3.83.75 \(\int \frac {1+3 x+101250 x^2 \log ^2(2)+x \log (x^3)}{x} \, dx\)

Optimal. Leaf size=18 \[ 50625 x^2 \log ^2(2)+\log (x)+x \log \left (x^3\right ) \]

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Rubi [A]  time = 0.02, antiderivative size = 18, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 2, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {14, 2295} \begin {gather*} x \log \left (x^3\right )+50625 x^2 \log ^2(2)+\log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(1 + 3*x + 101250*x^2*Log[2]^2 + x*Log[x^3])/x,x]

[Out]

50625*x^2*Log[2]^2 + Log[x] + x*Log[x^3]

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]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (\frac {1+3 x+101250 x^2 \log ^2(2)}{x}+\log \left (x^3\right )\right ) \, dx\\ &=\int \frac {1+3 x+101250 x^2 \log ^2(2)}{x} \, dx+\int \log \left (x^3\right ) \, dx\\ &=-3 x+x \log \left (x^3\right )+\int \left (3+\frac {1}{x}+101250 x \log ^2(2)\right ) \, dx\\ &=50625 x^2 \log ^2(2)+\log (x)+x \log \left (x^3\right )\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.00, size = 18, normalized size = 1.00 \begin {gather*} 50625 x^2 \log ^2(2)+\log (x)+x \log \left (x^3\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(1 + 3*x + 101250*x^2*Log[2]^2 + x*Log[x^3])/x,x]

[Out]

50625*x^2*Log[2]^2 + Log[x] + x*Log[x^3]

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fricas [A]  time = 0.72, size = 21, normalized size = 1.17 \begin {gather*} 50625 \, x^{2} \log \relax (2)^{2} + \frac {1}{3} \, {\left (3 \, x + 1\right )} \log \left (x^{3}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x*log(x^3)+101250*x^2*log(2)^2+3*x+1)/x,x, algorithm="fricas")

[Out]

50625*x^2*log(2)^2 + 1/3*(3*x + 1)*log(x^3)

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giac [A]  time = 0.16, size = 18, normalized size = 1.00 \begin {gather*} 50625 \, x^{2} \log \relax (2)^{2} + x \log \left (x^{3}\right ) + \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x*log(x^3)+101250*x^2*log(2)^2+3*x+1)/x,x, algorithm="giac")

[Out]

50625*x^2*log(2)^2 + x*log(x^3) + log(x)

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maple [A]  time = 0.04, size = 19, normalized size = 1.06

method	result	size
default	\(x \ln \left (x^{3}\right )+\ln \relax (x )+50625 x^{2} \ln \relax (2)^{2}\)	\(19\)
risch	\(x \ln \left (x^{3}\right )+\ln \relax (x )+50625 x^{2} \ln \relax (2)^{2}\)	\(19\)
norman	\(x \ln \left (x^{3}\right )+\frac {\ln \left (x^{3}\right )}{3}+50625 x^{2} \ln \relax (2)^{2}\)	\(23\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x*ln(x^3)+101250*x^2*ln(2)^2+3*x+1)/x,x,method=_RETURNVERBOSE)

[Out]

x*ln(x^3)+ln(x)+50625*x^2*ln(2)^2

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maxima [A]  time = 0.35, size = 18, normalized size = 1.00 \begin {gather*} 50625 \, x^{2} \log \relax (2)^{2} + x \log \left (x^{3}\right ) + \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x*log(x^3)+101250*x^2*log(2)^2+3*x+1)/x,x, algorithm="maxima")

[Out]

50625*x^2*log(2)^2 + x*log(x^3) + log(x)

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mupad [B]  time = 5.10, size = 22, normalized size = 1.22 \begin {gather*} \frac {\ln \left (x^3\right )}{3}+50625\,x^2\,{\ln \relax (2)}^2+x\,\ln \left (x^3\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((3*x + 101250*x^2*log(2)^2 + x*log(x^3) + 1)/x,x)

[Out]

log(x^3)/3 + 50625*x^2*log(2)^2 + x*log(x^3)

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sympy [A]  time = 0.12, size = 19, normalized size = 1.06 \begin {gather*} 50625 x^{2} \log {\relax (2 )}^{2} + x \log {\left (x^{3} \right )} + \log {\relax (x )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x*ln(x**3)+101250*x**2*ln(2)**2+3*x+1)/x,x)

[Out]

50625*x**2*log(2)**2 + x*log(x**3) + log(x)

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