\(\int \frac {4+2 x^3-4 x^6+e^5 (-4 x^2+2 x^5)}{x^5} \, dx\) [7951]

Optimal result

Integrand size = 31, antiderivative size = 22 \[ \int \frac {4+2 x^3-4 x^6+e^5 \left (-4 x^2+2 x^5\right )}{x^5} \, dx=-\left (e^5-\frac {1}{x^2}-x\right )^2-x^2 \]

[Out]

-x^2-(exp(5)-1/x^2-x)^2

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.36, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.032, Rules used = {14} \[ \int \frac {4+2 x^3-4 x^6+e^5 \left (-4 x^2+2 x^5\right )}{x^5} \, dx=-\frac {1}{x^4}-2 x^2+\frac {2 e^5}{x^2}+2 e^5 x-\frac {2}{x} \]

[In]

Int[(4 + 2*x^3 - 4*x^6 + E^5*(-4*x^2 + 2*x^5))/x^5,x]

[Out]

-x^(-4) + (2*E^5)/x^2 - 2/x + 2*E^5*x - 2*x^2

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}& = \int \left (2 e^5+\frac {4}{x^5}-\frac {4 e^5}{x^3}+\frac {2}{x^2}-4 x\right ) \, dx \\ & = -\frac {1}{x^4}+\frac {2 e^5}{x^2}-\frac {2}{x}+2 e^5 x-2 x^2 \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.45 \[ \int \frac {4+2 x^3-4 x^6+e^5 \left (-4 x^2+2 x^5\right )}{x^5} \, dx=2 \left (-\frac {1}{2 x^4}+\frac {e^5}{x^2}-\frac {1}{x}+e^5 x-x^2\right ) \]

[In]

Integrate[(4 + 2*x^3 - 4*x^6 + E^5*(-4*x^2 + 2*x^5))/x^5,x]

[Out]

2*(-1/2*1/x^4 + E^5/x^2 - x^(-1) + E^5*x - x^2)

Maple [A] (verified)

Time = 0.06 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.32

method	result	size
default	\(2 x \,{\mathrm e}^{5}-2 x^{2}-\frac {2}{x}+\frac {2 \,{\mathrm e}^{5}}{x^{2}}-\frac {1}{x^{4}}\)	\(29\)
risch	\(2 x \,{\mathrm e}^{5}-2 x^{2}+\frac {2 x^{2} {\mathrm e}^{5}-2 x^{3}-1}{x^{4}}\)	\(30\)
gosper	\(\frac {2 x^{5} {\mathrm e}^{5}-2 x^{6}+2 x^{2} {\mathrm e}^{5}-2 x^{3}-1}{x^{4}}\)	\(31\)
norman	\(\frac {2 x^{5} {\mathrm e}^{5}-2 x^{6}+2 x^{2} {\mathrm e}^{5}-2 x^{3}-1}{x^{4}}\)	\(31\)
parallelrisch	\(\frac {2 x^{5} {\mathrm e}^{5}-2 x^{6}+2 x^{2} {\mathrm e}^{5}-2 x^{3}-1}{x^{4}}\)	\(31\)

[In]

int(((2*x^5-4*x^2)*exp(5)-4*x^6+2*x^3+4)/x^5,x,method=_RETURNVERBOSE)

[Out]

2*x*exp(5)-2*x^2-2/x+2*exp(5)/x^2-1/x^4

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.27 \[ \int \frac {4+2 x^3-4 x^6+e^5 \left (-4 x^2+2 x^5\right )}{x^5} \, dx=-\frac {2 \, x^{6} + 2 \, x^{3} - 2 \, {\left (x^{5} + x^{2}\right )} e^{5} + 1}{x^{4}} \]

[In]

integrate(((2*x^5-4*x^2)*exp(5)-4*x^6+2*x^3+4)/x^5,x, algorithm="fricas")

[Out]

-(2*x^6 + 2*x^3 - 2*(x^5 + x^2)*e^5 + 1)/x^4

Sympy [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.32 \[ \int \frac {4+2 x^3-4 x^6+e^5 \left (-4 x^2+2 x^5\right )}{x^5} \, dx=- 2 x^{2} + 2 x e^{5} - \frac {2 x^{3} - 2 x^{2} e^{5} + 1}{x^{4}} \]

[In]

integrate(((2*x**5-4*x**2)*exp(5)-4*x**6+2*x**3+4)/x**5,x)

[Out]

-2*x**2 + 2*x*exp(5) - (2*x**3 - 2*x**2*exp(5) + 1)/x**4

Maxima [A] (verification not implemented)

none

Time = 0.18 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.36 \[ \int \frac {4+2 x^3-4 x^6+e^5 \left (-4 x^2+2 x^5\right )}{x^5} \, dx=-2 \, x^{2} + 2 \, x e^{5} - \frac {2 \, x^{3} - 2 \, x^{2} e^{5} + 1}{x^{4}} \]

[In]

integrate(((2*x^5-4*x^2)*exp(5)-4*x^6+2*x^3+4)/x^5,x, algorithm="maxima")

[Out]

-2*x^2 + 2*x*e^5 - (2*x^3 - 2*x^2*e^5 + 1)/x^4

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.36 \[ \int \frac {4+2 x^3-4 x^6+e^5 \left (-4 x^2+2 x^5\right )}{x^5} \, dx=-2 \, x^{2} + 2 \, x e^{5} - \frac {2 \, x^{3} - 2 \, x^{2} e^{5} + 1}{x^{4}} \]

[In]

integrate(((2*x^5-4*x^2)*exp(5)-4*x^6+2*x^3+4)/x^5,x, algorithm="giac")

[Out]

-2*x^2 + 2*x*e^5 - (2*x^3 - 2*x^2*e^5 + 1)/x^4

Mupad [B] (verification not implemented)

Time = 11.87 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.36 \[ \int \frac {4+2 x^3-4 x^6+e^5 \left (-4 x^2+2 x^5\right )}{x^5} \, dx=2\,x\,{\mathrm {e}}^5-\frac {2\,x^3-2\,{\mathrm {e}}^5\,x^2+1}{x^4}-2\,x^2 \]

[In]

int(-(exp(5)*(4*x^2 - 2*x^5) - 2*x^3 + 4*x^6 - 4)/x^5,x)

[Out]

2*x*exp(5) - (2*x^3 - 2*x^2*exp(5) + 1)/x^4 - 2*x^2