3.65.54 \(\int \frac {-300-750 x-900 \log (\frac {x^2}{3})}{x^3} \, dx\)

Optimal. Leaf size=24 \[ \frac {150 \left (4+5 x+3 \left (x^2+\log \left (\frac {x^2}{3}\right )\right )\right )}{x^2} \]

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Rubi [A]  time = 0.02, antiderivative size = 33, normalized size of antiderivative = 1.38, number of steps used = 4, number of rules used = 3, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {14, 37, 2304} \begin {gather*} \frac {75 (5 x+2)^2}{2 x^2}+\frac {450}{x^2}+\frac {450 \log \left (\frac {x^2}{3}\right )}{x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-300 - 750*x - 900*Log[x^2/3])/x^3,x]

[Out]

450/x^2 + (75*(2 + 5*x)^2)/(2*x^2) + (450*Log[x^2/3])/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 37

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n +
1))/((b*c - a*d)*(m + 1)), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[m + n + 2, 0] && NeQ
[m, -1]

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 {150 (2+5 x)}{x^3}-\frac {900 \log \left (\frac {x^2}{3}\right )}{x^3}\right ) \, dx\\ &=-\left (150 \int \frac {2+5 x}{x^3} \, dx\right )-900 \int \frac {\log \left (\frac {x^2}{3}\right )}{x^3} \, dx\\ &=\frac {450}{x^2}+\frac {75 (2+5 x)^2}{2 x^2}+\frac {450 \log \left (\frac {x^2}{3}\right )}{x^2}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.00, size = 24, normalized size = 1.00 \begin {gather*} \frac {600}{x^2}+\frac {750}{x}+\frac {450 \log \left (\frac {x^2}{3}\right )}{x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-300 - 750*x - 900*Log[x^2/3])/x^3,x]

[Out]

600/x^2 + 750/x + (450*Log[x^2/3])/x^2

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fricas [A]  time = 0.77, size = 18, normalized size = 0.75 \begin {gather*} \frac {150 \, {\left (5 \, x + 3 \, \log \left (\frac {1}{3} \, x^{2}\right ) + 4\right )}}{x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-900*log(1/3*x^2)-750*x-300)/x^3,x, algorithm="fricas")

[Out]

150*(5*x + 3*log(1/3*x^2) + 4)/x^2

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giac [A]  time = 0.19, size = 22, normalized size = 0.92 \begin {gather*} \frac {150 \, {\left (5 \, x + 4\right )}}{x^{2}} + \frac {450 \, \log \left (\frac {1}{3} \, x^{2}\right )}{x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-900*log(1/3*x^2)-750*x-300)/x^3,x, algorithm="giac")

[Out]

150*(5*x + 4)/x^2 + 450*log(1/3*x^2)/x^2

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maple [A]  time = 0.03, size = 18, normalized size = 0.75

method	result	size
norman	\(\frac {600+750 x +450 \ln \left (\frac {x^{2}}{3}\right )}{x^{2}}\)	\(18\)
risch	\(\frac {450 \ln \left (\frac {x^{2}}{3}\right )}{x^{2}}+\frac {750 x +600}{x^{2}}\)	\(23\)
default	\(-\frac {450 \ln \relax (3)}{x^{2}}+\frac {450 \ln \left (x^{2}\right )}{x^{2}}+\frac {600}{x^{2}}+\frac {750}{x}\)	\(28\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-900*ln(1/3*x^2)-750*x-300)/x^3,x,method=_RETURNVERBOSE)

[Out]

(600+750*x+450*ln(1/3*x^2))/x^2

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maxima [A]  time = 0.37, size = 22, normalized size = 0.92 \begin {gather*} \frac {750}{x} + \frac {450 \, \log \left (\frac {1}{3} \, x^{2}\right )}{x^{2}} + \frac {600}{x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-900*log(1/3*x^2)-750*x-300)/x^3,x, algorithm="maxima")

[Out]

750/x + 450*log(1/3*x^2)/x^2 + 600/x^2

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mupad [B]  time = 4.05, size = 16, normalized size = 0.67 \begin {gather*} \frac {150\,\left (5\,x+\ln \left (\frac {x^6}{27}\right )+4\right )}{x^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(750*x + 900*log(x^2/3) + 300)/x^3,x)

[Out]

(150*(5*x + log(x^6/27) + 4))/x^2

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sympy [A]  time = 0.11, size = 20, normalized size = 0.83 \begin {gather*} - \frac {- 750 x - 600}{x^{2}} + \frac {450 \log {\left (\frac {x^{2}}{3} \right )}}{x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-900*ln(1/3*x**2)-750*x-300)/x**3,x)

[Out]

-(-750*x - 600)/x**2 + 450*log(x**2/3)/x**2

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