3.78.53 \(\int \frac {1+8 x^5+4 x^6+(-1+2 x) \log (x)}{2 x^6+x \log (x)} \, dx\)

Optimal. Leaf size=20 \[ 2 (-25+x)+\log \left (x^4+\frac {\log (x)}{2 x}\right ) \]

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Rubi [A]  time = 0.25, antiderivative size = 17, normalized size of antiderivative = 0.85, number of steps used = 6, number of rules used = 4, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.121, Rules used = {2561, 6742, 43, 6684} \begin {gather*} \log \left (2 x^5+\log (x)\right )+2 x-\log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(1 + 8*x^5 + 4*x^6 + (-1 + 2*x)*Log[x])/(2*x^6 + x*Log[x]),x]

[Out]

2*x - Log[x] + Log[2*x^5 + Log[x]]

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 2561

Int[(u_.)*((a_.)*(x_)^(m_.) + Log[(c_.)*(x_)^(n_.)]^(q_.)*(b_.)*(x_)^(r_.))^(p_.), x_Symbol] :> Int[u*x^(p*r)*
(a*x^(m - r) + b*Log[c*x^n]^q)^p, x] /; FreeQ[{a, b, c, m, n, p, q, r}, x] && IntegerQ[p]

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

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Mathematica [A]  time = 0.09, size = 17, normalized size = 0.85 \begin {gather*} 2 x-\log (x)+\log \left (2 x^5+\log (x)\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(1 + 8*x^5 + 4*x^6 + (-1 + 2*x)*Log[x])/(2*x^6 + x*Log[x]),x]

[Out]

2*x - Log[x] + Log[2*x^5 + Log[x]]

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fricas [A]  time = 0.60, size = 17, normalized size = 0.85 \begin {gather*} 2 \, x + \log \left (2 \, x^{5} + \log \relax (x)\right ) - \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x-1)*log(x)+4*x^6+8*x^5+1)/(x*log(x)+2*x^6),x, algorithm="fricas")

[Out]

2*x + log(2*x^5 + log(x)) - log(x)

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giac [A]  time = 0.14, size = 17, normalized size = 0.85 \begin {gather*} 2 \, x + \log \left (2 \, x^{5} + \log \relax (x)\right ) - \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x-1)*log(x)+4*x^6+8*x^5+1)/(x*log(x)+2*x^6),x, algorithm="giac")

[Out]

2*x + log(2*x^5 + log(x)) - log(x)

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maple [A]  time = 0.02, size = 18, normalized size = 0.90

method	result	size
norman	\(-\ln \relax (x )+2 x +\ln \left (2 x^{5}+\ln \relax (x )\right )\)	\(18\)
risch	\(-\ln \relax (x )+2 x +\ln \left (2 x^{5}+\ln \relax (x )\right )\)	\(18\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((2*x-1)*ln(x)+4*x^6+8*x^5+1)/(x*ln(x)+2*x^6),x,method=_RETURNVERBOSE)

[Out]

-ln(x)+2*x+ln(2*x^5+ln(x))

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maxima [A]  time = 0.40, size = 17, normalized size = 0.85 \begin {gather*} 2 \, x + \log \left (2 \, x^{5} + \log \relax (x)\right ) - \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x-1)*log(x)+4*x^6+8*x^5+1)/(x*log(x)+2*x^6),x, algorithm="maxima")

[Out]

2*x + log(2*x^5 + log(x)) - log(x)

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mupad [B]  time = 5.12, size = 17, normalized size = 0.85 \begin {gather*} 2\,x-\ln \relax (x)+\ln \left (\ln \relax (x)+2\,x^5\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((log(x)*(2*x - 1) + 8*x^5 + 4*x^6 + 1)/(x*log(x) + 2*x^6),x)

[Out]

2*x - log(x) + log(log(x) + 2*x^5)

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sympy [A]  time = 0.17, size = 15, normalized size = 0.75 \begin {gather*} 2 x - \log {\relax (x )} + \log {\left (2 x^{5} + \log {\relax (x )} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x-1)*ln(x)+4*x**6+8*x**5+1)/(x*ln(x)+2*x**6),x)

[Out]

2*x - log(x) + log(2*x**5 + log(x))

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