3.47 \(\int \frac{(a+b x^2)^3}{x^4} \, dx\)

Optimal. Leaf size=37 \[ -\frac{3 a^2 b}{x}-\frac{a^3}{3 x^3}+3 a b^2 x+\frac{b^3 x^3}{3} \]

[Out]

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

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Rubi [A]  time = 0.0125237, antiderivative size = 37, 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} \[ -\frac{3 a^2 b}{x}-\frac{a^3}{3 x^3}+3 a b^2 x+\frac{b^3 x^3}{3} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^2)^3/x^4,x]

[Out]

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

Mathematica [A]  time = 0.0033825, size = 37, normalized size = 1. \[ -\frac{3 a^2 b}{x}-\frac{a^3}{3 x^3}+3 a b^2 x+\frac{b^3 x^3}{3} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^2)^3/x^4,x]

[Out]

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

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Maple [A]  time = 0.006, size = 34, normalized size = 0.9 \begin{align*} -{\frac{{a}^{3}}{3\,{x}^{3}}}-3\,{\frac{{a}^{2}b}{x}}+3\,a{b}^{2}x+{\frac{{b}^{3}{x}^{3}}{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^2+a)^3/x^4,x)

[Out]

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

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Maxima [A]  time = 1.24575, size = 46, normalized size = 1.24 \begin{align*} \frac{1}{3} \, b^{3} x^{3} + 3 \, a b^{2} x - \frac{9 \, a^{2} b x^{2} + a^{3}}{3 \, x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^3/x^4,x, algorithm="maxima")

[Out]

1/3*b^3*x^3 + 3*a*b^2*x - 1/3*(9*a^2*b*x^2 + a^3)/x^3

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Fricas [A]  time = 1.20161, size = 72, normalized size = 1.95 \begin{align*} \frac{b^{3} x^{6} + 9 \, a b^{2} x^{4} - 9 \, a^{2} b x^{2} - a^{3}}{3 \, x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^3/x^4,x, algorithm="fricas")

[Out]

1/3*(b^3*x^6 + 9*a*b^2*x^4 - 9*a^2*b*x^2 - a^3)/x^3

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Sympy [A]  time = 0.293818, size = 34, normalized size = 0.92 \begin{align*} 3 a b^{2} x + \frac{b^{3} x^{3}}{3} - \frac{a^{3} + 9 a^{2} b x^{2}}{3 x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**2+a)**3/x**4,x)

[Out]

3*a*b**2*x + b**3*x**3/3 - (a**3 + 9*a**2*b*x**2)/(3*x**3)

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Giac [A]  time = 2.78575, size = 46, normalized size = 1.24 \begin{align*} \frac{1}{3} \, b^{3} x^{3} + 3 \, a b^{2} x - \frac{9 \, a^{2} b x^{2} + a^{3}}{3 \, x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^3/x^4,x, algorithm="giac")

[Out]

1/3*b^3*x^3 + 3*a*b^2*x - 1/3*(9*a^2*b*x^2 + a^3)/x^3