3.71.55 \(\int \frac {-512 x+384 x^2+184 x^3+16 x^4+(512-384 x-248 x^2-24 x^3) \log (2 x)+(64 x+8 x^2) \log ^2(2 x)}{9 x} \, dx\)

Optimal. Leaf size=19 \[ \frac {4}{9} (8+x)^2 (-x+\log (2 x))^2 \]

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Rubi [B]  time = 0.14, antiderivative size = 87, normalized size of antiderivative = 4.58, number of steps used = 16, number of rules used = 9, integrand size = 62, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.145, Rules used = {12, 14, 2357, 2295, 2301, 2304, 2330, 2296, 2305} \begin {gather*} \frac {4 x^4}{9}+\frac {64 x^3}{9}-\frac {8}{9} x^3 \log (2 x)+\frac {256 x^2}{9}+\frac {4}{9} x^2 \log ^2(2 x)-\frac {128}{9} x^2 \log (2 x)+\frac {64}{9} x \log ^2(2 x)+\frac {256}{9} \log ^2(2 x)-\frac {512}{9} x \log (2 x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-512*x + 384*x^2 + 184*x^3 + 16*x^4 + (512 - 384*x - 248*x^2 - 24*x^3)*Log[2*x] + (64*x + 8*x^2)*Log[2*x]
^2)/(9*x),x]

[Out]

(256*x^2)/9 + (64*x^3)/9 + (4*x^4)/9 - (512*x*Log[2*x])/9 - (128*x^2*Log[2*x])/9 - (8*x^3*Log[2*x])/9 + (256*L
og[2*x]^2)/9 + (64*x*Log[2*x]^2)/9 + (4*x^2*Log[2*x]^2)/9

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]]

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 2296

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.), x_Symbol] :> Simp[x*(a + b*Log[c*x^n])^p, x] - Dist[b*n*p, In
t[(a + b*Log[c*x^n])^(p - 1), x], x] /; FreeQ[{a, b, c, n}, x] && GtQ[p, 0] && IntegerQ[2*p]

Rule 2301

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))/(x_), x_Symbol] :> Simp[(a + b*Log[c*x^n])^2/(2*b*n), x] /; FreeQ[{a
, b, 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 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 2330

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_) + (e_.)*(x_)^(r_.))^(q_.), x_Symbol] :> With[{u = Expand
Integrand[(a + b*Log[c*x^n])^p, (d + e*x^r)^q, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c, d, e, n, p, q, r}
, x] && IntegerQ[q] && (GtQ[q, 0] || (IGtQ[p, 0] && IntegerQ[r]))

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} &=\frac {1}{9} \int \frac {-512 x+384 x^2+184 x^3+16 x^4+\left (512-384 x-248 x^2-24 x^3\right ) \log (2 x)+\left (64 x+8 x^2\right ) \log ^2(2 x)}{x} \, dx\\ &=\frac {1}{9} \int \left (8 \left (-64+48 x+23 x^2+2 x^3\right )-\frac {8 (8+x) \left (-8+7 x+3 x^2\right ) \log (2 x)}{x}+8 (8+x) \log ^2(2 x)\right ) \, dx\\ &=\frac {8}{9} \int \left (-64+48 x+23 x^2+2 x^3\right ) \, dx-\frac {8}{9} \int \frac {(8+x) \left (-8+7 x+3 x^2\right ) \log (2 x)}{x} \, dx+\frac {8}{9} \int (8+x) \log ^2(2 x) \, dx\\ &=-\frac {512 x}{9}+\frac {64 x^2}{3}+\frac {184 x^3}{27}+\frac {4 x^4}{9}-\frac {8}{9} \int \left (48 \log (2 x)-\frac {64 \log (2 x)}{x}+31 x \log (2 x)+3 x^2 \log (2 x)\right ) \, dx+\frac {8}{9} \int \left (8 \log ^2(2 x)+x \log ^2(2 x)\right ) \, dx\\ &=-\frac {512 x}{9}+\frac {64 x^2}{3}+\frac {184 x^3}{27}+\frac {4 x^4}{9}+\frac {8}{9} \int x \log ^2(2 x) \, dx-\frac {8}{3} \int x^2 \log (2 x) \, dx+\frac {64}{9} \int \log ^2(2 x) \, dx-\frac {248}{9} \int x \log (2 x) \, dx-\frac {128}{3} \int \log (2 x) \, dx+\frac {512}{9} \int \frac {\log (2 x)}{x} \, dx\\ &=-\frac {128 x}{9}+\frac {254 x^2}{9}+\frac {64 x^3}{9}+\frac {4 x^4}{9}-\frac {128}{3} x \log (2 x)-\frac {124}{9} x^2 \log (2 x)-\frac {8}{9} x^3 \log (2 x)+\frac {256}{9} \log ^2(2 x)+\frac {64}{9} x \log ^2(2 x)+\frac {4}{9} x^2 \log ^2(2 x)-\frac {8}{9} \int x \log (2 x) \, dx-\frac {128}{9} \int \log (2 x) \, dx\\ &=\frac {256 x^2}{9}+\frac {64 x^3}{9}+\frac {4 x^4}{9}-\frac {512}{9} x \log (2 x)-\frac {128}{9} x^2 \log (2 x)-\frac {8}{9} x^3 \log (2 x)+\frac {256}{9} \log ^2(2 x)+\frac {64}{9} x \log ^2(2 x)+\frac {4}{9} x^2 \log ^2(2 x)\\ \end {aligned} \end {gather*}

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Mathematica [B]  time = 0.07, size = 77, normalized size = 4.05 \begin {gather*} \frac {8}{9} \left (32 x^2+8 x^3+\frac {x^4}{2}-64 x \log (2 x)-16 x^2 \log (2 x)-x^3 \log (2 x)+32 \log ^2(2 x)+8 x \log ^2(2 x)+\frac {1}{2} x^2 \log ^2(2 x)\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-512*x + 384*x^2 + 184*x^3 + 16*x^4 + (512 - 384*x - 248*x^2 - 24*x^3)*Log[2*x] + (64*x + 8*x^2)*Lo
g[2*x]^2)/(9*x),x]

[Out]

(8*(32*x^2 + 8*x^3 + x^4/2 - 64*x*Log[2*x] - 16*x^2*Log[2*x] - x^3*Log[2*x] + 32*Log[2*x]^2 + 8*x*Log[2*x]^2 +
(x^2*Log[2*x]^2)/2))/9

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fricas [B]  time = 0.91, size = 50, normalized size = 2.63 \begin {gather*} \frac {4}{9} \, x^{4} + \frac {64}{9} \, x^{3} + \frac {4}{9} \, {\left (x^{2} + 16 \, x + 64\right )} \log \left (2 \, x\right )^{2} + \frac {256}{9} \, x^{2} - \frac {8}{9} \, {\left (x^{3} + 16 \, x^{2} + 64 \, x\right )} \log \left (2 \, x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/9*((8*x^2+64*x)*log(2*x)^2+(-24*x^3-248*x^2-384*x+512)*log(2*x)+16*x^4+184*x^3+384*x^2-512*x)/x,x,
algorithm="fricas")

[Out]

4/9*x^4 + 64/9*x^3 + 4/9*(x^2 + 16*x + 64)*log(2*x)^2 + 256/9*x^2 - 8/9*(x^3 + 16*x^2 + 64*x)*log(2*x)

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giac [B]  time = 0.13, size = 50, normalized size = 2.63 \begin {gather*} \frac {4}{9} \, x^{4} + \frac {64}{9} \, x^{3} + \frac {4}{9} \, {\left (x^{2} + 16 \, x + 64\right )} \log \left (2 \, x\right )^{2} + \frac {256}{9} \, x^{2} - \frac {8}{9} \, {\left (x^{3} + 16 \, x^{2} + 64 \, x\right )} \log \left (2 \, x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/9*((8*x^2+64*x)*log(2*x)^2+(-24*x^3-248*x^2-384*x+512)*log(2*x)+16*x^4+184*x^3+384*x^2-512*x)/x,x,
algorithm="giac")

[Out]

4/9*x^4 + 64/9*x^3 + 4/9*(x^2 + 16*x + 64)*log(2*x)^2 + 256/9*x^2 - 8/9*(x^3 + 16*x^2 + 64*x)*log(2*x)

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maple [B]  time = 0.03, size = 55, normalized size = 2.89

method	result	size
risch	\(\frac {\left (4 x^{2}+64 x +256\right ) \ln \left (2 x \right )^{2}}{9}+\frac {\left (-8 x^{3}-128 x^{2}-512 x \right ) \ln \left (2 x \right )}{9}+\frac {4 x^{4}}{9}+\frac {64 x^{3}}{9}+\frac {256 x^{2}}{9}\)	\(55\)
derivativedivides	\(\frac {4 x^{2} \ln \left (2 x \right )^{2}}{9}-\frac {128 x^{2} \ln \left (2 x \right )}{9}+\frac {256 x^{2}}{9}-\frac {8 x^{3} \ln \left (2 x \right )}{9}+\frac {64 x^{3}}{9}+\frac {4 x^{4}}{9}+\frac {64 x \ln \left (2 x \right )^{2}}{9}-\frac {512 x \ln \left (2 x \right )}{9}+\frac {256 \ln \left (2 x \right )^{2}}{9}\)	\(70\)
default	\(\frac {4 x^{2} \ln \left (2 x \right )^{2}}{9}-\frac {128 x^{2} \ln \left (2 x \right )}{9}+\frac {256 x^{2}}{9}-\frac {8 x^{3} \ln \left (2 x \right )}{9}+\frac {64 x^{3}}{9}+\frac {4 x^{4}}{9}+\frac {64 x \ln \left (2 x \right )^{2}}{9}-\frac {512 x \ln \left (2 x \right )}{9}+\frac {256 \ln \left (2 x \right )^{2}}{9}\)	\(70\)
norman	\(\frac {4 x^{2} \ln \left (2 x \right )^{2}}{9}-\frac {128 x^{2} \ln \left (2 x \right )}{9}+\frac {256 x^{2}}{9}-\frac {8 x^{3} \ln \left (2 x \right )}{9}+\frac {64 x^{3}}{9}+\frac {4 x^{4}}{9}+\frac {64 x \ln \left (2 x \right )^{2}}{9}-\frac {512 x \ln \left (2 x \right )}{9}+\frac {256 \ln \left (2 x \right )^{2}}{9}\)	\(70\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/9*((8*x^2+64*x)*ln(2*x)^2+(-24*x^3-248*x^2-384*x+512)*ln(2*x)+16*x^4+184*x^3+384*x^2-512*x)/x,x,method=_
RETURNVERBOSE)

[Out]

1/9*(4*x^2+64*x+256)*ln(2*x)^2+1/9*(-8*x^3-128*x^2-512*x)*ln(2*x)+4/9*x^4+64/9*x^3+256/9*x^2

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maxima [B]  time = 0.38, size = 90, normalized size = 4.74 \begin {gather*} \frac {4}{9} \, x^{4} - \frac {8}{9} \, x^{3} \log \left (2 \, x\right ) + \frac {2}{9} \, {\left (2 \, \log \left (2 \, x\right )^{2} - 2 \, \log \left (2 \, x\right ) + 1\right )} x^{2} + \frac {64}{9} \, x^{3} - \frac {124}{9} \, x^{2} \log \left (2 \, x\right ) + \frac {64}{9} \, {\left (\log \left (2 \, x\right )^{2} - 2 \, \log \left (2 \, x\right ) + 2\right )} x + \frac {254}{9} \, x^{2} - \frac {128}{3} \, x \log \left (2 \, x\right ) + \frac {256}{9} \, \log \left (2 \, x\right )^{2} - \frac {128}{9} \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/9*((8*x^2+64*x)*log(2*x)^2+(-24*x^3-248*x^2-384*x+512)*log(2*x)+16*x^4+184*x^3+384*x^2-512*x)/x,x,
algorithm="maxima")

[Out]

4/9*x^4 - 8/9*x^3*log(2*x) + 2/9*(2*log(2*x)^2 - 2*log(2*x) + 1)*x^2 + 64/9*x^3 - 124/9*x^2*log(2*x) + 64/9*(l
og(2*x)^2 - 2*log(2*x) + 2)*x + 254/9*x^2 - 128/3*x*log(2*x) + 256/9*log(2*x)^2 - 128/9*x

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mupad [B]  time = 4.23, size = 17, normalized size = 0.89 \begin {gather*} \frac {4\,{\left (x-\ln \left (2\,x\right )\right )}^2\,{\left (x+8\right )}^2}{9} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((log(2*x)^2*(64*x + 8*x^2))/9 - (log(2*x)*(384*x + 248*x^2 + 24*x^3 - 512))/9 - (512*x)/9 + (128*x^2)/3 +
(184*x^3)/9 + (16*x^4)/9)/x,x)

[Out]

(4*(x - log(2*x))^2*(x + 8)^2)/9

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sympy [B]  time = 0.19, size = 66, normalized size = 3.47 \begin {gather*} \frac {4 x^{4}}{9} + \frac {64 x^{3}}{9} + \frac {256 x^{2}}{9} + \left (\frac {4 x^{2}}{9} + \frac {64 x}{9} + \frac {256}{9}\right ) \log {\left (2 x \right )}^{2} + \left (- \frac {8 x^{3}}{9} - \frac {128 x^{2}}{9} - \frac {512 x}{9}\right ) \log {\left (2 x \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/9*((8*x**2+64*x)*ln(2*x)**2+(-24*x**3-248*x**2-384*x+512)*ln(2*x)+16*x**4+184*x**3+384*x**2-512*x)
/x,x)

[Out]

4*x**4/9 + 64*x**3/9 + 256*x**2/9 + (4*x**2/9 + 64*x/9 + 256/9)*log(2*x)**2 + (-8*x**3/9 - 128*x**2/9 - 512*x/
9)*log(2*x)

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