3.13.66 \(\int \frac {-17860500-12026070 x-2452842 x^2-142884 x^3+(16074450+9406530 x+1468530 x^2+39690 x^3) \log (x)+(-3572100-1786050 x-198450 x^2) \log ^2(x)}{x^5} \, dx\)

Optimal. Leaf size=22 \[ \frac {3969 (3+x)^2 (-x+5 (-2+\log (x)))^2}{x^4} \]

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Rubi [B]  time = 0.24, antiderivative size = 76, normalized size of antiderivative = 3.45, number of steps used = 18, number of rules used = 4, integrand size = 52, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {14, 2357, 2304, 2305} \begin {gather*} \frac {3572100}{x^4}+\frac {893025 \log ^2(x)}{x^4}-\frac {3572100 \log (x)}{x^4}+\frac {3095820}{x^3}+\frac {595350 \log ^2(x)}{x^3}-\frac {2738610 \log (x)}{x^3}+\frac {908901}{x^2}+\frac {99225 \log ^2(x)}{x^2}-\frac {635040 \log (x)}{x^2}+\frac {103194}{x}-\frac {39690 \log (x)}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-17860500 - 12026070*x - 2452842*x^2 - 142884*x^3 + (16074450 + 9406530*x + 1468530*x^2 + 39690*x^3)*Log[
x] + (-3572100 - 1786050*x - 198450*x^2)*Log[x]^2)/x^5,x]

[Out]

3572100/x^4 + 3095820/x^3 + 908901/x^2 + 103194/x - (3572100*Log[x])/x^4 - (2738610*Log[x])/x^3 - (635040*Log[
x])/x^2 - (39690*Log[x])/x + (893025*Log[x]^2)/x^4 + (595350*Log[x]^2)/x^3 + (99225*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
]

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 2357

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*(RFx_), x_Symbol] :> With[{u = ExpandIntegrand[(a + b*Log[c*x^
n])^p, RFx, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c, n}, x] && RationalFunctionQ[RFx, x] && IGtQ[p, 0]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (-\frac {23814 \left (750+505 x+103 x^2+6 x^3\right )}{x^5}+\frac {39690 (3+x) \left (135+34 x+x^2\right ) \log (x)}{x^5}-\frac {198450 (3+x) (6+x) \log ^2(x)}{x^5}\right ) \, dx\\ &=-\left (23814 \int \frac {750+505 x+103 x^2+6 x^3}{x^5} \, dx\right )+39690 \int \frac {(3+x) \left (135+34 x+x^2\right ) \log (x)}{x^5} \, dx-198450 \int \frac {(3+x) (6+x) \log ^2(x)}{x^5} \, dx\\ &=-\left (23814 \int \left (\frac {750}{x^5}+\frac {505}{x^4}+\frac {103}{x^3}+\frac {6}{x^2}\right ) \, dx\right )+39690 \int \left (\frac {405 \log (x)}{x^5}+\frac {237 \log (x)}{x^4}+\frac {37 \log (x)}{x^3}+\frac {\log (x)}{x^2}\right ) \, dx-198450 \int \left (\frac {18 \log ^2(x)}{x^5}+\frac {9 \log ^2(x)}{x^4}+\frac {\log ^2(x)}{x^3}\right ) \, dx\\ &=\frac {4465125}{x^4}+\frac {4008690}{x^3}+\frac {1226421}{x^2}+\frac {142884}{x}+39690 \int \frac {\log (x)}{x^2} \, dx-198450 \int \frac {\log ^2(x)}{x^3} \, dx+1468530 \int \frac {\log (x)}{x^3} \, dx-1786050 \int \frac {\log ^2(x)}{x^4} \, dx-3572100 \int \frac {\log ^2(x)}{x^5} \, dx+9406530 \int \frac {\log (x)}{x^4} \, dx+16074450 \int \frac {\log (x)}{x^5} \, dx\\ &=\frac {27683775}{8 x^4}+\frac {2963520}{x^3}+\frac {1718577}{2 x^2}+\frac {103194}{x}-\frac {8037225 \log (x)}{2 x^4}-\frac {3135510 \log (x)}{x^3}-\frac {734265 \log (x)}{x^2}-\frac {39690 \log (x)}{x}+\frac {893025 \log ^2(x)}{x^4}+\frac {595350 \log ^2(x)}{x^3}+\frac {99225 \log ^2(x)}{x^2}-198450 \int \frac {\log (x)}{x^3} \, dx-1190700 \int \frac {\log (x)}{x^4} \, dx-1786050 \int \frac {\log (x)}{x^5} \, dx\\ &=\frac {3572100}{x^4}+\frac {3095820}{x^3}+\frac {908901}{x^2}+\frac {103194}{x}-\frac {3572100 \log (x)}{x^4}-\frac {2738610 \log (x)}{x^3}-\frac {635040 \log (x)}{x^2}-\frac {39690 \log (x)}{x}+\frac {893025 \log ^2(x)}{x^4}+\frac {595350 \log ^2(x)}{x^3}+\frac {99225 \log ^2(x)}{x^2}\\ \end {aligned} \end {gather*}

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Mathematica [B]  time = 0.07, size = 84, normalized size = 3.82 \begin {gather*} 7938 \left (\frac {450}{x^4}+\frac {390}{x^3}+\frac {229}{2 x^2}+\frac {13}{x}-\frac {450 \log (x)}{x^4}-\frac {345 \log (x)}{x^3}-\frac {80 \log (x)}{x^2}-\frac {5 \log (x)}{x}+\frac {225 \log ^2(x)}{2 x^4}+\frac {75 \log ^2(x)}{x^3}+\frac {25 \log ^2(x)}{2 x^2}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-17860500 - 12026070*x - 2452842*x^2 - 142884*x^3 + (16074450 + 9406530*x + 1468530*x^2 + 39690*x^3
)*Log[x] + (-3572100 - 1786050*x - 198450*x^2)*Log[x]^2)/x^5,x]

[Out]

7938*(450/x^4 + 390/x^3 + 229/(2*x^2) + 13/x - (450*Log[x])/x^4 - (345*Log[x])/x^3 - (80*Log[x])/x^2 - (5*Log[
x])/x + (225*Log[x]^2)/(2*x^4) + (75*Log[x]^2)/x^3 + (25*Log[x]^2)/(2*x^2))

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fricas [B]  time = 0.59, size = 51, normalized size = 2.32 \begin {gather*} \frac {3969 \, {\left (26 \, x^{3} + 25 \, {\left (x^{2} + 6 \, x + 9\right )} \log \relax (x)^{2} + 229 \, x^{2} - 10 \, {\left (x^{3} + 16 \, x^{2} + 69 \, x + 90\right )} \log \relax (x) + 780 \, x + 900\right )}}{x^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-198450*x^2-1786050*x-3572100)*log(x)^2+(39690*x^3+1468530*x^2+9406530*x+16074450)*log(x)-142884*x
^3-2452842*x^2-12026070*x-17860500)/x^5,x, algorithm="fricas")

[Out]

3969*(26*x^3 + 25*(x^2 + 6*x + 9)*log(x)^2 + 229*x^2 - 10*(x^3 + 16*x^2 + 69*x + 90)*log(x) + 780*x + 900)/x^4

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giac [B]  time = 0.32, size = 58, normalized size = 2.64 \begin {gather*} \frac {99225 \, {\left (x^{2} + 6 \, x + 9\right )} \log \relax (x)^{2}}{x^{4}} - \frac {39690 \, {\left (x^{3} + 16 \, x^{2} + 69 \, x + 90\right )} \log \relax (x)}{x^{4}} + \frac {3969 \, {\left (26 \, x^{3} + 229 \, x^{2} + 780 \, x + 900\right )}}{x^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-198450*x^2-1786050*x-3572100)*log(x)^2+(39690*x^3+1468530*x^2+9406530*x+16074450)*log(x)-142884*x
^3-2452842*x^2-12026070*x-17860500)/x^5,x, algorithm="giac")

[Out]

99225*(x^2 + 6*x + 9)*log(x)^2/x^4 - 39690*(x^3 + 16*x^2 + 69*x + 90)*log(x)/x^4 + 3969*(26*x^3 + 229*x^2 + 78
0*x + 900)/x^4

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maple [B]  time = 0.02, size = 59, normalized size = 2.68

method	result	size
risch	\(\frac {99225 \left (x^{2}+6 x +9\right ) \ln \relax (x )^{2}}{x^{4}}-\frac {39690 \left (x^{3}+16 x^{2}+69 x +90\right ) \ln \relax (x )}{x^{4}}+\frac {103194 x^{3}+908901 x^{2}+3095820 x +3572100}{x^{4}}\)	\(59\)
norman	\(\frac {3572100+3095820 x +908901 x^{2}+103194 x^{3}+893025 \ln \relax (x )^{2}-2738610 x \ln \relax (x )+595350 x \ln \relax (x )^{2}-635040 x^{2} \ln \relax (x )+99225 x^{2} \ln \relax (x )^{2}-39690 x^{3} \ln \relax (x )-3572100 \ln \relax (x )}{x^{4}}\)	\(65\)
default	\(\frac {99225 \ln \relax (x )^{2}}{x^{2}}-\frac {635040 \ln \relax (x )}{x^{2}}+\frac {908901}{x^{2}}-\frac {39690 \ln \relax (x )}{x}+\frac {103194}{x}+\frac {595350 \ln \relax (x )^{2}}{x^{3}}-\frac {2738610 \ln \relax (x )}{x^{3}}+\frac {3095820}{x^{3}}+\frac {893025 \ln \relax (x )^{2}}{x^{4}}-\frac {3572100 \ln \relax (x )}{x^{4}}+\frac {3572100}{x^{4}}\)	\(77\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-198450*x^2-1786050*x-3572100)*ln(x)^2+(39690*x^3+1468530*x^2+9406530*x+16074450)*ln(x)-142884*x^3-24528
42*x^2-12026070*x-17860500)/x^5,x,method=_RETURNVERBOSE)

[Out]

99225*(x^2+6*x+9)/x^4*ln(x)^2-39690*(x^3+16*x^2+69*x+90)/x^4*ln(x)+3969*(26*x^3+229*x^2+780*x+900)/x^4

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maxima [B]  time = 0.40, size = 100, normalized size = 4.55 \begin {gather*} -\frac {39690 \, \log \relax (x)}{x} + \frac {99225 \, {\left (2 \, \log \relax (x)^{2} + 2 \, \log \relax (x) + 1\right )}}{2 \, x^{2}} + \frac {103194}{x} - \frac {734265 \, \log \relax (x)}{x^{2}} + \frac {66150 \, {\left (9 \, \log \relax (x)^{2} + 6 \, \log \relax (x) + 2\right )}}{x^{3}} + \frac {1718577}{2 \, x^{2}} - \frac {3135510 \, \log \relax (x)}{x^{3}} + \frac {893025 \, {\left (8 \, \log \relax (x)^{2} + 4 \, \log \relax (x) + 1\right )}}{8 \, x^{4}} + \frac {2963520}{x^{3}} - \frac {8037225 \, \log \relax (x)}{2 \, x^{4}} + \frac {27683775}{8 \, x^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-198450*x^2-1786050*x-3572100)*log(x)^2+(39690*x^3+1468530*x^2+9406530*x+16074450)*log(x)-142884*x
^3-2452842*x^2-12026070*x-17860500)/x^5,x, algorithm="maxima")

[Out]

-39690*log(x)/x + 99225/2*(2*log(x)^2 + 2*log(x) + 1)/x^2 + 103194/x - 734265*log(x)/x^2 + 66150*(9*log(x)^2 +
6*log(x) + 2)/x^3 + 1718577/2/x^2 - 3135510*log(x)/x^3 + 893025/8*(8*log(x)^2 + 4*log(x) + 1)/x^4 + 2963520/x
^3 - 8037225/2*log(x)/x^4 + 27683775/8/x^4

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mupad [B]  time = 0.91, size = 62, normalized size = 2.82 \begin {gather*} \frac {x\,\left (893025\,{\ln \relax (x)}^2-3572100\,\ln \relax (x)+3572100\right )+x^3\,\left (99225\,{\ln \relax (x)}^2-635040\,\ln \relax (x)+908901\right )+x^2\,\left (595350\,{\ln \relax (x)}^2-2738610\,\ln \relax (x)+3095820\right )-x^4\,\left (39690\,\ln \relax (x)-103194\right )}{x^5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(12026070*x + log(x)^2*(1786050*x + 198450*x^2 + 3572100) + 2452842*x^2 + 142884*x^3 - log(x)*(9406530*x
+ 1468530*x^2 + 39690*x^3 + 16074450) + 17860500)/x^5,x)

[Out]

(x*(893025*log(x)^2 - 3572100*log(x) + 3572100) + x^3*(99225*log(x)^2 - 635040*log(x) + 908901) + x^2*(595350*
log(x)^2 - 2738610*log(x) + 3095820) - x^4*(39690*log(x) - 103194))/x^5

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sympy [B]  time = 0.23, size = 61, normalized size = 2.77 \begin {gather*} \frac {\left (99225 x^{2} + 595350 x + 893025\right ) \log {\relax (x )}^{2}}{x^{4}} - \frac {- 103194 x^{3} - 908901 x^{2} - 3095820 x - 3572100}{x^{4}} + \frac {\left (- 39690 x^{3} - 635040 x^{2} - 2738610 x - 3572100\right ) \log {\relax (x )}}{x^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-198450*x**2-1786050*x-3572100)*ln(x)**2+(39690*x**3+1468530*x**2+9406530*x+16074450)*ln(x)-142884
*x**3-2452842*x**2-12026070*x-17860500)/x**5,x)

[Out]

(99225*x**2 + 595350*x + 893025)*log(x)**2/x**4 - (-103194*x**3 - 908901*x**2 - 3095820*x - 3572100)/x**4 + (-
39690*x**3 - 635040*x**2 - 2738610*x - 3572100)*log(x)/x**4

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