3.3.40 \(\int x^2 (a+b x^3)^3 \, dx\) [240]

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

[Out]

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

________________________________________________________________________________________

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 )^4}{12 b} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^2*(a + b*x^3)^3,x]

[Out]

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

________________________________________________________________________________________

Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(43\) vs. \(2(16)=32\).
time = 0.00, size = 43, normalized size = 2.69 \begin {gather*} \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} \end {gather*}

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

________________________________________________________________________________________

Maple [A]
time = 0.14, size = 15, normalized size = 0.94

method	result	size
default	\(\frac {\left (b \,x^{3}+a \right )^{4}}{12 b}\)	\(15\)
gosper	\(\frac {1}{12} b^{3} x^{12}+\frac {1}{3} a \,b^{2} x^{9}+\frac {1}{2} a^{2} b \,x^{6}+\frac {1}{3} a^{3} x^{3}\)	\(36\)
norman	\(\frac {1}{12} b^{3} x^{12}+\frac {1}{3} a \,b^{2} x^{9}+\frac {1}{2} a^{2} b \,x^{6}+\frac {1}{3} a^{3} x^{3}\)	\(36\)
risch	\(\frac {b^{3} x^{12}}{12}+\frac {a \,b^{2} x^{9}}{3}+\frac {a^{2} b \,x^{6}}{2}+\frac {a^{3} x^{3}}{3}+\frac {a^{4}}{12 b}\)	\(44\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [A]
time = 0.30, size = 14, normalized size = 0.88 \begin {gather*} \frac {{\left (b x^{3} + a\right )}^{4}}{12 \, b} \end {gather*}

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

________________________________________________________________________________________

Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 35 vs. \(2 (14) = 28\).
time = 0.34, size = 35, normalized size = 2.19 \begin {gather*} \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} \end {gather*}

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

________________________________________________________________________________________

Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 36 vs. \(2 (10) = 20\).
time = 0.01, size = 36, normalized size = 2.25 \begin {gather*} \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} \end {gather*}

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

________________________________________________________________________________________

Giac [A]
time = 1.48, size = 14, normalized size = 0.88 \begin {gather*} \frac {{\left (b x^{3} + a\right )}^{4}}{12 \, b} \end {gather*}

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

________________________________________________________________________________________

Mupad [B]
time = 0.04, size = 35, normalized size = 2.19 \begin {gather*} \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} \end {gather*}

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

________________________________________________________________________________________