3.60.72 \(\int \frac {-8 x^2+(e^5+x^2) \log (5)}{8 x^2} \, dx\)

Optimal. Leaf size=29 \[ -5-x+\frac {1}{2} \left (1-\frac {e^5-x^2}{4 x}\right ) \log (5) \]

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Rubi [A]  time = 0.01, antiderivative size = 24, normalized size of antiderivative = 0.83, number of steps used = 3, number of rules used = 2, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {12, 14} \begin {gather*} -\frac {1}{8} x (8-\log (5))-\frac {e^5 \log (5)}{8 x} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/8*(x*(8 - Log[5])) - (E^5*Log[5])/(8*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}{8} \int \frac {-8 x^2+\left (e^5+x^2\right ) \log (5)}{x^2} \, dx\\ &=\frac {1}{8} \int \left (-8 \left (1-\frac {\log (5)}{8}\right )+\frac {e^5 \log (5)}{x^2}\right ) \, dx\\ &=-\frac {1}{8} x (8-\log (5))-\frac {e^5 \log (5)}{8 x}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.00, size = 22, normalized size = 0.76 \begin {gather*} \frac {1}{8} \left (-8 x-\frac {e^5 \log (5)}{x}+x \log (5)\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-8*x - (E^5*Log[5])/x + x*Log[5])/8

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fricas [A]  time = 0.66, size = 23, normalized size = 0.79 \begin {gather*} -\frac {8 \, x^{2} - {\left (x^{2} - e^{5}\right )} \log \relax (5)}{8 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/8*(8*x^2 - (x^2 - e^5)*log(5))/x

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giac [A]  time = 0.20, size = 18, normalized size = 0.62 \begin {gather*} \frac {1}{8} \, x \log \relax (5) - x - \frac {e^{5} \log \relax (5)}{8 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*x*log(5) - x - 1/8*e^5*log(5)/x

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maple [A]  time = 0.04, size = 19, normalized size = 0.66

method	result	size
default	\(\frac {x \ln \relax (5)}{8}-x -\frac {{\mathrm e}^{5} \ln \relax (5)}{8 x}\)	\(19\)
risch	\(\frac {x \ln \relax (5)}{8}-x -\frac {{\mathrm e}^{5} \ln \relax (5)}{8 x}\)	\(19\)
norman	\(\frac {\left (-1+\frac {\ln \relax (5)}{8}\right ) x^{2}-\frac {{\mathrm e}^{5} \ln \relax (5)}{8}}{x}\)	\(22\)
gosper	\(-\frac {-x^{2} \ln \relax (5)+{\mathrm e}^{5} \ln \relax (5)+8 x^{2}}{8 x}\)	\(24\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*x*ln(5)-x-1/8*exp(5)*ln(5)/x

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maxima [A]  time = 0.35, size = 17, normalized size = 0.59 \begin {gather*} \frac {1}{8} \, x {\left (\log \relax (5) - 8\right )} - \frac {e^{5} \log \relax (5)}{8 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*x*(log(5) - 8) - 1/8*e^5*log(5)/x

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mupad [B]  time = 4.16, size = 18, normalized size = 0.62 \begin {gather*} x\,\left (\frac {\ln \relax (5)}{8}-1\right )-\frac {{\mathrm {e}}^5\,\ln \relax (5)}{8\,x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((log(5)*(exp(5) + x^2))/8 - x^2)/x^2,x)

[Out]

x*(log(5)/8 - 1) - (exp(5)*log(5))/(8*x)

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sympy [A]  time = 0.09, size = 19, normalized size = 0.66 \begin {gather*} - \frac {x \left (8 - \log {\relax (5 )}\right )}{8} - \frac {e^{5} \log {\relax (5 )}}{8 x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-x*(8 - log(5))/8 - exp(5)*log(5)/(8*x)

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