3.103.21 \(\int \frac {3^{-\frac {2}{-5-x-x^2+\log (x)}} (-4050-1620 x-1782 x^2-324 x^3-162 x^4+(54-54 x-108 x^2) \log (3)+(1620+324 x+324 x^2) \log (x)-162 \log ^2(x))}{25 x^7+10 x^8+11 x^9+2 x^{10}+x^{11}+(-10 x^7-2 x^8-2 x^9) \log (x)+x^7 \log ^2(x)} \, dx\)

Optimal. Leaf size=24 \[ -3+\frac {3^{3+\frac {2}{5+x+x^2-\log (x)}}}{x^6} \]

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Rubi [B]  time = 0.26, antiderivative size = 104, normalized size of antiderivative = 4.33, number of steps used = 1, number of rules used = 1, integrand size = 124, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.008, Rules used = {2288} \begin {gather*} -\frac {\left (-2 x^2-x+1\right ) 3^{\frac {2}{x^2+x-\log (x)+5}+3} \left (x^2+x-\log (x)+5\right )^2}{\left (2 x-\frac {1}{x}+1\right ) \left (x^{11}+2 x^{10}+11 x^9+10 x^8+25 x^7+x^7 \log ^2(x)-2 \left (x^9+x^8+5 x^7\right ) \log (x)\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-4050 - 1620*x - 1782*x^2 - 324*x^3 - 162*x^4 + (54 - 54*x - 108*x^2)*Log[3] + (1620 + 324*x + 324*x^2)*L
og[x] - 162*Log[x]^2)/(3^(2/(-5 - x - x^2 + Log[x]))*(25*x^7 + 10*x^8 + 11*x^9 + 2*x^10 + x^11 + (-10*x^7 - 2*
x^8 - 2*x^9)*Log[x] + x^7*Log[x]^2)),x]

[Out]

-((3^(3 + 2/(5 + x + x^2 - Log[x]))*(1 - x - 2*x^2)*(5 + x + x^2 - Log[x])^2)/((1 - x^(-1) + 2*x)*(25*x^7 + 10
*x^8 + 11*x^9 + 2*x^10 + x^11 - 2*(5*x^7 + x^8 + x^9)*Log[x] + x^7*Log[x]^2)))

Rule 2288

Int[(y_.)*(F_)^(u_)*((v_) + (w_)), x_Symbol] :> With[{z = (v*y)/(Log[F]*D[u, x])}, Simp[F^u*z, x] /; EqQ[D[z,
x], w*y]] /; FreeQ[F, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=-\frac {3^{3+\frac {2}{5+x+x^2-\log (x)}} \left (1-x-2 x^2\right ) \left (5+x+x^2-\log (x)\right )^2}{\left (1-\frac {1}{x}+2 x\right ) \left (25 x^7+10 x^8+11 x^9+2 x^{10}+x^{11}-2 \left (5 x^7+x^8+x^9\right ) \log (x)+x^7 \log ^2(x)\right )}\\ \end {aligned} \end {gather*}

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Mathematica [F]  time = 3.50, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {3^{-\frac {2}{-5-x-x^2+\log (x)}} \left (-4050-1620 x-1782 x^2-324 x^3-162 x^4+\left (54-54 x-108 x^2\right ) \log (3)+\left (1620+324 x+324 x^2\right ) \log (x)-162 \log ^2(x)\right )}{25 x^7+10 x^8+11 x^9+2 x^{10}+x^{11}+\left (-10 x^7-2 x^8-2 x^9\right ) \log (x)+x^7 \log ^2(x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[(-4050 - 1620*x - 1782*x^2 - 324*x^3 - 162*x^4 + (54 - 54*x - 108*x^2)*Log[3] + (1620 + 324*x + 324*
x^2)*Log[x] - 162*Log[x]^2)/(3^(2/(-5 - x - x^2 + Log[x]))*(25*x^7 + 10*x^8 + 11*x^9 + 2*x^10 + x^11 + (-10*x^
7 - 2*x^8 - 2*x^9)*Log[x] + x^7*Log[x]^2)),x]

[Out]

Integrate[(-4050 - 1620*x - 1782*x^2 - 324*x^3 - 162*x^4 + (54 - 54*x - 108*x^2)*Log[3] + (1620 + 324*x + 324*
x^2)*Log[x] - 162*Log[x]^2)/(3^(2/(-5 - x - x^2 + Log[x]))*(25*x^7 + 10*x^8 + 11*x^9 + 2*x^10 + x^11 + (-10*x^
7 - 2*x^8 - 2*x^9)*Log[x] + x^7*Log[x]^2)), x]

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fricas [A]  time = 0.53, size = 21, normalized size = 0.88 \begin {gather*} \frac {27 \cdot 3^{\frac {2}{x^{2} + x - \log \relax (x) + 5}}}{x^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-162*log(x)^2+(324*x^2+324*x+1620)*log(x)+(-108*x^2-54*x+54)*log(3)-162*x^4-324*x^3-1782*x^2-1620*x
-4050)*exp(-log(3)/(log(x)-x^2-x-5))^2/(x^7*log(x)^2+(-2*x^9-2*x^8-10*x^7)*log(x)+x^11+2*x^10+11*x^9+10*x^8+25
*x^7),x, algorithm="fricas")

[Out]

27*3^(2/(x^2 + x - log(x) + 5))/x^6

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int -\frac {54 \, {\left (3 \, x^{4} + 6 \, x^{3} + 33 \, x^{2} + {\left (2 \, x^{2} + x - 1\right )} \log \relax (3) - 6 \, {\left (x^{2} + x + 5\right )} \log \relax (x) + 3 \, \log \relax (x)^{2} + 30 \, x + 75\right )} 3^{\frac {2}{x^{2} + x - \log \relax (x) + 5}}}{x^{11} + 2 \, x^{10} + 11 \, x^{9} + x^{7} \log \relax (x)^{2} + 10 \, x^{8} + 25 \, x^{7} - 2 \, {\left (x^{9} + x^{8} + 5 \, x^{7}\right )} \log \relax (x)}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-162*log(x)^2+(324*x^2+324*x+1620)*log(x)+(-108*x^2-54*x+54)*log(3)-162*x^4-324*x^3-1782*x^2-1620*x
-4050)*exp(-log(3)/(log(x)-x^2-x-5))^2/(x^7*log(x)^2+(-2*x^9-2*x^8-10*x^7)*log(x)+x^11+2*x^10+11*x^9+10*x^8+25
*x^7),x, algorithm="giac")

[Out]

integrate(-54*(3*x^4 + 6*x^3 + 33*x^2 + (2*x^2 + x - 1)*log(3) - 6*(x^2 + x + 5)*log(x) + 3*log(x)^2 + 30*x +
75)*3^(2/(x^2 + x - log(x) + 5))/(x^11 + 2*x^10 + 11*x^9 + x^7*log(x)^2 + 10*x^8 + 25*x^7 - 2*(x^9 + x^8 + 5*x
^7)*log(x)), x)

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maple [A]  time = 0.08, size = 24, normalized size = 1.00

method	result	size
risch	\(\frac {27 \,3^{-\frac {2}{\ln \relax (x )-x^{2}-x -5}}}{x^{6}}\)	\(24\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-162*ln(x)^2+(324*x^2+324*x+1620)*ln(x)+(-108*x^2-54*x+54)*ln(3)-162*x^4-324*x^3-1782*x^2-1620*x-4050)*ex
p(-ln(3)/(ln(x)-x^2-x-5))^2/(x^7*ln(x)^2+(-2*x^9-2*x^8-10*x^7)*ln(x)+x^11+2*x^10+11*x^9+10*x^8+25*x^7),x,metho
d=_RETURNVERBOSE)

[Out]

27/x^6*((1/3)^(1/(ln(x)-x^2-x-5)))^2

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} -\frac {27 \cdot 3^{\frac {2}{x^{2} + x - \log \relax (x) + 5}} \log \relax (3)}{2 \, x^{8} \log \relax (3) + x^{7} \log \relax (3) - x^{6} \log \relax (3)} + \frac {27 \cdot 3^{\frac {2}{x^{2} + x - \log \relax (x) + 5}} \log \relax (3)}{2 \, x^{7} \log \relax (3) + x^{6} \log \relax (3) - x^{5} \log \relax (3)} + \frac {54 \cdot 3^{\frac {2}{x^{2} + x - \log \relax (x) + 5}} \log \relax (3)}{2 \, x^{6} \log \relax (3) + x^{5} \log \relax (3) - x^{4} \log \relax (3)} + \frac {2025 \cdot 3^{\frac {2}{x^{2} + x - \log \relax (x) + 5}}}{2 \, x^{8} \log \relax (3) + x^{7} \log \relax (3) - x^{6} \log \relax (3)} + \frac {810 \cdot 3^{\frac {2}{x^{2} + x - \log \relax (x) + 5}}}{2 \, x^{7} \log \relax (3) + x^{6} \log \relax (3) - x^{5} \log \relax (3)} + \frac {891 \cdot 3^{\frac {2}{x^{2} + x - \log \relax (x) + 5}}}{2 \, x^{6} \log \relax (3) + x^{5} \log \relax (3) - x^{4} \log \relax (3)} + \frac {162 \cdot 3^{\frac {2}{x^{2} + x - \log \relax (x) + 5}}}{2 \, x^{5} \log \relax (3) + x^{4} \log \relax (3) - x^{3} \log \relax (3)} + \frac {81 \cdot 3^{\frac {2}{x^{2} + x - \log \relax (x) + 5}}}{2 \, x^{4} \log \relax (3) + x^{3} \log \relax (3) - x^{2} \log \relax (3)} - 2 \, \int -\frac {81 \, {\left (2 \, {\left (x^{2} + x + 5\right )} \log \relax (x) - \log \relax (x)^{2}\right )} 3^{\frac {2}{x^{2} + x - \log \relax (x) + 5}}}{x^{11} + 2 \, x^{10} + 11 \, x^{9} + x^{7} \log \relax (x)^{2} + 10 \, x^{8} + 25 \, x^{7} - 2 \, {\left (x^{9} + x^{8} + 5 \, x^{7}\right )} \log \relax (x)}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-162*log(x)^2+(324*x^2+324*x+1620)*log(x)+(-108*x^2-54*x+54)*log(3)-162*x^4-324*x^3-1782*x^2-1620*x
-4050)*exp(-log(3)/(log(x)-x^2-x-5))^2/(x^7*log(x)^2+(-2*x^9-2*x^8-10*x^7)*log(x)+x^11+2*x^10+11*x^9+10*x^8+25
*x^7),x, algorithm="maxima")

[Out]

-27*3^(2/(x^2 + x - log(x) + 5))*log(3)/(2*x^8*log(3) + x^7*log(3) - x^6*log(3)) + 27*3^(2/(x^2 + x - log(x) +
5))*log(3)/(2*x^7*log(3) + x^6*log(3) - x^5*log(3)) + 54*3^(2/(x^2 + x - log(x) + 5))*log(3)/(2*x^6*log(3) +
x^5*log(3) - x^4*log(3)) + 2025*3^(2/(x^2 + x - log(x) + 5))/(2*x^8*log(3) + x^7*log(3) - x^6*log(3)) + 810*3^
(2/(x^2 + x - log(x) + 5))/(2*x^7*log(3) + x^6*log(3) - x^5*log(3)) + 891*3^(2/(x^2 + x - log(x) + 5))/(2*x^6*
log(3) + x^5*log(3) - x^4*log(3)) + 162*3^(2/(x^2 + x - log(x) + 5))/(2*x^5*log(3) + x^4*log(3) - x^3*log(3))
+ 81*3^(2/(x^2 + x - log(x) + 5))/(2*x^4*log(3) + x^3*log(3) - x^2*log(3)) - 2*integrate(-81*(2*(x^2 + x + 5)*
log(x) - log(x)^2)*3^(2/(x^2 + x - log(x) + 5))/(x^11 + 2*x^10 + 11*x^9 + x^7*log(x)^2 + 10*x^8 + 25*x^7 - 2*(
x^9 + x^8 + 5*x^7)*log(x)), x)

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mupad [B]  time = 6.62, size = 21, normalized size = 0.88 \begin {gather*} \frac {27\,3^{\frac {2}{x-\ln \relax (x)+x^2+5}}}{x^6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp((2*log(3))/(x - log(x) + x^2 + 5))*(1620*x + log(3)*(54*x + 108*x^2 - 54) + 162*log(x)^2 - log(x)*(3
24*x + 324*x^2 + 1620) + 1782*x^2 + 324*x^3 + 162*x^4 + 4050))/(x^7*log(x)^2 - log(x)*(10*x^7 + 2*x^8 + 2*x^9)
+ 25*x^7 + 10*x^8 + 11*x^9 + 2*x^10 + x^11),x)

[Out]

(27*3^(2/(x - log(x) + x^2 + 5)))/x^6

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sympy [A]  time = 0.64, size = 20, normalized size = 0.83 \begin {gather*} \frac {27 e^{- \frac {2 \log {\relax (3 )}}{- x^{2} - x + \log {\relax (x )} - 5}}}{x^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-162*ln(x)**2+(324*x**2+324*x+1620)*ln(x)+(-108*x**2-54*x+54)*ln(3)-162*x**4-324*x**3-1782*x**2-162
0*x-4050)*exp(-ln(3)/(ln(x)-x**2-x-5))**2/(x**7*ln(x)**2+(-2*x**9-2*x**8-10*x**7)*ln(x)+x**11+2*x**10+11*x**9+
10*x**8+25*x**7),x)

[Out]

27*exp(-2*log(3)/(-x**2 - x + log(x) - 5))/x**6

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