3.291 \(\int x^2 (a+b x^3)^8 \, dx\)

Optimal. Leaf size=16 \[ \frac {\left (a+b x^3\right )^9}{27 b} \]

[Out]

1/27*(b*x^3+a)^9/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 = {261} \[ \frac {\left (a+b x^3\right )^9}{27 b} \]

Antiderivative was successfully verified.

[In]

Int[x^2*(a + b*x^3)^8,x]

[Out]

(a + b*x^3)^9/(27*b)

Rule 261

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 )^8 \, dx &=\frac {\left (a+b x^3\right )^9}{27 b}\\ \end {align*}

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Mathematica [B]  time = 0.00, size = 108, normalized size = 6.75 \[ \frac {a^8 x^3}{3}+\frac {4}{3} a^7 b x^6+\frac {28}{9} a^6 b^2 x^9+\frac {14}{3} a^5 b^3 x^{12}+\frac {14}{3} a^4 b^4 x^{15}+\frac {28}{9} a^3 b^5 x^{18}+\frac {4}{3} a^2 b^6 x^{21}+\frac {1}{3} a b^7 x^{24}+\frac {b^8 x^{27}}{27} \]

Antiderivative was successfully verified.

[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

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fricas [B]  time = 0.73, size = 90, normalized size = 5.62 \[ \frac {1}{27} x^{27} b^{8} + \frac {1}{3} x^{24} b^{7} a + \frac {4}{3} x^{21} b^{6} a^{2} + \frac {28}{9} x^{18} b^{5} a^{3} + \frac {14}{3} x^{15} b^{4} a^{4} + \frac {14}{3} x^{12} b^{3} a^{5} + \frac {28}{9} x^{9} b^{2} a^{6} + \frac {4}{3} x^{6} b a^{7} + \frac {1}{3} x^{3} a^{8} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.18, size = 14, normalized size = 0.88 \[ \frac {{\left (b x^{3} + a\right )}^{9}}{27 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/27*(b*x^3 + a)^9/b

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maple [B]  time = 0.00, size = 91, normalized size = 5.69 \[ \frac {1}{27} b^{8} x^{27}+\frac {1}{3} a \,b^{7} x^{24}+\frac {4}{3} a^{2} b^{6} x^{21}+\frac {28}{9} a^{3} b^{5} x^{18}+\frac {14}{3} a^{4} b^{4} x^{15}+\frac {14}{3} a^{5} b^{3} x^{12}+\frac {28}{9} a^{6} b^{2} x^{9}+\frac {4}{3} a^{7} b \,x^{6}+\frac {1}{3} a^{8} x^{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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

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maxima [A]  time = 1.27, size = 14, normalized size = 0.88 \[ \frac {{\left (b x^{3} + a\right )}^{9}}{27 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/27*(b*x^3 + a)^9/b

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mupad [B]  time = 0.09, size = 14, normalized size = 0.88 \[ \frac {{\left (b\,x^3+a\right )}^9}{27\,b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*(a + b*x^3)^8,x)

[Out]

(a + b*x^3)^9/(27*b)

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sympy [B]  time = 0.09, size = 105, normalized size = 6.56 \[ \frac {a^{8} x^{3}}{3} + \frac {4 a^{7} b x^{6}}{3} + \frac {28 a^{6} b^{2} x^{9}}{9} + \frac {14 a^{5} b^{3} x^{12}}{3} + \frac {14 a^{4} b^{4} x^{15}}{3} + \frac {28 a^{3} b^{5} x^{18}}{9} + \frac {4 a^{2} b^{6} x^{21}}{3} + \frac {a b^{7} x^{24}}{3} + \frac {b^{8} x^{27}}{27} \]

Verification of antiderivative is not currently implemented for this CAS.

[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

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