Integrand size = 107, antiderivative size = 25 \[ \int \frac {150+e (-6-4 x)+100 x+36 x^2+16 x^3}{5625 x^2+3750 x^3+1525 x^4+600 x^5+136 x^6+24 x^7+4 x^8+e^2 \left (9 x^2+6 x^3+x^4\right )+e \left (-450 x^2-300 x^3-86 x^4-24 x^5-4 x^6\right )} \, dx=-\frac {2}{x^2 (3+x) \left (\frac {25-e}{x}+2 x\right )} \] Output:
-2/x^2/(2*x+(25-exp(1))/x)/(3+x)
Time = 0.04 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.88 \[ \int \frac {150+e (-6-4 x)+100 x+36 x^2+16 x^3}{5625 x^2+3750 x^3+1525 x^4+600 x^5+136 x^6+24 x^7+4 x^8+e^2 \left (9 x^2+6 x^3+x^4\right )+e \left (-450 x^2-300 x^3-86 x^4-24 x^5-4 x^6\right )} \, dx=-\frac {2}{x (3+x) \left (25-e+2 x^2\right )} \] Input:
Integrate[(150 + E*(-6 - 4*x) + 100*x + 36*x^2 + 16*x^3)/(5625*x^2 + 3750* x^3 + 1525*x^4 + 600*x^5 + 136*x^6 + 24*x^7 + 4*x^8 + E^2*(9*x^2 + 6*x^3 + x^4) + E*(-450*x^2 - 300*x^3 - 86*x^4 - 24*x^5 - 4*x^6)),x]
Output:
-2/(x*(3 + x)*(25 - E + 2*x^2))
Result contains higher order function than in optimal. Order 3 vs. order 1 in optimal.
Time = 0.63 (sec) , antiderivative size = 143, normalized size of antiderivative = 5.72, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.028, Rules used = {2026, 2462, 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.
\(\displaystyle \int \frac {16 x^3+36 x^2+100 x+e (-4 x-6)+150}{4 x^8+24 x^7+136 x^6+600 x^5+1525 x^4+3750 x^3+5625 x^2+e^2 \left (x^4+6 x^3+9 x^2\right )+e \left (-4 x^6-24 x^5-86 x^4-300 x^3-450 x^2\right )} \, dx\) |
\(\Big \downarrow \) 2026 |
\(\displaystyle \int \frac {16 x^3+36 x^2+100 x+e (-4 x-6)+150}{x^2 \left (4 x^6+24 x^5+4 (34-e) x^4+24 (25-e) x^3+(25-e) (61-e) x^2+6 (25-e)^2 x+9 (25-e)^2\right )}dx\) |
\(\Big \downarrow \) 2462 |
\(\displaystyle \int \left (\frac {16 (x-3)}{(e-43) \left (-2 x^2+e-25\right )^2}+\frac {24}{(e-43) (e-25) \left (-2 x^2+e-25\right )}-\frac {2}{3 (e-25) x^2}+\frac {2}{3 (e-43) (x+3)^2}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {12 \sqrt {\frac {2}{25-e}} \arctan \left (\sqrt {\frac {2}{25-e}} x\right )}{1075-68 e+e^2}+\frac {12 \sqrt {2} \arctan \left (\sqrt {\frac {2}{25-e}} x\right )}{(25-e)^{3/2} (43-e)}+\frac {4 (6 x-e+25)}{\left (1075-68 e+e^2\right ) \left (2 x^2-e+25\right )}-\frac {2}{3 (25-e) x}+\frac {2}{3 (43-e) (x+3)}\) |
Input:
Int[(150 + E*(-6 - 4*x) + 100*x + 36*x^2 + 16*x^3)/(5625*x^2 + 3750*x^3 + 1525*x^4 + 600*x^5 + 136*x^6 + 24*x^7 + 4*x^8 + E^2*(9*x^2 + 6*x^3 + x^4) + E*(-450*x^2 - 300*x^3 - 86*x^4 - 24*x^5 - 4*x^6)),x]
Output:
-2/(3*(25 - E)*x) + 2/(3*(43 - E)*(3 + x)) + (4*(25 - E + 6*x))/((1075 - 6 8*E + E^2)*(25 - E + 2*x^2)) + (12*Sqrt[2]*ArcTan[Sqrt[2/(25 - E)]*x])/((2 5 - E)^(3/2)*(43 - E)) - (12*Sqrt[2/(25 - E)]*ArcTan[Sqrt[2/(25 - E)]*x])/ (1075 - 68*E + E^2)
Int[(Fx_.)*(Px_)^(p_.), x_Symbol] :> With[{r = Expon[Px, x, Min]}, Int[x^(p *r)*ExpandToSum[Px/x^r, x]^p*Fx, x] /; IGtQ[r, 0]] /; PolyQ[Px, x] && Integ erQ[p] && !MonomialQ[Px, x] && (ILtQ[p, 0] || !PolyQ[u, x])
Int[(u_.)*(Px_)^(p_), x_Symbol] :> With[{Qx = Factor[Px]}, Int[ExpandIntegr and[u*Qx^p, x], x] /; !SumQ[NonfreeFactors[Qx, x]]] /; PolyQ[Px, x] && GtQ [Expon[Px, x], 2] && !BinomialQ[Px, x] && !TrinomialQ[Px, x] && ILtQ[p, 0 ] && RationalFunctionQ[u, x]
Time = 0.71 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.88
method | result | size |
norman | \(\frac {2}{x \left (3+x \right ) \left (-2 x^{2}+{\mathrm e}-25\right )}\) | \(22\) |
gosper | \(\frac {2}{x \left (-2 x^{3}+x \,{\mathrm e}-6 x^{2}+3 \,{\mathrm e}-25 x -75\right )}\) | \(31\) |
risch | \(\frac {2}{x \left (-2 x^{3}+x \,{\mathrm e}-6 x^{2}+3 \,{\mathrm e}-25 x -75\right )}\) | \(31\) |
parallelrisch | \(\frac {2}{x \left (-2 x^{3}+x \,{\mathrm e}-6 x^{2}+3 \,{\mathrm e}-25 x -75\right )}\) | \(31\) |
default | \(-\frac {2 \left ({\mathrm e}^{3}-129 \,{\mathrm e}^{2}+5547 \,{\mathrm e}-79507\right )}{3 \left (1849-86 \,{\mathrm e}+{\mathrm e}^{2}\right )^{2} \left (3+x \right )}+\frac {\frac {2 \left (\frac {6 \left (154 \,{\mathrm e} \,{\mathrm e}^{3}-8772 \,{\mathrm e} \,{\mathrm e}^{2}-{\mathrm e}^{4} {\mathrm e}+874964 \,{\mathrm e}^{2}-16472 \,{\mathrm e}^{3}+204 \,{\mathrm e}^{4}-14884450 \,{\mathrm e}-{\mathrm e}^{5}+99383750\right ) x}{{\mathrm e}-25}+25244 \,{\mathrm e}^{3}-874964 \,{\mathrm e}^{2}-358 \,{\mathrm e}^{4}+14884450 \,{\mathrm e}+2 \,{\mathrm e}^{5}-99383750\right )}{-2 x^{2}+{\mathrm e}-25}+\frac {12 \left (154 \,{\mathrm e} \,{\mathrm e}^{3}-8772 \,{\mathrm e} \,{\mathrm e}^{2}-{\mathrm e}^{4} {\mathrm e}+8772 \,{\mathrm e}^{3}-154 \,{\mathrm e}^{4}+{\mathrm e}^{5}\right ) \arctan \left (\frac {2 x}{\sqrt {-2 \,{\mathrm e}+50}}\right )}{\left ({\mathrm e}-25\right ) \sqrt {-2 \,{\mathrm e}+50}}}{\left (1849-86 \,{\mathrm e}+{\mathrm e}^{2}\right )^{2} \left ({\mathrm e}^{2}-50 \,{\mathrm e}+625\right )}-\frac {2 \left (25-{\mathrm e}\right )}{\left (3 \,{\mathrm e}^{2}-150 \,{\mathrm e}+1875\right ) x}\) | \(234\) |
Input:
int(((-4*x-6)*exp(1)+16*x^3+36*x^2+100*x+150)/((x^4+6*x^3+9*x^2)*exp(1)^2+ (-4*x^6-24*x^5-86*x^4-300*x^3-450*x^2)*exp(1)+4*x^8+24*x^7+136*x^6+600*x^5 +1525*x^4+3750*x^3+5625*x^2),x,method=_RETURNVERBOSE)
Output:
2/x/(3+x)/(-2*x^2+exp(1)-25)
Time = 0.08 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.36 \[ \int \frac {150+e (-6-4 x)+100 x+36 x^2+16 x^3}{5625 x^2+3750 x^3+1525 x^4+600 x^5+136 x^6+24 x^7+4 x^8+e^2 \left (9 x^2+6 x^3+x^4\right )+e \left (-450 x^2-300 x^3-86 x^4-24 x^5-4 x^6\right )} \, dx=-\frac {2}{2 \, x^{4} + 6 \, x^{3} + 25 \, x^{2} - {\left (x^{2} + 3 \, x\right )} e + 75 \, x} \] Input:
integrate(((-4*x-6)*exp(1)+16*x^3+36*x^2+100*x+150)/((x^4+6*x^3+9*x^2)*exp (1)^2+(-4*x^6-24*x^5-86*x^4-300*x^3-450*x^2)*exp(1)+4*x^8+24*x^7+136*x^6+6 00*x^5+1525*x^4+3750*x^3+5625*x^2),x, algorithm="fricas")
Output:
-2/(2*x^4 + 6*x^3 + 25*x^2 - (x^2 + 3*x)*e + 75*x)
Time = 1.26 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.16 \[ \int \frac {150+e (-6-4 x)+100 x+36 x^2+16 x^3}{5625 x^2+3750 x^3+1525 x^4+600 x^5+136 x^6+24 x^7+4 x^8+e^2 \left (9 x^2+6 x^3+x^4\right )+e \left (-450 x^2-300 x^3-86 x^4-24 x^5-4 x^6\right )} \, dx=- \frac {2}{2 x^{4} + 6 x^{3} + x^{2} \cdot \left (25 - e\right ) + x \left (75 - 3 e\right )} \] Input:
integrate(((-4*x-6)*exp(1)+16*x**3+36*x**2+100*x+150)/((x**4+6*x**3+9*x**2 )*exp(1)**2+(-4*x**6-24*x**5-86*x**4-300*x**3-450*x**2)*exp(1)+4*x**8+24*x **7+136*x**6+600*x**5+1525*x**4+3750*x**3+5625*x**2),x)
Output:
-2/(2*x**4 + 6*x**3 + x**2*(25 - E) + x*(75 - 3*E))
Time = 0.04 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.24 \[ \int \frac {150+e (-6-4 x)+100 x+36 x^2+16 x^3}{5625 x^2+3750 x^3+1525 x^4+600 x^5+136 x^6+24 x^7+4 x^8+e^2 \left (9 x^2+6 x^3+x^4\right )+e \left (-450 x^2-300 x^3-86 x^4-24 x^5-4 x^6\right )} \, dx=-\frac {2}{2 \, x^{4} + 6 \, x^{3} - x^{2} {\left (e - 25\right )} - 3 \, x {\left (e - 25\right )}} \] Input:
integrate(((-4*x-6)*exp(1)+16*x^3+36*x^2+100*x+150)/((x^4+6*x^3+9*x^2)*exp (1)^2+(-4*x^6-24*x^5-86*x^4-300*x^3-450*x^2)*exp(1)+4*x^8+24*x^7+136*x^6+6 00*x^5+1525*x^4+3750*x^3+5625*x^2),x, algorithm="maxima")
Output:
-2/(2*x^4 + 6*x^3 - x^2*(e - 25) - 3*x*(e - 25))
Time = 0.12 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.36 \[ \int \frac {150+e (-6-4 x)+100 x+36 x^2+16 x^3}{5625 x^2+3750 x^3+1525 x^4+600 x^5+136 x^6+24 x^7+4 x^8+e^2 \left (9 x^2+6 x^3+x^4\right )+e \left (-450 x^2-300 x^3-86 x^4-24 x^5-4 x^6\right )} \, dx=-\frac {2}{2 \, x^{4} + 6 \, x^{3} + 25 \, x^{2} - {\left (x^{2} + 3 \, x\right )} e + 75 \, x} \] Input:
integrate(((-4*x-6)*exp(1)+16*x^3+36*x^2+100*x+150)/((x^4+6*x^3+9*x^2)*exp (1)^2+(-4*x^6-24*x^5-86*x^4-300*x^3-450*x^2)*exp(1)+4*x^8+24*x^7+136*x^6+6 00*x^5+1525*x^4+3750*x^3+5625*x^2),x, algorithm="giac")
Output:
-2/(2*x^4 + 6*x^3 + 25*x^2 - (x^2 + 3*x)*e + 75*x)
Time = 2.93 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.32 \[ \int \frac {150+e (-6-4 x)+100 x+36 x^2+16 x^3}{5625 x^2+3750 x^3+1525 x^4+600 x^5+136 x^6+24 x^7+4 x^8+e^2 \left (9 x^2+6 x^3+x^4\right )+e \left (-450 x^2-300 x^3-86 x^4-24 x^5-4 x^6\right )} \, dx=-\frac {2}{2\,x^4+6\,x^3+\left (25-\mathrm {e}\right )\,x^2+\left (75-3\,\mathrm {e}\right )\,x} \] Input:
int((100*x + 36*x^2 + 16*x^3 - exp(1)*(4*x + 6) + 150)/(exp(2)*(9*x^2 + 6* x^3 + x^4) - exp(1)*(450*x^2 + 300*x^3 + 86*x^4 + 24*x^5 + 4*x^6) + 5625*x ^2 + 3750*x^3 + 1525*x^4 + 600*x^5 + 136*x^6 + 24*x^7 + 4*x^8),x)
Output:
-2/(6*x^3 + 2*x^4 - x*(3*exp(1) - 75) - x^2*(exp(1) - 25))
Time = 0.17 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.12 \[ \int \frac {150+e (-6-4 x)+100 x+36 x^2+16 x^3}{5625 x^2+3750 x^3+1525 x^4+600 x^5+136 x^6+24 x^7+4 x^8+e^2 \left (9 x^2+6 x^3+x^4\right )+e \left (-450 x^2-300 x^3-86 x^4-24 x^5-4 x^6\right )} \, dx=\frac {2}{x \left (-2 x^{3}+e x -6 x^{2}+3 e -25 x -75\right )} \] Input:
int(((-4*x-6)*exp(1)+16*x^3+36*x^2+100*x+150)/((x^4+6*x^3+9*x^2)*exp(1)^2+ (-4*x^6-24*x^5-86*x^4-300*x^3-450*x^2)*exp(1)+4*x^8+24*x^7+136*x^6+600*x^5 +1525*x^4+3750*x^3+5625*x^2),x)
Output:
2/(x*(e*x + 3*e - 2*x**3 - 6*x**2 - 25*x - 75))