3.61 \(\int \frac{(a+b x)^2}{x^6} \, dx\)

Optimal. Leaf size=30 \[ -\frac{a^2}{5 x^5}-\frac{a b}{2 x^4}-\frac{b^2}{3 x^3} \]

[Out]

-a^2/(5*x^5) - (a*b)/(2*x^4) - b^2/(3*x^3)

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Rubi [A]  time = 0.0080537, 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} \[ -\frac{a^2}{5 x^5}-\frac{a b}{2 x^4}-\frac{b^2}{3 x^3} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^2/x^6,x]

[Out]

-a^2/(5*x^5) - (a*b)/(2*x^4) - b^2/(3*x^3)

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^6} \, dx &=\int \left (\frac{a^2}{x^6}+\frac{2 a b}{x^5}+\frac{b^2}{x^4}\right ) \, dx\\ &=-\frac{a^2}{5 x^5}-\frac{a b}{2 x^4}-\frac{b^2}{3 x^3}\\ \end{align*}

Mathematica [A]  time = 0.0078579, size = 30, normalized size = 1. \[ -\frac{a^2}{5 x^5}-\frac{a b}{2 x^4}-\frac{b^2}{3 x^3} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^2/x^6,x]

[Out]

-a^2/(5*x^5) - (a*b)/(2*x^4) - b^2/(3*x^3)

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Maple [A]  time = 0.005, size = 25, normalized size = 0.8 \begin{align*} -{\frac{{a}^{2}}{5\,{x}^{5}}}-{\frac{ab}{2\,{x}^{4}}}-{\frac{{b}^{2}}{3\,{x}^{3}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^2/x^6,x)

[Out]

-1/5*a^2/x^5-1/2*a*b/x^4-1/3*b^2/x^3

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Maxima [A]  time = 1.0103, size = 32, normalized size = 1.07 \begin{align*} -\frac{10 \, b^{2} x^{2} + 15 \, a b x + 6 \, a^{2}}{30 \, x^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^2/x^6,x, algorithm="maxima")

[Out]

-1/30*(10*b^2*x^2 + 15*a*b*x + 6*a^2)/x^5

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Fricas [A]  time = 1.49608, size = 58, normalized size = 1.93 \begin{align*} -\frac{10 \, b^{2} x^{2} + 15 \, a b x + 6 \, a^{2}}{30 \, x^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^2/x^6,x, algorithm="fricas")

[Out]

-1/30*(10*b^2*x^2 + 15*a*b*x + 6*a^2)/x^5

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Sympy [A]  time = 0.464478, size = 26, normalized size = 0.87 \begin{align*} - \frac{6 a^{2} + 15 a b x + 10 b^{2} x^{2}}{30 x^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**2/x**6,x)

[Out]

-(6*a**2 + 15*a*b*x + 10*b**2*x**2)/(30*x**5)

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Giac [A]  time = 1.19931, size = 32, normalized size = 1.07 \begin{align*} -\frac{10 \, b^{2} x^{2} + 15 \, a b x + 6 \, a^{2}}{30 \, x^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^2/x^6,x, algorithm="giac")

[Out]

-1/30*(10*b^2*x^2 + 15*a*b*x + 6*a^2)/x^5