3.10.18 \(\int \frac {32-4 x^3-2 x^4+2 x^4 \log (4 x)+2 x^4 \log ^2(4 x)}{x^3} \, dx\)

Optimal. Leaf size=27 \[ \frac {7}{2}-\frac {16}{x^2}-4 x-x^2+x^2 \log ^2(4 x) \]

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Rubi [A]  time = 0.05, antiderivative size = 24, normalized size of antiderivative = 0.89, number of steps used = 7, number of rules used = 3, integrand size = 36, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {14, 2304, 2305} \begin {gather*} -x^2-\frac {16}{x^2}+x^2 \log ^2(4 x)-4 x \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(32 - 4*x^3 - 2*x^4 + 2*x^4*Log[4*x] + 2*x^4*Log[4*x]^2)/x^3,x]

[Out]

-16/x^2 - 4*x - x^2 + x^2*Log[4*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]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (-\frac {2 \left (-16+2 x^3+x^4\right )}{x^3}+2 x \log (4 x)+2 x \log ^2(4 x)\right ) \, dx\\ &=-\left (2 \int \frac {-16+2 x^3+x^4}{x^3} \, dx\right )+2 \int x \log (4 x) \, dx+2 \int x \log ^2(4 x) \, dx\\ &=-\frac {x^2}{2}+x^2 \log (4 x)+x^2 \log ^2(4 x)-2 \int \left (2-\frac {16}{x^3}+x\right ) \, dx-2 \int x \log (4 x) \, dx\\ &=-\frac {16}{x^2}-4 x-x^2+x^2 \log ^2(4 x)\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.01, size = 24, normalized size = 0.89 \begin {gather*} -\frac {16}{x^2}-4 x-x^2+x^2 \log ^2(4 x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(32 - 4*x^3 - 2*x^4 + 2*x^4*Log[4*x] + 2*x^4*Log[4*x]^2)/x^3,x]

[Out]

-16/x^2 - 4*x - x^2 + x^2*Log[4*x]^2

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fricas [A]  time = 0.50, size = 26, normalized size = 0.96 \begin {gather*} \frac {x^{4} \log \left (4 \, x\right )^{2} - x^{4} - 4 \, x^{3} - 16}{x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x^4*log(4*x)^2+2*x^4*log(4*x)-2*x^4-4*x^3+32)/x^3,x, algorithm="fricas")

[Out]

(x^4*log(4*x)^2 - x^4 - 4*x^3 - 16)/x^2

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giac [A]  time = 0.30, size = 24, normalized size = 0.89 \begin {gather*} x^{2} \log \left (4 \, x\right )^{2} - x^{2} - 4 \, x - \frac {16}{x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x^4*log(4*x)^2+2*x^4*log(4*x)-2*x^4-4*x^3+32)/x^3,x, algorithm="giac")

[Out]

x^2*log(4*x)^2 - x^2 - 4*x - 16/x^2

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maple [A]  time = 0.02, size = 25, normalized size = 0.93

method	result	size
derivativedivides	\(x^{2} \ln \left (4 x \right )^{2}-x^{2}-4 x -\frac {16}{x^{2}}\)	\(25\)
default	\(x^{2} \ln \left (4 x \right )^{2}-x^{2}-4 x -\frac {16}{x^{2}}\)	\(25\)
norman	\(\frac {-16+x^{4} \ln \left (4 x \right )^{2}-4 x^{3}-x^{4}}{x^{2}}\)	\(27\)
risch	\(x^{2} \ln \left (4 x \right )^{2}-\frac {x^{4}+4 x^{3}+16}{x^{2}}\)	\(27\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*x^4*ln(4*x)^2+2*x^4*ln(4*x)-2*x^4-4*x^3+32)/x^3,x,method=_RETURNVERBOSE)

[Out]

x^2*ln(4*x)^2-x^2-4*x-16/x^2

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maxima [A]  time = 0.43, size = 43, normalized size = 1.59 \begin {gather*} \frac {1}{2} \, {\left (2 \, \log \left (4 \, x\right )^{2} - 2 \, \log \left (4 \, x\right ) + 1\right )} x^{2} + x^{2} \log \left (4 \, x\right ) - \frac {3}{2} \, x^{2} - 4 \, x - \frac {16}{x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x^4*log(4*x)^2+2*x^4*log(4*x)-2*x^4-4*x^3+32)/x^3,x, algorithm="maxima")

[Out]

1/2*(2*log(4*x)^2 - 2*log(4*x) + 1)*x^2 + x^2*log(4*x) - 3/2*x^2 - 4*x - 16/x^2

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mupad [B]  time = 0.69, size = 21, normalized size = 0.78 \begin {gather*} x^2\,\left ({\ln \left (4\,x\right )}^2-1\right )-4\,x-\frac {16}{x^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*x^4*log(4*x) - 4*x^3 - 2*x^4 + 2*x^4*log(4*x)^2 + 32)/x^3,x)

[Out]

x^2*(log(4*x)^2 - 1) - 4*x - 16/x^2

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sympy [A]  time = 0.12, size = 20, normalized size = 0.74 \begin {gather*} x^{2} \log {\left (4 x \right )}^{2} - x^{2} - 4 x - \frac {16}{x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x**4*ln(4*x)**2+2*x**4*ln(4*x)-2*x**4-4*x**3+32)/x**3,x)

[Out]

x**2*log(4*x)**2 - x**2 - 4*x - 16/x**2

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