3.55.30 \(\int \frac {2 x^6+(6 x^3+2 x^4) \log (2)+(-6 x^3 \log (2)+(18+12 x+2 x^2) \log ^2(2)) \log (x)+(-36-18 x-2 x^2) \log ^2(2) \log ^2(x)}{x^5} \, dx\)

Optimal. Leaf size=15 \[ \left (x+\frac {(3+x) \log (2) \log (x)}{x^2}\right )^2 \]

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Rubi [B]  time = 0.26, antiderivative size = 165, normalized size of antiderivative = 11.00, number of steps used = 18, number of rules used = 4, integrand size = 69, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.058, Rules used = {14, 2357, 2304, 2305} \begin {gather*} \frac {9 \log ^2(2) \log ^2(x)}{x^4}+\frac {9 \log ^2(2) \log (x)}{2 x^4}+\frac {9 \log ^2(2)}{8 x^4}-\frac {\log (2) \log (512) \log (x)}{2 x^4}-\frac {\log (2) \log (512)}{8 x^4}+\frac {6 \log ^2(2) \log ^2(x)}{x^3}+\frac {4 \log ^2(2) \log (x)}{x^3}+\frac {4 \log ^2(2)}{3 x^3}-\frac {2 \log (2) \log (64) \log (x)}{3 x^3}-\frac {2 \log (2) \log (64)}{9 x^3}+x^2+\frac {\log ^2(2) \log ^2(x)}{x^2}+2 \log (2) \log (x)+\frac {6 \log (2) \log (x)}{x}-\frac {2 \log (8)}{x}+\frac {6 \log (2)}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(2*x^6 + (6*x^3 + 2*x^4)*Log[2] + (-6*x^3*Log[2] + (18 + 12*x + 2*x^2)*Log[2]^2)*Log[x] + (-36 - 18*x - 2*
x^2)*Log[2]^2*Log[x]^2)/x^5,x]

[Out]

x^2 + (6*Log[2])/x + (9*Log[2]^2)/(8*x^4) + (4*Log[2]^2)/(3*x^3) - (2*Log[8])/x - (2*Log[2]*Log[64])/(9*x^3) -
(Log[2]*Log[512])/(8*x^4) + 2*Log[2]*Log[x] + (6*Log[2]*Log[x])/x + (9*Log[2]^2*Log[x])/(2*x^4) + (4*Log[2]^2
*Log[x])/x^3 - (2*Log[2]*Log[64]*Log[x])/(3*x^3) - (Log[2]*Log[512]*Log[x])/(2*x^4) + (9*Log[2]^2*Log[x]^2)/x^
4 + (6*Log[2]^2*Log[x]^2)/x^3 + (Log[2]^2*Log[x]^2)/x^2

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 2304

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[((d*x)^(m + 1)*(a + b*Log[c*x^
n]))/(d*(m + 1)), x] - Simp[(b*n*(d*x)^(m + 1))/(d*(m + 1)^2), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[m, -1
]

Rule 2305

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[((d*x)^(m + 1)*(a + b*Lo
g[c*x^n])^p)/(d*(m + 1)), x] - Dist[(b*n*p)/(m + 1), Int[(d*x)^m*(a + b*Log[c*x^n])^(p - 1), x], x] /; FreeQ[{
a, b, c, d, m, n}, x] && NeQ[m, -1] && GtQ[p, 0]

Rule 2357

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*(RFx_), x_Symbol] :> With[{u = ExpandIntegrand[(a + b*Log[c*x^
n])^p, RFx, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c, n}, x] && RationalFunctionQ[RFx, x] && IGtQ[p, 0]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (\frac {2 \left (x^3+x \log (2)+\log (8)\right )}{x^2}-\frac {2 \log (2) \left (3 x^3-x^2 \log (2)-x \log (64)-\log (512)\right ) \log (x)}{x^5}-\frac {2 (3+x) (6+x) \log ^2(2) \log ^2(x)}{x^5}\right ) \, dx\\ &=2 \int \frac {x^3+x \log (2)+\log (8)}{x^2} \, dx-(2 \log (2)) \int \frac {\left (3 x^3-x^2 \log (2)-x \log (64)-\log (512)\right ) \log (x)}{x^5} \, dx-\left (2 \log ^2(2)\right ) \int \frac {(3+x) (6+x) \log ^2(x)}{x^5} \, dx\\ &=2 \int \left (x+\frac {\log (2)}{x}+\frac {\log (8)}{x^2}\right ) \, dx-(2 \log (2)) \int \left (\frac {3 \log (x)}{x^2}-\frac {\log (2) \log (x)}{x^3}-\frac {\log (64) \log (x)}{x^4}-\frac {\log (512) \log (x)}{x^5}\right ) \, dx-\left (2 \log ^2(2)\right ) \int \left (\frac {18 \log ^2(x)}{x^5}+\frac {9 \log ^2(x)}{x^4}+\frac {\log ^2(x)}{x^3}\right ) \, dx\\ &=x^2-\frac {2 \log (8)}{x}+2 \log (2) \log (x)-(6 \log (2)) \int \frac {\log (x)}{x^2} \, dx+\left (2 \log ^2(2)\right ) \int \frac {\log (x)}{x^3} \, dx-\left (2 \log ^2(2)\right ) \int \frac {\log ^2(x)}{x^3} \, dx-\left (18 \log ^2(2)\right ) \int \frac {\log ^2(x)}{x^4} \, dx-\left (36 \log ^2(2)\right ) \int \frac {\log ^2(x)}{x^5} \, dx+(2 \log (2) \log (64)) \int \frac {\log (x)}{x^4} \, dx+(2 \log (2) \log (512)) \int \frac {\log (x)}{x^5} \, dx\\ &=x^2+\frac {6 \log (2)}{x}-\frac {\log ^2(2)}{2 x^2}-\frac {2 \log (8)}{x}-\frac {2 \log (2) \log (64)}{9 x^3}-\frac {\log (2) \log (512)}{8 x^4}+2 \log (2) \log (x)+\frac {6 \log (2) \log (x)}{x}-\frac {\log ^2(2) \log (x)}{x^2}-\frac {2 \log (2) \log (64) \log (x)}{3 x^3}-\frac {\log (2) \log (512) \log (x)}{2 x^4}+\frac {9 \log ^2(2) \log ^2(x)}{x^4}+\frac {6 \log ^2(2) \log ^2(x)}{x^3}+\frac {\log ^2(2) \log ^2(x)}{x^2}-\left (2 \log ^2(2)\right ) \int \frac {\log (x)}{x^3} \, dx-\left (12 \log ^2(2)\right ) \int \frac {\log (x)}{x^4} \, dx-\left (18 \log ^2(2)\right ) \int \frac {\log (x)}{x^5} \, dx\\ &=x^2+\frac {6 \log (2)}{x}+\frac {9 \log ^2(2)}{8 x^4}+\frac {4 \log ^2(2)}{3 x^3}-\frac {2 \log (8)}{x}-\frac {2 \log (2) \log (64)}{9 x^3}-\frac {\log (2) \log (512)}{8 x^4}+2 \log (2) \log (x)+\frac {6 \log (2) \log (x)}{x}+\frac {9 \log ^2(2) \log (x)}{2 x^4}+\frac {4 \log ^2(2) \log (x)}{x^3}-\frac {2 \log (2) \log (64) \log (x)}{3 x^3}-\frac {\log (2) \log (512) \log (x)}{2 x^4}+\frac {9 \log ^2(2) \log ^2(x)}{x^4}+\frac {6 \log ^2(2) \log ^2(x)}{x^3}+\frac {\log ^2(2) \log ^2(x)}{x^2}\\ \end {aligned} \end {gather*}

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Mathematica [B]  time = 0.03, size = 57, normalized size = 3.80 \begin {gather*} x^2+2 \log (2) \log (x)+\frac {2 \log (8) \log (x)}{x}+\frac {9 \log ^2(2) \log ^2(x)}{x^4}+\frac {6 \log ^2(2) \log ^2(x)}{x^3}+\frac {\log ^2(2) \log ^2(x)}{x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(2*x^6 + (6*x^3 + 2*x^4)*Log[2] + (-6*x^3*Log[2] + (18 + 12*x + 2*x^2)*Log[2]^2)*Log[x] + (-36 - 18*
x - 2*x^2)*Log[2]^2*Log[x]^2)/x^5,x]

[Out]

x^2 + 2*Log[2]*Log[x] + (2*Log[8]*Log[x])/x + (9*Log[2]^2*Log[x]^2)/x^4 + (6*Log[2]^2*Log[x]^2)/x^3 + (Log[2]^
2*Log[x]^2)/x^2

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fricas [B]  time = 0.78, size = 40, normalized size = 2.67 \begin {gather*} \frac {x^{6} + {\left (x^{2} + 6 \, x + 9\right )} \log \relax (2)^{2} \log \relax (x)^{2} + 2 \, {\left (x^{4} + 3 \, x^{3}\right )} \log \relax (2) \log \relax (x)}{x^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*x^2-18*x-36)*log(2)^2*log(x)^2+((2*x^2+12*x+18)*log(2)^2-6*x^3*log(2))*log(x)+(2*x^4+6*x^3)*log
(2)+2*x^6)/x^5,x, algorithm="fricas")

[Out]

(x^6 + (x^2 + 6*x + 9)*log(2)^2*log(x)^2 + 2*(x^4 + 3*x^3)*log(2)*log(x))/x^4

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giac [B]  time = 0.22, size = 49, normalized size = 3.27 \begin {gather*} x^{2} + 2 \, \log \relax (2) \log \relax (x) + \frac {6 \, \log \relax (2) \log \relax (x)}{x} + \frac {{\left (x^{2} \log \relax (2)^{2} + 6 \, x \log \relax (2)^{2} + 9 \, \log \relax (2)^{2}\right )} \log \relax (x)^{2}}{x^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*x^2-18*x-36)*log(2)^2*log(x)^2+((2*x^2+12*x+18)*log(2)^2-6*x^3*log(2))*log(x)+(2*x^4+6*x^3)*log
(2)+2*x^6)/x^5,x, algorithm="giac")

[Out]

x^2 + 2*log(2)*log(x) + 6*log(2)*log(x)/x + (x^2*log(2)^2 + 6*x*log(2)^2 + 9*log(2)^2)*log(x)^2/x^4

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maple [B]  time = 0.05, size = 40, normalized size = 2.67

method	result	size
risch	\(\frac {\ln \relax (2)^{2} \left (x^{2}+6 x +9\right ) \ln \relax (x )^{2}}{x^{4}}+\frac {6 \ln \relax (2) \ln \relax (x )}{x}+x^{2}+2 \ln \relax (2) \ln \relax (x )\)	\(40\)
norman	\(\frac {x^{6}+2 \ln \relax (2) x^{4} \ln \relax (x )+\ln \relax (2)^{2} \ln \relax (x )^{2} x^{2}+9 \ln \relax (2)^{2} \ln \relax (x )^{2}+6 x \ln \relax (2)^{2} \ln \relax (x )^{2}+6 \ln \relax (x ) \ln \relax (2) x^{3}}{x^{4}}\)	\(60\)
default	\(-2 \ln \relax (2)^{2} \left (-\frac {\ln \relax (x )^{2}}{2 x^{2}}-\frac {\ln \relax (x )}{2 x^{2}}-\frac {1}{4 x^{2}}\right )+x^{2}-18 \ln \relax (2)^{2} \left (-\frac {\ln \relax (x )^{2}}{3 x^{3}}-\frac {2 \ln \relax (x )}{9 x^{3}}-\frac {2}{27 x^{3}}\right )+2 \ln \relax (2)^{2} \left (-\frac {\ln \relax (x )}{2 x^{2}}-\frac {1}{4 x^{2}}\right )-6 \ln \relax (2) \left (-\frac {\ln \relax (x )}{x}-\frac {1}{x}\right )+2 \ln \relax (2) \ln \relax (x )-36 \ln \relax (2)^{2} \left (-\frac {\ln \relax (x )^{2}}{4 x^{4}}-\frac {\ln \relax (x )}{8 x^{4}}-\frac {1}{32 x^{4}}\right )+12 \ln \relax (2)^{2} \left (-\frac {\ln \relax (x )}{3 x^{3}}-\frac {1}{9 x^{3}}\right )-\frac {6 \ln \relax (2)}{x}+18 \ln \relax (2)^{2} \left (-\frac {\ln \relax (x )}{4 x^{4}}-\frac {1}{16 x^{4}}\right )\)	\(176\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-2*x^2-18*x-36)*ln(2)^2*ln(x)^2+((2*x^2+12*x+18)*ln(2)^2-6*x^3*ln(2))*ln(x)+(2*x^4+6*x^3)*ln(2)+2*x^6)/x
^5,x,method=_RETURNVERBOSE)

[Out]

ln(2)^2*(x^2+6*x+9)/x^4*ln(x)^2+6*ln(2)/x*ln(x)+x^2+2*ln(2)*ln(x)

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maxima [B]  time = 0.38, size = 145, normalized size = 9.67 \begin {gather*} -\frac {1}{2} \, {\left (\frac {2 \, \log \relax (x)}{x^{2}} + \frac {1}{x^{2}}\right )} \log \relax (2)^{2} - \frac {4}{3} \, {\left (\frac {3 \, \log \relax (x)}{x^{3}} + \frac {1}{x^{3}}\right )} \log \relax (2)^{2} - \frac {9}{8} \, {\left (\frac {4 \, \log \relax (x)}{x^{4}} + \frac {1}{x^{4}}\right )} \log \relax (2)^{2} + x^{2} + 6 \, {\left (\frac {\log \relax (x)}{x} + \frac {1}{x}\right )} \log \relax (2) + 2 \, \log \relax (2) \log \relax (x) + \frac {{\left (2 \, \log \relax (x)^{2} + 2 \, \log \relax (x) + 1\right )} \log \relax (2)^{2}}{2 \, x^{2}} - \frac {6 \, \log \relax (2)}{x} + \frac {2 \, {\left (9 \, \log \relax (x)^{2} + 6 \, \log \relax (x) + 2\right )} \log \relax (2)^{2}}{3 \, x^{3}} + \frac {9 \, {\left (8 \, \log \relax (x)^{2} + 4 \, \log \relax (x) + 1\right )} \log \relax (2)^{2}}{8 \, x^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*x^2-18*x-36)*log(2)^2*log(x)^2+((2*x^2+12*x+18)*log(2)^2-6*x^3*log(2))*log(x)+(2*x^4+6*x^3)*log
(2)+2*x^6)/x^5,x, algorithm="maxima")

[Out]

-1/2*(2*log(x)/x^2 + 1/x^2)*log(2)^2 - 4/3*(3*log(x)/x^3 + 1/x^3)*log(2)^2 - 9/8*(4*log(x)/x^4 + 1/x^4)*log(2)
^2 + x^2 + 6*(log(x)/x + 1/x)*log(2) + 2*log(2)*log(x) + 1/2*(2*log(x)^2 + 2*log(x) + 1)*log(2)^2/x^2 - 6*log(
2)/x + 2/3*(9*log(x)^2 + 6*log(x) + 2)*log(2)^2/x^3 + 9/8*(8*log(x)^2 + 4*log(x) + 1)*log(2)^2/x^4

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mupad [B]  time = 3.41, size = 60, normalized size = 4.00 \begin {gather*} \frac {x^7+\ln \relax (4)\,x^5\,\ln \relax (x)+\ln \left (64\right )\,x^4\,\ln \relax (x)+{\ln \relax (2)}^2\,x^3\,{\ln \relax (x)}^2+6\,{\ln \relax (2)}^2\,x^2\,{\ln \relax (x)}^2+9\,{\ln \relax (2)}^2\,x\,{\ln \relax (x)}^2}{x^5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((log(2)*(6*x^3 + 2*x^4) + 2*x^6 + log(x)*(log(2)^2*(12*x + 2*x^2 + 18) - 6*x^3*log(2)) - log(2)^2*log(x)^2
*(18*x + 2*x^2 + 36))/x^5,x)

[Out]

(x^7 + 6*x^2*log(2)^2*log(x)^2 + x^3*log(2)^2*log(x)^2 + x^5*log(4)*log(x) + x^4*log(64)*log(x) + 9*x*log(2)^2
*log(x)^2)/x^5

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sympy [B]  time = 0.25, size = 53, normalized size = 3.53 \begin {gather*} x^{2} + 2 \log {\relax (2 )} \log {\relax (x )} + \frac {6 \log {\relax (2 )} \log {\relax (x )}}{x} + \frac {\left (x^{2} \log {\relax (2 )}^{2} + 6 x \log {\relax (2 )}^{2} + 9 \log {\relax (2 )}^{2}\right ) \log {\relax (x )}^{2}}{x^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*x**2-18*x-36)*ln(2)**2*ln(x)**2+((2*x**2+12*x+18)*ln(2)**2-6*x**3*ln(2))*ln(x)+(2*x**4+6*x**3)*
ln(2)+2*x**6)/x**5,x)

[Out]

x**2 + 2*log(2)*log(x) + 6*log(2)*log(x)/x + (x**2*log(2)**2 + 6*x*log(2)**2 + 9*log(2)**2)*log(x)**2/x**4

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