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

Optimal. Leaf size=25 \[ -\frac{a^2}{x}+a b x^2+\frac{b^2 x^5}{5} \]

[Out]

-(a^2/x) + a*b*x^2 + (b^2*x^5)/5

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Rubi [A]  time = 0.009013, antiderivative size = 25, 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} \[ -\frac{a^2}{x}+a b x^2+\frac{b^2 x^5}{5} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^3)^2/x^2,x]

[Out]

-(a^2/x) + a*b*x^2 + (b^2*x^5)/5

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

Mathematica [A]  time = 0.0008208, size = 25, normalized size = 1. \[ -\frac{a^2}{x}+a b x^2+\frac{b^2 x^5}{5} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^3)^2/x^2,x]

[Out]

-(a^2/x) + a*b*x^2 + (b^2*x^5)/5

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Maple [A]  time = 0.003, size = 24, normalized size = 1. \begin{align*} -{\frac{{a}^{2}}{x}}+ab{x}^{2}+{\frac{{b}^{2}{x}^{5}}{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.966987, size = 31, normalized size = 1.24 \begin{align*} \frac{1}{5} \, b^{2} x^{5} + a b x^{2} - \frac{a^{2}}{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/5*b^2*x^5 + a*b*x^2 - a^2/x

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Fricas [A]  time = 1.61463, size = 50, normalized size = 2. \begin{align*} \frac{b^{2} x^{6} + 5 \, a b x^{3} - 5 \, a^{2}}{5 \, x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/5*(b^2*x^6 + 5*a*b*x^3 - 5*a^2)/x

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Sympy [A]  time = 0.305865, size = 19, normalized size = 0.76 \begin{align*} - \frac{a^{2}}{x} + a b x^{2} + \frac{b^{2} x^{5}}{5} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-a**2/x + a*b*x**2 + b**2*x**5/5

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Giac [A]  time = 1.14241, size = 31, normalized size = 1.24 \begin{align*} \frac{1}{5} \, b^{2} x^{5} + a b x^{2} - \frac{a^{2}}{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/5*b^2*x^5 + a*b*x^2 - a^2/x