\(\int \frac {a+b x^3}{x^5} \, dx\) [214]

Optimal result

Integrand size = 11, antiderivative size = 15 \[ \int \frac {a+b x^3}{x^5} \, dx=-\frac {a}{4 x^4}-\frac {b}{x} \]

[Out]

-1/4*a/x^4-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} \[ \int \frac {a+b x^3}{x^5} \, dx=-\frac {a}{4 x^4}-\frac {b}{x} \]

[In]

Int[(a + b*x^3)/x^5,x]

[Out]

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

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.00 \[ \int \frac {a+b x^3}{x^5} \, dx=-\frac {a}{4 x^4}-\frac {b}{x} \]

[In]

Integrate[(a + b*x^3)/x^5,x]

[Out]

-1/4*a/x^4 - b/x

Maple [A] (verified)

Time = 0.02 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.93

method	result	size
gosper	\(-\frac {4 b \,x^{3}+a}{4 x^{4}}\)	\(14\)
default	\(-\frac {a}{4 x^{4}}-\frac {b}{x}\)	\(14\)
norman	\(\frac {-b \,x^{3}-\frac {a}{4}}{x^{4}}\)	\(15\)
risch	\(\frac {-b \,x^{3}-\frac {a}{4}}{x^{4}}\)	\(15\)
parallelrisch	\(\frac {-4 b \,x^{3}-a}{4 x^{4}}\)	\(16\)

[In]

int((b*x^3+a)/x^5,x,method=_RETURNVERBOSE)

[Out]

-1/4*(4*b*x^3+a)/x^4

Fricas [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.87 \[ \int \frac {a+b x^3}{x^5} \, dx=-\frac {4 \, b x^{3} + a}{4 \, x^{4}} \]

[In]

integrate((b*x^3+a)/x^5,x, algorithm="fricas")

[Out]

-1/4*(4*b*x^3 + a)/x^4

Sympy [A] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.93 \[ \int \frac {a+b x^3}{x^5} \, dx=\frac {- a - 4 b x^{3}}{4 x^{4}} \]

[In]

integrate((b*x**3+a)/x**5,x)

[Out]

(-a - 4*b*x**3)/(4*x**4)

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.87 \[ \int \frac {a+b x^3}{x^5} \, dx=-\frac {4 \, b x^{3} + a}{4 \, x^{4}} \]

[In]

integrate((b*x^3+a)/x^5,x, algorithm="maxima")

[Out]

-1/4*(4*b*x^3 + a)/x^4

Giac [A] (verification not implemented)

none

Time = 0.36 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.87 \[ \int \frac {a+b x^3}{x^5} \, dx=-\frac {4 \, b x^{3} + a}{4 \, x^{4}} \]

[In]

integrate((b*x^3+a)/x^5,x, algorithm="giac")

[Out]

-1/4*(4*b*x^3 + a)/x^4

Mupad [B] (verification not implemented)

Time = 0.02 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.87 \[ \int \frac {a+b x^3}{x^5} \, dx=-\frac {4\,b\,x^3+a}{4\,x^4} \]

[In]

int((a + b*x^3)/x^5,x)

[Out]

-(a + 4*b*x^3)/(4*x^4)