3.61.23 \(\int \frac {e^3-x+4 x^2+(-8 x^2-24 x^3-16 x^4) \log (3)}{x} \, dx\)

Optimal. Leaf size=26 \[ -x+2 x^2-4 \left (x+x^2\right )^2 \log (3)+e^3 \log (x) \]

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Rubi [A]  time = 0.01, antiderivative size = 35, normalized size of antiderivative = 1.35, number of steps used = 2, number of rules used = 1, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.029, Rules used = {14} \begin {gather*} -4 x^4 \log (3)-8 x^3 \log (3)+2 x^2 (1-\log (9))-x+e^3 \log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(E^3 - x + 4*x^2 + (-8*x^2 - 24*x^3 - 16*x^4)*Log[3])/x,x]

[Out]

-x - 8*x^3*Log[3] - 4*x^4*Log[3] + 2*x^2*(1 - Log[9]) + E^3*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]]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (-1+\frac {e^3}{x}-24 x^2 \log (3)-16 x^3 \log (3)+4 x (1-\log (9))\right ) \, dx\\ &=-x-8 x^3 \log (3)-4 x^4 \log (3)+2 x^2 (1-\log (9))+e^3 \log (x)\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.01, size = 36, normalized size = 1.38 \begin {gather*} -x+2 x^2-4 x^2 \log (3)-8 x^3 \log (3)-4 x^4 \log (3)+e^3 \log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^3 - x + 4*x^2 + (-8*x^2 - 24*x^3 - 16*x^4)*Log[3])/x,x]

[Out]

-x + 2*x^2 - 4*x^2*Log[3] - 8*x^3*Log[3] - 4*x^4*Log[3] + E^3*Log[x]

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fricas [A]  time = 0.56, size = 30, normalized size = 1.15 \begin {gather*} 2 \, x^{2} - 4 \, {\left (x^{4} + 2 \, x^{3} + x^{2}\right )} \log \relax (3) + e^{3} \log \relax (x) - x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-16*x^4-24*x^3-8*x^2)*log(3)+exp(3)+4*x^2-x)/x,x, algorithm="fricas")

[Out]

2*x^2 - 4*(x^4 + 2*x^3 + x^2)*log(3) + e^3*log(x) - x

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giac [A]  time = 0.17, size = 36, normalized size = 1.38 \begin {gather*} -4 \, x^{4} \log \relax (3) - 8 \, x^{3} \log \relax (3) - 4 \, x^{2} \log \relax (3) + 2 \, x^{2} + e^{3} \log \left ({\left | x \right |}\right ) - x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-16*x^4-24*x^3-8*x^2)*log(3)+exp(3)+4*x^2-x)/x,x, algorithm="giac")

[Out]

-4*x^4*log(3) - 8*x^3*log(3) - 4*x^2*log(3) + 2*x^2 + e^3*log(abs(x)) - x

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maple [A]  time = 0.05, size = 34, normalized size = 1.31

method	result	size
norman	\(\left (-4 \ln \relax (3)+2\right ) x^{2}-x -8 x^{3} \ln \relax (3)-4 x^{4} \ln \relax (3)+\ln \relax (x ) {\mathrm e}^{3}\)	\(34\)
default	\(-4 x^{4} \ln \relax (3)-8 x^{3} \ln \relax (3)-4 x^{2} \ln \relax (3)+2 x^{2}-x +\ln \relax (x ) {\mathrm e}^{3}\)	\(36\)
risch	\(-4 x^{4} \ln \relax (3)-8 x^{3} \ln \relax (3)-4 x^{2} \ln \relax (3)+2 x^{2}-x +\ln \relax (x ) {\mathrm e}^{3}\)	\(36\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-16*x^4-24*x^3-8*x^2)*ln(3)+exp(3)+4*x^2-x)/x,x,method=_RETURNVERBOSE)

[Out]

(-4*ln(3)+2)*x^2-x-8*x^3*ln(3)-4*x^4*ln(3)+ln(x)*exp(3)

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maxima [A]  time = 0.35, size = 34, normalized size = 1.31 \begin {gather*} -4 \, x^{4} \log \relax (3) - 8 \, x^{3} \log \relax (3) - 2 \, x^{2} {\left (2 \, \log \relax (3) - 1\right )} + e^{3} \log \relax (x) - x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-16*x^4-24*x^3-8*x^2)*log(3)+exp(3)+4*x^2-x)/x,x, algorithm="maxima")

[Out]

-4*x^4*log(3) - 8*x^3*log(3) - 2*x^2*(2*log(3) - 1) + e^3*log(x) - x

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mupad [B]  time = 0.06, size = 34, normalized size = 1.31 \begin {gather*} {\mathrm {e}}^3\,\ln \relax (x)-x^2\,\left (4\,\ln \relax (3)-2\right )-8\,x^3\,\ln \relax (3)-4\,x^4\,\ln \relax (3)-x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(x - exp(3) + log(3)*(8*x^2 + 24*x^3 + 16*x^4) - 4*x^2)/x,x)

[Out]

exp(3)*log(x) - x^2*(4*log(3) - 2) - 8*x^3*log(3) - 4*x^4*log(3) - x

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sympy [A]  time = 0.11, size = 34, normalized size = 1.31 \begin {gather*} - 4 x^{4} \log {\relax (3 )} - 8 x^{3} \log {\relax (3 )} - x^{2} \left (-2 + 4 \log {\relax (3 )}\right ) - x + e^{3} \log {\relax (x )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-16*x**4-24*x**3-8*x**2)*ln(3)+exp(3)+4*x**2-x)/x,x)

[Out]

-4*x**4*log(3) - 8*x**3*log(3) - x**2*(-2 + 4*log(3)) - x + exp(3)*log(x)

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