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

Optimal. Leaf size=26 \[ -\frac{a^2}{2 x^2}+2 a b x+\frac{b^2 x^4}{4} \]

[Out]

-a^2/(2*x^2) + 2*a*b*x + (b^2*x^4)/4

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Rubi [A]  time = 0.0087787, antiderivative size = 26, 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} \[ -\frac{a^2}{2 x^2}+2 a b x+\frac{b^2 x^4}{4} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^3)^2/x^3,x]

[Out]

-a^2/(2*x^2) + 2*a*b*x + (b^2*x^4)/4

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

Mathematica [A]  time = 0.0006522, size = 26, normalized size = 1. \[ -\frac{a^2}{2 x^2}+2 a b x+\frac{b^2 x^4}{4} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^3)^2/x^3,x]

[Out]

-a^2/(2*x^2) + 2*a*b*x + (b^2*x^4)/4

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Maple [A]  time = 0.004, size = 23, normalized size = 0.9 \begin{align*} -{\frac{{a}^{2}}{2\,{x}^{2}}}+2\,xab+{\frac{{b}^{2}{x}^{4}}{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.01241, size = 30, normalized size = 1.15 \begin{align*} \frac{1}{4} \, b^{2} x^{4} + 2 \, a b x - \frac{a^{2}}{2 \, x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*b^2*x^4 + 2*a*b*x - 1/2*a^2/x^2

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Fricas [A]  time = 1.63114, size = 53, normalized size = 2.04 \begin{align*} \frac{b^{2} x^{6} + 8 \, a b x^{3} - 2 \, a^{2}}{4 \, x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*(b^2*x^6 + 8*a*b*x^3 - 2*a^2)/x^2

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Sympy [A]  time = 0.266472, size = 22, normalized size = 0.85 \begin{align*} - \frac{a^{2}}{2 x^{2}} + 2 a b x + \frac{b^{2} x^{4}}{4} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.64387, size = 30, normalized size = 1.15 \begin{align*} \frac{1}{4} \, b^{2} x^{4} + 2 \, a b x - \frac{a^{2}}{2 \, x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*b^2*x^4 + 2*a*b*x - 1/2*a^2/x^2