3.82.63 \(\int \frac {16777216+16809984 x+7364624 x^2+1842184 x^3+287745 x^4+28744 x^5+1794 x^6+64 x^7+x^8+(-33554432-16809984 x+1842184 x^3+575490 x^4+86232 x^5+7176 x^6+320 x^7+6 x^8) \log (x)}{x^3} \, dx\)

Optimal. Leaf size=22 \[ \log (2)+\left (4+\frac {(-8-x)^4}{x}+x\right )^2 \log (x) \]

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Rubi [B]  time = 0.20, antiderivative size = 58, normalized size of antiderivative = 2.64, number of steps used = 14, number of rules used = 5, integrand size = 80, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {14, 6742, 2357, 2295, 2304} \begin {gather*} x^6 \log (x)+64 x^5 \log (x)+1794 x^4 \log (x)+28744 x^3 \log (x)+287745 x^2 \log (x)+\frac {16777216 \log (x)}{x^2}+1842184 x \log (x)+7364624 \log (x)+\frac {16809984 \log (x)}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(16777216 + 16809984*x + 7364624*x^2 + 1842184*x^3 + 287745*x^4 + 28744*x^5 + 1794*x^6 + 64*x^7 + x^8 + (-
33554432 - 16809984*x + 1842184*x^3 + 575490*x^4 + 86232*x^5 + 7176*x^6 + 320*x^7 + 6*x^8)*Log[x])/x^3,x]

[Out]

7364624*Log[x] + (16777216*Log[x])/x^2 + (16809984*Log[x])/x + 1842184*x*Log[x] + 287745*x^2*Log[x] + 28744*x^
3*Log[x] + 1794*x^4*Log[x] + 64*x^5*Log[x] + x^6*Log[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]]

Rule 2295

Int[Log[(c_.)*(x_)^(n_.)], x_Symbol] :> Simp[x*Log[c*x^n], x] - Simp[n*x, x] /; FreeQ[{c, n}, x]

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 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]

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 \left (\frac {\left (4096+2052 x+385 x^2+32 x^3+x^4\right )^2}{x^3}+\frac {2 \left (4096+2052 x+385 x^2+32 x^3+x^4\right ) \left (-4096+385 x^2+64 x^3+3 x^4\right ) \log (x)}{x^3}\right ) \, dx\\ &=2 \int \frac {\left (4096+2052 x+385 x^2+32 x^3+x^4\right ) \left (-4096+385 x^2+64 x^3+3 x^4\right ) \log (x)}{x^3} \, dx+\int \frac {\left (4096+2052 x+385 x^2+32 x^3+x^4\right )^2}{x^3} \, dx\\ &=2 \int \left (921092 \log (x)-\frac {16777216 \log (x)}{x^3}-\frac {8404992 \log (x)}{x^2}+287745 x \log (x)+43116 x^2 \log (x)+3588 x^3 \log (x)+160 x^4 \log (x)+3 x^5 \log (x)\right ) \, dx+\int \left (1842184+\frac {16777216}{x^3}+\frac {16809984}{x^2}+\frac {7364624}{x}+287745 x+28744 x^2+1794 x^3+64 x^4+x^5\right ) \, dx\\ &=-\frac {8388608}{x^2}-\frac {16809984}{x}+1842184 x+\frac {287745 x^2}{2}+\frac {28744 x^3}{3}+\frac {897 x^4}{2}+\frac {64 x^5}{5}+\frac {x^6}{6}+7364624 \log (x)+6 \int x^5 \log (x) \, dx+320 \int x^4 \log (x) \, dx+7176 \int x^3 \log (x) \, dx+86232 \int x^2 \log (x) \, dx+575490 \int x \log (x) \, dx+1842184 \int \log (x) \, dx-16809984 \int \frac {\log (x)}{x^2} \, dx-33554432 \int \frac {\log (x)}{x^3} \, dx\\ &=7364624 \log (x)+\frac {16777216 \log (x)}{x^2}+\frac {16809984 \log (x)}{x}+1842184 x \log (x)+287745 x^2 \log (x)+28744 x^3 \log (x)+1794 x^4 \log (x)+64 x^5 \log (x)+x^6 \log (x)\\ \end {aligned} \end {gather*}

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Mathematica [B]  time = 0.01, size = 58, normalized size = 2.64 \begin {gather*} 7364624 \log (x)+\frac {16777216 \log (x)}{x^2}+\frac {16809984 \log (x)}{x}+1842184 x \log (x)+287745 x^2 \log (x)+28744 x^3 \log (x)+1794 x^4 \log (x)+64 x^5 \log (x)+x^6 \log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(16777216 + 16809984*x + 7364624*x^2 + 1842184*x^3 + 287745*x^4 + 28744*x^5 + 1794*x^6 + 64*x^7 + x^
8 + (-33554432 - 16809984*x + 1842184*x^3 + 575490*x^4 + 86232*x^5 + 7176*x^6 + 320*x^7 + 6*x^8)*Log[x])/x^3,x
]

[Out]

7364624*Log[x] + (16777216*Log[x])/x^2 + (16809984*Log[x])/x + 1842184*x*Log[x] + 287745*x^2*Log[x] + 28744*x^
3*Log[x] + 1794*x^4*Log[x] + 64*x^5*Log[x] + x^6*Log[x]

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fricas [B]  time = 0.87, size = 44, normalized size = 2.00 \begin {gather*} \frac {{\left (x^{8} + 64 \, x^{7} + 1794 \, x^{6} + 28744 \, x^{5} + 287745 \, x^{4} + 1842184 \, x^{3} + 7364624 \, x^{2} + 16809984 \, x + 16777216\right )} \log \relax (x)}{x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((6*x^8+320*x^7+7176*x^6+86232*x^5+575490*x^4+1842184*x^3-16809984*x-33554432)*log(x)+x^8+64*x^7+179
4*x^6+28744*x^5+287745*x^4+1842184*x^3+7364624*x^2+16809984*x+16777216)/x^3,x, algorithm="fricas")

[Out]

(x^8 + 64*x^7 + 1794*x^6 + 28744*x^5 + 287745*x^4 + 1842184*x^3 + 7364624*x^2 + 16809984*x + 16777216)*log(x)/
x^2

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giac [B]  time = 0.18, size = 45, normalized size = 2.05 \begin {gather*} {\left (x^{6} + 64 \, x^{5} + 1794 \, x^{4} + 28744 \, x^{3} + 287745 \, x^{2} + 1842184 \, x + \frac {32768 \, {\left (513 \, x + 512\right )}}{x^{2}}\right )} \log \relax (x) + 7364624 \, \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((6*x^8+320*x^7+7176*x^6+86232*x^5+575490*x^4+1842184*x^3-16809984*x-33554432)*log(x)+x^8+64*x^7+179
4*x^6+28744*x^5+287745*x^4+1842184*x^3+7364624*x^2+16809984*x+16777216)/x^3,x, algorithm="giac")

[Out]

(x^6 + 64*x^5 + 1794*x^4 + 28744*x^3 + 287745*x^2 + 1842184*x + 32768*(513*x + 512)/x^2)*log(x) + 7364624*log(
x)

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maple [A]  time = 0.03, size = 45, normalized size = 2.05

method	result	size
risch	\(\frac {\left (x^{8}+64 x^{7}+1794 x^{6}+28744 x^{5}+287745 x^{4}+1842184 x^{3}[phone] x [phone]\right ) \ln \relax (x )}{x^{2}}+7364624 \ln \relax (x )\)	\(45\)
default	\(287745 x^{2} \ln \relax (x )+\frac {16809984 \ln \relax (x )}{x}+7364624 \ln \relax (x )+28744 x^{3} \ln \relax (x )+x^{6} \ln \relax (x )+\frac {16777216 \ln \relax (x )}{x^{2}}+64 x^{5} \ln \relax (x )+1794 x^{4} \ln \relax (x )+1842184 x \ln \relax (x )\)	\(59\)
norman	\(\frac {x^{8} \ln \relax (x )+7364624 x^{2} \ln \relax (x )[phone] x \ln \relax (x )+1842184 x^{3} \ln \relax (x )+287745 x^{4} \ln \relax (x )+28744 x^{5} \ln \relax (x )+1794 x^{6} \ln \relax (x )+64 x^{7} \ln \relax (x )[phone] \ln \relax (x )}{x^{2}}\)	\(63\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((6*x^8+320*x^7+7176*x^6+86232*x^5+575490*x^4+1842184*x^3-16809984*x-33554432)*ln(x)+x^8+64*x^7+1794*x^6+2
8744*x^5+287745*x^4+1842184*x^3+7364624*x^2+16809984*x+16777216)/x^3,x,method=_RETURNVERBOSE)

[Out]

(x^8+64*x^7+1794*x^6+28744*x^5+287745*x^4+1842184*x^3+16809984*x+16777216)/x^2*ln(x)+7364624*ln(x)

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maxima [B]  time = 0.37, size = 58, normalized size = 2.64 \begin {gather*} x^{6} \log \relax (x) + 64 \, x^{5} \log \relax (x) + 1794 \, x^{4} \log \relax (x) + 28744 \, x^{3} \log \relax (x) + 287745 \, x^{2} \log \relax (x) + 1842184 \, x \log \relax (x) + \frac {16809984 \, \log \relax (x)}{x} + \frac {16777216 \, \log \relax (x)}{x^{2}} + 7364624 \, \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((6*x^8+320*x^7+7176*x^6+86232*x^5+575490*x^4+1842184*x^3-16809984*x-33554432)*log(x)+x^8+64*x^7+179
4*x^6+28744*x^5+287745*x^4+1842184*x^3+7364624*x^2+16809984*x+16777216)/x^3,x, algorithm="maxima")

[Out]

x^6*log(x) + 64*x^5*log(x) + 1794*x^4*log(x) + 28744*x^3*log(x) + 287745*x^2*log(x) + 1842184*x*log(x) + 16809
984*log(x)/x + 16777216*log(x)/x^2 + 7364624*log(x)

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mupad [B]  time = 5.62, size = 26, normalized size = 1.18 \begin {gather*} \frac {\ln \relax (x)\,{\left (x^4+32\,x^3+385\,x^2+2052\,x+4096\right )}^2}{x^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((16809984*x + log(x)*(1842184*x^3 - 16809984*x + 575490*x^4 + 86232*x^5 + 7176*x^6 + 320*x^7 + 6*x^8 - 335
54432) + 7364624*x^2 + 1842184*x^3 + 287745*x^4 + 28744*x^5 + 1794*x^6 + 64*x^7 + x^8 + 16777216)/x^3,x)

[Out]

(log(x)*(2052*x + 385*x^2 + 32*x^3 + x^4 + 4096)^2)/x^2

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sympy [B]  time = 0.18, size = 44, normalized size = 2.00 \begin {gather*} 7364624 \log {\relax (x )} + \frac {\left (x^{8} + 64 x^{7} + 1794 x^{6} + 28744 x^{5} + 287745 x^{4} + 1842184 x^{3} + 16809984 x + 16777216\right ) \log {\relax (x )}}{x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((6*x**8+320*x**7+7176*x**6+86232*x**5+575490*x**4+1842184*x**3-16809984*x-33554432)*ln(x)+x**8+64*x
**7+1794*x**6+28744*x**5+287745*x**4+1842184*x**3+7364624*x**2+16809984*x+16777216)/x**3,x)

[Out]

7364624*log(x) + (x**8 + 64*x**7 + 1794*x**6 + 28744*x**5 + 287745*x**4 + 1842184*x**3 + 16809984*x + 16777216
)*log(x)/x**2

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