3.14.24 \(\int \frac {15+15 x+2 x^2+6 x^3+x^7-5 x^8+3 x^9+(4 x^7-3 x^8) \log (x)}{-5 x-2 x^3-x^9+x^8 \log (x)} \, dx\)

Optimal. Leaf size=27 \[ -3 x+\log \left (\frac {2+\frac {5}{x^2}-x^5 (-x+\log (x))}{x}\right ) \]

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Rubi [F]  time = 0.83, 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 {15+15 x+2 x^2+6 x^3+x^7-5 x^8+3 x^9+\left (4 x^7-3 x^8\right ) \log (x)}{-5 x-2 x^3-x^9+x^8 \log (x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(15 + 15*x + 2*x^2 + 6*x^3 + x^7 - 5*x^8 + 3*x^9 + (4*x^7 - 3*x^8)*Log[x])/(-5*x - 2*x^3 - x^9 + x^8*Log[x
]),x]

[Out]

-3*x + 4*Log[x] - 35*Defer[Int][1/(x*(5 + 2*x^2 + x^8 - x^7*Log[x])), x] - 10*Defer[Int][x/(5 + 2*x^2 + x^8 -
x^7*Log[x]), x] - Defer[Int][x^6/(5 + 2*x^2 + x^8 - x^7*Log[x]), x] + Defer[Int][x^7/(5 + 2*x^2 + x^8 - x^7*Lo
g[x]), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (\frac {4-3 x}{x}+\frac {-35-10 x^2-x^7+x^8}{x \left (5+2 x^2+x^8-x^7 \log (x)\right )}\right ) \, dx\\ &=\int \frac {4-3 x}{x} \, dx+\int \frac {-35-10 x^2-x^7+x^8}{x \left (5+2 x^2+x^8-x^7 \log (x)\right )} \, dx\\ &=\int \left (-3+\frac {4}{x}\right ) \, dx+\int \left (-\frac {35}{x \left (5+2 x^2+x^8-x^7 \log (x)\right )}-\frac {10 x}{5+2 x^2+x^8-x^7 \log (x)}-\frac {x^6}{5+2 x^2+x^8-x^7 \log (x)}+\frac {x^7}{5+2 x^2+x^8-x^7 \log (x)}\right ) \, dx\\ &=-3 x+4 \log (x)-10 \int \frac {x}{5+2 x^2+x^8-x^7 \log (x)} \, dx-35 \int \frac {1}{x \left (5+2 x^2+x^8-x^7 \log (x)\right )} \, dx-\int \frac {x^6}{5+2 x^2+x^8-x^7 \log (x)} \, dx+\int \frac {x^7}{5+2 x^2+x^8-x^7 \log (x)} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.19, size = 26, normalized size = 0.96 \begin {gather*} -3 x-3 \log (x)+\log \left (5+2 x^2+x^8-x^7 \log (x)\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(15 + 15*x + 2*x^2 + 6*x^3 + x^7 - 5*x^8 + 3*x^9 + (4*x^7 - 3*x^8)*Log[x])/(-5*x - 2*x^3 - x^9 + x^8
*Log[x]),x]

[Out]

-3*x - 3*Log[x] + Log[5 + 2*x^2 + x^8 - x^7*Log[x]]

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fricas [A]  time = 0.97, size = 31, normalized size = 1.15 \begin {gather*} -3 \, x + 4 \, \log \relax (x) + \log \left (-\frac {x^{8} - x^{7} \log \relax (x) + 2 \, x^{2} + 5}{x^{7}}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-3*x^8+4*x^7)*log(x)+3*x^9-5*x^8+x^7+6*x^3+2*x^2+15*x+15)/(x^8*log(x)-x^9-2*x^3-5*x),x, algorithm=
"fricas")

[Out]

-3*x + 4*log(x) + log(-(x^8 - x^7*log(x) + 2*x^2 + 5)/x^7)

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giac [A]  time = 0.33, size = 26, normalized size = 0.96 \begin {gather*} -3 \, x + \log \left (x^{8} - x^{7} \log \relax (x) + 2 \, x^{2} + 5\right ) - 3 \, \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-3*x^8+4*x^7)*log(x)+3*x^9-5*x^8+x^7+6*x^3+2*x^2+15*x+15)/(x^8*log(x)-x^9-2*x^3-5*x),x, algorithm=
"giac")

[Out]

-3*x + log(x^8 - x^7*log(x) + 2*x^2 + 5) - 3*log(x)

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maple [A]  time = 0.03, size = 27, normalized size = 1.00

method	result	size
norman	\(-3 x -3 \ln \relax (x )+\ln \left (x^{8}-x^{7} \ln \relax (x )+2 x^{2}+5\right )\)	\(27\)
risch	\(-3 x +4 \ln \relax (x )+\ln \left (\ln \relax (x )-\frac {x^{8}+2 x^{2}+5}{x^{7}}\right )\)	\(28\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-3*x^8+4*x^7)*ln(x)+3*x^9-5*x^8+x^7+6*x^3+2*x^2+15*x+15)/(x^8*ln(x)-x^9-2*x^3-5*x),x,method=_RETURNVERBO
SE)

[Out]

-3*x-3*ln(x)+ln(x^8-x^7*ln(x)+2*x^2+5)

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maxima [A]  time = 0.46, size = 31, normalized size = 1.15 \begin {gather*} -3 \, x + 4 \, \log \relax (x) + \log \left (-\frac {x^{8} - x^{7} \log \relax (x) + 2 \, x^{2} + 5}{x^{7}}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-3*x^8+4*x^7)*log(x)+3*x^9-5*x^8+x^7+6*x^3+2*x^2+15*x+15)/(x^8*log(x)-x^9-2*x^3-5*x),x, algorithm=
"maxima")

[Out]

-3*x + 4*log(x) + log(-(x^8 - x^7*log(x) + 2*x^2 + 5)/x^7)

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mupad [B]  time = 1.09, size = 30, normalized size = 1.11 \begin {gather*} \ln \left (\frac {2\,x^2-x^7\,\ln \relax (x)+x^8+5}{x^7}\right )-3\,x+4\,\ln \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(15*x + log(x)*(4*x^7 - 3*x^8) + 2*x^2 + 6*x^3 + x^7 - 5*x^8 + 3*x^9 + 15)/(5*x - x^8*log(x) + 2*x^3 + x^
9),x)

[Out]

log((2*x^2 - x^7*log(x) + x^8 + 5)/x^7) - 3*x + 4*log(x)

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sympy [A]  time = 0.22, size = 27, normalized size = 1.00 \begin {gather*} - 3 x + 4 \log {\relax (x )} + \log {\left (\log {\relax (x )} + \frac {- x^{8} - 2 x^{2} - 5}{x^{7}} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-3*x**8+4*x**7)*ln(x)+3*x**9-5*x**8+x**7+6*x**3+2*x**2+15*x+15)/(x**8*ln(x)-x**9-2*x**3-5*x),x)

[Out]

-3*x + 4*log(x) + log(log(x) + (-x**8 - 2*x**2 - 5)/x**7)

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