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

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

[Out]

-(a + b*x)^2/(2*a*x^2)

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Rubi [A]  time = 0.0017435, antiderivative size = 17, normalized size of antiderivative = 1., 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} \[ -\frac{(a+b x)^2}{2 a x^2} \]

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*}

Mathematica [A]  time = 0.0015521, size = 15, normalized size = 0.88 \[ -\frac{a}{2 x^2}-\frac{b}{x} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)/x^3,x]

[Out]

-a/(2*x^2) - b/x

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Maple [A]  time = 0.004, size = 14, normalized size = 0.8 \begin{align*} -{\frac{a}{2\,{x}^{2}}}-{\frac{b}{x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.0543, size = 15, normalized size = 0.88 \begin{align*} -\frac{2 \, b x + a}{2 \, x^{2}} \end{align*}

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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Fricas [A]  time = 1.55443, size = 30, normalized size = 1.76 \begin{align*} -\frac{2 \, b x + a}{2 \, x^{2}} \end{align*}

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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Sympy [A]  time = 0.383781, size = 12, normalized size = 0.71 \begin{align*} - \frac{a + 2 b x}{2 x^{2}} \end{align*}

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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Giac [A]  time = 1.1712, size = 15, normalized size = 0.88 \begin{align*} -\frac{2 \, b x + a}{2 \, x^{2}} \end{align*}

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