\(\int \frac {-90+49 x-4 x^3}{9 e x^3} \, dx\) [6795]

Optimal result

Integrand size = 20, antiderivative size = 31 \[ \int \frac {-90+49 x-4 x^3}{9 e x^3} \, dx=1+\frac {-\frac {1}{9} (7-2 x)^2+\frac {5}{x}+\frac {x}{3}}{e x} \]

[Out]

1+(1/3*x-1/3*(-2*x+7)*(-2/3*x+7/3)+5/x)/x/exp(1)

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.87, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {12, 14} \[ \int \frac {-90+49 x-4 x^3}{9 e x^3} \, dx=\frac {5}{e x^2}-\frac {4 x}{9 e}-\frac {49}{9 e x} \]

[In]

Int[(-90 + 49*x - 4*x^3)/(9*E*x^3),x]

[Out]

5/(E*x^2) - 49/(9*E*x) - (4*x)/(9*E)

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, 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]]

Rubi steps \begin{align*} \text {integral}& = \frac {\int \frac {-90+49 x-4 x^3}{x^3} \, dx}{9 e} \\ & = \frac {\int \left (-4-\frac {90}{x^3}+\frac {49}{x^2}\right ) \, dx}{9 e} \\ & = \frac {5}{e x^2}-\frac {49}{9 e x}-\frac {4 x}{9 e} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.68 \[ \int \frac {-90+49 x-4 x^3}{9 e x^3} \, dx=-\frac {-\frac {45}{x^2}+\frac {49}{x}+4 x}{9 e} \]

[In]

Integrate[(-90 + 49*x - 4*x^3)/(9*E*x^3),x]

[Out]

-1/9*(-45/x^2 + 49/x + 4*x)/E

Maple [A] (verified)

Time = 0.12 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.61

method	result	size
risch	\(-\frac {4 \,{\mathrm e}^{-1} x}{9}+\frac {{\mathrm e}^{-1} \left (-49 x +45\right )}{9 x^{2}}\)	\(19\)
gosper	\(-\frac {{\mathrm e}^{-1} \left (4 x^{3}+49 x -45\right )}{9 x^{2}}\)	\(20\)
parallelrisch	\(-\frac {{\mathrm e}^{-1} \left (4 x^{3}+49 x -45\right )}{9 x^{2}}\)	\(20\)
default	\(\frac {{\mathrm e}^{-1} \left (-4 x -\frac {49}{x}+\frac {45}{x^{2}}\right )}{9}\)	\(21\)
norman	\(\frac {5 \,{\mathrm e}^{-1}-\frac {49 \,{\mathrm e}^{-1} x}{9}-\frac {4 \,{\mathrm e}^{-1} x^{3}}{9}}{x^{2}}\)	\(28\)

[In]

int(1/9*(-4*x^3+49*x-90)/x^3/exp(1),x,method=_RETURNVERBOSE)

[Out]

-4/9*exp(-1)*x+1/9*exp(-1)*(-49*x+45)/x^2

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.55 \[ \int \frac {-90+49 x-4 x^3}{9 e x^3} \, dx=-\frac {{\left (4 \, x^{3} + 49 \, x - 45\right )} e^{\left (-1\right )}}{9 \, x^{2}} \]

[In]

integrate(1/9*(-4*x^3+49*x-90)/x^3/exp(1),x, algorithm="fricas")

[Out]

-1/9*(4*x^3 + 49*x - 45)*e^(-1)/x^2

Sympy [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.55 \[ \int \frac {-90+49 x-4 x^3}{9 e x^3} \, dx=\frac {- 4 x - \frac {49 x - 45}{x^{2}}}{9 e} \]

[In]

integrate(1/9*(-4*x**3+49*x-90)/x**3/exp(1),x)

[Out]

(-4*x - (49*x - 45)/x**2)*exp(-1)/9

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.55 \[ \int \frac {-90+49 x-4 x^3}{9 e x^3} \, dx=-\frac {1}{9} \, {\left (4 \, x + \frac {49 \, x - 45}{x^{2}}\right )} e^{\left (-1\right )} \]

[In]

integrate(1/9*(-4*x^3+49*x-90)/x^3/exp(1),x, algorithm="maxima")

[Out]

-1/9*(4*x + (49*x - 45)/x^2)*e^(-1)

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.55 \[ \int \frac {-90+49 x-4 x^3}{9 e x^3} \, dx=-\frac {1}{9} \, {\left (4 \, x + \frac {49 \, x - 45}{x^{2}}\right )} e^{\left (-1\right )} \]

[In]

integrate(1/9*(-4*x^3+49*x-90)/x^3/exp(1),x, algorithm="giac")

[Out]

-1/9*(4*x + (49*x - 45)/x^2)*e^(-1)

Mupad [B] (verification not implemented)

Time = 12.50 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.55 \[ \int \frac {-90+49 x-4 x^3}{9 e x^3} \, dx=-\frac {{\mathrm {e}}^{-1}\,\left (4\,x^3+49\,x-45\right )}{9\,x^2} \]

[In]

int(-(exp(-1)*((4*x^3)/9 - (49*x)/9 + 10))/x^3,x)

[Out]

-(exp(-1)*(49*x + 4*x^3 - 45))/(9*x^2)