\(\int \frac {-45000-2400 x^3+3528 x^4-2016 x^5+256 x^6}{625 x^3} \, dx\) [1024]

Optimal result

Integrand size = 29, antiderivative size = 25 \[ \int \frac {-45000-2400 x^3+3528 x^4-2016 x^5+256 x^6}{625 x^3} \, dx=4 \left (3+\left (\frac {3}{x}+x-\frac {4}{25} \left (x+x^2\right )\right )^2\right )+\log (2) \]

[Out]

4*(3/x+21/25*x-4/25*x^2)^2+12+ln(2)

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.28, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.069, Rules used = {12, 14} \[ \int \frac {-45000-2400 x^3+3528 x^4-2016 x^5+256 x^6}{625 x^3} \, dx=\frac {64 x^4}{625}-\frac {672 x^3}{625}+\frac {1764 x^2}{625}+\frac {36}{x^2}-\frac {96 x}{25} \]

[In]

Int[(-45000 - 2400*x^3 + 3528*x^4 - 2016*x^5 + 256*x^6)/(625*x^3),x]

[Out]

36/x^2 - (96*x)/25 + (1764*x^2)/625 - (672*x^3)/625 + (64*x^4)/625

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 {1}{625} \int \frac {-45000-2400 x^3+3528 x^4-2016 x^5+256 x^6}{x^3} \, dx \\ & = \frac {1}{625} \int \left (-2400-\frac {45000}{x^3}+3528 x-2016 x^2+256 x^3\right ) \, dx \\ & = \frac {36}{x^2}-\frac {96 x}{25}+\frac {1764 x^2}{625}-\frac {672 x^3}{625}+\frac {64 x^4}{625} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.28 \[ \int \frac {-45000-2400 x^3+3528 x^4-2016 x^5+256 x^6}{625 x^3} \, dx=\frac {8}{625} \left (\frac {5625}{2 x^2}-300 x+\frac {441 x^2}{2}-84 x^3+8 x^4\right ) \]

[In]

Integrate[(-45000 - 2400*x^3 + 3528*x^4 - 2016*x^5 + 256*x^6)/(625*x^3),x]

[Out]

(8*(5625/(2*x^2) - 300*x + (441*x^2)/2 - 84*x^3 + 8*x^4))/625

Maple [A] (verified)

Time = 0.02 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00

method	result	size
default	\(\frac {64 x^{4}}{625}-\frac {672 x^{3}}{625}+\frac {1764 x^{2}}{625}-\frac {96 x}{25}+\frac {36}{x^{2}}\)	\(25\)
risch	\(\frac {64 x^{4}}{625}-\frac {672 x^{3}}{625}+\frac {1764 x^{2}}{625}-\frac {96 x}{25}+\frac {36}{x^{2}}\)	\(25\)
norman	\(\frac {36-\frac {96}{25} x^{3}+\frac {1764}{625} x^{4}-\frac {672}{625} x^{5}+\frac {64}{625} x^{6}}{x^{2}}\)	\(27\)
gosper	\(\frac {36-\frac {96}{25} x^{3}+\frac {1764}{625} x^{4}-\frac {672}{625} x^{5}+\frac {64}{625} x^{6}}{x^{2}}\)	\(28\)
parallelrisch	\(\frac {64 x^{6}-672 x^{5}+1764 x^{4}-2400 x^{3}+22500}{625 x^{2}}\)	\(28\)

[In]

int(1/625*(256*x^6-2016*x^5+3528*x^4-2400*x^3-45000)/x^3,x,method=_RETURNVERBOSE)

[Out]

64/625*x^4-672/625*x^3+1764/625*x^2-96/25*x+36/x^2

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.08 \[ \int \frac {-45000-2400 x^3+3528 x^4-2016 x^5+256 x^6}{625 x^3} \, dx=\frac {4 \, {\left (16 \, x^{6} - 168 \, x^{5} + 441 \, x^{4} - 600 \, x^{3} + 5625\right )}}{625 \, x^{2}} \]

[In]

integrate(1/625*(256*x^6-2016*x^5+3528*x^4-2400*x^3-45000)/x^3,x, algorithm="fricas")

[Out]

4/625*(16*x^6 - 168*x^5 + 441*x^4 - 600*x^3 + 5625)/x^2

Sympy [A] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.16 \[ \int \frac {-45000-2400 x^3+3528 x^4-2016 x^5+256 x^6}{625 x^3} \, dx=\frac {64 x^{4}}{625} - \frac {672 x^{3}}{625} + \frac {1764 x^{2}}{625} - \frac {96 x}{25} + \frac {36}{x^{2}} \]

[In]

integrate(1/625*(256*x**6-2016*x**5+3528*x**4-2400*x**3-45000)/x**3,x)

[Out]

64*x**4/625 - 672*x**3/625 + 1764*x**2/625 - 96*x/25 + 36/x**2

Maxima [A] (verification not implemented)

none

Time = 0.18 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.96 \[ \int \frac {-45000-2400 x^3+3528 x^4-2016 x^5+256 x^6}{625 x^3} \, dx=\frac {64}{625} \, x^{4} - \frac {672}{625} \, x^{3} + \frac {1764}{625} \, x^{2} - \frac {96}{25} \, x + \frac {36}{x^{2}} \]

[In]

integrate(1/625*(256*x^6-2016*x^5+3528*x^4-2400*x^3-45000)/x^3,x, algorithm="maxima")

[Out]

64/625*x^4 - 672/625*x^3 + 1764/625*x^2 - 96/25*x + 36/x^2

Giac [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.96 \[ \int \frac {-45000-2400 x^3+3528 x^4-2016 x^5+256 x^6}{625 x^3} \, dx=\frac {64}{625} \, x^{4} - \frac {672}{625} \, x^{3} + \frac {1764}{625} \, x^{2} - \frac {96}{25} \, x + \frac {36}{x^{2}} \]

[In]

integrate(1/625*(256*x^6-2016*x^5+3528*x^4-2400*x^3-45000)/x^3,x, algorithm="giac")

[Out]

64/625*x^4 - 672/625*x^3 + 1764/625*x^2 - 96/25*x + 36/x^2

Mupad [B] (verification not implemented)

Time = 9.33 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.96 \[ \int \frac {-45000-2400 x^3+3528 x^4-2016 x^5+256 x^6}{625 x^3} \, dx=\frac {36}{x^2}-\frac {96\,x}{25}+\frac {1764\,x^2}{625}-\frac {672\,x^3}{625}+\frac {64\,x^4}{625} \]

[In]

int(-((96*x^3)/25 - (3528*x^4)/625 + (2016*x^5)/625 - (256*x^6)/625 + 72)/x^3,x)

[Out]

36/x^2 - (96*x)/25 + (1764*x^2)/625 - (672*x^3)/625 + (64*x^4)/625