3.60 \(\int \frac{(a+b x)^2}{x^5} \, dx\)

Optimal. Leaf size=30 \[ -\frac{a^2}{4 x^4}-\frac{2 a b}{3 x^3}-\frac{b^2}{2 x^2} \]

[Out]

-a^2/(4*x^4) - (2*a*b)/(3*x^3) - b^2/(2*x^2)

________________________________________________________________________________________

Rubi [A]  time = 0.0078972, antiderivative size = 30, normalized size of antiderivative = 1., 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 = {43} \[ -\frac{a^2}{4 x^4}-\frac{2 a b}{3 x^3}-\frac{b^2}{2 x^2} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^2/x^5,x]

[Out]

-a^2/(4*x^4) - (2*a*b)/(3*x^3) - b^2/(2*x^2)

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int \frac{(a+b x)^2}{x^5} \, dx &=\int \left (\frac{a^2}{x^5}+\frac{2 a b}{x^4}+\frac{b^2}{x^3}\right ) \, dx\\ &=-\frac{a^2}{4 x^4}-\frac{2 a b}{3 x^3}-\frac{b^2}{2 x^2}\\ \end{align*}

Mathematica [A]  time = 0.0052963, size = 30, normalized size = 1. \[ -\frac{a^2}{4 x^4}-\frac{2 a b}{3 x^3}-\frac{b^2}{2 x^2} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^2/x^5,x]

[Out]

-a^2/(4*x^4) - (2*a*b)/(3*x^3) - b^2/(2*x^2)

________________________________________________________________________________________

Maple [A]  time = 0.005, size = 25, normalized size = 0.8 \begin{align*} -{\frac{{a}^{2}}{4\,{x}^{4}}}-{\frac{2\,ab}{3\,{x}^{3}}}-{\frac{{b}^{2}}{2\,{x}^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^2/x^5,x)

[Out]

-1/4*a^2/x^4-2/3*a*b/x^3-1/2*b^2/x^2

________________________________________________________________________________________

Maxima [A]  time = 1.02448, size = 32, normalized size = 1.07 \begin{align*} -\frac{6 \, b^{2} x^{2} + 8 \, a b x + 3 \, a^{2}}{12 \, x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^2/x^5,x, algorithm="maxima")

[Out]

-1/12*(6*b^2*x^2 + 8*a*b*x + 3*a^2)/x^4

________________________________________________________________________________________

Fricas [A]  time = 1.50164, size = 55, normalized size = 1.83 \begin{align*} -\frac{6 \, b^{2} x^{2} + 8 \, a b x + 3 \, a^{2}}{12 \, x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^2/x^5,x, algorithm="fricas")

[Out]

-1/12*(6*b^2*x^2 + 8*a*b*x + 3*a^2)/x^4

________________________________________________________________________________________

Sympy [A]  time = 0.355127, size = 26, normalized size = 0.87 \begin{align*} - \frac{3 a^{2} + 8 a b x + 6 b^{2} x^{2}}{12 x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**2/x**5,x)

[Out]

-(3*a**2 + 8*a*b*x + 6*b**2*x**2)/(12*x**4)

________________________________________________________________________________________

Giac [A]  time = 1.16497, size = 32, normalized size = 1.07 \begin{align*} -\frac{6 \, b^{2} x^{2} + 8 \, a b x + 3 \, a^{2}}{12 \, x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^2/x^5,x, algorithm="giac")

[Out]

-1/12*(6*b^2*x^2 + 8*a*b*x + 3*a^2)/x^4