\(\int \frac {-1200+5280 x-5776 x^2-149 x^4+1440 x^5+300 x^6}{16 x^4} \, dx\) [428]

Optimal result

Integrand size = 32, antiderivative size = 33 \[ \int \frac {-1200+5280 x-5776 x^2-149 x^4+1440 x^5+300 x^6}{16 x^4} \, dx=-\frac {\left (\frac {1}{4} (-2+x)+x\right )^2 \left (5-4 \left (4-\frac {5}{x}+x\right )^2\right )}{x}+\log (4) \]

[Out]

2*ln(2)-(5-2*(4+x-5/x)*(8+2*x-10/x))*(5/4*x-1/2)^2/x

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {12, 14} \[ \int \frac {-1200+5280 x-5776 x^2-149 x^4+1440 x^5+300 x^6}{16 x^4} \, dx=\frac {25 x^3}{4}+\frac {25}{x^3}+45 x^2-\frac {165}{x^2}-\frac {149 x}{16}+\frac {361}{x} \]

[In]

Int[(-1200 + 5280*x - 5776*x^2 - 149*x^4 + 1440*x^5 + 300*x^6)/(16*x^4),x]

[Out]

25/x^3 - 165/x^2 + 361/x - (149*x)/16 + 45*x^2 + (25*x^3)/4

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}{16} \int \frac {-1200+5280 x-5776 x^2-149 x^4+1440 x^5+300 x^6}{x^4} \, dx \\ & = \frac {1}{16} \int \left (-149-\frac {1200}{x^4}+\frac {5280}{x^3}-\frac {5776}{x^2}+1440 x+300 x^2\right ) \, dx \\ & = \frac {25}{x^3}-\frac {165}{x^2}+\frac {361}{x}-\frac {149 x}{16}+45 x^2+\frac {25 x^3}{4} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.00 \[ \int \frac {-1200+5280 x-5776 x^2-149 x^4+1440 x^5+300 x^6}{16 x^4} \, dx=\frac {25}{x^3}-\frac {165}{x^2}+\frac {361}{x}-\frac {149 x}{16}+45 x^2+\frac {25 x^3}{4} \]

[In]

Integrate[(-1200 + 5280*x - 5776*x^2 - 149*x^4 + 1440*x^5 + 300*x^6)/(16*x^4),x]

[Out]

25/x^3 - 165/x^2 + 361/x - (149*x)/16 + 45*x^2 + (25*x^3)/4

Maple [A] (verified)

Time = 0.02 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.91

method	result	size
default	\(\frac {25 x^{3}}{4}+45 x^{2}-\frac {149 x}{16}+\frac {361}{x}-\frac {165}{x^{2}}+\frac {25}{x^{3}}\)	\(30\)
norman	\(\frac {25-165 x +361 x^{2}-\frac {149}{16} x^{4}+45 x^{5}+\frac {25}{4} x^{6}}{x^{3}}\)	\(30\)
risch	\(\frac {25 x^{3}}{4}+45 x^{2}-\frac {149 x}{16}+\frac {5776 x^{2}-2640 x +400}{16 x^{3}}\)	\(30\)
gosper	\(\frac {100 x^{6}+720 x^{5}-149 x^{4}+5776 x^{2}-2640 x +400}{16 x^{3}}\)	\(31\)
parallelrisch	\(\frac {100 x^{6}+720 x^{5}-149 x^{4}+5776 x^{2}-2640 x +400}{16 x^{3}}\)	\(31\)

[In]

int(1/16*(300*x^6+1440*x^5-149*x^4-5776*x^2+5280*x-1200)/x^4,x,method=_RETURNVERBOSE)

[Out]

25/4*x^3+45*x^2-149/16*x+361/x-165/x^2+25/x^3

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.91 \[ \int \frac {-1200+5280 x-5776 x^2-149 x^4+1440 x^5+300 x^6}{16 x^4} \, dx=\frac {100 \, x^{6} + 720 \, x^{5} - 149 \, x^{4} + 5776 \, x^{2} - 2640 \, x + 400}{16 \, x^{3}} \]

[In]

integrate(1/16*(300*x^6+1440*x^5-149*x^4-5776*x^2+5280*x-1200)/x^4,x, algorithm="fricas")

[Out]

1/16*(100*x^6 + 720*x^5 - 149*x^4 + 5776*x^2 - 2640*x + 400)/x^3

Sympy [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.94 \[ \int \frac {-1200+5280 x-5776 x^2-149 x^4+1440 x^5+300 x^6}{16 x^4} \, dx=\frac {25 x^{3}}{4} + 45 x^{2} - \frac {149 x}{16} + \frac {5776 x^{2} - 2640 x + 400}{16 x^{3}} \]

[In]

integrate(1/16*(300*x**6+1440*x**5-149*x**4-5776*x**2+5280*x-1200)/x**4,x)

[Out]

25*x**3/4 + 45*x**2 - 149*x/16 + (5776*x**2 - 2640*x + 400)/(16*x**3)

Maxima [A] (verification not implemented)

none

Time = 0.18 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.85 \[ \int \frac {-1200+5280 x-5776 x^2-149 x^4+1440 x^5+300 x^6}{16 x^4} \, dx=\frac {25}{4} \, x^{3} + 45 \, x^{2} - \frac {149}{16} \, x + \frac {361 \, x^{2} - 165 \, x + 25}{x^{3}} \]

[In]

integrate(1/16*(300*x^6+1440*x^5-149*x^4-5776*x^2+5280*x-1200)/x^4,x, algorithm="maxima")

[Out]

25/4*x^3 + 45*x^2 - 149/16*x + (361*x^2 - 165*x + 25)/x^3

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.85 \[ \int \frac {-1200+5280 x-5776 x^2-149 x^4+1440 x^5+300 x^6}{16 x^4} \, dx=\frac {25}{4} \, x^{3} + 45 \, x^{2} - \frac {149}{16} \, x + \frac {361 \, x^{2} - 165 \, x + 25}{x^{3}} \]

[In]

integrate(1/16*(300*x^6+1440*x^5-149*x^4-5776*x^2+5280*x-1200)/x^4,x, algorithm="giac")

[Out]

25/4*x^3 + 45*x^2 - 149/16*x + (361*x^2 - 165*x + 25)/x^3

Mupad [B] (verification not implemented)

Time = 8.24 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.85 \[ \int \frac {-1200+5280 x-5776 x^2-149 x^4+1440 x^5+300 x^6}{16 x^4} \, dx=\frac {361\,x^2-165\,x+25}{x^3}-\frac {149\,x}{16}+45\,x^2+\frac {25\,x^3}{4} \]

[In]

int((330*x - 361*x^2 - (149*x^4)/16 + 90*x^5 + (75*x^6)/4 - 75)/x^4,x)

[Out]

(361*x^2 - 165*x + 25)/x^3 - (149*x)/16 + 45*x^2 + (25*x^3)/4