3.48 \(\int \frac{(a+b x^2)^3}{x^6} \, dx\)

Optimal. Leaf size=34 \[ -\frac{a^2 b}{x^3}-\frac{a^3}{5 x^5}-\frac{3 a b^2}{x}+b^3 x \]

[Out]

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

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Rubi [A]  time = 0.0128706, antiderivative size = 34, 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} \[ -\frac{a^2 b}{x^3}-\frac{a^3}{5 x^5}-\frac{3 a b^2}{x}+b^3 x \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^2)^3/x^6,x]

[Out]

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

Mathematica [A]  time = 0.0047768, size = 34, normalized size = 1. \[ -\frac{a^2 b}{x^3}-\frac{a^3}{5 x^5}-\frac{3 a b^2}{x}+b^3 x \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^2)^3/x^6,x]

[Out]

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

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Maple [A]  time = 0.004, size = 33, normalized size = 1. \begin{align*} -{\frac{{a}^{3}}{5\,{x}^{5}}}-{\frac{{a}^{2}b}{{x}^{3}}}-3\,{\frac{a{b}^{2}}{x}}+{b}^{3}x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^2+a)^3/x^6,x)

[Out]

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

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Maxima [A]  time = 1.6791, size = 45, normalized size = 1.32 \begin{align*} b^{3} x - \frac{15 \, a b^{2} x^{4} + 5 \, a^{2} b x^{2} + a^{3}}{5 \, x^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^3/x^6,x, algorithm="maxima")

[Out]

b^3*x - 1/5*(15*a*b^2*x^4 + 5*a^2*b*x^2 + a^3)/x^5

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Fricas [A]  time = 1.19691, size = 76, normalized size = 2.24 \begin{align*} \frac{5 \, b^{3} x^{6} - 15 \, a b^{2} x^{4} - 5 \, a^{2} b x^{2} - a^{3}}{5 \, x^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^3/x^6,x, algorithm="fricas")

[Out]

1/5*(5*b^3*x^6 - 15*a*b^2*x^4 - 5*a^2*b*x^2 - a^3)/x^5

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Sympy [A]  time = 0.341248, size = 32, normalized size = 0.94 \begin{align*} b^{3} x - \frac{a^{3} + 5 a^{2} b x^{2} + 15 a b^{2} x^{4}}{5 x^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**2+a)**3/x**6,x)

[Out]

b**3*x - (a**3 + 5*a**2*b*x**2 + 15*a*b**2*x**4)/(5*x**5)

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Giac [A]  time = 2.33084, size = 45, normalized size = 1.32 \begin{align*} b^{3} x - \frac{15 \, a b^{2} x^{4} + 5 \, a^{2} b x^{2} + a^{3}}{5 \, x^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^3/x^6,x, algorithm="giac")

[Out]

b^3*x - 1/5*(15*a*b^2*x^4 + 5*a^2*b*x^2 + a^3)/x^5