3.263 \(\int x^2 (a+b x^3)^5 \, dx\)

Optimal. Leaf size=16 \[ \frac{\left (a+b x^3\right )^6}{18 b} \]

[Out]

(a + b*x^3)^6/(18*b)

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Rubi [A]  time = 0.0028552, antiderivative size = 16, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {261} \[ \frac{\left (a+b x^3\right )^6}{18 b} \]

Antiderivative was successfully verified.

[In]

Int[x^2*(a + b*x^3)^5,x]

[Out]

(a + b*x^3)^6/(18*b)

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 x^2 \left (a+b x^3\right )^5 \, dx &=\frac{\left (a+b x^3\right )^6}{18 b}\\ \end{align*}

Mathematica [B]  time = 0.0021107, size = 69, normalized size = 4.31 \[ \frac{5}{6} a^2 b^3 x^{12}+\frac{10}{9} a^3 b^2 x^9+\frac{5}{6} a^4 b x^6+\frac{a^5 x^3}{3}+\frac{1}{3} a b^4 x^{15}+\frac{b^5 x^{18}}{18} \]

Antiderivative was successfully verified.

[In]

Integrate[x^2*(a + b*x^3)^5,x]

[Out]

(a^5*x^3)/3 + (5*a^4*b*x^6)/6 + (10*a^3*b^2*x^9)/9 + (5*a^2*b^3*x^12)/6 + (a*b^4*x^15)/3 + (b^5*x^18)/18

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Maple [B]  time = 0.002, size = 58, normalized size = 3.6 \begin{align*}{\frac{{b}^{5}{x}^{18}}{18}}+{\frac{a{b}^{4}{x}^{15}}{3}}+{\frac{5\,{a}^{2}{b}^{3}{x}^{12}}{6}}+{\frac{10\,{a}^{3}{b}^{2}{x}^{9}}{9}}+{\frac{5\,{a}^{4}b{x}^{6}}{6}}+{\frac{{a}^{5}{x}^{3}}{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*(b*x^3+a)^5,x)

[Out]

1/18*b^5*x^18+1/3*a*b^4*x^15+5/6*a^2*b^3*x^12+10/9*a^3*b^2*x^9+5/6*a^4*b*x^6+1/3*a^5*x^3

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Maxima [A]  time = 0.992131, size = 19, normalized size = 1.19 \begin{align*} \frac{{\left (b x^{3} + a\right )}^{6}}{18 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(b*x^3+a)^5,x, algorithm="maxima")

[Out]

1/18*(b*x^3 + a)^6/b

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Fricas [B]  time = 1.42604, size = 135, normalized size = 8.44 \begin{align*} \frac{1}{18} x^{18} b^{5} + \frac{1}{3} x^{15} b^{4} a + \frac{5}{6} x^{12} b^{3} a^{2} + \frac{10}{9} x^{9} b^{2} a^{3} + \frac{5}{6} x^{6} b a^{4} + \frac{1}{3} x^{3} a^{5} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(b*x^3+a)^5,x, algorithm="fricas")

[Out]

1/18*x^18*b^5 + 1/3*x^15*b^4*a + 5/6*x^12*b^3*a^2 + 10/9*x^9*b^2*a^3 + 5/6*x^6*b*a^4 + 1/3*x^3*a^5

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Sympy [B]  time = 0.072683, size = 65, normalized size = 4.06 \begin{align*} \frac{a^{5} x^{3}}{3} + \frac{5 a^{4} b x^{6}}{6} + \frac{10 a^{3} b^{2} x^{9}}{9} + \frac{5 a^{2} b^{3} x^{12}}{6} + \frac{a b^{4} x^{15}}{3} + \frac{b^{5} x^{18}}{18} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**2*(b*x**3+a)**5,x)

[Out]

a**5*x**3/3 + 5*a**4*b*x**6/6 + 10*a**3*b**2*x**9/9 + 5*a**2*b**3*x**12/6 + a*b**4*x**15/3 + b**5*x**18/18

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Giac [A]  time = 1.15885, size = 19, normalized size = 1.19 \begin{align*} \frac{{\left (b x^{3} + a\right )}^{6}}{18 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(b*x^3+a)^5,x, algorithm="giac")

[Out]

1/18*(b*x^3 + a)^6/b