\(\int \frac {a+b x^2+c x^4}{x^7} \, dx\) [823]

   Optimal result
   Rubi [A] (verified)
   Mathematica [A] (verified)
   Maple [A] (verified)
   Fricas [A] (verification not implemented)
   Sympy [A] (verification not implemented)
   Maxima [A] (verification not implemented)
   Giac [A] (verification not implemented)
   Mupad [B] (verification not implemented)

Optimal result

Integrand size = 16, antiderivative size = 25 \[ \int \frac {a+b x^2+c x^4}{x^7} \, dx=-\frac {a}{6 x^6}-\frac {b}{4 x^4}-\frac {c}{2 x^2} \]

[Out]

-1/6*a/x^6-1/4*b/x^4-1/2*c/x^2

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {14} \[ \int \frac {a+b x^2+c x^4}{x^7} \, dx=-\frac {a}{6 x^6}-\frac {b}{4 x^4}-\frac {c}{2 x^2} \]

[In]

Int[(a + b*x^2 + c*x^4)/x^7,x]

[Out]

-1/6*a/x^6 - b/(4*x^4) - c/(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 (\frac {a}{x^7}+\frac {b}{x^5}+\frac {c}{x^3}\right ) \, dx \\ & = -\frac {a}{6 x^6}-\frac {b}{4 x^4}-\frac {c}{2 x^2} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00 \[ \int \frac {a+b x^2+c x^4}{x^7} \, dx=-\frac {a}{6 x^6}-\frac {b}{4 x^4}-\frac {c}{2 x^2} \]

[In]

Integrate[(a + b*x^2 + c*x^4)/x^7,x]

[Out]

-1/6*a/x^6 - b/(4*x^4) - c/(2*x^2)

Maple [A] (verified)

Time = 0.02 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.80

method result size
default \(-\frac {a}{6 x^{6}}-\frac {b}{4 x^{4}}-\frac {c}{2 x^{2}}\) \(20\)
norman \(\frac {-\frac {1}{2} c \,x^{4}-\frac {1}{4} b \,x^{2}-\frac {1}{6} a}{x^{6}}\) \(21\)
risch \(\frac {-\frac {1}{2} c \,x^{4}-\frac {1}{4} b \,x^{2}-\frac {1}{6} a}{x^{6}}\) \(21\)
gosper \(-\frac {6 c \,x^{4}+3 b \,x^{2}+2 a}{12 x^{6}}\) \(22\)
parallelrisch \(\frac {-6 c \,x^{4}-3 b \,x^{2}-2 a}{12 x^{6}}\) \(22\)

[In]

int((c*x^4+b*x^2+a)/x^7,x,method=_RETURNVERBOSE)

[Out]

-1/6*a/x^6-1/4*b/x^4-1/2*c/x^2

Fricas [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.84 \[ \int \frac {a+b x^2+c x^4}{x^7} \, dx=-\frac {6 \, c x^{4} + 3 \, b x^{2} + 2 \, a}{12 \, x^{6}} \]

[In]

integrate((c*x^4+b*x^2+a)/x^7,x, algorithm="fricas")

[Out]

-1/12*(6*c*x^4 + 3*b*x^2 + 2*a)/x^6

Sympy [A] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.88 \[ \int \frac {a+b x^2+c x^4}{x^7} \, dx=\frac {- 2 a - 3 b x^{2} - 6 c x^{4}}{12 x^{6}} \]

[In]

integrate((c*x**4+b*x**2+a)/x**7,x)

[Out]

(-2*a - 3*b*x**2 - 6*c*x**4)/(12*x**6)

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.84 \[ \int \frac {a+b x^2+c x^4}{x^7} \, dx=-\frac {6 \, c x^{4} + 3 \, b x^{2} + 2 \, a}{12 \, x^{6}} \]

[In]

integrate((c*x^4+b*x^2+a)/x^7,x, algorithm="maxima")

[Out]

-1/12*(6*c*x^4 + 3*b*x^2 + 2*a)/x^6

Giac [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.84 \[ \int \frac {a+b x^2+c x^4}{x^7} \, dx=-\frac {6 \, c x^{4} + 3 \, b x^{2} + 2 \, a}{12 \, x^{6}} \]

[In]

integrate((c*x^4+b*x^2+a)/x^7,x, algorithm="giac")

[Out]

-1/12*(6*c*x^4 + 3*b*x^2 + 2*a)/x^6

Mupad [B] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.84 \[ \int \frac {a+b x^2+c x^4}{x^7} \, dx=-\frac {\frac {c\,x^4}{2}+\frac {b\,x^2}{4}+\frac {a}{6}}{x^6} \]

[In]

int((a + b*x^2 + c*x^4)/x^7,x)

[Out]

-(a/6 + (b*x^2)/4 + (c*x^4)/2)/x^6