3.57.56 \(\int \frac {(-250 x^2+50 x^3) \log (x) \log (3+x)+(375 x+50 x^2-25 x^3+(-750 x-250 x^2) \log (x)) \log ^2(3+x)}{(-54000+14400 x+4320 x^2-1728 x^3+144 x^4) \log ^2(x)} \, dx\)

Optimal. Leaf size=24 \[ \frac {25 x^2 \log ^2(3+x)}{144 (5-x)^2 \log (x)} \]

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Rubi [F]  time = 4.04, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {\left (-250 x^2+50 x^3\right ) \log (x) \log (3+x)+\left (375 x+50 x^2-25 x^3+\left (-750 x-250 x^2\right ) \log (x)\right ) \log ^2(3+x)}{\left (-54000+14400 x+4320 x^2-1728 x^3+144 x^4\right ) \log ^2(x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[((-250*x^2 + 50*x^3)*Log[x]*Log[3 + x] + (375*x + 50*x^2 - 25*x^3 + (-750*x - 250*x^2)*Log[x])*Log[3 + x]^
2)/((-54000 + 14400*x + 4320*x^2 - 1728*x^3 + 144*x^4)*Log[x]^2),x]

[Out]

(625*Defer[Int][Log[3 + x]/((-5 + x)^2*Log[x]), x])/576 + (1375*Defer[Int][Log[3 + x]/((-5 + x)*Log[x]), x])/4
608 + (25*Defer[Int][Log[3 + x]/((3 + x)*Log[x]), x])/512 - (125*Defer[Int][Log[3 + x]^2/((-5 + x)^2*Log[x]^2)
, x])/144 - (25*Defer[Int][Log[3 + x]^2/((-5 + x)*Log[x]^2), x])/144 - (625*Defer[Int][Log[3 + x]^2/((-5 + x)^
3*Log[x]), x])/72 - (125*Defer[Int][Log[3 + x]^2/((-5 + x)^2*Log[x]), x])/72

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {25 x \log (3+x) \left (\left (-15-2 x+x^2\right ) \log (3+x)-2 \log (x) ((-5+x) x-5 (3+x) \log (3+x))\right )}{144 (5-x)^3 (3+x) \log ^2(x)} \, dx\\ &=\frac {25}{144} \int \frac {x \log (3+x) \left (\left (-15-2 x+x^2\right ) \log (3+x)-2 \log (x) ((-5+x) x-5 (3+x) \log (3+x))\right )}{(5-x)^3 (3+x) \log ^2(x)} \, dx\\ &=\frac {25}{144} \int \left (\frac {2 x^2 \log (3+x)}{(-5+x)^2 (3+x) \log (x)}-\frac {x (-5+x+10 \log (x)) \log ^2(3+x)}{(-5+x)^3 \log ^2(x)}\right ) \, dx\\ &=-\left (\frac {25}{144} \int \frac {x (-5+x+10 \log (x)) \log ^2(3+x)}{(-5+x)^3 \log ^2(x)} \, dx\right )+\frac {25}{72} \int \frac {x^2 \log (3+x)}{(-5+x)^2 (3+x) \log (x)} \, dx\\ &=-\left (\frac {25}{144} \int \left (\frac {5 (-5+x+10 \log (x)) \log ^2(3+x)}{(-5+x)^3 \log ^2(x)}+\frac {(-5+x+10 \log (x)) \log ^2(3+x)}{(-5+x)^2 \log ^2(x)}\right ) \, dx\right )+\frac {25}{72} \int \left (\frac {25 \log (3+x)}{8 (-5+x)^2 \log (x)}+\frac {55 \log (3+x)}{64 (-5+x) \log (x)}+\frac {9 \log (3+x)}{64 (3+x) \log (x)}\right ) \, dx\\ &=\frac {25}{512} \int \frac {\log (3+x)}{(3+x) \log (x)} \, dx-\frac {25}{144} \int \frac {(-5+x+10 \log (x)) \log ^2(3+x)}{(-5+x)^2 \log ^2(x)} \, dx+\frac {1375 \int \frac {\log (3+x)}{(-5+x) \log (x)} \, dx}{4608}-\frac {125}{144} \int \frac {(-5+x+10 \log (x)) \log ^2(3+x)}{(-5+x)^3 \log ^2(x)} \, dx+\frac {625}{576} \int \frac {\log (3+x)}{(-5+x)^2 \log (x)} \, dx\\ &=\frac {25}{512} \int \frac {\log (3+x)}{(3+x) \log (x)} \, dx-\frac {25}{144} \int \left (-\frac {5 \log ^2(3+x)}{(-5+x)^2 \log ^2(x)}+\frac {x \log ^2(3+x)}{(-5+x)^2 \log ^2(x)}+\frac {10 \log ^2(3+x)}{(-5+x)^2 \log (x)}\right ) \, dx+\frac {1375 \int \frac {\log (3+x)}{(-5+x) \log (x)} \, dx}{4608}-\frac {125}{144} \int \left (-\frac {5 \log ^2(3+x)}{(-5+x)^3 \log ^2(x)}+\frac {x \log ^2(3+x)}{(-5+x)^3 \log ^2(x)}+\frac {10 \log ^2(3+x)}{(-5+x)^3 \log (x)}\right ) \, dx+\frac {625}{576} \int \frac {\log (3+x)}{(-5+x)^2 \log (x)} \, dx\\ &=\frac {25}{512} \int \frac {\log (3+x)}{(3+x) \log (x)} \, dx-\frac {25}{144} \int \frac {x \log ^2(3+x)}{(-5+x)^2 \log ^2(x)} \, dx+\frac {1375 \int \frac {\log (3+x)}{(-5+x) \log (x)} \, dx}{4608}+\frac {125}{144} \int \frac {\log ^2(3+x)}{(-5+x)^2 \log ^2(x)} \, dx-\frac {125}{144} \int \frac {x \log ^2(3+x)}{(-5+x)^3 \log ^2(x)} \, dx+\frac {625}{576} \int \frac {\log (3+x)}{(-5+x)^2 \log (x)} \, dx-\frac {125}{72} \int \frac {\log ^2(3+x)}{(-5+x)^2 \log (x)} \, dx+\frac {625}{144} \int \frac {\log ^2(3+x)}{(-5+x)^3 \log ^2(x)} \, dx-\frac {625}{72} \int \frac {\log ^2(3+x)}{(-5+x)^3 \log (x)} \, dx\\ &=\frac {25}{512} \int \frac {\log (3+x)}{(3+x) \log (x)} \, dx-\frac {25}{144} \int \left (\frac {5 \log ^2(3+x)}{(-5+x)^2 \log ^2(x)}+\frac {\log ^2(3+x)}{(-5+x) \log ^2(x)}\right ) \, dx+\frac {1375 \int \frac {\log (3+x)}{(-5+x) \log (x)} \, dx}{4608}+\frac {125}{144} \int \frac {\log ^2(3+x)}{(-5+x)^2 \log ^2(x)} \, dx-\frac {125}{144} \int \left (\frac {5 \log ^2(3+x)}{(-5+x)^3 \log ^2(x)}+\frac {\log ^2(3+x)}{(-5+x)^2 \log ^2(x)}\right ) \, dx+\frac {625}{576} \int \frac {\log (3+x)}{(-5+x)^2 \log (x)} \, dx-\frac {125}{72} \int \frac {\log ^2(3+x)}{(-5+x)^2 \log (x)} \, dx+\frac {625}{144} \int \frac {\log ^2(3+x)}{(-5+x)^3 \log ^2(x)} \, dx-\frac {625}{72} \int \frac {\log ^2(3+x)}{(-5+x)^3 \log (x)} \, dx\\ &=\frac {25}{512} \int \frac {\log (3+x)}{(3+x) \log (x)} \, dx-\frac {25}{144} \int \frac {\log ^2(3+x)}{(-5+x) \log ^2(x)} \, dx+\frac {1375 \int \frac {\log (3+x)}{(-5+x) \log (x)} \, dx}{4608}-\frac {125}{144} \int \frac {\log ^2(3+x)}{(-5+x)^2 \log ^2(x)} \, dx+\frac {625}{576} \int \frac {\log (3+x)}{(-5+x)^2 \log (x)} \, dx-\frac {125}{72} \int \frac {\log ^2(3+x)}{(-5+x)^2 \log (x)} \, dx-\frac {625}{72} \int \frac {\log ^2(3+x)}{(-5+x)^3 \log (x)} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.26, size = 22, normalized size = 0.92 \begin {gather*} \frac {25 x^2 \log ^2(3+x)}{144 (-5+x)^2 \log (x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((-250*x^2 + 50*x^3)*Log[x]*Log[3 + x] + (375*x + 50*x^2 - 25*x^3 + (-750*x - 250*x^2)*Log[x])*Log[3
+ x]^2)/((-54000 + 14400*x + 4320*x^2 - 1728*x^3 + 144*x^4)*Log[x]^2),x]

[Out]

(25*x^2*Log[3 + x]^2)/(144*(-5 + x)^2*Log[x])

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fricas [A]  time = 0.63, size = 25, normalized size = 1.04 \begin {gather*} \frac {25 \, x^{2} \log \left (x + 3\right )^{2}}{144 \, {\left (x^{2} - 10 \, x + 25\right )} \log \relax (x)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-250*x^2-750*x)*log(x)-25*x^3+50*x^2+375*x)*log(3+x)^2+(50*x^3-250*x^2)*log(x)*log(3+x))/(144*x^4
-1728*x^3+4320*x^2+14400*x-54000)/log(x)^2,x, algorithm="fricas")

[Out]

25/144*x^2*log(x + 3)^2/((x^2 - 10*x + 25)*log(x))

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giac [A]  time = 0.21, size = 29, normalized size = 1.21 \begin {gather*} \frac {25 \, x^{2} \log \left (x + 3\right )^{2}}{144 \, {\left (x^{2} \log \relax (x) - 10 \, x \log \relax (x) + 25 \, \log \relax (x)\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-250*x^2-750*x)*log(x)-25*x^3+50*x^2+375*x)*log(3+x)^2+(50*x^3-250*x^2)*log(x)*log(3+x))/(144*x^4
-1728*x^3+4320*x^2+14400*x-54000)/log(x)^2,x, algorithm="giac")

[Out]

25/144*x^2*log(x + 3)^2/(x^2*log(x) - 10*x*log(x) + 25*log(x))

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maple [A]  time = 0.04, size = 26, normalized size = 1.08

method	result	size
risch	\(\frac {25 x^{2} \ln \left (3+x \right )^{2}}{144 \left (x^{2}-10 x +25\right ) \ln \relax (x )}\)	\(26\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((-250*x^2-750*x)*ln(x)-25*x^3+50*x^2+375*x)*ln(3+x)^2+(50*x^3-250*x^2)*ln(x)*ln(3+x))/(144*x^4-1728*x^3+
4320*x^2+14400*x-54000)/ln(x)^2,x,method=_RETURNVERBOSE)

[Out]

25/144*x^2/(x^2-10*x+25)/ln(x)*ln(3+x)^2

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maxima [A]  time = 0.44, size = 25, normalized size = 1.04 \begin {gather*} \frac {25 \, x^{2} \log \left (x + 3\right )^{2}}{144 \, {\left (x^{2} - 10 \, x + 25\right )} \log \relax (x)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-250*x^2-750*x)*log(x)-25*x^3+50*x^2+375*x)*log(3+x)^2+(50*x^3-250*x^2)*log(x)*log(3+x))/(144*x^4
-1728*x^3+4320*x^2+14400*x-54000)/log(x)^2,x, algorithm="maxima")

[Out]

25/144*x^2*log(x + 3)^2/((x^2 - 10*x + 25)*log(x))

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mupad [B]  time = 3.68, size = 20, normalized size = 0.83 \begin {gather*} \frac {25\,x^2\,{\ln \left (x+3\right )}^2}{144\,\ln \relax (x)\,{\left (x-5\right )}^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((log(x + 3)^2*(375*x - log(x)*(750*x + 250*x^2) + 50*x^2 - 25*x^3) - log(x + 3)*log(x)*(250*x^2 - 50*x^3))
/(log(x)^2*(14400*x + 4320*x^2 - 1728*x^3 + 144*x^4 - 54000)),x)

[Out]

(25*x^2*log(x + 3)^2)/(144*log(x)*(x - 5)^2)

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sympy [A]  time = 0.44, size = 31, normalized size = 1.29 \begin {gather*} \frac {25 x^{2} \log {\left (x + 3 \right )}^{2}}{144 x^{2} \log {\relax (x )} - 1440 x \log {\relax (x )} + 3600 \log {\relax (x )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-250*x**2-750*x)*ln(x)-25*x**3+50*x**2+375*x)*ln(3+x)**2+(50*x**3-250*x**2)*ln(x)*ln(3+x))/(144*x
**4-1728*x**3+4320*x**2+14400*x-54000)/ln(x)**2,x)

[Out]

25*x**2*log(x + 3)**2/(144*x**2*log(x) - 1440*x*log(x) + 3600*log(x))

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