3.11.79 \(\int \frac {-378535936+187392 x^2-1152 x^3+3 x^4+(179306496-27648 x^2+128 x^3) \log (5)+(-31195136+1536 x^2) \log ^2(5)+2359296 \log ^3(5)-65536 \log ^4(5)}{65536 x^2} \, dx\)

Optimal. Leaf size=25 \[ x \left (1+\frac {\left (-5+\left (9-\frac {x}{16}-\log (5)\right )^2\right )^2}{x^2}\right ) \]

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Rubi [B]  time = 0.04, antiderivative size = 52, normalized size of antiderivative = 2.08, number of steps used = 3, number of rules used = 2, integrand size = 63, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.032, Rules used = {12, 14} \begin {gather*} \frac {x^3}{65536}-\frac {x^2 (9-\log (5))}{1024}+\frac {3}{128} x \left (122+\log ^2(5)-18 \log (5)\right )+\frac {\left (76+\log ^2(5)-18 \log (5)\right )^2}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-378535936 + 187392*x^2 - 1152*x^3 + 3*x^4 + (179306496 - 27648*x^2 + 128*x^3)*Log[5] + (-31195136 + 1536
*x^2)*Log[5]^2 + 2359296*Log[5]^3 - 65536*Log[5]^4)/(65536*x^2),x]

[Out]

x^3/65536 - (x^2*(9 - Log[5]))/1024 + (76 - 18*Log[5] + Log[5]^2)^2/x + (3*x*(122 - 18*Log[5] + Log[5]^2))/128

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]]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {\int \frac {-378535936+187392 x^2-1152 x^3+3 x^4+\left (179306496-27648 x^2+128 x^3\right ) \log (5)+\left (-31195136+1536 x^2\right ) \log ^2(5)+2359296 \log ^3(5)-65536 \log ^4(5)}{x^2} \, dx}{65536}\\ &=\frac {\int \left (3 x^2+128 x (-9+\log (5))-\frac {65536 \left (76-18 \log (5)+\log ^2(5)\right )^2}{x^2}+1536 \left (122-18 \log (5)+\log ^2(5)\right )\right ) \, dx}{65536}\\ &=\frac {x^3}{65536}-\frac {x^2 (9-\log (5))}{1024}+\frac {\left (76-18 \log (5)+\log ^2(5)\right )^2}{x}+\frac {3}{128} x \left (122-18 \log (5)+\log ^2(5)\right )\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.02, size = 50, normalized size = 2.00 \begin {gather*} \frac {x^3}{65536}+\frac {x^2 (-9+\log (5))}{1024}+\frac {\left (76-18 \log (5)+\log ^2(5)\right )^2}{x}+\frac {3}{128} x \left (122-18 \log (5)+\log ^2(5)\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-378535936 + 187392*x^2 - 1152*x^3 + 3*x^4 + (179306496 - 27648*x^2 + 128*x^3)*Log[5] + (-31195136
+ 1536*x^2)*Log[5]^2 + 2359296*Log[5]^3 - 65536*Log[5]^4)/(65536*x^2),x]

[Out]

x^3/65536 + (x^2*(-9 + Log[5]))/1024 + (76 - 18*Log[5] + Log[5]^2)^2/x + (3*x*(122 - 18*Log[5] + Log[5]^2))/12
8

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fricas [B]  time = 0.98, size = 59, normalized size = 2.36 \begin {gather*} \frac {x^{4} + 65536 \, \log \relax (5)^{4} - 576 \, x^{3} + 512 \, {\left (3 \, x^{2} + 60928\right )} \log \relax (5)^{2} - 2359296 \, \log \relax (5)^{3} + 187392 \, x^{2} + 64 \, {\left (x^{3} - 432 \, x^{2} - 2801664\right )} \log \relax (5) + 378535936}{65536 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/65536*(-65536*log(5)^4+2359296*log(5)^3+(1536*x^2-31195136)*log(5)^2+(128*x^3-27648*x^2+179306496)
*log(5)+3*x^4-1152*x^3+187392*x^2-378535936)/x^2,x, algorithm="fricas")

[Out]

1/65536*(x^4 + 65536*log(5)^4 - 576*x^3 + 512*(3*x^2 + 60928)*log(5)^2 - 2359296*log(5)^3 + 187392*x^2 + 64*(x
^3 - 432*x^2 - 2801664)*log(5) + 378535936)/x

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giac [B]  time = 0.18, size = 59, normalized size = 2.36 \begin {gather*} \frac {1}{65536} \, x^{3} + \frac {1}{1024} \, x^{2} \log \relax (5) + \frac {3}{128} \, x \log \relax (5)^{2} - \frac {9}{1024} \, x^{2} - \frac {27}{64} \, x \log \relax (5) + \frac {183}{64} \, x + \frac {\log \relax (5)^{4} - 36 \, \log \relax (5)^{3} + 476 \, \log \relax (5)^{2} - 2736 \, \log \relax (5) + 5776}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/65536*(-65536*log(5)^4+2359296*log(5)^3+(1536*x^2-31195136)*log(5)^2+(128*x^3-27648*x^2+179306496)
*log(5)+3*x^4-1152*x^3+187392*x^2-378535936)/x^2,x, algorithm="giac")

[Out]

1/65536*x^3 + 1/1024*x^2*log(5) + 3/128*x*log(5)^2 - 9/1024*x^2 - 27/64*x*log(5) + 183/64*x + (log(5)^4 - 36*l
og(5)^3 + 476*log(5)^2 - 2736*log(5) + 5776)/x

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maple [B]  time = 0.07, size = 58, normalized size = 2.32

method	result	size
norman	\(\frac {\left (\frac {\ln \relax (5)}{1024}-\frac {9}{1024}\right ) x^{3}+\left (\frac {3 \ln \relax (5)^{2}}{128}-\frac {27 \ln \relax (5)}{64}+\frac {183}{64}\right ) x^{2}+\frac {x^{4}}{65536}+\ln \relax (5)^{4}-36 \ln \relax (5)^{3}+476 \ln \relax (5)^{2}-2736 \ln \relax (5)+5776}{x}\)	\(58\)
default	\(\frac {3 x \ln \relax (5)^{2}}{128}+\frac {x^{2} \ln \relax (5)}{1024}+\frac {x^{3}}{65536}-\frac {27 x \ln \relax (5)}{64}-\frac {9 x^{2}}{1024}+\frac {183 x}{64}-\frac {-65536 \ln \relax (5)^{4}+2359296 \ln \relax (5)^{3}-31195136 \ln \relax (5)^{2}[phone] \ln \relax (5)-378535936}{65536 x}\)	\(63\)
gosper	\(\frac {65536 \ln \relax (5)^{4}+1536 x^{2} \ln \relax (5)^{2}+64 x^{3} \ln \relax (5)+x^{4}-2359296 \ln \relax (5)^{3}-27648 x^{2} \ln \relax (5)-576 x^{3}[phone] \ln \relax (5)^{2}+187392 x^{2}-179306496 \ln \relax (5)[phone]}{65536 x}\)	\(66\)
risch	\(\frac {3 x \ln \relax (5)^{2}}{128}+\frac {x^{2} \ln \relax (5)}{1024}+\frac {x^{3}}{65536}-\frac {27 x \ln \relax (5)}{64}-\frac {9 x^{2}}{1024}+\frac {183 x}{64}+\frac {\ln \relax (5)^{4}}{x}-\frac {36 \ln \relax (5)^{3}}{x}+\frac {476 \ln \relax (5)^{2}}{x}-\frac {2736 \ln \relax (5)}{x}+\frac {5776}{x}\)	\(72\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/65536*(-65536*ln(5)^4+2359296*ln(5)^3+(1536*x^2-31195136)*ln(5)^2+(128*x^3-27648*x^2+179306496)*ln(5)+3*
x^4-1152*x^3+187392*x^2-378535936)/x^2,x,method=_RETURNVERBOSE)

[Out]

((1/1024*ln(5)-9/1024)*x^3+(3/128*ln(5)^2-27/64*ln(5)+183/64)*x^2+1/65536*x^4+ln(5)^4-36*ln(5)^3+476*ln(5)^2-2
736*ln(5)+5776)/x

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maxima [B]  time = 0.37, size = 54, normalized size = 2.16 \begin {gather*} \frac {1}{65536} \, x^{3} + \frac {1}{1024} \, x^{2} {\left (\log \relax (5) - 9\right )} + \frac {3}{128} \, {\left (\log \relax (5)^{2} - 18 \, \log \relax (5) + 122\right )} x + \frac {\log \relax (5)^{4} - 36 \, \log \relax (5)^{3} + 476 \, \log \relax (5)^{2} - 2736 \, \log \relax (5) + 5776}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/65536*(-65536*log(5)^4+2359296*log(5)^3+(1536*x^2-31195136)*log(5)^2+(128*x^3-27648*x^2+179306496)
*log(5)+3*x^4-1152*x^3+187392*x^2-378535936)/x^2,x, algorithm="maxima")

[Out]

1/65536*x^3 + 1/1024*x^2*(log(5) - 9) + 3/128*(log(5)^2 - 18*log(5) + 122)*x + (log(5)^4 - 36*log(5)^3 + 476*l
og(5)^2 - 2736*log(5) + 5776)/x

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mupad [B]  time = 0.73, size = 56, normalized size = 2.24 \begin {gather*} x\,\left (\frac {3\,{\ln \relax (5)}^2}{128}-\frac {27\,\ln \relax (5)}{64}+\frac {183}{64}\right )+x^2\,\left (\frac {\ln \relax (5)}{1024}-\frac {9}{1024}\right )+\frac {476\,{\ln \relax (5)}^2-2736\,\ln \relax (5)-36\,{\ln \relax (5)}^3+{\ln \relax (5)}^4+5776}{x}+\frac {x^3}{65536} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((log(5)*(128*x^3 - 27648*x^2 + 179306496))/65536 + (log(5)^2*(1536*x^2 - 31195136))/65536 + 36*log(5)^3 -
log(5)^4 + (183*x^2)/64 - (9*x^3)/512 + (3*x^4)/65536 - 5776)/x^2,x)

[Out]

x*((3*log(5)^2)/128 - (27*log(5))/64 + 183/64) + x^2*(log(5)/1024 - 9/1024) + (476*log(5)^2 - 2736*log(5) - 36
*log(5)^3 + log(5)^4 + 5776)/x + x^3/65536

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sympy [B]  time = 0.18, size = 63, normalized size = 2.52 \begin {gather*} \frac {x^{3}}{65536} + \frac {x^{2} \left (-576 + 64 \log {\relax (5 )}\right )}{65536} + \frac {x \left (- 27648 \log {\relax (5 )} + 1536 \log {\relax (5 )}^{2} + 187392\right )}{65536} + \frac {- 179306496 \log {\relax (5 )} - 2359296 \log {\relax (5 )}^{3} + 65536 \log {\relax (5 )}^{4} + 31195136 \log {\relax (5 )}^{2} + 378535936}{65536 x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/65536*(-65536*ln(5)**4+2359296*ln(5)**3+(1536*x**2-31195136)*ln(5)**2+(128*x**3-27648*x**2+1793064
96)*ln(5)+3*x**4-1152*x**3+187392*x**2-378535936)/x**2,x)

[Out]

x**3/65536 + x**2*(-576 + 64*log(5))/65536 + x*(-27648*log(5) + 1536*log(5)**2 + 187392)/65536 + (-179306496*l
og(5) - 2359296*log(5)**3 + 65536*log(5)**4 + 31195136*log(5)**2 + 378535936)/(65536*x)

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