3.49 \(\int \frac{(a+b x^2)^3}{x^8} \, dx\)

Optimal. Leaf size=39 \[ -\frac{3 a^2 b}{5 x^5}-\frac{a^3}{7 x^7}-\frac{a b^2}{x^3}-\frac{b^3}{x} \]

[Out]

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

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Rubi [A]  time = 0.0134047, antiderivative size = 39, 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} \[ -\frac{3 a^2 b}{5 x^5}-\frac{a^3}{7 x^7}-\frac{a b^2}{x^3}-\frac{b^3}{x} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^2)^3/x^8,x]

[Out]

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

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

Mathematica [A]  time = 0.0035532, size = 39, normalized size = 1. \[ -\frac{3 a^2 b}{5 x^5}-\frac{a^3}{7 x^7}-\frac{a b^2}{x^3}-\frac{b^3}{x} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^2)^3/x^8,x]

[Out]

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

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Maple [A]  time = 0.004, size = 36, normalized size = 0.9 \begin{align*} -{\frac{{a}^{3}}{7\,{x}^{7}}}-{\frac{3\,{a}^{2}b}{5\,{x}^{5}}}-{\frac{a{b}^{2}}{{x}^{3}}}-{\frac{{b}^{3}}{x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^2+a)^3/x^8,x)

[Out]

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

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Maxima [A]  time = 1.9276, size = 50, normalized size = 1.28 \begin{align*} -\frac{35 \, b^{3} x^{6} + 35 \, a b^{2} x^{4} + 21 \, a^{2} b x^{2} + 5 \, a^{3}}{35 \, x^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^3/x^8,x, algorithm="maxima")

[Out]

-1/35*(35*b^3*x^6 + 35*a*b^2*x^4 + 21*a^2*b*x^2 + 5*a^3)/x^7

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Fricas [A]  time = 1.26732, size = 84, normalized size = 2.15 \begin{align*} -\frac{35 \, b^{3} x^{6} + 35 \, a b^{2} x^{4} + 21 \, a^{2} b x^{2} + 5 \, a^{3}}{35 \, x^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^3/x^8,x, algorithm="fricas")

[Out]

-1/35*(35*b^3*x^6 + 35*a*b^2*x^4 + 21*a^2*b*x^2 + 5*a^3)/x^7

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Sympy [A]  time = 0.386656, size = 39, normalized size = 1. \begin{align*} - \frac{5 a^{3} + 21 a^{2} b x^{2} + 35 a b^{2} x^{4} + 35 b^{3} x^{6}}{35 x^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**2+a)**3/x**8,x)

[Out]

-(5*a**3 + 21*a**2*b*x**2 + 35*a*b**2*x**4 + 35*b**3*x**6)/(35*x**7)

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Giac [A]  time = 2.53245, size = 50, normalized size = 1.28 \begin{align*} -\frac{35 \, b^{3} x^{6} + 35 \, a b^{2} x^{4} + 21 \, a^{2} b x^{2} + 5 \, a^{3}}{35 \, x^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^3/x^8,x, algorithm="giac")

[Out]

-1/35*(35*b^3*x^6 + 35*a*b^2*x^4 + 21*a^2*b*x^2 + 5*a^3)/x^7