∫ e2x(120−480x+400x2−40x4+e2∕3(−8+24x−24x2+8x3))________________________________________ 64000−72000x+3000x2+14625x3−375x4−1125x5−125x6+e2(−1+3x−3x2+x3)+e4∕3(120−285x+195x2−15x3−15x4)+e2∕3(−4800+8400x−3075x2−975x3+375x4+75x5) dx [1522]

3.16.22 $\int \frac {e^{2 x} (120-480 x+400 x^2-40 x^4+e^{2/3} (-8+24 x-24 x^2+8 x^3))}{64000-72000 x+3000 x^2+14625 x^3-375 x^4-1125 x^5-125 x^6+e^2 (-1+3 x-3 x^2+x^3)+e^{4/3} (120-285 x+195 x^2-15 x^3-15 x^4)+e^{2/3} (-4800+8400 x-3075 x^2-975 x^3+375 x^4+75 x^5)} \, dx$ [1522]

Optimal. Leaf size=32 \[ \frac {4 e^{2 x}}{\left (e^{2/3}-5 \left (3-\frac {5-x}{-1+x}+x\right )\right )^2} \]

[Out]

4*exp(x)^2/(exp(2/3)-15+5*(5-x)/(-1+x)-5*x)^2

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Rubi [C] Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
time = 5.09, antiderivative size = 3777, normalized size of antiderivative = 118.03, number of steps used = 56, number of rules used = 6, integrand size = 148, $\frac {\text {number of rules}}{\text {integrand size}}$ = 0.041, Rules used = {6820, 12, 6874, 2208, 2209, 2300} \begin {gather*} \text {Too large to display} \end {gather*}

Antiderivative was successfully veriﬁed.

[In]

Int[(E^(2*x)*(120 - 480*x + 400*x^2 - 40*x^4 + E^(2/3)*(-8 + 24*x - 24*x^2 + 8*x^3)))/(64000 - 72000*x + 3000*

x^2 + 14625*x^3 - 375*x^4 - 1125*x^5 - 125*x^6 + E^2*(-1 + 3*x - 3*x^2 + x^3) + E^(4/3)*(120 - 285*x + 195*x^2

- 15*x^3 - 15*x^4) + E^(2/3)*(-4800 + 8400*x - 3075*x^2 - 975*x^3 + 375*x^4 + 75*x^5)),x]

[Out]

(80*(70 - 1325/E^(2/3) - E^(2/3))*E^(2/3 + 2*x))/((1025 - 50*E^(2/3) + E^(4/3))^(3/2)*(15 - E^(2/3) - Sqrt[102

5 - 50*E^(2/3) + E^(4/3)] + 10*x)^2) - (8*E^(2*x)*(825 - 50*E^(2/3) + E^(4/3))*(15 - E^(2/3) - Sqrt[1025 - 50*

E^(2/3) + E^(4/3)]))/((1025 - 50*E^(2/3) + E^(4/3))^(3/2)*(15 - E^(2/3) - Sqrt[1025 - 50*E^(2/3) + E^(4/3)] +

10*x)^2) - (240*(70 - 1325/E^(2/3) - E^(2/3))*E^(2/3 + 2*x))/((1025 - 50*E^(2/3) + E^(4/3))^2*(15 - E^(2/3) -

Sqrt[1025 - 50*E^(2/3) + E^(4/3)] + 10*x)) + (16*(70 - 1325/E^(2/3) - E^(2/3))*E^(2/3 + 2*x))/((1025 - 50*E^(2

/3) + E^(4/3))^(3/2)*(15 - E^(2/3) - Sqrt[1025 - 50*E^(2/3) + E^(4/3)] + 10*x)) - (16*(75 - 2*E^(2/3))*E^(2*x)

)/((1025 - 50*E^(2/3) + E^(4/3))*(15 - E^(2/3) - Sqrt[1025 - 50*E^(2/3) + E^(4/3)] + 10*x)) + (8*E^(2*x)*(825

- 50*E^(2/3) + E^(4/3))*(45 - 3*E^(2/3) - Sqrt[1025 - 50*E^(2/3) + E^(4/3)]))/((1025 - 50*E^(2/3) + E^(4/3))^2

*(15 - E^(2/3) - Sqrt[1025 - 50*E^(2/3) + E^(4/3)] + 10*x)) - (8*E^(2*x)*(825 - 50*E^(2/3) + E^(4/3))*(15 - E^

(2/3) - Sqrt[1025 - 50*E^(2/3) + E^(4/3)]))/(5*(1025 - 50*E^(2/3) + E^(4/3))^(3/2)*(15 - E^(2/3) - Sqrt[1025 -

50*E^(2/3) + E^(4/3)] + 10*x)) - (8*(30 - E^(2/3))*E^(2*x)*(15 - E^(2/3) - Sqrt[1025 - 50*E^(2/3) + E^(4/3)])

)/(5*(1025 - 50*E^(2/3) + E^(4/3))*(15 - E^(2/3) - Sqrt[1025 - 50*E^(2/3) + E^(4/3)] + 10*x)) - (80*(70 - 1325

/E^(2/3) - E^(2/3))*E^(2/3 + 2*x))/((1025 - 50*E^(2/3) + E^(4/3))^(3/2)*(15 - E^(2/3) + Sqrt[1025 - 50*E^(2/3)

+ E^(4/3)] + 10*x)^2) + (8*E^(2*x)*(825 - 50*E^(2/3) + E^(4/3))*(15 - E^(2/3) + Sqrt[1025 - 50*E^(2/3) + E^(4

/3)]))/((1025 - 50*E^(2/3) + E^(4/3))^(3/2)*(15 - E^(2/3) + Sqrt[1025 - 50*E^(2/3) + E^(4/3)] + 10*x)^2) - (24

0*(70 - 1325/E^(2/3) - E^(2/3))*E^(2/3 + 2*x))/((1025 - 50*E^(2/3) + E^(4/3))^2*(15 - E^(2/3) + Sqrt[1025 - 50

*E^(2/3) + E^(4/3)] + 10*x)) - (16*(70 - 1325/E^(2/3) - E^(2/3))*E^(2/3 + 2*x))/((1025 - 50*E^(2/3) + E^(4/3))

^(3/2)*(15 - E^(2/3) + Sqrt[1025 - 50*E^(2/3) + E^(4/3)] + 10*x)) - (16*(75 - 2*E^(2/3))*E^(2*x))/((1025 - 50*

E^(2/3) + E^(4/3))*(15 - E^(2/3) + Sqrt[1025 - 50*E^(2/3) + E^(4/3)] + 10*x)) + (8*E^(2*x)*(825 - 50*E^(2/3) +

E^(4/3))*(45 - 3*E^(2/3) + Sqrt[1025 - 50*E^(2/3) + E^(4/3)]))/((1025 - 50*E^(2/3) + E^(4/3))^2*(15 - E^(2/3)

+ Sqrt[1025 - 50*E^(2/3) + E^(4/3)] + 10*x)) + (8*E^(2*x)*(825 - 50*E^(2/3) + E^(4/3))*(15 - E^(2/3) + Sqrt[1

025 - 50*E^(2/3) + E^(4/3)]))/(5*(1025 - 50*E^(2/3) + E^(4/3))^(3/2)*(15 - E^(2/3) + Sqrt[1025 - 50*E^(2/3) +

E^(4/3)] + 10*x)) - (8*(30 - E^(2/3))*E^(2*x)*(15 - E^(2/3) + Sqrt[1025 - 50*E^(2/3) + E^(4/3)]))/(5*(1025 - 5

0*E^(2/3) + E^(4/3))*(15 - E^(2/3) + Sqrt[1025 - 50*E^(2/3) + E^(4/3)] + 10*x)) + (240*E^((-15 + E^(2/3) + Sqr

t[1025 - 50*E^(2/3) + E^(4/3)])/5)*(1325 - 70*E^(2/3) + E^(4/3))*ExpIntegralEi[(15 - E^(2/3) - Sqrt[1025 - 50*

E^(2/3) + E^(4/3)] + 10*x)/5])/(1025 - 50*E^(2/3) + E^(4/3))^(5/2) + (24*(15 - E^(2/3))*E^((-15 + E^(2/3) + Sq

rt[1025 - 50*E^(2/3) + E^(4/3)])/5)*(825 - 50*E^(2/3) + E^(4/3))*ExpIntegralEi[(15 - E^(2/3) - Sqrt[1025 - 50*

E^(2/3) + E^(4/3)] + 10*x)/5])/(1025 - 50*E^(2/3) + E^(4/3))^(5/2) - (48*E^((-15 + E^(2/3) + Sqrt[1025 - 50*E^

(2/3) + E^(4/3)])/5)*(1325 - 70*E^(2/3) + E^(4/3))*ExpIntegralEi[(15 - E^(2/3) - Sqrt[1025 - 50*E^(2/3) + E^(4

/3)] + 10*x)/5])/(1025 - 50*E^(2/3) + E^(4/3))^2 - (16*(75 - 2*E^(2/3))*E^((-15 + E^(2/3) + Sqrt[1025 - 50*E^(

2/3) + E^(4/3)])/5)*ExpIntegralEi[(15 - E^(2/3) - Sqrt[1025 - 50*E^(2/3) + E^(4/3)] + 10*x)/5])/(1025 - 50*E^(

2/3) + E^(4/3))^(3/2) - (8*(15 - E^(2/3))*(30 - E^(2/3))*E^((-15 + E^(2/3) + Sqrt[1025 - 50*E^(2/3) + E^(4/3)]

)/5)*ExpIntegralEi[(15 - E^(2/3) - Sqrt[1025 - 50*E^(2/3) + E^(4/3)] + 10*x)/5])/(5*(1025 - 50*E^(2/3) + E^(4/

3))^(3/2)) + (16*E^((-15 + E^(2/3) + Sqrt[1025 - 50*E^(2/3) + E^(4/3)])/5)*(1325 - 70*E^(2/3) + E^(4/3))*ExpIn

tegralEi[(15 - E^(2/3) - Sqrt[1025 - 50*E^(2/3) + E^(4/3)] + 10*x)/5])/(5*(1025 - 50*E^(2/3) + E^(4/3))^(3/2))

+ (16*(75 - 2*E^(2/3))*E^((-15 + E^(2/3) + Sqrt[1025 - 50*E^(2/3) + E^(4/3)])/5)*ExpIntegralEi[(15 - E^(2/3)

- Sqrt[1025 - 50*E^(2/3) + E^(4/3)] + 10*x)/5])/(5*(1025 - 50*E^(2/3) + E^(4/3))) + (8*E^((-15 + E^(2/3) + Sqr

t[1025 - 50*E^(2/3) + E^(4/3)])/5)*ExpIntegralEi[(15 - E^(2/3) - Sqrt[1025 - 50*E^(2/3) + E^(4/3)] + 10*x)/5])

/(5*Sqrt[1025 - 50*E^(2/3) + E^(4/3)]) - (8*E^((-15 + E^(2/3) + Sqrt[1025 - 50*E^(2/3) + E^(4/3)])/5)*(825 - 5

0*E^(2/3) + E^(4/3))*(45 - 3*E^(2/3) - Sqrt[1025 - 50*E^(2/3) + E^(4/3)])*ExpIntegralEi[(15 - E^(2/3) - Sqrt[1

025 - 50*E^(2/3) + E^(4/3)] + 10*x)/5])/(5*(1025 - 50*E^(2/3) + E^(4/3))^2) + (8*E^((-15 + E^(2/3) + Sqrt[1025

- 50*E^(2/3) + E^(4/3)])/5)*(825 - 50*E^(2/3) + E^(4/3))*(15 - E^(2/3) - Sqrt[1025 - 50*E^(2/3) + E^(4/3)])*E

xpIntegralEi[(15 - E^(2/3) - Sqrt[1025 - 50*E^(2/3) + E^(4/3)] + 10*x)/5])/(25*(1025 - 50*E^(2/3) + E^(4/3))^(

3/2)) + (8*(30 - E^(2/3))*E^((-15 + E^(2/3) + Sqrt[1025 - 50*E^(2/3) + E^(4/3)])/5)*(15 - E^(2/3) - Sqrt[1025

- 50*E^(2/3) + E^(4/3)])*ExpIntegralEi[(15 - E^(2/3) - Sqrt[1025 - 50*E^(2/3) + E^(4/3)] + 10*x)/5])/(25*(1025

- 50*E^(2/3) + E^(4/3))) - (240*E^((-15 + E^(2...

Rule 12

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

Rule 2208

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] && !TrueQ[$UseGamm

Rule 2209

Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[(F^(g*(e - c*(f/d)))/d)*ExpInteg

ralEi[f*g*(c + d*x)*(Log[F]/d)], x] /; FreeQ[{F, c, d, e, f, g}, x] && !TrueQ[$UseGamma]

Rule 2300

Int[(F_)^((g_.)*((d_.) + (e_.)*(x_))^(n_.))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Int[ExpandIntegr

and[F^(g*(d + e*x)^n), 1/(a + b*x + c*x^2), x], x] /; FreeQ[{F, a, b, c, d, e, g, n}, x]

Rule 6820

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

Rule 6874

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

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {8 e^{2 x} (1-x) \left (15-e^{2/3}-\left (45-2 e^{2/3}\right ) x+\left (5-e^{2/3}\right ) x^2+5 x^3\right )}{\left (40-e^{2/3}-\left (15-e^{2/3}\right ) x-5 x^2\right )^3} \, dx\\ &=8 \int \frac {e^{2 x} (1-x) \left (15-e^{2/3}-\left (45-2 e^{2/3}\right ) x+\left (5-e^{2/3}\right ) x^2+5 x^3\right )}{\left (40-e^{2/3}-\left (15-e^{2/3}\right ) x-5 x^2\right )^3} \, dx\\ &=8 \int \left (\frac {e^{2 x} \left (-1325+70 e^{2/3}-e^{4/3}+\left (825-50 e^{2/3}+e^{4/3}\right ) x\right )}{5 \left (40-e^{2/3}-\left (15-e^{2/3}\right ) x-5 x^2\right )^3}+\frac {e^{2 x} \left (75-2 e^{2/3}-\left (30-e^{2/3}\right ) x\right )}{5 \left (40-e^{2/3}-\left (15-e^{2/3}\right ) x-5 x^2\right )^2}+\frac {e^{2 x}}{5 \left (-40+e^{2/3}+\left (15-e^{2/3}\right ) x+5 x^2\right )}\right ) \, dx\\ &=\frac {8}{5} \int \frac {e^{2 x} \left (-1325+70 e^{2/3}-e^{4/3}+\left (825-50 e^{2/3}+e^{4/3}\right ) x\right )}{\left (40-e^{2/3}-\left (15-e^{2/3}\right ) x-5 x^2\right )^3} \, dx+\frac {8}{5} \int \frac {e^{2 x} \left (75-2 e^{2/3}-\left (30-e^{2/3}\right ) x\right )}{\left (40-e^{2/3}-\left (15-e^{2/3}\right ) x-5 x^2\right )^2} \, dx+\frac {8}{5} \int \frac {e^{2 x}}{-40+e^{2/3}+\left (15-e^{2/3}\right ) x+5 x^2} \, dx\\ &=\frac {8}{5} \int \left (-\frac {10 e^{2 x}}{\sqrt {1025-50 e^{2/3}+e^{4/3}} \left (-15+e^{2/3}+\sqrt {1025-50 e^{2/3}+e^{4/3}}-10 x\right )}-\frac {10 e^{2 x}}{\sqrt {1025-50 e^{2/3}+e^{4/3}} \left (15-e^{2/3}+\sqrt {1025-50 e^{2/3}+e^{4/3}}+10 x\right )}\right ) \, dx+\frac {8}{5} \int \left (\frac {70 e^{\frac {2}{3}+2 x} \left (1+\frac {-1325-e^{4/3}}{70 e^{2/3}}\right )}{\left (40-e^{2/3}-\left (15-e^{2/3}\right ) x-5 x^2\right )^3}+\frac {e^{2 x} \left (825-50 e^{2/3}+e^{4/3}\right ) x}{\left (40-e^{2/3}-\left (15-e^{2/3}\right ) x-5 x^2\right )^3}\right ) \, dx+\frac {8}{5} \int \left (\frac {75 \left (1-\frac {2 e^{2/3}}{75}\right ) e^{2 x}}{\left (40-e^{2/3}-\left (15-e^{2/3}\right ) x-5 x^2\right )^2}+\frac {\left (-30+e^{2/3}\right ) e^{2 x} x}{\left (40-e^{2/3}-\left (15-e^{2/3}\right ) x-5 x^2\right )^2}\right ) \, dx\\ &=\frac {1}{5} \left (8 \left (75-2 e^{2/3}\right )\right ) \int \frac {e^{2 x}}{\left (40-e^{2/3}-\left (15-e^{2/3}\right ) x-5 x^2\right )^2} \, dx-\frac {1}{5} \left (8 \left (30-e^{2/3}\right )\right ) \int \frac {e^{2 x} x}{\left (40-e^{2/3}-\left (15-e^{2/3}\right ) x-5 x^2\right )^2} \, dx+\frac {1}{5} \left (8 \left (70-\frac {1325}{e^{2/3}}-e^{2/3}\right )\right ) \int \frac {e^{\frac {2}{3}+2 x}}{\left (40-e^{2/3}-\left (15-e^{2/3}\right ) x-5 x^2\right )^3} \, dx+\frac {1}{5} \left (8 \left (825-50 e^{2/3}+e^{4/3}\right )\right ) \int \frac {e^{2 x} x}{\left (40-e^{2/3}-\left (15-e^{2/3}\right ) x-5 x^2\right )^3} \, dx-\frac {16 \int \frac {e^{2 x}}{-15+e^{2/3}+\sqrt {1025-50 e^{2/3}+e^{4/3}}-10 x} \, dx}{\sqrt {1025-50 e^{2/3}+e^{4/3}}}-\frac {16 \int \frac {e^{2 x}}{15-e^{2/3}+\sqrt {1025-50 e^{2/3}+e^{4/3}}+10 x} \, dx}{\sqrt {1025-50 e^{2/3}+e^{4/3}}}\\ &=\text {Rest of rules removed due to large latex content} \end {aligned} \end {gather*}

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Mathematica [A]
time = 5.70, size = 34, normalized size = 1.06 \begin {gather*} \frac {4 e^{2 x} (-1+x)^2}{\left (e^{2/3} (-1+x)-5 \left (-8+3 x+x^2\right )\right )^2} \end {gather*}

Antiderivative was successfully veriﬁed.

[In]

Integrate[(E^(2*x)*(120 - 480*x + 400*x^2 - 40*x^4 + E^(2/3)*(-8 + 24*x - 24*x^2 + 8*x^3)))/(64000 - 72000*x +

3000*x^2 + 14625*x^3 - 375*x^4 - 1125*x^5 - 125*x^6 + E^2*(-1 + 3*x - 3*x^2 + x^3) + E^(4/3)*(120 - 285*x + 1

95*x^2 - 15*x^3 - 15*x^4) + E^(2/3)*(-4800 + 8400*x - 3075*x^2 - 975*x^3 + 375*x^4 + 75*x^5)),x]

[Out]

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

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Maple [C] Result contains higher order function than in optimal. Order 4 vs. order 3.
time = 0.27, size = 10281, normalized size = 321.28


method	result	size

norman	$\frac {4 \,{\mathrm e}^{2 x}-8 x \,{\mathrm e}^{2 x}+4 \,{\mathrm e}^{2 x} x^{2}}{\left ({\mathrm e}^{\frac {2}{3}} x -5 x^{2}-{\mathrm e}^{\frac {2}{3}}-15 x +40\right )^{2}}$	$45$
gosper	$\frac {4 \left (x^{2}-2 x +1\right ) {\mathrm e}^{2 x}}{{\mathrm e}^{\frac {4}{3}} x^{2}-10 x^{3} {\mathrm e}^{\frac {2}{3}}+25 x^{4}-2 x \,{\mathrm e}^{\frac {4}{3}}-20 x^{2} {\mathrm e}^{\frac {2}{3}}+150 x^{3}+{\mathrm e}^{\frac {4}{3}}+110 \,{\mathrm e}^{\frac {2}{3}} x -175 x^{2}-80 \,{\mathrm e}^{\frac {2}{3}}-1200 x +1600}$	$79$
default	$\text {Expression too large to display}$	$10281$

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(((8*x^3-24*x^2+24*x-8)*exp(2/3)-40*x^4+400*x^2-480*x+120)*exp(x)^2/((x^3-3*x^2+3*x-1)*exp(2/3)^3+(-15*x^4-

15*x^3+195*x^2-285*x+120)*exp(2/3)^2+(75*x^5+375*x^4-975*x^3-3075*x^2+8400*x-4800)*exp(2/3)-125*x^6-1125*x^5-3

75*x^4+14625*x^3+3000*x^2-72000*x+64000),x,method=_RETURNVERBOSE)

[Out]

result too large to display

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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: RuntimeError} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(((8*x^3-24*x^2+24*x-8)*exp(2/3)-40*x^4+400*x^2-480*x+120)*exp(x)^2/((x^3-3*x^2+3*x-1)*exp(2/3)^3+(-1

5*x^4-15*x^3+195*x^2-285*x+120)*exp(2/3)^2+(75*x^5+375*x^4-975*x^3-3075*x^2+8400*x-4800)*exp(2/3)-125*x^6-1125

*x^5-375*x^4+14625*x^3+3000*x^2-72000*x+64000),x, algorithm="maxima")

[Out]

Exception raised: RuntimeError >> ECL says: THROW: The catch RAT-ERR is undefined.

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 64 vs. $2 (27) = 54$.
time = 0.36, size = 64, normalized size = 2.00 \begin {gather*} \frac {4 \, {\left (x^{2} - 2 \, x + 1\right )} e^{\left (2 \, x\right )}}{25 \, x^{4} + 150 \, x^{3} - 175 \, x^{2} + {\left (x^{2} - 2 \, x + 1\right )} e^{\frac {4}{3}} - 10 \, {\left (x^{3} + 2 \, x^{2} - 11 \, x + 8\right )} e^{\frac {2}{3}} - 1200 \, x + 1600} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(((8*x^3-24*x^2+24*x-8)*exp(2/3)-40*x^4+400*x^2-480*x+120)*exp(x)^2/((x^3-3*x^2+3*x-1)*exp(2/3)^3+(-1

5*x^4-15*x^3+195*x^2-285*x+120)*exp(2/3)^2+(75*x^5+375*x^4-975*x^3-3075*x^2+8400*x-4800)*exp(2/3)-125*x^6-1125

*x^5-375*x^4+14625*x^3+3000*x^2-72000*x+64000),x, algorithm="fricas")

[Out]

4*(x^2 - 2*x + 1)*e^(2*x)/(25*x^4 + 150*x^3 - 175*x^2 + (x^2 - 2*x + 1)*e^(4/3) - 10*(x^3 + 2*x^2 - 11*x + 8)*

e^(2/3) - 1200*x + 1600)

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Sympy [F(-1)] Timed out
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(((8*x**3-24*x**2+24*x-8)*exp(2/3)-40*x**4+400*x**2-480*x+120)*exp(x)**2/((x**3-3*x**2+3*x-1)*exp(2/3

)**3+(-15*x**4-15*x**3+195*x**2-285*x+120)*exp(2/3)**2+(75*x**5+375*x**4-975*x**3-3075*x**2+8400*x-4800)*exp(2

/3)-125*x**6-1125*x**5-375*x**4+14625*x**3+3000*x**2-72000*x+64000),x)

[Out]

Timed out

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 80 vs. $2 (27) = 54$.
time = 0.43, size = 80, normalized size = 2.50 \begin {gather*} \frac {8 \, {\left (x^{2} e^{\left (2 \, x\right )} - 2 \, x e^{\left (2 \, x\right )} + e^{\left (2 \, x\right )}\right )}}{25 \, x^{4} - 10 \, x^{3} e^{\frac {2}{3}} + 150 \, x^{3} + x^{2} e^{\frac {4}{3}} - 20 \, x^{2} e^{\frac {2}{3}} - 175 \, x^{2} - 2 \, x e^{\frac {4}{3}} + 110 \, x e^{\frac {2}{3}} - 1200 \, x + e^{\frac {4}{3}} - 80 \, e^{\frac {2}{3}} + 1600} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(((8*x^3-24*x^2+24*x-8)*exp(2/3)-40*x^4+400*x^2-480*x+120)*exp(x)^2/((x^3-3*x^2+3*x-1)*exp(2/3)^3+(-1

5*x^4-15*x^3+195*x^2-285*x+120)*exp(2/3)^2+(75*x^5+375*x^4-975*x^3-3075*x^2+8400*x-4800)*exp(2/3)-125*x^6-1125

*x^5-375*x^4+14625*x^3+3000*x^2-72000*x+64000),x, algorithm="giac")

[Out]

8*(x^2*e^(2*x) - 2*x*e^(2*x) + e^(2*x))/(25*x^4 - 10*x^3*e^(2/3) + 150*x^3 + x^2*e^(4/3) - 20*x^2*e^(2/3) - 17

5*x^2 - 2*x*e^(4/3) + 110*x*e^(2/3) - 1200*x + e^(4/3) - 80*e^(2/3) + 1600)

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Mupad [B]
time = 1.79, size = 68, normalized size = 2.12 \begin {gather*} -\frac {{\mathrm {e}}^{2\,x}\,\left (\frac {4\,x^2}{25}-\frac {8\,x}{25}+\frac {4}{25}\right )}{-x^4+\left (\frac {2\,{\mathrm {e}}^{2/3}}{5}-6\right )\,x^3+\left (\frac {4\,{\mathrm {e}}^{2/3}}{5}-\frac {{\mathrm {e}}^{4/3}}{25}+7\right )\,x^2+\left (\frac {2\,{\mathrm {e}}^{4/3}}{25}-\frac {22\,{\mathrm {e}}^{2/3}}{5}+48\right )\,x-\frac {{\left ({\mathrm {e}}^{2/3}-40\right )}^2}{25}} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(2*x)*(exp(2/3)*(24*x - 24*x^2 + 8*x^3 - 8) - 480*x + 400*x^2 - 40*x^4 + 120))/(72000*x - exp(2)*(3*x

- 3*x^2 + x^3 - 1) + exp(4/3)*(285*x - 195*x^2 + 15*x^3 + 15*x^4 - 120) - exp(2/3)*(8400*x - 3075*x^2 - 975*x

^3 + 375*x^4 + 75*x^5 - 4800) - 3000*x^2 - 14625*x^3 + 375*x^4 + 1125*x^5 + 125*x^6 - 64000),x)

[Out]

-(exp(2*x)*((4*x^2)/25 - (8*x)/25 + 4/25))/(x^2*((4*exp(2/3))/5 - exp(4/3)/25 + 7) - (exp(2/3) - 40)^2/25 + x^

3*((2*exp(2/3))/5 - 6) + x*((2*exp(4/3))/25 - (22*exp(2/3))/5 + 48) - x^4)

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3.16.22 \(\int \frac {e^{2 x} (120-480 x+400 x^2-40 x^4+e^{2/3} (-8+24 x-24 x^2+8 x^3))}{64000-72000 x+3000 x^2+14625 x^3-375 x^4-1125 x^5-125 x^6+e^2 (-1+3 x-3 x^2+x^3)+e^{4/3} (120-285 x+195 x^2-15 x^3-15 x^4)+e^{2/3} (-4800+8400 x-3075 x^2-975 x^3+375 x^4+75 x^5)} \, dx\) [1522]