3.46.39 \(\int \frac {-1800+900 x-300 x^2+330 x^3+132 x^4+(-150 x^3-170 x^4) \log (\frac {4}{x})+50 x^4 \log ^2(\frac {4}{x})}{25 x^3} \, dx\)

Optimal. Leaf size=29 \[ \left (3+x-\frac {6-x^2 \left (\frac {1}{5}-\log \left (\frac {4}{x}\right )\right )}{x}\right )^2 \]

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Rubi [B]  time = 0.07, antiderivative size = 68, normalized size of antiderivative = 2.34, number of steps used = 9, number of rules used = 5, integrand size = 58, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.086, Rules used = {12, 14, 2313, 2305, 2304} \begin {gather*} \frac {36 x^2}{25}+\frac {36}{x^2}+x^2 \log ^2\left (\frac {4}{x}\right )+x^2 \log \left (\frac {4}{x}\right )-\frac {1}{5} \left (17 x^2+30 x\right ) \log \left (\frac {4}{x}\right )+\frac {36 x}{5}-\frac {36}{x}-12 \log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-1800 + 900*x - 300*x^2 + 330*x^3 + 132*x^4 + (-150*x^3 - 170*x^4)*Log[4/x] + 50*x^4*Log[4/x]^2)/(25*x^3)
,x]

[Out]

36/x^2 - 36/x + (36*x)/5 + (36*x^2)/25 + x^2*Log[4/x] - ((30*x + 17*x^2)*Log[4/x])/5 + x^2*Log[4/x]^2 - 12*Log
[x]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, 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]]

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
]

Rule 2305

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

Rule 2313

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((d_) + (e_.)*(x_)^(r_.))^(q_.), x_Symbol] :> With[{u = IntHide[(d +
e*x^r)^q, x]}, Simp[u*(a + b*Log[c*x^n]), x] - Dist[b*n, Int[SimplifyIntegrand[u/x, x], x], x]] /; FreeQ[{a,
b, c, d, e, n, r}, x] && IGtQ[q, 0]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{25} \int \frac {-1800+900 x-300 x^2+330 x^3+132 x^4+\left (-150 x^3-170 x^4\right ) \log \left (\frac {4}{x}\right )+50 x^4 \log ^2\left (\frac {4}{x}\right )}{x^3} \, dx\\ &=\frac {1}{25} \int \left (\frac {6 \left (-300+150 x-50 x^2+55 x^3+22 x^4\right )}{x^3}-10 (15+17 x) \log \left (\frac {4}{x}\right )+50 x \log ^2\left (\frac {4}{x}\right )\right ) \, dx\\ &=\frac {6}{25} \int \frac {-300+150 x-50 x^2+55 x^3+22 x^4}{x^3} \, dx-\frac {2}{5} \int (15+17 x) \log \left (\frac {4}{x}\right ) \, dx+2 \int x \log ^2\left (\frac {4}{x}\right ) \, dx\\ &=-\frac {1}{5} \left (30 x+17 x^2\right ) \log \left (\frac {4}{x}\right )+x^2 \log ^2\left (\frac {4}{x}\right )+\frac {6}{25} \int \left (55-\frac {300}{x^3}+\frac {150}{x^2}-\frac {50}{x}+22 x\right ) \, dx-\frac {2}{5} \int \left (15+\frac {17 x}{2}\right ) \, dx+2 \int x \log \left (\frac {4}{x}\right ) \, dx\\ &=\frac {36}{x^2}-\frac {36}{x}+\frac {36 x}{5}+\frac {36 x^2}{25}+x^2 \log \left (\frac {4}{x}\right )-\frac {1}{5} \left (30 x+17 x^2\right ) \log \left (\frac {4}{x}\right )+x^2 \log ^2\left (\frac {4}{x}\right )-12 \log (x)\\ \end {aligned} \end {gather*}

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Mathematica [B]  time = 0.02, size = 61, normalized size = 2.10 \begin {gather*} \frac {36}{x^2}-\frac {36}{x}+\frac {36 x}{5}+\frac {36 x^2}{25}-6 x \log \left (\frac {4}{x}\right )-\frac {12}{5} x^2 \log \left (\frac {4}{x}\right )+x^2 \log ^2\left (\frac {4}{x}\right )-12 \log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-1800 + 900*x - 300*x^2 + 330*x^3 + 132*x^4 + (-150*x^3 - 170*x^4)*Log[4/x] + 50*x^4*Log[4/x]^2)/(2
5*x^3),x]

[Out]

36/x^2 - 36/x + (36*x)/5 + (36*x^2)/25 - 6*x*Log[4/x] - (12*x^2*Log[4/x])/5 + x^2*Log[4/x]^2 - 12*Log[x]

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fricas [A]  time = 0.54, size = 57, normalized size = 1.97 \begin {gather*} \frac {25 \, x^{4} \log \left (\frac {4}{x}\right )^{2} + 36 \, x^{4} + 180 \, x^{3} - 30 \, {\left (2 \, x^{4} + 5 \, x^{3} - 10 \, x^{2}\right )} \log \left (\frac {4}{x}\right ) - 900 \, x + 900}{25 \, x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/25*(50*x^4*log(4/x)^2+(-170*x^4-150*x^3)*log(4/x)+132*x^4+330*x^3-300*x^2+900*x-1800)/x^3,x, algor
ithm="fricas")

[Out]

1/25*(25*x^4*log(4/x)^2 + 36*x^4 + 180*x^3 - 30*(2*x^4 + 5*x^3 - 10*x^2)*log(4/x) - 900*x + 900)/x^2

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giac [B]  time = 0.15, size = 61, normalized size = 2.10 \begin {gather*} -\frac {6}{5} \, x^{2} {\left (\frac {5}{x} + 2\right )} \log \left (\frac {4}{x}\right ) + x^{2} \log \left (\frac {4}{x}\right )^{2} + \frac {36}{25} \, x^{2} {\left (\frac {5}{x} + 1\right )} - \frac {36}{x} + \frac {36}{x^{2}} + 12 \, \log \left (\frac {4}{x}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/25*(50*x^4*log(4/x)^2+(-170*x^4-150*x^3)*log(4/x)+132*x^4+330*x^3-300*x^2+900*x-1800)/x^3,x, algor
ithm="giac")

[Out]

-6/5*x^2*(5/x + 2)*log(4/x) + x^2*log(4/x)^2 + 36/25*x^2*(5/x + 1) - 36/x + 36/x^2 + 12*log(4/x)

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maple [B]  time = 0.05, size = 58, normalized size = 2.00

method	result	size
risch	\(x^{2} \ln \left (\frac {4}{x}\right )^{2}+\frac {\left (-60 x^{2}-150 x \right ) \ln \left (\frac {4}{x}\right )}{25}-\frac {12 \left (-3 x^{4}+25 x^{2} \ln \relax (x )-15 x^{3}+75 x -75\right )}{25 x^{2}}\)	\(58\)
derivativedivides	\(\frac {36}{x^{2}}-\frac {36}{x}+x^{2} \ln \left (\frac {4}{x}\right )^{2}-\frac {12 x^{2} \ln \left (\frac {4}{x}\right )}{5}+\frac {36 x^{2}}{25}-6 x \ln \left (\frac {4}{x}\right )+\frac {36 x}{5}+12 \ln \left (\frac {4}{x}\right )\)	\(60\)
default	\(\frac {36}{x^{2}}-\frac {36}{x}+x^{2} \ln \left (\frac {4}{x}\right )^{2}-\frac {12 x^{2} \ln \left (\frac {4}{x}\right )}{5}+\frac {36 x^{2}}{25}-6 x \ln \left (\frac {4}{x}\right )+\frac {36 x}{5}+12 \ln \left (\frac {4}{x}\right )\)	\(60\)
norman	\(\frac {36+x^{4} \ln \left (\frac {4}{x}\right )^{2}+12 x^{2} \ln \left (\frac {4}{x}\right )-36 x +\frac {36 x^{3}}{5}+\frac {36 x^{4}}{25}-6 x^{3} \ln \left (\frac {4}{x}\right )-\frac {12 x^{4} \ln \left (\frac {4}{x}\right )}{5}}{x^{2}}\)	\(65\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/25*(50*x^4*ln(4/x)^2+(-170*x^4-150*x^3)*ln(4/x)+132*x^4+330*x^3-300*x^2+900*x-1800)/x^3,x,method=_RETURN
VERBOSE)

[Out]

x^2*ln(4/x)^2+1/25*(-60*x^2-150*x)*ln(4/x)-12/25*(-3*x^4+25*x^2*ln(x)-15*x^3+75*x-75)/x^2

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maxima [A]  time = 0.39, size = 55, normalized size = 1.90 \begin {gather*} x^{2} \log \left (\frac {4}{x}\right )^{2} - \frac {12}{5} \, x^{2} \log \left (\frac {4}{x}\right ) + \frac {36}{25} \, x^{2} - 6 \, x \log \left (\frac {4}{x}\right ) + \frac {36}{5} \, x - \frac {36}{x} + \frac {36}{x^{2}} - 12 \, \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/25*(50*x^4*log(4/x)^2+(-170*x^4-150*x^3)*log(4/x)+132*x^4+330*x^3-300*x^2+900*x-1800)/x^3,x, algor
ithm="maxima")

[Out]

x^2*log(4/x)^2 - 12/5*x^2*log(4/x) + 36/25*x^2 - 6*x*log(4/x) + 36/5*x - 36/x + 36/x^2 - 12*log(x)

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mupad [B]  time = 3.11, size = 55, normalized size = 1.90 \begin {gather*} 12\,\ln \left (\frac {1}{x}\right )-x\,\left (6\,\ln \left (\frac {4}{x}\right )-\frac {36}{5}\right )+x^2\,\left ({\ln \left (\frac {4}{x}\right )}^2-\frac {12\,\ln \left (\frac {4}{x}\right )}{5}+\frac {36}{25}\right )+\frac {36\,x-36\,x^2}{x^3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((36*x + 2*x^4*log(4/x)^2 - (log(4/x)*(150*x^3 + 170*x^4))/25 - 12*x^2 + (66*x^3)/5 + (132*x^4)/25 - 72)/x^
3,x)

[Out]

12*log(1/x) - x*(6*log(4/x) - 36/5) + x^2*(log(4/x)^2 - (12*log(4/x))/5 + 36/25) + (36*x - 36*x^2)/x^3

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sympy [B]  time = 0.18, size = 53, normalized size = 1.83 \begin {gather*} x^{2} \log {\left (\frac {4}{x} \right )}^{2} + \frac {36 x^{2}}{25} + \frac {36 x}{5} + \left (- \frac {12 x^{2}}{5} - 6 x\right ) \log {\left (\frac {4}{x} \right )} - 12 \log {\relax (x )} + \frac {900 - 900 x}{25 x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/25*(50*x**4*ln(4/x)**2+(-170*x**4-150*x**3)*ln(4/x)+132*x**4+330*x**3-300*x**2+900*x-1800)/x**3,x)

[Out]

x**2*log(4/x)**2 + 36*x**2/25 + 36*x/5 + (-12*x**2/5 - 6*x)*log(4/x) - 12*log(x) + (900 - 900*x)/(25*x**2)

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