3.16.55 \(\int \frac {a+\frac {b}{x}}{x^3} \, dx\) [1555]

Optimal. Leaf size=17 \[ -\frac {b}{3 x^3}-\frac {a}{2 x^2} \]

[Out]

-1/3*b/x^3-1/2*a/x^2

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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} \begin {gather*} -\frac {a}{2 x^2}-\frac {b}{3 x^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b/x)/x^3,x]

[Out]

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

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Mathematica [A]
time = 0.00, size = 17, normalized size = 1.00 \begin {gather*} -\frac {b}{3 x^3}-\frac {a}{2 x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b/x)/x^3,x]

[Out]

-1/3*b/x^3 - a/(2*x^2)

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Maple [A]
time = 0.01, size = 14, normalized size = 0.82

method	result	size
norman	\(\frac {-\frac {a x}{2}-\frac {b}{3}}{x^{3}}\)	\(13\)
risch	\(\frac {-\frac {a x}{2}-\frac {b}{3}}{x^{3}}\)	\(13\)
gosper	\(-\frac {3 a x +2 b}{6 x^{3}}\)	\(14\)
default	\(-\frac {b}{3 x^{3}}-\frac {a}{2 x^{2}}\)	\(14\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+1/x*b)/x^3,x,method=_RETURNVERBOSE)

[Out]

-1/3*b/x^3-1/2*a/x^2

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Maxima [A]
time = 0.29, size = 13, normalized size = 0.76 \begin {gather*} -\frac {3 \, a x + 2 \, b}{6 \, x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/6*(3*a*x + 2*b)/x^3

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Fricas [A]
time = 0.36, size = 13, normalized size = 0.76 \begin {gather*} -\frac {3 \, a x + 2 \, b}{6 \, x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/6*(3*a*x + 2*b)/x^3

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Sympy [A]
time = 0.03, size = 14, normalized size = 0.82 \begin {gather*} \frac {- 3 a x - 2 b}{6 x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(-3*a*x - 2*b)/(6*x**3)

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Giac [A]
time = 1.59, size = 13, normalized size = 0.76 \begin {gather*} -\frac {3 \, a x + 2 \, b}{6 \, x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/6*(3*a*x + 2*b)/x^3

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Mupad [B]
time = 0.02, size = 13, normalized size = 0.76 \begin {gather*} -\frac {2\,b+3\,a\,x}{6\,x^3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b/x)/x^3,x)

[Out]

-(2*b + 3*a*x)/(6*x^3)

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