3.444 \(\int x^8 (a^2+2 a b x^2+b^2 x^4)^3 \, dx\)

Optimal. Leaf size=82 \[ \frac{15}{17} a^2 b^4 x^{17}+\frac{4}{3} a^3 b^3 x^{15}+\frac{15}{13} a^4 b^2 x^{13}+\frac{6}{11} a^5 b x^{11}+\frac{a^6 x^9}{9}+\frac{6}{19} a b^5 x^{19}+\frac{b^6 x^{21}}{21} \]

[Out]

(a^6*x^9)/9 + (6*a^5*b*x^11)/11 + (15*a^4*b^2*x^13)/13 + (4*a^3*b^3*x^15)/3 + (15*a^2*b^4*x^17)/17 + (6*a*b^5*
x^19)/19 + (b^6*x^21)/21

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Rubi [A]  time = 0.0456947, antiderivative size = 82, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083, Rules used = {28, 270} \[ \frac{15}{17} a^2 b^4 x^{17}+\frac{4}{3} a^3 b^3 x^{15}+\frac{15}{13} a^4 b^2 x^{13}+\frac{6}{11} a^5 b x^{11}+\frac{a^6 x^9}{9}+\frac{6}{19} a b^5 x^{19}+\frac{b^6 x^{21}}{21} \]

Antiderivative was successfully verified.

[In]

Int[x^8*(a^2 + 2*a*b*x^2 + b^2*x^4)^3,x]

[Out]

(a^6*x^9)/9 + (6*a^5*b*x^11)/11 + (15*a^4*b^2*x^13)/13 + (4*a^3*b^3*x^15)/3 + (15*a^2*b^4*x^17)/17 + (6*a*b^5*
x^19)/19 + (b^6*x^21)/21

Rule 28

Int[(u_.)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Dist[1/c^p, Int[u*(b/2 + c*x^n)^(2*
p), x], x] /; FreeQ[{a, b, c, n}, x] && EqQ[n2, 2*n] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]

Rule 270

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*(a + b*x^n)^p,
x], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0]

Rubi steps

\begin{align*} \int x^8 \left (a^2+2 a b x^2+b^2 x^4\right )^3 \, dx &=\frac{\int x^8 \left (a b+b^2 x^2\right )^6 \, dx}{b^6}\\ &=\frac{\int \left (a^6 b^6 x^8+6 a^5 b^7 x^{10}+15 a^4 b^8 x^{12}+20 a^3 b^9 x^{14}+15 a^2 b^{10} x^{16}+6 a b^{11} x^{18}+b^{12} x^{20}\right ) \, dx}{b^6}\\ &=\frac{a^6 x^9}{9}+\frac{6}{11} a^5 b x^{11}+\frac{15}{13} a^4 b^2 x^{13}+\frac{4}{3} a^3 b^3 x^{15}+\frac{15}{17} a^2 b^4 x^{17}+\frac{6}{19} a b^5 x^{19}+\frac{b^6 x^{21}}{21}\\ \end{align*}

Mathematica [A]  time = 0.0031235, size = 82, normalized size = 1. \[ \frac{15}{17} a^2 b^4 x^{17}+\frac{4}{3} a^3 b^3 x^{15}+\frac{15}{13} a^4 b^2 x^{13}+\frac{6}{11} a^5 b x^{11}+\frac{a^6 x^9}{9}+\frac{6}{19} a b^5 x^{19}+\frac{b^6 x^{21}}{21} \]

Antiderivative was successfully verified.

[In]

Integrate[x^8*(a^2 + 2*a*b*x^2 + b^2*x^4)^3,x]

[Out]

(a^6*x^9)/9 + (6*a^5*b*x^11)/11 + (15*a^4*b^2*x^13)/13 + (4*a^3*b^3*x^15)/3 + (15*a^2*b^4*x^17)/17 + (6*a*b^5*
x^19)/19 + (b^6*x^21)/21

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Maple [A]  time = 0.042, size = 69, normalized size = 0.8 \begin{align*}{\frac{{a}^{6}{x}^{9}}{9}}+{\frac{6\,{a}^{5}b{x}^{11}}{11}}+{\frac{15\,{a}^{4}{b}^{2}{x}^{13}}{13}}+{\frac{4\,{a}^{3}{b}^{3}{x}^{15}}{3}}+{\frac{15\,{a}^{2}{b}^{4}{x}^{17}}{17}}+{\frac{6\,a{b}^{5}{x}^{19}}{19}}+{\frac{{b}^{6}{x}^{21}}{21}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^8*(b^2*x^4+2*a*b*x^2+a^2)^3,x)

[Out]

1/9*a^6*x^9+6/11*a^5*b*x^11+15/13*a^4*b^2*x^13+4/3*a^3*b^3*x^15+15/17*a^2*b^4*x^17+6/19*a*b^5*x^19+1/21*b^6*x^
21

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Maxima [A]  time = 0.980238, size = 92, normalized size = 1.12 \begin{align*} \frac{1}{21} \, b^{6} x^{21} + \frac{6}{19} \, a b^{5} x^{19} + \frac{15}{17} \, a^{2} b^{4} x^{17} + \frac{4}{3} \, a^{3} b^{3} x^{15} + \frac{15}{13} \, a^{4} b^{2} x^{13} + \frac{6}{11} \, a^{5} b x^{11} + \frac{1}{9} \, a^{6} x^{9} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^8*(b^2*x^4+2*a*b*x^2+a^2)^3,x, algorithm="maxima")

[Out]

1/21*b^6*x^21 + 6/19*a*b^5*x^19 + 15/17*a^2*b^4*x^17 + 4/3*a^3*b^3*x^15 + 15/13*a^4*b^2*x^13 + 6/11*a^5*b*x^11
+ 1/9*a^6*x^9

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Fricas [A]  time = 1.72178, size = 170, normalized size = 2.07 \begin{align*} \frac{1}{21} x^{21} b^{6} + \frac{6}{19} x^{19} b^{5} a + \frac{15}{17} x^{17} b^{4} a^{2} + \frac{4}{3} x^{15} b^{3} a^{3} + \frac{15}{13} x^{13} b^{2} a^{4} + \frac{6}{11} x^{11} b a^{5} + \frac{1}{9} x^{9} a^{6} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^8*(b^2*x^4+2*a*b*x^2+a^2)^3,x, algorithm="fricas")

[Out]

1/21*x^21*b^6 + 6/19*x^19*b^5*a + 15/17*x^17*b^4*a^2 + 4/3*x^15*b^3*a^3 + 15/13*x^13*b^2*a^4 + 6/11*x^11*b*a^5
+ 1/9*x^9*a^6

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Sympy [A]  time = 0.082118, size = 80, normalized size = 0.98 \begin{align*} \frac{a^{6} x^{9}}{9} + \frac{6 a^{5} b x^{11}}{11} + \frac{15 a^{4} b^{2} x^{13}}{13} + \frac{4 a^{3} b^{3} x^{15}}{3} + \frac{15 a^{2} b^{4} x^{17}}{17} + \frac{6 a b^{5} x^{19}}{19} + \frac{b^{6} x^{21}}{21} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**8*(b**2*x**4+2*a*b*x**2+a**2)**3,x)

[Out]

a**6*x**9/9 + 6*a**5*b*x**11/11 + 15*a**4*b**2*x**13/13 + 4*a**3*b**3*x**15/3 + 15*a**2*b**4*x**17/17 + 6*a*b*
*5*x**19/19 + b**6*x**21/21

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Giac [A]  time = 1.17663, size = 92, normalized size = 1.12 \begin{align*} \frac{1}{21} \, b^{6} x^{21} + \frac{6}{19} \, a b^{5} x^{19} + \frac{15}{17} \, a^{2} b^{4} x^{17} + \frac{4}{3} \, a^{3} b^{3} x^{15} + \frac{15}{13} \, a^{4} b^{2} x^{13} + \frac{6}{11} \, a^{5} b x^{11} + \frac{1}{9} \, a^{6} x^{9} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^8*(b^2*x^4+2*a*b*x^2+a^2)^3,x, algorithm="giac")

[Out]

1/21*b^6*x^21 + 6/19*a*b^5*x^19 + 15/17*a^2*b^4*x^17 + 4/3*a^3*b^3*x^15 + 15/13*a^4*b^2*x^13 + 6/11*a^5*b*x^11
+ 1/9*a^6*x^9