3.83.57 \(\int \frac {384+320 e^3+88 e^6+8 e^9-1000 e^{24} x^3+e^8 (-1760 x-920 e^3 x-120 e^6 x)+e^{16} (2400 x^2+600 e^3 x^2)}{-64-144 x-108 x^2-27 x^3+e^3 (-48-120 x-99 x^2-27 x^3)+e^6 (-12-33 x-30 x^2-9 x^3)+e^9 (-1-3 x-3 x^2-x^3)+e^{24} (125 x^3+375 x^4+375 x^5+125 x^6)+e^8 (240 x+600 x^2+495 x^3+135 x^4+e^6 (15 x+45 x^2+45 x^3+15 x^4)+e^3 (120 x+330 x^2+300 x^3+90 x^4))+e^{16} (-300 x^2-825 x^3-750 x^4-225 x^5+e^3 (-75 x^2-225 x^3-225 x^4-75 x^5))} \, dx\)

Optimal. Leaf size=23 \[ \frac {4}{\left (1+x-\frac {x}{4+e^3-5 e^8 x}\right )^2} \]

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Rubi [B]  time = 0.78, antiderivative size = 248, normalized size of antiderivative = 10.78, number of steps used = 9, number of rules used = 5, integrand size = 282, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.018, Rules used = {2074, 638, 614, 618, 206} \begin {gather*} \frac {60 e^8 \left (5+e^3+5 e^8\right ) \left (-10 e^8 x-5 e^8+e^3+3\right )}{\left (9+6 e^3+e^6+50 e^8+10 e^{11}+25 e^{16}\right ) \left (-5 e^8 x^2+\left (3+e^3-5 e^8\right ) x+e^3+4\right )}-\frac {40 e^8 \left (-15 e^8 \left (5+e^3+5 e^8\right ) x-25 e^{16}+5 e^{11}+10 e^8+2 e^6+15 e^3+27\right )}{\left (20 e^8 \left (4+e^3\right )+\left (3+e^3-5 e^8\right )^2\right ) \left (-5 e^8 x^2+\left (3+e^3-5 e^8\right ) x+e^3+4\right )}+\frac {4 \left (\left (4+e^3\right ) \left (4+e^3+5 e^8\right )-5 e^8 \left (5+e^3+5 e^8\right ) x\right )}{\left (-5 e^8 x^2+\left (3+e^3-5 e^8\right ) x+e^3+4\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(384 + 320*E^3 + 88*E^6 + 8*E^9 - 1000*E^24*x^3 + E^8*(-1760*x - 920*E^3*x - 120*E^6*x) + E^16*(2400*x^2 +
600*E^3*x^2))/(-64 - 144*x - 108*x^2 - 27*x^3 + E^3*(-48 - 120*x - 99*x^2 - 27*x^3) + E^6*(-12 - 33*x - 30*x^
2 - 9*x^3) + E^9*(-1 - 3*x - 3*x^2 - x^3) + E^24*(125*x^3 + 375*x^4 + 375*x^5 + 125*x^6) + E^8*(240*x + 600*x^
2 + 495*x^3 + 135*x^4 + E^6*(15*x + 45*x^2 + 45*x^3 + 15*x^4) + E^3*(120*x + 330*x^2 + 300*x^3 + 90*x^4)) + E^
16*(-300*x^2 - 825*x^3 - 750*x^4 - 225*x^5 + E^3*(-75*x^2 - 225*x^3 - 225*x^4 - 75*x^5))),x]

[Out]

(4*((4 + E^3)*(4 + E^3 + 5*E^8) - 5*E^8*(5 + E^3 + 5*E^8)*x))/(4 + E^3 + (3 + E^3 - 5*E^8)*x - 5*E^8*x^2)^2 +
(60*E^8*(5 + E^3 + 5*E^8)*(3 + E^3 - 5*E^8 - 10*E^8*x))/((9 + 6*E^3 + E^6 + 50*E^8 + 10*E^11 + 25*E^16)*(4 + E
^3 + (3 + E^3 - 5*E^8)*x - 5*E^8*x^2)) - (40*E^8*(27 + 15*E^3 + 2*E^6 + 10*E^8 + 5*E^11 - 25*E^16 - 15*E^8*(5
+ E^3 + 5*E^8)*x))/((20*E^8*(4 + E^3) + (3 + E^3 - 5*E^8)^2)*(4 + E^3 + (3 + E^3 - 5*E^8)*x - 5*E^8*x^2))

Rule 206

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

Rule 614

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 618

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 638

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)*(a + b*x + c*x^2)^(p + 1))/((p + 1)*(b^2 - 4*a*c)), 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 2074

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 {gather*} \begin {aligned} \text {integral} &=\int \left (\frac {8 \left (-\left (\left (4+e^3\right ) \left (12+7 e^3+e^6+45 e^8+10 e^{11}+25 e^{16}\right )\right )+5 e^8 \left (17+8 e^3+e^6+50 e^8+10 e^{11}+25 e^{16}\right ) x\right )}{\left (4+e^3+\left (3+e^3-5 e^8\right ) x-5 e^8 x^2\right )^3}+\frac {40 e^8 \left (9+2 e^3+5 e^8-5 e^8 x\right )}{\left (4+e^3+\left (3+e^3-5 e^8\right ) x-5 e^8 x^2\right )^2}\right ) \, dx\\ &=8 \int \frac {-\left (\left (4+e^3\right ) \left (12+7 e^3+e^6+45 e^8+10 e^{11}+25 e^{16}\right )\right )+5 e^8 \left (17+8 e^3+e^6+50 e^8+10 e^{11}+25 e^{16}\right ) x}{\left (4+e^3+\left (3+e^3-5 e^8\right ) x-5 e^8 x^2\right )^3} \, dx+\left (40 e^8\right ) \int \frac {9+2 e^3+5 e^8-5 e^8 x}{\left (4+e^3+\left (3+e^3-5 e^8\right ) x-5 e^8 x^2\right )^2} \, dx\\ &=\frac {4 \left (\left (4+e^3\right ) \left (4+e^3+5 e^8\right )-5 e^8 \left (5+e^3+5 e^8\right ) x\right )}{\left (4+e^3+\left (3+e^3-5 e^8\right ) x-5 e^8 x^2\right )^2}-\frac {40 e^8 \left (27+15 e^3+2 e^6+10 e^8+5 e^{11}-25 e^{16}-15 e^8 \left (5+e^3+5 e^8\right ) x\right )}{\left (20 e^8 \left (4+e^3\right )+\left (3+e^3-5 e^8\right )^2\right ) \left (4+e^3+\left (3+e^3-5 e^8\right ) x-5 e^8 x^2\right )}-\left (60 e^8 \left (5+e^3+5 e^8\right )\right ) \int \frac {1}{\left (4+e^3+\left (3+e^3-5 e^8\right ) x-5 e^8 x^2\right )^2} \, dx+\frac {\left (600 e^{16} \left (5+e^3+5 e^8\right )\right ) \int \frac {1}{4+e^3+\left (3+e^3-5 e^8\right ) x-5 e^8 x^2} \, dx}{9+6 e^3+e^6+50 e^8+10 e^{11}+25 e^{16}}\\ &=\frac {4 \left (\left (4+e^3\right ) \left (4+e^3+5 e^8\right )-5 e^8 \left (5+e^3+5 e^8\right ) x\right )}{\left (4+e^3+\left (3+e^3-5 e^8\right ) x-5 e^8 x^2\right )^2}+\frac {60 e^8 \left (5+e^3+5 e^8\right ) \left (3+e^3-5 e^8-10 e^8 x\right )}{\left (20 e^8 \left (4+e^3\right )+\left (3+e^3-5 e^8\right )^2\right ) \left (4+e^3+\left (3+e^3-5 e^8\right ) x-5 e^8 x^2\right )}-\frac {40 e^8 \left (27+15 e^3+2 e^6+10 e^8+5 e^{11}-25 e^{16}-15 e^8 \left (5+e^3+5 e^8\right ) x\right )}{\left (20 e^8 \left (4+e^3\right )+\left (3+e^3-5 e^8\right )^2\right ) \left (4+e^3+\left (3+e^3-5 e^8\right ) x-5 e^8 x^2\right )}-\frac {\left (1200 e^{16} \left (5+e^3+5 e^8\right )\right ) \operatorname {Subst}\left (\int \frac {1}{20 e^8 \left (4+e^3\right )+\left (3+e^3-5 e^8\right )^2-x^2} \, dx,x,3+e^3-5 e^8-10 e^8 x\right )}{9+6 e^3+e^6+50 e^8+10 e^{11}+25 e^{16}}-\frac {\left (600 e^{16} \left (5+e^3+5 e^8\right )\right ) \int \frac {1}{4+e^3+\left (3+e^3-5 e^8\right ) x-5 e^8 x^2} \, dx}{20 e^8 \left (4+e^3\right )+\left (3+e^3-5 e^8\right )^2}\\ &=\frac {4 \left (\left (4+e^3\right ) \left (4+e^3+5 e^8\right )-5 e^8 \left (5+e^3+5 e^8\right ) x\right )}{\left (4+e^3+\left (3+e^3-5 e^8\right ) x-5 e^8 x^2\right )^2}+\frac {60 e^8 \left (5+e^3+5 e^8\right ) \left (3+e^3-5 e^8-10 e^8 x\right )}{\left (20 e^8 \left (4+e^3\right )+\left (3+e^3-5 e^8\right )^2\right ) \left (4+e^3+\left (3+e^3-5 e^8\right ) x-5 e^8 x^2\right )}-\frac {40 e^8 \left (27+15 e^3+2 e^6+10 e^8+5 e^{11}-25 e^{16}-15 e^8 \left (5+e^3+5 e^8\right ) x\right )}{\left (20 e^8 \left (4+e^3\right )+\left (3+e^3-5 e^8\right )^2\right ) \left (4+e^3+\left (3+e^3-5 e^8\right ) x-5 e^8 x^2\right )}-\frac {1200 e^{16} \left (5+e^3+5 e^8\right ) \tanh ^{-1}\left (\frac {3+e^3-5 e^8 (1+2 x)}{\sqrt {9+6 e^3+e^6+50 e^8+10 e^{11}+25 e^{16}}}\right )}{\left (9+6 e^3+e^6+50 e^8+10 e^{11}+25 e^{16}\right )^{3/2}}+\frac {\left (1200 e^{16} \left (5+e^3+5 e^8\right )\right ) \operatorname {Subst}\left (\int \frac {1}{20 e^8 \left (4+e^3\right )+\left (3+e^3-5 e^8\right )^2-x^2} \, dx,x,3+e^3-5 e^8-10 e^8 x\right )}{20 e^8 \left (4+e^3\right )+\left (3+e^3-5 e^8\right )^2}\\ &=\frac {4 \left (\left (4+e^3\right ) \left (4+e^3+5 e^8\right )-5 e^8 \left (5+e^3+5 e^8\right ) x\right )}{\left (4+e^3+\left (3+e^3-5 e^8\right ) x-5 e^8 x^2\right )^2}+\frac {60 e^8 \left (5+e^3+5 e^8\right ) \left (3+e^3-5 e^8-10 e^8 x\right )}{\left (20 e^8 \left (4+e^3\right )+\left (3+e^3-5 e^8\right )^2\right ) \left (4+e^3+\left (3+e^3-5 e^8\right ) x-5 e^8 x^2\right )}-\frac {40 e^8 \left (27+15 e^3+2 e^6+10 e^8+5 e^{11}-25 e^{16}-15 e^8 \left (5+e^3+5 e^8\right ) x\right )}{\left (20 e^8 \left (4+e^3\right )+\left (3+e^3-5 e^8\right )^2\right ) \left (4+e^3+\left (3+e^3-5 e^8\right ) x-5 e^8 x^2\right )}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.05, size = 38, normalized size = 1.65 \begin {gather*} \frac {4 \left (4+e^3-5 e^8 x\right )^2}{\left (4+3 x+e^3 (1+x)-5 e^8 x (1+x)\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(384 + 320*E^3 + 88*E^6 + 8*E^9 - 1000*E^24*x^3 + E^8*(-1760*x - 920*E^3*x - 120*E^6*x) + E^16*(2400
*x^2 + 600*E^3*x^2))/(-64 - 144*x - 108*x^2 - 27*x^3 + E^3*(-48 - 120*x - 99*x^2 - 27*x^3) + E^6*(-12 - 33*x -
30*x^2 - 9*x^3) + E^9*(-1 - 3*x - 3*x^2 - x^3) + E^24*(125*x^3 + 375*x^4 + 375*x^5 + 125*x^6) + E^8*(240*x +
600*x^2 + 495*x^3 + 135*x^4 + E^6*(15*x + 45*x^2 + 45*x^3 + 15*x^4) + E^3*(120*x + 330*x^2 + 300*x^3 + 90*x^4)
) + E^16*(-300*x^2 - 825*x^3 - 750*x^4 - 225*x^5 + E^3*(-75*x^2 - 225*x^3 - 225*x^4 - 75*x^5))),x]

[Out]

(4*(4 + E^3 - 5*E^8*x)^2)/(4 + 3*x + E^3*(1 + x) - 5*E^8*x*(1 + x))^2

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fricas [B]  time = 0.69, size = 112, normalized size = 4.87 \begin {gather*} \frac {4 \, {\left (25 \, x^{2} e^{16} - 10 \, x e^{11} - 40 \, x e^{8} + e^{6} + 8 \, e^{3} + 16\right )}}{9 \, x^{2} + 25 \, {\left (x^{4} + 2 \, x^{3} + x^{2}\right )} e^{16} - 10 \, {\left (x^{3} + 2 \, x^{2} + x\right )} e^{11} - 10 \, {\left (3 \, x^{3} + 7 \, x^{2} + 4 \, x\right )} e^{8} + {\left (x^{2} + 2 \, x + 1\right )} e^{6} + 2 \, {\left (3 \, x^{2} + 7 \, x + 4\right )} e^{3} + 24 \, x + 16} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-1000*x^3*exp(4)^6+(600*x^2*exp(3)+2400*x^2)*exp(4)^4+(-120*x*exp(3)^2-920*x*exp(3)-1760*x)*exp(4)^
2+8*exp(3)^3+88*exp(3)^2+320*exp(3)+384)/((125*x^6+375*x^5+375*x^4+125*x^3)*exp(4)^6+((-75*x^5-225*x^4-225*x^3
-75*x^2)*exp(3)-225*x^5-750*x^4-825*x^3-300*x^2)*exp(4)^4+((15*x^4+45*x^3+45*x^2+15*x)*exp(3)^2+(90*x^4+300*x^
3+330*x^2+120*x)*exp(3)+135*x^4+495*x^3+600*x^2+240*x)*exp(4)^2+(-x^3-3*x^2-3*x-1)*exp(3)^3+(-9*x^3-30*x^2-33*
x-12)*exp(3)^2+(-27*x^3-99*x^2-120*x-48)*exp(3)-27*x^3-108*x^2-144*x-64),x, algorithm="fricas")

[Out]

4*(25*x^2*e^16 - 10*x*e^11 - 40*x*e^8 + e^6 + 8*e^3 + 16)/(9*x^2 + 25*(x^4 + 2*x^3 + x^2)*e^16 - 10*(x^3 + 2*x
^2 + x)*e^11 - 10*(3*x^3 + 7*x^2 + 4*x)*e^8 + (x^2 + 2*x + 1)*e^6 + 2*(3*x^2 + 7*x + 4)*e^3 + 24*x + 16)

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giac [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-1000*x^3*exp(4)^6+(600*x^2*exp(3)+2400*x^2)*exp(4)^4+(-120*x*exp(3)^2-920*x*exp(3)-1760*x)*exp(4)^
2+8*exp(3)^3+88*exp(3)^2+320*exp(3)+384)/((125*x^6+375*x^5+375*x^4+125*x^3)*exp(4)^6+((-75*x^5-225*x^4-225*x^3
-75*x^2)*exp(3)-225*x^5-750*x^4-825*x^3-300*x^2)*exp(4)^4+((15*x^4+45*x^3+45*x^2+15*x)*exp(3)^2+(90*x^4+300*x^
3+330*x^2+120*x)*exp(3)+135*x^4+495*x^3+600*x^2+240*x)*exp(4)^2+(-x^3-3*x^2-3*x-1)*exp(3)^3+(-9*x^3-30*x^2-33*
x-12)*exp(3)^2+(-27*x^3-99*x^2-120*x-48)*exp(3)-27*x^3-108*x^2-144*x-64),x, algorithm="giac")

[Out]

Timed out

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maple [B]  time = 0.90, size = 66, normalized size = 2.87

method	result	size
norman	\(\frac {-40 \,{\mathrm e}^{8} \left (4+{\mathrm e}^{3}\right ) x +100 x^{2} {\mathrm e}^{16}+64+4 \,{\mathrm e}^{6}+32 \,{\mathrm e}^{3}}{\left (5 x^{2} {\mathrm e}^{8}+5 x \,{\mathrm e}^{8}-x \,{\mathrm e}^{3}-{\mathrm e}^{3}-3 x -4\right )^{2}}\)	\(66\)
risch	\(\frac {\frac {64}{25}+4 x^{2} {\mathrm e}^{16}-\frac {8 \,{\mathrm e}^{8} \left (4+{\mathrm e}^{3}\right ) x}{5}+\frac {4 \,{\mathrm e}^{6}}{25}+\frac {32 \,{\mathrm e}^{3}}{25}}{x^{4} {\mathrm e}^{16}+2 x^{3} {\mathrm e}^{16}+x^{2} {\mathrm e}^{16}-\frac {2 x^{3} {\mathrm e}^{11}}{5}-\frac {4 x^{2} {\mathrm e}^{11}}{5}-\frac {2 x \,{\mathrm e}^{11}}{5}-\frac {6 x^{3} {\mathrm e}^{8}}{5}-\frac {14 x^{2} {\mathrm e}^{8}}{5}-\frac {8 x \,{\mathrm e}^{8}}{5}+\frac {x^{2} {\mathrm e}^{6}}{25}+\frac {2 x \,{\mathrm e}^{6}}{25}+\frac {{\mathrm e}^{6}}{25}+\frac {6 x^{2} {\mathrm e}^{3}}{25}+\frac {14 x \,{\mathrm e}^{3}}{25}+\frac {8 \,{\mathrm e}^{3}}{25}+\frac {9 x^{2}}{25}+\frac {24 x}{25}+\frac {16}{25}}\)	\(129\)
gosper	\(\frac {100 x^{2} {\mathrm e}^{16}-40 x \,{\mathrm e}^{3} {\mathrm e}^{8}-160 x \,{\mathrm e}^{8}+4 \,{\mathrm e}^{6}+32 \,{\mathrm e}^{3}+64}{25 x^{4} {\mathrm e}^{16}+50 x^{3} {\mathrm e}^{16}+25 x^{2} {\mathrm e}^{16}-10 x^{3} {\mathrm e}^{3} {\mathrm e}^{8}-20 x^{2} {\mathrm e}^{3} {\mathrm e}^{8}-30 x^{3} {\mathrm e}^{8}-10 x \,{\mathrm e}^{3} {\mathrm e}^{8}-70 x^{2} {\mathrm e}^{8}+x^{2} {\mathrm e}^{6}-40 x \,{\mathrm e}^{8}+2 x \,{\mathrm e}^{6}+6 x^{2} {\mathrm e}^{3}+{\mathrm e}^{6}+14 x \,{\mathrm e}^{3}+9 x^{2}+8 \,{\mathrm e}^{3}+24 x +16}\)	\(168\)
default	\(-\frac {8 \left (\munderset {\textit {\_R} =\RootOf \left (125 \textit {\_Z}^{6} {\mathrm e}^{24}-\left (225 \,{\mathrm e}^{16}-375 \,{\mathrm e}^{24}+75 \,{\mathrm e}^{19}\right ) \textit {\_Z}^{5}-\left (750 \,{\mathrm e}^{16}-375 \,{\mathrm e}^{24}-15 \,{\mathrm e}^{14}-90 \,{\mathrm e}^{11}+225 \,{\mathrm e}^{19}-135 \,{\mathrm e}^{8}\right ) \textit {\_Z}^{4}-\left (9 \,{\mathrm e}^{6}+{\mathrm e}^{9}+825 \,{\mathrm e}^{16}-125 \,{\mathrm e}^{24}-45 \,{\mathrm e}^{14}-300 \,{\mathrm e}^{11}+225 \,{\mathrm e}^{19}+27 \,{\mathrm e}^{3}-495 \,{\mathrm e}^{8}+27\right ) \textit {\_Z}^{3}-\left (30 \,{\mathrm e}^{6}+3 \,{\mathrm e}^{9}+300 \,{\mathrm e}^{16}-45 \,{\mathrm e}^{14}-330 \,{\mathrm e}^{11}+75 \,{\mathrm e}^{19}+99 \,{\mathrm e}^{3}-600 \,{\mathrm e}^{8}+108\right ) \textit {\_Z}^{2}-\left (33 \,{\mathrm e}^{6}+3 \,{\mathrm e}^{9}-15 \,{\mathrm e}^{14}-120 \,{\mathrm e}^{11}+120 \,{\mathrm e}^{3}-240 \,{\mathrm e}^{8}+144\right ) \textit {\_Z} -64-{\mathrm e}^{9}-12 \,{\mathrm e}^{6}-48 \,{\mathrm e}^{3}\right )}{\sum }\frac {\left (48-125 \textit {\_R}^{3} {\mathrm e}^{24}+75 \left (4 \,{\mathrm e}^{16}+{\mathrm e}^{19}\right ) \textit {\_R}^{2}+5 \left (-3 \,{\mathrm e}^{14}-23 \,{\mathrm e}^{11}-44 \,{\mathrm e}^{8}\right ) \textit {\_R} +{\mathrm e}^{9}+11 \,{\mathrm e}^{6}+40 \,{\mathrm e}^{3}\right ) \ln \left (x -\textit {\_R} \right )}{48+72 \textit {\_R} -80 \,{\mathrm e}^{8}+{\mathrm e}^{9}+11 \,{\mathrm e}^{6}+40 \,{\mathrm e}^{3}-5 \,{\mathrm e}^{14}+27 \textit {\_R}^{2} {\mathrm e}^{3}-40 \,{\mathrm e}^{11}-400 \textit {\_R} \,{\mathrm e}^{8}-180 \textit {\_R}^{3} {\mathrm e}^{8}-495 \textit {\_R}^{2} {\mathrm e}^{8}+20 \textit {\_R} \,{\mathrm e}^{6}+27 \textit {\_R}^{2}+66 \textit {\_R} \,{\mathrm e}^{3}-500 \textit {\_R}^{3} {\mathrm e}^{24}+9 \textit {\_R}^{2} {\mathrm e}^{6}+825 \textit {\_R}^{2} {\mathrm e}^{16}+2 \textit {\_R} \,{\mathrm e}^{9}+200 \textit {\_R} \,{\mathrm e}^{16}+\textit {\_R}^{2} {\mathrm e}^{9}+1000 \textit {\_R}^{3} {\mathrm e}^{16}-625 \,{\mathrm e}^{24} \textit {\_R}^{4}-250 \,{\mathrm e}^{24} \textit {\_R}^{5}-45 \textit {\_R}^{2} {\mathrm e}^{14}+125 \textit {\_R}^{4} {\mathrm e}^{19}+300 \textit {\_R}^{3} {\mathrm e}^{19}-20 \textit {\_R}^{3} {\mathrm e}^{14}-220 \textit {\_R} \,{\mathrm e}^{11}+225 \textit {\_R}^{2} {\mathrm e}^{19}-30 \textit {\_R} \,{\mathrm e}^{14}-120 \textit {\_R}^{3} {\mathrm e}^{11}+50 \textit {\_R} \,{\mathrm e}^{19}+375 \textit {\_R}^{4} {\mathrm e}^{16}-125 \textit {\_R}^{2} {\mathrm e}^{24}-300 \textit {\_R}^{2} {\mathrm e}^{11}}\right )}{3}\)	\(445\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-1000*x^3*exp(4)^6+(600*x^2*exp(3)+2400*x^2)*exp(4)^4+(-120*x*exp(3)^2-920*x*exp(3)-1760*x)*exp(4)^2+8*ex
p(3)^3+88*exp(3)^2+320*exp(3)+384)/((125*x^6+375*x^5+375*x^4+125*x^3)*exp(4)^6+((-75*x^5-225*x^4-225*x^3-75*x^
2)*exp(3)-225*x^5-750*x^4-825*x^3-300*x^2)*exp(4)^4+((15*x^4+45*x^3+45*x^2+15*x)*exp(3)^2+(90*x^4+300*x^3+330*
x^2+120*x)*exp(3)+135*x^4+495*x^3+600*x^2+240*x)*exp(4)^2+(-x^3-3*x^2-3*x-1)*exp(3)^3+(-9*x^3-30*x^2-33*x-12)*
exp(3)^2+(-27*x^3-99*x^2-120*x-48)*exp(3)-27*x^3-108*x^2-144*x-64),x,method=_RETURNVERBOSE)

[Out]

(-40*exp(4)^2*(4+exp(3))*x+100*x^2*exp(4)^4+64+4*exp(3)^2+32*exp(3))/(5*x^2*exp(4)^2+5*x*exp(4)^2-x*exp(3)-exp
(3)-3*x-4)^2

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maxima [B]  time = 0.58, size = 107, normalized size = 4.65 \begin {gather*} \frac {4 \, {\left (25 \, x^{2} e^{16} - 10 \, x {\left (e^{11} + 4 \, e^{8}\right )} + e^{6} + 8 \, e^{3} + 16\right )}}{25 \, x^{4} e^{16} + 10 \, x^{3} {\left (5 \, e^{16} - e^{11} - 3 \, e^{8}\right )} + x^{2} {\left (25 \, e^{16} - 20 \, e^{11} - 70 \, e^{8} + e^{6} + 6 \, e^{3} + 9\right )} - 2 \, x {\left (5 \, e^{11} + 20 \, e^{8} - e^{6} - 7 \, e^{3} - 12\right )} + e^{6} + 8 \, e^{3} + 16} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-1000*x^3*exp(4)^6+(600*x^2*exp(3)+2400*x^2)*exp(4)^4+(-120*x*exp(3)^2-920*x*exp(3)-1760*x)*exp(4)^
2+8*exp(3)^3+88*exp(3)^2+320*exp(3)+384)/((125*x^6+375*x^5+375*x^4+125*x^3)*exp(4)^6+((-75*x^5-225*x^4-225*x^3
-75*x^2)*exp(3)-225*x^5-750*x^4-825*x^3-300*x^2)*exp(4)^4+((15*x^4+45*x^3+45*x^2+15*x)*exp(3)^2+(90*x^4+300*x^
3+330*x^2+120*x)*exp(3)+135*x^4+495*x^3+600*x^2+240*x)*exp(4)^2+(-x^3-3*x^2-3*x-1)*exp(3)^3+(-9*x^3-30*x^2-33*
x-12)*exp(3)^2+(-27*x^3-99*x^2-120*x-48)*exp(3)-27*x^3-108*x^2-144*x-64),x, algorithm="maxima")

[Out]

4*(25*x^2*e^16 - 10*x*(e^11 + 4*e^8) + e^6 + 8*e^3 + 16)/(25*x^4*e^16 + 10*x^3*(5*e^16 - e^11 - 3*e^8) + x^2*(
25*e^16 - 20*e^11 - 70*e^8 + e^6 + 6*e^3 + 9) - 2*x*(5*e^11 + 20*e^8 - e^6 - 7*e^3 - 12) + e^6 + 8*e^3 + 16)

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mupad [B]  time = 0.57, size = 92, normalized size = 4.00 \begin {gather*} \frac {4\,{\left ({\mathrm {e}}^3-5\,x\,{\mathrm {e}}^8+4\right )}^2}{25\,{\mathrm {e}}^{16}\,x^4+\left (50\,{\mathrm {e}}^{16}-10\,{\mathrm {e}}^{11}-30\,{\mathrm {e}}^8\right )\,x^3+\left (6\,{\mathrm {e}}^3+{\mathrm {e}}^6-70\,{\mathrm {e}}^8-20\,{\mathrm {e}}^{11}+25\,{\mathrm {e}}^{16}+9\right )\,x^2+\left (14\,{\mathrm {e}}^3+2\,{\mathrm {e}}^6-40\,{\mathrm {e}}^8-10\,{\mathrm {e}}^{11}+24\right )\,x+8\,{\mathrm {e}}^3+{\mathrm {e}}^6+16} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(320*exp(3) + 88*exp(6) + 8*exp(9) - exp(8)*(1760*x + 920*x*exp(3) + 120*x*exp(6)) - 1000*x^3*exp(24) + e
xp(16)*(600*x^2*exp(3) + 2400*x^2) + 384)/(144*x + exp(9)*(3*x + 3*x^2 + x^3 + 1) + exp(6)*(33*x + 30*x^2 + 9*
x^3 + 12) + exp(3)*(120*x + 99*x^2 + 27*x^3 + 48) + 108*x^2 + 27*x^3 - exp(8)*(240*x + exp(6)*(15*x + 45*x^2 +
45*x^3 + 15*x^4) + exp(3)*(120*x + 330*x^2 + 300*x^3 + 90*x^4) + 600*x^2 + 495*x^3 + 135*x^4) + exp(16)*(300*
x^2 + 825*x^3 + 750*x^4 + 225*x^5 + exp(3)*(75*x^2 + 225*x^3 + 225*x^4 + 75*x^5)) - exp(24)*(125*x^3 + 375*x^4
+ 375*x^5 + 125*x^6) + 64),x)

[Out]

(4*(exp(3) - 5*x*exp(8) + 4)^2)/(8*exp(3) + exp(6) + x^2*(6*exp(3) + exp(6) - 70*exp(8) - 20*exp(11) + 25*exp(
16) + 9) + x*(14*exp(3) + 2*exp(6) - 40*exp(8) - 10*exp(11) + 24) + 25*x^4*exp(16) - x^3*(30*exp(8) + 10*exp(1
1) - 50*exp(16)) + 16)

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sympy [B]  time = 10.01, size = 122, normalized size = 5.30 \begin {gather*} - \frac {- 100 x^{2} e^{16} + x \left (160 e^{8} + 40 e^{11}\right ) - 4 e^{6} - 32 e^{3} - 64}{25 x^{4} e^{16} + x^{3} \left (- 10 e^{11} - 30 e^{8} + 50 e^{16}\right ) + x^{2} \left (- 20 e^{11} - 70 e^{8} + 9 + 6 e^{3} + e^{6} + 25 e^{16}\right ) + x \left (- 10 e^{11} - 40 e^{8} + 24 + 14 e^{3} + 2 e^{6}\right ) + 16 + 8 e^{3} + e^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-1000*x**3*exp(4)**6+(600*x**2*exp(3)+2400*x**2)*exp(4)**4+(-120*x*exp(3)**2-920*x*exp(3)-1760*x)*e
xp(4)**2+8*exp(3)**3+88*exp(3)**2+320*exp(3)+384)/((125*x**6+375*x**5+375*x**4+125*x**3)*exp(4)**6+((-75*x**5-
225*x**4-225*x**3-75*x**2)*exp(3)-225*x**5-750*x**4-825*x**3-300*x**2)*exp(4)**4+((15*x**4+45*x**3+45*x**2+15*
x)*exp(3)**2+(90*x**4+300*x**3+330*x**2+120*x)*exp(3)+135*x**4+495*x**3+600*x**2+240*x)*exp(4)**2+(-x**3-3*x**
2-3*x-1)*exp(3)**3+(-9*x**3-30*x**2-33*x-12)*exp(3)**2+(-27*x**3-99*x**2-120*x-48)*exp(3)-27*x**3-108*x**2-144
*x-64),x)

[Out]

-(-100*x**2*exp(16) + x*(160*exp(8) + 40*exp(11)) - 4*exp(6) - 32*exp(3) - 64)/(25*x**4*exp(16) + x**3*(-10*ex
p(11) - 30*exp(8) + 50*exp(16)) + x**2*(-20*exp(11) - 70*exp(8) + 9 + 6*exp(3) + exp(6) + 25*exp(16)) + x*(-10
*exp(11) - 40*exp(8) + 24 + 14*exp(3) + 2*exp(6)) + 16 + 8*exp(3) + exp(6))

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