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]

(a + b*x^3)^9/(27*b)

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Rubi [A]  time = 0.0028053, antiderivative size = 16, normalized size of antiderivative = 1., 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*}

Mathematica [B]  time = 0.0026088, size = 108, normalized size = 6.75 \[ \frac{4}{3} a^2 b^6 x^{21}+\frac{28}{9} a^3 b^5 x^{18}+\frac{14}{3} a^4 b^4 x^{15}+\frac{14}{3} a^5 b^3 x^{12}+\frac{28}{9} a^6 b^2 x^9+\frac{4}{3} a^7 b x^6+\frac{a^8 x^3}{3}+\frac{1}{3} a b^7 x^{24}+\frac{b^8 x^{27}}{27} \]

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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Maple [B]  time = 0.001, size = 91, normalized size = 5.7 \begin{align*}{\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\,{a}^{6}{b}^{2}{x}^{9}}{9}}+{\frac{4\,{a}^{7}b{x}^{6}}{3}}+{\frac{{a}^{8}{x}^{3}}{3}} \end{align*}

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 = 0.950321, size = 19, normalized size = 1.19 \begin{align*} \frac{{\left (b x^{3} + a\right )}^{9}}{27 \, b} \end{align*}

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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Fricas [B]  time = 1.48985, size = 216, normalized size = 13.5 \begin{align*} \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} \end{align*}

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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Sympy [B]  time = 0.099723, size = 105, normalized size = 6.56 \begin{align*} \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} \end{align*}

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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Giac [A]  time = 1.11054, size = 19, normalized size = 1.19 \begin{align*} \frac{{\left (b x^{3} + a\right )}^{9}}{27 \, b} \end{align*}

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