\(\int \frac {10800-3258 x^2-e^x x^2+954 x^3-81 x^4}{x^2} \, dx\) [8259]

Optimal result

Integrand size = 29, antiderivative size = 33 \[ \int \frac {10800-3258 x^2-e^x x^2+954 x^3-81 x^4}{x^2} \, dx=-e^x+3 \left (-1+3 \left (1-\frac {3 (-5+x)^2 (-4+x)^2}{x^2}-x\right ) x\right ) \]

[Out]

3*x*(3-3*x-9*(-5+x)^2/x^2*(x-4)^2)-3-exp(x)

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.73, number of steps used = 5, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.069, Rules used = {14, 2225} \[ \int \frac {10800-3258 x^2-e^x x^2+954 x^3-81 x^4}{x^2} \, dx=-27 x^3+477 x^2-3258 x-e^x-\frac {10800}{x} \]

[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 2225

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{align*} \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{align*}

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.73 \[ \int \frac {10800-3258 x^2-e^x x^2+954 x^3-81 x^4}{x^2} \, dx=-e^x-\frac {10800}{x}-3258 x+477 x^2-27 x^3 \]

[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

Maple [A] (verified)

Time = 0.06 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.73

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

[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)

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.79 \[ \int \frac {10800-3258 x^2-e^x x^2+954 x^3-81 x^4}{x^2} \, dx=-\frac {27 \, x^{4} - 477 \, x^{3} + 3258 \, x^{2} + x e^{x} + 10800}{x} \]

[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

Sympy [A] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.58 \[ \int \frac {10800-3258 x^2-e^x x^2+954 x^3-81 x^4}{x^2} \, dx=- 27 x^{3} + 477 x^{2} - 3258 x - e^{x} - \frac {10800}{x} \]

[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

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.70 \[ \int \frac {10800-3258 x^2-e^x x^2+954 x^3-81 x^4}{x^2} \, dx=-27 \, x^{3} + 477 \, x^{2} - 3258 \, x - \frac {10800}{x} - e^{x} \]

[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

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.79 \[ \int \frac {10800-3258 x^2-e^x x^2+954 x^3-81 x^4}{x^2} \, dx=-\frac {27 \, x^{4} - 477 \, x^{3} + 3258 \, x^{2} + x e^{x} + 10800}{x} \]

[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

Mupad [B] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.70 \[ \int \frac {10800-3258 x^2-e^x x^2+954 x^3-81 x^4}{x^2} \, dx=477\,x^2-{\mathrm {e}}^x-\frac {10800}{x}-3258\,x-27\,x^3 \]

[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