3.3.20 \(\int x^2 (a+b x^3)^2 \, dx\) [220]

Optimal. Leaf size=16 \[ \frac {\left (a+b x^3\right )^3}{9 b} \]

[Out]

1/9*(b*x^3+a)^3/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 = {267} \begin {gather*} \frac {\left (a+b x^3\right )^3}{9 b} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^2*(a + b*x^3)^2,x]

[Out]

(a + b*x^3)^3/(9*b)

Rule 267

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 )^2 \, dx &=\frac {\left (a+b x^3\right )^3}{9 b}\\ \end {align*}

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Mathematica [A]
time = 0.00, size = 30, normalized size = 1.88 \begin {gather*} \frac {a^2 x^3}{3}+\frac {1}{3} a b x^6+\frac {b^2 x^9}{9} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^2*(a + b*x^3)^2,x]

[Out]

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

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Maple [A]
time = 0.13, size = 15, normalized size = 0.94

method	result	size
default	\(\frac {\left (b \,x^{3}+a \right )^{3}}{9 b}\)	\(15\)
gosper	\(\frac {1}{9} b^{2} x^{9}+\frac {1}{3} a b \,x^{6}+\frac {1}{3} a^{2} x^{3}\)	\(25\)
norman	\(\frac {1}{9} b^{2} x^{9}+\frac {1}{3} a b \,x^{6}+\frac {1}{3} a^{2} x^{3}\)	\(25\)
risch	\(\frac {b^{2} x^{9}}{9}+\frac {a b \,x^{6}}{3}+\frac {a^{2} x^{3}}{3}+\frac {a^{3}}{9 b}\)	\(33\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/9*(b*x^3+a)^3/b

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Maxima [A]
time = 0.29, size = 14, normalized size = 0.88 \begin {gather*} \frac {{\left (b x^{3} + a\right )}^{3}}{9 \, b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/9*(b*x^3 + a)^3/b

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Fricas [A]
time = 0.34, size = 24, normalized size = 1.50 \begin {gather*} \frac {1}{9} \, b^{2} x^{9} + \frac {1}{3} \, a b x^{6} + \frac {1}{3} \, a^{2} x^{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 24 vs. \(2 (10) = 20\).
time = 0.01, size = 24, normalized size = 1.50 \begin {gather*} \frac {a^{2} x^{3}}{3} + \frac {a b x^{6}}{3} + \frac {b^{2} x^{9}}{9} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]
time = 2.47, size = 14, normalized size = 0.88 \begin {gather*} \frac {{\left (b x^{3} + a\right )}^{3}}{9 \, b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/9*(b*x^3 + a)^3/b

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Mupad [B]
time = 0.03, size = 24, normalized size = 1.50 \begin {gather*} \frac {a^2\,x^3}{3}+\frac {a\,b\,x^6}{3}+\frac {b^2\,x^9}{9} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*(a + b*x^3)^2,x)

[Out]

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

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