3.100.70 \(\int \frac {-4 x^2+4900 x^5-125 x^6+(196-8 x+6125 x^4-150 x^5) \log (5)}{x^2+2 x \log (5)+\log ^2(5)} \, dx\)

Optimal. Leaf size=22 \[ 2+\frac {(49-x) x \left (4+25 x^4\right )}{x+\log (5)} \]

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Rubi [B]  time = 0.12, antiderivative size = 80, normalized size of antiderivative = 3.64, number of steps used = 3, number of rules used = 2, integrand size = 50, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.040, Rules used = {27, 1850} \begin {gather*} -25 x^5+25 x^4 (49+\log (5))-25 x^3 \log (5) (49+\log (5))+25 x^2 \log ^2(5) (49+\log (5))-x \left (4+25 \log ^3(5) (49+\log (5))\right )-\frac {\log (5) \left (196+25 \log ^5(5)+1225 \log ^4(5)+\log (625)\right )}{x+\log (5)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-4*x^2 + 4900*x^5 - 125*x^6 + (196 - 8*x + 6125*x^4 - 150*x^5)*Log[5])/(x^2 + 2*x*Log[5] + Log[5]^2),x]

[Out]

-25*x^5 + 25*x^4*(49 + Log[5]) - 25*x^3*Log[5]*(49 + Log[5]) + 25*x^2*Log[5]^2*(49 + Log[5]) - x*(4 + 25*Log[5
]^3*(49 + Log[5])) - (Log[5]*(196 + 1225*Log[5]^4 + 25*Log[5]^5 + Log[625]))/(x + Log[5])

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 {-4 x^2+4900 x^5-125 x^6+\left (196-8 x+6125 x^4-150 x^5\right ) \log (5)}{(x+\log (5))^2} \, dx\\ &=\int \left (-125 x^4+100 x^3 (49+\log (5))-75 x^2 \log (5) (49+\log (5))+50 x \log ^2(5) (49+\log (5))-4 \left (1+\frac {25}{4} \log ^3(5) (49+\log (5))\right )+\frac {\log (5) \left (196+1225 \log ^4(5)+25 \log ^5(5)+\log (625)\right )}{(x+\log (5))^2}\right ) \, dx\\ &=-25 x^5+25 x^4 (49+\log (5))-25 x^3 \log (5) (49+\log (5))+25 x^2 \log ^2(5) (49+\log (5))-x \left (4+25 \log ^3(5) (49+\log (5))\right )-\frac {\log (5) \left (196+1225 \log ^4(5)+25 \log ^5(5)+\log (625)\right )}{x+\log (5)}\\ \end {aligned} \end {gather*}

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Mathematica [B]  time = 0.05, size = 89, normalized size = 4.05 \begin {gather*} \frac {1225 x^5-25 x^6+x^2 \left (-4-375 \log ^4(5)+125 \log ^3(5) \log (125)\right )-x \log (5) \left (4+625 \log ^4(5)-25 \log ^3(5) (-49+8 \log (125))\right )-\log (5) \left (196+625 \log ^5(5)-25 \log ^4(5) (-49+8 \log (125))+\log (625)\right )}{x+\log (5)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-4*x^2 + 4900*x^5 - 125*x^6 + (196 - 8*x + 6125*x^4 - 150*x^5)*Log[5])/(x^2 + 2*x*Log[5] + Log[5]^2
),x]

[Out]

(1225*x^5 - 25*x^6 + x^2*(-4 - 375*Log[5]^4 + 125*Log[5]^3*Log[125]) - x*Log[5]*(4 + 625*Log[5]^4 - 25*Log[5]^
3*(-49 + 8*Log[125])) - Log[5]*(196 + 625*Log[5]^5 - 25*Log[5]^4*(-49 + 8*Log[125]) + Log[625]))/(x + Log[5])

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fricas [B]  time = 1.03, size = 59, normalized size = 2.68 \begin {gather*} -\frac {25 \, x^{6} + 25 \, {\left (x + 49\right )} \log \relax (5)^{5} + 25 \, \log \relax (5)^{6} - 1225 \, x^{5} + 1225 \, x \log \relax (5)^{4} + 4 \, x^{2} + 4 \, {\left (x + 49\right )} \log \relax (5) + 4 \, \log \relax (5)^{2}}{x + \log \relax (5)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-150*x^5+6125*x^4-8*x+196)*log(5)-125*x^6+4900*x^5-4*x^2)/(log(5)^2+2*x*log(5)+x^2),x, algorithm="
fricas")

[Out]

-(25*x^6 + 25*(x + 49)*log(5)^5 + 25*log(5)^6 - 1225*x^5 + 1225*x*log(5)^4 + 4*x^2 + 4*(x + 49)*log(5) + 4*log
(5)^2)/(x + log(5))

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giac [B]  time = 0.17, size = 100, normalized size = 4.55 \begin {gather*} -25 \, x^{5} + 25 \, x^{4} \log \relax (5) - 25 \, x^{3} \log \relax (5)^{2} + 25 \, x^{2} \log \relax (5)^{3} - 25 \, x \log \relax (5)^{4} + 1225 \, x^{4} - 1225 \, x^{3} \log \relax (5) + 1225 \, x^{2} \log \relax (5)^{2} - 1225 \, x \log \relax (5)^{3} - 4 \, x - \frac {25 \, \log \relax (5)^{6} + 1225 \, \log \relax (5)^{5} + 4 \, \log \relax (5)^{2} + 196 \, \log \relax (5)}{x + \log \relax (5)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-150*x^5+6125*x^4-8*x+196)*log(5)-125*x^6+4900*x^5-4*x^2)/(log(5)^2+2*x*log(5)+x^2),x, algorithm="
giac")

[Out]

-25*x^5 + 25*x^4*log(5) - 25*x^3*log(5)^2 + 25*x^2*log(5)^3 - 25*x*log(5)^4 + 1225*x^4 - 1225*x^3*log(5) + 122
5*x^2*log(5)^2 - 1225*x*log(5)^3 - 4*x - (25*log(5)^6 + 1225*log(5)^5 + 4*log(5)^2 + 196*log(5))/(x + log(5))

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maple [A]  time = 0.16, size = 28, normalized size = 1.27

method	result	size
norman	\(\frac {-25 x^{6}+1225 x^{5}-4 x^{2}-196 \ln \relax (5)}{\ln \relax (5)+x}\)	\(28\)
gosper	\(-\frac {25 x^{6}-1225 x^{5}+4 x^{2}+196 \ln \relax (5)}{\ln \relax (5)+x}\)	\(29\)
default	\(-25 x \ln \relax (5)^{4}+25 x^{2} \ln \relax (5)^{3}-25 x^{3} \ln \relax (5)^{2}+25 x^{4} \ln \relax (5)-25 x^{5}-1225 \ln \relax (5)^{3} x +1225 x^{2} \ln \relax (5)^{2}-1225 x^{3} \ln \relax (5)+1225 x^{4}-4 x -\frac {\ln \relax (5) \left (25 \ln \relax (5)^{5}+1225 \ln \relax (5)^{4}+4 \ln \relax (5)+196\right )}{\ln \relax (5)+x}\)	\(98\)
risch	\(-25 x \ln \relax (5)^{4}+25 x^{2} \ln \relax (5)^{3}-25 x^{3} \ln \relax (5)^{2}+25 x^{4} \ln \relax (5)-25 x^{5}-1225 \ln \relax (5)^{3} x +1225 x^{2} \ln \relax (5)^{2}-1225 x^{3} \ln \relax (5)+1225 x^{4}-4 x -\frac {25 \ln \relax (5)^{6}}{\ln \relax (5)+x}-\frac {1225 \ln \relax (5)^{5}}{\ln \relax (5)+x}-\frac {4 \ln \relax (5)^{2}}{\ln \relax (5)+x}-\frac {196 \ln \relax (5)}{\ln \relax (5)+x}\)	\(116\)
meijerg	\(\left (-150 \ln \relax (5)+4900\right ) \ln \relax (5)^{4} \left (-\frac {x \left (-\frac {3 x^{4}}{\ln \relax (5)^{4}}+\frac {5 x^{3}}{\ln \relax (5)^{3}}-\frac {10 x^{2}}{\ln \relax (5)^{2}}+\frac {30 x}{\ln \relax (5)}+60\right )}{12 \ln \relax (5) \left (1+\frac {x}{\ln \relax (5)}\right )}+5 \ln \left (1+\frac {x}{\ln \relax (5)}\right )\right )+6125 \ln \relax (5)^{4} \left (\frac {x \left (\frac {5 x^{3}}{\ln \relax (5)^{3}}-\frac {10 x^{2}}{\ln \relax (5)^{2}}+\frac {30 x}{\ln \relax (5)}+60\right )}{15 \ln \relax (5) \left (1+\frac {x}{\ln \relax (5)}\right )}-4 \ln \left (1+\frac {x}{\ln \relax (5)}\right )\right )-8 \ln \relax (5) \left (-\frac {x}{\ln \relax (5) \left (1+\frac {x}{\ln \relax (5)}\right )}+\ln \left (1+\frac {x}{\ln \relax (5)}\right )\right )-125 \ln \relax (5)^{5} \left (\frac {x \left (\frac {14 x^{5}}{\ln \relax (5)^{5}}-\frac {21 x^{4}}{\ln \relax (5)^{4}}+\frac {35 x^{3}}{\ln \relax (5)^{3}}-\frac {70 x^{2}}{\ln \relax (5)^{2}}+\frac {210 x}{\ln \relax (5)}+420\right )}{70 \ln \relax (5) \left (1+\frac {x}{\ln \relax (5)}\right )}-6 \ln \left (1+\frac {x}{\ln \relax (5)}\right )\right )-4 \ln \relax (5) \left (\frac {x \left (\frac {3 x}{\ln \relax (5)}+6\right )}{3 \ln \relax (5) \left (1+\frac {x}{\ln \relax (5)}\right )}-2 \ln \left (1+\frac {x}{\ln \relax (5)}\right )\right )+\frac {196 x}{\ln \relax (5) \left (1+\frac {x}{\ln \relax (5)}\right )}\)	\(310\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-150*x^5+6125*x^4-8*x+196)*ln(5)-125*x^6+4900*x^5-4*x^2)/(ln(5)^2+2*x*ln(5)+x^2),x,method=_RETURNVERBOSE
)

[Out]

(-25*x^6+1225*x^5-4*x^2-196*ln(5))/(ln(5)+x)

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maxima [B]  time = 0.36, size = 93, normalized size = 4.23 \begin {gather*} -25 \, x^{5} + 25 \, x^{4} {\left (\log \relax (5) + 49\right )} - 25 \, {\left (\log \relax (5)^{2} + 49 \, \log \relax (5)\right )} x^{3} + 25 \, {\left (\log \relax (5)^{3} + 49 \, \log \relax (5)^{2}\right )} x^{2} - {\left (25 \, \log \relax (5)^{4} + 1225 \, \log \relax (5)^{3} + 4\right )} x - \frac {25 \, \log \relax (5)^{6} + 1225 \, \log \relax (5)^{5} + 4 \, \log \relax (5)^{2} + 196 \, \log \relax (5)}{x + \log \relax (5)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-150*x^5+6125*x^4-8*x+196)*log(5)-125*x^6+4900*x^5-4*x^2)/(log(5)^2+2*x*log(5)+x^2),x, algorithm="
maxima")

[Out]

-25*x^5 + 25*x^4*(log(5) + 49) - 25*(log(5)^2 + 49*log(5))*x^3 + 25*(log(5)^3 + 49*log(5)^2)*x^2 - (25*log(5)^
4 + 1225*log(5)^3 + 4)*x - (25*log(5)^6 + 1225*log(5)^5 + 4*log(5)^2 + 196*log(5))/(x + log(5))

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mupad [B]  time = 7.14, size = 185, normalized size = 8.41 \begin {gather*} x^4\,\left (25\,\ln \relax (5)+1225\right )-x\,\left ({\ln \relax (5)}^2\,\left (6125\,\ln \relax (5)+125\,{\ln \relax (5)}^2-2\,\ln \relax (5)\,\left (100\,\ln \relax (5)+4900\right )\right )-2\,\ln \relax (5)\,\left (2\,\ln \relax (5)\,\left (6125\,\ln \relax (5)+125\,{\ln \relax (5)}^2-2\,\ln \relax (5)\,\left (100\,\ln \relax (5)+4900\right )\right )+{\ln \relax (5)}^2\,\left (100\,\ln \relax (5)+4900\right )\right )+4\right )-x^2\,\left (\ln \relax (5)\,\left (6125\,\ln \relax (5)+125\,{\ln \relax (5)}^2-2\,\ln \relax (5)\,\left (100\,\ln \relax (5)+4900\right )\right )+\frac {{\ln \relax (5)}^2\,\left (100\,\ln \relax (5)+4900\right )}{2}\right )+x^3\,\left (\frac {6125\,\ln \relax (5)}{3}+\frac {125\,{\ln \relax (5)}^2}{3}-\frac {2\,\ln \relax (5)\,\left (100\,\ln \relax (5)+4900\right )}{3}\right )-\frac {196\,\ln \relax (5)+\ln \relax (5)\,\ln \left (625\right )+1225\,{\ln \relax (5)}^5+25\,{\ln \relax (5)}^6}{x+\ln \relax (5)}-25\,x^5 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(log(5)*(8*x - 6125*x^4 + 150*x^5 - 196) + 4*x^2 - 4900*x^5 + 125*x^6)/(2*x*log(5) + log(5)^2 + x^2),x)

[Out]

x^4*(25*log(5) + 1225) - x*(log(5)^2*(6125*log(5) + 125*log(5)^2 - 2*log(5)*(100*log(5) + 4900)) - 2*log(5)*(2
*log(5)*(6125*log(5) + 125*log(5)^2 - 2*log(5)*(100*log(5) + 4900)) + log(5)^2*(100*log(5) + 4900)) + 4) - x^2
*(log(5)*(6125*log(5) + 125*log(5)^2 - 2*log(5)*(100*log(5) + 4900)) + (log(5)^2*(100*log(5) + 4900))/2) + x^3
*((6125*log(5))/3 + (125*log(5)^2)/3 - (2*log(5)*(100*log(5) + 4900))/3) - (196*log(5) + log(5)*log(625) + 122
5*log(5)^5 + 25*log(5)^6)/(x + log(5)) - 25*x^5

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sympy [B]  time = 0.31, size = 99, normalized size = 4.50 \begin {gather*} - 25 x^{5} - x^{4} \left (-1225 - 25 \log {\relax (5 )}\right ) - x^{3} \left (25 \log {\relax (5 )}^{2} + 1225 \log {\relax (5 )}\right ) - x^{2} \left (- 1225 \log {\relax (5 )}^{2} - 25 \log {\relax (5 )}^{3}\right ) - x \left (4 + 25 \log {\relax (5 )}^{4} + 1225 \log {\relax (5 )}^{3}\right ) - \frac {4 \log {\relax (5 )}^{2} + 196 \log {\relax (5 )} + 25 \log {\relax (5 )}^{6} + 1225 \log {\relax (5 )}^{5}}{x + \log {\relax (5 )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-150*x**5+6125*x**4-8*x+196)*ln(5)-125*x**6+4900*x**5-4*x**2)/(ln(5)**2+2*x*ln(5)+x**2),x)

[Out]

-25*x**5 - x**4*(-1225 - 25*log(5)) - x**3*(25*log(5)**2 + 1225*log(5)) - x**2*(-1225*log(5)**2 - 25*log(5)**3
) - x*(4 + 25*log(5)**4 + 1225*log(5)**3) - (4*log(5)**2 + 196*log(5) + 25*log(5)**6 + 1225*log(5)**5)/(x + lo
g(5))

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