3.15.94 \(\int \frac {-3+24 x+4 x^2}{3 x} \, dx\)

Optimal. Leaf size=18 \[ -2-3 x+\left (11+\frac {2 x}{3}\right ) x-\log (x) \]

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Rubi [A]  time = 0.00, antiderivative size = 15, normalized size of antiderivative = 0.83, number of steps used = 3, number of rules used = 2, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {12, 14} \begin {gather*} \frac {2 x^2}{3}+8 x-\log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-3 + 24*x + 4*x^2)/(3*x),x]

[Out]

8*x + (2*x^2)/3 - Log[x]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, 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]]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{3} \int \frac {-3+24 x+4 x^2}{x} \, dx\\ &=\frac {1}{3} \int \left (24-\frac {3}{x}+4 x\right ) \, dx\\ &=8 x+\frac {2 x^2}{3}-\log (x)\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.00, size = 15, normalized size = 0.83 \begin {gather*} 8 x+\frac {2 x^2}{3}-\log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-3 + 24*x + 4*x^2)/(3*x),x]

[Out]

8*x + (2*x^2)/3 - Log[x]

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fricas [A]  time = 0.88, size = 13, normalized size = 0.72 \begin {gather*} \frac {2}{3} \, x^{2} + 8 \, x - \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/3*(4*x^2+24*x-3)/x,x, algorithm="fricas")

[Out]

2/3*x^2 + 8*x - log(x)

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giac [A]  time = 0.15, size = 14, normalized size = 0.78 \begin {gather*} \frac {2}{3} \, x^{2} + 8 \, x - \log \left ({\left | x \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/3*(4*x^2+24*x-3)/x,x, algorithm="giac")

[Out]

2/3*x^2 + 8*x - log(abs(x))

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maple [A]  time = 0.02, size = 14, normalized size = 0.78

method	result	size
default	\(\frac {2 x^{2}}{3}+8 x -\ln \relax (x )\)	\(14\)
norman	\(\frac {2 x^{2}}{3}+8 x -\ln \relax (x )\)	\(14\)
risch	\(\frac {2 x^{2}}{3}+8 x -\ln \relax (x )\)	\(14\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/3*(4*x^2+24*x-3)/x,x,method=_RETURNVERBOSE)

[Out]

2/3*x^2+8*x-ln(x)

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maxima [A]  time = 0.38, size = 13, normalized size = 0.72 \begin {gather*} \frac {2}{3} \, x^{2} + 8 \, x - \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/3*(4*x^2+24*x-3)/x,x, algorithm="maxima")

[Out]

2/3*x^2 + 8*x - log(x)

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mupad [B]  time = 0.02, size = 13, normalized size = 0.72 \begin {gather*} 8\,x-\ln \relax (x)+\frac {2\,x^2}{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((8*x + (4*x^2)/3 - 1)/x,x)

[Out]

8*x - log(x) + (2*x^2)/3

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sympy [A]  time = 0.07, size = 12, normalized size = 0.67 \begin {gather*} \frac {2 x^{2}}{3} + 8 x - \log {\relax (x )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/3*(4*x**2+24*x-3)/x,x)

[Out]

2*x**2/3 + 8*x - log(x)

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