3.56.51 \(\int \frac {-5670-3240 x-2430 \log (x)}{8 x^7+12 x^8+6 x^9+x^{10}+(12 x^7+12 x^8+3 x^9) \log (x)+(6 x^7+3 x^8) \log ^2(x)+x^7 \log ^3(x)} \, dx\)

Optimal. Leaf size=12 \[ \frac {405}{x^6 (2+x+\log (x))^2} \]

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Rubi [F]  time = 0.45, 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 {-5670-3240 x-2430 \log (x)}{8 x^7+12 x^8+6 x^9+x^{10}+\left (12 x^7+12 x^8+3 x^9\right ) \log (x)+\left (6 x^7+3 x^8\right ) \log ^2(x)+x^7 \log ^3(x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-5670 - 3240*x - 2430*Log[x])/(8*x^7 + 12*x^8 + 6*x^9 + x^10 + (12*x^7 + 12*x^8 + 3*x^9)*Log[x] + (6*x^7
+ 3*x^8)*Log[x]^2 + x^7*Log[x]^3),x]

[Out]

-810*Defer[Int][1/(x^7*(2 + x + Log[x])^3), x] - 810*Defer[Int][1/(x^6*(2 + x + Log[x])^3), x] - 2430*Defer[In
t][1/(x^7*(2 + x + Log[x])^2), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {810 (-7-4 x-3 \log (x))}{x^7 (2+x+\log (x))^3} \, dx\\ &=810 \int \frac {-7-4 x-3 \log (x)}{x^7 (2+x+\log (x))^3} \, dx\\ &=810 \int \left (\frac {-1-x}{x^7 (2+x+\log (x))^3}-\frac {3}{x^7 (2+x+\log (x))^2}\right ) \, dx\\ &=810 \int \frac {-1-x}{x^7 (2+x+\log (x))^3} \, dx-2430 \int \frac {1}{x^7 (2+x+\log (x))^2} \, dx\\ &=810 \int \left (-\frac {1}{x^7 (2+x+\log (x))^3}-\frac {1}{x^6 (2+x+\log (x))^3}\right ) \, dx-2430 \int \frac {1}{x^7 (2+x+\log (x))^2} \, dx\\ &=-\left (810 \int \frac {1}{x^7 (2+x+\log (x))^3} \, dx\right )-810 \int \frac {1}{x^6 (2+x+\log (x))^3} \, dx-2430 \int \frac {1}{x^7 (2+x+\log (x))^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.44, size = 12, normalized size = 1.00 \begin {gather*} \frac {405}{x^6 (2+x+\log (x))^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-5670 - 3240*x - 2430*Log[x])/(8*x^7 + 12*x^8 + 6*x^9 + x^10 + (12*x^7 + 12*x^8 + 3*x^9)*Log[x] + (
6*x^7 + 3*x^8)*Log[x]^2 + x^7*Log[x]^3),x]

[Out]

405/(x^6*(2 + x + Log[x])^2)

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fricas [B]  time = 0.45, size = 39, normalized size = 3.25 \begin {gather*} \frac {405}{x^{8} + x^{6} \log \relax (x)^{2} + 4 \, x^{7} + 4 \, x^{6} + 2 \, {\left (x^{7} + 2 \, x^{6}\right )} \log \relax (x)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2430*log(x)-3240*x-5670)/(x^7*log(x)^3+(3*x^8+6*x^7)*log(x)^2+(3*x^9+12*x^8+12*x^7)*log(x)+x^10+6*
x^9+12*x^8+8*x^7),x, algorithm="fricas")

[Out]

405/(x^8 + x^6*log(x)^2 + 4*x^7 + 4*x^6 + 2*(x^7 + 2*x^6)*log(x))

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giac [B]  time = 0.22, size = 63, normalized size = 5.25 \begin {gather*} \frac {405 \, {\left (x + 1\right )}}{x^{9} + 2 \, x^{8} \log \relax (x) + x^{7} \log \relax (x)^{2} + 5 \, x^{8} + 6 \, x^{7} \log \relax (x) + x^{6} \log \relax (x)^{2} + 8 \, x^{7} + 4 \, x^{6} \log \relax (x) + 4 \, x^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2430*log(x)-3240*x-5670)/(x^7*log(x)^3+(3*x^8+6*x^7)*log(x)^2+(3*x^9+12*x^8+12*x^7)*log(x)+x^10+6*
x^9+12*x^8+8*x^7),x, algorithm="giac")

[Out]

405*(x + 1)/(x^9 + 2*x^8*log(x) + x^7*log(x)^2 + 5*x^8 + 6*x^7*log(x) + x^6*log(x)^2 + 8*x^7 + 4*x^6*log(x) +
4*x^6)

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maple [A]  time = 0.02, size = 13, normalized size = 1.08

method	result	size
risch	\(\frac {405}{\left (x +\ln \relax (x )+2\right )^{2} x^{6}}\)	\(13\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-2430*ln(x)-3240*x-5670)/(x^7*ln(x)^3+(3*x^8+6*x^7)*ln(x)^2+(3*x^9+12*x^8+12*x^7)*ln(x)+x^10+6*x^9+12*x^8
+8*x^7),x,method=_RETURNVERBOSE)

[Out]

405/(x+ln(x)+2)^2/x^6

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maxima [B]  time = 0.40, size = 39, normalized size = 3.25 \begin {gather*} \frac {405}{x^{8} + x^{6} \log \relax (x)^{2} + 4 \, x^{7} + 4 \, x^{6} + 2 \, {\left (x^{7} + 2 \, x^{6}\right )} \log \relax (x)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2430*log(x)-3240*x-5670)/(x^7*log(x)^3+(3*x^8+6*x^7)*log(x)^2+(3*x^9+12*x^8+12*x^7)*log(x)+x^10+6*
x^9+12*x^8+8*x^7),x, algorithm="maxima")

[Out]

405/(x^8 + x^6*log(x)^2 + 4*x^7 + 4*x^6 + 2*(x^7 + 2*x^6)*log(x))

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mupad [F]  time = 0.00, size = -1, normalized size = -0.08 \begin {gather*} \int -\frac {3240\,x+2430\,\ln \relax (x)+5670}{\ln \relax (x)\,\left (3\,x^9+12\,x^8+12\,x^7\right )+{\ln \relax (x)}^2\,\left (3\,x^8+6\,x^7\right )+x^7\,{\ln \relax (x)}^3+8\,x^7+12\,x^8+6\,x^9+x^{10}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(3240*x + 2430*log(x) + 5670)/(log(x)*(12*x^7 + 12*x^8 + 3*x^9) + log(x)^2*(6*x^7 + 3*x^8) + x^7*log(x)^3
+ 8*x^7 + 12*x^8 + 6*x^9 + x^10),x)

[Out]

int(-(3240*x + 2430*log(x) + 5670)/(log(x)*(12*x^7 + 12*x^8 + 3*x^9) + log(x)^2*(6*x^7 + 3*x^8) + x^7*log(x)^3
+ 8*x^7 + 12*x^8 + 6*x^9 + x^10), x)

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sympy [B]  time = 0.13, size = 36, normalized size = 3.00 \begin {gather*} \frac {405}{x^{8} + 4 x^{7} + x^{6} \log {\relax (x )}^{2} + 4 x^{6} + \left (2 x^{7} + 4 x^{6}\right ) \log {\relax (x )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2430*ln(x)-3240*x-5670)/(x**7*ln(x)**3+(3*x**8+6*x**7)*ln(x)**2+(3*x**9+12*x**8+12*x**7)*ln(x)+x**
10+6*x**9+12*x**8+8*x**7),x)

[Out]

405/(x**8 + 4*x**7 + x**6*log(x)**2 + 4*x**6 + (2*x**7 + 4*x**6)*log(x))

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