3.59 \(\int \frac{(a+b x)^2}{x^4} \, dx\)

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

[Out]

-(a + b*x)^3/(3*a*x^3)

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Rubi [A]  time = 0.0015496, antiderivative size = 17, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {37} \[ -\frac{(a+b x)^3}{3 a x^3} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^2/x^4,x]

[Out]

-(a + b*x)^3/(3*a*x^3)

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

Mathematica [A]  time = 0.0073309, size = 26, normalized size = 1.53 \[ -\frac{a^2}{3 x^3}-\frac{a b}{x^2}-\frac{b^2}{x} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^2/x^4,x]

[Out]

-a^2/(3*x^3) - (a*b)/x^2 - b^2/x

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Maple [A]  time = 0.005, size = 25, normalized size = 1.5 \begin{align*} -{\frac{{a}^{2}}{3\,{x}^{3}}}-{\frac{ab}{{x}^{2}}}-{\frac{{b}^{2}}{x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^2/x^4,x)

[Out]

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

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Maxima [A]  time = 1.00007, size = 30, normalized size = 1.76 \begin{align*} -\frac{3 \, b^{2} x^{2} + 3 \, a b x + a^{2}}{3 \, x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^2/x^4,x, algorithm="maxima")

[Out]

-1/3*(3*b^2*x^2 + 3*a*b*x + a^2)/x^3

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Fricas [A]  time = 1.46217, size = 51, normalized size = 3. \begin{align*} -\frac{3 \, b^{2} x^{2} + 3 \, a b x + a^{2}}{3 \, x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^2/x^4,x, algorithm="fricas")

[Out]

-1/3*(3*b^2*x^2 + 3*a*b*x + a^2)/x^3

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Sympy [A]  time = 0.369987, size = 24, normalized size = 1.41 \begin{align*} - \frac{a^{2} + 3 a b x + 3 b^{2} x^{2}}{3 x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**2/x**4,x)

[Out]

-(a**2 + 3*a*b*x + 3*b**2*x**2)/(3*x**3)

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Giac [A]  time = 1.12523, size = 30, normalized size = 1.76 \begin{align*} -\frac{3 \, b^{2} x^{2} + 3 \, a b x + a^{2}}{3 \, x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^2/x^4,x, algorithm="giac")

[Out]

-1/3*(3*b^2*x^2 + 3*a*b*x + a^2)/x^3