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

Optimal. Leaf size=16 \[ \frac{\left (a+b x^2\right )^3}{6 b} \]

[Out]

(a + b*x^2)^3/(6*b)

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Rubi [A]  time = 0.0070228, antiderivative size = 16, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {1584, 261} \[ \frac{\left (a+b x^2\right )^3}{6 b} \]

Antiderivative was successfully verified.

[In]

Int[(a*x + b*x^3)^2/x,x]

[Out]

(a + b*x^2)^3/(6*b)

Rule 1584

Int[(u_.)*(x_)^(m_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.))^(n_.), x_Symbol] :> Int[u*x^(m + n*p)*(a + b*x^(q -
p))^n, x] /; FreeQ[{a, b, m, p, q}, x] && IntegerQ[n] && PosQ[q - p]

Rule 261

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a + b*x^n)^(p + 1)/(b*n*(p + 1)), x] /; FreeQ
[{a, b, m, n, p}, x] && EqQ[m, n - 1] && NeQ[p, -1]

Rubi steps

\begin{align*} \int \frac{\left (a x+b x^3\right )^2}{x} \, dx &=\int x \left (a+b x^2\right )^2 \, dx\\ &=\frac{\left (a+b x^2\right )^3}{6 b}\\ \end{align*}

Mathematica [A]  time = 0.0016032, size = 16, normalized size = 1. \[ \frac{\left (a+b x^2\right )^3}{6 b} \]

Antiderivative was successfully verified.

[In]

Integrate[(a*x + b*x^3)^2/x,x]

[Out]

(a + b*x^2)^3/(6*b)

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Maple [A]  time = 0.002, size = 25, normalized size = 1.6 \begin{align*}{\frac{{b}^{2}{x}^{6}}{6}}+{\frac{ab{x}^{4}}{2}}+{\frac{{a}^{2}{x}^{2}}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*b^2*x^6+1/2*a*b*x^4+1/2*a^2*x^2

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Maxima [A]  time = 1.10286, size = 32, normalized size = 2. \begin{align*} \frac{1}{6} \, b^{2} x^{6} + \frac{1}{2} \, a b x^{4} + \frac{1}{2} \, a^{2} x^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*b^2*x^6 + 1/2*a*b*x^4 + 1/2*a^2*x^2

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Fricas [A]  time = 1.34173, size = 55, normalized size = 3.44 \begin{align*} \frac{1}{6} \, b^{2} x^{6} + \frac{1}{2} \, a b x^{4} + \frac{1}{2} \, a^{2} x^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*b^2*x^6 + 1/2*a*b*x^4 + 1/2*a^2*x^2

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Sympy [B]  time = 0.067713, size = 24, normalized size = 1.5 \begin{align*} \frac{a^{2} x^{2}}{2} + \frac{a b x^{4}}{2} + \frac{b^{2} x^{6}}{6} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a**2*x**2/2 + a*b*x**4/2 + b**2*x**6/6

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Giac [A]  time = 1.17046, size = 32, normalized size = 2. \begin{align*} \frac{1}{6} \, b^{2} x^{6} + \frac{1}{2} \, a b x^{4} + \frac{1}{2} \, a^{2} x^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*b^2*x^6 + 1/2*a*b*x^4 + 1/2*a^2*x^2