3.7.25 \(\int \frac {-320+82 x+16 x^2-3 x^3+(128-52 x+8 x^2) \log (x)-4 x \log ^2(x)}{x} \, dx\)

Optimal. Leaf size=17 \[ (16-x) \left (9+(5+x-2 \log (x))^2\right ) \]

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Rubi [B]  time = 0.07, antiderivative size = 43, normalized size of antiderivative = 2.53, number of steps used = 11, number of rules used = 6, integrand size = 39, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {14, 2357, 2295, 2301, 2304, 2296} \begin {gather*} -x^3+6 x^2+4 x^2 \log (x)+126 x-4 x \log ^2(x)+64 \log ^2(x)-44 x \log (x)-320 \log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-320 + 82*x + 16*x^2 - 3*x^3 + (128 - 52*x + 8*x^2)*Log[x] - 4*x*Log[x]^2)/x,x]

[Out]

126*x + 6*x^2 - x^3 - 320*Log[x] - 44*x*Log[x] + 4*x^2*Log[x] + 64*Log[x]^2 - 4*x*Log[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 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 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 {-320+82 x+16 x^2-3 x^3}{x}+\frac {4 \left (32-13 x+2 x^2\right ) \log (x)}{x}-4 \log ^2(x)\right ) \, dx\\ &=4 \int \frac {\left (32-13 x+2 x^2\right ) \log (x)}{x} \, dx-4 \int \log ^2(x) \, dx+\int \frac {-320+82 x+16 x^2-3 x^3}{x} \, dx\\ &=-4 x \log ^2(x)+4 \int \left (-13 \log (x)+\frac {32 \log (x)}{x}+2 x \log (x)\right ) \, dx+8 \int \log (x) \, dx+\int \left (82-\frac {320}{x}+16 x-3 x^2\right ) \, dx\\ &=74 x+8 x^2-x^3-320 \log (x)+8 x \log (x)-4 x \log ^2(x)+8 \int x \log (x) \, dx-52 \int \log (x) \, dx+128 \int \frac {\log (x)}{x} \, dx\\ &=126 x+6 x^2-x^3-320 \log (x)-44 x \log (x)+4 x^2 \log (x)+64 \log ^2(x)-4 x \log ^2(x)\\ \end {aligned} \end {gather*}

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Mathematica [B]  time = 0.01, size = 43, normalized size = 2.53 \begin {gather*} 126 x+6 x^2-x^3-320 \log (x)-44 x \log (x)+4 x^2 \log (x)+64 \log ^2(x)-4 x \log ^2(x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-320 + 82*x + 16*x^2 - 3*x^3 + (128 - 52*x + 8*x^2)*Log[x] - 4*x*Log[x]^2)/x,x]

[Out]

126*x + 6*x^2 - x^3 - 320*Log[x] - 44*x*Log[x] + 4*x^2*Log[x] + 64*Log[x]^2 - 4*x*Log[x]^2

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fricas [B]  time = 0.96, size = 35, normalized size = 2.06 \begin {gather*} -x^{3} - 4 \, {\left (x - 16\right )} \log \relax (x)^{2} + 6 \, x^{2} + 4 \, {\left (x^{2} - 11 \, x - 80\right )} \log \relax (x) + 126 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-4*x*log(x)^2+(8*x^2-52*x+128)*log(x)-3*x^3+16*x^2+82*x-320)/x,x, algorithm="fricas")

[Out]

-x^3 - 4*(x - 16)*log(x)^2 + 6*x^2 + 4*(x^2 - 11*x - 80)*log(x) + 126*x

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giac [B]  time = 0.40, size = 38, normalized size = 2.24 \begin {gather*} -x^{3} - 4 \, {\left (x - 16\right )} \log \relax (x)^{2} + 6 \, x^{2} + 4 \, {\left (x^{2} - 11 \, x\right )} \log \relax (x) + 126 \, x - 320 \, \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-4*x*log(x)^2+(8*x^2-52*x+128)*log(x)-3*x^3+16*x^2+82*x-320)/x,x, algorithm="giac")

[Out]

-x^3 - 4*(x - 16)*log(x)^2 + 6*x^2 + 4*(x^2 - 11*x)*log(x) + 126*x - 320*log(x)

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maple [B]  time = 0.03, size = 41, normalized size = 2.41

method	result	size
risch	\(\left (-4 x +64\right ) \ln \relax (x )^{2}+\left (4 x^{2}-44 x \right ) \ln \relax (x )-x^{3}+6 x^{2}+126 x -320 \ln \relax (x )\)	\(41\)
default	\(-4 x \ln \relax (x )^{2}-44 x \ln \relax (x )+126 x +4 x^{2} \ln \relax (x )+6 x^{2}-x^{3}+64 \ln \relax (x )^{2}-320 \ln \relax (x )\)	\(44\)
norman	\(-4 x \ln \relax (x )^{2}-44 x \ln \relax (x )+126 x +4 x^{2} \ln \relax (x )+6 x^{2}-x^{3}+64 \ln \relax (x )^{2}-320 \ln \relax (x )\)	\(44\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-4*x*ln(x)^2+(8*x^2-52*x+128)*ln(x)-3*x^3+16*x^2+82*x-320)/x,x,method=_RETURNVERBOSE)

[Out]

(-4*x+64)*ln(x)^2+(4*x^2-44*x)*ln(x)-x^3+6*x^2+126*x-320*ln(x)

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maxima [B]  time = 0.39, size = 49, normalized size = 2.88 \begin {gather*} -x^{3} + 4 \, x^{2} \log \relax (x) - 4 \, {\left (\log \relax (x)^{2} - 2 \, \log \relax (x) + 2\right )} x + 6 \, x^{2} - 52 \, x \log \relax (x) + 64 \, \log \relax (x)^{2} + 134 \, x - 320 \, \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-4*x*log(x)^2+(8*x^2-52*x+128)*log(x)-3*x^3+16*x^2+82*x-320)/x,x, algorithm="maxima")

[Out]

-x^3 + 4*x^2*log(x) - 4*(log(x)^2 - 2*log(x) + 2)*x + 6*x^2 - 52*x*log(x) + 64*log(x)^2 + 134*x - 320*log(x)

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mupad [B]  time = 0.53, size = 43, normalized size = 2.53 \begin {gather*} -x^3+4\,x^2\,\ln \relax (x)+6\,x^2-4\,x\,{\ln \relax (x)}^2-44\,x\,\ln \relax (x)+126\,x+64\,{\ln \relax (x)}^2-320\,\ln \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((82*x - 4*x*log(x)^2 + log(x)*(8*x^2 - 52*x + 128) + 16*x^2 - 3*x^3 - 320)/x,x)

[Out]

126*x - 320*log(x) - 4*x*log(x)^2 + 4*x^2*log(x) + 64*log(x)^2 - 44*x*log(x) + 6*x^2 - x^3

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sympy [B]  time = 0.15, size = 37, normalized size = 2.18 \begin {gather*} - x^{3} + 6 x^{2} + 126 x + \left (64 - 4 x\right ) \log {\relax (x )}^{2} + \left (4 x^{2} - 44 x\right ) \log {\relax (x )} - 320 \log {\relax (x )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-4*x*ln(x)**2+(8*x**2-52*x+128)*ln(x)-3*x**3+16*x**2+82*x-320)/x,x)

[Out]

-x**3 + 6*x**2 + 126*x + (64 - 4*x)*log(x)**2 + (4*x**2 - 44*x)*log(x) - 320*log(x)

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