3.76.41 \(\int \frac {7500 x+9000 x^2+e^2 (-25000-30000 x) \log ^3(x)-7500 e^2 x \log ^4(x)+25000 e^4 \log ^7(x)}{-216 x^4+540 e^2 x^3 \log ^4(x)-450 e^4 x^2 \log ^8(x)+125 e^6 x \log ^{12}(x)} \, dx\)

Optimal. Leaf size=22 \[ \left (-5+\frac {5}{-\frac {6 x}{5}+e^2 \log ^4(x)}\right )^2 \]

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Rubi [A]  time = 0.54, antiderivative size = 35, normalized size of antiderivative = 1.59, number of steps used = 6, number of rules used = 4, integrand size = 84, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.048, Rules used = {6688, 12, 6742, 2544} \begin {gather*} \frac {250}{6 x-5 e^2 \log ^4(x)}+\frac {625}{\left (6 x-5 e^2 \log ^4(x)\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(7500*x + 9000*x^2 + E^2*(-25000 - 30000*x)*Log[x]^3 - 7500*E^2*x*Log[x]^4 + 25000*E^4*Log[x]^7)/(-216*x^4
+ 540*E^2*x^3*Log[x]^4 - 450*E^4*x^2*Log[x]^8 + 125*E^6*x*Log[x]^12),x]

[Out]

625/(6*x - 5*E^2*Log[x]^4)^2 + 250/(6*x - 5*E^2*Log[x]^4)

Rule 12

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

Rule 2544

Int[((Log[(c_.)*(x_)^(n_.)]^(q_)*(b_.) + (a_.)*(x_)^(m_.))^(p_.)*(Log[(c_.)*(x_)^(n_.)]^(r_.)*(e_.) + (d_.)*(x
_)^(m_.)))/(x_), x_Symbol] :> Simp[(e*(a*x^m + b*Log[c*x^n]^q)^(p + 1))/(b*n*q*(p + 1)), x] /; FreeQ[{a, b, c,
d, e, m, n, p, q, r}, x] && EqQ[r, q - 1] && NeQ[p, -1] && EqQ[a*e*m - b*d*n*q, 0]

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} &=\int \frac {500 \left (-3 x (5+6 x)+10 e^2 (5+6 x) \log ^3(x)+15 e^2 x \log ^4(x)-50 e^4 \log ^7(x)\right )}{x \left (6 x-5 e^2 \log ^4(x)\right )^3} \, dx\\ &=500 \int \frac {-3 x (5+6 x)+10 e^2 (5+6 x) \log ^3(x)+15 e^2 x \log ^4(x)-50 e^4 \log ^7(x)}{x \left (6 x-5 e^2 \log ^4(x)\right )^3} \, dx\\ &=500 \int \left (-\frac {5 \left (3 x-10 e^2 \log ^3(x)\right )}{x \left (6 x-5 e^2 \log ^4(x)\right )^3}+\frac {-3 x+10 e^2 \log ^3(x)}{x \left (6 x-5 e^2 \log ^4(x)\right )^2}\right ) \, dx\\ &=500 \int \frac {-3 x+10 e^2 \log ^3(x)}{x \left (6 x-5 e^2 \log ^4(x)\right )^2} \, dx-2500 \int \frac {3 x-10 e^2 \log ^3(x)}{x \left (6 x-5 e^2 \log ^4(x)\right )^3} \, dx\\ &=\frac {625}{\left (6 x-5 e^2 \log ^4(x)\right )^2}+\frac {250}{6 x-5 e^2 \log ^4(x)}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.33, size = 31, normalized size = 1.41 \begin {gather*} -\frac {125 \left (-5-12 x+10 e^2 \log ^4(x)\right )}{\left (6 x-5 e^2 \log ^4(x)\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(7500*x + 9000*x^2 + E^2*(-25000 - 30000*x)*Log[x]^3 - 7500*E^2*x*Log[x]^4 + 25000*E^4*Log[x]^7)/(-2
16*x^4 + 540*E^2*x^3*Log[x]^4 - 450*E^4*x^2*Log[x]^8 + 125*E^6*x*Log[x]^12),x]

[Out]

(-125*(-5 - 12*x + 10*E^2*Log[x]^4))/(6*x - 5*E^2*Log[x]^4)^2

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fricas [A]  time = 0.54, size = 40, normalized size = 1.82 \begin {gather*} -\frac {125 \, {\left (10 \, e^{2} \log \relax (x)^{4} - 12 \, x - 5\right )}}{25 \, e^{4} \log \relax (x)^{8} - 60 \, x e^{2} \log \relax (x)^{4} + 36 \, x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((25000*exp(1)^4*log(x)^7-7500*x*exp(1)^2*log(x)^4+(-30000*x-25000)*exp(1)^2*log(x)^3+9000*x^2+7500*x
)/(125*x*exp(1)^6*log(x)^12-450*x^2*exp(1)^4*log(x)^8+540*x^3*exp(1)^2*log(x)^4-216*x^4),x, algorithm="fricas"
)

[Out]

-125*(10*e^2*log(x)^4 - 12*x - 5)/(25*e^4*log(x)^8 - 60*x*e^2*log(x)^4 + 36*x^2)

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giac [A]  time = 1.29, size = 40, normalized size = 1.82 \begin {gather*} -\frac {250 \, {\left (10 \, e^{2} \log \relax (x)^{4} - 12 \, x - 5\right )}}{25 \, e^{4} \log \relax (x)^{8} - 60 \, x e^{2} \log \relax (x)^{4} + 36 \, x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((25000*exp(1)^4*log(x)^7-7500*x*exp(1)^2*log(x)^4+(-30000*x-25000)*exp(1)^2*log(x)^3+9000*x^2+7500*x
)/(125*x*exp(1)^6*log(x)^12-450*x^2*exp(1)^4*log(x)^8+540*x^3*exp(1)^2*log(x)^4-216*x^4),x, algorithm="giac")

[Out]

-250*(10*e^2*log(x)^4 - 12*x - 5)/(25*e^4*log(x)^8 - 60*x*e^2*log(x)^4 + 36*x^2)

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maple [A]  time = 0.04, size = 30, normalized size = 1.36

method	result	size
risch	\(-\frac {125 \left (10 \,{\mathrm e}^{2} \ln \relax (x )^{4}-12 x -5\right )}{\left (5 \,{\mathrm e}^{2} \ln \relax (x )^{4}-6 x \right )^{2}}\)	\(30\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((25000*exp(1)^4*ln(x)^7-7500*x*exp(1)^2*ln(x)^4+(-30000*x-25000)*exp(1)^2*ln(x)^3+9000*x^2+7500*x)/(125*x*
exp(1)^6*ln(x)^12-450*x^2*exp(1)^4*ln(x)^8+540*x^3*exp(1)^2*ln(x)^4-216*x^4),x,method=_RETURNVERBOSE)

[Out]

-125*(10*exp(2)*ln(x)^4-12*x-5)/(5*exp(2)*ln(x)^4-6*x)^2

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maxima [A]  time = 0.40, size = 40, normalized size = 1.82 \begin {gather*} -\frac {125 \, {\left (10 \, e^{2} \log \relax (x)^{4} - 12 \, x - 5\right )}}{25 \, e^{4} \log \relax (x)^{8} - 60 \, x e^{2} \log \relax (x)^{4} + 36 \, x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((25000*exp(1)^4*log(x)^7-7500*x*exp(1)^2*log(x)^4+(-30000*x-25000)*exp(1)^2*log(x)^3+9000*x^2+7500*x
)/(125*x*exp(1)^6*log(x)^12-450*x^2*exp(1)^4*log(x)^8+540*x^3*exp(1)^2*log(x)^4-216*x^4),x, algorithm="maxima"
)

[Out]

-125*(10*e^2*log(x)^4 - 12*x - 5)/(25*e^4*log(x)^8 - 60*x*e^2*log(x)^4 + 36*x^2)

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mupad [B]  time = 5.73, size = 29, normalized size = 1.32 \begin {gather*} \frac {125\,\left (-10\,{\mathrm {e}}^2\,{\ln \relax (x)}^4+12\,x+5\right )}{{\left (6\,x-5\,{\mathrm {e}}^2\,{\ln \relax (x)}^4\right )}^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(7500*x + 25000*exp(4)*log(x)^7 + 9000*x^2 - exp(2)*log(x)^3*(30000*x + 25000) - 7500*x*exp(2)*log(x)^4)/
(216*x^4 - 125*x*exp(6)*log(x)^12 - 540*x^3*exp(2)*log(x)^4 + 450*x^2*exp(4)*log(x)^8),x)

[Out]

(125*(12*x - 10*exp(2)*log(x)^4 + 5))/(6*x - 5*exp(2)*log(x)^4)^2

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sympy [B]  time = 0.18, size = 41, normalized size = 1.86 \begin {gather*} \frac {1500 x - 1250 e^{2} \log {\relax (x )}^{4} + 625}{36 x^{2} - 60 x e^{2} \log {\relax (x )}^{4} + 25 e^{4} \log {\relax (x )}^{8}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((25000*exp(1)**4*ln(x)**7-7500*x*exp(1)**2*ln(x)**4+(-30000*x-25000)*exp(1)**2*ln(x)**3+9000*x**2+75
00*x)/(125*x*exp(1)**6*ln(x)**12-450*x**2*exp(1)**4*ln(x)**8+540*x**3*exp(1)**2*ln(x)**4-216*x**4),x)

[Out]

(1500*x - 1250*exp(2)*log(x)**4 + 625)/(36*x**2 - 60*x*exp(2)*log(x)**4 + 25*exp(4)*log(x)**8)

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