\(\int \frac {-100+(-20 x^2+50 x^3) \log ^2(3)+(-10 x^5+54 x^6-30 x^7+4 x^8) \log ^4(3)}{x^5 \log ^4(3)} \, dx\) [5913]

Optimal result

Integrand size = 52, antiderivative size = 26 \[ \int \frac {-100+\left (-20 x^2+50 x^3\right ) \log ^2(3)+\left (-10 x^5+54 x^6-30 x^7+4 x^8\right ) \log ^4(3)}{x^5 \log ^4(3)} \, dx=\log \left (3 e^{\left (-3+(-2+x)^2-x+\frac {5}{x^2 \log ^2(3)}\right )^2}\right ) \]

[Out]

ln(3*exp(((-2+x)^2-x+5/ln(3)^2/x^2-3)^2))

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.69, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.038, Rules used = {12, 14} \[ \int \frac {-100+\left (-20 x^2+50 x^3\right ) \log ^2(3)+\left (-10 x^5+54 x^6-30 x^7+4 x^8\right ) \log ^4(3)}{x^5 \log ^4(3)} \, dx=x^4+\frac {25}{x^4 \log ^4(3)}-10 x^3+27 x^2+\frac {10}{x^2 \log ^2(3)}-10 x-\frac {50}{x \log ^2(3)} \]

[In]

Int[(-100 + (-20*x^2 + 50*x^3)*Log[3]^2 + (-10*x^5 + 54*x^6 - 30*x^7 + 4*x^8)*Log[3]^4)/(x^5*Log[3]^4),x]

[Out]

-10*x + 27*x^2 - 10*x^3 + x^4 + 25/(x^4*Log[3]^4) + 10/(x^2*Log[3]^2) - 50/(x*Log[3]^2)

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]]

Rubi steps \begin{align*} \text {integral}& = \frac {\int \frac {-100+\left (-20 x^2+50 x^3\right ) \log ^2(3)+\left (-10 x^5+54 x^6-30 x^7+4 x^8\right ) \log ^4(3)}{x^5} \, dx}{\log ^4(3)} \\ & = \frac {\int \left (-\frac {100}{x^5}-\frac {20 \log ^2(3)}{x^3}+\frac {50 \log ^2(3)}{x^2}-10 \log ^4(3)+54 x \log ^4(3)-30 x^2 \log ^4(3)+4 x^3 \log ^4(3)\right ) \, dx}{\log ^4(3)} \\ & = -10 x+27 x^2-10 x^3+x^4+\frac {25}{x^4 \log ^4(3)}+\frac {10}{x^2 \log ^2(3)}-\frac {50}{x \log ^2(3)} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.69 \[ \int \frac {-100+\left (-20 x^2+50 x^3\right ) \log ^2(3)+\left (-10 x^5+54 x^6-30 x^7+4 x^8\right ) \log ^4(3)}{x^5 \log ^4(3)} \, dx=-10 x+27 x^2-10 x^3+x^4+\frac {25}{x^4 \log ^4(3)}+\frac {10}{x^2 \log ^2(3)}-\frac {50}{x \log ^2(3)} \]

[In]

Integrate[(-100 + (-20*x^2 + 50*x^3)*Log[3]^2 + (-10*x^5 + 54*x^6 - 30*x^7 + 4*x^8)*Log[3]^4)/(x^5*Log[3]^4),x
]

[Out]

-10*x + 27*x^2 - 10*x^3 + x^4 + 25/(x^4*Log[3]^4) + 10/(x^2*Log[3]^2) - 50/(x*Log[3]^2)

Maple [A] (verified)

Time = 0.12 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.54

method	result	size
risch	\(\left (x^{2}-5 x +1\right )^{2}+\frac {-50 x^{3} \ln \left (3\right )^{2}+10 x^{2} \ln \left (3\right )^{2}+25}{\ln \left (3\right )^{4} x^{4}}\)	\(40\)
default	\(\frac {x^{4} \ln \left (3\right )^{4}-10 \ln \left (3\right )^{4} x^{3}+27 x^{2} \ln \left (3\right )^{4}-10 x \ln \left (3\right )^{4}+\frac {25}{x^{4}}-\frac {50 \ln \left (3\right )^{2}}{x}+\frac {10 \ln \left (3\right )^{2}}{x^{2}}}{\ln \left (3\right )^{4}}\)	\(63\)
gosper	\(\frac {x^{8} \ln \left (3\right )^{4}-10 \ln \left (3\right )^{4} x^{7}+27 x^{6} \ln \left (3\right )^{4}-10 x^{5} \ln \left (3\right )^{4}-50 x^{3} \ln \left (3\right )^{2}+10 x^{2} \ln \left (3\right )^{2}+25}{\ln \left (3\right )^{4} x^{4}}\)	\(64\)
parallelrisch	\(\frac {x^{8} \ln \left (3\right )^{4}-10 \ln \left (3\right )^{4} x^{7}+27 x^{6} \ln \left (3\right )^{4}-10 x^{5} \ln \left (3\right )^{4}-50 x^{3} \ln \left (3\right )^{2}+10 x^{2} \ln \left (3\right )^{2}+25}{\ln \left (3\right )^{4} x^{4}}\)	\(64\)
norman	\(\frac {\ln \left (3\right )^{3} x^{8}+\frac {25}{\ln \left (3\right )}+10 x^{2} \ln \left (3\right )-50 x^{3} \ln \left (3\right )-10 \ln \left (3\right )^{3} x^{5}+27 \ln \left (3\right )^{3} x^{6}-10 \ln \left (3\right )^{3} x^{7}}{x^{4} \ln \left (3\right )^{3}}\)	\(65\)

[In]

int(((4*x^8-30*x^7+54*x^6-10*x^5)*ln(3)^4+(50*x^3-20*x^2)*ln(3)^2-100)/x^5/ln(3)^4,x,method=_RETURNVERBOSE)

[Out]

(x^2-5*x+1)^2+1/ln(3)^4*(-50*x^3*ln(3)^2+10*x^2*ln(3)^2+25)/x^4

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 51 vs. \(2 (25) = 50\).

Time = 0.25 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.96 \[ \int \frac {-100+\left (-20 x^2+50 x^3\right ) \log ^2(3)+\left (-10 x^5+54 x^6-30 x^7+4 x^8\right ) \log ^4(3)}{x^5 \log ^4(3)} \, dx=\frac {{\left (x^{8} - 10 \, x^{7} + 27 \, x^{6} - 10 \, x^{5}\right )} \log \left (3\right )^{4} - 10 \, {\left (5 \, x^{3} - x^{2}\right )} \log \left (3\right )^{2} + 25}{x^{4} \log \left (3\right )^{4}} \]

[In]

integrate(((4*x^8-30*x^7+54*x^6-10*x^5)*log(3)^4+(50*x^3-20*x^2)*log(3)^2-100)/x^5/log(3)^4,x, algorithm="fric
as")

[Out]

((x^8 - 10*x^7 + 27*x^6 - 10*x^5)*log(3)^4 - 10*(5*x^3 - x^2)*log(3)^2 + 25)/(x^4*log(3)^4)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 66 vs. \(2 (24) = 48\).

Time = 0.13 (sec) , antiderivative size = 66, normalized size of antiderivative = 2.54 \[ \int \frac {-100+\left (-20 x^2+50 x^3\right ) \log ^2(3)+\left (-10 x^5+54 x^6-30 x^7+4 x^8\right ) \log ^4(3)}{x^5 \log ^4(3)} \, dx=\frac {x^{4} \log {\left (3 \right )}^{4} - 10 x^{3} \log {\left (3 \right )}^{4} + 27 x^{2} \log {\left (3 \right )}^{4} - 10 x \log {\left (3 \right )}^{4} + \frac {- 50 x^{3} \log {\left (3 \right )}^{2} + 10 x^{2} \log {\left (3 \right )}^{2} + 25}{x^{4}}}{\log {\left (3 \right )}^{4}} \]

[In]

integrate(((4*x**8-30*x**7+54*x**6-10*x**5)*ln(3)**4+(50*x**3-20*x**2)*ln(3)**2-100)/x**5/ln(3)**4,x)

[Out]

(x**4*log(3)**4 - 10*x**3*log(3)**4 + 27*x**2*log(3)**4 - 10*x*log(3)**4 + (-50*x**3*log(3)**2 + 10*x**2*log(3
)**2 + 25)/x**4)/log(3)**4

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 64 vs. \(2 (25) = 50\).

Time = 0.18 (sec) , antiderivative size = 64, normalized size of antiderivative = 2.46 \[ \int \frac {-100+\left (-20 x^2+50 x^3\right ) \log ^2(3)+\left (-10 x^5+54 x^6-30 x^7+4 x^8\right ) \log ^4(3)}{x^5 \log ^4(3)} \, dx=\frac {x^{4} \log \left (3\right )^{4} - 10 \, x^{3} \log \left (3\right )^{4} + 27 \, x^{2} \log \left (3\right )^{4} - 10 \, x \log \left (3\right )^{4} - \frac {5 \, {\left (10 \, x^{3} \log \left (3\right )^{2} - 2 \, x^{2} \log \left (3\right )^{2} - 5\right )}}{x^{4}}}{\log \left (3\right )^{4}} \]

[In]

integrate(((4*x^8-30*x^7+54*x^6-10*x^5)*log(3)^4+(50*x^3-20*x^2)*log(3)^2-100)/x^5/log(3)^4,x, algorithm="maxi
ma")

[Out]

(x^4*log(3)^4 - 10*x^3*log(3)^4 + 27*x^2*log(3)^4 - 10*x*log(3)^4 - 5*(10*x^3*log(3)^2 - 2*x^2*log(3)^2 - 5)/x
^4)/log(3)^4

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 64 vs. \(2 (25) = 50\).

Time = 0.27 (sec) , antiderivative size = 64, normalized size of antiderivative = 2.46 \[ \int \frac {-100+\left (-20 x^2+50 x^3\right ) \log ^2(3)+\left (-10 x^5+54 x^6-30 x^7+4 x^8\right ) \log ^4(3)}{x^5 \log ^4(3)} \, dx=\frac {x^{4} \log \left (3\right )^{4} - 10 \, x^{3} \log \left (3\right )^{4} + 27 \, x^{2} \log \left (3\right )^{4} - 10 \, x \log \left (3\right )^{4} - \frac {5 \, {\left (10 \, x^{3} \log \left (3\right )^{2} - 2 \, x^{2} \log \left (3\right )^{2} - 5\right )}}{x^{4}}}{\log \left (3\right )^{4}} \]

[In]

integrate(((4*x^8-30*x^7+54*x^6-10*x^5)*log(3)^4+(50*x^3-20*x^2)*log(3)^2-100)/x^5/log(3)^4,x, algorithm="giac
")

[Out]

(x^4*log(3)^4 - 10*x^3*log(3)^4 + 27*x^2*log(3)^4 - 10*x*log(3)^4 - 5*(10*x^3*log(3)^2 - 2*x^2*log(3)^2 - 5)/x
^4)/log(3)^4

Mupad [B] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.73 \[ \int \frac {-100+\left (-20 x^2+50 x^3\right ) \log ^2(3)+\left (-10 x^5+54 x^6-30 x^7+4 x^8\right ) \log ^4(3)}{x^5 \log ^4(3)} \, dx=27\,x^2-10\,x-10\,x^3+x^4+\frac {-50\,{\ln \left (3\right )}^2\,x^3+10\,{\ln \left (3\right )}^2\,x^2+25}{x^4\,{\ln \left (3\right )}^4} \]

[In]

int(-(log(3)^4*(10*x^5 - 54*x^6 + 30*x^7 - 4*x^8) + log(3)^2*(20*x^2 - 50*x^3) + 100)/(x^5*log(3)^4),x)

[Out]

27*x^2 - 10*x - 10*x^3 + x^4 + (10*x^2*log(3)^2 - 50*x^3*log(3)^2 + 25)/(x^4*log(3)^4)