3.16.56 \(\int \frac {-x^2 \log (3)+(2 x+2 x^2) \log ^2(3)+(81 x^2+(-162 x-324 x^2) \log (3)+(81+324 x+324 x^2) \log ^2(3)) \log (5)}{(81 x^2+(-162 x-324 x^2) \log (3)+(81+324 x+324 x^2) \log ^2(3)) \log (5)} \, dx\)

Optimal. Leaf size=27 \[ x+\frac {x^2}{81 \left (1+2 x-\frac {x}{\log (3)}\right ) \log (5)} \]

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Rubi [B]  time = 0.17, antiderivative size = 80, normalized size of antiderivative = 2.96, number of steps used = 7, number of rules used = 5, integrand size = 98, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.051, Rules used = {12, 1984, 27, 6, 683} \begin {gather*} \frac {\log ^2(3) (\log (3)+162 \log (3) \log (5)-81 \log (5) \log (9))}{81 \log (5) (1-\log (9))^2 (\log (3)-x (1-\log (9)))}-\frac {x (\log (3)-81 \log (5)+162 \log (3) \log (5))}{81 \log (5) (1-\log (9))} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-(x^2*Log[3]) + (2*x + 2*x^2)*Log[3]^2 + (81*x^2 + (-162*x - 324*x^2)*Log[3] + (81 + 324*x + 324*x^2)*Log
[3]^2)*Log[5])/((81*x^2 + (-162*x - 324*x^2)*Log[3] + (81 + 324*x + 324*x^2)*Log[3]^2)*Log[5]),x]

[Out]

-1/81*(x*(Log[3] - 81*Log[5] + 162*Log[3]*Log[5]))/(Log[5]*(1 - Log[9])) + (Log[3]^2*(Log[3] + 162*Log[3]*Log[
5] - 81*Log[5]*Log[9]))/(81*Log[5]*(Log[3] - x*(1 - Log[9]))*(1 - Log[9])^2)

Rule 6

Int[(u_.)*((w_.) + (a_.)*(v_) + (b_.)*(v_))^(p_.), x_Symbol] :> Int[u*((a + b)*v + w)^p, x] /; FreeQ[{a, b}, x
] &&  !FreeQ[v, x]

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 683

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d +
e*x)^m*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[2*c*d - b*e,
0] && IGtQ[p, 0] &&  !(EqQ[m, 3] && NeQ[p, 1])

Rule 1984

Int[(u_)^(p_.)*(v_)^(q_.), x_Symbol] :> Int[ExpandToSum[u, x]^p*ExpandToSum[v, x]^q, x] /; FreeQ[{p, q}, x] &&
QuadraticQ[{u, v}, x] &&  !QuadraticMatchQ[{u, v}, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {\int \frac {-x^2 \log (3)+\left (2 x+2 x^2\right ) \log ^2(3)+\left (81 x^2+\left (-162 x-324 x^2\right ) \log (3)+\left (81+324 x+324 x^2\right ) \log ^2(3)\right ) \log (5)}{81 x^2+\left (-162 x-324 x^2\right ) \log (3)+\left (81+324 x+324 x^2\right ) \log ^2(3)} \, dx}{\log (5)}\\ &=\frac {\int \frac {81 \log ^2(3) \log (5)+2 x \log (3) (\log (3)-81 \log (5)+162 \log (3) \log (5))-x^2 (\log (3)-81 \log (5)+162 \log (3) \log (5)) (1-\log (9))}{81 \log ^2(3)-162 x \log (3) (1-\log (9))+81 x^2 (1-\log (9))^2} \, dx}{\log (5)}\\ &=\frac {\int \frac {81 \log ^2(3) \log (5)+2 x \log (3) (\log (3)-81 \log (5)+162 \log (3) \log (5))-x^2 (\log (3)-81 \log (5)+162 \log (3) \log (5)) (1-\log (9))}{81 (-x+\log (3)+x \log (9))^2} \, dx}{\log (5)}\\ &=\frac {\int \frac {81 \log ^2(3) \log (5)+2 x \log (3) (\log (3)-81 \log (5)+162 \log (3) \log (5))-x^2 (\log (3)-81 \log (5)+162 \log (3) \log (5)) (1-\log (9))}{81 (\log (3)+x (-1+\log (9)))^2} \, dx}{\log (5)}\\ &=\frac {\int \frac {81 \log ^2(3) \log (5)+2 x \log (3) (\log (3)-81 \log (5)+162 \log (3) \log (5))-x^2 (\log (3)-81 \log (5)+162 \log (3) \log (5)) (1-\log (9))}{(\log (3)+x (-1+\log (9)))^2} \, dx}{81 \log (5)}\\ &=\frac {\int \left (\frac {\log (3)-81 \log (5)+162 \log (3) \log (5)}{-1+\log (9)}-\frac {\log ^2(3) (\log (3)+162 \log (3) \log (5)-81 \log (5) \log (9))}{(\log (3)+x (-1+\log (9)))^2 (-1+\log (9))}\right ) \, dx}{81 \log (5)}\\ &=-\frac {x (\log (3)-81 \log (5)+162 \log (3) \log (5))}{81 \log (5) (1-\log (9))}+\frac {\log ^2(3) (\log (3)+162 \log (3) \log (5)-81 \log (5) \log (9))}{81 \log (5) (\log (3)-x (1-\log (9))) (1-\log (9))^2}\\ \end {aligned} \end {gather*}

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Mathematica [B]  time = 0.06, size = 65, normalized size = 2.41 \begin {gather*} \frac {x (\log (3)-81 \log (5)+162 \log (3) \log (5)) (-1+\log (9))+\frac {\log ^2(3) (\log (3)+162 \log (3) \log (5)-81 \log (5) \log (9))}{\log (3)+x (-1+\log (9))}}{81 \log (5) (-1+\log (9))^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-(x^2*Log[3]) + (2*x + 2*x^2)*Log[3]^2 + (81*x^2 + (-162*x - 324*x^2)*Log[3] + (81 + 324*x + 324*x^
2)*Log[3]^2)*Log[5])/((81*x^2 + (-162*x - 324*x^2)*Log[3] + (81 + 324*x + 324*x^2)*Log[3]^2)*Log[5]),x]

[Out]

(x*(Log[3] - 81*Log[5] + 162*Log[3]*Log[5])*(-1 + Log[9]) + (Log[3]^2*(Log[3] + 162*Log[3]*Log[5] - 81*Log[5]*
Log[9]))/(Log[3] + x*(-1 + Log[9])))/(81*Log[5]*(-1 + Log[9])^2)

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fricas [B]  time = 0.70, size = 123, normalized size = 4.56 \begin {gather*} \frac {{\left (4 \, x^{2} + 2 \, x + 1\right )} \log \relax (3)^{3} + x^{2} \log \relax (3) - {\left (4 \, x^{2} + x\right )} \log \relax (3)^{2} + 81 \, {\left (4 \, {\left (2 \, x^{2} + x\right )} \log \relax (3)^{3} - 4 \, {\left (3 \, x^{2} + x\right )} \log \relax (3)^{2} - x^{2} + {\left (6 \, x^{2} + x\right )} \log \relax (3)\right )} \log \relax (5)}{81 \, {\left (4 \, {\left (2 \, x + 1\right )} \log \relax (3)^{3} - 4 \, {\left (3 \, x + 1\right )} \log \relax (3)^{2} + {\left (6 \, x + 1\right )} \log \relax (3) - x\right )} \log \relax (5)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((324*x^2+324*x+81)*log(3)^2+(-324*x^2-162*x)*log(3)+81*x^2)*log(5)+(2*x^2+2*x)*log(3)^2-x^2*log(3)
)/((324*x^2+324*x+81)*log(3)^2+(-324*x^2-162*x)*log(3)+81*x^2)/log(5),x, algorithm="fricas")

[Out]

1/81*((4*x^2 + 2*x + 1)*log(3)^3 + x^2*log(3) - (4*x^2 + x)*log(3)^2 + 81*(4*(2*x^2 + x)*log(3)^3 - 4*(3*x^2 +
x)*log(3)^2 - x^2 + (6*x^2 + x)*log(3))*log(5))/((4*(2*x + 1)*log(3)^3 - 4*(3*x + 1)*log(3)^2 + (6*x + 1)*log
(3) - x)*log(5))

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giac [B]  time = 0.19, size = 88, normalized size = 3.26 \begin {gather*} \frac {\frac {\log \relax (3)^{3}}{{\left (2 \, x \log \relax (3) - x + \log \relax (3)\right )} {\left (4 \, \log \relax (3)^{2} - 4 \, \log \relax (3) + 1\right )}} + \frac {324 \, x \log \relax (5) \log \relax (3)^{2} - 324 \, x \log \relax (5) \log \relax (3) + 2 \, x \log \relax (3)^{2} + 81 \, x \log \relax (5) - x \log \relax (3)}{4 \, \log \relax (3)^{2} - 4 \, \log \relax (3) + 1}}{81 \, \log \relax (5)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((324*x^2+324*x+81)*log(3)^2+(-324*x^2-162*x)*log(3)+81*x^2)*log(5)+(2*x^2+2*x)*log(3)^2-x^2*log(3)
)/((324*x^2+324*x+81)*log(3)^2+(-324*x^2-162*x)*log(3)+81*x^2)/log(5),x, algorithm="giac")

[Out]

1/81*(log(3)^3/((2*x*log(3) - x + log(3))*(4*log(3)^2 - 4*log(3) + 1)) + (324*x*log(5)*log(3)^2 - 324*x*log(5)
*log(3) + 2*x*log(3)^2 + 81*x*log(5) - x*log(3))/(4*log(3)^2 - 4*log(3) + 1))/log(5)

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maple [A]  time = 0.59, size = 42, normalized size = 1.56

method	result	size
norman	\(\frac {x \ln \relax (3)+\frac {\left (162 \ln \relax (3) \ln \relax (5)-81 \ln \relax (5)+\ln \relax (3)\right ) x^{2}}{81 \ln \relax (5)}}{2 x \ln \relax (3)+\ln \relax (3)-x}\)	\(42\)
gosper	\(\frac {x \left (162 x \ln \relax (3) \ln \relax (5)+81 \ln \relax (3) \ln \relax (5)-81 x \ln \relax (5)+x \ln \relax (3)\right )}{81 \ln \relax (5) \left (2 x \ln \relax (3)+\ln \relax (3)-x \right )}\)	\(44\)
default	\(\frac {\frac {\left (162 \ln \relax (3) \ln \relax (5)-81 \ln \relax (5)+\ln \relax (3)\right ) x}{162 \ln \relax (3)-81}+\frac {\ln \relax (3)^{3}}{81 \left (2 \ln \relax (3)-1\right )^{2} \left (2 x \ln \relax (3)+\ln \relax (3)-x \right )}}{\ln \relax (5)}\)	\(58\)
risch	\(\frac {162 x \ln \relax (3)}{162 \ln \relax (3)-81}-\frac {81 x}{162 \ln \relax (3)-81}+\frac {x \ln \relax (3)}{\ln \relax (5) \left (162 \ln \relax (3)-81\right )}+\frac {\ln \relax (3)^{3}}{2 \ln \relax (5) \left (2 \ln \relax (3)-1\right ) \left (162 \ln \relax (3)-81\right ) \left (x \ln \relax (3)+\frac {\ln \relax (3)}{2}-\frac {x}{2}\right )}\)	\(82\)
meijerg	\(\frac {81 \left (2 \ln \relax (3)-1\right )^{2} x}{\left (324 \ln \relax (3)^{2}-324 \ln \relax (3)+81\right ) \left (1+\frac {x \left (2 \ln \relax (3)-1\right )}{\ln \relax (3)}\right )}+\frac {\left (324 \ln \relax (5) \ln \relax (3)^{2}-324 \ln \relax (3) \ln \relax (5)+2 \ln \relax (3)^{2}+81 \ln \relax (5)-\ln \relax (3)\right ) \ln \relax (3) \left (\frac {x \left (2 \ln \relax (3)-1\right ) \left (\frac {3 x \left (2 \ln \relax (3)-1\right )}{\ln \relax (3)}+6\right )}{3 \ln \relax (3) \left (1+\frac {x \left (2 \ln \relax (3)-1\right )}{\ln \relax (3)}\right )}-2 \ln \left (1+\frac {x \left (2 \ln \relax (3)-1\right )}{\ln \relax (3)}\right )\right )}{\left (324 \ln \relax (3)^{2}-324 \ln \relax (3)+81\right ) \ln \relax (5) \left (2 \ln \relax (3)-1\right )}+\frac {\left (324 \ln \relax (5) \ln \relax (3)^{2}-162 \ln \relax (3) \ln \relax (5)+2 \ln \relax (3)^{2}\right ) \left (-\frac {x \left (2 \ln \relax (3)-1\right )}{\ln \relax (3) \left (1+\frac {x \left (2 \ln \relax (3)-1\right )}{\ln \relax (3)}\right )}+\ln \left (1+\frac {x \left (2 \ln \relax (3)-1\right )}{\ln \relax (3)}\right )\right )}{\left (324 \ln \relax (3)^{2}-324 \ln \relax (3)+81\right ) \ln \relax (5)}\)	\(248\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((324*x^2+324*x+81)*ln(3)^2+(-324*x^2-162*x)*ln(3)+81*x^2)*ln(5)+(2*x^2+2*x)*ln(3)^2-x^2*ln(3))/((324*x^2
+324*x+81)*ln(3)^2+(-324*x^2-162*x)*ln(3)+81*x^2)/ln(5),x,method=_RETURNVERBOSE)

[Out]

(x*ln(3)+1/81*(162*ln(3)*ln(5)-81*ln(5)+ln(3))/ln(5)*x^2)/(2*x*ln(3)+ln(3)-x)

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maxima [B]  time = 0.40, size = 72, normalized size = 2.67 \begin {gather*} \frac {\frac {\log \relax (3)^{3}}{4 \, \log \relax (3)^{3} + {\left (8 \, \log \relax (3)^{3} - 12 \, \log \relax (3)^{2} + 6 \, \log \relax (3) - 1\right )} x - 4 \, \log \relax (3)^{2} + \log \relax (3)} + \frac {{\left (81 \, {\left (2 \, \log \relax (3) - 1\right )} \log \relax (5) + \log \relax (3)\right )} x}{2 \, \log \relax (3) - 1}}{81 \, \log \relax (5)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((324*x^2+324*x+81)*log(3)^2+(-324*x^2-162*x)*log(3)+81*x^2)*log(5)+(2*x^2+2*x)*log(3)^2-x^2*log(3)
)/((324*x^2+324*x+81)*log(3)^2+(-324*x^2-162*x)*log(3)+81*x^2)/log(5),x, algorithm="maxima")

[Out]

1/81*(log(3)^3/(4*log(3)^3 + (8*log(3)^3 - 12*log(3)^2 + 6*log(3) - 1)*x - 4*log(3)^2 + log(3)) + (81*(2*log(3
) - 1)*log(5) + log(3))*x/(2*log(3) - 1))/log(5)

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mupad [B]  time = 0.25, size = 82, normalized size = 3.04 \begin {gather*} \frac {{\ln \relax (3)}^3}{\left (\ln \relax (9)-1\right )\,\left (x\,\left (81\,\ln \relax (5)-162\,\ln \relax (5)\,\ln \relax (9)+81\,\ln \relax (5)\,{\ln \relax (9)}^2\right )-81\,\ln \relax (3)\,\ln \relax (5)+81\,\ln \relax (3)\,\ln \relax (5)\,\ln \relax (9)\right )}+\frac {x\,\left (2\,\ln \relax (3)-1\right )\,\left (\ln \relax (3)-81\,\ln \relax (5)+162\,\ln \relax (3)\,\ln \relax (5)\right )}{81\,\ln \relax (5)\,{\left (\ln \relax (9)-1\right )}^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((log(5)*(log(3)^2*(324*x + 324*x^2 + 81) - log(3)*(162*x + 324*x^2) + 81*x^2) + log(3)^2*(2*x + 2*x^2) - x
^2*log(3))/(log(5)*(log(3)^2*(324*x + 324*x^2 + 81) - log(3)*(162*x + 324*x^2) + 81*x^2)),x)

[Out]

log(3)^3/((log(9) - 1)*(x*(81*log(5) - 162*log(5)*log(9) + 81*log(5)*log(9)^2) - 81*log(3)*log(5) + 81*log(3)*
log(5)*log(9))) + (x*(2*log(3) - 1)*(log(3) - 81*log(5) + 162*log(3)*log(5)))/(81*log(5)*(log(9) - 1)^2)

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sympy [B]  time = 0.44, size = 128, normalized size = 4.74 \begin {gather*} x \left (- \frac {81 \log {\relax (5 )}}{- 81 \log {\relax (5 )} + 162 \log {\relax (3 )} \log {\relax (5 )}} + \frac {\log {\relax (3 )}}{- 81 \log {\relax (5 )} + 162 \log {\relax (3 )} \log {\relax (5 )}} + \frac {162 \log {\relax (3 )} \log {\relax (5 )}}{- 81 \log {\relax (5 )} + 162 \log {\relax (3 )} \log {\relax (5 )}}\right ) + \frac {\log {\relax (3 )}^{3}}{x \left (- 972 \log {\relax (3 )}^{2} \log {\relax (5 )} - 81 \log {\relax (5 )} + 486 \log {\relax (3 )} \log {\relax (5 )} + 648 \log {\relax (3 )}^{3} \log {\relax (5 )}\right ) - 324 \log {\relax (3 )}^{2} \log {\relax (5 )} + 81 \log {\relax (3 )} \log {\relax (5 )} + 324 \log {\relax (3 )}^{3} \log {\relax (5 )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((324*x**2+324*x+81)*ln(3)**2+(-324*x**2-162*x)*ln(3)+81*x**2)*ln(5)+(2*x**2+2*x)*ln(3)**2-x**2*ln(
3))/((324*x**2+324*x+81)*ln(3)**2+(-324*x**2-162*x)*ln(3)+81*x**2)/ln(5),x)

[Out]

x*(-81*log(5)/(-81*log(5) + 162*log(3)*log(5)) + log(3)/(-81*log(5) + 162*log(3)*log(5)) + 162*log(3)*log(5)/(
-81*log(5) + 162*log(3)*log(5))) + log(3)**3/(x*(-972*log(3)**2*log(5) - 81*log(5) + 486*log(3)*log(5) + 648*l
og(3)**3*log(5)) - 324*log(3)**2*log(5) + 81*log(3)*log(5) + 324*log(3)**3*log(5))

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