3.5.87 \(\int \frac {-1-3 x+3 x^2-21 x^7+3 x^8+x \log (2 x)}{3 x^2+3 x^8+x \log (2 x)} \, dx\)

Optimal. Leaf size=22 \[ 3+x-\log \left (x \left (1+x^6\right )+\frac {1}{3} \log (2 x)\right ) \]

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Rubi [A]  time = 0.17, antiderivative size = 18, normalized size of antiderivative = 0.82, number of steps used = 3, number of rules used = 2, integrand size = 46, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.043, Rules used = {6742, 6684} \begin {gather*} x-\log \left (3 x^7+3 x+\log (2 x)\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-1 - 3*x + 3*x^2 - 21*x^7 + 3*x^8 + x*Log[2*x])/(3*x^2 + 3*x^8 + x*Log[2*x]),x]

[Out]

x - Log[3*x + 3*x^7 + Log[2*x]]

Rule 6684

Int[(u_)/(y_), x_Symbol] :> With[{q = DerivativeDivides[y, u, x]}, Simp[q*Log[RemoveContent[y, x]], x] /;  !Fa
lseQ[q]]

Rule 6742

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (1+\frac {-1-3 x-21 x^7}{x \left (3 x+3 x^7+\log (2 x)\right )}\right ) \, dx\\ &=x+\int \frac {-1-3 x-21 x^7}{x \left (3 x+3 x^7+\log (2 x)\right )} \, dx\\ &=x-\log \left (3 x+3 x^7+\log (2 x)\right )\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.15, size = 18, normalized size = 0.82 \begin {gather*} x-\log \left (3 x+3 x^7+\log (2 x)\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-1 - 3*x + 3*x^2 - 21*x^7 + 3*x^8 + x*Log[2*x])/(3*x^2 + 3*x^8 + x*Log[2*x]),x]

[Out]

x - Log[3*x + 3*x^7 + Log[2*x]]

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fricas [A]  time = 0.87, size = 18, normalized size = 0.82 \begin {gather*} x - \log \left (3 \, x^{7} + 3 \, x + \log \left (2 \, x\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x*log(2*x)+3*x^8-21*x^7+3*x^2-3*x-1)/(x*log(2*x)+3*x^8+3*x^2),x, algorithm="fricas")

[Out]

x - log(3*x^7 + 3*x + log(2*x))

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giac [A]  time = 0.41, size = 20, normalized size = 0.91 \begin {gather*} x - \log \left (-3 \, x^{7} - 3 \, x - \log \left (2 \, x\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x*log(2*x)+3*x^8-21*x^7+3*x^2-3*x-1)/(x*log(2*x)+3*x^8+3*x^2),x, algorithm="giac")

[Out]

x - log(-3*x^7 - 3*x - log(2*x))

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maple [A]  time = 0.04, size = 19, normalized size = 0.86

method	result	size
norman	\(x -\ln \left (3 x^{7}+\ln \left (2 x \right )+3 x \right )\)	\(19\)
risch	\(x -\ln \left (3 x^{7}+\ln \left (2 x \right )+3 x \right )\)	\(19\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x*ln(2*x)+3*x^8-21*x^7+3*x^2-3*x-1)/(x*ln(2*x)+3*x^8+3*x^2),x,method=_RETURNVERBOSE)

[Out]

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

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maxima [A]  time = 0.71, size = 18, normalized size = 0.82 \begin {gather*} x - \log \left (3 \, x^{7} + 3 \, x + \log \relax (2) + \log \relax (x)\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x*log(2*x)+3*x^8-21*x^7+3*x^2-3*x-1)/(x*log(2*x)+3*x^8+3*x^2),x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 0.56, size = 18, normalized size = 0.82 \begin {gather*} x-\ln \left (3\,x+\ln \left (2\,x\right )+3\,x^7\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(3*x - x*log(2*x) - 3*x^2 + 21*x^7 - 3*x^8 + 1)/(x*log(2*x) + 3*x^2 + 3*x^8),x)

[Out]

x - log(3*x + log(2*x) + 3*x^7)

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sympy [A]  time = 0.15, size = 15, normalized size = 0.68 \begin {gather*} x - \log {\left (3 x^{7} + 3 x + \log {\left (2 x \right )} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x*ln(2*x)+3*x**8-21*x**7+3*x**2-3*x-1)/(x*ln(2*x)+3*x**8+3*x**2),x)

[Out]

x - log(3*x**7 + 3*x + log(2*x))

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