3.45.29 \(\int \frac {30 x+10 x^2+(384+544 x+264 x^2+54 x^3+4 x^4) \log (3 x+x^2)}{-60 x-35 x^2-5 x^3+(192 x+208 x^2+84 x^3+15 x^4+x^5) \log ^2(3 x+x^2)} \, dx\)

Optimal. Leaf size=23 \[ 2+\log \left (\frac {5}{(4+x)^2}-\log ^2\left (3 x+x^2\right )\right ) \]

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Rubi [F]  time = 37.08, 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 {30 x+10 x^2+\left (384+544 x+264 x^2+54 x^3+4 x^4\right ) \log \left (3 x+x^2\right )}{-60 x-35 x^2-5 x^3+\left (192 x+208 x^2+84 x^3+15 x^4+x^5\right ) \log ^2\left (3 x+x^2\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(30*x + 10*x^2 + (384 + 544*x + 264*x^2 + 54*x^3 + 4*x^4)*Log[3*x + x^2])/(-60*x - 35*x^2 - 5*x^3 + (192*x
+ 208*x^2 + 84*x^3 + 15*x^4 + x^5)*Log[3*x + x^2]^2),x]

[Out]

10*Defer[Int][1/((-4 - x)*(5 - (4 + x)^2*Log[x*(3 + x)]^2)), x] + 2*Defer[Int][Log[x*(3 + x)]/((-3 - x)*(5 - (
4 + x)^2*Log[x*(3 + x)]^2)), x] + 26*Defer[Int][Log[x*(3 + x)]/(-5 + (4 + x)^2*Log[x*(3 + x)]^2), x] + 32*Defe
r[Int][Log[x*(3 + x)]/(x*(-5 + (4 + x)^2*Log[x*(3 + x)]^2)), x] + 4*Defer[Int][(x*Log[x*(3 + x)])/(-5 + (4 + x
)^2*Log[x*(3 + x)]^2), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-10 x (3+x)-2 (4+x)^3 (3+2 x) \log (x (3+x))}{x \left (12+7 x+x^2\right ) \left (5-(4+x)^2 \log ^2(x (3+x))\right )} \, dx\\ &=\int \left (\frac {15 x+5 x^2+192 \log (x (3+x))+272 x \log (x (3+x))+132 x^2 \log (x (3+x))+27 x^3 \log (x (3+x))+2 x^4 \log (x (3+x))}{6 x \left (-5+16 \log ^2(x (3+x))+8 x \log ^2(x (3+x))+x^2 \log ^2(x (3+x))\right )}-\frac {2 \left (15 x+5 x^2+192 \log (x (3+x))+272 x \log (x (3+x))+132 x^2 \log (x (3+x))+27 x^3 \log (x (3+x))+2 x^4 \log (x (3+x))\right )}{3 (3+x) \left (-5+16 \log ^2(x (3+x))+8 x \log ^2(x (3+x))+x^2 \log ^2(x (3+x))\right )}+\frac {15 x+5 x^2+192 \log (x (3+x))+272 x \log (x (3+x))+132 x^2 \log (x (3+x))+27 x^3 \log (x (3+x))+2 x^4 \log (x (3+x))}{2 (4+x) \left (-5+16 \log ^2(x (3+x))+8 x \log ^2(x (3+x))+x^2 \log ^2(x (3+x))\right )}\right ) \, dx\\ &=\frac {1}{6} \int \frac {15 x+5 x^2+192 \log (x (3+x))+272 x \log (x (3+x))+132 x^2 \log (x (3+x))+27 x^3 \log (x (3+x))+2 x^4 \log (x (3+x))}{x \left (-5+16 \log ^2(x (3+x))+8 x \log ^2(x (3+x))+x^2 \log ^2(x (3+x))\right )} \, dx+\frac {1}{2} \int \frac {15 x+5 x^2+192 \log (x (3+x))+272 x \log (x (3+x))+132 x^2 \log (x (3+x))+27 x^3 \log (x (3+x))+2 x^4 \log (x (3+x))}{(4+x) \left (-5+16 \log ^2(x (3+x))+8 x \log ^2(x (3+x))+x^2 \log ^2(x (3+x))\right )} \, dx-\frac {2}{3} \int \frac {15 x+5 x^2+192 \log (x (3+x))+272 x \log (x (3+x))+132 x^2 \log (x (3+x))+27 x^3 \log (x (3+x))+2 x^4 \log (x (3+x))}{(3+x) \left (-5+16 \log ^2(x (3+x))+8 x \log ^2(x (3+x))+x^2 \log ^2(x (3+x))\right )} \, dx\\ &=\frac {1}{6} \int \frac {-5 x (3+x)-(4+x)^3 (3+2 x) \log (x (3+x))}{x \left (5-(4+x)^2 \log ^2(x (3+x))\right )} \, dx+\frac {1}{2} \int \frac {-5 x (3+x)-(4+x)^3 (3+2 x) \log (x (3+x))}{(4+x) \left (5-(4+x)^2 \log ^2(x (3+x))\right )} \, dx-\frac {2}{3} \int \frac {-5 x (3+x)-(4+x)^3 (3+2 x) \log (x (3+x))}{(3+x) \left (5-(4+x)^2 \log ^2(x (3+x))\right )} \, dx\\ &=\text {Rest of rules removed due to large latex content} \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.88, size = 44, normalized size = 1.91 \begin {gather*} -2 \log (4+x)+\log \left (5-16 \log ^2(x (3+x))-8 x \log ^2(x (3+x))-x^2 \log ^2(x (3+x))\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(30*x + 10*x^2 + (384 + 544*x + 264*x^2 + 54*x^3 + 4*x^4)*Log[3*x + x^2])/(-60*x - 35*x^2 - 5*x^3 +
(192*x + 208*x^2 + 84*x^3 + 15*x^4 + x^5)*Log[3*x + x^2]^2),x]

[Out]

-2*Log[4 + x] + Log[5 - 16*Log[x*(3 + x)]^2 - 8*x*Log[x*(3 + x)]^2 - x^2*Log[x*(3 + x)]^2]

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fricas [A]  time = 0.55, size = 33, normalized size = 1.43 \begin {gather*} \log \left (\frac {{\left (x^{2} + 8 \, x + 16\right )} \log \left (x^{2} + 3 \, x\right )^{2} - 5}{x^{2} + 8 \, x + 16}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*x^4+54*x^3+264*x^2+544*x+384)*log(x^2+3*x)+10*x^2+30*x)/((x^5+15*x^4+84*x^3+208*x^2+192*x)*log(x
^2+3*x)^2-5*x^3-35*x^2-60*x),x, algorithm="fricas")

[Out]

log(((x^2 + 8*x + 16)*log(x^2 + 3*x)^2 - 5)/(x^2 + 8*x + 16))

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giac [B]  time = 0.27, size = 49, normalized size = 2.13 \begin {gather*} \log \left (x^{2} \log \left (x^{2} + 3 \, x\right )^{2} + 8 \, x \log \left (x^{2} + 3 \, x\right )^{2} + 16 \, \log \left (x^{2} + 3 \, x\right )^{2} - 5\right ) - 2 \, \log \left (x + 4\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*x^4+54*x^3+264*x^2+544*x+384)*log(x^2+3*x)+10*x^2+30*x)/((x^5+15*x^4+84*x^3+208*x^2+192*x)*log(x
^2+3*x)^2-5*x^3-35*x^2-60*x),x, algorithm="giac")

[Out]

log(x^2*log(x^2 + 3*x)^2 + 8*x*log(x^2 + 3*x)^2 + 16*log(x^2 + 3*x)^2 - 5) - 2*log(x + 4)

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maple [A]  time = 0.09, size = 25, normalized size = 1.09

method	result	size
risch	\(\ln \left (\ln \left (x^{2}+3 x \right )^{2}-\frac {5}{x^{2}+8 x +16}\right )\)	\(25\)
norman	\(-2 \ln \left (4+x \right )+\ln \left (\ln \left (x^{2}+3 x \right )^{2} x^{2}+8 \ln \left (x^{2}+3 x \right )^{2} x +16 \ln \left (x^{2}+3 x \right )^{2}-5\right )\)	\(50\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((4*x^4+54*x^3+264*x^2+544*x+384)*ln(x^2+3*x)+10*x^2+30*x)/((x^5+15*x^4+84*x^3+208*x^2+192*x)*ln(x^2+3*x)^
2-5*x^3-35*x^2-60*x),x,method=_RETURNVERBOSE)

[Out]

ln(ln(x^2+3*x)^2-5/(x^2+8*x+16))

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maxima [B]  time = 0.41, size = 58, normalized size = 2.52 \begin {gather*} \log \left (\frac {{\left (x^{2} + 8 \, x + 16\right )} \log \left (x + 3\right )^{2} + 2 \, {\left (x^{2} + 8 \, x + 16\right )} \log \left (x + 3\right ) \log \relax (x) + {\left (x^{2} + 8 \, x + 16\right )} \log \relax (x)^{2} - 5}{x^{2} + 8 \, x + 16}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*x^4+54*x^3+264*x^2+544*x+384)*log(x^2+3*x)+10*x^2+30*x)/((x^5+15*x^4+84*x^3+208*x^2+192*x)*log(x
^2+3*x)^2-5*x^3-35*x^2-60*x),x, algorithm="maxima")

[Out]

log(((x^2 + 8*x + 16)*log(x + 3)^2 + 2*(x^2 + 8*x + 16)*log(x + 3)*log(x) + (x^2 + 8*x + 16)*log(x)^2 - 5)/(x^
2 + 8*x + 16))

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mupad [F]  time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} -\int \frac {30\,x+\ln \left (x^2+3\,x\right )\,\left (4\,x^4+54\,x^3+264\,x^2+544\,x+384\right )+10\,x^2}{60\,x+35\,x^2+5\,x^3-{\ln \left (x^2+3\,x\right )}^2\,\left (x^5+15\,x^4+84\,x^3+208\,x^2+192\,x\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(30*x + log(3*x + x^2)*(544*x + 264*x^2 + 54*x^3 + 4*x^4 + 384) + 10*x^2)/(60*x + 35*x^2 + 5*x^3 - log(3*
x + x^2)^2*(192*x + 208*x^2 + 84*x^3 + 15*x^4 + x^5)),x)

[Out]

-int((30*x + log(3*x + x^2)*(544*x + 264*x^2 + 54*x^3 + 4*x^4 + 384) + 10*x^2)/(60*x + 35*x^2 + 5*x^3 - log(3*
x + x^2)^2*(192*x + 208*x^2 + 84*x^3 + 15*x^4 + x^5)), x)

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sympy [A]  time = 0.55, size = 20, normalized size = 0.87 \begin {gather*} \log {\left (\log {\left (x^{2} + 3 x \right )}^{2} - \frac {5}{x^{2} + 8 x + 16} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*x**4+54*x**3+264*x**2+544*x+384)*ln(x**2+3*x)+10*x**2+30*x)/((x**5+15*x**4+84*x**3+208*x**2+192*
x)*ln(x**2+3*x)**2-5*x**3-35*x**2-60*x),x)

[Out]

log(log(x**2 + 3*x)**2 - 5/(x**2 + 8*x + 16))

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