3.74.98 \(\int \frac {125+125 x^2-125 \log (x)}{2 x^2} \, dx\)

Optimal. Leaf size=18 \[ \frac {1}{2} \left (\log (3)+\frac {125 \left (x^2+\log (x)\right )}{x}\right ) \]

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Rubi [A]  time = 0.02, antiderivative size = 15, normalized size of antiderivative = 0.83, number of steps used = 6, number of rules used = 3, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {12, 14, 2304} \begin {gather*} \frac {125 x}{2}+\frac {125 \log (x)}{2 x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(125 + 125*x^2 - 125*Log[x])/(2*x^2),x]

[Out]

(125*x)/2 + (125*Log[x])/(2*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]]

Rule 2304

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[((d*x)^(m + 1)*(a + b*Log[c*x^
n]))/(d*(m + 1)), x] - Simp[(b*n*(d*x)^(m + 1))/(d*(m + 1)^2), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[m, -1
]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{2} \int \frac {125+125 x^2-125 \log (x)}{x^2} \, dx\\ &=\frac {1}{2} \int \left (\frac {125 \left (1+x^2\right )}{x^2}-\frac {125 \log (x)}{x^2}\right ) \, dx\\ &=\frac {125}{2} \int \frac {1+x^2}{x^2} \, dx-\frac {125}{2} \int \frac {\log (x)}{x^2} \, dx\\ &=\frac {125}{2 x}+\frac {125 \log (x)}{2 x}+\frac {125}{2} \int \left (1+\frac {1}{x^2}\right ) \, dx\\ &=\frac {125 x}{2}+\frac {125 \log (x)}{2 x}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.00, size = 15, normalized size = 0.83 \begin {gather*} \frac {125 x}{2}+\frac {125 \log (x)}{2 x} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(125 + 125*x^2 - 125*Log[x])/(2*x^2),x]

[Out]

(125*x)/2 + (125*Log[x])/(2*x)

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fricas [A]  time = 0.57, size = 11, normalized size = 0.61 \begin {gather*} \frac {125 \, {\left (x^{2} + \log \relax (x)\right )}}{2 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/2*(-125*log(x)+125*x^2+125)/x^2,x, algorithm="fricas")

[Out]

125/2*(x^2 + log(x))/x

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giac [A]  time = 0.13, size = 11, normalized size = 0.61 \begin {gather*} \frac {125}{2} \, x + \frac {125 \, \log \relax (x)}{2 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/2*(-125*log(x)+125*x^2+125)/x^2,x, algorithm="giac")

[Out]

125/2*x + 125/2*log(x)/x

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maple [A]  time = 0.01, size = 12, normalized size = 0.67

method	result	size
default	\(\frac {125 \ln \relax (x )}{2 x}+\frac {125 x}{2}\)	\(12\)
risch	\(\frac {125 \ln \relax (x )}{2 x}+\frac {125 x}{2}\)	\(12\)
norman	\(\frac {\frac {125 x^{2}}{2}+\frac {125 \ln \relax (x )}{2}}{x}\)	\(15\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/2*(-125*ln(x)+125*x^2+125)/x^2,x,method=_RETURNVERBOSE)

[Out]

125/2*ln(x)/x+125/2*x

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maxima [A]  time = 0.36, size = 11, normalized size = 0.61 \begin {gather*} \frac {125}{2} \, x + \frac {125 \, \log \relax (x)}{2 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/2*(-125*log(x)+125*x^2+125)/x^2,x, algorithm="maxima")

[Out]

125/2*x + 125/2*log(x)/x

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mupad [B]  time = 5.24, size = 11, normalized size = 0.61 \begin {gather*} \frac {125\,\left (\ln \relax (x)+x^2\right )}{2\,x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((125*x^2)/2 - (125*log(x))/2 + 125/2)/x^2,x)

[Out]

(125*(log(x) + x^2))/(2*x)

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sympy [A]  time = 0.13, size = 12, normalized size = 0.67 \begin {gather*} \frac {125 x}{2} + \frac {125 \log {\relax (x )}}{2 x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/2*(-125*ln(x)+125*x**2+125)/x**2,x)

[Out]

125*x/2 + 125*log(x)/(2*x)

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