6+102*x^5-105*x^4+6*x^3-3*x^2)/(289*x^4-34*x^3 +35*x^2-2*x+1),x)
Output:
(e**3 - 17*x**5 + x**4 - x**3)/(17*x**2 - x + 1)