3.78.77 \(\int \frac {1}{12} (15+12 e^x+548 x-348 x^2+48 x^3-8 x \log (x)) \, dx\)

Optimal. Leaf size=33 \[ e^x+\frac {5 x}{4}-x^2 \left (2-(-5+x)^2+\frac {1}{3} (-x+\log (x))\right ) \]

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Rubi [A]  time = 0.01, antiderivative size = 33, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.103, Rules used = {12, 2194, 2304} \begin {gather*} x^4-\frac {29 x^3}{3}+23 x^2-\frac {1}{3} x^2 \log (x)+\frac {5 x}{4}+e^x \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(15 + 12*E^x + 548*x - 348*x^2 + 48*x^3 - 8*x*Log[x])/12,x]

[Out]

E^x + (5*x)/4 + 23*x^2 - (29*x^3)/3 + x^4 - (x^2*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 2194

Int[((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.), x_Symbol] :> Simp[(F^(c*(a + b*x)))^n/(b*c*n*Log[F]), x] /; Fre
eQ[{F, a, b, c, n}, x]

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}{12} \int \left (15+12 e^x+548 x-348 x^2+48 x^3-8 x \log (x)\right ) \, dx\\ &=\frac {5 x}{4}+\frac {137 x^2}{6}-\frac {29 x^3}{3}+x^4-\frac {2}{3} \int x \log (x) \, dx+\int e^x \, dx\\ &=e^x+\frac {5 x}{4}+23 x^2-\frac {29 x^3}{3}+x^4-\frac {1}{3} x^2 \log (x)\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.01, size = 33, normalized size = 1.00 \begin {gather*} e^x+\frac {5 x}{4}+23 x^2-\frac {29 x^3}{3}+x^4-\frac {1}{3} x^2 \log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(15 + 12*E^x + 548*x - 348*x^2 + 48*x^3 - 8*x*Log[x])/12,x]

[Out]

E^x + (5*x)/4 + 23*x^2 - (29*x^3)/3 + x^4 - (x^2*Log[x])/3

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fricas [A]  time = 0.82, size = 26, normalized size = 0.79 \begin {gather*} x^{4} - \frac {29}{3} \, x^{3} - \frac {1}{3} \, x^{2} \log \relax (x) + 23 \, x^{2} + \frac {5}{4} \, x + e^{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-2/3*x*log(x)+exp(x)+4*x^3-29*x^2+137/3*x+5/4,x, algorithm="fricas")

[Out]

x^4 - 29/3*x^3 - 1/3*x^2*log(x) + 23*x^2 + 5/4*x + e^x

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giac [A]  time = 0.22, size = 26, normalized size = 0.79 \begin {gather*} x^{4} - \frac {29}{3} \, x^{3} - \frac {1}{3} \, x^{2} \log \relax (x) + 23 \, x^{2} + \frac {5}{4} \, x + e^{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-2/3*x*log(x)+exp(x)+4*x^3-29*x^2+137/3*x+5/4,x, algorithm="giac")

[Out]

x^4 - 29/3*x^3 - 1/3*x^2*log(x) + 23*x^2 + 5/4*x + e^x

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maple [A]  time = 0.02, size = 27, normalized size = 0.82

method	result	size
default	\(\frac {5 x}{4}+23 x^{2}-\frac {29 x^{3}}{3}+x^{4}-\frac {x^{2} \ln \relax (x )}{3}+{\mathrm e}^{x}\)	\(27\)
norman	\(\frac {5 x}{4}+23 x^{2}-\frac {29 x^{3}}{3}+x^{4}-\frac {x^{2} \ln \relax (x )}{3}+{\mathrm e}^{x}\)	\(27\)
risch	\(\frac {5 x}{4}+23 x^{2}-\frac {29 x^{3}}{3}+x^{4}-\frac {x^{2} \ln \relax (x )}{3}+{\mathrm e}^{x}\)	\(27\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-2/3*x*ln(x)+exp(x)+4*x^3-29*x^2+137/3*x+5/4,x,method=_RETURNVERBOSE)

[Out]

5/4*x+23*x^2-29/3*x^3+x^4-1/3*x^2*ln(x)+exp(x)

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maxima [A]  time = 0.38, size = 26, normalized size = 0.79 \begin {gather*} x^{4} - \frac {29}{3} \, x^{3} - \frac {1}{3} \, x^{2} \log \relax (x) + 23 \, x^{2} + \frac {5}{4} \, x + e^{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-2/3*x*log(x)+exp(x)+4*x^3-29*x^2+137/3*x+5/4,x, algorithm="maxima")

[Out]

x^4 - 29/3*x^3 - 1/3*x^2*log(x) + 23*x^2 + 5/4*x + e^x

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mupad [B]  time = 5.23, size = 26, normalized size = 0.79 \begin {gather*} \frac {5\,x}{4}+{\mathrm {e}}^x-\frac {x^2\,\ln \relax (x)}{3}+23\,x^2-\frac {29\,x^3}{3}+x^4 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((137*x)/3 + exp(x) - (2*x*log(x))/3 - 29*x^2 + 4*x^3 + 5/4,x)

[Out]

(5*x)/4 + exp(x) - (x^2*log(x))/3 + 23*x^2 - (29*x^3)/3 + x^4

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sympy [A]  time = 0.26, size = 31, normalized size = 0.94 \begin {gather*} x^{4} - \frac {29 x^{3}}{3} - \frac {x^{2} \log {\relax (x )}}{3} + 23 x^{2} + \frac {5 x}{4} + e^{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-2/3*x*ln(x)+exp(x)+4*x**3-29*x**2+137/3*x+5/4,x)

[Out]

x**4 - 29*x**3/3 - x**2*log(x)/3 + 23*x**2 + 5*x/4 + exp(x)

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