3.98.87 \(\int \frac {5 e^3-7 x^3-3 x^3 \log (x)}{5 x} \, dx\)

Optimal. Leaf size=18 \[ 629+\left (e^3-\frac {x^3}{5}\right ) (2+\log (x)) \]

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Rubi [A]  time = 0.02, antiderivative size = 23, normalized size of antiderivative = 1.28, number of steps used = 6, number of rules used = 3, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {12, 14, 2304} \begin {gather*} -\frac {2 x^3}{5}-\frac {1}{5} x^3 \log (x)+e^3 \log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(5*E^3 - 7*x^3 - 3*x^3*Log[x])/(5*x),x]

[Out]

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

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}{5} \int \frac {5 e^3-7 x^3-3 x^3 \log (x)}{x} \, dx\\ &=\frac {1}{5} \int \left (\frac {5 e^3-7 x^3}{x}-3 x^2 \log (x)\right ) \, dx\\ &=\frac {1}{5} \int \frac {5 e^3-7 x^3}{x} \, dx-\frac {3}{5} \int x^2 \log (x) \, dx\\ &=\frac {x^3}{15}-\frac {1}{5} x^3 \log (x)+\frac {1}{5} \int \left (\frac {5 e^3}{x}-7 x^2\right ) \, dx\\ &=-\frac {2 x^3}{5}+e^3 \log (x)-\frac {1}{5} x^3 \log (x)\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.00, size = 23, normalized size = 1.28 \begin {gather*} -\frac {2 x^3}{5}+e^3 \log (x)-\frac {1}{5} x^3 \log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(5*E^3 - 7*x^3 - 3*x^3*Log[x])/(5*x),x]

[Out]

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

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fricas [A]  time = 1.01, size = 18, normalized size = 1.00 \begin {gather*} -\frac {2}{5} \, x^{3} - \frac {1}{5} \, {\left (x^{3} - 5 \, e^{3}\right )} \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/5*(-3*x^3*log(x)+5*exp(3)-7*x^3)/x,x, algorithm="fricas")

[Out]

-2/5*x^3 - 1/5*(x^3 - 5*e^3)*log(x)

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giac [A]  time = 0.22, size = 18, normalized size = 1.00 \begin {gather*} -\frac {1}{5} \, x^{3} \log \relax (x) - \frac {2}{5} \, x^{3} + e^{3} \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/5*(-3*x^3*log(x)+5*exp(3)-7*x^3)/x,x, algorithm="giac")

[Out]

-1/5*x^3*log(x) - 2/5*x^3 + e^3*log(x)

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maple [A]  time = 0.03, size = 19, normalized size = 1.06

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/5*(-3*x^3*ln(x)+5*exp(3)-7*x^3)/x,x,method=_RETURNVERBOSE)

[Out]

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

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maxima [A]  time = 0.35, size = 18, normalized size = 1.00 \begin {gather*} -\frac {1}{5} \, x^{3} \log \relax (x) - \frac {2}{5} \, x^{3} + e^{3} \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/5*(-3*x^3*log(x)+5*exp(3)-7*x^3)/x,x, algorithm="maxima")

[Out]

-1/5*x^3*log(x) - 2/5*x^3 + e^3*log(x)

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mupad [B]  time = 5.68, size = 18, normalized size = 1.00 \begin {gather*} {\mathrm {e}}^3\,\ln \relax (x)-\frac {x^3\,\ln \relax (x)}{5}-\frac {2\,x^3}{5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-((3*x^3*log(x))/5 - exp(3) + (7*x^3)/5)/x,x)

[Out]

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

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sympy [A]  time = 0.13, size = 20, normalized size = 1.11 \begin {gather*} - \frac {x^{3} \log {\relax (x )}}{5} - \frac {2 x^{3}}{5} + e^{3} \log {\relax (x )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/5*(-3*x**3*ln(x)+5*exp(3)-7*x**3)/x,x)

[Out]

-x**3*log(x)/5 - 2*x**3/5 + exp(3)*log(x)

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