\(\int \frac {-2800-1560 x-4260 x^2-1200 x^3+600 x^4-60 x^6-10 x^7+e^3 (720+720 x+1200 x^2+600 x^3-30 x^5-5 x^6)}{-64+96 x-40 x^3+6 x^5+x^6} \, dx\) [7857]

Optimal result

Integrand size = 80, antiderivative size = 29 \[ \int \frac {-2800-1560 x-4260 x^2-1200 x^3+600 x^4-60 x^6-10 x^7+e^3 \left (720+720 x+1200 x^2+600 x^3-30 x^5-5 x^6\right )}{-64+96 x-40 x^3+6 x^5+x^6} \, dx=4+5 \left (4-e^3-x\right ) \left (x+\frac {100}{\left (5-\frac {25}{(1+x)^2}\right )^2}\right ) \]

[Out]

4+5*(x+100/(5-25/(1+x)^2)^2)*(4-exp(3)-x)

Rubi [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(77\) vs. \(2(29)=58\).

Time = 0.14 (sec) , antiderivative size = 77, normalized size of antiderivative = 2.66, number of steps used = 11, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {2099, 652, 628, 632, 212} \[ \int \frac {-2800-1560 x-4260 x^2-1200 x^3+600 x^4-60 x^6-10 x^7+e^3 \left (720+720 x+1200 x^2+600 x^3-30 x^5-5 x^6\right )}{-64+96 x-40 x^3+6 x^5+x^6} \, dx=-5 x^2-\frac {50 \left (-7 x-4 e^3+13\right )}{-x^2-2 x+4}-\frac {150 (x+1)}{-x^2-2 x+4}+\frac {500 \left (-x-e^3+4\right )}{\left (-x^2-2 x+4\right )^2}-5 e^3 x \]

[In]

Int[(-2800 - 1560*x - 4260*x^2 - 1200*x^3 + 600*x^4 - 60*x^6 - 10*x^7 + E^3*(720 + 720*x + 1200*x^2 + 600*x^3
- 30*x^5 - 5*x^6))/(-64 + 96*x - 40*x^3 + 6*x^5 + x^6),x]

[Out]

-5*E^3*x - 5*x^2 + (500*(4 - E^3 - x))/(4 - 2*x - x^2)^2 - (50*(13 - 4*E^3 - 7*x))/(4 - 2*x - x^2) - (150*(1 +
x))/(4 - 2*x - x^2)

Rule 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 628

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(b + 2*c*x)*((a + b*x + c*x^2)^(p + 1)/((p + 1
)*(b^2 - 4*a*c))), x] - Dist[2*c*((2*p + 3)/((p + 1)*(b^2 - 4*a*c))), Int[(a + b*x + c*x^2)^(p + 1), x], x] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && LtQ[p, -1] && NeQ[p, -3/2] && IntegerQ[4*p]

Rule 632

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 652

Int[((d_.) + (e_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((b*d - 2*a*e + (2*c*d -
b*e)*x)/((p + 1)*(b^2 - 4*a*c)))*(a + b*x + c*x^2)^(p + 1), x] - Dist[(2*p + 3)*((2*c*d - b*e)/((p + 1)*(b^2 -
4*a*c))), Int[(a + b*x + c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[2*c*d - b*e, 0] && NeQ[b^
2 - 4*a*c, 0] && LtQ[p, -1] && NeQ[p, -3/2]

Rule 2099

Int[(P_)^(p_)*(Q_)^(q_.), x_Symbol] :> With[{PP = Factor[P]}, Int[ExpandIntegrand[PP^p*Q^q, x], x] /;  !SumQ[N
onfreeFactors[PP, x]]] /; FreeQ[q, x] && PolyQ[P, x] && PolyQ[Q, x] && IntegerQ[p] && NeQ[P, x]

Rubi steps \begin{align*} \text {integral}& = \int \left (-5 e^3-10 x+\frac {2000 \left (-e^3+\left (5-e^3\right ) x\right )}{\left (4-2 x-x^2\right )^3}+\frac {100 \left (15+4 e^3-4 \left (5-e^3\right ) x\right )}{\left (4-2 x-x^2\right )^2}+\frac {200}{-4+2 x+x^2}\right ) \, dx \\ & = -5 e^3 x-5 x^2+100 \int \frac {15+4 e^3-4 \left (5-e^3\right ) x}{\left (4-2 x-x^2\right )^2} \, dx+200 \int \frac {1}{-4+2 x+x^2} \, dx+2000 \int \frac {-e^3+\left (5-e^3\right ) x}{\left (4-2 x-x^2\right )^3} \, dx \\ & = -5 e^3 x-5 x^2+\frac {500 \left (4-e^3-x\right )}{\left (4-2 x-x^2\right )^2}-\frac {50 \left (13-4 e^3-7 x\right )}{4-2 x-x^2}+350 \int \frac {1}{4-2 x-x^2} \, dx-400 \text {Subst}\left (\int \frac {1}{20-x^2} \, dx,x,2+2 x\right )-1500 \int \frac {1}{\left (4-2 x-x^2\right )^2} \, dx \\ & = -5 e^3 x-5 x^2+\frac {500 \left (4-e^3-x\right )}{\left (4-2 x-x^2\right )^2}-\frac {50 \left (13-4 e^3-7 x\right )}{4-2 x-x^2}-\frac {150 (1+x)}{4-2 x-x^2}-40 \sqrt {5} \text {arctanh}\left (\frac {1+x}{\sqrt {5}}\right )-150 \int \frac {1}{4-2 x-x^2} \, dx-700 \text {Subst}\left (\int \frac {1}{20-x^2} \, dx,x,-2-2 x\right ) \\ & = -5 e^3 x-5 x^2+\frac {500 \left (4-e^3-x\right )}{\left (4-2 x-x^2\right )^2}-\frac {50 \left (13-4 e^3-7 x\right )}{4-2 x-x^2}-\frac {150 (1+x)}{4-2 x-x^2}+30 \sqrt {5} \text {arctanh}\left (\frac {1+x}{\sqrt {5}}\right )+300 \text {Subst}\left (\int \frac {1}{20-x^2} \, dx,x,-2-2 x\right ) \\ & = -5 e^3 x-5 x^2+\frac {500 \left (4-e^3-x\right )}{\left (4-2 x-x^2\right )^2}-\frac {50 \left (13-4 e^3-7 x\right )}{4-2 x-x^2}-\frac {150 (1+x)}{4-2 x-x^2} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.62 \[ \int \frac {-2800-1560 x-4260 x^2-1200 x^3+600 x^4-60 x^6-10 x^7+e^3 \left (720+720 x+1200 x^2+600 x^3-30 x^5-5 x^6\right )}{-64+96 x-40 x^3+6 x^5+x^6} \, dx=-5 \left (e^3 x+x^2+\frac {100 \left (-4+e^3+x\right )}{\left (-4+2 x+x^2\right )^2}+\frac {40 \left (-4+e^3+x\right )}{-4+2 x+x^2}\right ) \]

[In]

Integrate[(-2800 - 1560*x - 4260*x^2 - 1200*x^3 + 600*x^4 - 60*x^6 - 10*x^7 + E^3*(720 + 720*x + 1200*x^2 + 60
0*x^3 - 30*x^5 - 5*x^6))/(-64 + 96*x - 40*x^3 + 6*x^5 + x^6),x]

[Out]

-5*(E^3*x + x^2 + (100*(-4 + E^3 + x))/(-4 + 2*x + x^2)^2 + (40*(-4 + E^3 + x))/(-4 + 2*x + x^2))

Maple [A] (verified)

Time = 0.08 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.83

method	result	size
default	\(-5 x^{2}-5 x \,{\mathrm e}^{3}+\frac {-200 x^{3}+100 \left (-2 \,{\mathrm e}^{3}+4\right ) x^{2}+100 \left (19-4 \,{\mathrm e}^{3}\right ) x -1200+300 \,{\mathrm e}^{3}}{\left (x^{2}+2 x -4\right )^{2}}\)	\(53\)
norman	\(\frac {\left (-20-5 \,{\mathrm e}^{3}\right ) x^{5}+\left (-200+100 \,{\mathrm e}^{3}\right ) x^{3}+\left (400-200 \,{\mathrm e}^{3}\right ) x^{2}+\left (2220-800 \,{\mathrm e}^{3}\right ) x -5 x^{6}-1520+620 \,{\mathrm e}^{3}}{\left (x^{2}+2 x -4\right )^{2}}\)	\(61\)
risch	\(-5 x \,{\mathrm e}^{3}-5 x^{2}+\frac {-200 x^{3}+\left (400-200 \,{\mathrm e}^{3}\right ) x^{2}+\left (1900-400 \,{\mathrm e}^{3}\right ) x -1200+300 \,{\mathrm e}^{3}}{x^{4}+4 x^{3}-4 x^{2}-16 x +16}\)	\(62\)
gosper	\(-\frac {5 \left (x^{5} {\mathrm e}^{3}+x^{6}+4 x^{5}-20 x^{3} {\mathrm e}^{3}+40 x^{2} {\mathrm e}^{3}+40 x^{3}+160 x \,{\mathrm e}^{3}-80 x^{2}-124 \,{\mathrm e}^{3}-444 x +304\right )}{x^{4}+4 x^{3}-4 x^{2}-16 x +16}\)	\(75\)
parallelrisch	\(-\frac {5 x^{5} {\mathrm e}^{3}+5 x^{6}+20 x^{5}+1520-100 x^{3} {\mathrm e}^{3}+200 x^{2} {\mathrm e}^{3}+200 x^{3}+800 x \,{\mathrm e}^{3}-400 x^{2}-620 \,{\mathrm e}^{3}-2220 x}{x^{4}+4 x^{3}-4 x^{2}-16 x +16}\)	\(78\)

[In]

int(((-5*x^6-30*x^5+600*x^3+1200*x^2+720*x+720)*exp(3)-10*x^7-60*x^6+600*x^4-1200*x^3-4260*x^2-1560*x-2800)/(x
^6+6*x^5-40*x^3+96*x-64),x,method=_RETURNVERBOSE)

[Out]

-5*x^2-5*x*exp(3)+100*(-2*x^3+(-2*exp(3)+4)*x^2+(19-4*exp(3))*x-12+3*exp(3))/(x^2+2*x-4)^2

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 76 vs. \(2 (24) = 48\).

Time = 0.22 (sec) , antiderivative size = 76, normalized size of antiderivative = 2.62 \[ \int \frac {-2800-1560 x-4260 x^2-1200 x^3+600 x^4-60 x^6-10 x^7+e^3 \left (720+720 x+1200 x^2+600 x^3-30 x^5-5 x^6\right )}{-64+96 x-40 x^3+6 x^5+x^6} \, dx=-\frac {5 \, {\left (x^{6} + 4 \, x^{5} - 4 \, x^{4} + 24 \, x^{3} - 64 \, x^{2} + {\left (x^{5} + 4 \, x^{4} - 4 \, x^{3} + 24 \, x^{2} + 96 \, x - 60\right )} e^{3} - 380 \, x + 240\right )}}{x^{4} + 4 \, x^{3} - 4 \, x^{2} - 16 \, x + 16} \]

[In]

integrate(((-5*x^6-30*x^5+600*x^3+1200*x^2+720*x+720)*exp(3)-10*x^7-60*x^6+600*x^4-1200*x^3-4260*x^2-1560*x-28
00)/(x^6+6*x^5-40*x^3+96*x-64),x, algorithm="fricas")

[Out]

-5*(x^6 + 4*x^5 - 4*x^4 + 24*x^3 - 64*x^2 + (x^5 + 4*x^4 - 4*x^3 + 24*x^2 + 96*x - 60)*e^3 - 380*x + 240)/(x^4
+ 4*x^3 - 4*x^2 - 16*x + 16)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 61 vs. \(2 (22) = 44\).

Time = 0.61 (sec) , antiderivative size = 61, normalized size of antiderivative = 2.10 \[ \int \frac {-2800-1560 x-4260 x^2-1200 x^3+600 x^4-60 x^6-10 x^7+e^3 \left (720+720 x+1200 x^2+600 x^3-30 x^5-5 x^6\right )}{-64+96 x-40 x^3+6 x^5+x^6} \, dx=- 5 x^{2} - 5 x e^{3} - \frac {200 x^{3} + x^{2} \left (-400 + 200 e^{3}\right ) + x \left (-1900 + 400 e^{3}\right ) - 300 e^{3} + 1200}{x^{4} + 4 x^{3} - 4 x^{2} - 16 x + 16} \]

[In]

integrate(((-5*x**6-30*x**5+600*x**3+1200*x**2+720*x+720)*exp(3)-10*x**7-60*x**6+600*x**4-1200*x**3-4260*x**2-
1560*x-2800)/(x**6+6*x**5-40*x**3+96*x-64),x)

[Out]

-5*x**2 - 5*x*exp(3) - (200*x**3 + x**2*(-400 + 200*exp(3)) + x*(-1900 + 400*exp(3)) - 300*exp(3) + 1200)/(x**
4 + 4*x**3 - 4*x**2 - 16*x + 16)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 61 vs. \(2 (24) = 48\).

Time = 0.19 (sec) , antiderivative size = 61, normalized size of antiderivative = 2.10 \[ \int \frac {-2800-1560 x-4260 x^2-1200 x^3+600 x^4-60 x^6-10 x^7+e^3 \left (720+720 x+1200 x^2+600 x^3-30 x^5-5 x^6\right )}{-64+96 x-40 x^3+6 x^5+x^6} \, dx=-5 \, x^{2} - 5 \, x e^{3} - \frac {100 \, {\left (2 \, x^{3} + 2 \, x^{2} {\left (e^{3} - 2\right )} + x {\left (4 \, e^{3} - 19\right )} - 3 \, e^{3} + 12\right )}}{x^{4} + 4 \, x^{3} - 4 \, x^{2} - 16 \, x + 16} \]

[In]

integrate(((-5*x^6-30*x^5+600*x^3+1200*x^2+720*x+720)*exp(3)-10*x^7-60*x^6+600*x^4-1200*x^3-4260*x^2-1560*x-28
00)/(x^6+6*x^5-40*x^3+96*x-64),x, algorithm="maxima")

[Out]

-5*x^2 - 5*x*e^3 - 100*(2*x^3 + 2*x^2*(e^3 - 2) + x*(4*e^3 - 19) - 3*e^3 + 12)/(x^4 + 4*x^3 - 4*x^2 - 16*x + 1
6)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 54 vs. \(2 (24) = 48\).

Time = 0.27 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.86 \[ \int \frac {-2800-1560 x-4260 x^2-1200 x^3+600 x^4-60 x^6-10 x^7+e^3 \left (720+720 x+1200 x^2+600 x^3-30 x^5-5 x^6\right )}{-64+96 x-40 x^3+6 x^5+x^6} \, dx=-5 \, x^{2} - 5 \, x e^{3} - \frac {100 \, {\left (2 \, x^{3} + 2 \, x^{2} e^{3} - 4 \, x^{2} + 4 \, x e^{3} - 19 \, x - 3 \, e^{3} + 12\right )}}{{\left (x^{2} + 2 \, x - 4\right )}^{2}} \]

[In]

integrate(((-5*x^6-30*x^5+600*x^3+1200*x^2+720*x+720)*exp(3)-10*x^7-60*x^6+600*x^4-1200*x^3-4260*x^2-1560*x-28
00)/(x^6+6*x^5-40*x^3+96*x-64),x, algorithm="giac")

[Out]

-5*x^2 - 5*x*e^3 - 100*(2*x^3 + 2*x^2*e^3 - 4*x^2 + 4*x*e^3 - 19*x - 3*e^3 + 12)/(x^2 + 2*x - 4)^2

Mupad [B] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 62, normalized size of antiderivative = 2.14 \[ \int \frac {-2800-1560 x-4260 x^2-1200 x^3+600 x^4-60 x^6-10 x^7+e^3 \left (720+720 x+1200 x^2+600 x^3-30 x^5-5 x^6\right )}{-64+96 x-40 x^3+6 x^5+x^6} \, dx=-5\,x\,{\mathrm {e}}^3-5\,x^2-\frac {200\,x^3+\left (200\,{\mathrm {e}}^3-400\right )\,x^2+\left (400\,{\mathrm {e}}^3-1900\right )\,x-300\,{\mathrm {e}}^3+1200}{x^4+4\,x^3-4\,x^2-16\,x+16} \]

[In]

int(-(1560*x - exp(3)*(720*x + 1200*x^2 + 600*x^3 - 30*x^5 - 5*x^6 + 720) + 4260*x^2 + 1200*x^3 - 600*x^4 + 60
*x^6 + 10*x^7 + 2800)/(96*x - 40*x^3 + 6*x^5 + x^6 - 64),x)

[Out]

- 5*x*exp(3) - 5*x^2 - (x^2*(200*exp(3) - 400) - 300*exp(3) + 200*x^3 + x*(400*exp(3) - 1900) + 1200)/(4*x^3 -
4*x^2 - 16*x + x^4 + 16)