3.1.49 \(\int \frac {a+b x}{x^3} \, dx\)

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

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Rubi [A]  time = 0.00, antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {37} \begin {gather*} -\frac {(a+b x)^2}{2 a x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x)/x^3,x]

[Out]

-(a + b*x)^2/(2*a*x^2)

Rule 37

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n +
1))/((b*c - a*d)*(m + 1)), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[m + n + 2, 0] && NeQ
[m, -1]

Rubi steps

\begin {align*} \int \frac {a+b x}{x^3} \, dx &=-\frac {(a+b x)^2}{2 a x^2}\\ \end {align*}

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Mathematica [A]  time = 0.00, size = 15, normalized size = 0.88 \begin {gather*} -\frac {a}{2 x^2}-\frac {b}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)/x^3,x]

[Out]

-1/2*a/x^2 - b/x

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {a+b x}{x^3} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[(a + b*x)/x^3,x]

[Out]

IntegrateAlgebraic[(a + b*x)/x^3, x]

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fricas [A]  time = 1.32, size = 11, normalized size = 0.65 \begin {gather*} -\frac {2 \, b x + a}{2 \, x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(2*b*x + a)/x^2

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giac [A]  time = 1.22, size = 11, normalized size = 0.65 \begin {gather*} -\frac {2 \, b x + a}{2 \, x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(2*b*x + a)/x^2

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maple [A]  time = 0.00, size = 14, normalized size = 0.82 \begin {gather*} -\frac {b}{x}-\frac {a}{2 x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)/x^3,x)

[Out]

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

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maxima [A]  time = 1.12, size = 11, normalized size = 0.65 \begin {gather*} -\frac {2 \, b x + a}{2 \, x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(2*b*x + a)/x^2

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mupad [B]  time = 0.02, size = 11, normalized size = 0.65 \begin {gather*} -\frac {a+2\,b\,x}{2\,x^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x)/x^3,x)

[Out]

-(a + 2*b*x)/(2*x^2)

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sympy [A]  time = 0.11, size = 12, normalized size = 0.71 \begin {gather*} \frac {- a - 2 b x}{2 x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(-a - 2*b*x)/(2*x**2)

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