3.40.16 \(\int \frac {-3+4 x+6 x^2-11 x^3+(x^2-2 x^3) \log (x)}{x^2} \, dx\)

Optimal. Leaf size=30 \[ -\frac {-3+x}{x}+5 \left (x-x^2\right )-\left (-4-x+x^2\right ) \log (x) \]

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Rubi [A]  time = 0.03, antiderivative size = 28, normalized size of antiderivative = 0.93, number of steps used = 6, number of rules used = 2, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.065, Rules used = {14, 2313} \begin {gather*} -5 x^2+\left (x-x^2\right ) \log (x)+5 x+\frac {3}{x}+4 \log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-3 + 4*x + 6*x^2 - 11*x^3 + (x^2 - 2*x^3)*Log[x])/x^2,x]

[Out]

3/x + 5*x - 5*x^2 + 4*Log[x] + (x - x^2)*Log[x]

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rule 2313

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((d_) + (e_.)*(x_)^(r_.))^(q_.), x_Symbol] :> With[{u = IntHide[(d +
e*x^r)^q, x]}, Simp[u*(a + b*Log[c*x^n]), x] - Dist[b*n, Int[SimplifyIntegrand[u/x, x], x], x]] /; FreeQ[{a,
b, c, d, e, n, r}, x] && IGtQ[q, 0]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (\frac {-3+4 x+6 x^2-11 x^3}{x^2}-(-1+2 x) \log (x)\right ) \, dx\\ &=\int \frac {-3+4 x+6 x^2-11 x^3}{x^2} \, dx-\int (-1+2 x) \log (x) \, dx\\ &=\left (x-x^2\right ) \log (x)+\int \left (6-\frac {3}{x^2}+\frac {4}{x}-11 x\right ) \, dx+\int (-1+x) \, dx\\ &=\frac {3}{x}+5 x-5 x^2+4 \log (x)+\left (x-x^2\right ) \log (x)\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.01, size = 29, normalized size = 0.97 \begin {gather*} \frac {3}{x}+5 x-5 x^2+4 \log (x)+x \log (x)-x^2 \log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-3 + 4*x + 6*x^2 - 11*x^3 + (x^2 - 2*x^3)*Log[x])/x^2,x]

[Out]

3/x + 5*x - 5*x^2 + 4*Log[x] + x*Log[x] - x^2*Log[x]

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fricas [A]  time = 1.04, size = 32, normalized size = 1.07 \begin {gather*} -\frac {5 \, x^{3} - 5 \, x^{2} + {\left (x^{3} - x^{2} - 4 \, x\right )} \log \relax (x) - 3}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*x^3+x^2)*log(x)-11*x^3+6*x^2+4*x-3)/x^2,x, algorithm="fricas")

[Out]

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

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giac [A]  time = 0.16, size = 29, normalized size = 0.97 \begin {gather*} -5 \, x^{2} - {\left (x^{2} - x\right )} \log \relax (x) + 5 \, x + \frac {3}{x} + 4 \, \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*x^3+x^2)*log(x)-11*x^3+6*x^2+4*x-3)/x^2,x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.02, size = 30, normalized size = 1.00

method	result	size
default	\(-x^{2} \ln \relax (x )-5 x^{2}+x \ln \relax (x )+5 x +4 \ln \relax (x )+\frac {3}{x}\)	\(30\)
risch	\(\left (-x^{2}+x \right ) \ln \relax (x )+\frac {-5 x^{3}+4 x \ln \relax (x )+5 x^{2}+3}{x}\)	\(33\)
norman	\(\frac {3+x^{2} \ln \relax (x )+4 x \ln \relax (x )+5 x^{2}-5 x^{3}-x^{3} \ln \relax (x )}{x}\)	\(35\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-2*x^3+x^2)*ln(x)-11*x^3+6*x^2+4*x-3)/x^2,x,method=_RETURNVERBOSE)

[Out]

-x^2*ln(x)-5*x^2+x*ln(x)+5*x+4*ln(x)+3/x

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maxima [A]  time = 0.34, size = 29, normalized size = 0.97 \begin {gather*} -x^{2} \log \relax (x) - 5 \, x^{2} + x \log \relax (x) + 5 \, x + \frac {3}{x} + 4 \, \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*x^3+x^2)*log(x)-11*x^3+6*x^2+4*x-3)/x^2,x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 2.21, size = 25, normalized size = 0.83 \begin {gather*} 4\,\ln \relax (x)+x\,\left (\ln \relax (x)+5\right )-x^2\,\left (\ln \relax (x)+5\right )+\frac {3}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((4*x + log(x)*(x^2 - 2*x^3) + 6*x^2 - 11*x^3 - 3)/x^2,x)

[Out]

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

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sympy [A]  time = 0.13, size = 24, normalized size = 0.80 \begin {gather*} - 5 x^{2} + 5 x + \left (- x^{2} + x\right ) \log {\relax (x )} + 4 \log {\relax (x )} + \frac {3}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*x**3+x**2)*ln(x)-11*x**3+6*x**2+4*x-3)/x**2,x)

[Out]

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

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