\(\int \frac {a+b x^3}{x^7} \, dx\) [216]

Optimal result

Integrand size = 11, antiderivative size = 17 \[ \int \frac {a+b x^3}{x^7} \, dx=-\frac {a}{6 x^6}-\frac {b}{3 x^3} \]

[Out]

-1/6*a/x^6-1/3*b/x^3

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 17, 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} \[ \int \frac {a+b x^3}{x^7} \, dx=-\frac {a}{6 x^6}-\frac {b}{3 x^3} \]

[In]

Int[(a + b*x^3)/x^7,x]

[Out]

-1/6*a/x^6 - b/(3*x^3)

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^4}\right ) \, dx \\ & = -\frac {a}{6 x^6}-\frac {b}{3 x^3} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.00 \[ \int \frac {a+b x^3}{x^7} \, dx=-\frac {a}{6 x^6}-\frac {b}{3 x^3} \]

[In]

Integrate[(a + b*x^3)/x^7,x]

[Out]

-1/6*a/x^6 - b/(3*x^3)

Maple [A] (verified)

Time = 0.03 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.82

method	result	size
gosper	\(-\frac {2 b \,x^{3}+a}{6 x^{6}}\)	\(14\)
default	\(-\frac {a}{6 x^{6}}-\frac {b}{3 x^{3}}\)	\(14\)
norman	\(\frac {-\frac {b \,x^{3}}{3}-\frac {a}{6}}{x^{6}}\)	\(15\)
risch	\(\frac {-\frac {b \,x^{3}}{3}-\frac {a}{6}}{x^{6}}\)	\(15\)
parallelrisch	\(\frac {-2 b \,x^{3}-a}{6 x^{6}}\)	\(16\)

[In]

int((b*x^3+a)/x^7,x,method=_RETURNVERBOSE)

[Out]

-1/6*(2*b*x^3+a)/x^6

Fricas [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.76 \[ \int \frac {a+b x^3}{x^7} \, dx=-\frac {2 \, b x^{3} + a}{6 \, x^{6}} \]

[In]

integrate((b*x^3+a)/x^7,x, algorithm="fricas")

[Out]

-1/6*(2*b*x^3 + a)/x^6

Sympy [A] (verification not implemented)

Time = 0.06 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.82 \[ \int \frac {a+b x^3}{x^7} \, dx=\frac {- a - 2 b x^{3}}{6 x^{6}} \]

[In]

integrate((b*x**3+a)/x**7,x)

[Out]

(-a - 2*b*x**3)/(6*x**6)

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.76 \[ \int \frac {a+b x^3}{x^7} \, dx=-\frac {2 \, b x^{3} + a}{6 \, x^{6}} \]

[In]

integrate((b*x^3+a)/x^7,x, algorithm="maxima")

[Out]

-1/6*(2*b*x^3 + a)/x^6

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.76 \[ \int \frac {a+b x^3}{x^7} \, dx=-\frac {2 \, b x^{3} + a}{6 \, x^{6}} \]

[In]

integrate((b*x^3+a)/x^7,x, algorithm="giac")

[Out]

-1/6*(2*b*x^3 + a)/x^6

Mupad [B] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.76 \[ \int \frac {a+b x^3}{x^7} \, dx=-\frac {2\,b\,x^3+a}{6\,x^6} \]

[In]

int((a + b*x^3)/x^7,x)

[Out]

-(a + 2*b*x^3)/(6*x^6)