\(\int \frac {a+b x^4}{x^6} \, dx\) [615]

Optimal result

Integrand size = 11, antiderivative size = 15 \[ \int \frac {a+b x^4}{x^6} \, dx=-\frac {a}{5 x^5}-\frac {b}{x} \]

[Out]

-1/5*a/x^5-b/x

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {14} \[ \int \frac {a+b x^4}{x^6} \, dx=-\frac {a}{5 x^5}-\frac {b}{x} \]

[In]

Int[(a + b*x^4)/x^6,x]

[Out]

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

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.00 \[ \int \frac {a+b x^4}{x^6} \, dx=-\frac {a}{5 x^5}-\frac {b}{x} \]

[In]

Integrate[(a + b*x^4)/x^6,x]

[Out]

-1/5*a/x^5 - b/x

Maple [A] (verified)

Time = 0.03 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.93

method	result	size
gosper	\(-\frac {5 b \,x^{4}+a}{5 x^{5}}\)	\(14\)
default	\(-\frac {a}{5 x^{5}}-\frac {b}{x}\)	\(14\)
norman	\(\frac {-b \,x^{4}-\frac {a}{5}}{x^{5}}\)	\(15\)
risch	\(\frac {-b \,x^{4}-\frac {a}{5}}{x^{5}}\)	\(15\)
parallelrisch	\(\frac {-5 b \,x^{4}-a}{5 x^{5}}\)	\(16\)

[In]

int((b*x^4+a)/x^6,x,method=_RETURNVERBOSE)

[Out]

-1/5*(5*b*x^4+a)/x^5

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.87 \[ \int \frac {a+b x^4}{x^6} \, dx=-\frac {5 \, b x^{4} + a}{5 \, x^{5}} \]

[In]

integrate((b*x^4+a)/x^6,x, algorithm="fricas")

[Out]

-1/5*(5*b*x^4 + a)/x^5

Sympy [A] (verification not implemented)

Time = 0.06 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.93 \[ \int \frac {a+b x^4}{x^6} \, dx=\frac {- a - 5 b x^{4}}{5 x^{5}} \]

[In]

integrate((b*x**4+a)/x**6,x)

[Out]

(-a - 5*b*x**4)/(5*x**5)

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.87 \[ \int \frac {a+b x^4}{x^6} \, dx=-\frac {5 \, b x^{4} + a}{5 \, x^{5}} \]

[In]

integrate((b*x^4+a)/x^6,x, algorithm="maxima")

[Out]

-1/5*(5*b*x^4 + a)/x^5

Giac [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.87 \[ \int \frac {a+b x^4}{x^6} \, dx=-\frac {5 \, b x^{4} + a}{5 \, x^{5}} \]

[In]

integrate((b*x^4+a)/x^6,x, algorithm="giac")

[Out]

-1/5*(5*b*x^4 + a)/x^5

Mupad [B] (verification not implemented)

Time = 0.02 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.87 \[ \int \frac {a+b x^4}{x^6} \, dx=-\frac {5\,b\,x^4+a}{5\,x^5} \]

[In]

int((a + b*x^4)/x^6,x)

[Out]

-(a + 5*b*x^4)/(5*x^5)