3.54.90 \(\int \frac {-3 x^2+12 x^4+16 x^5+5 x^6+(-6 x+16 x^3+20 x^4+6 x^5) \log (5)-3 \log ^2(5)}{3 x^2+6 x \log (5)+3 \log ^2(5)} \, dx\)

Optimal. Leaf size=26 \[ -x+\frac {x^2 \left (2 x+x^2\right )^2}{3 (x+\log (5))} \]

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Rubi [B]  time = 0.12, antiderivative size = 96, normalized size of antiderivative = 3.69, number of steps used = 4, number of rules used = 3, integrand size = 69, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.043, Rules used = {27, 12, 1850} \begin {gather*} \frac {x^5}{3}+\frac {1}{3} x^4 (4-\log (5))+\frac {1}{3} x^3 (2-\log (5))^2-\frac {1}{3} x^2 (2-\log (5))^2 \log (5)+\frac {(2-\log (5))^2 \log ^4(5)}{3 (x+\log (5))}-\frac {1}{3} x \left (3-(2-\log (5))^2 \log ^2(5)\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-3*x^2 + 12*x^4 + 16*x^5 + 5*x^6 + (-6*x + 16*x^3 + 20*x^4 + 6*x^5)*Log[5] - 3*Log[5]^2)/(3*x^2 + 6*x*Log
[5] + 3*Log[5]^2),x]

[Out]

x^5/3 + (x^3*(2 - Log[5])^2)/3 + (x^4*(4 - Log[5]))/3 - (x^2*(2 - Log[5])^2*Log[5])/3 + ((2 - Log[5])^2*Log[5]
^4)/(3*(x + Log[5])) - (x*(3 - (2 - Log[5])^2*Log[5]^2))/3

Rule 12

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

Rule 27

Int[(u_.)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[u*Cancel[(b/2 + c*x)^(2*p)/c^p], x] /; Fr
eeQ[{a, b, c}, x] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]

Rule 1850

Int[(Pq_)*((a_) + (b_.)*(x_)^(n_.))^(p_.), x_Symbol] :> Int[ExpandIntegrand[Pq*(a + b*x^n)^p, x], x] /; FreeQ[
{a, b, n}, x] && PolyQ[Pq, x] && (IGtQ[p, 0] || EqQ[n, 1])

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-3 x^2+12 x^4+16 x^5+5 x^6+\left (-6 x+16 x^3+20 x^4+6 x^5\right ) \log (5)-3 \log ^2(5)}{3 (x+\log (5))^2} \, dx\\ &=\frac {1}{3} \int \frac {-3 x^2+12 x^4+16 x^5+5 x^6+\left (-6 x+16 x^3+20 x^4+6 x^5\right ) \log (5)-3 \log ^2(5)}{(x+\log (5))^2} \, dx\\ &=\frac {1}{3} \int \left (5 x^4-4 x^3 (-4+\log (5))+3 x^2 (-2+\log (5))^2-2 x (-2+\log (5))^2 \log (5)-\frac {(-2+\log (5))^2 \log ^4(5)}{(x+\log (5))^2}-3 \left (1-\frac {1}{3} (-2+\log (5))^2 \log ^2(5)\right )\right ) \, dx\\ &=\frac {x^5}{3}+\frac {1}{3} x^3 (2-\log (5))^2+\frac {1}{3} x^4 (4-\log (5))-\frac {1}{3} x^2 (2-\log (5))^2 \log (5)+\frac {(2-\log (5))^2 \log ^4(5)}{3 (x+\log (5))}-\frac {1}{3} x \left (3-(2-\log (5))^2 \log ^2(5)\right )\\ \end {aligned} \end {gather*}

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Mathematica [B]  time = 0.06, size = 111, normalized size = 4.27 \begin {gather*} \frac {24 x^5+6 x^6+x^4 \left (24+15 \log ^2(5)-5 \log (5) \log (125)\right )+6 x^2 \left (-3+15 \log ^4(5)-5 \log ^3(5) \log (125)\right )+6 \log ^4(5) \left (4+25 \log ^2(5)-4 \log (5) (1+\log (15625))\right )+6 x \log (5) \left (-3+4 \log ^2(5)+25 \log ^4(5)-4 \log ^3(5) (1+\log (15625))\right )}{18 (x+\log (5))} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-3*x^2 + 12*x^4 + 16*x^5 + 5*x^6 + (-6*x + 16*x^3 + 20*x^4 + 6*x^5)*Log[5] - 3*Log[5]^2)/(3*x^2 + 6
*x*Log[5] + 3*Log[5]^2),x]

[Out]

(24*x^5 + 6*x^6 + x^4*(24 + 15*Log[5]^2 - 5*Log[5]*Log[125]) + 6*x^2*(-3 + 15*Log[5]^4 - 5*Log[5]^3*Log[125])
+ 6*Log[5]^4*(4 + 25*Log[5]^2 - 4*Log[5]*(1 + Log[15625])) + 6*x*Log[5]*(-3 + 4*Log[5]^2 + 25*Log[5]^4 - 4*Log
[5]^3*(1 + Log[15625])))/(18*(x + Log[5]))

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fricas [B]  time = 0.46, size = 60, normalized size = 2.31 \begin {gather*} \frac {x^{6} + {\left (x - 4\right )} \log \relax (5)^{5} + \log \relax (5)^{6} + 4 \, x^{5} - 4 \, {\left (x - 1\right )} \log \relax (5)^{4} + 4 \, x^{4} + 4 \, x \log \relax (5)^{3} - 3 \, x^{2} - 3 \, x \log \relax (5)}{3 \, {\left (x + \log \relax (5)\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-3*log(5)^2+(6*x^5+20*x^4+16*x^3-6*x)*log(5)+5*x^6+16*x^5+12*x^4-3*x^2)/(3*log(5)^2+6*x*log(5)+3*x^
2),x, algorithm="fricas")

[Out]

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

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giac [B]  time = 0.19, size = 113, normalized size = 4.35 \begin {gather*} \frac {1}{3} \, x^{5} - \frac {1}{3} \, x^{4} \log \relax (5) + \frac {1}{3} \, x^{3} \log \relax (5)^{2} - \frac {1}{3} \, x^{2} \log \relax (5)^{3} + \frac {1}{3} \, x \log \relax (5)^{4} + \frac {4}{3} \, x^{4} - \frac {4}{3} \, x^{3} \log \relax (5) + \frac {4}{3} \, x^{2} \log \relax (5)^{2} - \frac {4}{3} \, x \log \relax (5)^{3} + \frac {4}{3} \, x^{3} - \frac {4}{3} \, x^{2} \log \relax (5) + \frac {4}{3} \, x \log \relax (5)^{2} - x + \frac {\log \relax (5)^{6} - 4 \, \log \relax (5)^{5} + 4 \, \log \relax (5)^{4}}{3 \, {\left (x + \log \relax (5)\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-3*log(5)^2+(6*x^5+20*x^4+16*x^3-6*x)*log(5)+5*x^6+16*x^5+12*x^4-3*x^2)/(3*log(5)^2+6*x*log(5)+3*x^
2),x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.28, size = 33, normalized size = 1.27

method	result	size
norman	\(\frac {-x^{2}+\frac {4 x^{4}}{3}+\frac {4 x^{5}}{3}+\frac {x^{6}}{3}+\ln \relax (5)^{2}}{\ln \relax (5)+x}\)	\(33\)
gosper	\(\frac {x^{6}+4 x^{5}+4 x^{4}+3 \ln \relax (5)^{2}-3 x^{2}}{3 \ln \relax (5)+3 x}\)	\(34\)
default	\(\frac {x \ln \relax (5)^{4}}{3}-\frac {x^{2} \ln \relax (5)^{3}}{3}+\frac {x^{3} \ln \relax (5)^{2}}{3}-\frac {x^{4} \ln \relax (5)}{3}+\frac {x^{5}}{3}-\frac {4 \ln \relax (5)^{3} x}{3}+\frac {4 x^{2} \ln \relax (5)^{2}}{3}-\frac {4 x^{3} \ln \relax (5)}{3}+\frac {4 x^{4}}{3}+\frac {4 x \ln \relax (5)^{2}}{3}-\frac {4 x^{2} \ln \relax (5)}{3}+\frac {4 x^{3}}{3}-x +\frac {\ln \relax (5)^{4} \left (\ln \relax (5)^{2}-4 \ln \relax (5)+4\right )}{3 \ln \relax (5)+3 x}\)	\(111\)
risch	\(\frac {x \ln \relax (5)^{4}}{3}-\frac {x^{2} \ln \relax (5)^{3}}{3}+\frac {x^{3} \ln \relax (5)^{2}}{3}-\frac {x^{4} \ln \relax (5)}{3}+\frac {x^{5}}{3}-\frac {4 \ln \relax (5)^{3} x}{3}+\frac {4 x^{2} \ln \relax (5)^{2}}{3}-\frac {4 x^{3} \ln \relax (5)}{3}+\frac {4 x^{4}}{3}+\frac {4 x \ln \relax (5)^{2}}{3}-\frac {4 x^{2} \ln \relax (5)}{3}+\frac {4 x^{3}}{3}-x +\frac {\ln \relax (5)^{6}}{3 \ln \relax (5)+3 x}-\frac {4 \ln \relax (5)^{5}}{3 \left (\ln \relax (5)+x \right )}+\frac {4 \ln \relax (5)^{4}}{3 \left (\ln \relax (5)+x \right )}\)	\(125\)
meijerg	\(-\frac {x}{1+\frac {x}{\ln \relax (5)}}+\ln \relax (5)^{4} \left (2 \ln \relax (5)+\frac {16}{3}\right ) \left (-\frac {x \left (-\frac {3 x^{4}}{\ln \relax (5)^{4}}+\frac {5 x^{3}}{\ln \relax (5)^{3}}-\frac {10 x^{2}}{\ln \relax (5)^{2}}+\frac {30 x}{\ln \relax (5)}+60\right )}{12 \ln \relax (5) \left (1+\frac {x}{\ln \relax (5)}\right )}+5 \ln \left (1+\frac {x}{\ln \relax (5)}\right )\right )+\ln \relax (5)^{3} \left (\frac {20 \ln \relax (5)}{3}+4\right ) \left (\frac {x \left (\frac {5 x^{3}}{\ln \relax (5)^{3}}-\frac {10 x^{2}}{\ln \relax (5)^{2}}+\frac {30 x}{\ln \relax (5)}+60\right )}{15 \ln \relax (5) \left (1+\frac {x}{\ln \relax (5)}\right )}-4 \ln \left (1+\frac {x}{\ln \relax (5)}\right )\right )+\frac {16 \ln \relax (5)^{3} \left (-\frac {x \left (-\frac {2 x^{2}}{\ln \relax (5)^{2}}+\frac {6 x}{\ln \relax (5)}+12\right )}{4 \ln \relax (5) \left (1+\frac {x}{\ln \relax (5)}\right )}+3 \ln \left (1+\frac {x}{\ln \relax (5)}\right )\right )}{3}-2 \ln \relax (5) \left (-\frac {x}{\ln \relax (5) \left (1+\frac {x}{\ln \relax (5)}\right )}+\ln \left (1+\frac {x}{\ln \relax (5)}\right )\right )+\frac {5 \ln \relax (5)^{5} \left (\frac {x \left (\frac {14 x^{5}}{\ln \relax (5)^{5}}-\frac {21 x^{4}}{\ln \relax (5)^{4}}+\frac {35 x^{3}}{\ln \relax (5)^{3}}-\frac {70 x^{2}}{\ln \relax (5)^{2}}+\frac {210 x}{\ln \relax (5)}+420\right )}{70 \ln \relax (5) \left (1+\frac {x}{\ln \relax (5)}\right )}-6 \ln \left (1+\frac {x}{\ln \relax (5)}\right )\right )}{3}-\ln \relax (5) \left (\frac {x \left (\frac {3 x}{\ln \relax (5)}+6\right )}{3 \ln \relax (5) \left (1+\frac {x}{\ln \relax (5)}\right )}-2 \ln \left (1+\frac {x}{\ln \relax (5)}\right )\right )\)	\(364\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-3*ln(5)^2+(6*x^5+20*x^4+16*x^3-6*x)*ln(5)+5*x^6+16*x^5+12*x^4-3*x^2)/(3*ln(5)^2+6*x*ln(5)+3*x^2),x,metho
d=_RETURNVERBOSE)

[Out]

(-x^2+4/3*x^4+4/3*x^5+1/3*x^6+ln(5)^2)/(ln(5)+x)

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maxima [B]  time = 0.36, size = 96, normalized size = 3.69 \begin {gather*} \frac {1}{3} \, x^{5} - \frac {1}{3} \, x^{4} {\left (\log \relax (5) - 4\right )} + \frac {1}{3} \, {\left (\log \relax (5)^{2} - 4 \, \log \relax (5) + 4\right )} x^{3} - \frac {1}{3} \, {\left (\log \relax (5)^{3} - 4 \, \log \relax (5)^{2} + 4 \, \log \relax (5)\right )} x^{2} + \frac {1}{3} \, {\left (\log \relax (5)^{4} - 4 \, \log \relax (5)^{3} + 4 \, \log \relax (5)^{2} - 3\right )} x + \frac {\log \relax (5)^{6} - 4 \, \log \relax (5)^{5} + 4 \, \log \relax (5)^{4}}{3 \, {\left (x + \log \relax (5)\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-3*log(5)^2+(6*x^5+20*x^4+16*x^3-6*x)*log(5)+5*x^6+16*x^5+12*x^4-3*x^2)/(3*log(5)^2+6*x*log(5)+3*x^
2),x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 0.12, size = 196, normalized size = 7.54 \begin {gather*} \frac {4\,{\ln \relax (5)}^4-4\,{\ln \relax (5)}^5+{\ln \relax (5)}^6}{3\,x+3\,\ln \relax (5)}-x\,\left (2\,\ln \relax (5)\,\left (\frac {16\,\ln \relax (5)}{3}-2\,\ln \relax (5)\,\left (\frac {20\,\ln \relax (5)}{3}-\frac {5\,{\ln \relax (5)}^2}{3}+2\,\ln \relax (5)\,\left (\frac {4\,\ln \relax (5)}{3}-\frac {16}{3}\right )+4\right )+{\ln \relax (5)}^2\,\left (\frac {4\,\ln \relax (5)}{3}-\frac {16}{3}\right )\right )+{\ln \relax (5)}^2\,\left (\frac {20\,\ln \relax (5)}{3}-\frac {5\,{\ln \relax (5)}^2}{3}+2\,\ln \relax (5)\,\left (\frac {4\,\ln \relax (5)}{3}-\frac {16}{3}\right )+4\right )+1\right )-x^4\,\left (\frac {\ln \relax (5)}{3}-\frac {4}{3}\right )+x^3\,\left (\frac {20\,\ln \relax (5)}{9}-\frac {5\,{\ln \relax (5)}^2}{9}+\frac {2\,\ln \relax (5)\,\left (\frac {4\,\ln \relax (5)}{3}-\frac {16}{3}\right )}{3}+\frac {4}{3}\right )+x^2\,\left (\frac {8\,\ln \relax (5)}{3}-\ln \relax (5)\,\left (\frac {20\,\ln \relax (5)}{3}-\frac {5\,{\ln \relax (5)}^2}{3}+2\,\ln \relax (5)\,\left (\frac {4\,\ln \relax (5)}{3}-\frac {16}{3}\right )+4\right )+\frac {{\ln \relax (5)}^2\,\left (\frac {4\,\ln \relax (5)}{3}-\frac {16}{3}\right )}{2}\right )+\frac {x^5}{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((log(5)*(16*x^3 - 6*x + 20*x^4 + 6*x^5) - 3*log(5)^2 - 3*x^2 + 12*x^4 + 16*x^5 + 5*x^6)/(6*x*log(5) + 3*lo
g(5)^2 + 3*x^2),x)

[Out]

(4*log(5)^4 - 4*log(5)^5 + log(5)^6)/(3*x + 3*log(5)) - x*(2*log(5)*((16*log(5))/3 - 2*log(5)*((20*log(5))/3 -
(5*log(5)^2)/3 + 2*log(5)*((4*log(5))/3 - 16/3) + 4) + log(5)^2*((4*log(5))/3 - 16/3)) + log(5)^2*((20*log(5)
)/3 - (5*log(5)^2)/3 + 2*log(5)*((4*log(5))/3 - 16/3) + 4) + 1) - x^4*(log(5)/3 - 4/3) + x^3*((20*log(5))/9 -
(5*log(5)^2)/9 + (2*log(5)*((4*log(5))/3 - 16/3))/3 + 4/3) + x^2*((8*log(5))/3 - log(5)*((20*log(5))/3 - (5*lo
g(5)^2)/3 + 2*log(5)*((4*log(5))/3 - 16/3) + 4) + (log(5)^2*((4*log(5))/3 - 16/3))/2) + x^5/3

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sympy [B]  time = 0.34, size = 116, normalized size = 4.46 \begin {gather*} \frac {x^{5}}{3} + x^{4} \left (\frac {4}{3} - \frac {\log {\relax (5 )}}{3}\right ) + x^{3} \left (- \frac {4 \log {\relax (5 )}}{3} + \frac {\log {\relax (5 )}^{2}}{3} + \frac {4}{3}\right ) + x^{2} \left (- \frac {4 \log {\relax (5 )}}{3} - \frac {\log {\relax (5 )}^{3}}{3} + \frac {4 \log {\relax (5 )}^{2}}{3}\right ) + x \left (- \frac {4 \log {\relax (5 )}^{3}}{3} - 1 + \frac {\log {\relax (5 )}^{4}}{3} + \frac {4 \log {\relax (5 )}^{2}}{3}\right ) + \frac {- 4 \log {\relax (5 )}^{5} + \log {\relax (5 )}^{6} + 4 \log {\relax (5 )}^{4}}{3 x + 3 \log {\relax (5 )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-3*ln(5)**2+(6*x**5+20*x**4+16*x**3-6*x)*ln(5)+5*x**6+16*x**5+12*x**4-3*x**2)/(3*ln(5)**2+6*x*ln(5)
+3*x**2),x)

[Out]

x**5/3 + x**4*(4/3 - log(5)/3) + x**3*(-4*log(5)/3 + log(5)**2/3 + 4/3) + x**2*(-4*log(5)/3 - log(5)**3/3 + 4*
log(5)**2/3) + x*(-4*log(5)**3/3 - 1 + log(5)**4/3 + 4*log(5)**2/3) + (-4*log(5)**5 + log(5)**6 + 4*log(5)**4)
/(3*x + 3*log(5))

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