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

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

[Out]

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

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Rubi [A]  time = 0.0129008, 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} \[ 3 a^2 b x-\frac{a^3}{x}+a b^2 x^3+\frac{b^3 x^5}{5} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^2)^3/x^2,x]

[Out]

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

Mathematica [A]  time = 0.003336, size = 34, normalized size = 1. \[ 3 a^2 b x-\frac{a^3}{x}+a b^2 x^3+\frac{b^3 x^5}{5} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^2)^3/x^2,x]

[Out]

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

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Maple [A]  time = 0.003, size = 33, normalized size = 1. \begin{align*} -{\frac{{a}^{3}}{x}}+3\,{a}^{2}bx+a{b}^{2}{x}^{3}+{\frac{{b}^{3}{x}^{5}}{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.00988, size = 43, normalized size = 1.26 \begin{align*} \frac{1}{5} \, b^{3} x^{5} + a b^{2} x^{3} + 3 \, a^{2} b x - \frac{a^{3}}{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.16299, size = 73, normalized size = 2.15 \begin{align*} \frac{b^{3} x^{6} + 5 \, a b^{2} x^{4} + 15 \, a^{2} b x^{2} - 5 \, a^{3}}{5 \, x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.256705, size = 29, normalized size = 0.85 \begin{align*} - \frac{a^{3}}{x} + 3 a^{2} b x + a b^{2} x^{3} + \frac{b^{3} x^{5}}{5} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 2.36333, size = 43, normalized size = 1.26 \begin{align*} \frac{1}{5} \, b^{3} x^{5} + a b^{2} x^{3} + 3 \, a^{2} b x - \frac{a^{3}}{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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