3.81.97 \(\int \frac {-1344-60 e^8 x^2+e^4 (-480 x+96 x^2)+e^x (576+144 x-96 x^2+e^4 (240 x+72 x^2-24 x^3)+e^8 (30 x^2+15 x^3))}{25 e^{11} x^2+e^7 (200 x-80 x^2)+e^3 (400-320 x+64 x^2)} \, dx\)

Optimal. Leaf size=30 \[ \frac {3 \left (-4+e^x\right ) (1+x)}{e^3 \left (5-\frac {8}{e^4+\frac {4}{x}}\right )} \]

________________________________________________________________________________________

Rubi [B]  time = 0.70, antiderivative size = 147, normalized size of antiderivative = 4.90, number of steps used = 14, number of rules used = 9, integrand size = 109, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {6688, 12, 6742, 683, 2199, 2194, 2176, 2177, 2178} \begin {gather*} -\frac {3 e^{x+1} x}{8-5 e^4}+\frac {12 e x}{8-5 e^4}+\frac {96 \left (28-5 e^4\right ) e^{x-3}}{\left (8-5 e^4\right )^2 \left (20-\left (8-5 e^4\right ) x\right )}-\frac {384 \left (28-5 e^4\right )}{e^3 \left (8-5 e^4\right )^2 \left (20-\left (8-5 e^4\right ) x\right )}-\frac {6 \left (16+8 e^4-5 e^8\right ) e^{x-3}}{\left (8-5 e^4\right )^2}+\frac {3 e^{x+1}}{8-5 e^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-1344 - 60*E^8*x^2 + E^4*(-480*x + 96*x^2) + E^x*(576 + 144*x - 96*x^2 + E^4*(240*x + 72*x^2 - 24*x^3) +
E^8*(30*x^2 + 15*x^3)))/(25*E^11*x^2 + E^7*(200*x - 80*x^2) + E^3*(400 - 320*x + 64*x^2)),x]

[Out]

(3*E^(1 + x))/(8 - 5*E^4) - (6*E^(-3 + x)*(16 + 8*E^4 - 5*E^8))/(8 - 5*E^4)^2 + (12*E*x)/(8 - 5*E^4) - (3*E^(1
+ x)*x)/(8 - 5*E^4) - (384*(28 - 5*E^4))/(E^3*(8 - 5*E^4)^2*(20 - (8 - 5*E^4)*x)) + (96*E^(-3 + x)*(28 - 5*E^
4))/((8 - 5*E^4)^2*(20 - (8 - 5*E^4)*x))

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 683

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

Rule 2176

Int[((b_.)*(F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[((c + d*x)^m
*(b*F^(g*(e + f*x)))^n)/(f*g*n*Log[F]), x] - Dist[(d*m)/(f*g*n*Log[F]), Int[(c + d*x)^(m - 1)*(b*F^(g*(e + f*x
)))^n, x], x] /; FreeQ[{F, b, c, d, e, f, g, n}, x] && GtQ[m, 0] && IntegerQ[2*m] &&  !$UseGamma === True

Rule 2177

Int[((b_.)*(F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_), x_Symbol] :> Simp[((c + d*x)^(m
+ 1)*(b*F^(g*(e + f*x)))^n)/(d*(m + 1)), x] - Dist[(f*g*n*Log[F])/(d*(m + 1)), Int[(c + d*x)^(m + 1)*(b*F^(g*
(e + f*x)))^n, x], x] /; FreeQ[{F, b, c, d, e, f, g, n}, x] && LtQ[m, -1] && IntegerQ[2*m] &&  !$UseGamma ===
True

Rule 2178

Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[(F^(g*(e - (c*f)/d))*ExpIntegral
Ei[(f*g*(c + d*x)*Log[F])/d])/d, x] /; FreeQ[{F, c, d, e, f, g}, x] &&  !$UseGamma === True

Rule 2194

Int[((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.), x_Symbol] :> Simp[(F^(c*(a + b*x)))^n/(b*c*n*Log[F]), x] /; Fre
eQ[{F, a, b, c, n}, x]

Rule 2199

Int[(F_)^((c_.)*(v_))*(u_)^(m_.)*(w_), x_Symbol] :> Int[ExpandIntegrand[F^(c*ExpandToSum[v, x]), w*NormalizePo
werOfLinear[u, x]^m, x], x] /; FreeQ[{F, c}, x] && PolynomialQ[w, x] && LinearQ[v, x] && PowerOfLinearQ[u, x]
&& IntegerQ[m] &&  !$UseGamma === True

Rule 6688

Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; SimplerIntegrandQ[v, u, x]]

Rule 6742

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {3 \left (-448+32 e^4 (-5+x) x-20 e^8 x^2+5 e^{8+x} x^2 (2+x)+e^x \left (192+48 x-32 x^2\right )-8 e^{4+x} x \left (-10-3 x+x^2\right )\right )}{e^3 \left (20+\left (-8+5 e^4\right ) x\right )^2} \, dx\\ &=\frac {3 \int \frac {-448+32 e^4 (-5+x) x-20 e^8 x^2+5 e^{8+x} x^2 (2+x)+e^x \left (192+48 x-32 x^2\right )-8 e^{4+x} x \left (-10-3 x+x^2\right )}{\left (20+\left (-8+5 e^4\right ) x\right )^2} \, dx}{e^3}\\ &=\frac {3 \int \left (\frac {4 \left (-112-40 e^4 x+e^4 \left (8-5 e^4\right ) x^2\right )}{\left (20-\left (8-5 e^4\right ) x\right )^2}+\frac {e^x \left (192+16 \left (3+5 e^4\right ) x-2 \left (16-12 e^4-5 e^8\right ) x^2-e^4 \left (8-5 e^4\right ) x^3\right )}{\left (20-\left (8-5 e^4\right ) x\right )^2}\right ) \, dx}{e^3}\\ &=\frac {3 \int \frac {e^x \left (192+16 \left (3+5 e^4\right ) x-2 \left (16-12 e^4-5 e^8\right ) x^2-e^4 \left (8-5 e^4\right ) x^3\right )}{\left (20-\left (8-5 e^4\right ) x\right )^2} \, dx}{e^3}+\frac {12 \int \frac {-112-40 e^4 x+e^4 \left (8-5 e^4\right ) x^2}{\left (20-\left (8-5 e^4\right ) x\right )^2} \, dx}{e^3}\\ &=\frac {3 \int \left (\frac {2 e^x \left (-16-8 e^4+5 e^8\right )}{\left (-8+5 e^4\right )^2}+\frac {e^{4+x} x}{-8+5 e^4}+\frac {32 e^x \left (28-5 e^4\right )}{\left (8-5 e^4\right ) \left (20-\left (8-5 e^4\right ) x\right )^2}+\frac {32 e^x \left (28-5 e^4\right )}{\left (8-5 e^4\right )^2 \left (20-\left (8-5 e^4\right ) x\right )}\right ) \, dx}{e^3}+\frac {12 \int \left (-\frac {e^4}{-8+5 e^4}+\frac {32 \left (-28+5 e^4\right )}{\left (8-5 e^4\right ) \left (20-\left (8-5 e^4\right ) x\right )^2}\right ) \, dx}{e^3}\\ &=\frac {12 e x}{8-5 e^4}-\frac {384 \left (28-5 e^4\right )}{e^3 \left (8-5 e^4\right )^2 \left (20-\left (8-5 e^4\right ) x\right )}-\frac {3 \int e^{4+x} x \, dx}{e^3 \left (8-5 e^4\right )}+\frac {\left (96 \left (28-5 e^4\right )\right ) \int \frac {e^x}{20-\left (8-5 e^4\right ) x} \, dx}{e^3 \left (8-5 e^4\right )^2}+\frac {\left (96 \left (28-5 e^4\right )\right ) \int \frac {e^x}{\left (20-\left (8-5 e^4\right ) x\right )^2} \, dx}{e^3 \left (8-5 e^4\right )}-\frac {\left (6 \left (16+8 e^4-5 e^8\right )\right ) \int e^x \, dx}{e^3 \left (8-5 e^4\right )^2}\\ &=-\frac {6 e^{-3+x} \left (16+8 e^4-5 e^8\right )}{\left (8-5 e^4\right )^2}+\frac {12 e x}{8-5 e^4}-\frac {3 e^{1+x} x}{8-5 e^4}-\frac {384 \left (28-5 e^4\right )}{e^3 \left (8-5 e^4\right )^2 \left (20-\left (8-5 e^4\right ) x\right )}+\frac {96 e^{-3+x} \left (28-5 e^4\right )}{\left (8-5 e^4\right )^2 \left (20-\left (8-5 e^4\right ) x\right )}-\frac {96 e^{-3+\frac {20}{8-5 e^4}} \left (28-5 e^4\right ) \text {Ei}\left (-\frac {20-\left (8-5 e^4\right ) x}{8-5 e^4}\right )}{\left (8-5 e^4\right )^3}+\frac {3 \int e^{4+x} \, dx}{e^3 \left (8-5 e^4\right )}-\frac {\left (96 \left (28-5 e^4\right )\right ) \int \frac {e^x}{20+\left (-8+5 e^4\right ) x} \, dx}{e^3 \left (8-5 e^4\right )^2}\\ &=\frac {3 e^{1+x}}{8-5 e^4}-\frac {6 e^{-3+x} \left (16+8 e^4-5 e^8\right )}{\left (8-5 e^4\right )^2}+\frac {12 e x}{8-5 e^4}-\frac {3 e^{1+x} x}{8-5 e^4}-\frac {384 \left (28-5 e^4\right )}{e^3 \left (8-5 e^4\right )^2 \left (20-\left (8-5 e^4\right ) x\right )}+\frac {96 e^{-3+x} \left (28-5 e^4\right )}{\left (8-5 e^4\right )^2 \left (20-\left (8-5 e^4\right ) x\right )}\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [B]  time = 0.37, size = 114, normalized size = 3.80 \begin {gather*} \frac {3 \left (-3584-100 e^{12} x^2+256 e^x (1+x)+25 e^{12+x} x (1+x)+80 e^8 x (-5+4 x)+64 e^{4+x} \left (-5-4 x+x^2\right )-128 e^4 \left (-5-5 x+2 x^2\right )-20 e^{8+x} \left (-5-x+4 x^2\right )\right )}{e^3 \left (8-5 e^4\right )^2 \left (20+\left (-8+5 e^4\right ) x\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-1344 - 60*E^8*x^2 + E^4*(-480*x + 96*x^2) + E^x*(576 + 144*x - 96*x^2 + E^4*(240*x + 72*x^2 - 24*x
^3) + E^8*(30*x^2 + 15*x^3)))/(25*E^11*x^2 + E^7*(200*x - 80*x^2) + E^3*(400 - 320*x + 64*x^2)),x]

[Out]

(3*(-3584 - 100*E^12*x^2 + 256*E^x*(1 + x) + 25*E^(12 + x)*x*(1 + x) + 80*E^8*x*(-5 + 4*x) + 64*E^(4 + x)*(-5
- 4*x + x^2) - 128*E^4*(-5 - 5*x + 2*x^2) - 20*E^(8 + x)*(-5 - x + 4*x^2)))/(E^3*(8 - 5*E^4)^2*(20 + (-8 + 5*E
^4)*x))

________________________________________________________________________________________

fricas [B]  time = 0.61, size = 117, normalized size = 3.90 \begin {gather*} -\frac {3 \, {\left (100 \, x^{2} e^{12} - 80 \, {\left (4 \, x^{2} - 5 \, x\right )} e^{8} + 128 \, {\left (2 \, x^{2} - 5 \, x - 5\right )} e^{4} - {\left (25 \, {\left (x^{2} + x\right )} e^{12} - 20 \, {\left (4 \, x^{2} - x - 5\right )} e^{8} + 64 \, {\left (x^{2} - 4 \, x - 5\right )} e^{4} + 256 \, x + 256\right )} e^{x} + 3584\right )}}{125 \, x e^{15} - 100 \, {\left (6 \, x - 5\right )} e^{11} + 320 \, {\left (3 \, x - 5\right )} e^{7} - 256 \, {\left (2 \, x - 5\right )} e^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((15*x^3+30*x^2)*exp(4)^2+(-24*x^3+72*x^2+240*x)*exp(4)-96*x^2+144*x+576)*exp(x)-60*x^2*exp(4)^2+(9
6*x^2-480*x)*exp(4)-1344)/(25*x^2*exp(3)*exp(4)^2+(-80*x^2+200*x)*exp(3)*exp(4)+(64*x^2-320*x+400)*exp(3)),x,
algorithm="fricas")

[Out]

-3*(100*x^2*e^12 - 80*(4*x^2 - 5*x)*e^8 + 128*(2*x^2 - 5*x - 5)*e^4 - (25*(x^2 + x)*e^12 - 20*(4*x^2 - x - 5)*
e^8 + 64*(x^2 - 4*x - 5)*e^4 + 256*x + 256)*e^x + 3584)/(125*x*e^15 - 100*(6*x - 5)*e^11 + 320*(3*x - 5)*e^7 -
256*(2*x - 5)*e^3)

________________________________________________________________________________________

giac [B]  time = 0.17, size = 237, normalized size = 7.90 \begin {gather*} -\frac {3 \, {\left (2500 \, x^{2} e^{20} - 16000 \, x^{2} e^{16} + 38400 \, x^{2} e^{12} - 40960 \, x^{2} e^{8} + 16384 \, x^{2} e^{4} - 625 \, x^{2} e^{\left (x + 20\right )} + 4000 \, x^{2} e^{\left (x + 16\right )} - 9600 \, x^{2} e^{\left (x + 12\right )} + 10240 \, x^{2} e^{\left (x + 8\right )} - 4096 \, x^{2} e^{\left (x + 4\right )} + 10000 \, x e^{16} - 48000 \, x e^{12} + 76800 \, x e^{8} - 40960 \, x e^{4} - 625 \, x e^{\left (x + 20\right )} + 1500 \, x e^{\left (x + 16\right )} + 6400 \, x e^{\left (x + 12\right )} - 28160 \, x e^{\left (x + 8\right )} + 36864 \, x e^{\left (x + 4\right )} - 16384 \, x e^{x} - 16000 \, e^{12} + 140800 \, e^{8} - 327680 \, e^{4} - 2500 \, e^{\left (x + 16\right )} + 16000 \, e^{\left (x + 12\right )} - 38400 \, e^{\left (x + 8\right )} + 40960 \, e^{\left (x + 4\right )} - 16384 \, e^{x} + 229376\right )}}{3125 \, x e^{23} - 25000 \, x e^{19} + 80000 \, x e^{15} - 128000 \, x e^{11} + 102400 \, x e^{7} - 32768 \, x e^{3} + 12500 \, e^{19} - 80000 \, e^{15} + 192000 \, e^{11} - 204800 \, e^{7} + 81920 \, e^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((15*x^3+30*x^2)*exp(4)^2+(-24*x^3+72*x^2+240*x)*exp(4)-96*x^2+144*x+576)*exp(x)-60*x^2*exp(4)^2+(9
6*x^2-480*x)*exp(4)-1344)/(25*x^2*exp(3)*exp(4)^2+(-80*x^2+200*x)*exp(3)*exp(4)+(64*x^2-320*x+400)*exp(3)),x,
algorithm="giac")

[Out]

-3*(2500*x^2*e^20 - 16000*x^2*e^16 + 38400*x^2*e^12 - 40960*x^2*e^8 + 16384*x^2*e^4 - 625*x^2*e^(x + 20) + 400
0*x^2*e^(x + 16) - 9600*x^2*e^(x + 12) + 10240*x^2*e^(x + 8) - 4096*x^2*e^(x + 4) + 10000*x*e^16 - 48000*x*e^1
2 + 76800*x*e^8 - 40960*x*e^4 - 625*x*e^(x + 20) + 1500*x*e^(x + 16) + 6400*x*e^(x + 12) - 28160*x*e^(x + 8) +
36864*x*e^(x + 4) - 16384*x*e^x - 16000*e^12 + 140800*e^8 - 327680*e^4 - 2500*e^(x + 16) + 16000*e^(x + 12) -
38400*e^(x + 8) + 40960*e^(x + 4) - 16384*e^x + 229376)/(3125*x*e^23 - 25000*x*e^19 + 80000*x*e^15 - 128000*x
*e^11 + 102400*x*e^7 - 32768*x*e^3 + 12500*e^19 - 80000*e^15 + 192000*e^11 - 204800*e^7 + 81920*e^3)

________________________________________________________________________________________

maple [B]  time = 10.35, size = 67, normalized size = 2.23

method	result	size
norman	\(\frac {12 \,{\mathrm e}^{-3} {\mathrm e}^{x}-\frac {336 x \,{\mathrm e}^{-3}}{5}-12 \,{\mathrm e}^{-3} {\mathrm e}^{4} x^{2}+3 \left (4+{\mathrm e}^{4}\right ) {\mathrm e}^{-3} x \,{\mathrm e}^{x}+3 \,{\mathrm e}^{-3} {\mathrm e}^{4} x^{2} {\mathrm e}^{x}}{5 x \,{\mathrm e}^{4}-8 x +20}\)	\(67\)
risch	\(-\frac {12 \,{\mathrm e} x}{5 \,{\mathrm e}^{4}-8}+\frac {1920 \,{\mathrm e}^{-3} {\mathrm e}^{4}}{\left (5 \,{\mathrm e}^{4}-8\right )^{2} \left (5 x \,{\mathrm e}^{4}-8 x +20\right )}-\frac {10752 \,{\mathrm e}^{-3}}{\left (5 \,{\mathrm e}^{4}-8\right )^{2} \left (5 x \,{\mathrm e}^{4}-8 x +20\right )}+\frac {3 \left (x^{2} {\mathrm e}^{4}+x \,{\mathrm e}^{4}+4 x +4\right ) {\mathrm e}^{x -3}}{5 x \,{\mathrm e}^{4}-8 x +20}\)	\(98\)
default	\(-\frac {336 \,{\mathrm e}^{-3} x}{5 \left (5 x \,{\mathrm e}^{4}-8 x +20\right )}-\frac {576 \,{\mathrm e}^{x} {\mathrm e}^{-3}}{\left (5 \,{\mathrm e}^{4}-8\right ) \left (5 x \,{\mathrm e}^{4}-8 x +20\right )}-\frac {576 \,{\mathrm e}^{-3} {\mathrm e}^{-\frac {20}{5 \,{\mathrm e}^{4}-8}} \expIntegralEi \left (1, -x -\frac {20}{5 \,{\mathrm e}^{4}-8}\right )}{\left (5 \,{\mathrm e}^{4}-8\right )^{2}}+\frac {480 \,{\mathrm e}^{4} {\mathrm e}^{-3} x}{\left (5 \,{\mathrm e}^{4}-8\right ) \left (5 x \,{\mathrm e}^{4}-8 x +20\right )}-\frac {480 \,{\mathrm e}^{-3} {\mathrm e}^{4} \ln \left (5 x \,{\mathrm e}^{4}-8 x +20\right )}{25 \,{\mathrm e}^{8}-80 \,{\mathrm e}^{4}+64}+\frac {\frac {3840 \,{\mathrm e}^{4} {\mathrm e}^{-3} x}{25 \,{\mathrm e}^{8}-80 \,{\mathrm e}^{4}+64}+\frac {96 \,{\mathrm e}^{4} {\mathrm e}^{-3} x^{2}}{5 \,{\mathrm e}^{4}-8}}{5 x \,{\mathrm e}^{4}-8 x +20}-\frac {3840 \,{\mathrm e}^{-3} {\mathrm e}^{4} \ln \left (5 x \,{\mathrm e}^{4}-8 x +20\right )}{125 \,{\mathrm e}^{12}-600 \,{\mathrm e}^{8}+960 \,{\mathrm e}^{4}-512}+\frac {-\frac {2400 \,{\mathrm e}^{8} {\mathrm e}^{-3} x}{25 \,{\mathrm e}^{8}-80 \,{\mathrm e}^{4}+64}-\frac {60 \,{\mathrm e}^{8} {\mathrm e}^{-3} x^{2}}{5 \,{\mathrm e}^{4}-8}}{5 x \,{\mathrm e}^{4}-8 x +20}+\frac {2400 \,{\mathrm e}^{-3} {\mathrm e}^{8} \ln \left (5 x \,{\mathrm e}^{4}-8 x +20\right )}{125 \,{\mathrm e}^{12}-600 \,{\mathrm e}^{8}+960 \,{\mathrm e}^{4}-512}+\frac {2880 \,{\mathrm e}^{x} {\mathrm e}^{-3}}{\left (25 \,{\mathrm e}^{8}-80 \,{\mathrm e}^{4}+64\right ) \left (5 x \,{\mathrm e}^{4}-8 x +20\right )}-\frac {144 \left (5 \,{\mathrm e}^{4}-28\right ) {\mathrm e}^{-3} {\mathrm e}^{-\frac {20}{5 \,{\mathrm e}^{4}-8}} \expIntegralEi \left (1, -x -\frac {20}{5 \,{\mathrm e}^{4}-8}\right )}{\left (25 \,{\mathrm e}^{8}-80 \,{\mathrm e}^{4}+64\right ) \left (5 \,{\mathrm e}^{4}-8\right )}-\frac {96 \,{\mathrm e}^{x}}{25 \,{\mathrm e}^{3} {\mathrm e}^{8}-80 \,{\mathrm e}^{3} {\mathrm e}^{4}+64 \,{\mathrm e}^{3}}+\frac {38400 \,{\mathrm e}^{x} {\mathrm e}^{-3}}{\left (5 \,{\mathrm e}^{4}-8\right ) \left (25 \,{\mathrm e}^{8}-80 \,{\mathrm e}^{4}+64\right ) \left (5 x \,{\mathrm e}^{4}-8 x +20\right )}-\frac {3840 \left (5 \,{\mathrm e}^{4}-18\right ) {\mathrm e}^{-\frac {20}{5 \,{\mathrm e}^{4}-8}} \expIntegralEi \left (1, -x -\frac {20}{5 \,{\mathrm e}^{4}-8}\right )}{\left (5 \,{\mathrm e}^{4}-8\right )^{2} \left (25 \,{\mathrm e}^{3} {\mathrm e}^{8}-80 \,{\mathrm e}^{3} {\mathrm e}^{4}+64 \,{\mathrm e}^{3}\right )}+240 \,{\mathrm e}^{4} \left (\frac {20 \,{\mathrm e}^{x} {\mathrm e}^{-3}}{\left (25 \,{\mathrm e}^{8}-80 \,{\mathrm e}^{4}+64\right ) \left (5 x \,{\mathrm e}^{4}-8 x +20\right )}-\frac {\left (5 \,{\mathrm e}^{4}-28\right ) {\mathrm e}^{-3} {\mathrm e}^{-\frac {20}{5 \,{\mathrm e}^{4}-8}} \expIntegralEi \left (1, -x -\frac {20}{5 \,{\mathrm e}^{4}-8}\right )}{\left (25 \,{\mathrm e}^{8}-80 \,{\mathrm e}^{4}+64\right ) \left (5 \,{\mathrm e}^{4}-8\right )}\right )+72 \,{\mathrm e}^{4} \left (\frac {{\mathrm e}^{x}}{25 \,{\mathrm e}^{3} {\mathrm e}^{8}-80 \,{\mathrm e}^{3} {\mathrm e}^{4}+64 \,{\mathrm e}^{3}}-\frac {400 \,{\mathrm e}^{x} {\mathrm e}^{-3}}{\left (5 \,{\mathrm e}^{4}-8\right ) \left (25 \,{\mathrm e}^{8}-80 \,{\mathrm e}^{4}+64\right ) \left (5 x \,{\mathrm e}^{4}-8 x +20\right )}+\frac {40 \left (5 \,{\mathrm e}^{4}-18\right ) {\mathrm e}^{-\frac {20}{5 \,{\mathrm e}^{4}-8}} \expIntegralEi \left (1, -x -\frac {20}{5 \,{\mathrm e}^{4}-8}\right )}{\left (5 \,{\mathrm e}^{4}-8\right )^{2} \left (25 \,{\mathrm e}^{3} {\mathrm e}^{8}-80 \,{\mathrm e}^{3} {\mathrm e}^{4}+64 \,{\mathrm e}^{3}\right )}\right )-24 \,{\mathrm e}^{4} \left (\frac {\left (5 x \,{\mathrm e}^{4}-5 \,{\mathrm e}^{4}-8 x -32\right ) {\mathrm e}^{x}}{\left (25 \,{\mathrm e}^{3} {\mathrm e}^{8}-80 \,{\mathrm e}^{3} {\mathrm e}^{4}+64 \,{\mathrm e}^{3}\right ) \left (5 \,{\mathrm e}^{4}-8\right )}+\frac {8000 \,{\mathrm e}^{x} {\mathrm e}^{-3}}{\left (5 \,{\mathrm e}^{4}-8\right )^{2} \left (25 \,{\mathrm e}^{8}-80 \,{\mathrm e}^{4}+64\right ) \left (5 x \,{\mathrm e}^{4}-8 x +20\right )}-\frac {400 \left (15 \,{\mathrm e}^{4}-44\right ) {\mathrm e}^{-\frac {20}{5 \,{\mathrm e}^{4}-8}} \expIntegralEi \left (1, -x -\frac {20}{5 \,{\mathrm e}^{4}-8}\right )}{\left (5 \,{\mathrm e}^{4}-8\right )^{3} \left (25 \,{\mathrm e}^{3} {\mathrm e}^{8}-80 \,{\mathrm e}^{3} {\mathrm e}^{4}+64 \,{\mathrm e}^{3}\right )}\right )+30 \,{\mathrm e}^{8} \left (\frac {{\mathrm e}^{x}}{25 \,{\mathrm e}^{3} {\mathrm e}^{8}-80 \,{\mathrm e}^{3} {\mathrm e}^{4}+64 \,{\mathrm e}^{3}}-\frac {400 \,{\mathrm e}^{x} {\mathrm e}^{-3}}{\left (5 \,{\mathrm e}^{4}-8\right ) \left (25 \,{\mathrm e}^{8}-80 \,{\mathrm e}^{4}+64\right ) \left (5 x \,{\mathrm e}^{4}-8 x +20\right )}+\frac {40 \left (5 \,{\mathrm e}^{4}-18\right ) {\mathrm e}^{-\frac {20}{5 \,{\mathrm e}^{4}-8}} \expIntegralEi \left (1, -x -\frac {20}{5 \,{\mathrm e}^{4}-8}\right )}{\left (5 \,{\mathrm e}^{4}-8\right )^{2} \left (25 \,{\mathrm e}^{3} {\mathrm e}^{8}-80 \,{\mathrm e}^{3} {\mathrm e}^{4}+64 \,{\mathrm e}^{3}\right )}\right )+15 \,{\mathrm e}^{8} \left (\frac {\left (5 x \,{\mathrm e}^{4}-5 \,{\mathrm e}^{4}-8 x -32\right ) {\mathrm e}^{x}}{\left (25 \,{\mathrm e}^{3} {\mathrm e}^{8}-80 \,{\mathrm e}^{3} {\mathrm e}^{4}+64 \,{\mathrm e}^{3}\right ) \left (5 \,{\mathrm e}^{4}-8\right )}+\frac {8000 \,{\mathrm e}^{x} {\mathrm e}^{-3}}{\left (5 \,{\mathrm e}^{4}-8\right )^{2} \left (25 \,{\mathrm e}^{8}-80 \,{\mathrm e}^{4}+64\right ) \left (5 x \,{\mathrm e}^{4}-8 x +20\right )}-\frac {400 \left (15 \,{\mathrm e}^{4}-44\right ) {\mathrm e}^{-\frac {20}{5 \,{\mathrm e}^{4}-8}} \expIntegralEi \left (1, -x -\frac {20}{5 \,{\mathrm e}^{4}-8}\right )}{\left (5 \,{\mathrm e}^{4}-8\right )^{3} \left (25 \,{\mathrm e}^{3} {\mathrm e}^{8}-80 \,{\mathrm e}^{3} {\mathrm e}^{4}+64 \,{\mathrm e}^{3}\right )}\right )\)	\(1252\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((15*x^3+30*x^2)*exp(4)^2+(-24*x^3+72*x^2+240*x)*exp(4)-96*x^2+144*x+576)*exp(x)-60*x^2*exp(4)^2+(96*x^2-
480*x)*exp(4)-1344)/(25*x^2*exp(3)*exp(4)^2+(-80*x^2+200*x)*exp(3)*exp(4)+(64*x^2-320*x+400)*exp(3)),x,method=
_RETURNVERBOSE)

[Out]

(12*exp(x)/exp(3)-336/5*x/exp(3)-12/exp(3)*exp(4)*x^2+3*(4+exp(4))/exp(3)*x*exp(x)+3/exp(3)*exp(4)*x^2*exp(x))
/(5*x*exp(4)-8*x+20)

________________________________________________________________________________________

maxima [B]  time = 0.49, size = 329, normalized size = 10.97 \begin {gather*} -60 \, {\left (\frac {x}{25 \, e^{11} - 80 \, e^{7} + 64 \, e^{3}} - \frac {40 \, \log \left (x {\left (5 \, e^{4} - 8\right )} + 20\right )}{125 \, e^{15} - 600 \, e^{11} + 960 \, e^{7} - 512 \, e^{3}} - \frac {400}{x {\left (625 \, e^{19} - 4000 \, e^{15} + 9600 \, e^{11} - 10240 \, e^{7} + 4096 \, e^{3}\right )} + 2500 \, e^{15} - 12000 \, e^{11} + 19200 \, e^{7} - 10240 \, e^{3}}\right )} e^{8} + 96 \, {\left (\frac {x}{25 \, e^{11} - 80 \, e^{7} + 64 \, e^{3}} - \frac {40 \, \log \left (x {\left (5 \, e^{4} - 8\right )} + 20\right )}{125 \, e^{15} - 600 \, e^{11} + 960 \, e^{7} - 512 \, e^{3}} - \frac {400}{x {\left (625 \, e^{19} - 4000 \, e^{15} + 9600 \, e^{11} - 10240 \, e^{7} + 4096 \, e^{3}\right )} + 2500 \, e^{15} - 12000 \, e^{11} + 19200 \, e^{7} - 10240 \, e^{3}}\right )} e^{4} - 480 \, {\left (\frac {\log \left (x {\left (5 \, e^{4} - 8\right )} + 20\right )}{25 \, e^{11} - 80 \, e^{7} + 64 \, e^{3}} + \frac {20}{x {\left (125 \, e^{15} - 600 \, e^{11} + 960 \, e^{7} - 512 \, e^{3}\right )} + 500 \, e^{11} - 1600 \, e^{7} + 1280 \, e^{3}}\right )} e^{4} + \frac {3 \, {\left (x^{2} e^{4} + x {\left (e^{4} + 4\right )} + 4\right )} e^{x}}{x {\left (5 \, e^{7} - 8 \, e^{3}\right )} + 20 \, e^{3}} + \frac {1344}{x {\left (25 \, e^{11} - 80 \, e^{7} + 64 \, e^{3}\right )} + 100 \, e^{7} - 160 \, e^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((15*x^3+30*x^2)*exp(4)^2+(-24*x^3+72*x^2+240*x)*exp(4)-96*x^2+144*x+576)*exp(x)-60*x^2*exp(4)^2+(9
6*x^2-480*x)*exp(4)-1344)/(25*x^2*exp(3)*exp(4)^2+(-80*x^2+200*x)*exp(3)*exp(4)+(64*x^2-320*x+400)*exp(3)),x,
algorithm="maxima")

[Out]

-60*(x/(25*e^11 - 80*e^7 + 64*e^3) - 40*log(x*(5*e^4 - 8) + 20)/(125*e^15 - 600*e^11 + 960*e^7 - 512*e^3) - 40
0/(x*(625*e^19 - 4000*e^15 + 9600*e^11 - 10240*e^7 + 4096*e^3) + 2500*e^15 - 12000*e^11 + 19200*e^7 - 10240*e^
3))*e^8 + 96*(x/(25*e^11 - 80*e^7 + 64*e^3) - 40*log(x*(5*e^4 - 8) + 20)/(125*e^15 - 600*e^11 + 960*e^7 - 512*
e^3) - 400/(x*(625*e^19 - 4000*e^15 + 9600*e^11 - 10240*e^7 + 4096*e^3) + 2500*e^15 - 12000*e^11 + 19200*e^7 -
10240*e^3))*e^4 - 480*(log(x*(5*e^4 - 8) + 20)/(25*e^11 - 80*e^7 + 64*e^3) + 20/(x*(125*e^15 - 600*e^11 + 960
*e^7 - 512*e^3) + 500*e^11 - 1600*e^7 + 1280*e^3))*e^4 + 3*(x^2*e^4 + x*(e^4 + 4) + 4)*e^x/(x*(5*e^7 - 8*e^3)
+ 20*e^3) + 1344/(x*(25*e^11 - 80*e^7 + 64*e^3) + 100*e^7 - 160*e^3)

________________________________________________________________________________________

mupad [B]  time = 7.29, size = 130, normalized size = 4.33 \begin {gather*} \frac {384\,\left (5\,{\mathrm {e}}^4-28\right )}{\left (5\,{\mathrm {e}}^4-8\right )\,\left (100\,{\mathrm {e}}^7-160\,{\mathrm {e}}^3+x\,\left (64\,{\mathrm {e}}^3-80\,{\mathrm {e}}^7+25\,{\mathrm {e}}^{11}\right )\right )}-\frac {12\,x\,\mathrm {e}}{5\,{\mathrm {e}}^4-8}-\frac {{\mathrm {e}}^x\,\left (\frac {3\,{\mathrm {e}}^4\,x^2}{8\,{\mathrm {e}}^3-5\,{\mathrm {e}}^7}+\frac {\left (3\,{\mathrm {e}}^4+12\right )\,x}{8\,{\mathrm {e}}^3-5\,{\mathrm {e}}^7}+\frac {12}{8\,{\mathrm {e}}^3-5\,{\mathrm {e}}^7}\right )}{x-\frac {20\,{\mathrm {e}}^3}{8\,{\mathrm {e}}^3-5\,{\mathrm {e}}^7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(4)*(480*x - 96*x^2) + 60*x^2*exp(8) - exp(x)*(144*x + exp(4)*(240*x + 72*x^2 - 24*x^3) + exp(8)*(30*
x^2 + 15*x^3) - 96*x^2 + 576) + 1344)/(exp(7)*(200*x - 80*x^2) + exp(3)*(64*x^2 - 320*x + 400) + 25*x^2*exp(11
)),x)

[Out]

(384*(5*exp(4) - 28))/((5*exp(4) - 8)*(100*exp(7) - 160*exp(3) + x*(64*exp(3) - 80*exp(7) + 25*exp(11)))) - (1
2*x*exp(1))/(5*exp(4) - 8) - (exp(x)*(12/(8*exp(3) - 5*exp(7)) + (x*(3*exp(4) + 12))/(8*exp(3) - 5*exp(7)) + (
3*x^2*exp(4))/(8*exp(3) - 5*exp(7))))/(x - (20*exp(3))/(8*exp(3) - 5*exp(7)))

________________________________________________________________________________________

sympy [B]  time = 0.57, size = 99, normalized size = 3.30 \begin {gather*} - \frac {12 e x}{-8 + 5 e^{4}} + \frac {-10752 + 1920 e^{4}}{x \left (- 600 e^{11} - 512 e^{3} + 960 e^{7} + 125 e^{15}\right ) - 1600 e^{7} + 1280 e^{3} + 500 e^{11}} + \frac {\left (3 x^{2} e^{4} + 12 x + 3 x e^{4} + 12\right ) e^{x}}{- 8 x e^{3} + 5 x e^{7} + 20 e^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((15*x**3+30*x**2)*exp(4)**2+(-24*x**3+72*x**2+240*x)*exp(4)-96*x**2+144*x+576)*exp(x)-60*x**2*exp(
4)**2+(96*x**2-480*x)*exp(4)-1344)/(25*x**2*exp(3)*exp(4)**2+(-80*x**2+200*x)*exp(3)*exp(4)+(64*x**2-320*x+400
)*exp(3)),x)

[Out]

-12*E*x/(-8 + 5*exp(4)) + (-10752 + 1920*exp(4))/(x*(-600*exp(11) - 512*exp(3) + 960*exp(7) + 125*exp(15)) - 1
600*exp(7) + 1280*exp(3) + 500*exp(11)) + (3*x**2*exp(4) + 12*x + 3*x*exp(4) + 12)*exp(x)/(-8*x*exp(3) + 5*x*e
xp(7) + 20*exp(3))

________________________________________________________________________________________