3.3.63 \(\int x^2 (a+b x^3)^5 \, dx\) [263]

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

[Out]

1/18*(b*x^3+a)^6/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 )^6}{18 b} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^2*(a + b*x^3)^5,x]

[Out]

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

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(69\) vs. \(2(16)=32\).
time = 0.00, size = 69, normalized size = 4.31 \begin {gather*} \frac {a^5 x^3}{3}+\frac {5}{6} a^4 b x^6+\frac {10}{9} a^3 b^2 x^9+\frac {5}{6} a^2 b^3 x^{12}+\frac {1}{3} a b^4 x^{15}+\frac {b^5 x^{18}}{18} \end {gather*}

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

method	result	size
default	\(\frac {\left (b \,x^{3}+a \right )^{6}}{18 b}\)	\(15\)
gosper	\(\frac {1}{3} a^{5} x^{3}+\frac {5}{6} a^{4} b \,x^{6}+\frac {10}{9} a^{3} b^{2} x^{9}+\frac {5}{6} a^{2} b^{3} x^{12}+\frac {1}{3} a \,b^{4} x^{15}+\frac {1}{18} b^{5} x^{18}\)	\(58\)
norman	\(\frac {1}{3} a^{5} x^{3}+\frac {5}{6} a^{4} b \,x^{6}+\frac {10}{9} a^{3} b^{2} x^{9}+\frac {5}{6} a^{2} b^{3} x^{12}+\frac {1}{3} a \,b^{4} x^{15}+\frac {1}{18} b^{5} x^{18}\)	\(58\)
risch	\(\frac {b^{5} x^{18}}{18}+\frac {a \,b^{4} x^{15}}{3}+\frac {5 a^{2} b^{3} x^{12}}{6}+\frac {10 a^{3} b^{2} x^{9}}{9}+\frac {5 a^{4} b \,x^{6}}{6}+\frac {a^{5} x^{3}}{3}+\frac {a^{6}}{18 b}\)	\(66\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/18*(b*x^3+a)^6/b

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Maxima [A]
time = 0.30, size = 14, normalized size = 0.88 \begin {gather*} \frac {{\left (b x^{3} + a\right )}^{6}}{18 \, b} \end {gather*}

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] Leaf count of result is larger than twice the leaf count of optimal. 57 vs. \(2 (14) = 28\).
time = 0.33, size = 57, normalized size = 3.56 \begin {gather*} \frac {1}{18} \, b^{5} x^{18} + \frac {1}{3} \, a b^{4} x^{15} + \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 {1}{3} \, a^{5} x^{3} \end {gather*}

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*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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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 65 vs. \(2 (10) = 20\).
time = 0.01, size = 65, normalized size = 4.06 \begin {gather*} \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 {gather*}

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 = 1.16, size = 14, normalized size = 0.88 \begin {gather*} \frac {{\left (b x^{3} + a\right )}^{6}}{18 \, b} \end {gather*}

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

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Mupad [B]
time = 0.03, size = 57, normalized size = 3.56 \begin {gather*} \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 {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*(a + b*x^3)^5,x)

[Out]

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

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