\(\int x^2 (a+b x^3)^8 \, dx\) [291]

Optimal result

Integrand size = 13, antiderivative size = 16 \[ \int x^2 \left (a+b x^3\right )^8 \, dx=\frac {\left (a+b x^3\right )^9}{27 b} \]

[Out]

1/27*(b*x^3+a)^9/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} \[ \int x^2 \left (a+b x^3\right )^8 \, dx=\frac {\left (a+b x^3\right )^9}{27 b} \]

[In]

Int[x^2*(a + b*x^3)^8,x]

[Out]

(a + b*x^3)^9/(27*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 )^9}{27 b} \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(108\) vs. \(2(16)=32\).

Time = 0.00 (sec) , antiderivative size = 108, normalized size of antiderivative = 6.75 \[ \int x^2 \left (a+b x^3\right )^8 \, dx=\frac {a^8 x^3}{3}+\frac {4}{3} a^7 b x^6+\frac {28}{9} a^6 b^2 x^9+\frac {14}{3} a^5 b^3 x^{12}+\frac {14}{3} a^4 b^4 x^{15}+\frac {28}{9} a^3 b^5 x^{18}+\frac {4}{3} a^2 b^6 x^{21}+\frac {1}{3} a b^7 x^{24}+\frac {b^8 x^{27}}{27} \]

[In]

Integrate[x^2*(a + b*x^3)^8,x]

[Out]

(a^8*x^3)/3 + (4*a^7*b*x^6)/3 + (28*a^6*b^2*x^9)/9 + (14*a^5*b^3*x^12)/3 + (14*a^4*b^4*x^15)/3 + (28*a^3*b^5*x
^18)/9 + (4*a^2*b^6*x^21)/3 + (a*b^7*x^24)/3 + (b^8*x^27)/27

Maple [A] (verified)

Time = 3.65 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.94

method	result	size
default	\(\frac {\left (b \,x^{3}+a \right )^{9}}{27 b}\)	\(15\)
gosper	\(\frac {1}{3} x^{3} a^{8}+\frac {4}{3} a^{7} b \,x^{6}+\frac {28}{9} x^{9} b^{2} a^{6}+\frac {14}{3} a^{5} b^{3} x^{12}+\frac {14}{3} a^{4} b^{4} x^{15}+\frac {28}{9} a^{3} b^{5} x^{18}+\frac {4}{3} a^{2} b^{6} x^{21}+\frac {1}{3} a \,b^{7} x^{24}+\frac {1}{27} b^{8} x^{27}\)	\(91\)
norman	\(\frac {1}{3} x^{3} a^{8}+\frac {4}{3} a^{7} b \,x^{6}+\frac {28}{9} x^{9} b^{2} a^{6}+\frac {14}{3} a^{5} b^{3} x^{12}+\frac {14}{3} a^{4} b^{4} x^{15}+\frac {28}{9} a^{3} b^{5} x^{18}+\frac {4}{3} a^{2} b^{6} x^{21}+\frac {1}{3} a \,b^{7} x^{24}+\frac {1}{27} b^{8} x^{27}\)	\(91\)
parallelrisch	\(\frac {1}{3} x^{3} a^{8}+\frac {4}{3} a^{7} b \,x^{6}+\frac {28}{9} x^{9} b^{2} a^{6}+\frac {14}{3} a^{5} b^{3} x^{12}+\frac {14}{3} a^{4} b^{4} x^{15}+\frac {28}{9} a^{3} b^{5} x^{18}+\frac {4}{3} a^{2} b^{6} x^{21}+\frac {1}{3} a \,b^{7} x^{24}+\frac {1}{27} b^{8} x^{27}\)	\(91\)
risch	\(\frac {b^{8} x^{27}}{27}+\frac {a \,b^{7} x^{24}}{3}+\frac {4 a^{2} b^{6} x^{21}}{3}+\frac {28 a^{3} b^{5} x^{18}}{9}+\frac {14 a^{4} b^{4} x^{15}}{3}+\frac {14 a^{5} b^{3} x^{12}}{3}+\frac {28 x^{9} b^{2} a^{6}}{9}+\frac {4 a^{7} b \,x^{6}}{3}+\frac {x^{3} a^{8}}{3}+\frac {a^{9}}{27 b}\)	\(99\)

[In]

int(x^2*(b*x^3+a)^8,x,method=_RETURNVERBOSE)

[Out]

1/27*(b*x^3+a)^9/b

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 90 vs. \(2 (14) = 28\).

Time = 0.28 (sec) , antiderivative size = 90, normalized size of antiderivative = 5.62 \[ \int x^2 \left (a+b x^3\right )^8 \, dx=\frac {1}{27} \, b^{8} x^{27} + \frac {1}{3} \, a b^{7} x^{24} + \frac {4}{3} \, a^{2} b^{6} x^{21} + \frac {28}{9} \, a^{3} b^{5} x^{18} + \frac {14}{3} \, a^{4} b^{4} x^{15} + \frac {14}{3} \, a^{5} b^{3} x^{12} + \frac {28}{9} \, a^{6} b^{2} x^{9} + \frac {4}{3} \, a^{7} b x^{6} + \frac {1}{3} \, a^{8} x^{3} \]

[In]

integrate(x^2*(b*x^3+a)^8,x, algorithm="fricas")

[Out]

1/27*b^8*x^27 + 1/3*a*b^7*x^24 + 4/3*a^2*b^6*x^21 + 28/9*a^3*b^5*x^18 + 14/3*a^4*b^4*x^15 + 14/3*a^5*b^3*x^12
+ 28/9*a^6*b^2*x^9 + 4/3*a^7*b*x^6 + 1/3*a^8*x^3

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 105 vs. \(2 (10) = 20\).

Time = 0.03 (sec) , antiderivative size = 105, normalized size of antiderivative = 6.56 \[ \int x^2 \left (a+b x^3\right )^8 \, dx=\frac {a^{8} x^{3}}{3} + \frac {4 a^{7} b x^{6}}{3} + \frac {28 a^{6} b^{2} x^{9}}{9} + \frac {14 a^{5} b^{3} x^{12}}{3} + \frac {14 a^{4} b^{4} x^{15}}{3} + \frac {28 a^{3} b^{5} x^{18}}{9} + \frac {4 a^{2} b^{6} x^{21}}{3} + \frac {a b^{7} x^{24}}{3} + \frac {b^{8} x^{27}}{27} \]

[In]

integrate(x**2*(b*x**3+a)**8,x)

[Out]

a**8*x**3/3 + 4*a**7*b*x**6/3 + 28*a**6*b**2*x**9/9 + 14*a**5*b**3*x**12/3 + 14*a**4*b**4*x**15/3 + 28*a**3*b*
*5*x**18/9 + 4*a**2*b**6*x**21/3 + a*b**7*x**24/3 + b**8*x**27/27

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.88 \[ \int x^2 \left (a+b x^3\right )^8 \, dx=\frac {{\left (b x^{3} + a\right )}^{9}}{27 \, b} \]

[In]

integrate(x^2*(b*x^3+a)^8,x, algorithm="maxima")

[Out]

1/27*(b*x^3 + a)^9/b

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.88 \[ \int x^2 \left (a+b x^3\right )^8 \, dx=\frac {{\left (b x^{3} + a\right )}^{9}}{27 \, b} \]

[In]

integrate(x^2*(b*x^3+a)^8,x, algorithm="giac")

[Out]

1/27*(b*x^3 + a)^9/b

Mupad [B] (verification not implemented)

Time = 5.69 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.88 \[ \int x^2 \left (a+b x^3\right )^8 \, dx=\frac {{\left (b\,x^3+a\right )}^9}{27\,b} \]

[In]

int(x^2*(a + b*x^3)^8,x)

[Out]

(a + b*x^3)^9/(27*b)