3.31.95 \(\int \frac {-7680000+e^{4 x} (-3+2 x)+e^{3 x} (-480+240 x)+e^{2 x} (-28800+9600 x)+e^x (-768000+128000 x)}{5000 x^7} \, dx\)

Optimal. Leaf size=15 \[ \frac {\left (-4-\frac {e^x}{10}\right )^4}{x^6} \]

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Rubi [B]  time = 0.15, antiderivative size = 52, normalized size of antiderivative = 3.47, number of steps used = 7, number of rules used = 3, integrand size = 51, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {12, 14, 2197} \begin {gather*} \frac {128 e^x}{5 x^6}+\frac {24 e^{2 x}}{25 x^6}+\frac {2 e^{3 x}}{125 x^6}+\frac {e^{4 x}}{10000 x^6}+\frac {256}{x^6} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-7680000 + E^(4*x)*(-3 + 2*x) + E^(3*x)*(-480 + 240*x) + E^(2*x)*(-28800 + 9600*x) + E^x*(-768000 + 12800
0*x))/(5000*x^7),x]

[Out]

256/x^6 + (128*E^x)/(5*x^6) + (24*E^(2*x))/(25*x^6) + (2*E^(3*x))/(125*x^6) + E^(4*x)/(10000*x^6)

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]]

Rule 2197

Int[(F_)^((c_.)*(v_))*(u_)^(m_.)*(w_), x_Symbol] :> With[{b = Coefficient[v, x, 1], d = Coefficient[u, x, 0],
e = Coefficient[u, x, 1], f = Coefficient[w, x, 0], g = Coefficient[w, x, 1]}, Simp[(g*u^(m + 1)*F^(c*v))/(b*c
*e*Log[F]), x] /; EqQ[e*g*(m + 1) - b*c*(e*f - d*g)*Log[F], 0]] /; FreeQ[{F, c, m}, x] && LinearQ[{u, v, w}, x
]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {\int \frac {-7680000+e^{4 x} (-3+2 x)+e^{3 x} (-480+240 x)+e^{2 x} (-28800+9600 x)+e^x (-768000+128000 x)}{x^7} \, dx}{5000}\\ &=\frac {\int \left (-\frac {7680000}{x^7}+\frac {128000 e^x (-6+x)}{x^7}+\frac {9600 e^{2 x} (-3+x)}{x^7}+\frac {240 e^{3 x} (-2+x)}{x^7}+\frac {e^{4 x} (-3+2 x)}{x^7}\right ) \, dx}{5000}\\ &=\frac {256}{x^6}+\frac {\int \frac {e^{4 x} (-3+2 x)}{x^7} \, dx}{5000}+\frac {6}{125} \int \frac {e^{3 x} (-2+x)}{x^7} \, dx+\frac {48}{25} \int \frac {e^{2 x} (-3+x)}{x^7} \, dx+\frac {128}{5} \int \frac {e^x (-6+x)}{x^7} \, dx\\ &=\frac {256}{x^6}+\frac {128 e^x}{5 x^6}+\frac {24 e^{2 x}}{25 x^6}+\frac {2 e^{3 x}}{125 x^6}+\frac {e^{4 x}}{10000 x^6}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.07, size = 14, normalized size = 0.93 \begin {gather*} \frac {\left (40+e^x\right )^4}{10000 x^6} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-7680000 + E^(4*x)*(-3 + 2*x) + E^(3*x)*(-480 + 240*x) + E^(2*x)*(-28800 + 9600*x) + E^x*(-768000 +
128000*x))/(5000*x^7),x]

[Out]

(40 + E^x)^4/(10000*x^6)

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fricas [B]  time = 0.88, size = 27, normalized size = 1.80 \begin {gather*} \frac {e^{\left (4 \, x\right )} + 160 \, e^{\left (3 \, x\right )} + 9600 \, e^{\left (2 \, x\right )} + 256000 \, e^{x} + 2560000}{10000 \, x^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/5000*((2*x-3)*exp(x)^4+(240*x-480)*exp(x)^3+(9600*x-28800)*exp(x)^2+(128000*x-768000)*exp(x)-76800
00)/x^7,x, algorithm="fricas")

[Out]

1/10000*(e^(4*x) + 160*e^(3*x) + 9600*e^(2*x) + 256000*e^x + 2560000)/x^6

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giac [B]  time = 0.35, size = 27, normalized size = 1.80 \begin {gather*} \frac {e^{\left (4 \, x\right )} + 160 \, e^{\left (3 \, x\right )} + 9600 \, e^{\left (2 \, x\right )} + 256000 \, e^{x} + 2560000}{10000 \, x^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/5000*((2*x-3)*exp(x)^4+(240*x-480)*exp(x)^3+(9600*x-28800)*exp(x)^2+(128000*x-768000)*exp(x)-76800
00)/x^7,x, algorithm="giac")

[Out]

1/10000*(e^(4*x) + 160*e^(3*x) + 9600*e^(2*x) + 256000*e^x + 2560000)/x^6

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maple [B]  time = 0.04, size = 41, normalized size = 2.73

method	result	size
default	\(\frac {256}{x^{6}}+\frac {24 \,{\mathrm e}^{2 x}}{25 x^{6}}+\frac {{\mathrm e}^{4 x}}{10000 x^{6}}+\frac {2 \,{\mathrm e}^{3 x}}{125 x^{6}}+\frac {128 \,{\mathrm e}^{x}}{5 x^{6}}\)	\(41\)
risch	\(\frac {256}{x^{6}}+\frac {24 \,{\mathrm e}^{2 x}}{25 x^{6}}+\frac {{\mathrm e}^{4 x}}{10000 x^{6}}+\frac {2 \,{\mathrm e}^{3 x}}{125 x^{6}}+\frac {128 \,{\mathrm e}^{x}}{5 x^{6}}\)	\(41\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/5000*((2*x-3)*exp(x)^4+(240*x-480)*exp(x)^3+(9600*x-28800)*exp(x)^2+(128000*x-768000)*exp(x)-7680000)/x^
7,x,method=_RETURNVERBOSE)

[Out]

256/x^6+24/25*exp(x)^2/x^6+1/10000/x^6*exp(x)^4+2/125/x^6*exp(x)^3+128/5/x^6*exp(x)

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maxima [C]  time = 0.90, size = 62, normalized size = 4.13 \begin {gather*} \frac {256}{x^{6}} + \frac {128}{5} \, \Gamma \left (-5, -x\right ) + \frac {1536}{25} \, \Gamma \left (-5, -2 \, x\right ) + \frac {1458}{125} \, \Gamma \left (-5, -3 \, x\right ) + \frac {256}{625} \, \Gamma \left (-5, -4 \, x\right ) + \frac {768}{5} \, \Gamma \left (-6, -x\right ) + \frac {9216}{25} \, \Gamma \left (-6, -2 \, x\right ) + \frac {8748}{125} \, \Gamma \left (-6, -3 \, x\right ) + \frac {1536}{625} \, \Gamma \left (-6, -4 \, x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/5000*((2*x-3)*exp(x)^4+(240*x-480)*exp(x)^3+(9600*x-28800)*exp(x)^2+(128000*x-768000)*exp(x)-76800
00)/x^7,x, algorithm="maxima")

[Out]

256/x^6 + 128/5*gamma(-5, -x) + 1536/25*gamma(-5, -2*x) + 1458/125*gamma(-5, -3*x) + 256/625*gamma(-5, -4*x) +
768/5*gamma(-6, -x) + 9216/25*gamma(-6, -2*x) + 8748/125*gamma(-6, -3*x) + 1536/625*gamma(-6, -4*x)

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mupad [B]  time = 0.08, size = 11, normalized size = 0.73 \begin {gather*} \frac {{\left ({\mathrm {e}}^x+40\right )}^4}{10000\,x^6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((exp(x)*(128000*x - 768000))/5000 + (exp(4*x)*(2*x - 3))/5000 + (exp(3*x)*(240*x - 480))/5000 + (exp(2*x)
*(9600*x - 28800))/5000 - 1536)/x^7,x)

[Out]

(exp(x) + 40)^4/(10000*x^6)

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sympy [B]  time = 0.16, size = 48, normalized size = 3.20 \begin {gather*} \frac {256}{x^{6}} + \frac {15625 x^{18} e^{4 x} + 2500000 x^{18} e^{3 x} + 150000000 x^{18} e^{2 x} + 4000000000 x^{18} e^{x}}{156250000 x^{24}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/5000*((2*x-3)*exp(x)**4+(240*x-480)*exp(x)**3+(9600*x-28800)*exp(x)**2+(128000*x-768000)*exp(x)-76
80000)/x**7,x)

[Out]

256/x**6 + (15625*x**18*exp(4*x) + 2500000*x**18*exp(3*x) + 150000000*x**18*exp(2*x) + 4000000000*x**18*exp(x)
)/(156250000*x**24)

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