\(\int \frac {-192+144 x+104 x^2-32 x^3-2 x^6}{x^5} \, dx\) [6151]

Optimal result

Integrand size = 24, antiderivative size = 29 \[ \int \frac {-192+144 x+104 x^2-32 x^3-2 x^6}{x^5} \, dx=-x^2+\frac {1}{3} \left (1+4 \left (\left (-4-\frac {3 (-2+x)}{x^2}\right )^2+\log (4)\right )\right ) \]

[Out]

1/3+4/3*(-4-3*(-2+x)/x^2)^2+8/3*ln(2)-x^2

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.90, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.042, Rules used = {14} \[ \int \frac {-192+144 x+104 x^2-32 x^3-2 x^6}{x^5} \, dx=\frac {48}{x^4}-\frac {48}{x^3}-x^2-\frac {52}{x^2}+\frac {32}{x} \]

[In]

Int[(-192 + 144*x + 104*x^2 - 32*x^3 - 2*x^6)/x^5,x]

[Out]

48/x^4 - 48/x^3 - 52/x^2 + 32/x - 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 (-\frac {192}{x^5}+\frac {144}{x^4}+\frac {104}{x^3}-\frac {32}{x^2}-2 x\right ) \, dx \\ & = \frac {48}{x^4}-\frac {48}{x^3}-\frac {52}{x^2}+\frac {32}{x}-x^2 \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.03 \[ \int \frac {-192+144 x+104 x^2-32 x^3-2 x^6}{x^5} \, dx=-2 \left (-\frac {24}{x^4}+\frac {24}{x^3}+\frac {26}{x^2}-\frac {16}{x}+\frac {x^2}{2}\right ) \]

[In]

Integrate[(-192 + 144*x + 104*x^2 - 32*x^3 - 2*x^6)/x^5,x]

[Out]

-2*(-24/x^4 + 24/x^3 + 26/x^2 - 16/x + x^2/2)

Maple [A] (verified)

Time = 0.04 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.83

method	result	size
gosper	\(-\frac {x^{6}-32 x^{3}+52 x^{2}+48 x -48}{x^{4}}\)	\(24\)
parallelrisch	\(-\frac {x^{6}-32 x^{3}+52 x^{2}+48 x -48}{x^{4}}\)	\(24\)
norman	\(\frac {-x^{6}+32 x^{3}-52 x^{2}-48 x +48}{x^{4}}\)	\(25\)
risch	\(-x^{2}+\frac {32 x^{3}-52 x^{2}-48 x +48}{x^{4}}\)	\(26\)
default	\(-x^{2}+\frac {48}{x^{4}}-\frac {48}{x^{3}}+\frac {32}{x}-\frac {52}{x^{2}}\)	\(27\)

[In]

int((-2*x^6-32*x^3+104*x^2+144*x-192)/x^5,x,method=_RETURNVERBOSE)

[Out]

-(x^6-32*x^3+52*x^2+48*x-48)/x^4

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.79 \[ \int \frac {-192+144 x+104 x^2-32 x^3-2 x^6}{x^5} \, dx=-\frac {x^{6} - 32 \, x^{3} + 52 \, x^{2} + 48 \, x - 48}{x^{4}} \]

[In]

integrate((-2*x^6-32*x^3+104*x^2+144*x-192)/x^5,x, algorithm="fricas")

[Out]

-(x^6 - 32*x^3 + 52*x^2 + 48*x - 48)/x^4

Sympy [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.76 \[ \int \frac {-192+144 x+104 x^2-32 x^3-2 x^6}{x^5} \, dx=- x^{2} - \frac {- 32 x^{3} + 52 x^{2} + 48 x - 48}{x^{4}} \]

[In]

integrate((-2*x**6-32*x**3+104*x**2+144*x-192)/x**5,x)

[Out]

-x**2 - (-32*x**3 + 52*x**2 + 48*x - 48)/x**4

Maxima [A] (verification not implemented)

none

Time = 0.18 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.90 \[ \int \frac {-192+144 x+104 x^2-32 x^3-2 x^6}{x^5} \, dx=-x^{2} + \frac {4 \, {\left (8 \, x^{3} - 13 \, x^{2} - 12 \, x + 12\right )}}{x^{4}} \]

[In]

integrate((-2*x^6-32*x^3+104*x^2+144*x-192)/x^5,x, algorithm="maxima")

[Out]

-x^2 + 4*(8*x^3 - 13*x^2 - 12*x + 12)/x^4

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.90 \[ \int \frac {-192+144 x+104 x^2-32 x^3-2 x^6}{x^5} \, dx=-x^{2} + \frac {4 \, {\left (8 \, x^{3} - 13 \, x^{2} - 12 \, x + 12\right )}}{x^{4}} \]

[In]

integrate((-2*x^6-32*x^3+104*x^2+144*x-192)/x^5,x, algorithm="giac")

[Out]

-x^2 + 4*(8*x^3 - 13*x^2 - 12*x + 12)/x^4

Mupad [B] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.90 \[ \int \frac {-192+144 x+104 x^2-32 x^3-2 x^6}{x^5} \, dx=-x^2-\frac {-32\,x^3+52\,x^2+48\,x-48}{x^4} \]

[In]

int(-(32*x^3 - 104*x^2 - 144*x + 2*x^6 + 192)/x^5,x)

[Out]

- x^2 - (48*x + 52*x^2 - 32*x^3 - 48)/x^4