3.83.59 \(\int \frac {10800-3258 x^2-e^x x^2+954 x^3-81 x^4}{x^2} \, dx\)

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

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Rubi [A]  time = 0.02, antiderivative size = 24, normalized size of antiderivative = 0.73, 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*} -27 x^3+477 x^2-3258 x-e^x-\frac {10800}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(10800 - 3258*x^2 - E^x*x^2 + 954*x^3 - 81*x^4)/x^2,x]

[Out]

-E^x - 10800/x - 3258*x + 477*x^2 - 27*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 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 (-e^x-\frac {9 \left (-1200+362 x^2-106 x^3+9 x^4\right )}{x^2}\right ) \, dx\\ &=-\left (9 \int \frac {-1200+362 x^2-106 x^3+9 x^4}{x^2} \, dx\right )-\int e^x \, dx\\ &=-e^x-9 \int \left (362-\frac {1200}{x^2}-106 x+9 x^2\right ) \, dx\\ &=-e^x-\frac {10800}{x}-3258 x+477 x^2-27 x^3\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.02, size = 24, normalized size = 0.73 \begin {gather*} -e^x-\frac {10800}{x}-3258 x+477 x^2-27 x^3 \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(10800 - 3258*x^2 - E^x*x^2 + 954*x^3 - 81*x^4)/x^2,x]

[Out]

-E^x - 10800/x - 3258*x + 477*x^2 - 27*x^3

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fricas [A]  time = 1.08, size = 26, normalized size = 0.79 \begin {gather*} -\frac {27 \, x^{4} - 477 \, x^{3} + 3258 \, x^{2} + x e^{x} + 10800}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-exp(x)*x^2-81*x^4+954*x^3-3258*x^2+10800)/x^2,x, algorithm="fricas")

[Out]

-(27*x^4 - 477*x^3 + 3258*x^2 + x*e^x + 10800)/x

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giac [A]  time = 1.98, size = 26, normalized size = 0.79 \begin {gather*} -\frac {27 \, x^{4} - 477 \, x^{3} + 3258 \, x^{2} + x e^{x} + 10800}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-exp(x)*x^2-81*x^4+954*x^3-3258*x^2+10800)/x^2,x, algorithm="giac")

[Out]

-(27*x^4 - 477*x^3 + 3258*x^2 + x*e^x + 10800)/x

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maple [A]  time = 0.02, size = 24, normalized size = 0.73

method	result	size
default	\(477 x^{2}-3258 x -\frac {10800}{x}-27 x^{3}-{\mathrm e}^{x}\)	\(24\)
risch	\(477 x^{2}-3258 x -\frac {10800}{x}-27 x^{3}-{\mathrm e}^{x}\)	\(24\)
norman	\(\frac {-10800-3258 x^{2}+477 x^{3}-27 x^{4}-{\mathrm e}^{x} x}{x}\)	\(27\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-exp(x)*x^2-81*x^4+954*x^3-3258*x^2+10800)/x^2,x,method=_RETURNVERBOSE)

[Out]

477*x^2-3258*x-10800/x-27*x^3-exp(x)

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maxima [A]  time = 0.50, size = 23, normalized size = 0.70 \begin {gather*} -27 \, x^{3} + 477 \, x^{2} - 3258 \, x - \frac {10800}{x} - e^{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-exp(x)*x^2-81*x^4+954*x^3-3258*x^2+10800)/x^2,x, algorithm="maxima")

[Out]

-27*x^3 + 477*x^2 - 3258*x - 10800/x - e^x

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mupad [B]  time = 0.06, size = 23, normalized size = 0.70 \begin {gather*} 477\,x^2-{\mathrm {e}}^x-\frac {10800}{x}-3258\,x-27\,x^3 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(x^2*exp(x) + 3258*x^2 - 954*x^3 + 81*x^4 - 10800)/x^2,x)

[Out]

477*x^2 - exp(x) - 10800/x - 3258*x - 27*x^3

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sympy [A]  time = 0.11, size = 19, normalized size = 0.58 \begin {gather*} - 27 x^{3} + 477 x^{2} - 3258 x - e^{x} - \frac {10800}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-exp(x)*x**2-81*x**4+954*x**3-3258*x**2+10800)/x**2,x)

[Out]

-27*x**3 + 477*x**2 - 3258*x - exp(x) - 10800/x

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