3.9.27 \(\int \frac {e^{15} (250-1000 x)+e^{20} (-x+4 x^2)+(e^{10} (127500-10000 x)-500 e^{15} x) \log (x^2)+(1875000 e^5-2500 e^{10} x) \log ^2(x^2)+6250000 \log ^3(x^2)}{32 e^{20} x} \, dx\)

Optimal. Leaf size=26 \[ \frac {1}{16} \left (6+x-\left (\frac {5}{2}+\frac {25 \log \left (x^2\right )}{e^5}\right )^2\right )^2 \]

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Rubi [B]  time = 0.20, antiderivative size = 131, normalized size of antiderivative = 5.04, number of steps used = 15, number of rules used = 8, integrand size = 81, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.099, Rules used = {12, 14, 2346, 2301, 2295, 2302, 30, 2296} \begin {gather*} \frac {x^2}{16}+\frac {390625 \log ^4\left (x^2\right )}{16 e^{20}}+\frac {78125 \log ^3\left (x^2\right )}{8 e^{15}}-\frac {625 x \log ^2\left (x^2\right )}{8 e^{10}}+\frac {31875 \log ^2\left (x^2\right )}{32 e^{10}}-\frac {125 \left (20+e^5\right ) x \log \left (x^2\right )}{8 e^{10}}+\frac {625 x \log \left (x^2\right )}{2 e^{10}}-\frac {\left (1000+e^5\right ) x}{32 e^5}+\frac {125 \left (20+e^5\right ) x}{4 e^{10}}-\frac {625 x}{e^{10}}+\frac {125 \log (x)}{16 e^5} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(E^15*(250 - 1000*x) + E^20*(-x + 4*x^2) + (E^10*(127500 - 10000*x) - 500*E^15*x)*Log[x^2] + (1875000*E^5
- 2500*E^10*x)*Log[x^2]^2 + 6250000*Log[x^2]^3)/(32*E^20*x),x]

[Out]

(-625*x)/E^10 + (125*(20 + E^5)*x)/(4*E^10) - ((1000 + E^5)*x)/(32*E^5) + x^2/16 + (125*Log[x])/(16*E^5) + (62
5*x*Log[x^2])/(2*E^10) - (125*(20 + E^5)*x*Log[x^2])/(8*E^10) + (31875*Log[x^2]^2)/(32*E^10) - (625*x*Log[x^2]
^2)/(8*E^10) + (78125*Log[x^2]^3)/(8*E^15) + (390625*Log[x^2]^4)/(16*E^20)

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 30

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

Rule 2295

Int[Log[(c_.)*(x_)^(n_.)], x_Symbol] :> Simp[x*Log[c*x^n], x] - Simp[n*x, x] /; FreeQ[{c, n}, x]

Rule 2296

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.), x_Symbol] :> Simp[x*(a + b*Log[c*x^n])^p, x] - Dist[b*n*p, In
t[(a + b*Log[c*x^n])^(p - 1), x], x] /; FreeQ[{a, b, c, n}, x] && GtQ[p, 0] && IntegerQ[2*p]

Rule 2301

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))/(x_), x_Symbol] :> Simp[(a + b*Log[c*x^n])^2/(2*b*n), x] /; FreeQ[{a
, b, c, n}, 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 2346

Int[(((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_) + (e_.)*(x_))^(q_.))/(x_), x_Symbol] :> Dist[d, Int[((d
+ e*x)^(q - 1)*(a + b*Log[c*x^n])^p)/x, x], x] + Dist[e, Int[(d + e*x)^(q - 1)*(a + b*Log[c*x^n])^p, x], x] /
; FreeQ[{a, b, c, d, e, n}, x] && IGtQ[p, 0] && GtQ[q, 0] && IntegerQ[2*q]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {\int \frac {e^{15} (250-1000 x)+e^{20} \left (-x+4 x^2\right )+\left (e^{10} (127500-10000 x)-500 e^{15} x\right ) \log \left (x^2\right )+\left (1875000 e^5-2500 e^{10} x\right ) \log ^2\left (x^2\right )+6250000 \log ^3\left (x^2\right )}{x} \, dx}{32 e^{20}}\\ &=\frac {\int \left (\frac {e^{15} \left (250-\left (1000+e^5\right ) x+4 e^5 x^2\right )}{x}+\frac {500 e^{10} \left (255-\left (20+e^5\right ) x\right ) \log \left (x^2\right )}{x}-\frac {2500 e^5 \left (-750+e^5 x\right ) \log ^2\left (x^2\right )}{x}+\frac {6250000 \log ^3\left (x^2\right )}{x}\right ) \, dx}{32 e^{20}}\\ &=\frac {390625 \int \frac {\log ^3\left (x^2\right )}{x} \, dx}{2 e^{20}}-\frac {625 \int \frac {\left (-750+e^5 x\right ) \log ^2\left (x^2\right )}{x} \, dx}{8 e^{15}}+\frac {125 \int \frac {\left (255+\left (-20-e^5\right ) x\right ) \log \left (x^2\right )}{x} \, dx}{8 e^{10}}+\frac {\int \frac {250-\left (1000+e^5\right ) x+4 e^5 x^2}{x} \, dx}{32 e^5}\\ &=\frac {390625 \operatorname {Subst}\left (\int x^3 \, dx,x,\log \left (x^2\right )\right )}{4 e^{20}}+\frac {234375 \int \frac {\log ^2\left (x^2\right )}{x} \, dx}{4 e^{15}}-\frac {625 \int \log ^2\left (x^2\right ) \, dx}{8 e^{10}}+\frac {31875 \int \frac {\log \left (x^2\right )}{x} \, dx}{8 e^{10}}+\frac {\int \left (-1000-e^5+\frac {250}{x}+4 e^5 x\right ) \, dx}{32 e^5}-\frac {\left (125 \left (20+e^5\right )\right ) \int \log \left (x^2\right ) \, dx}{8 e^{10}}\\ &=\frac {125 \left (20+e^5\right ) x}{4 e^{10}}-\frac {\left (1000+e^5\right ) x}{32 e^5}+\frac {x^2}{16}+\frac {125 \log (x)}{16 e^5}-\frac {125 \left (20+e^5\right ) x \log \left (x^2\right )}{8 e^{10}}+\frac {31875 \log ^2\left (x^2\right )}{32 e^{10}}-\frac {625 x \log ^2\left (x^2\right )}{8 e^{10}}+\frac {390625 \log ^4\left (x^2\right )}{16 e^{20}}+\frac {234375 \operatorname {Subst}\left (\int x^2 \, dx,x,\log \left (x^2\right )\right )}{8 e^{15}}+\frac {625 \int \log \left (x^2\right ) \, dx}{2 e^{10}}\\ &=-\frac {625 x}{e^{10}}+\frac {125 \left (20+e^5\right ) x}{4 e^{10}}-\frac {\left (1000+e^5\right ) x}{32 e^5}+\frac {x^2}{16}+\frac {125 \log (x)}{16 e^5}+\frac {625 x \log \left (x^2\right )}{2 e^{10}}-\frac {125 \left (20+e^5\right ) x \log \left (x^2\right )}{8 e^{10}}+\frac {31875 \log ^2\left (x^2\right )}{32 e^{10}}-\frac {625 x \log ^2\left (x^2\right )}{8 e^{10}}+\frac {78125 \log ^3\left (x^2\right )}{8 e^{15}}+\frac {390625 \log ^4\left (x^2\right )}{16 e^{20}}\\ \end {aligned} \end {gather*}

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Mathematica [B]  time = 0.07, size = 81, normalized size = 3.12 \begin {gather*} \frac {-e^{20} x+2 e^{20} x^2+250 e^{15} \log (x)-500 e^{15} x \log \left (x^2\right )+31875 e^{10} \log ^2\left (x^2\right )-2500 e^{10} x \log ^2\left (x^2\right )+312500 e^5 \log ^3\left (x^2\right )+781250 \log ^4\left (x^2\right )}{32 e^{20}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^15*(250 - 1000*x) + E^20*(-x + 4*x^2) + (E^10*(127500 - 10000*x) - 500*E^15*x)*Log[x^2] + (187500
0*E^5 - 2500*E^10*x)*Log[x^2]^2 + 6250000*Log[x^2]^3)/(32*E^20*x),x]

[Out]

(-(E^20*x) + 2*E^20*x^2 + 250*E^15*Log[x] - 500*E^15*x*Log[x^2] + 31875*E^10*Log[x^2]^2 - 2500*E^10*x*Log[x^2]
^2 + 312500*E^5*Log[x^2]^3 + 781250*Log[x^2]^4)/(32*E^20)

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fricas [B]  time = 0.66, size = 64, normalized size = 2.46 \begin {gather*} -\frac {1}{32} \, {\left (625 \, {\left (4 \, x - 51\right )} e^{10} \log \left (x^{2}\right )^{2} - 312500 \, e^{5} \log \left (x^{2}\right )^{3} - 781250 \, \log \left (x^{2}\right )^{4} + 125 \, {\left (4 \, x - 1\right )} e^{15} \log \left (x^{2}\right ) - {\left (2 \, x^{2} - x\right )} e^{20}\right )} e^{\left (-20\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/32*(6250000*log(x^2)^3+(-2500*x*exp(5)^2+1875000*exp(5))*log(x^2)^2+(-500*x*exp(5)^3+(-10000*x+127
500)*exp(5)^2)*log(x^2)+(4*x^2-x)*exp(5)^4+(-1000*x+250)*exp(5)^3)/x/exp(5)^4,x, algorithm="fricas")

[Out]

-1/32*(625*(4*x - 51)*e^10*log(x^2)^2 - 312500*e^5*log(x^2)^3 - 781250*log(x^2)^4 + 125*(4*x - 1)*e^15*log(x^2
) - (2*x^2 - x)*e^20)*e^(-20)

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giac [B]  time = 0.34, size = 70, normalized size = 2.69 \begin {gather*} -\frac {1}{32} \, {\left (2500 \, x e^{10} \log \left (x^{2}\right )^{2} - 312500 \, e^{5} \log \left (x^{2}\right )^{3} - 781250 \, \log \left (x^{2}\right )^{4} - 2 \, x^{2} e^{20} + 500 \, x e^{15} \log \left (x^{2}\right ) - 31875 \, e^{10} \log \left (x^{2}\right )^{2} + x e^{20} - 250 \, e^{15} \log \relax (x)\right )} e^{\left (-20\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/32*(6250000*log(x^2)^3+(-2500*x*exp(5)^2+1875000*exp(5))*log(x^2)^2+(-500*x*exp(5)^3+(-10000*x+127
500)*exp(5)^2)*log(x^2)+(4*x^2-x)*exp(5)^4+(-1000*x+250)*exp(5)^3)/x/exp(5)^4,x, algorithm="giac")

[Out]

-1/32*(2500*x*e^10*log(x^2)^2 - 312500*e^5*log(x^2)^3 - 781250*log(x^2)^4 - 2*x^2*e^20 + 500*x*e^15*log(x^2) -
31875*e^10*log(x^2)^2 + x*e^20 - 250*e^15*log(x))*e^(-20)

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maple [A]  time = 0.09, size = 74, normalized size = 2.85

method	result	size
risch	\(\frac {x^{2}}{16}-\frac {x}{32}+\frac {125 \ln \relax (x ) {\mathrm e}^{-5}}{16}+\frac {390625 \,{\mathrm e}^{-20} \ln \left (x^{2}\right )^{4}}{16}-\frac {625 \,{\mathrm e}^{-10} \ln \left (x^{2}\right )^{2} x}{8}+\frac {78125 \,{\mathrm e}^{-15} \ln \left (x^{2}\right )^{3}}{8}-\frac {125 \,{\mathrm e}^{-5} \ln \left (x^{2}\right ) x}{8}-\frac {31875 \,{\mathrm e}^{-10} \ln \relax (x )^{2}}{8}+\frac {31875 \,{\mathrm e}^{-10} \ln \relax (x ) \ln \left (x^{2}\right )}{8}\)	\(74\)
norman	\(\left (\frac {125 \,{\mathrm e}^{10} \ln \left (x^{2}\right )}{32}+\frac {78125 \ln \left (x^{2}\right )^{3}}{8}-\frac {x \,{\mathrm e}^{15}}{32}+\frac {x^{2} {\mathrm e}^{15}}{16}+\frac {390625 \,{\mathrm e}^{-5} \ln \left (x^{2}\right )^{4}}{16}+\frac {31875 \,{\mathrm e}^{5} \ln \left (x^{2}\right )^{2}}{32}-\frac {625 x \,{\mathrm e}^{5} \ln \left (x^{2}\right )^{2}}{8}-\frac {125 \,{\mathrm e}^{10} \ln \left (x^{2}\right ) x}{8}\right ) {\mathrm e}^{-15}\)	\(85\)
default	\(\frac {{\mathrm e}^{-20} \left (2 x^{2} {\mathrm e}^{20}-x \,{\mathrm e}^{20}+250 \,{\mathrm e}^{15} \ln \relax (x )+781250 \ln \left (x^{2}\right )^{4}-2500 \,{\mathrm e}^{10} \ln \left (x^{2}\right )^{2} x +312500 \,{\mathrm e}^{5} \ln \left (x^{2}\right )^{3}-500 \,{\mathrm e}^{15} \ln \left (x^{2}\right ) x -127500 \,{\mathrm e}^{10} \ln \relax (x )^{2}+127500 \,{\mathrm e}^{10} \ln \relax (x ) \ln \left (x^{2}\right )\right )}{32}\)	\(96\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/32*(6250000*ln(x^2)^3+(-2500*x*exp(5)^2+1875000*exp(5))*ln(x^2)^2+(-500*x*exp(5)^3+(-10000*x+127500)*exp
(5)^2)*ln(x^2)+(4*x^2-x)*exp(5)^4+(-1000*x+250)*exp(5)^3)/x/exp(5)^4,x,method=_RETURNVERBOSE)

[Out]

1/16*x^2-1/32*x+125/16*ln(x)*exp(-5)+390625/16*exp(-20)*ln(x^2)^4-625/8*exp(-10)*ln(x^2)^2*x+78125/8*exp(-15)*
ln(x^2)^3-125/8*exp(-5)*ln(x^2)*x-31875/8*exp(-10)*ln(x)^2+31875/8*exp(-10)*ln(x)*ln(x^2)

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maxima [B]  time = 0.37, size = 80, normalized size = 3.08 \begin {gather*} -\frac {1}{32} \, {\left (2500 \, x e^{10} \log \left (x^{2}\right )^{2} - 312500 \, e^{5} \log \left (x^{2}\right )^{3} - 781250 \, \log \left (x^{2}\right )^{4} - 2 \, x^{2} e^{20} - 31875 \, e^{10} \log \left (x^{2}\right )^{2} + x e^{20} + 500 \, {\left (x \log \left (x^{2}\right ) - 2 \, x\right )} e^{15} + 1000 \, x e^{15} - 250 \, e^{15} \log \relax (x)\right )} e^{\left (-20\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/32*(6250000*log(x^2)^3+(-2500*x*exp(5)^2+1875000*exp(5))*log(x^2)^2+(-500*x*exp(5)^3+(-10000*x+127
500)*exp(5)^2)*log(x^2)+(4*x^2-x)*exp(5)^4+(-1000*x+250)*exp(5)^3)/x/exp(5)^4,x, algorithm="maxima")

[Out]

-1/32*(2500*x*e^10*log(x^2)^2 - 312500*e^5*log(x^2)^3 - 781250*log(x^2)^4 - 2*x^2*e^20 - 31875*e^10*log(x^2)^2
+ x*e^20 + 500*(x*log(x^2) - 2*x)*e^15 + 1000*x*e^15 - 250*e^15*log(x))*e^(-20)

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mupad [B]  time = 0.69, size = 67, normalized size = 2.58 \begin {gather*} \frac {x^2}{16}-\frac {625\,{\mathrm {e}}^{-10}\,x\,{\ln \left (x^2\right )}^2}{8}-\frac {125\,{\mathrm {e}}^{-5}\,x\,\ln \left (x^2\right )}{8}-\frac {x}{32}+\frac {390625\,{\mathrm {e}}^{-20}\,{\ln \left (x^2\right )}^4}{16}+\frac {78125\,{\mathrm {e}}^{-15}\,{\ln \left (x^2\right )}^3}{8}+\frac {31875\,{\mathrm {e}}^{-10}\,{\ln \left (x^2\right )}^2}{32}+\frac {125\,{\mathrm {e}}^{-5}\,\ln \left (x^2\right )}{32} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(-20)*((log(x^2)*(500*x*exp(15) + exp(10)*(10000*x - 127500)))/32 - (log(x^2)^2*(1875000*exp(5) - 250
0*x*exp(10)))/32 - (390625*log(x^2)^3)/2 + (exp(20)*(x - 4*x^2))/32 + (exp(15)*(1000*x - 250))/32))/x,x)

[Out]

(125*log(x^2)*exp(-5))/32 - x/32 + (31875*log(x^2)^2*exp(-10))/32 + (78125*log(x^2)^3*exp(-15))/8 + (390625*lo
g(x^2)^4*exp(-20))/16 + x^2/16 - (125*x*log(x^2)*exp(-5))/8 - (625*x*log(x^2)^2*exp(-10))/8

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sympy [B]  time = 0.27, size = 80, normalized size = 3.08 \begin {gather*} - \frac {125 x \log {\left (x^{2} \right )}}{8 e^{5}} + \frac {\left (31875 - 2500 x\right ) \log {\left (x^{2} \right )}^{2}}{32 e^{10}} + \frac {2 x^{2} e^{5} - x e^{5} + 250 \log {\relax (x )}}{32 e^{5}} + \frac {390625 \log {\left (x^{2} \right )}^{4}}{16 e^{20}} + \frac {78125 \log {\left (x^{2} \right )}^{3}}{8 e^{15}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/32*(6250000*ln(x**2)**3+(-2500*x*exp(5)**2+1875000*exp(5))*ln(x**2)**2+(-500*x*exp(5)**3+(-10000*x
+127500)*exp(5)**2)*ln(x**2)+(4*x**2-x)*exp(5)**4+(-1000*x+250)*exp(5)**3)/x/exp(5)**4,x)

[Out]

-125*x*exp(-5)*log(x**2)/8 + (31875 - 2500*x)*exp(-10)*log(x**2)**2/32 + (2*x**2*exp(5) - x*exp(5) + 250*log(x
))*exp(-5)/32 + 390625*exp(-20)*log(x**2)**4/16 + 78125*exp(-15)*log(x**2)**3/8

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