3.63 \(\int \frac{(a+b x)^2}{x^8} \, dx\)

Optimal. Leaf size=30 \[ -\frac{a^2}{7 x^7}-\frac{a b}{3 x^6}-\frac{b^2}{5 x^5} \]

[Out]

-a^2/(7*x^7) - (a*b)/(3*x^6) - b^2/(5*x^5)

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Rubi [A]  time = 0.0081628, 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} \[ -\frac{a^2}{7 x^7}-\frac{a b}{3 x^6}-\frac{b^2}{5 x^5} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^2/x^8,x]

[Out]

-a^2/(7*x^7) - (a*b)/(3*x^6) - b^2/(5*x^5)

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^8} \, dx &=\int \left (\frac{a^2}{x^8}+\frac{2 a b}{x^7}+\frac{b^2}{x^6}\right ) \, dx\\ &=-\frac{a^2}{7 x^7}-\frac{a b}{3 x^6}-\frac{b^2}{5 x^5}\\ \end{align*}

Mathematica [A]  time = 0.0109021, size = 30, normalized size = 1. \[ -\frac{a^2}{7 x^7}-\frac{a b}{3 x^6}-\frac{b^2}{5 x^5} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^2/x^8,x]

[Out]

-a^2/(7*x^7) - (a*b)/(3*x^6) - b^2/(5*x^5)

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Maple [A]  time = 0.005, size = 25, normalized size = 0.8 \begin{align*} -{\frac{{a}^{2}}{7\,{x}^{7}}}-{\frac{ab}{3\,{x}^{6}}}-{\frac{{b}^{2}}{5\,{x}^{5}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^2/x^8,x)

[Out]

-1/7*a^2/x^7-1/3*a*b/x^6-1/5*b^2/x^5

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Maxima [A]  time = 1.07029, size = 32, normalized size = 1.07 \begin{align*} -\frac{21 \, b^{2} x^{2} + 35 \, a b x + 15 \, a^{2}}{105 \, x^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^2/x^8,x, algorithm="maxima")

[Out]

-1/105*(21*b^2*x^2 + 35*a*b*x + 15*a^2)/x^7

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Fricas [A]  time = 1.48804, size = 61, normalized size = 2.03 \begin{align*} -\frac{21 \, b^{2} x^{2} + 35 \, a b x + 15 \, a^{2}}{105 \, x^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^2/x^8,x, algorithm="fricas")

[Out]

-1/105*(21*b^2*x^2 + 35*a*b*x + 15*a^2)/x^7

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Sympy [A]  time = 0.462055, size = 26, normalized size = 0.87 \begin{align*} - \frac{15 a^{2} + 35 a b x + 21 b^{2} x^{2}}{105 x^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**2/x**8,x)

[Out]

-(15*a**2 + 35*a*b*x + 21*b**2*x**2)/(105*x**7)

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Giac [A]  time = 1.18508, size = 32, normalized size = 1.07 \begin{align*} -\frac{21 \, b^{2} x^{2} + 35 \, a b x + 15 \, a^{2}}{105 \, x^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^2/x^8,x, algorithm="giac")

[Out]

-1/105*(21*b^2*x^2 + 35*a*b*x + 15*a^2)/x^7