3.65.25 \(\int \frac {(-150-50 x) \log (3)}{9000+5400 x+900 x^2+(600+360 x+60 x^2) \log ^2(3)+(10+6 x+x^2) \log ^4(3)+((-600-360 x-60 x^2) \log (3)+(-20-12 x-2 x^2) \log ^3(3)) \log (10+6 x+x^2)+(10+6 x+x^2) \log ^2(3) \log ^2(10+6 x+x^2)} \, dx\)

Optimal. Leaf size=27 \[ \frac {5}{-6+\frac {1}{5} \log (3) (-\log (3)+\log (5+(1+x) (5+x)))} \]

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Rubi [A]  time = 0.24, antiderivative size = 29, normalized size of antiderivative = 1.07, number of steps used = 4, number of rules used = 3, integrand size = 111, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.027, Rules used = {12, 6688, 6686} \begin {gather*} -\frac {50 \log (3)}{\log (9) \left (-\log (3) \log \left (x^2+6 x+10\right )+30+\log ^2(3)\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((-150 - 50*x)*Log[3])/(9000 + 5400*x + 900*x^2 + (600 + 360*x + 60*x^2)*Log[3]^2 + (10 + 6*x + x^2)*Log[3
]^4 + ((-600 - 360*x - 60*x^2)*Log[3] + (-20 - 12*x - 2*x^2)*Log[3]^3)*Log[10 + 6*x + x^2] + (10 + 6*x + x^2)*
Log[3]^2*Log[10 + 6*x + x^2]^2),x]

[Out]

(-50*Log[3])/(Log[9]*(30 + Log[3]^2 - Log[3]*Log[10 + 6*x + x^2]))

Rule 12

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

Rule 6686

Int[(u_)*(y_)^(m_.), x_Symbol] :> With[{q = DerivativeDivides[y, u, x]}, Simp[(q*y^(m + 1))/(m + 1), x] /;  !F
alseQ[q]] /; FreeQ[m, x] && NeQ[m, -1]

Rule 6688

Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; SimplerIntegrandQ[v, u, x]]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\log (3) \int \frac {-150-50 x}{9000+5400 x+900 x^2+\left (600+360 x+60 x^2\right ) \log ^2(3)+\left (10+6 x+x^2\right ) \log ^4(3)+\left (\left (-600-360 x-60 x^2\right ) \log (3)+\left (-20-12 x-2 x^2\right ) \log ^3(3)\right ) \log \left (10+6 x+x^2\right )+\left (10+6 x+x^2\right ) \log ^2(3) \log ^2\left (10+6 x+x^2\right )} \, dx\\ &=\log (3) \int \frac {50 (-3-x)}{\left (10+6 x+x^2\right ) \left (30 \left (1+\frac {\log ^2(3)}{30}\right )-\log (3) \log \left (10+6 x+x^2\right )\right )^2} \, dx\\ &=(50 \log (3)) \int \frac {-3-x}{\left (10+6 x+x^2\right ) \left (30 \left (1+\frac {\log ^2(3)}{30}\right )-\log (3) \log \left (10+6 x+x^2\right )\right )^2} \, dx\\ &=-\frac {50 \log (3)}{\log (9) \left (30+\log ^2(3)-\log (3) \log \left (10+6 x+x^2\right )\right )}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.02, size = 29, normalized size = 1.07 \begin {gather*} -\frac {50 \log (3)}{\log (9) \left (30+\log ^2(3)-\log (3) \log \left (10+6 x+x^2\right )\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((-150 - 50*x)*Log[3])/(9000 + 5400*x + 900*x^2 + (600 + 360*x + 60*x^2)*Log[3]^2 + (10 + 6*x + x^2)
*Log[3]^4 + ((-600 - 360*x - 60*x^2)*Log[3] + (-20 - 12*x - 2*x^2)*Log[3]^3)*Log[10 + 6*x + x^2] + (10 + 6*x +
x^2)*Log[3]^2*Log[10 + 6*x + x^2]^2),x]

[Out]

(-50*Log[3])/(Log[9]*(30 + Log[3]^2 - Log[3]*Log[10 + 6*x + x^2]))

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fricas [A]  time = 0.74, size = 23, normalized size = 0.85 \begin {gather*} -\frac {25}{\log \relax (3)^{2} - \log \relax (3) \log \left (x^{2} + 6 \, x + 10\right ) + 30} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-50*x-150)*log(3)/((x^2+6*x+10)*log(3)^2*log(x^2+6*x+10)^2+((-2*x^2-12*x-20)*log(3)^3+(-60*x^2-360*
x-600)*log(3))*log(x^2+6*x+10)+(x^2+6*x+10)*log(3)^4+(60*x^2+360*x+600)*log(3)^2+900*x^2+5400*x+9000),x, algor
ithm="fricas")

[Out]

-25/(log(3)^2 - log(3)*log(x^2 + 6*x + 10) + 30)

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giac [A]  time = 0.29, size = 30, normalized size = 1.11 \begin {gather*} -\frac {25 \, \log \relax (3)}{\log \relax (3)^{3} - \log \relax (3)^{2} \log \left (x^{2} + 6 \, x + 10\right ) + 30 \, \log \relax (3)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-50*x-150)*log(3)/((x^2+6*x+10)*log(3)^2*log(x^2+6*x+10)^2+((-2*x^2-12*x-20)*log(3)^3+(-60*x^2-360*
x-600)*log(3))*log(x^2+6*x+10)+(x^2+6*x+10)*log(3)^4+(60*x^2+360*x+600)*log(3)^2+900*x^2+5400*x+9000),x, algor
ithm="giac")

[Out]

-25*log(3)/(log(3)^3 - log(3)^2*log(x^2 + 6*x + 10) + 30*log(3))

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maple [A]  time = 0.20, size = 24, normalized size = 0.89

method	result	size
norman	\(-\frac {25}{\ln \relax (3)^{2}-\ln \relax (3) \ln \left (x^{2}+6 x +10\right )+30}\)	\(24\)
risch	\(-\frac {25}{\ln \relax (3)^{2}-\ln \relax (3) \ln \left (x^{2}+6 x +10\right )+30}\)	\(24\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-50*x-150)*ln(3)/((x^2+6*x+10)*ln(3)^2*ln(x^2+6*x+10)^2+((-2*x^2-12*x-20)*ln(3)^3+(-60*x^2-360*x-600)*ln(
3))*ln(x^2+6*x+10)+(x^2+6*x+10)*ln(3)^4+(60*x^2+360*x+600)*ln(3)^2+900*x^2+5400*x+9000),x,method=_RETURNVERBOS
E)

[Out]

-25/(ln(3)^2-ln(3)*ln(x^2+6*x+10)+30)

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maxima [A]  time = 0.51, size = 30, normalized size = 1.11 \begin {gather*} -\frac {25 \, \log \relax (3)}{\log \relax (3)^{3} - \log \relax (3)^{2} \log \left (x^{2} + 6 \, x + 10\right ) + 30 \, \log \relax (3)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-50*x-150)*log(3)/((x^2+6*x+10)*log(3)^2*log(x^2+6*x+10)^2+((-2*x^2-12*x-20)*log(3)^3+(-60*x^2-360*
x-600)*log(3))*log(x^2+6*x+10)+(x^2+6*x+10)*log(3)^4+(60*x^2+360*x+600)*log(3)^2+900*x^2+5400*x+9000),x, algor
ithm="maxima")

[Out]

-25*log(3)/(log(3)^3 - log(3)^2*log(x^2 + 6*x + 10) + 30*log(3))

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mupad [B]  time = 5.15, size = 30, normalized size = 1.11 \begin {gather*} \frac {25}{\ln \relax (3)\,\left (\ln \left (x^2+6\,x+10\right )-\frac {{\ln \relax (3)}^2+30}{\ln \relax (3)}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(log(3)*(50*x + 150))/(5400*x - log(6*x + x^2 + 10)*(log(3)*(360*x + 60*x^2 + 600) + log(3)^3*(12*x + 2*x
^2 + 20)) + log(3)^4*(6*x + x^2 + 10) + log(3)^2*(360*x + 60*x^2 + 600) + 900*x^2 + log(3)^2*log(6*x + x^2 + 1
0)^2*(6*x + x^2 + 10) + 9000),x)

[Out]

25/(log(3)*(log(6*x + x^2 + 10) - (log(3)^2 + 30)/log(3)))

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sympy [A]  time = 0.20, size = 20, normalized size = 0.74 \begin {gather*} \frac {25}{\log {\relax (3 )} \log {\left (x^{2} + 6 x + 10 \right )} - 30 - \log {\relax (3 )}^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-50*x-150)*ln(3)/((x**2+6*x+10)*ln(3)**2*ln(x**2+6*x+10)**2+((-2*x**2-12*x-20)*ln(3)**3+(-60*x**2-3
60*x-600)*ln(3))*ln(x**2+6*x+10)+(x**2+6*x+10)*ln(3)**4+(60*x**2+360*x+600)*ln(3)**2+900*x**2+5400*x+9000),x)

[Out]

25/(log(3)*log(x**2 + 6*x + 10) - 30 - log(3)**2)

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