3.248 \(\int \frac{(a+b x)^3}{x^{10}} \, dx\)

Optimal. Leaf size=43 \[ -\frac{3 a^2 b}{8 x^8}-\frac{a^3}{9 x^9}-\frac{3 a b^2}{7 x^7}-\frac{b^3}{6 x^6} \]

[Out]

-a^3/(9*x^9) - (3*a^2*b)/(8*x^8) - (3*a*b^2)/(7*x^7) - b^3/(6*x^6)

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Rubi [A]  time = 0.0140346, antiderivative size = 43, 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} \[ -\frac{3 a^2 b}{8 x^8}-\frac{a^3}{9 x^9}-\frac{3 a b^2}{7 x^7}-\frac{b^3}{6 x^6} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^3/x^10,x]

[Out]

-a^3/(9*x^9) - (3*a^2*b)/(8*x^8) - (3*a*b^2)/(7*x^7) - b^3/(6*x^6)

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)^3}{x^{10}} \, dx &=\int \left (\frac{a^3}{x^{10}}+\frac{3 a^2 b}{x^9}+\frac{3 a b^2}{x^8}+\frac{b^3}{x^7}\right ) \, dx\\ &=-\frac{a^3}{9 x^9}-\frac{3 a^2 b}{8 x^8}-\frac{3 a b^2}{7 x^7}-\frac{b^3}{6 x^6}\\ \end{align*}

Mathematica [A]  time = 0.0036274, size = 43, normalized size = 1. \[ -\frac{3 a^2 b}{8 x^8}-\frac{a^3}{9 x^9}-\frac{3 a b^2}{7 x^7}-\frac{b^3}{6 x^6} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^3/x^10,x]

[Out]

-a^3/(9*x^9) - (3*a^2*b)/(8*x^8) - (3*a*b^2)/(7*x^7) - b^3/(6*x^6)

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Maple [A]  time = 0.005, size = 36, normalized size = 0.8 \begin{align*} -{\frac{{a}^{3}}{9\,{x}^{9}}}-{\frac{3\,{a}^{2}b}{8\,{x}^{8}}}-{\frac{3\,{b}^{2}a}{7\,{x}^{7}}}-{\frac{{b}^{3}}{6\,{x}^{6}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^3/x^10,x)

[Out]

-1/9*a^3/x^9-3/8*a^2*b/x^8-3/7*a*b^2/x^7-1/6*b^3/x^6

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Maxima [A]  time = 1.10647, size = 47, normalized size = 1.09 \begin{align*} -\frac{84 \, b^{3} x^{3} + 216 \, a b^{2} x^{2} + 189 \, a^{2} b x + 56 \, a^{3}}{504 \, x^{9}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^3/x^10,x, algorithm="maxima")

[Out]

-1/504*(84*b^3*x^3 + 216*a*b^2*x^2 + 189*a^2*b*x + 56*a^3)/x^9

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Fricas [A]  time = 1.70502, size = 86, normalized size = 2. \begin{align*} -\frac{84 \, b^{3} x^{3} + 216 \, a b^{2} x^{2} + 189 \, a^{2} b x + 56 \, a^{3}}{504 \, x^{9}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^3/x^10,x, algorithm="fricas")

[Out]

-1/504*(84*b^3*x^3 + 216*a*b^2*x^2 + 189*a^2*b*x + 56*a^3)/x^9

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Sympy [A]  time = 0.540495, size = 37, normalized size = 0.86 \begin{align*} - \frac{56 a^{3} + 189 a^{2} b x + 216 a b^{2} x^{2} + 84 b^{3} x^{3}}{504 x^{9}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**3/x**10,x)

[Out]

-(56*a**3 + 189*a**2*b*x + 216*a*b**2*x**2 + 84*b**3*x**3)/(504*x**9)

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Giac [A]  time = 1.14324, size = 47, normalized size = 1.09 \begin{align*} -\frac{84 \, b^{3} x^{3} + 216 \, a b^{2} x^{2} + 189 \, a^{2} b x + 56 \, a^{3}}{504 \, x^{9}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^3/x^10,x, algorithm="giac")

[Out]

-1/504*(84*b^3*x^3 + 216*a*b^2*x^2 + 189*a^2*b*x + 56*a^3)/x^9