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

Optimal. Leaf size=16 \[ \frac {\left (a+b x^3\right )^4}{12 b} \]

[Out]

1/12*(b*x^3+a)^4/b

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Rubi [A]  time = 0.00, antiderivative size = 16, normalized size of antiderivative = 1.00, 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} \[ \frac {\left (a+b x^3\right )^4}{12 b} \]

Antiderivative was successfully verified.

[In]

Int[x^2*(a + b*x^3)^3,x]

[Out]

(a + b*x^3)^4/(12*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 )^3 \, dx &=\frac {\left (a+b x^3\right )^4}{12 b}\\ \end {align*}

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Mathematica [B]  time = 0.00, size = 43, normalized size = 2.69 \[ \frac {a^3 x^3}{3}+\frac {1}{2} a^2 b x^6+\frac {1}{3} a b^2 x^9+\frac {b^3 x^{12}}{12} \]

Antiderivative was successfully verified.

[In]

Integrate[x^2*(a + b*x^3)^3,x]

[Out]

(a^3*x^3)/3 + (a^2*b*x^6)/2 + (a*b^2*x^9)/3 + (b^3*x^12)/12

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fricas [B]  time = 0.78, size = 35, normalized size = 2.19 \[ \frac {1}{12} x^{12} b^{3} + \frac {1}{3} x^{9} b^{2} a + \frac {1}{2} x^{6} b a^{2} + \frac {1}{3} x^{3} a^{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12*x^12*b^3 + 1/3*x^9*b^2*a + 1/2*x^6*b*a^2 + 1/3*x^3*a^3

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giac [A]  time = 0.15, size = 14, normalized size = 0.88 \[ \frac {{\left (b x^{3} + a\right )}^{4}}{12 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12*(b*x^3 + a)^4/b

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maple [B]  time = 0.00, size = 36, normalized size = 2.25 \[ \frac {1}{12} b^{3} x^{12}+\frac {1}{3} a \,b^{2} x^{9}+\frac {1}{2} a^{2} b \,x^{6}+\frac {1}{3} a^{3} x^{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12*b^3*x^12+1/3*a*b^2*x^9+1/2*a^2*b*x^6+1/3*a^3*x^3

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maxima [A]  time = 1.27, size = 14, normalized size = 0.88 \[ \frac {{\left (b x^{3} + a\right )}^{4}}{12 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12*(b*x^3 + a)^4/b

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mupad [B]  time = 0.04, size = 35, normalized size = 2.19 \[ \frac {a^3\,x^3}{3}+\frac {a^2\,b\,x^6}{2}+\frac {a\,b^2\,x^9}{3}+\frac {b^3\,x^{12}}{12} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*(a + b*x^3)^3,x)

[Out]

(a^3*x^3)/3 + (b^3*x^12)/12 + (a^2*b*x^6)/2 + (a*b^2*x^9)/3

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sympy [B]  time = 0.08, size = 36, normalized size = 2.25 \[ \frac {a^{3} x^{3}}{3} + \frac {a^{2} b x^{6}}{2} + \frac {a b^{2} x^{9}}{3} + \frac {b^{3} x^{12}}{12} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a**3*x**3/3 + a**2*b*x**6/2 + a*b**2*x**9/3 + b**3*x**12/12

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