3.249 \(\int \frac{(a+b x)^2}{x^{10}} \, dx\)

Optimal. Leaf size=30 \[ -\frac{a^2}{9 x^9}-\frac{a b}{4 x^8}-\frac{b^2}{7 x^7} \]

[Out]

-a^2/(9*x^9) - (a*b)/(4*x^8) - b^2/(7*x^7)

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Rubi [A]  time = 0.009093, 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} \[ -\frac{a^2}{9 x^9}-\frac{a b}{4 x^8}-\frac{b^2}{7 x^7} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^2/x^10,x]

[Out]

-a^2/(9*x^9) - (a*b)/(4*x^8) - b^2/(7*x^7)

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^{10}} \, dx &=\int \left (\frac{a^2}{x^{10}}+\frac{2 a b}{x^9}+\frac{b^2}{x^8}\right ) \, dx\\ &=-\frac{a^2}{9 x^9}-\frac{a b}{4 x^8}-\frac{b^2}{7 x^7}\\ \end{align*}

Mathematica [A]  time = 0.0077923, size = 30, normalized size = 1. \[ -\frac{a^2}{9 x^9}-\frac{a b}{4 x^8}-\frac{b^2}{7 x^7} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^2/x^10,x]

[Out]

-a^2/(9*x^9) - (a*b)/(4*x^8) - b^2/(7*x^7)

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Maple [A]  time = 0.005, size = 25, normalized size = 0.8 \begin{align*} -{\frac{{a}^{2}}{9\,{x}^{9}}}-{\frac{ab}{4\,{x}^{8}}}-{\frac{{b}^{2}}{7\,{x}^{7}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^2/x^10,x)

[Out]

-1/9*a^2/x^9-1/4*a*b/x^8-1/7*b^2/x^7

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Maxima [A]  time = 1.11856, size = 32, normalized size = 1.07 \begin{align*} -\frac{36 \, b^{2} x^{2} + 63 \, a b x + 28 \, a^{2}}{252 \, x^{9}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^2/x^10,x, algorithm="maxima")

[Out]

-1/252*(36*b^2*x^2 + 63*a*b*x + 28*a^2)/x^9

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Fricas [A]  time = 1.74829, size = 61, normalized size = 2.03 \begin{align*} -\frac{36 \, b^{2} x^{2} + 63 \, a b x + 28 \, a^{2}}{252 \, x^{9}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^2/x^10,x, algorithm="fricas")

[Out]

-1/252*(36*b^2*x^2 + 63*a*b*x + 28*a^2)/x^9

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Sympy [A]  time = 0.51904, size = 26, normalized size = 0.87 \begin{align*} - \frac{28 a^{2} + 63 a b x + 36 b^{2} x^{2}}{252 x^{9}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**2/x**10,x)

[Out]

-(28*a**2 + 63*a*b*x + 36*b**2*x**2)/(252*x**9)

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Giac [A]  time = 1.20263, size = 32, normalized size = 1.07 \begin{align*} -\frac{36 \, b^{2} x^{2} + 63 \, a b x + 28 \, a^{2}}{252 \, x^{9}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^2/x^10,x, algorithm="giac")

[Out]

-1/252*(36*b^2*x^2 + 63*a*b*x + 28*a^2)/x^9