3.74.66 \(\int \frac {540-90 x-12 x^2+9 x^3+120 \log (x^2)}{x^3} \, dx\)

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

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Rubi [A]  time = 0.03, antiderivative size = 27, normalized size of antiderivative = 1.12, number of steps used = 5, number of rules used = 2, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {14, 2304} \begin {gather*} -\frac {330}{x^2}-\frac {60 \log \left (x^2\right )}{x^2}+9 x+\frac {90}{x}-12 \log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(540 - 90*x - 12*x^2 + 9*x^3 + 120*Log[x^2])/x^3,x]

[Out]

-330/x^2 + 90/x + 9*x - 12*Log[x] - (60*Log[x^2])/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
]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (\frac {3 \left (180-30 x-4 x^2+3 x^3\right )}{x^3}+\frac {120 \log \left (x^2\right )}{x^3}\right ) \, dx\\ &=3 \int \frac {180-30 x-4 x^2+3 x^3}{x^3} \, dx+120 \int \frac {\log \left (x^2\right )}{x^3} \, dx\\ &=-\frac {60}{x^2}-\frac {60 \log \left (x^2\right )}{x^2}+3 \int \left (3+\frac {180}{x^3}-\frac {30}{x^2}-\frac {4}{x}\right ) \, dx\\ &=-\frac {330}{x^2}+\frac {90}{x}+9 x-12 \log (x)-\frac {60 \log \left (x^2\right )}{x^2}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.00, size = 27, normalized size = 1.12 \begin {gather*} -\frac {330}{x^2}+\frac {90}{x}+9 x-12 \log (x)-\frac {60 \log \left (x^2\right )}{x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(540 - 90*x - 12*x^2 + 9*x^3 + 120*Log[x^2])/x^3,x]

[Out]

-330/x^2 + 90/x + 9*x - 12*Log[x] - (60*Log[x^2])/x^2

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fricas [A]  time = 0.81, size = 26, normalized size = 1.08 \begin {gather*} \frac {3 \, {\left (3 \, x^{3} - 2 \, {\left (x^{2} + 10\right )} \log \left (x^{2}\right ) + 30 \, x - 110\right )}}{x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((120*log(x^2)+9*x^3-12*x^2-90*x+540)/x^3,x, algorithm="fricas")

[Out]

3*(3*x^3 - 2*(x^2 + 10)*log(x^2) + 30*x - 110)/x^2

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giac [A]  time = 0.16, size = 27, normalized size = 1.12 \begin {gather*} 9 \, x + \frac {30 \, {\left (3 \, x - 11\right )}}{x^{2}} - \frac {60 \, \log \left (x^{2}\right )}{x^{2}} - 12 \, \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((120*log(x^2)+9*x^3-12*x^2-90*x+540)/x^3,x, algorithm="giac")

[Out]

9*x + 30*(3*x - 11)/x^2 - 60*log(x^2)/x^2 - 12*log(x)

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maple [A]  time = 0.03, size = 26, normalized size = 1.08

method	result	size
norman	\(\frac {-330+90 x +9 x^{3}-60 \ln \left (x^{2}\right )}{x^{2}}-12 \ln \relax (x )\)	\(26\)
default	\(9 x -12 \ln \relax (x )+\frac {90}{x}-\frac {330}{x^{2}}-\frac {60 \ln \left (x^{2}\right )}{x^{2}}\)	\(28\)
risch	\(-\frac {60 \ln \left (x^{2}\right )}{x^{2}}-\frac {3 \left (4 x^{2} \ln \relax (x )-3 x^{3}-30 x +110\right )}{x^{2}}\)	\(33\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((120*ln(x^2)+9*x^3-12*x^2-90*x+540)/x^3,x,method=_RETURNVERBOSE)

[Out]

(-330+90*x+9*x^3-60*ln(x^2))/x^2-12*ln(x)

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maxima [A]  time = 0.36, size = 27, normalized size = 1.12 \begin {gather*} 9 \, x + \frac {90}{x} - \frac {60 \, \log \left (x^{2}\right )}{x^{2}} - \frac {330}{x^{2}} - 12 \, \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((120*log(x^2)+9*x^3-12*x^2-90*x+540)/x^3,x, algorithm="maxima")

[Out]

9*x + 90/x - 60*log(x^2)/x^2 - 330/x^2 - 12*log(x)

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mupad [B]  time = 4.46, size = 26, normalized size = 1.08 \begin {gather*} 9\,x-6\,\ln \left (x^2\right )-\frac {60\,\ln \left (x^2\right )-90\,x+330}{x^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((120*log(x^2) - 90*x - 12*x^2 + 9*x^3 + 540)/x^3,x)

[Out]

9*x - 6*log(x^2) - (60*log(x^2) - 90*x + 330)/x^2

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sympy [A]  time = 0.13, size = 26, normalized size = 1.08 \begin {gather*} 9 x - 12 \log {\relax (x )} + \frac {90 x - 330}{x^{2}} - \frac {60 \log {\left (x^{2} \right )}}{x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((120*ln(x**2)+9*x**3-12*x**2-90*x+540)/x**3,x)

[Out]

9*x - 12*log(x) + (90*x - 330)/x**2 - 60*log(x**2)/x**2

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