3.1556 \(\int \frac {a+\frac {b}{x}}{x^4} \, dx\)

Optimal. Leaf size=17 \[ -\frac {a}{3 x^3}-\frac {b}{4 x^4} \]

[Out]

-1/4*b/x^4-1/3*a/x^3

________________________________________________________________________________________

Rubi [A]  time = 0.00, antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {14} \[ -\frac {a}{3 x^3}-\frac {b}{4 x^4} \]

Antiderivative was successfully verified.

[In]

Int[(a + b/x)/x^4,x]

[Out]

-b/(4*x^4) - a/(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*} \int \frac {a+\frac {b}{x}}{x^4} \, dx &=\int \left (\frac {b}{x^5}+\frac {a}{x^4}\right ) \, dx\\ &=-\frac {b}{4 x^4}-\frac {a}{3 x^3}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.00, size = 17, normalized size = 1.00 \[ -\frac {a}{3 x^3}-\frac {b}{4 x^4} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b/x)/x^4,x]

[Out]

-1/4*b/x^4 - a/(3*x^3)

________________________________________________________________________________________

fricas [A]  time = 0.90, size = 13, normalized size = 0.76 \[ -\frac {4 \, a x + 3 \, b}{12 \, x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/12*(4*a*x + 3*b)/x^4

________________________________________________________________________________________

giac [A]  time = 0.15, size = 13, normalized size = 0.76 \[ -\frac {4 \, a x + 3 \, b}{12 \, x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/12*(4*a*x + 3*b)/x^4

________________________________________________________________________________________

maple [A]  time = 0.00, size = 14, normalized size = 0.82 \[ -\frac {a}{3 x^{3}}-\frac {b}{4 x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b/x)/x^4,x)

[Out]

-1/4*b/x^4-1/3*a/x^3

________________________________________________________________________________________

maxima [A]  time = 1.05, size = 13, normalized size = 0.76 \[ -\frac {4 \, a x + 3 \, b}{12 \, x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/12*(4*a*x + 3*b)/x^4

________________________________________________________________________________________

mupad [B]  time = 0.02, size = 13, normalized size = 0.76 \[ -\frac {3\,b+4\,a\,x}{12\,x^4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b/x)/x^4,x)

[Out]

-(3*b + 4*a*x)/(12*x^4)

________________________________________________________________________________________

sympy [A]  time = 0.13, size = 14, normalized size = 0.82 \[ \frac {- 4 a x - 3 b}{12 x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(-4*a*x - 3*b)/(12*x**4)

________________________________________________________________________________________