3.227 \(\int \frac{(a+b x^3)^2}{x^5} \, dx\)

Optimal. Leaf size=28 \[ -\frac{a^2}{4 x^4}-\frac{2 a b}{x}+\frac{b^2 x^2}{2} \]

[Out]

-a^2/(4*x^4) - (2*a*b)/x + (b^2*x^2)/2

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Rubi [A]  time = 0.0094556, antiderivative size = 28, 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} \[ -\frac{a^2}{4 x^4}-\frac{2 a b}{x}+\frac{b^2 x^2}{2} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^3)^2/x^5,x]

[Out]

-a^2/(4*x^4) - (2*a*b)/x + (b^2*x^2)/2

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

Mathematica [A]  time = 0.0008494, size = 28, normalized size = 1. \[ -\frac{a^2}{4 x^4}-\frac{2 a b}{x}+\frac{b^2 x^2}{2} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^3)^2/x^5,x]

[Out]

-a^2/(4*x^4) - (2*a*b)/x + (b^2*x^2)/2

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Maple [A]  time = 0.005, size = 25, normalized size = 0.9 \begin{align*} -{\frac{{a}^{2}}{4\,{x}^{4}}}-2\,{\frac{ab}{x}}+{\frac{{b}^{2}{x}^{2}}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*a^2/x^4-2*a*b/x+1/2*b^2*x^2

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Maxima [A]  time = 0.952324, size = 34, normalized size = 1.21 \begin{align*} \frac{1}{2} \, b^{2} x^{2} - \frac{8 \, a b x^{3} + a^{2}}{4 \, x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^2/x^5,x, algorithm="maxima")

[Out]

1/2*b^2*x^2 - 1/4*(8*a*b*x^3 + a^2)/x^4

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Fricas [A]  time = 1.70124, size = 53, normalized size = 1.89 \begin{align*} \frac{2 \, b^{2} x^{6} - 8 \, a b x^{3} - a^{2}}{4 \, x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^2/x^5,x, algorithm="fricas")

[Out]

1/4*(2*b^2*x^6 - 8*a*b*x^3 - a^2)/x^4

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Sympy [A]  time = 0.361735, size = 24, normalized size = 0.86 \begin{align*} \frac{b^{2} x^{2}}{2} - \frac{a^{2} + 8 a b x^{3}}{4 x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**3+a)**2/x**5,x)

[Out]

b**2*x**2/2 - (a**2 + 8*a*b*x**3)/(4*x**4)

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Giac [A]  time = 1.09944, size = 34, normalized size = 1.21 \begin{align*} \frac{1}{2} \, b^{2} x^{2} - \frac{8 \, a b x^{3} + a^{2}}{4 \, x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^2/x^5,x, algorithm="giac")

[Out]

1/2*b^2*x^2 - 1/4*(8*a*b*x^3 + a^2)/x^4