\(\int \frac {3 x+4 x^2+x^3+(64 x+32 x^2) \log ^2(x) \log ^3(3+4 x+x^2)+(24+32 x+8 x^2) \log (x) \log ^4(3+4 x+x^2)}{3 x+4 x^2+x^3} \, dx\) [264]

Optimal result

Integrand size = 76, antiderivative size = 18 \[ \int \frac {3 x+4 x^2+x^3+\left (64 x+32 x^2\right ) \log ^2(x) \log ^3\left (3+4 x+x^2\right )+\left (24+32 x+8 x^2\right ) \log (x) \log ^4\left (3+4 x+x^2\right )}{3 x+4 x^2+x^3} \, dx=x+4 \log ^2(x) \log ^4((1+x) (3+x)) \]

[Out]

4*ln((1+x)*(3+x))^4*ln(x)^2+x

Rubi [F]

\[ \int \frac {3 x+4 x^2+x^3+\left (64 x+32 x^2\right ) \log ^2(x) \log ^3\left (3+4 x+x^2\right )+\left (24+32 x+8 x^2\right ) \log (x) \log ^4\left (3+4 x+x^2\right )}{3 x+4 x^2+x^3} \, dx=\int \frac {3 x+4 x^2+x^3+\left (64 x+32 x^2\right ) \log ^2(x) \log ^3\left (3+4 x+x^2\right )+\left (24+32 x+8 x^2\right ) \log (x) \log ^4\left (3+4 x+x^2\right )}{3 x+4 x^2+x^3} \, dx \]

[In]

Int[(3*x + 4*x^2 + x^3 + (64*x + 32*x^2)*Log[x]^2*Log[3 + 4*x + x^2]^3 + (24 + 32*x + 8*x^2)*Log[x]*Log[3 + 4*
x + x^2]^4)/(3*x + 4*x^2 + x^3),x]

[Out]

x + 16*Defer[Int][(Log[x]^2*Log[3 + 4*x + x^2]^3)/(1 + x), x] + 16*Defer[Int][(Log[x]^2*Log[3 + 4*x + x^2]^3)/
(3 + x), x] + 8*Defer[Int][(Log[x]*Log[3 + 4*x + x^2]^4)/x, x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {3 x+4 x^2+x^3+\left (64 x+32 x^2\right ) \log ^2(x) \log ^3\left (3+4 x+x^2\right )+\left (24+32 x+8 x^2\right ) \log (x) \log ^4\left (3+4 x+x^2\right )}{x \left (3+4 x+x^2\right )} \, dx \\ & = \int \left (1+\frac {32 (2+x) \log ^2(x) \log ^3\left (3+4 x+x^2\right )}{3+4 x+x^2}+\frac {8 \log (x) \log ^4\left (3+4 x+x^2\right )}{x}\right ) \, dx \\ & = x+8 \int \frac {\log (x) \log ^4\left (3+4 x+x^2\right )}{x} \, dx+32 \int \frac {(2+x) \log ^2(x) \log ^3\left (3+4 x+x^2\right )}{3+4 x+x^2} \, dx \\ & = x+8 \int \frac {\log (x) \log ^4\left (3+4 x+x^2\right )}{x} \, dx+32 \int \left (\frac {\log ^2(x) \log ^3\left (3+4 x+x^2\right )}{2 (1+x)}+\frac {\log ^2(x) \log ^3\left (3+4 x+x^2\right )}{2 (3+x)}\right ) \, dx \\ & = x+8 \int \frac {\log (x) \log ^4\left (3+4 x+x^2\right )}{x} \, dx+16 \int \frac {\log ^2(x) \log ^3\left (3+4 x+x^2\right )}{1+x} \, dx+16 \int \frac {\log ^2(x) \log ^3\left (3+4 x+x^2\right )}{3+x} \, dx \\ \end{align*}

Mathematica [F]

\[ \int \frac {3 x+4 x^2+x^3+\left (64 x+32 x^2\right ) \log ^2(x) \log ^3\left (3+4 x+x^2\right )+\left (24+32 x+8 x^2\right ) \log (x) \log ^4\left (3+4 x+x^2\right )}{3 x+4 x^2+x^3} \, dx=\int \frac {3 x+4 x^2+x^3+\left (64 x+32 x^2\right ) \log ^2(x) \log ^3\left (3+4 x+x^2\right )+\left (24+32 x+8 x^2\right ) \log (x) \log ^4\left (3+4 x+x^2\right )}{3 x+4 x^2+x^3} \, dx \]

[In]

Integrate[(3*x + 4*x^2 + x^3 + (64*x + 32*x^2)*Log[x]^2*Log[3 + 4*x + x^2]^3 + (24 + 32*x + 8*x^2)*Log[x]*Log[
3 + 4*x + x^2]^4)/(3*x + 4*x^2 + x^3),x]

[Out]

Integrate[(3*x + 4*x^2 + x^3 + (64*x + 32*x^2)*Log[x]^2*Log[3 + 4*x + x^2]^3 + (24 + 32*x + 8*x^2)*Log[x]*Log[
3 + 4*x + x^2]^4)/(3*x + 4*x^2 + x^3), x]

Maple [A] (verified)

Time = 0.69 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.11

method	result	size
risch	\(4 \ln \left (x \right )^{2} \ln \left (x^{2}+4 x +3\right )^{4}+x\)	\(20\)

[In]

int(((8*x^2+32*x+24)*ln(x)*ln(x^2+4*x+3)^4+(32*x^2+64*x)*ln(x)^2*ln(x^2+4*x+3)^3+x^3+4*x^2+3*x)/(x^3+4*x^2+3*x
),x,method=_RETURNVERBOSE)

[Out]

4*ln(x)^2*ln(x^2+4*x+3)^4+x

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.06 \[ \int \frac {3 x+4 x^2+x^3+\left (64 x+32 x^2\right ) \log ^2(x) \log ^3\left (3+4 x+x^2\right )+\left (24+32 x+8 x^2\right ) \log (x) \log ^4\left (3+4 x+x^2\right )}{3 x+4 x^2+x^3} \, dx=4 \, \log \left (x^{2} + 4 \, x + 3\right )^{4} \log \left (x\right )^{2} + x \]

[In]

integrate(((8*x^2+32*x+24)*log(x)*log(x^2+4*x+3)^4+(32*x^2+64*x)*log(x)^2*log(x^2+4*x+3)^3+x^3+4*x^2+3*x)/(x^3
+4*x^2+3*x),x, algorithm="fricas")

[Out]

4*log(x^2 + 4*x + 3)^4*log(x)^2 + x

Sympy [A] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.06 \[ \int \frac {3 x+4 x^2+x^3+\left (64 x+32 x^2\right ) \log ^2(x) \log ^3\left (3+4 x+x^2\right )+\left (24+32 x+8 x^2\right ) \log (x) \log ^4\left (3+4 x+x^2\right )}{3 x+4 x^2+x^3} \, dx=x + 4 \log {\left (x \right )}^{2} \log {\left (x^{2} + 4 x + 3 \right )}^{4} \]

[In]

integrate(((8*x**2+32*x+24)*ln(x)*ln(x**2+4*x+3)**4+(32*x**2+64*x)*ln(x)**2*ln(x**2+4*x+3)**3+x**3+4*x**2+3*x)
/(x**3+4*x**2+3*x),x)

[Out]

x + 4*log(x)**2*log(x**2 + 4*x + 3)**4

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 76 vs. \(2 (18) = 36\).

Time = 0.33 (sec) , antiderivative size = 76, normalized size of antiderivative = 4.22 \[ \int \frac {3 x+4 x^2+x^3+\left (64 x+32 x^2\right ) \log ^2(x) \log ^3\left (3+4 x+x^2\right )+\left (24+32 x+8 x^2\right ) \log (x) \log ^4\left (3+4 x+x^2\right )}{3 x+4 x^2+x^3} \, dx=4 \, \log \left (x + 3\right )^{4} \log \left (x\right )^{2} + 16 \, \log \left (x + 3\right )^{3} \log \left (x + 1\right ) \log \left (x\right )^{2} + 24 \, \log \left (x + 3\right )^{2} \log \left (x + 1\right )^{2} \log \left (x\right )^{2} + 16 \, \log \left (x + 3\right ) \log \left (x + 1\right )^{3} \log \left (x\right )^{2} + 4 \, \log \left (x + 1\right )^{4} \log \left (x\right )^{2} + x \]

[In]

integrate(((8*x^2+32*x+24)*log(x)*log(x^2+4*x+3)^4+(32*x^2+64*x)*log(x)^2*log(x^2+4*x+3)^3+x^3+4*x^2+3*x)/(x^3
+4*x^2+3*x),x, algorithm="maxima")

[Out]

4*log(x + 3)^4*log(x)^2 + 16*log(x + 3)^3*log(x + 1)*log(x)^2 + 24*log(x + 3)^2*log(x + 1)^2*log(x)^2 + 16*log
(x + 3)*log(x + 1)^3*log(x)^2 + 4*log(x + 1)^4*log(x)^2 + x

Giac [A] (verification not implemented)

none

Time = 0.81 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.06 \[ \int \frac {3 x+4 x^2+x^3+\left (64 x+32 x^2\right ) \log ^2(x) \log ^3\left (3+4 x+x^2\right )+\left (24+32 x+8 x^2\right ) \log (x) \log ^4\left (3+4 x+x^2\right )}{3 x+4 x^2+x^3} \, dx=4 \, \log \left (x^{2} + 4 \, x + 3\right )^{4} \log \left (x\right )^{2} + x \]

[In]

integrate(((8*x^2+32*x+24)*log(x)*log(x^2+4*x+3)^4+(32*x^2+64*x)*log(x)^2*log(x^2+4*x+3)^3+x^3+4*x^2+3*x)/(x^3
+4*x^2+3*x),x, algorithm="giac")

[Out]

4*log(x^2 + 4*x + 3)^4*log(x)^2 + x

Mupad [B] (verification not implemented)

Time = 8.10 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.06 \[ \int \frac {3 x+4 x^2+x^3+\left (64 x+32 x^2\right ) \log ^2(x) \log ^3\left (3+4 x+x^2\right )+\left (24+32 x+8 x^2\right ) \log (x) \log ^4\left (3+4 x+x^2\right )}{3 x+4 x^2+x^3} \, dx=4\,{\ln \left (x^2+4\,x+3\right )}^4\,{\ln \left (x\right )}^2+x \]

[In]

int((3*x + 4*x^2 + x^3 + log(4*x + x^2 + 3)^4*log(x)*(32*x + 8*x^2 + 24) + log(4*x + x^2 + 3)^3*log(x)^2*(64*x
+ 32*x^2))/(3*x + 4*x^2 + x^3),x)

[Out]

x + 4*log(4*x + x^2 + 3)^4*log(x)^2