3.15.1 \(\int \frac {-2 e^5 x-384 x^2-48 x^3+36 x^2 \log (x)}{3 e^5} \, dx\)

Optimal. Leaf size=26 \[ 3-x \left (\frac {x}{3}+\frac {4 x^2 (11+x-\log (x))}{e^5}\right ) \]

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Rubi [A]  time = 0.02, antiderivative size = 34, normalized size of antiderivative = 1.31, number of steps used = 3, number of rules used = 2, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.065, Rules used = {12, 2304} \begin {gather*} -\frac {4 x^4}{e^5}-\frac {44 x^3}{e^5}+\frac {4 x^3 \log (x)}{e^5}-\frac {x^2}{3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-2*E^5*x - 384*x^2 - 48*x^3 + 36*x^2*Log[x])/(3*E^5),x]

[Out]

-1/3*x^2 - (44*x^3)/E^5 - (4*x^4)/E^5 + (4*x^3*Log[x])/E^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 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 {\int \left (-2 e^5 x-384 x^2-48 x^3+36 x^2 \log (x)\right ) \, dx}{3 e^5}\\ &=-\frac {x^2}{3}-\frac {128 x^3}{3 e^5}-\frac {4 x^4}{e^5}+\frac {12 \int x^2 \log (x) \, dx}{e^5}\\ &=-\frac {x^2}{3}-\frac {44 x^3}{e^5}-\frac {4 x^4}{e^5}+\frac {4 x^3 \log (x)}{e^5}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.01, size = 35, normalized size = 1.35 \begin {gather*} -\frac {2 \left (\frac {e^5 x^2}{2}+66 x^3+6 x^4-6 x^3 \log (x)\right )}{3 e^5} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-2*E^5*x - 384*x^2 - 48*x^3 + 36*x^2*Log[x])/(3*E^5),x]

[Out]

(-2*((E^5*x^2)/2 + 66*x^3 + 6*x^4 - 6*x^3*Log[x]))/(3*E^5)

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fricas [A]  time = 0.60, size = 28, normalized size = 1.08 \begin {gather*} -\frac {1}{3} \, {\left (12 \, x^{4} - 12 \, x^{3} \log \relax (x) + 132 \, x^{3} + x^{2} e^{5}\right )} e^{\left (-5\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/3*(36*x^2*log(x)-2*x*exp(5)-48*x^3-384*x^2)/exp(5),x, algorithm="fricas")

[Out]

-1/3*(12*x^4 - 12*x^3*log(x) + 132*x^3 + x^2*e^5)*e^(-5)

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giac [A]  time = 0.35, size = 28, normalized size = 1.08 \begin {gather*} -\frac {1}{3} \, {\left (12 \, x^{4} - 12 \, x^{3} \log \relax (x) + 132 \, x^{3} + x^{2} e^{5}\right )} e^{\left (-5\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/3*(36*x^2*log(x)-2*x*exp(5)-48*x^3-384*x^2)/exp(5),x, algorithm="giac")

[Out]

-1/3*(12*x^4 - 12*x^3*log(x) + 132*x^3 + x^2*e^5)*e^(-5)

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maple [A]  time = 0.04, size = 30, normalized size = 1.15

method	result	size
risch	\(-\frac {x^{2}}{3}-44 \,{\mathrm e}^{-5} x^{3}-4 \,{\mathrm e}^{-5} x^{4}+4 \,{\mathrm e}^{-5} x^{3} \ln \relax (x )\)	\(30\)
default	\(\frac {2 \,{\mathrm e}^{-5} \left (-6 x^{4}-66 x^{3}-\frac {x^{2} {\mathrm e}^{5}}{2}+6 x^{3} \ln \relax (x )\right )}{3}\)	\(32\)
norman	\(-\frac {x^{2}}{3}-44 \,{\mathrm e}^{-5} x^{3}-4 \,{\mathrm e}^{-5} x^{4}+4 \,{\mathrm e}^{-5} x^{3} \ln \relax (x )\)	\(36\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/3*(36*x^2*ln(x)-2*x*exp(5)-48*x^3-384*x^2)/exp(5),x,method=_RETURNVERBOSE)

[Out]

-1/3*x^2-44*exp(-5)*x^3-4*exp(-5)*x^4+4*exp(-5)*x^3*ln(x)

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maxima [A]  time = 0.41, size = 28, normalized size = 1.08 \begin {gather*} -\frac {1}{3} \, {\left (12 \, x^{4} - 12 \, x^{3} \log \relax (x) + 132 \, x^{3} + x^{2} e^{5}\right )} e^{\left (-5\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/3*(36*x^2*log(x)-2*x*exp(5)-48*x^3-384*x^2)/exp(5),x, algorithm="maxima")

[Out]

-1/3*(12*x^4 - 12*x^3*log(x) + 132*x^3 + x^2*e^5)*e^(-5)

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mupad [B]  time = 0.98, size = 23, normalized size = 0.88 \begin {gather*} -\frac {x^2\,{\mathrm {e}}^{-5}\,\left (132\,x+{\mathrm {e}}^5-12\,x\,\ln \relax (x)+12\,x^2\right )}{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-exp(-5)*((2*x*exp(5))/3 - 12*x^2*log(x) + 128*x^2 + 16*x^3),x)

[Out]

-(x^2*exp(-5)*(132*x + exp(5) - 12*x*log(x) + 12*x^2))/3

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sympy [A]  time = 0.13, size = 32, normalized size = 1.23 \begin {gather*} - \frac {4 x^{4}}{e^{5}} + \frac {4 x^{3} \log {\relax (x )}}{e^{5}} - \frac {44 x^{3}}{e^{5}} - \frac {x^{2}}{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/3*(36*x**2*ln(x)-2*x*exp(5)-48*x**3-384*x**2)/exp(5),x)

[Out]

-4*x**4*exp(-5) + 4*x**3*exp(-5)*log(x) - 44*x**3*exp(-5) - x**2/3

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