3.74.4 \(\int \frac {3 x^2+x^{.\frac {5}{3}/x} (5-5 \log (x))}{3 x^2} \, dx\)

Optimal. Leaf size=12 \[ -2+x+x^{\left .\frac {5}{3}\right /x} \]

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Rubi [F]  time = 0.09, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {3 x^2+x^{\left .\frac {5}{3}\right /x} (5-5 \log (x))}{3 x^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

x + (5*Defer[Int][x^(-2 + 5/(3*x)), x])/3 - (5*Log[x]*Defer[Int][x^(-2 + 5/(3*x)), x])/3 + (5*Defer[Int][Defer
[Int][x^(-2 + 5/(3*x)), x]/x, x])/3

Rubi steps

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

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Mathematica [A]  time = 0.03, size = 11, normalized size = 0.92 \begin {gather*} x+x^{\left .\frac {5}{3}\right /x} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

x + x^(5/(3*x))

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fricas [A]  time = 0.63, size = 9, normalized size = 0.75 \begin {gather*} x + x^{\frac {5}{3 \, x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/3*((-5*log(x)+5)*exp(5/3*log(x)/x)+3*x^2)/x^2,x, algorithm="fricas")

[Out]

x + x^(5/3/x)

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giac [A]  time = 0.21, size = 9, normalized size = 0.75 \begin {gather*} x + x^{\frac {5}{3 \, x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/3*((-5*log(x)+5)*exp(5/3*log(x)/x)+3*x^2)/x^2,x, algorithm="giac")

[Out]

x + x^(5/3/x)

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maple [A]  time = 0.04, size = 10, normalized size = 0.83

method	result	size
risch	\(x +x^{\frac {5}{3 x}}\)	\(10\)
default	\(x +{\mathrm e}^{\frac {5 \ln \relax (x )}{3 x}}\)	\(11\)
norman	\(\frac {x^{2}+x \,{\mathrm e}^{\frac {5 \ln \relax (x )}{3 x}}}{x}\)	\(19\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/3*((-5*ln(x)+5)*exp(5/3*ln(x)/x)+3*x^2)/x^2,x,method=_RETURNVERBOSE)

[Out]

x+x^(5/3/x)

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maxima [A]  time = 0.41, size = 9, normalized size = 0.75 \begin {gather*} x + x^{\frac {5}{3 \, x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/3*((-5*log(x)+5)*exp(5/3*log(x)/x)+3*x^2)/x^2,x, algorithm="maxima")

[Out]

x + x^(5/3/x)

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mupad [B]  time = 4.48, size = 9, normalized size = 0.75 \begin {gather*} x+x^{\frac {5}{3\,x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-((exp((5*log(x))/(3*x))*(5*log(x) - 5))/3 - x^2)/x^2,x)

[Out]

x + x^(5/(3*x))

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sympy [A]  time = 0.30, size = 10, normalized size = 0.83 \begin {gather*} x + e^{\frac {5 \log {\relax (x )}}{3 x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/3*((-5*ln(x)+5)*exp(5/3*ln(x)/x)+3*x**2)/x**2,x)

[Out]

x + exp(5*log(x)/(3*x))

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