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

Optimal. Leaf size=26 \[ \log \left (3 e^{\left (-3+(-2+x)^2-x+\frac {5}{x^2 \log ^2(3)}\right )^2}\right ) \]

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Rubi [A]  time = 0.03, antiderivative size = 44, normalized size of antiderivative = 1.69, number of steps used = 3, number of rules used = 2, integrand size = 52, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.038, Rules used = {12, 14} \begin {gather*} 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)} \end {gather*}

Antiderivative was successfully verified.

[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 {gather*} \begin {aligned} \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 {aligned} \end {gather*}

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Mathematica [A]  time = 0.01, size = 44, normalized size = 1.69 \begin {gather*} -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 {gather*}

Antiderivative was successfully verified.

[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)

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fricas [B]  time = 0.72, size = 51, normalized size = 1.96 \begin {gather*} \frac {{\left (x^{8} - 10 \, x^{7} + 27 \, x^{6} - 10 \, x^{5}\right )} \log \relax (3)^{4} - 10 \, {\left (5 \, x^{3} - x^{2}\right )} \log \relax (3)^{2} + 25}{x^{4} \log \relax (3)^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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giac [B]  time = 0.21, size = 64, normalized size = 2.46 \begin {gather*} \frac {x^{4} \log \relax (3)^{4} - 10 \, x^{3} \log \relax (3)^{4} + 27 \, x^{2} \log \relax (3)^{4} - 10 \, x \log \relax (3)^{4} - \frac {5 \, {\left (10 \, x^{3} \log \relax (3)^{2} - 2 \, x^{2} \log \relax (3)^{2} - 5\right )}}{x^{4}}}{\log \relax (3)^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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maple [A]  time = 0.10, size = 46, normalized size = 1.77

method	result	size
risch	\(x^{4}-10 x^{3}+27 x^{2}-10 x +\frac {-50 x^{3} \ln \relax (3)^{2}+10 x^{2} \ln \relax (3)^{2}+25}{\ln \relax (3)^{4} x^{4}}\)	\(46\)
default	\(\frac {x^{4} \ln \relax (3)^{4}-10 \ln \relax (3)^{4} x^{3}+27 x^{2} \ln \relax (3)^{4}-10 x \ln \relax (3)^{4}+\frac {25}{x^{4}}-\frac {50 \ln \relax (3)^{2}}{x}+\frac {10 \ln \relax (3)^{2}}{x^{2}}}{\ln \relax (3)^{4}}\)	\(63\)
gosper	\(\frac {x^{8} \ln \relax (3)^{4}-10 \ln \relax (3)^{4} x^{7}+27 x^{6} \ln \relax (3)^{4}-10 x^{5} \ln \relax (3)^{4}-50 x^{3} \ln \relax (3)^{2}+10 x^{2} \ln \relax (3)^{2}+25}{\ln \relax (3)^{4} x^{4}}\)	\(64\)
norman	\(\frac {\ln \relax (3)^{3} x^{8}+\frac {25}{\ln \relax (3)}+10 x^{2} \ln \relax (3)-50 x^{3} \ln \relax (3)-10 \ln \relax (3)^{3} x^{5}+27 \ln \relax (3)^{3} x^{6}-10 \ln \relax (3)^{3} x^{7}}{x^{4} \ln \relax (3)^{3}}\)	\(65\)

Verification of antiderivative is not currently implemented for this CAS.

[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^4-10*x^3+27*x^2-10*x+1/ln(3)^4*(-50*x^3*ln(3)^2+10*x^2*ln(3)^2+25)/x^4

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maxima [B]  time = 0.36, size = 64, normalized size = 2.46 \begin {gather*} \frac {x^{4} \log \relax (3)^{4} - 10 \, x^{3} \log \relax (3)^{4} + 27 \, x^{2} \log \relax (3)^{4} - 10 \, x \log \relax (3)^{4} - \frac {5 \, {\left (10 \, x^{3} \log \relax (3)^{2} - 2 \, x^{2} \log \relax (3)^{2} - 5\right )}}{x^{4}}}{\log \relax (3)^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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mupad [B]  time = 0.07, size = 45, normalized size = 1.73 \begin {gather*} 27\,x^2-10\,x-10\,x^3+x^4+\frac {-50\,{\ln \relax (3)}^2\,x^3+10\,{\ln \relax (3)}^2\,x^2+25}{x^4\,{\ln \relax (3)}^4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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sympy [B]  time = 0.23, size = 66, normalized size = 2.54 \begin {gather*} \frac {x^{4} \log {\relax (3 )}^{4} - 10 x^{3} \log {\relax (3 )}^{4} + 27 x^{2} \log {\relax (3 )}^{4} - 10 x \log {\relax (3 )}^{4} + \frac {- 50 x^{3} \log {\relax (3 )}^{2} + 10 x^{2} \log {\relax (3 )}^{2} + 25}{x^{4}}}{\log {\relax (3 )}^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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