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

Optimal result

Integrand size = 13, antiderivative size = 16 \[ \int x^2 \left (a+b x^3\right )^2 \, dx=\frac {\left (a+b x^3\right )^3}{9 b} \]

[Out]

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

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {267} \[ \int x^2 \left (a+b x^3\right )^2 \, dx=\frac {\left (a+b x^3\right )^3}{9 b} \]

[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*} \text {integral}& = \frac {\left (a+b x^3\right )^3}{9 b} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.88 \[ \int x^2 \left (a+b x^3\right )^2 \, dx=\frac {a^2 x^3}{3}+\frac {1}{3} a b x^6+\frac {b^2 x^9}{9} \]

[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

Maple [A] (verified)

Time = 3.64 (sec) , antiderivative size = 15, normalized size of antiderivative = 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	\(\frac {1}{9} b^{2} x^{9}+\frac {1}{3} a b \,x^{6}+\frac {1}{3} a^{2} x^{3}\)	\(25\)
parallelrisch	\(\frac {1}{9} b^{2} x^{9}+\frac {1}{3} a b \,x^{6}+\frac {1}{3} a^{2} x^{3}\)	\(25\)
risch	\(\frac {b^{2} x^{9}}{9}+\frac {a b \,x^{6}}{3}+\frac {a^{2} x^{3}}{3}+\frac {a^{3}}{9 b}\)	\(33\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.50 \[ \int x^2 \left (a+b x^3\right )^2 \, dx=\frac {1}{9} \, b^{2} x^{9} + \frac {1}{3} \, a b x^{6} + \frac {1}{3} \, a^{2} x^{3} \]

[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

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 24 vs. \(2 (10) = 20\).

Time = 0.02 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.50 \[ \int x^2 \left (a+b x^3\right )^2 \, dx=\frac {a^{2} x^{3}}{3} + \frac {a b x^{6}}{3} + \frac {b^{2} x^{9}}{9} \]

[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

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.88 \[ \int x^2 \left (a+b x^3\right )^2 \, dx=\frac {{\left (b x^{3} + a\right )}^{3}}{9 \, b} \]

[In]

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

[Out]

1/9*(b*x^3 + a)^3/b

Giac [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.88 \[ \int x^2 \left (a+b x^3\right )^2 \, dx=\frac {{\left (b x^{3} + a\right )}^{3}}{9 \, b} \]

[In]

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

[Out]

1/9*(b*x^3 + a)^3/b

Mupad [B] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.50 \[ \int x^2 \left (a+b x^3\right )^2 \, dx=\frac {a^2\,x^3}{3}+\frac {a\,b\,x^6}{3}+\frac {b^2\,x^9}{9} \]

[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