3.37.99 \(\int \frac {-1+25 x^2+6 x^3+16 x^2 \log (x)}{3 x^2} \, dx\)

Optimal. Leaf size=22 \[ \frac {1}{3 x}-x+x \left (4+x+\frac {16 \log (x)}{3}\right ) \]

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Rubi [A]  time = 0.02, antiderivative size = 21, normalized size of antiderivative = 0.95, number of steps used = 6, number of rules used = 3, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {12, 14, 2295} \begin {gather*} x^2+3 x+\frac {1}{3 x}+\frac {16}{3} x \log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-1 + 25*x^2 + 6*x^3 + 16*x^2*Log[x])/(3*x^2),x]

[Out]

1/(3*x) + 3*x + x^2 + (16*x*Log[x])/3

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]]

Rule 2295

Int[Log[(c_.)*(x_)^(n_.)], x_Symbol] :> Simp[x*Log[c*x^n], x] - Simp[n*x, x] /; FreeQ[{c, n}, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{3} \int \frac {-1+25 x^2+6 x^3+16 x^2 \log (x)}{x^2} \, dx\\ &=\frac {1}{3} \int \left (\frac {-1+25 x^2+6 x^3}{x^2}+16 \log (x)\right ) \, dx\\ &=\frac {1}{3} \int \frac {-1+25 x^2+6 x^3}{x^2} \, dx+\frac {16}{3} \int \log (x) \, dx\\ &=-\frac {16 x}{3}+\frac {16}{3} x \log (x)+\frac {1}{3} \int \left (25-\frac {1}{x^2}+6 x\right ) \, dx\\ &=\frac {1}{3 x}+3 x+x^2+\frac {16}{3} x \log (x)\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.01, size = 21, normalized size = 0.95 \begin {gather*} \frac {1}{3 x}+3 x+x^2+\frac {16}{3} x \log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-1 + 25*x^2 + 6*x^3 + 16*x^2*Log[x])/(3*x^2),x]

[Out]

1/(3*x) + 3*x + x^2 + (16*x*Log[x])/3

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fricas [A]  time = 0.67, size = 24, normalized size = 1.09 \begin {gather*} \frac {3 \, x^{3} + 16 \, x^{2} \log \relax (x) + 9 \, x^{2} + 1}{3 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/3*(16*x^2*log(x)+6*x^3+25*x^2-1)/x^2,x, algorithm="fricas")

[Out]

1/3*(3*x^3 + 16*x^2*log(x) + 9*x^2 + 1)/x

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giac [A]  time = 0.13, size = 17, normalized size = 0.77 \begin {gather*} x^{2} + \frac {16}{3} \, x \log \relax (x) + 3 \, x + \frac {1}{3 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/3*(16*x^2*log(x)+6*x^3+25*x^2-1)/x^2,x, algorithm="giac")

[Out]

x^2 + 16/3*x*log(x) + 3*x + 1/3/x

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maple [A]  time = 0.01, size = 18, normalized size = 0.82

method	result	size
default	\(\frac {16 x \ln \relax (x )}{3}+3 x +x^{2}+\frac {1}{3 x}\)	\(18\)
norman	\(\frac {\frac {1}{3}+x^{3}+3 x^{2}+\frac {16 x^{2} \ln \relax (x )}{3}}{x}\)	\(22\)
risch	\(\frac {16 x \ln \relax (x )}{3}+\frac {3 x^{3}+9 x^{2}+1}{3 x}\)	\(24\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/3*(16*x^2*ln(x)+6*x^3+25*x^2-1)/x^2,x,method=_RETURNVERBOSE)

[Out]

16/3*x*ln(x)+3*x+x^2+1/3/x

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maxima [A]  time = 0.37, size = 17, normalized size = 0.77 \begin {gather*} x^{2} + \frac {16}{3} \, x \log \relax (x) + 3 \, x + \frac {1}{3 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/3*(16*x^2*log(x)+6*x^3+25*x^2-1)/x^2,x, algorithm="maxima")

[Out]

x^2 + 16/3*x*log(x) + 3*x + 1/3/x

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mupad [B]  time = 2.20, size = 17, normalized size = 0.77 \begin {gather*} x\,\left (\frac {16\,\ln \relax (x)}{3}+3\right )+\frac {1}{3\,x}+x^2 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((16*x^2*log(x))/3 + (25*x^2)/3 + 2*x^3 - 1/3)/x^2,x)

[Out]

x*((16*log(x))/3 + 3) + 1/(3*x) + x^2

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sympy [A]  time = 0.11, size = 19, normalized size = 0.86 \begin {gather*} x^{2} + \frac {16 x \log {\relax (x )}}{3} + 3 x + \frac {1}{3 x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/3*(16*x**2*ln(x)+6*x**3+25*x**2-1)/x**2,x)

[Out]

x**2 + 16*x*log(x)/3 + 3*x + 1/(3*x)

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