3.87.95 \(\int \frac {e^{16} (-10000-8000 x-2400 x^2-320 x^3-16 x^4)+e^{16} (8000 x+4800 x^2+960 x^3+64 x^4) \log (\frac {x}{25})}{625 x \log ^2(\frac {x}{25})} \, dx\)

Optimal. Leaf size=22 \[ \frac {16 e^{16} \left (1+\frac {x}{5}\right )^4}{\log \left (\frac {x}{25}\right )} \]

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Rubi [B]  time = 0.61, antiderivative size = 84, normalized size of antiderivative = 3.82, number of steps used = 29, number of rules used = 12, integrand size = 69, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {12, 6688, 6742, 2353, 2297, 2298, 2302, 30, 2306, 2309, 2178, 2330} \begin {gather*} \frac {16 e^{16} x^4}{625 \log \left (\frac {x}{25}\right )}+\frac {64 e^{16} x^3}{125 \log \left (\frac {x}{25}\right )}+\frac {96 e^{16} x^2}{25 \log \left (\frac {x}{25}\right )}+\frac {64 e^{16} x}{5 \log \left (\frac {x}{25}\right )}+\frac {16 e^{16}}{\log \left (\frac {x}{25}\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(E^16*(-10000 - 8000*x - 2400*x^2 - 320*x^3 - 16*x^4) + E^16*(8000*x + 4800*x^2 + 960*x^3 + 64*x^4)*Log[x/
25])/(625*x*Log[x/25]^2),x]

[Out]

(16*E^16)/Log[x/25] + (64*E^16*x)/(5*Log[x/25]) + (96*E^16*x^2)/(25*Log[x/25]) + (64*E^16*x^3)/(125*Log[x/25])
+ (16*E^16*x^4)/(625*Log[x/25])

Rule 12

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

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

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 2297

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_), x_Symbol] :> Simp[(x*(a + b*Log[c*x^n])^(p + 1))/(b*n*(p + 1))
, x] - Dist[1/(b*n*(p + 1)), Int[(a + b*Log[c*x^n])^(p + 1), x], x] /; FreeQ[{a, b, c, n}, x] && LtQ[p, -1] &&
IntegerQ[2*p]

Rule 2298

Int[Log[(c_.)*(x_)]^(-1), x_Symbol] :> Simp[LogIntegral[c*x]/c, x] /; FreeQ[c, x]

Rule 2302

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)/(x_), x_Symbol] :> Dist[1/(b*n), Subst[Int[x^p, x], x, a + b*L
og[c*x^n]], x] /; FreeQ[{a, b, c, n, p}, x]

Rule 2306

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[((d*x)^(m + 1)*(a + b*Log
[c*x^n])^(p + 1))/(b*d*n*(p + 1)), x] - Dist[(m + 1)/(b*n*(p + 1)), Int[(d*x)^m*(a + b*Log[c*x^n])^(p + 1), x]
, x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[m, -1] && LtQ[p, -1]

Rule 2309

Int[((a_.) + Log[(c_.)*(x_)]*(b_.))^(p_)*(x_)^(m_.), x_Symbol] :> Dist[1/c^(m + 1), Subst[Int[E^((m + 1)*x)*(a
+ b*x)^p, x], x, Log[c*x]], x] /; FreeQ[{a, b, c, p}, x] && IntegerQ[m]

Rule 2330

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_) + (e_.)*(x_)^(r_.))^(q_.), x_Symbol] :> With[{u = Expand
Integrand[(a + b*Log[c*x^n])^p, (d + e*x^r)^q, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c, d, e, n, p, q, r}
, x] && IntegerQ[q] && (GtQ[q, 0] || (IGtQ[p, 0] && IntegerQ[r]))

Rule 2353

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^(r_.))^(q_.), x_Symbol]
:> With[{u = ExpandIntegrand[(a + b*Log[c*x^n])^p, (f*x)^m*(d + e*x^r)^q, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[
{a, b, c, d, e, f, m, n, p, q, r}, x] && IntegerQ[q] && (GtQ[q, 0] || (IGtQ[p, 0] && IntegerQ[m] && IntegerQ[r
]))

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} &=\frac {1}{625} \int \frac {e^{16} \left (-10000-8000 x-2400 x^2-320 x^3-16 x^4\right )+e^{16} \left (8000 x+4800 x^2+960 x^3+64 x^4\right ) \log \left (\frac {x}{25}\right )}{x \log ^2\left (\frac {x}{25}\right )} \, dx\\ &=\frac {1}{625} \int \frac {16 e^{16} (5+x)^3 \left (-5-x+4 x \log \left (\frac {x}{25}\right )\right )}{x \log ^2\left (\frac {x}{25}\right )} \, dx\\ &=\frac {1}{625} \left (16 e^{16}\right ) \int \frac {(5+x)^3 \left (-5-x+4 x \log \left (\frac {x}{25}\right )\right )}{x \log ^2\left (\frac {x}{25}\right )} \, dx\\ &=\frac {1}{625} \left (16 e^{16}\right ) \int \left (-\frac {(5+x)^4}{x \log ^2\left (\frac {x}{25}\right )}+\frac {4 (5+x)^3}{\log \left (\frac {x}{25}\right )}\right ) \, dx\\ &=-\left (\frac {1}{625} \left (16 e^{16}\right ) \int \frac {(5+x)^4}{x \log ^2\left (\frac {x}{25}\right )} \, dx\right )+\frac {1}{625} \left (64 e^{16}\right ) \int \frac {(5+x)^3}{\log \left (\frac {x}{25}\right )} \, dx\\ &=-\left (\frac {1}{625} \left (16 e^{16}\right ) \int \left (\frac {500}{\log ^2\left (\frac {x}{25}\right )}+\frac {625}{x \log ^2\left (\frac {x}{25}\right )}+\frac {150 x}{\log ^2\left (\frac {x}{25}\right )}+\frac {20 x^2}{\log ^2\left (\frac {x}{25}\right )}+\frac {x^3}{\log ^2\left (\frac {x}{25}\right )}\right ) \, dx\right )+\frac {1}{625} \left (64 e^{16}\right ) \int \left (\frac {125}{\log \left (\frac {x}{25}\right )}+\frac {75 x}{\log \left (\frac {x}{25}\right )}+\frac {15 x^2}{\log \left (\frac {x}{25}\right )}+\frac {x^3}{\log \left (\frac {x}{25}\right )}\right ) \, dx\\ &=-\left (\frac {1}{625} \left (16 e^{16}\right ) \int \frac {x^3}{\log ^2\left (\frac {x}{25}\right )} \, dx\right )+\frac {1}{625} \left (64 e^{16}\right ) \int \frac {x^3}{\log \left (\frac {x}{25}\right )} \, dx-\frac {1}{125} \left (64 e^{16}\right ) \int \frac {x^2}{\log ^2\left (\frac {x}{25}\right )} \, dx+\frac {1}{125} \left (192 e^{16}\right ) \int \frac {x^2}{\log \left (\frac {x}{25}\right )} \, dx-\frac {1}{25} \left (96 e^{16}\right ) \int \frac {x}{\log ^2\left (\frac {x}{25}\right )} \, dx+\frac {1}{25} \left (192 e^{16}\right ) \int \frac {x}{\log \left (\frac {x}{25}\right )} \, dx-\frac {1}{5} \left (64 e^{16}\right ) \int \frac {1}{\log ^2\left (\frac {x}{25}\right )} \, dx+\frac {1}{5} \left (64 e^{16}\right ) \int \frac {1}{\log \left (\frac {x}{25}\right )} \, dx-\left (16 e^{16}\right ) \int \frac {1}{x \log ^2\left (\frac {x}{25}\right )} \, dx\\ &=\frac {64 e^{16} x}{5 \log \left (\frac {x}{25}\right )}+\frac {96 e^{16} x^2}{25 \log \left (\frac {x}{25}\right )}+\frac {64 e^{16} x^3}{125 \log \left (\frac {x}{25}\right )}+\frac {16 e^{16} x^4}{625 \log \left (\frac {x}{25}\right )}+320 e^{16} \text {li}\left (\frac {x}{25}\right )-\frac {1}{625} \left (64 e^{16}\right ) \int \frac {x^3}{\log \left (\frac {x}{25}\right )} \, dx-\frac {1}{125} \left (192 e^{16}\right ) \int \frac {x^2}{\log \left (\frac {x}{25}\right )} \, dx-\frac {1}{25} \left (192 e^{16}\right ) \int \frac {x}{\log \left (\frac {x}{25}\right )} \, dx-\frac {1}{5} \left (64 e^{16}\right ) \int \frac {1}{\log \left (\frac {x}{25}\right )} \, dx-\left (16 e^{16}\right ) \operatorname {Subst}\left (\int \frac {1}{x^2} \, dx,x,\log \left (\frac {x}{25}\right )\right )+\left (4800 e^{16}\right ) \operatorname {Subst}\left (\int \frac {e^{2 x}}{x} \, dx,x,\log \left (\frac {x}{25}\right )\right )+\left (24000 e^{16}\right ) \operatorname {Subst}\left (\int \frac {e^{3 x}}{x} \, dx,x,\log \left (\frac {x}{25}\right )\right )+\left (40000 e^{16}\right ) \operatorname {Subst}\left (\int \frac {e^{4 x}}{x} \, dx,x,\log \left (\frac {x}{25}\right )\right )\\ &=4800 e^{16} \text {Ei}\left (2 \log \left (\frac {x}{25}\right )\right )+24000 e^{16} \text {Ei}\left (3 \log \left (\frac {x}{25}\right )\right )+40000 e^{16} \text {Ei}\left (4 \log \left (\frac {x}{25}\right )\right )+\frac {16 e^{16}}{\log \left (\frac {x}{25}\right )}+\frac {64 e^{16} x}{5 \log \left (\frac {x}{25}\right )}+\frac {96 e^{16} x^2}{25 \log \left (\frac {x}{25}\right )}+\frac {64 e^{16} x^3}{125 \log \left (\frac {x}{25}\right )}+\frac {16 e^{16} x^4}{625 \log \left (\frac {x}{25}\right )}-\left (4800 e^{16}\right ) \operatorname {Subst}\left (\int \frac {e^{2 x}}{x} \, dx,x,\log \left (\frac {x}{25}\right )\right )-\left (24000 e^{16}\right ) \operatorname {Subst}\left (\int \frac {e^{3 x}}{x} \, dx,x,\log \left (\frac {x}{25}\right )\right )-\left (40000 e^{16}\right ) \operatorname {Subst}\left (\int \frac {e^{4 x}}{x} \, dx,x,\log \left (\frac {x}{25}\right )\right )\\ &=\frac {16 e^{16}}{\log \left (\frac {x}{25}\right )}+\frac {64 e^{16} x}{5 \log \left (\frac {x}{25}\right )}+\frac {96 e^{16} x^2}{25 \log \left (\frac {x}{25}\right )}+\frac {64 e^{16} x^3}{125 \log \left (\frac {x}{25}\right )}+\frac {16 e^{16} x^4}{625 \log \left (\frac {x}{25}\right )}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.14, size = 20, normalized size = 0.91 \begin {gather*} \frac {16 e^{16} (5+x)^4}{625 \log \left (\frac {x}{25}\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^16*(-10000 - 8000*x - 2400*x^2 - 320*x^3 - 16*x^4) + E^16*(8000*x + 4800*x^2 + 960*x^3 + 64*x^4)*
Log[x/25])/(625*x*Log[x/25]^2),x]

[Out]

(16*E^16*(5 + x)^4)/(625*Log[x/25])

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fricas [A]  time = 0.60, size = 28, normalized size = 1.27 \begin {gather*} \frac {16 \, {\left (x^{4} + 20 \, x^{3} + 150 \, x^{2} + 500 \, x + 625\right )} e^{16}}{625 \, \log \left (\frac {1}{25} \, x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/625*((64*x^4+960*x^3+4800*x^2+8000*x)*exp(4)^4*log(1/25*x)+(-16*x^4-320*x^3-2400*x^2-8000*x-10000)
*exp(4)^4)/x/log(1/25*x)^2,x, algorithm="fricas")

[Out]

16/625*(x^4 + 20*x^3 + 150*x^2 + 500*x + 625)*e^16/log(1/25*x)

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giac [B]  time = 0.17, size = 38, normalized size = 1.73 \begin {gather*} \frac {16 \, {\left (x^{4} e^{16} + 20 \, x^{3} e^{16} + 150 \, x^{2} e^{16} + 500 \, x e^{16} + 625 \, e^{16}\right )}}{625 \, \log \left (\frac {1}{25} \, x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/625*((64*x^4+960*x^3+4800*x^2+8000*x)*exp(4)^4*log(1/25*x)+(-16*x^4-320*x^3-2400*x^2-8000*x-10000)
*exp(4)^4)/x/log(1/25*x)^2,x, algorithm="giac")

[Out]

16/625*(x^4*e^16 + 20*x^3*e^16 + 150*x^2*e^16 + 500*x*e^16 + 625*e^16)/log(1/25*x)

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maple [A]  time = 0.08, size = 29, normalized size = 1.32

method	result	size
risch	\(\frac {16 \,{\mathrm e}^{16} \left (x^{4}+20 x^{3}+150 x^{2}+500 x +625\right )}{625 \ln \left (\frac {x}{25}\right )}\)	\(29\)
norman	\(\frac {16 \,{\mathrm e}^{16}+\frac {64 x \,{\mathrm e}^{16}}{5}+\frac {96 x^{2} {\mathrm e}^{16}}{25}+\frac {64 x^{3} {\mathrm e}^{16}}{125}+\frac {16 x^{4} {\mathrm e}^{16}}{625}}{\ln \left (\frac {x}{25}\right )}\)	\(49\)
derivativedivides	\(-40000 \,{\mathrm e}^{16} \expIntegralEi \left (1, -4 \ln \left (\frac {x}{25}\right )\right )-24000 \,{\mathrm e}^{16} \expIntegralEi \left (1, -3 \ln \left (\frac {x}{25}\right )\right )-10000 \,{\mathrm e}^{16} \left (-\frac {x^{4}}{390625 \ln \left (\frac {x}{25}\right )}-4 \expIntegralEi \left (1, -4 \ln \left (\frac {x}{25}\right )\right )\right )-4800 \,{\mathrm e}^{16} \expIntegralEi \left (1, -2 \ln \left (\frac {x}{25}\right )\right )-8000 \,{\mathrm e}^{16} \left (-\frac {x^{3}}{15625 \ln \left (\frac {x}{25}\right )}-3 \expIntegralEi \left (1, -3 \ln \left (\frac {x}{25}\right )\right )\right )-320 \,{\mathrm e}^{16} \expIntegralEi \left (1, -\ln \left (\frac {x}{25}\right )\right )-2400 \,{\mathrm e}^{16} \left (-\frac {x^{2}}{625 \ln \left (\frac {x}{25}\right )}-2 \expIntegralEi \left (1, -2 \ln \left (\frac {x}{25}\right )\right )\right )-320 \,{\mathrm e}^{16} \left (-\frac {x}{25 \ln \left (\frac {x}{25}\right )}-\expIntegralEi \left (1, -\ln \left (\frac {x}{25}\right )\right )\right )+\frac {16 \,{\mathrm e}^{16}}{\ln \left (\frac {x}{25}\right )}\)	\(180\)
default	\(-40000 \,{\mathrm e}^{16} \expIntegralEi \left (1, -4 \ln \left (\frac {x}{25}\right )\right )-24000 \,{\mathrm e}^{16} \expIntegralEi \left (1, -3 \ln \left (\frac {x}{25}\right )\right )-10000 \,{\mathrm e}^{16} \left (-\frac {x^{4}}{390625 \ln \left (\frac {x}{25}\right )}-4 \expIntegralEi \left (1, -4 \ln \left (\frac {x}{25}\right )\right )\right )-4800 \,{\mathrm e}^{16} \expIntegralEi \left (1, -2 \ln \left (\frac {x}{25}\right )\right )-8000 \,{\mathrm e}^{16} \left (-\frac {x^{3}}{15625 \ln \left (\frac {x}{25}\right )}-3 \expIntegralEi \left (1, -3 \ln \left (\frac {x}{25}\right )\right )\right )-320 \,{\mathrm e}^{16} \expIntegralEi \left (1, -\ln \left (\frac {x}{25}\right )\right )-2400 \,{\mathrm e}^{16} \left (-\frac {x^{2}}{625 \ln \left (\frac {x}{25}\right )}-2 \expIntegralEi \left (1, -2 \ln \left (\frac {x}{25}\right )\right )\right )-320 \,{\mathrm e}^{16} \left (-\frac {x}{25 \ln \left (\frac {x}{25}\right )}-\expIntegralEi \left (1, -\ln \left (\frac {x}{25}\right )\right )\right )+\frac {16 \,{\mathrm e}^{16}}{\ln \left (\frac {x}{25}\right )}\)	\(180\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/625*((64*x^4+960*x^3+4800*x^2+8000*x)*exp(4)^4*ln(1/25*x)+(-16*x^4-320*x^3-2400*x^2-8000*x-10000)*exp(4)
^4)/x/ln(1/25*x)^2,x,method=_RETURNVERBOSE)

[Out]

16/625*exp(16)*(x^4+20*x^3+150*x^2+500*x+625)/ln(1/25*x)

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maxima [C]  time = 0.41, size = 101, normalized size = 4.59 \begin {gather*} 40000 \, {\rm Ei}\left (4 \, \log \left (\frac {1}{25} \, x\right )\right ) e^{16} + 24000 \, {\rm Ei}\left (3 \, \log \left (\frac {1}{25} \, x\right )\right ) e^{16} + 4800 \, {\rm Ei}\left (2 \, \log \left (\frac {1}{25} \, x\right )\right ) e^{16} + 320 \, {\rm Ei}\left (\log \left (\frac {1}{25} \, x\right )\right ) e^{16} - 320 \, e^{16} \Gamma \left (-1, -\log \left (\frac {1}{25} \, x\right )\right ) - 4800 \, e^{16} \Gamma \left (-1, -2 \, \log \left (\frac {1}{25} \, x\right )\right ) - 24000 \, e^{16} \Gamma \left (-1, -3 \, \log \left (\frac {1}{25} \, x\right )\right ) - 40000 \, e^{16} \Gamma \left (-1, -4 \, \log \left (\frac {1}{25} \, x\right )\right ) + \frac {16 \, e^{16}}{\log \left (\frac {1}{25} \, x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/625*((64*x^4+960*x^3+4800*x^2+8000*x)*exp(4)^4*log(1/25*x)+(-16*x^4-320*x^3-2400*x^2-8000*x-10000)
*exp(4)^4)/x/log(1/25*x)^2,x, algorithm="maxima")

[Out]

40000*Ei(4*log(1/25*x))*e^16 + 24000*Ei(3*log(1/25*x))*e^16 + 4800*Ei(2*log(1/25*x))*e^16 + 320*Ei(log(1/25*x)
)*e^16 - 320*e^16*gamma(-1, -log(1/25*x)) - 4800*e^16*gamma(-1, -2*log(1/25*x)) - 24000*e^16*gamma(-1, -3*log(
1/25*x)) - 40000*e^16*gamma(-1, -4*log(1/25*x)) + 16*e^16/log(1/25*x)

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mupad [B]  time = 5.43, size = 47, normalized size = 2.14 \begin {gather*} \frac {16\,{\mathrm {e}}^{16}\,x^6+320\,{\mathrm {e}}^{16}\,x^5+2400\,{\mathrm {e}}^{16}\,x^4+8000\,{\mathrm {e}}^{16}\,x^3+10000\,{\mathrm {e}}^{16}\,x^2}{625\,x^2\,\ln \left (\frac {x}{25}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-((exp(16)*(8000*x + 2400*x^2 + 320*x^3 + 16*x^4 + 10000))/625 - (log(x/25)*exp(16)*(8000*x + 4800*x^2 + 9
60*x^3 + 64*x^4))/625)/(x*log(x/25)^2),x)

[Out]

(10000*x^2*exp(16) + 8000*x^3*exp(16) + 2400*x^4*exp(16) + 320*x^5*exp(16) + 16*x^6*exp(16))/(625*x^2*log(x/25
))

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sympy [B]  time = 0.14, size = 42, normalized size = 1.91 \begin {gather*} \frac {16 x^{4} e^{16} + 320 x^{3} e^{16} + 2400 x^{2} e^{16} + 8000 x e^{16} + 10000 e^{16}}{625 \log {\left (\frac {x}{25} \right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/625*((64*x**4+960*x**3+4800*x**2+8000*x)*exp(4)**4*ln(1/25*x)+(-16*x**4-320*x**3-2400*x**2-8000*x-
10000)*exp(4)**4)/x/ln(1/25*x)**2,x)

[Out]

(16*x**4*exp(16) + 320*x**3*exp(16) + 2400*x**2*exp(16) + 8000*x*exp(16) + 10000*exp(16))/(625*log(x/25))

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