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

Optimal. Leaf size=26 \[ a^2 \log (x)+\frac{2}{3} a b x^3+\frac{b^2 x^6}{6} \]

[Out]

(2*a*b*x^3)/3 + (b^2*x^6)/6 + a^2*Log[x]

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Rubi [A]  time = 0.0120073, antiderivative size = 26, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {266, 43} \[ a^2 \log (x)+\frac{2}{3} a b x^3+\frac{b^2 x^6}{6} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^3)^2/x,x]

[Out]

(2*a*b*x^3)/3 + (b^2*x^6)/6 + a^2*Log[x]

Rule 266

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

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int \frac{\left (a+b x^3\right )^2}{x} \, dx &=\frac{1}{3} \operatorname{Subst}\left (\int \frac{(a+b x)^2}{x} \, dx,x,x^3\right )\\ &=\frac{1}{3} \operatorname{Subst}\left (\int \left (2 a b+\frac{a^2}{x}+b^2 x\right ) \, dx,x,x^3\right )\\ &=\frac{2}{3} a b x^3+\frac{b^2 x^6}{6}+a^2 \log (x)\\ \end{align*}

Mathematica [A]  time = 0.0007127, size = 26, normalized size = 1. \[ a^2 \log (x)+\frac{2}{3} a b x^3+\frac{b^2 x^6}{6} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^3)^2/x,x]

[Out]

(2*a*b*x^3)/3 + (b^2*x^6)/6 + a^2*Log[x]

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Maple [A]  time = 0.002, size = 23, normalized size = 0.9 \begin{align*}{\frac{2\,{x}^{3}ab}{3}}+{\frac{{b}^{2}{x}^{6}}{6}}+{a}^{2}\ln \left ( x \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3*x^3*a*b+1/6*b^2*x^6+a^2*ln(x)

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Maxima [A]  time = 0.965412, size = 34, normalized size = 1.31 \begin{align*} \frac{1}{6} \, b^{2} x^{6} + \frac{2}{3} \, a b x^{3} + \frac{1}{3} \, a^{2} \log \left (x^{3}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*b^2*x^6 + 2/3*a*b*x^3 + 1/3*a^2*log(x^3)

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Fricas [A]  time = 1.6821, size = 54, normalized size = 2.08 \begin{align*} \frac{1}{6} \, b^{2} x^{6} + \frac{2}{3} \, a b x^{3} + a^{2} \log \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*b^2*x^6 + 2/3*a*b*x^3 + a^2*log(x)

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Sympy [A]  time = 0.330352, size = 24, normalized size = 0.92 \begin{align*} a^{2} \log{\left (x \right )} + \frac{2 a b x^{3}}{3} + \frac{b^{2} x^{6}}{6} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a**2*log(x) + 2*a*b*x**3/3 + b**2*x**6/6

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Giac [A]  time = 1.13771, size = 31, normalized size = 1.19 \begin{align*} \frac{1}{6} \, b^{2} x^{6} + \frac{2}{3} \, a b x^{3} + a^{2} \log \left ({\left | x \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*b^2*x^6 + 2/3*a*b*x^3 + a^2*log(abs(x))