3.13.94 \(\int \frac {-3-6 x-108 x^2-27 x^4+e^x (-27 x^2+27 x^3)+9 \log (x)}{x^4} \, dx\)

Optimal. Leaf size=28 \[ \frac {3 \left (9 \left (4+e^x-x^2\right )+\frac {x-\log (x)}{x^2}\right )}{x} \]

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Rubi [A]  time = 0.07, antiderivative size = 29, normalized size of antiderivative = 1.04, number of steps used = 8, number of rules used = 3, integrand size = 38, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.079, Rules used = {14, 2197, 2304} \begin {gather*} -\frac {3 \log (x)}{x^3}+\frac {3}{x^2}-27 x+\frac {27 e^x}{x}+\frac {108}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-3 - 6*x - 108*x^2 - 27*x^4 + E^x*(-27*x^2 + 27*x^3) + 9*Log[x])/x^4,x]

[Out]

3/x^2 + 108/x + (27*E^x)/x - 27*x - (3*Log[x])/x^3

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 2197

Int[(F_)^((c_.)*(v_))*(u_)^(m_.)*(w_), x_Symbol] :> With[{b = Coefficient[v, x, 1], d = Coefficient[u, x, 0],
e = Coefficient[u, x, 1], f = Coefficient[w, x, 0], g = Coefficient[w, x, 1]}, Simp[(g*u^(m + 1)*F^(c*v))/(b*c
*e*Log[F]), x] /; EqQ[e*g*(m + 1) - b*c*(e*f - d*g)*Log[F], 0]] /; FreeQ[{F, c, m}, x] && LinearQ[{u, v, w}, 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} &=\int \left (\frac {27 e^x (-1+x)}{x^2}-\frac {3 \left (1+2 x+36 x^2+9 x^4-3 \log (x)\right )}{x^4}\right ) \, dx\\ &=-\left (3 \int \frac {1+2 x+36 x^2+9 x^4-3 \log (x)}{x^4} \, dx\right )+27 \int \frac {e^x (-1+x)}{x^2} \, dx\\ &=\frac {27 e^x}{x}-3 \int \left (\frac {1+2 x+36 x^2+9 x^4}{x^4}-\frac {3 \log (x)}{x^4}\right ) \, dx\\ &=\frac {27 e^x}{x}-3 \int \frac {1+2 x+36 x^2+9 x^4}{x^4} \, dx+9 \int \frac {\log (x)}{x^4} \, dx\\ &=-\frac {1}{x^3}+\frac {27 e^x}{x}-\frac {3 \log (x)}{x^3}-3 \int \left (9+\frac {1}{x^4}+\frac {2}{x^3}+\frac {36}{x^2}\right ) \, dx\\ &=\frac {3}{x^2}+\frac {108}{x}+\frac {27 e^x}{x}-27 x-\frac {3 \log (x)}{x^3}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.04, size = 25, normalized size = 0.89 \begin {gather*} -\frac {3 \left (x \left (-1-9 \left (4+e^x\right ) x+9 x^3\right )+\log (x)\right )}{x^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-3 - 6*x - 108*x^2 - 27*x^4 + E^x*(-27*x^2 + 27*x^3) + 9*Log[x])/x^4,x]

[Out]

(-3*(x*(-1 - 9*(4 + E^x)*x + 9*x^3) + Log[x]))/x^3

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fricas [A]  time = 0.67, size = 28, normalized size = 1.00 \begin {gather*} -\frac {3 \, {\left (9 \, x^{4} - 9 \, x^{2} e^{x} - 36 \, x^{2} - x + \log \relax (x)\right )}}{x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((9*log(x)+(27*x^3-27*x^2)*exp(x)-27*x^4-108*x^2-6*x-3)/x^4,x, algorithm="fricas")

[Out]

-3*(9*x^4 - 9*x^2*e^x - 36*x^2 - x + log(x))/x^3

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giac [A]  time = 0.32, size = 28, normalized size = 1.00 \begin {gather*} -\frac {3 \, {\left (9 \, x^{4} - 9 \, x^{2} e^{x} - 36 \, x^{2} - x + \log \relax (x)\right )}}{x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((9*log(x)+(27*x^3-27*x^2)*exp(x)-27*x^4-108*x^2-6*x-3)/x^4,x, algorithm="giac")

[Out]

-3*(9*x^4 - 9*x^2*e^x - 36*x^2 - x + log(x))/x^3

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maple [A]  time = 0.03, size = 29, normalized size = 1.04

method	result	size
default	\(-\frac {3 \ln \relax (x )}{x^{3}}-27 x +\frac {108}{x}+\frac {3}{x^{2}}+\frac {27 \,{\mathrm e}^{x}}{x}\)	\(29\)
risch	\(-\frac {3 \ln \relax (x )}{x^{3}}-\frac {3 \left (9 x^{3}-9 \,{\mathrm e}^{x} x -36 x -1\right )}{x^{2}}\)	\(29\)
norman	\(\frac {3 x +108 x^{2}-27 x^{4}+27 \,{\mathrm e}^{x} x^{2}-3 \ln \relax (x )}{x^{3}}\)	\(30\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((9*ln(x)+(27*x^3-27*x^2)*exp(x)-27*x^4-108*x^2-6*x-3)/x^4,x,method=_RETURNVERBOSE)

[Out]

-3*ln(x)/x^3-27*x+108/x+3/x^2+27*exp(x)/x

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maxima [C]  time = 0.44, size = 32, normalized size = 1.14 \begin {gather*} -27 \, x + \frac {108}{x} + \frac {3}{x^{2}} - \frac {3 \, \log \relax (x)}{x^{3}} + 27 \, {\rm Ei}\relax (x) - 27 \, \Gamma \left (-1, -x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((9*log(x)+(27*x^3-27*x^2)*exp(x)-27*x^4-108*x^2-6*x-3)/x^4,x, algorithm="maxima")

[Out]

-27*x + 108/x + 3/x^2 - 3*log(x)/x^3 + 27*Ei(x) - 27*gamma(-1, -x)

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mupad [B]  time = 0.95, size = 26, normalized size = 0.93 \begin {gather*} \frac {3\,x-3\,\ln \relax (x)+x^2\,\left (27\,{\mathrm {e}}^x+108\right )}{x^3}-27\,x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(6*x - 9*log(x) + exp(x)*(27*x^2 - 27*x^3) + 108*x^2 + 27*x^4 + 3)/x^4,x)

[Out]

(3*x - 3*log(x) + x^2*(27*exp(x) + 108))/x^3 - 27*x

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sympy [A]  time = 0.29, size = 27, normalized size = 0.96 \begin {gather*} - 27 x + \frac {27 e^{x}}{x} - \frac {- 108 x - 3}{x^{2}} - \frac {3 \log {\relax (x )}}{x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((9*ln(x)+(27*x**3-27*x**2)*exp(x)-27*x**4-108*x**2-6*x-3)/x**4,x)

[Out]

-27*x + 27*exp(x)/x - (-108*x - 3)/x**2 - 3*log(x)/x**3

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