3.37.3 \(\int \frac {-54 x^2+27 x^4-3 \log (5)}{x^2} \, dx\)

Optimal. Leaf size=20 \[ 3 \left (-24 x+3 x \left (2+x^2\right )+\frac {\log (5)}{x}\right ) \]

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Rubi [A]  time = 0.01, antiderivative size = 16, normalized size of antiderivative = 0.80, number of steps used = 2, number of rules used = 1, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {14} \begin {gather*} 9 x^3-54 x+\frac {3 \log (5)}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-54*x^2 + 27*x^4 - 3*Log[5])/x^2,x]

[Out]

-54*x + 9*x^3 + (3*Log[5])/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 (-54+27 x^2-\frac {3 \log (5)}{x^2}\right ) \, dx\\ &=-54 x+9 x^3+\frac {3 \log (5)}{x}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.00, size = 16, normalized size = 0.80 \begin {gather*} -54 x+9 x^3+\frac {3 \log (5)}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-54*x^2 + 27*x^4 - 3*Log[5])/x^2,x]

[Out]

-54*x + 9*x^3 + (3*Log[5])/x

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fricas [A]  time = 0.83, size = 18, normalized size = 0.90 \begin {gather*} \frac {3 \, {\left (3 \, x^{4} - 18 \, x^{2} + \log \relax (5)\right )}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-3*log(5)+27*x^4-54*x^2)/x^2,x, algorithm="fricas")

[Out]

3*(3*x^4 - 18*x^2 + log(5))/x

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giac [A]  time = 0.17, size = 16, normalized size = 0.80 \begin {gather*} 9 \, x^{3} - 54 \, x + \frac {3 \, \log \relax (5)}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-3*log(5)+27*x^4-54*x^2)/x^2,x, algorithm="giac")

[Out]

9*x^3 - 54*x + 3*log(5)/x

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maple [A]  time = 0.03, size = 17, normalized size = 0.85

method	result	size
default	\(9 x^{3}-54 x +\frac {3 \ln \relax (5)}{x}\)	\(17\)
risch	\(9 x^{3}-54 x +\frac {3 \ln \relax (5)}{x}\)	\(17\)
gosper	\(\frac {-54 x^{2}+9 x^{4}+3 \ln \relax (5)}{x}\)	\(19\)
norman	\(\frac {-54 x^{2}+9 x^{4}+3 \ln \relax (5)}{x}\)	\(20\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-3*ln(5)+27*x^4-54*x^2)/x^2,x,method=_RETURNVERBOSE)

[Out]

9*x^3-54*x+3*ln(5)/x

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maxima [A]  time = 0.38, size = 16, normalized size = 0.80 \begin {gather*} 9 \, x^{3} - 54 \, x + \frac {3 \, \log \relax (5)}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-3*log(5)+27*x^4-54*x^2)/x^2,x, algorithm="maxima")

[Out]

9*x^3 - 54*x + 3*log(5)/x

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mupad [B]  time = 0.04, size = 16, normalized size = 0.80 \begin {gather*} \frac {3\,\ln \relax (5)}{x}-54\,x+9\,x^3 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(3*log(5) + 54*x^2 - 27*x^4)/x^2,x)

[Out]

(3*log(5))/x - 54*x + 9*x^3

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sympy [A]  time = 0.08, size = 14, normalized size = 0.70 \begin {gather*} 9 x^{3} - 54 x + \frac {3 \log {\relax (5 )}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-3*ln(5)+27*x**4-54*x**2)/x**2,x)

[Out]

9*x**3 - 54*x + 3*log(5)/x

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