3.27.11 \(\int \frac {5000 x^2+2000 e x^2+200 e^2 x^2-3125 x^3-2500 x^5+e^5 (-20000-640 e^3-32 e^4+37500 x-15625 x^2-10000 x^3+10000 x^6+e (-16000+15000 x-4000 x^3)+e^2 (-4800+1500 x-400 x^3))}{5000 e^5 x^5} \, dx\)

Optimal. Leaf size=32 \[ \left (-\frac {(5+e)^2}{25 x^2}-x+\frac {5+\frac {x}{e^5}}{4 x}\right )^2 \]

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Rubi [B]  time = 0.08, antiderivative size = 88, normalized size of antiderivative = 2.75, number of steps used = 5, number of rules used = 3, integrand size = 100, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.030, Rules used = {6, 12, 14} \begin {gather*} \frac {(5+e)^4}{625 x^4}-\frac {(5+e)^2}{10 x^3}+x^2-\frac {200+80 e+8 e^2-625 e^5}{400 e^5 x^2}-\frac {x}{2 e^5}+\frac {125+400 e^5+160 e^6+16 e^7}{200 e^5 x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(5000*x^2 + 2000*E*x^2 + 200*E^2*x^2 - 3125*x^3 - 2500*x^5 + E^5*(-20000 - 640*E^3 - 32*E^4 + 37500*x - 15
625*x^2 - 10000*x^3 + 10000*x^6 + E*(-16000 + 15000*x - 4000*x^3) + E^2*(-4800 + 1500*x - 400*x^3)))/(5000*E^5
*x^5),x]

[Out]

(5 + E)^4/(625*x^4) - (5 + E)^2/(10*x^3) - (200 + 80*E + 8*E^2 - 625*E^5)/(400*E^5*x^2) + (125 + 400*E^5 + 160
*E^6 + 16*E^7)/(200*E^5*x) - x/(2*E^5) + x^2

Rule 6

Int[(u_.)*((w_.) + (a_.)*(v_) + (b_.)*(v_))^(p_.), x_Symbol] :> Int[u*((a + b)*v + w)^p, x] /; FreeQ[{a, b}, x
] &&  !FreeQ[v, 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 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {200 e^2 x^2+(5000+2000 e) x^2-3125 x^3-2500 x^5+e^5 \left (-20000-640 e^3-32 e^4+37500 x-15625 x^2-10000 x^3+10000 x^6+e \left (-16000+15000 x-4000 x^3\right )+e^2 \left (-4800+1500 x-400 x^3\right )\right )}{5000 e^5 x^5} \, dx\\ &=\int \frac {\left (5000+2000 e+200 e^2\right ) x^2-3125 x^3-2500 x^5+e^5 \left (-20000-640 e^3-32 e^4+37500 x-15625 x^2-10000 x^3+10000 x^6+e \left (-16000+15000 x-4000 x^3\right )+e^2 \left (-4800+1500 x-400 x^3\right )\right )}{5000 e^5 x^5} \, dx\\ &=\frac {\int \frac {\left (5000+2000 e+200 e^2\right ) x^2-3125 x^3-2500 x^5+e^5 \left (-20000-640 e^3-32 e^4+37500 x-15625 x^2-10000 x^3+10000 x^6+e \left (-16000+15000 x-4000 x^3\right )+e^2 \left (-4800+1500 x-400 x^3\right )\right )}{x^5} \, dx}{5000 e^5}\\ &=\frac {\int \left (-2500-\frac {32 e^5 (5+e)^4}{x^5}+\frac {1500 e^5 (5+e)^2}{x^4}-\frac {25 \left (-200-80 e-8 e^2+625 e^5\right )}{x^3}-\frac {25 \left (125+400 e^5+160 e^6+16 e^7\right )}{x^2}+10000 e^5 x\right ) \, dx}{5000 e^5}\\ &=\frac {(5+e)^4}{625 x^4}-\frac {(5+e)^2}{10 x^3}-\frac {200+80 e+8 e^2-625 e^5}{400 e^5 x^2}+\frac {125+400 e^5+160 e^6+16 e^7}{200 e^5 x}-\frac {x}{2 e^5}+x^2\\ \end {aligned} \end {gather*}

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Mathematica [B]  time = 0.04, size = 111, normalized size = 3.47 \begin {gather*} -\frac {-\frac {8 \left (625 e^5+500 e^6+150 e^7+20 e^8+e^9\right )}{x^4}+\frac {500 \left (25 e^5+10 e^6+e^7\right )}{x^3}-\frac {25 \left (-200-80 e-8 e^2+625 e^5\right )}{2 x^2}-\frac {25 \left (125+400 e^5+160 e^6+16 e^7\right )}{x}+2500 x-5000 e^5 x^2}{5000 e^5} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(5000*x^2 + 2000*E*x^2 + 200*E^2*x^2 - 3125*x^3 - 2500*x^5 + E^5*(-20000 - 640*E^3 - 32*E^4 + 37500*
x - 15625*x^2 - 10000*x^3 + 10000*x^6 + E*(-16000 + 15000*x - 4000*x^3) + E^2*(-4800 + 1500*x - 400*x^3)))/(50
00*E^5*x^5),x]

[Out]

-1/5000*((-8*(625*E^5 + 500*E^6 + 150*E^7 + 20*E^8 + E^9))/x^4 + (500*(25*E^5 + 10*E^6 + E^7))/x^3 - (25*(-200
- 80*E - 8*E^2 + 625*E^5))/(2*x^2) - (25*(125 + 400*E^5 + 160*E^6 + 16*E^7))/x + 2500*x - 5000*E^5*x^2)/E^5

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fricas [B]  time = 0.60, size = 97, normalized size = 3.03 \begin {gather*} -\frac {{\left (5000 \, x^{5} - 6250 \, x^{3} + 200 \, x^{2} e^{2} + 2000 \, x^{2} e + 5000 \, x^{2} - 200 \, {\left (4 \, x^{3} - 5 \, x + 12\right )} e^{7} - 2000 \, {\left (4 \, x^{3} - 5 \, x + 4\right )} e^{6} - 625 \, {\left (16 \, x^{6} + 32 \, x^{3} + 25 \, x^{2} - 40 \, x + 16\right )} e^{5} - 16 \, e^{9} - 320 \, e^{8}\right )} e^{\left (-5\right )}}{10000 \, x^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/5000*((-32*exp(1)^4-640*exp(1)^3+(-400*x^3+1500*x-4800)*exp(1)^2+(-4000*x^3+15000*x-16000)*exp(1)+
10000*x^6-10000*x^3-15625*x^2+37500*x-20000)*exp(5)+200*x^2*exp(1)^2+2000*x^2*exp(1)-2500*x^5-3125*x^3+5000*x^
2)/x^5/exp(5),x, algorithm="fricas")

[Out]

-1/10000*(5000*x^5 - 6250*x^3 + 200*x^2*e^2 + 2000*x^2*e + 5000*x^2 - 200*(4*x^3 - 5*x + 12)*e^7 - 2000*(4*x^3
- 5*x + 4)*e^6 - 625*(16*x^6 + 32*x^3 + 25*x^2 - 40*x + 16)*e^5 - 16*e^9 - 320*e^8)*e^(-5)/x^4

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giac [B]  time = 0.32, size = 107, normalized size = 3.34 \begin {gather*} \frac {1}{10000} \, {\left (10000 \, x^{2} e^{5} - 5000 \, x + \frac {800 \, x^{3} e^{7} + 8000 \, x^{3} e^{6} + 20000 \, x^{3} e^{5} + 6250 \, x^{3} + 15625 \, x^{2} e^{5} - 200 \, x^{2} e^{2} - 2000 \, x^{2} e - 5000 \, x^{2} - 1000 \, x e^{7} - 10000 \, x e^{6} - 25000 \, x e^{5} + 16 \, e^{9} + 320 \, e^{8} + 2400 \, e^{7} + 8000 \, e^{6} + 10000 \, e^{5}}{x^{4}}\right )} e^{\left (-5\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/5000*((-32*exp(1)^4-640*exp(1)^3+(-400*x^3+1500*x-4800)*exp(1)^2+(-4000*x^3+15000*x-16000)*exp(1)+
10000*x^6-10000*x^3-15625*x^2+37500*x-20000)*exp(5)+200*x^2*exp(1)^2+2000*x^2*exp(1)-2500*x^5-3125*x^3+5000*x^
2)/x^5/exp(5),x, algorithm="giac")

[Out]

1/10000*(10000*x^2*e^5 - 5000*x + (800*x^3*e^7 + 8000*x^3*e^6 + 20000*x^3*e^5 + 6250*x^3 + 15625*x^2*e^5 - 200
*x^2*e^2 - 2000*x^2*e - 5000*x^2 - 1000*x*e^7 - 10000*x*e^6 - 25000*x*e^5 + 16*e^9 + 320*e^8 + 2400*e^7 + 8000
*e^6 + 10000*e^5)/x^4)*e^(-5)

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maple [B]  time = 0.09, size = 89, normalized size = 2.78

method	result	size
risch	\(x^{2}-\frac {{\mathrm e}^{-5} x}{2}+\frac {{\mathrm e}^{-5} \left (\left (10000 \,{\mathrm e}^{5}+400 \,{\mathrm e}^{7}+4000 \,{\mathrm e}^{6}+3125\right ) x^{3}+\left (\frac {15625 \,{\mathrm e}^{5}}{2}-100 \,{\mathrm e}^{2}-1000 \,{\mathrm e}-2500\right ) x^{2}+\left (-500 \,{\mathrm e}^{7}-5000 \,{\mathrm e}^{6}-12500 \,{\mathrm e}^{5}\right ) x +8 \,{\mathrm e}^{9}+160 \,{\mathrm e}^{8}+1200 \,{\mathrm e}^{7}+4000 \,{\mathrm e}^{6}+5000 \,{\mathrm e}^{5}\right )}{5000 x^{4}}\)	\(89\)
default	\(\frac {{\mathrm e}^{-5} \left (-2500 x +5000 x^{2} {\mathrm e}^{5}-\frac {37500 \,{\mathrm e}^{5}+1500 \,{\mathrm e}^{7}+15000 \,{\mathrm e}^{6}}{3 x^{3}}-\frac {-15625 \,{\mathrm e}^{5}+200 \,{\mathrm e}^{2}+2000 \,{\mathrm e}+5000}{2 x^{2}}-\frac {-10000 \,{\mathrm e}^{5}-400 \,{\mathrm e}^{7}-4000 \,{\mathrm e}^{6}-3125}{x}-\frac {-20000 \,{\mathrm e}^{5}-4800 \,{\mathrm e}^{7}-32 \,{\mathrm e}^{9}-16000 \,{\mathrm e}^{6}-640 \,{\mathrm e}^{8}}{4 x^{4}}\right )}{5000}\)	\(100\)
norman	\(\frac {x^{6}+\left (-\frac {{\mathrm e}^{2}}{10}-{\mathrm e}-\frac {5}{2}\right ) x -\frac {{\mathrm e}^{-5} x^{5}}{2}+\frac {\left (-8 \,{\mathrm e}^{2}+625 \,{\mathrm e}^{5}-80 \,{\mathrm e}-200\right ) {\mathrm e}^{-5} x^{2}}{400}+\frac {\left (16 \,{\mathrm e}^{2} {\mathrm e}^{5}+160 \,{\mathrm e} \,{\mathrm e}^{5}+400 \,{\mathrm e}^{5}+125\right ) {\mathrm e}^{-5} x^{3}}{200}+\frac {{\mathrm e}^{4}}{625}+\frac {4 \,{\mathrm e}^{3}}{125}+\frac {6 \,{\mathrm e}^{2}}{25}+\frac {4 \,{\mathrm e}}{5}+1}{x^{4}}\)	\(109\)
gosper	\(\frac {\left (10000 x^{6} {\mathrm e}^{5}+800 \,{\mathrm e}^{5} {\mathrm e}^{2} x^{3}+16 \,{\mathrm e}^{4} {\mathrm e}^{5}+8000 \,{\mathrm e}^{5} {\mathrm e} x^{3}-5000 x^{5}+320 \,{\mathrm e}^{3} {\mathrm e}^{5}-1000 \,{\mathrm e}^{5} {\mathrm e}^{2} x +20000 x^{3} {\mathrm e}^{5}-200 x^{2} {\mathrm e}^{2}+2400 \,{\mathrm e}^{2} {\mathrm e}^{5}-10000 x \,{\mathrm e} \,{\mathrm e}^{5}+15625 x^{2} {\mathrm e}^{5}-2000 x^{2} {\mathrm e}+6250 x^{3}+8000 \,{\mathrm e} \,{\mathrm e}^{5}-25000 x \,{\mathrm e}^{5}-5000 x^{2}+10000 \,{\mathrm e}^{5}\right ) {\mathrm e}^{-5}}{10000 x^{4}}\)	\(138\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/5000*((-32*exp(1)^4-640*exp(1)^3+(-400*x^3+1500*x-4800)*exp(1)^2+(-4000*x^3+15000*x-16000)*exp(1)+10000*
x^6-10000*x^3-15625*x^2+37500*x-20000)*exp(5)+200*x^2*exp(1)^2+2000*x^2*exp(1)-2500*x^5-3125*x^3+5000*x^2)/x^5
/exp(5),x,method=_RETURNVERBOSE)

[Out]

x^2-1/2*exp(-5)*x+1/5000*exp(-5)*((10000*exp(5)+400*exp(7)+4000*exp(6)+3125)*x^3+(15625/2*exp(5)-100*exp(2)-10
00*exp(1)-2500)*x^2+(-500*exp(7)-5000*exp(6)-12500*exp(5))*x+8*exp(9)+160*exp(8)+1200*exp(7)+4000*exp(6)+5000*
exp(5))/x^4

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maxima [B]  time = 0.65, size = 92, normalized size = 2.88 \begin {gather*} \frac {1}{10000} \, {\left (10000 \, x^{2} e^{5} - 5000 \, x + \frac {50 \, x^{3} {\left (16 \, e^{7} + 160 \, e^{6} + 400 \, e^{5} + 125\right )} + 25 \, x^{2} {\left (625 \, e^{5} - 8 \, e^{2} - 80 \, e - 200\right )} - 1000 \, x {\left (e^{7} + 10 \, e^{6} + 25 \, e^{5}\right )} + 16 \, e^{9} + 320 \, e^{8} + 2400 \, e^{7} + 8000 \, e^{6} + 10000 \, e^{5}}{x^{4}}\right )} e^{\left (-5\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/5000*((-32*exp(1)^4-640*exp(1)^3+(-400*x^3+1500*x-4800)*exp(1)^2+(-4000*x^3+15000*x-16000)*exp(1)+
10000*x^6-10000*x^3-15625*x^2+37500*x-20000)*exp(5)+200*x^2*exp(1)^2+2000*x^2*exp(1)-2500*x^5-3125*x^3+5000*x^
2)/x^5/exp(5),x, algorithm="maxima")

[Out]

1/10000*(10000*x^2*e^5 - 5000*x + (50*x^3*(16*e^7 + 160*e^6 + 400*e^5 + 125) + 25*x^2*(625*e^5 - 8*e^2 - 80*e
- 200) - 1000*x*(e^7 + 10*e^6 + 25*e^5) + 16*e^9 + 320*e^8 + 2400*e^7 + 8000*e^6 + 10000*e^5)/x^4)*e^(-5)

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mupad [B]  time = 1.69, size = 90, normalized size = 2.81 \begin {gather*} x^2-\frac {x\,{\mathrm {e}}^{-5}}{2}+\frac {{\mathrm {e}}^{-5}\,\left (\left (10000\,{\mathrm {e}}^5+4000\,{\mathrm {e}}^6+400\,{\mathrm {e}}^7+3125\right )\,x^3+\left (\frac {15625\,{\mathrm {e}}^5}{2}-100\,{\mathrm {e}}^2-1000\,\mathrm {e}-2500\right )\,x^2+\left (-12500\,{\mathrm {e}}^5-5000\,{\mathrm {e}}^6-500\,{\mathrm {e}}^7\right )\,x+5000\,{\mathrm {e}}^5+4000\,{\mathrm {e}}^6+1200\,{\mathrm {e}}^7+160\,{\mathrm {e}}^8+8\,{\mathrm {e}}^9\right )}{5000\,x^4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(-5)*((exp(5)*(640*exp(3) - 37500*x + 32*exp(4) + exp(2)*(400*x^3 - 1500*x + 4800) + exp(1)*(4000*x^3
- 15000*x + 16000) + 15625*x^2 + 10000*x^3 - 10000*x^6 + 20000))/5000 - (2*x^2*exp(1))/5 - (x^2*exp(2))/25 -
x^2 + (5*x^3)/8 + x^5/2))/x^5,x)

[Out]

x^2 - (x*exp(-5))/2 + (exp(-5)*(5000*exp(5) + 4000*exp(6) + 1200*exp(7) + 160*exp(8) + 8*exp(9) - x*(12500*exp
(5) + 5000*exp(6) + 500*exp(7)) + x^3*(10000*exp(5) + 4000*exp(6) + 400*exp(7) + 3125) - x^2*(1000*exp(1) + 10
0*exp(2) - (15625*exp(5))/2 + 2500)))/(5000*x^4)

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sympy [B]  time = 4.14, size = 105, normalized size = 3.28 \begin {gather*} \frac {5000 x^{2} e^{5} - 2500 x + \frac {x^{3} \left (6250 + 800 e^{7} + 20000 e^{5} + 8000 e^{6}\right ) + x^{2} \left (- 2000 e - 5000 - 200 e^{2} + 15625 e^{5}\right ) + x \left (- 10000 e^{6} - 25000 e^{5} - 1000 e^{7}\right ) + 16 e^{9} + 320 e^{8} + 10000 e^{5} + 2400 e^{7} + 8000 e^{6}}{2 x^{4}}}{5000 e^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/5000*((-32*exp(1)**4-640*exp(1)**3+(-400*x**3+1500*x-4800)*exp(1)**2+(-4000*x**3+15000*x-16000)*ex
p(1)+10000*x**6-10000*x**3-15625*x**2+37500*x-20000)*exp(5)+200*x**2*exp(1)**2+2000*x**2*exp(1)-2500*x**5-3125
*x**3+5000*x**2)/x**5/exp(5),x)

[Out]

(5000*x**2*exp(5) - 2500*x + (x**3*(6250 + 800*exp(7) + 20000*exp(5) + 8000*exp(6)) + x**2*(-2000*E - 5000 - 2
00*exp(2) + 15625*exp(5)) + x*(-10000*exp(6) - 25000*exp(5) - 1000*exp(7)) + 16*exp(9) + 320*exp(8) + 10000*ex
p(5) + 2400*exp(7) + 8000*exp(6))/(2*x**4))*exp(-5)/5000

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