3.6.65 \(\int \frac {-x^3-6 e^{8+x} x^3+e^8 (432+72 x)}{x^3} \, dx\)

Optimal. Leaf size=26 \[ -4+e^2-x-6 e^8 \left (e^x+\frac {(6+x)^2}{x^2}\right ) \]

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Rubi [A]  time = 0.03, antiderivative size = 27, normalized size of antiderivative = 1.04, number of steps used = 5, number of rules used = 2, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.069, Rules used = {14, 2194} \begin {gather*} -\frac {216 e^8}{x^2}-x-6 e^{x+8}-\frac {72 e^8}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-x^3 - 6*E^(8 + x)*x^3 + E^8*(432 + 72*x))/x^3,x]

[Out]

-6*E^(8 + x) - (216*E^8)/x^2 - (72*E^8)/x - 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 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]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (-6 e^{8+x}+\frac {432 e^8+72 e^8 x-x^3}{x^3}\right ) \, dx\\ &=-\left (6 \int e^{8+x} \, dx\right )+\int \frac {432 e^8+72 e^8 x-x^3}{x^3} \, dx\\ &=-6 e^{8+x}+\int \left (-1+\frac {432 e^8}{x^3}+\frac {72 e^8}{x^2}\right ) \, dx\\ &=-6 e^{8+x}-\frac {216 e^8}{x^2}-\frac {72 e^8}{x}-x\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.01, size = 27, normalized size = 1.04 \begin {gather*} -6 e^{8+x}-\frac {216 e^8}{x^2}-\frac {72 e^8}{x}-x \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-x^3 - 6*E^(8 + x)*x^3 + E^8*(432 + 72*x))/x^3,x]

[Out]

-6*E^(8 + x) - (216*E^8)/x^2 - (72*E^8)/x - x

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fricas [A]  time = 0.88, size = 25, normalized size = 0.96 \begin {gather*} -\frac {x^{3} + 6 \, x^{2} e^{\left (x + 8\right )} + 72 \, {\left (x + 3\right )} e^{8}}{x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-6*x^3*exp(8)*exp(x)+(72*x+432)*exp(8)-x^3)/x^3,x, algorithm="fricas")

[Out]

-(x^3 + 6*x^2*e^(x + 8) + 72*(x + 3)*e^8)/x^2

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giac [A]  time = 0.29, size = 27, normalized size = 1.04 \begin {gather*} -\frac {x^{3} + 6 \, x^{2} e^{\left (x + 8\right )} + 72 \, x e^{8} + 216 \, e^{8}}{x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-6*x^3*exp(8)*exp(x)+(72*x+432)*exp(8)-x^3)/x^3,x, algorithm="giac")

[Out]

-(x^3 + 6*x^2*e^(x + 8) + 72*x*e^8 + 216*e^8)/x^2

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maple [A]  time = 0.03, size = 25, normalized size = 0.96

method	result	size
default	\(-x -\frac {216 \,{\mathrm e}^{8}}{x^{2}}-\frac {72 \,{\mathrm e}^{8}}{x}-6 \,{\mathrm e}^{8} {\mathrm e}^{x}\)	\(25\)
risch	\(-x +\frac {-72 x \,{\mathrm e}^{8}-216 \,{\mathrm e}^{8}}{x^{2}}-6 \,{\mathrm e}^{x +8}\)	\(25\)
norman	\(\frac {-x^{3}-72 x \,{\mathrm e}^{8}-6 x^{2} {\mathrm e}^{8} {\mathrm e}^{x}-216 \,{\mathrm e}^{8}}{x^{2}}\)	\(29\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-6*x^3*exp(8)*exp(x)+(72*x+432)*exp(8)-x^3)/x^3,x,method=_RETURNVERBOSE)

[Out]

-x-216/x^2*exp(8)-72*exp(8)/x-6*exp(8)*exp(x)

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maxima [A]  time = 0.77, size = 24, normalized size = 0.92 \begin {gather*} -x - \frac {72 \, e^{8}}{x} - \frac {216 \, e^{8}}{x^{2}} - 6 \, e^{\left (x + 8\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-6*x^3*exp(8)*exp(x)+(72*x+432)*exp(8)-x^3)/x^3,x, algorithm="maxima")

[Out]

-x - 72*e^8/x - 216*e^8/x^2 - 6*e^(x + 8)

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mupad [B]  time = 0.08, size = 25, normalized size = 0.96 \begin {gather*} -x-6\,{\mathrm {e}}^{x+8}-\frac {216\,{\mathrm {e}}^8+72\,x\,{\mathrm {e}}^8}{x^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(x^3 - exp(8)*(72*x + 432) + 6*x^3*exp(8)*exp(x))/x^3,x)

[Out]

- x - 6*exp(x + 8) - (216*exp(8) + 72*x*exp(8))/x^2

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sympy [A]  time = 0.15, size = 26, normalized size = 1.00 \begin {gather*} - x - 6 e^{8} e^{x} - \frac {72 x e^{8} + 216 e^{8}}{x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-6*x**3*exp(8)*exp(x)+(72*x+432)*exp(8)-x**3)/x**3,x)

[Out]

-x - 6*exp(8)*exp(x) - (72*x*exp(8) + 216*exp(8))/x**2

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