3.43.54 \(\int \frac {(-9-13 x-3 x^2) \log ^2(5)+(-6-2 x) \log ^2(5) \log (-3 x-x^2)}{3 x^3+7 x^4+5 x^5+x^6+(6 x^3+8 x^4+2 x^5) \log (-3 x-x^2)+(3 x^3+x^4) \log ^2(-3 x-x^2)} \, dx\)

Optimal. Leaf size=21 \[ \frac {\log ^2(5)}{x^2 (1+x+\log ((-3-x) x))} \]

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Rubi [F]  time = 0.82, 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 (-9-13 x-3 x^2\right ) \log ^2(5)+(-6-2 x) \log ^2(5) \log \left (-3 x-x^2\right )}{3 x^3+7 x^4+5 x^5+x^6+\left (6 x^3+8 x^4+2 x^5\right ) \log \left (-3 x-x^2\right )+\left (3 x^3+x^4\right ) \log ^2\left (-3 x-x^2\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[((-9 - 13*x - 3*x^2)*Log[5]^2 + (-6 - 2*x)*Log[5]^2*Log[-3*x - x^2])/(3*x^3 + 7*x^4 + 5*x^5 + x^6 + (6*x^3
+ 8*x^4 + 2*x^5)*Log[-3*x - x^2] + (3*x^3 + x^4)*Log[-3*x - x^2]^2),x]

[Out]

-(Log[5]^2*Defer[Int][1/(x^3*(1 + x + Log[-(x*(3 + x))])^2), x]) - (4*Log[5]^2*Defer[Int][1/(x^2*(1 + x + Log[
-(x*(3 + x))])^2), x])/3 + (Log[5]^2*Defer[Int][1/(x*(1 + x + Log[-(x*(3 + x))])^2), x])/9 - (Log[5]^2*Defer[I
nt][1/((3 + x)*(1 + x + Log[-(x*(3 + x))])^2), x])/9 - 2*Log[5]^2*Defer[Int][1/(x^3*(1 + x + Log[-(x*(3 + x))]
)), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {\log ^2(5) \left (-9-13 x-3 x^2-2 (3+x) \log (-x (3+x))\right )}{x^3 (3+x) (1+x+\log (-x (3+x)))^2} \, dx\\ &=\log ^2(5) \int \frac {-9-13 x-3 x^2-2 (3+x) \log (-x (3+x))}{x^3 (3+x) (1+x+\log (-x (3+x)))^2} \, dx\\ &=\log ^2(5) \int \left (\frac {-3-5 x-x^2}{x^3 (3+x) (1+x+\log (-x (3+x)))^2}-\frac {2}{x^3 (1+x+\log (-x (3+x)))}\right ) \, dx\\ &=\log ^2(5) \int \frac {-3-5 x-x^2}{x^3 (3+x) (1+x+\log (-x (3+x)))^2} \, dx-\left (2 \log ^2(5)\right ) \int \frac {1}{x^3 (1+x+\log (-x (3+x)))} \, dx\\ &=\log ^2(5) \int \left (-\frac {1}{x^3 (1+x+\log (-x (3+x)))^2}-\frac {4}{3 x^2 (1+x+\log (-x (3+x)))^2}+\frac {1}{9 x (1+x+\log (-x (3+x)))^2}-\frac {1}{9 (3+x) (1+x+\log (-x (3+x)))^2}\right ) \, dx-\left (2 \log ^2(5)\right ) \int \frac {1}{x^3 (1+x+\log (-x (3+x)))} \, dx\\ &=\frac {1}{9} \log ^2(5) \int \frac {1}{x (1+x+\log (-x (3+x)))^2} \, dx-\frac {1}{9} \log ^2(5) \int \frac {1}{(3+x) (1+x+\log (-x (3+x)))^2} \, dx-\log ^2(5) \int \frac {1}{x^3 (1+x+\log (-x (3+x)))^2} \, dx-\frac {1}{3} \left (4 \log ^2(5)\right ) \int \frac {1}{x^2 (1+x+\log (-x (3+x)))^2} \, dx-\left (2 \log ^2(5)\right ) \int \frac {1}{x^3 (1+x+\log (-x (3+x)))} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.61, size = 20, normalized size = 0.95 \begin {gather*} \frac {\log ^2(5)}{x^2 (1+x+\log (-x (3+x)))} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((-9 - 13*x - 3*x^2)*Log[5]^2 + (-6 - 2*x)*Log[5]^2*Log[-3*x - x^2])/(3*x^3 + 7*x^4 + 5*x^5 + x^6 +
(6*x^3 + 8*x^4 + 2*x^5)*Log[-3*x - x^2] + (3*x^3 + x^4)*Log[-3*x - x^2]^2),x]

[Out]

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

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fricas [A]  time = 0.52, size = 28, normalized size = 1.33 \begin {gather*} \frac {\log \relax (5)^{2}}{x^{3} + x^{2} \log \left (-x^{2} - 3 \, x\right ) + x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*x-6)*log(5)^2*log(-x^2-3*x)+(-3*x^2-13*x-9)*log(5)^2)/((x^4+3*x^3)*log(-x^2-3*x)^2+(2*x^5+8*x^4
+6*x^3)*log(-x^2-3*x)+x^6+5*x^5+7*x^4+3*x^3),x, algorithm="fricas")

[Out]

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

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giac [A]  time = 0.22, size = 28, normalized size = 1.33 \begin {gather*} \frac {\log \relax (5)^{2}}{x^{3} + x^{2} \log \left (-x^{2} - 3 \, x\right ) + x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*x-6)*log(5)^2*log(-x^2-3*x)+(-3*x^2-13*x-9)*log(5)^2)/((x^4+3*x^3)*log(-x^2-3*x)^2+(2*x^5+8*x^4
+6*x^3)*log(-x^2-3*x)+x^6+5*x^5+7*x^4+3*x^3),x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.10, size = 24, normalized size = 1.14

method	result	size
norman	\(\frac {\ln \relax (5)^{2}}{x^{2} \left (\ln \left (-x^{2}-3 x \right )+x +1\right )}\)	\(24\)
risch	\(\frac {\ln \relax (5)^{2}}{x^{2} \left (\ln \left (-x^{2}-3 x \right )+x +1\right )}\)	\(24\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-2*x-6)*ln(5)^2*ln(-x^2-3*x)+(-3*x^2-13*x-9)*ln(5)^2)/((x^4+3*x^3)*ln(-x^2-3*x)^2+(2*x^5+8*x^4+6*x^3)*ln
(-x^2-3*x)+x^6+5*x^5+7*x^4+3*x^3),x,method=_RETURNVERBOSE)

[Out]

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

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maxima [A]  time = 0.46, size = 30, normalized size = 1.43 \begin {gather*} \frac {\log \relax (5)^{2}}{x^{3} + x^{2} \log \relax (x) + x^{2} \log \left (-x - 3\right ) + x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*x-6)*log(5)^2*log(-x^2-3*x)+(-3*x^2-13*x-9)*log(5)^2)/((x^4+3*x^3)*log(-x^2-3*x)^2+(2*x^5+8*x^4
+6*x^3)*log(-x^2-3*x)+x^6+5*x^5+7*x^4+3*x^3),x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 3.25, size = 23, normalized size = 1.10 \begin {gather*} \frac {{\ln \relax (5)}^2}{x^2\,\left (x+\ln \left (-x^2-3\,x\right )+1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(log(5)^2*(13*x + 3*x^2 + 9) + log(5)^2*log(- 3*x - x^2)*(2*x + 6))/(log(- 3*x - x^2)^2*(3*x^3 + x^4) + l
og(- 3*x - x^2)*(6*x^3 + 8*x^4 + 2*x^5) + 3*x^3 + 7*x^4 + 5*x^5 + x^6),x)

[Out]

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

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sympy [A]  time = 0.18, size = 24, normalized size = 1.14 \begin {gather*} \frac {\log {\relax (5 )}^{2}}{x^{3} + x^{2} \log {\left (- x^{2} - 3 x \right )} + x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*x-6)*ln(5)**2*ln(-x**2-3*x)+(-3*x**2-13*x-9)*ln(5)**2)/((x**4+3*x**3)*ln(-x**2-3*x)**2+(2*x**5+
8*x**4+6*x**3)*ln(-x**2-3*x)+x**6+5*x**5+7*x**4+3*x**3),x)

[Out]

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

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