3.32.54 \(\int \frac {9+55 x^2-42 x^3-9 x^4-8 x^5-5 x^6+(-36 x^4+16 x^5+20 x^6+24 x^7) \log (5)+(24 x^5-10 x^6-12 x^7-42 x^8) \log ^2(5)+32 x^9 \log ^3(5)-9 x^{10} \log ^4(5)}{25 x^2} \, dx\)

Optimal. Leaf size=33 \[ x \left (2-x-\frac {1}{25} \left (1-\frac {3}{x}+x+\left (-x+x^2 \log (5)\right )^2\right )^2\right ) \]

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Rubi [B]  time = 0.09, antiderivative size = 117, normalized size of antiderivative = 3.55, number of steps used = 3, number of rules used = 2, integrand size = 102, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.020, Rules used = {12, 14} \begin {gather*} -\frac {1}{25} x^9 \log ^4(5)+\frac {4}{25} x^8 \log ^3(5)-\frac {6}{25} x^7 \log ^2(5)+\frac {2}{25} x^6 (2-\log (5)) \log (5)-\frac {1}{25} x^5 \left (1+2 \log ^2(5)-\log (625)\right )-\frac {2}{25} x^4 \left (1-3 \log ^2(5)-\log (25)\right )-\frac {3}{25} x^3 (1+\log (625))-\frac {21 x^2}{25}+\frac {11 x}{5}-\frac {9}{25 x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(9 + 55*x^2 - 42*x^3 - 9*x^4 - 8*x^5 - 5*x^6 + (-36*x^4 + 16*x^5 + 20*x^6 + 24*x^7)*Log[5] + (24*x^5 - 10*
x^6 - 12*x^7 - 42*x^8)*Log[5]^2 + 32*x^9*Log[5]^3 - 9*x^10*Log[5]^4)/(25*x^2),x]

[Out]

-9/(25*x) + (11*x)/5 - (21*x^2)/25 + (2*x^6*(2 - Log[5])*Log[5])/25 - (6*x^7*Log[5]^2)/25 + (4*x^8*Log[5]^3)/2
5 - (x^9*Log[5]^4)/25 - (2*x^4*(1 - 3*Log[5]^2 - Log[25]))/25 - (x^5*(1 + 2*Log[5]^2 - Log[625]))/25 - (3*x^3*
(1 + Log[625]))/25

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 {1}{25} \int \frac {9+55 x^2-42 x^3-9 x^4-8 x^5-5 x^6+\left (-36 x^4+16 x^5+20 x^6+24 x^7\right ) \log (5)+\left (24 x^5-10 x^6-12 x^7-42 x^8\right ) \log ^2(5)+32 x^9 \log ^3(5)-9 x^{10} \log ^4(5)}{x^2} \, dx\\ &=\frac {1}{25} \int \left (55+\frac {9}{x^2}-42 x-12 x^5 (-2+\log (5)) \log (5)-42 x^6 \log ^2(5)+32 x^7 \log ^3(5)-9 x^8 \log ^4(5)-5 x^4 \left (1-4 \log (5)+2 \log ^2(5)\right )+8 x^3 \left (-1+3 \log ^2(5)+\log (25)\right )-9 x^2 (1+\log (625))\right ) \, dx\\ &=-\frac {9}{25 x}+\frac {11 x}{5}-\frac {21 x^2}{25}+\frac {2}{25} x^6 (2-\log (5)) \log (5)-\frac {6}{25} x^7 \log ^2(5)+\frac {4}{25} x^8 \log ^3(5)-\frac {1}{25} x^9 \log ^4(5)-\frac {2}{25} x^4 \left (1-3 \log ^2(5)-\log (25)\right )-\frac {1}{25} x^5 \left (1+2 \log ^2(5)-\log (625)\right )-\frac {3}{25} x^3 (1+\log (625))\\ \end {aligned} \end {gather*}

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Mathematica [B]  time = 0.05, size = 97, normalized size = 2.94 \begin {gather*} \frac {1}{25} \left (-\frac {9}{x}+55 x-21 x^2-2 x^6 (-2+\log (5)) \log (5)-6 x^7 \log ^2(5)+4 x^8 \log ^3(5)-x^9 \log ^4(5)-x^5 \left (1-4 \log (5)+2 \log ^2(5)\right )+2 x^4 \left (-1+3 \log ^2(5)+\log (25)\right )-3 x^3 (1+\log (625))\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(9 + 55*x^2 - 42*x^3 - 9*x^4 - 8*x^5 - 5*x^6 + (-36*x^4 + 16*x^5 + 20*x^6 + 24*x^7)*Log[5] + (24*x^5
- 10*x^6 - 12*x^7 - 42*x^8)*Log[5]^2 + 32*x^9*Log[5]^3 - 9*x^10*Log[5]^4)/(25*x^2),x]

[Out]

(-9/x + 55*x - 21*x^2 - 2*x^6*(-2 + Log[5])*Log[5] - 6*x^7*Log[5]^2 + 4*x^8*Log[5]^3 - x^9*Log[5]^4 - x^5*(1 -
4*Log[5] + 2*Log[5]^2) + 2*x^4*(-1 + 3*Log[5]^2 + Log[25]) - 3*x^3*(1 + Log[625]))/25

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fricas [B]  time = 0.59, size = 89, normalized size = 2.70 \begin {gather*} -\frac {x^{10} \log \relax (5)^{4} - 4 \, x^{9} \log \relax (5)^{3} + x^{6} + 2 \, x^{5} + 3 \, x^{4} + 21 \, x^{3} + 2 \, {\left (3 \, x^{8} + x^{7} + x^{6} - 3 \, x^{5}\right )} \log \relax (5)^{2} - 55 \, x^{2} - 4 \, {\left (x^{7} + x^{6} + x^{5} - 3 \, x^{4}\right )} \log \relax (5) + 9}{25 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/25*(-9*x^10*log(5)^4+32*x^9*log(5)^3+(-42*x^8-12*x^7-10*x^6+24*x^5)*log(5)^2+(24*x^7+20*x^6+16*x^5
-36*x^4)*log(5)-5*x^6-8*x^5-9*x^4-42*x^3+55*x^2+9)/x^2,x, algorithm="fricas")

[Out]

-1/25*(x^10*log(5)^4 - 4*x^9*log(5)^3 + x^6 + 2*x^5 + 3*x^4 + 21*x^3 + 2*(3*x^8 + x^7 + x^6 - 3*x^5)*log(5)^2
- 55*x^2 - 4*(x^7 + x^6 + x^5 - 3*x^4)*log(5) + 9)/x

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giac [B]  time = 0.20, size = 111, normalized size = 3.36 \begin {gather*} -\frac {1}{25} \, x^{9} \log \relax (5)^{4} + \frac {4}{25} \, x^{8} \log \relax (5)^{3} - \frac {6}{25} \, x^{7} \log \relax (5)^{2} - \frac {2}{25} \, x^{6} \log \relax (5)^{2} + \frac {4}{25} \, x^{6} \log \relax (5) - \frac {2}{25} \, x^{5} \log \relax (5)^{2} + \frac {4}{25} \, x^{5} \log \relax (5) + \frac {6}{25} \, x^{4} \log \relax (5)^{2} - \frac {1}{25} \, x^{5} + \frac {4}{25} \, x^{4} \log \relax (5) - \frac {2}{25} \, x^{4} - \frac {12}{25} \, x^{3} \log \relax (5) - \frac {3}{25} \, x^{3} - \frac {21}{25} \, x^{2} + \frac {11}{5} \, x - \frac {9}{25 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/25*(-9*x^10*log(5)^4+32*x^9*log(5)^3+(-42*x^8-12*x^7-10*x^6+24*x^5)*log(5)^2+(24*x^7+20*x^6+16*x^5
-36*x^4)*log(5)-5*x^6-8*x^5-9*x^4-42*x^3+55*x^2+9)/x^2,x, algorithm="giac")

[Out]

-1/25*x^9*log(5)^4 + 4/25*x^8*log(5)^3 - 6/25*x^7*log(5)^2 - 2/25*x^6*log(5)^2 + 4/25*x^6*log(5) - 2/25*x^5*lo
g(5)^2 + 4/25*x^5*log(5) + 6/25*x^4*log(5)^2 - 1/25*x^5 + 4/25*x^4*log(5) - 2/25*x^4 - 12/25*x^3*log(5) - 3/25
*x^3 - 21/25*x^2 + 11/5*x - 9/25/x

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maple [A]  time = 0.08, size = 101, normalized size = 3.06

method	result	size
norman	\(\frac {-\frac {9}{25}+\left (-\frac {2 \ln \relax (5)^{2}}{25}+\frac {4 \ln \relax (5)}{25}\right ) x^{7}+\left (-\frac {12 \ln \relax (5)}{25}-\frac {3}{25}\right ) x^{4}+\left (-\frac {2 \ln \relax (5)^{2}}{25}+\frac {4 \ln \relax (5)}{25}-\frac {1}{25}\right ) x^{6}+\left (\frac {6 \ln \relax (5)^{2}}{25}+\frac {4 \ln \relax (5)}{25}-\frac {2}{25}\right ) x^{5}+\frac {11 x^{2}}{5}-\frac {21 x^{3}}{25}-\frac {6 x^{8} \ln \relax (5)^{2}}{25}+\frac {4 x^{9} \ln \relax (5)^{3}}{25}-\frac {x^{10} \ln \relax (5)^{4}}{25}}{x}\)	\(101\)
gosper	\(-\frac {x^{10} \ln \relax (5)^{4}-4 x^{9} \ln \relax (5)^{3}+6 x^{8} \ln \relax (5)^{2}+2 x^{7} \ln \relax (5)^{2}+2 x^{6} \ln \relax (5)^{2}-4 x^{7} \ln \relax (5)-6 x^{5} \ln \relax (5)^{2}-4 x^{6} \ln \relax (5)-4 x^{5} \ln \relax (5)+x^{6}+12 x^{4} \ln \relax (5)+2 x^{5}+3 x^{4}+21 x^{3}-55 x^{2}+9}{25 x}\)	\(112\)
default	\(-\frac {x^{9} \ln \relax (5)^{4}}{25}+\frac {4 x^{8} \ln \relax (5)^{3}}{25}-\frac {6 x^{7} \ln \relax (5)^{2}}{25}-\frac {2 x^{6} \ln \relax (5)^{2}}{25}-\frac {2 x^{5} \ln \relax (5)^{2}}{25}+\frac {4 x^{6} \ln \relax (5)}{25}+\frac {6 x^{4} \ln \relax (5)^{2}}{25}+\frac {4 x^{5} \ln \relax (5)}{25}+\frac {4 x^{4} \ln \relax (5)}{25}-\frac {x^{5}}{25}-\frac {12 x^{3} \ln \relax (5)}{25}-\frac {2 x^{4}}{25}-\frac {3 x^{3}}{25}-\frac {21 x^{2}}{25}+\frac {11 x}{5}-\frac {9}{25 x}\)	\(112\)
risch	\(-\frac {x^{9} \ln \relax (5)^{4}}{25}+\frac {4 x^{8} \ln \relax (5)^{3}}{25}-\frac {6 x^{7} \ln \relax (5)^{2}}{25}-\frac {2 x^{6} \ln \relax (5)^{2}}{25}-\frac {2 x^{5} \ln \relax (5)^{2}}{25}+\frac {4 x^{6} \ln \relax (5)}{25}+\frac {6 x^{4} \ln \relax (5)^{2}}{25}+\frac {4 x^{5} \ln \relax (5)}{25}+\frac {4 x^{4} \ln \relax (5)}{25}-\frac {x^{5}}{25}-\frac {12 x^{3} \ln \relax (5)}{25}-\frac {2 x^{4}}{25}-\frac {3 x^{3}}{25}-\frac {21 x^{2}}{25}+\frac {11 x}{5}-\frac {9}{25 x}\)	\(112\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/25*(-9*x^10*ln(5)^4+32*x^9*ln(5)^3+(-42*x^8-12*x^7-10*x^6+24*x^5)*ln(5)^2+(24*x^7+20*x^6+16*x^5-36*x^4)*
ln(5)-5*x^6-8*x^5-9*x^4-42*x^3+55*x^2+9)/x^2,x,method=_RETURNVERBOSE)

[Out]

(-9/25+(-2/25*ln(5)^2+4/25*ln(5))*x^7+(-12/25*ln(5)-3/25)*x^4+(-2/25*ln(5)^2+4/25*ln(5)-1/25)*x^6+(6/25*ln(5)^
2+4/25*ln(5)-2/25)*x^5+11/5*x^2-21/25*x^3-6/25*x^8*ln(5)^2+4/25*x^9*ln(5)^3-1/25*x^10*ln(5)^4)/x

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maxima [B]  time = 0.37, size = 100, normalized size = 3.03 \begin {gather*} -\frac {1}{25} \, x^{9} \log \relax (5)^{4} + \frac {4}{25} \, x^{8} \log \relax (5)^{3} - \frac {6}{25} \, x^{7} \log \relax (5)^{2} - \frac {2}{25} \, {\left (\log \relax (5)^{2} - 2 \, \log \relax (5)\right )} x^{6} - \frac {1}{25} \, {\left (2 \, \log \relax (5)^{2} - 4 \, \log \relax (5) + 1\right )} x^{5} + \frac {2}{25} \, {\left (3 \, \log \relax (5)^{2} + 2 \, \log \relax (5) - 1\right )} x^{4} - \frac {3}{25} \, x^{3} {\left (4 \, \log \relax (5) + 1\right )} - \frac {21}{25} \, x^{2} + \frac {11}{5} \, x - \frac {9}{25 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/25*(-9*x^10*log(5)^4+32*x^9*log(5)^3+(-42*x^8-12*x^7-10*x^6+24*x^5)*log(5)^2+(24*x^7+20*x^6+16*x^5
-36*x^4)*log(5)-5*x^6-8*x^5-9*x^4-42*x^3+55*x^2+9)/x^2,x, algorithm="maxima")

[Out]

-1/25*x^9*log(5)^4 + 4/25*x^8*log(5)^3 - 6/25*x^7*log(5)^2 - 2/25*(log(5)^2 - 2*log(5))*x^6 - 1/25*(2*log(5)^2
- 4*log(5) + 1)*x^5 + 2/25*(3*log(5)^2 + 2*log(5) - 1)*x^4 - 3/25*x^3*(4*log(5) + 1) - 21/25*x^2 + 11/5*x - 9
/25/x

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mupad [B]  time = 1.88, size = 100, normalized size = 3.03 \begin {gather*} \frac {11\,x}{5}-\frac {6\,x^7\,{\ln \relax (5)}^2}{25}+\frac {4\,x^8\,{\ln \relax (5)}^3}{25}-\frac {x^9\,{\ln \relax (5)}^4}{25}-x^3\,\left (\frac {12\,\ln \relax (5)}{25}+\frac {3}{25}\right )+x^6\,\left (\frac {4\,\ln \relax (5)}{25}-\frac {2\,{\ln \relax (5)}^2}{25}\right )-x^5\,\left (\frac {2\,{\ln \relax (5)}^2}{25}-\frac {4\,\ln \relax (5)}{25}+\frac {1}{25}\right )+x^4\,\left (\frac {4\,\ln \relax (5)}{25}+\frac {6\,{\ln \relax (5)}^2}{25}-\frac {2}{25}\right )-\frac {9}{25\,x}-\frac {21\,x^2}{25} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-((9*x^10*log(5)^4)/25 - (32*x^9*log(5)^3)/25 - (log(5)*(16*x^5 - 36*x^4 + 20*x^6 + 24*x^7))/25 + (log(5)^
2*(10*x^6 - 24*x^5 + 12*x^7 + 42*x^8))/25 - (11*x^2)/5 + (42*x^3)/25 + (9*x^4)/25 + (8*x^5)/25 + x^6/5 - 9/25)
/x^2,x)

[Out]

(11*x)/5 - (6*x^7*log(5)^2)/25 + (4*x^8*log(5)^3)/25 - (x^9*log(5)^4)/25 - x^3*((12*log(5))/25 + 3/25) + x^6*(
(4*log(5))/25 - (2*log(5)^2)/25) - x^5*((2*log(5)^2)/25 - (4*log(5))/25 + 1/25) + x^4*((4*log(5))/25 + (6*log(
5)^2)/25 - 2/25) - 9/(25*x) - (21*x^2)/25

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sympy [B]  time = 0.21, size = 116, normalized size = 3.52 \begin {gather*} - \frac {x^{9} \log {\relax (5 )}^{4}}{25} + \frac {4 x^{8} \log {\relax (5 )}^{3}}{25} - \frac {6 x^{7} \log {\relax (5 )}^{2}}{25} - \frac {x^{6} \left (- 4 \log {\relax (5 )} + 2 \log {\relax (5 )}^{2}\right )}{25} - \frac {x^{5} \left (- 4 \log {\relax (5 )} + 1 + 2 \log {\relax (5 )}^{2}\right )}{25} - \frac {x^{4} \left (- 6 \log {\relax (5 )}^{2} - 4 \log {\relax (5 )} + 2\right )}{25} - \frac {x^{3} \left (3 + 12 \log {\relax (5 )}\right )}{25} - \frac {21 x^{2}}{25} + \frac {11 x}{5} - \frac {9}{25 x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/25*(-9*x**10*ln(5)**4+32*x**9*ln(5)**3+(-42*x**8-12*x**7-10*x**6+24*x**5)*ln(5)**2+(24*x**7+20*x**
6+16*x**5-36*x**4)*ln(5)-5*x**6-8*x**5-9*x**4-42*x**3+55*x**2+9)/x**2,x)

[Out]

-x**9*log(5)**4/25 + 4*x**8*log(5)**3/25 - 6*x**7*log(5)**2/25 - x**6*(-4*log(5) + 2*log(5)**2)/25 - x**5*(-4*
log(5) + 1 + 2*log(5)**2)/25 - x**4*(-6*log(5)**2 - 4*log(5) + 2)/25 - x**3*(3 + 12*log(5))/25 - 21*x**2/25 +
11*x/5 - 9/(25*x)

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