3.50 \(\int \frac{(a+b x^2)^3}{x^{10}} \, dx\)

Optimal. Leaf size=43 \[ -\frac{3 a^2 b}{7 x^7}-\frac{a^3}{9 x^9}-\frac{3 a b^2}{5 x^5}-\frac{b^3}{3 x^3} \]

[Out]

-a^3/(9*x^9) - (3*a^2*b)/(7*x^7) - (3*a*b^2)/(5*x^5) - b^3/(3*x^3)

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Rubi [A]  time = 0.0134588, antiderivative size = 43, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {270} \[ -\frac{3 a^2 b}{7 x^7}-\frac{a^3}{9 x^9}-\frac{3 a b^2}{5 x^5}-\frac{b^3}{3 x^3} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^2)^3/x^10,x]

[Out]

-a^3/(9*x^9) - (3*a^2*b)/(7*x^7) - (3*a*b^2)/(5*x^5) - b^3/(3*x^3)

Rule 270

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*(a + b*x^n)^p,
x], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0]

Rubi steps

\begin{align*} \int \frac{\left (a+b x^2\right )^3}{x^{10}} \, dx &=\int \left (\frac{a^3}{x^{10}}+\frac{3 a^2 b}{x^8}+\frac{3 a b^2}{x^6}+\frac{b^3}{x^4}\right ) \, dx\\ &=-\frac{a^3}{9 x^9}-\frac{3 a^2 b}{7 x^7}-\frac{3 a b^2}{5 x^5}-\frac{b^3}{3 x^3}\\ \end{align*}

Mathematica [A]  time = 0.0039683, size = 43, normalized size = 1. \[ -\frac{3 a^2 b}{7 x^7}-\frac{a^3}{9 x^9}-\frac{3 a b^2}{5 x^5}-\frac{b^3}{3 x^3} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^2)^3/x^10,x]

[Out]

-a^3/(9*x^9) - (3*a^2*b)/(7*x^7) - (3*a*b^2)/(5*x^5) - b^3/(3*x^3)

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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}{7\,{x}^{7}}}-{\frac{3\,a{b}^{2}}{5\,{x}^{5}}}-{\frac{{b}^{3}}{3\,{x}^{3}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^2+a)^3/x^10,x)

[Out]

-1/9*a^3/x^9-3/7*a^2*b/x^7-3/5*a*b^2/x^5-1/3*b^3/x^3

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Maxima [A]  time = 2.28645, size = 50, normalized size = 1.16 \begin{align*} -\frac{105 \, b^{3} x^{6} + 189 \, a b^{2} x^{4} + 135 \, a^{2} b x^{2} + 35 \, a^{3}}{315 \, x^{9}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^3/x^10,x, algorithm="maxima")

[Out]

-1/315*(105*b^3*x^6 + 189*a*b^2*x^4 + 135*a^2*b*x^2 + 35*a^3)/x^9

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Fricas [A]  time = 1.26863, size = 90, normalized size = 2.09 \begin{align*} -\frac{105 \, b^{3} x^{6} + 189 \, a b^{2} x^{4} + 135 \, a^{2} b x^{2} + 35 \, a^{3}}{315 \, x^{9}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^3/x^10,x, algorithm="fricas")

[Out]

-1/315*(105*b^3*x^6 + 189*a*b^2*x^4 + 135*a^2*b*x^2 + 35*a^3)/x^9

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Sympy [A]  time = 0.449378, size = 39, normalized size = 0.91 \begin{align*} - \frac{35 a^{3} + 135 a^{2} b x^{2} + 189 a b^{2} x^{4} + 105 b^{3} x^{6}}{315 x^{9}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**2+a)**3/x**10,x)

[Out]

-(35*a**3 + 135*a**2*b*x**2 + 189*a*b**2*x**4 + 105*b**3*x**6)/(315*x**9)

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Giac [A]  time = 1.81699, size = 50, normalized size = 1.16 \begin{align*} -\frac{105 \, b^{3} x^{6} + 189 \, a b^{2} x^{4} + 135 \, a^{2} b x^{2} + 35 \, a^{3}}{315 \, x^{9}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^3/x^10,x, algorithm="giac")

[Out]

-1/315*(105*b^3*x^6 + 189*a*b^2*x^4 + 135*a^2*b*x^2 + 35*a^3)/x^9