3.446 \(\int x^6 (a^2+2 a b x^2+b^2 x^4)^3 \, dx\)

Optimal. Leaf size=79 \[ a^2 b^4 x^{15}+\frac{20}{13} a^3 b^3 x^{13}+\frac{15}{11} a^4 b^2 x^{11}+\frac{2}{3} a^5 b x^9+\frac{a^6 x^7}{7}+\frac{6}{17} a b^5 x^{17}+\frac{b^6 x^{19}}{19} \]

[Out]

(a^6*x^7)/7 + (2*a^5*b*x^9)/3 + (15*a^4*b^2*x^11)/11 + (20*a^3*b^3*x^13)/13 + a^2*b^4*x^15 + (6*a*b^5*x^17)/17
+ (b^6*x^19)/19

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Rubi [A]  time = 0.0380232, antiderivative size = 79, 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} \[ a^2 b^4 x^{15}+\frac{20}{13} a^3 b^3 x^{13}+\frac{15}{11} a^4 b^2 x^{11}+\frac{2}{3} a^5 b x^9+\frac{a^6 x^7}{7}+\frac{6}{17} a b^5 x^{17}+\frac{b^6 x^{19}}{19} \]

Antiderivative was successfully verified.

[In]

Int[x^6*(a^2 + 2*a*b*x^2 + b^2*x^4)^3,x]

[Out]

(a^6*x^7)/7 + (2*a^5*b*x^9)/3 + (15*a^4*b^2*x^11)/11 + (20*a^3*b^3*x^13)/13 + a^2*b^4*x^15 + (6*a*b^5*x^17)/17
+ (b^6*x^19)/19

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

Mathematica [A]  time = 0.0024519, size = 79, normalized size = 1. \[ a^2 b^4 x^{15}+\frac{20}{13} a^3 b^3 x^{13}+\frac{15}{11} a^4 b^2 x^{11}+\frac{2}{3} a^5 b x^9+\frac{a^6 x^7}{7}+\frac{6}{17} a b^5 x^{17}+\frac{b^6 x^{19}}{19} \]

Antiderivative was successfully verified.

[In]

Integrate[x^6*(a^2 + 2*a*b*x^2 + b^2*x^4)^3,x]

[Out]

(a^6*x^7)/7 + (2*a^5*b*x^9)/3 + (15*a^4*b^2*x^11)/11 + (20*a^3*b^3*x^13)/13 + a^2*b^4*x^15 + (6*a*b^5*x^17)/17
+ (b^6*x^19)/19

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Maple [A]  time = 0.043, size = 68, normalized size = 0.9 \begin{align*}{\frac{{a}^{6}{x}^{7}}{7}}+{\frac{2\,{a}^{5}b{x}^{9}}{3}}+{\frac{15\,{a}^{4}{b}^{2}{x}^{11}}{11}}+{\frac{20\,{a}^{3}{b}^{3}{x}^{13}}{13}}+{a}^{2}{b}^{4}{x}^{15}+{\frac{6\,a{b}^{5}{x}^{17}}{17}}+{\frac{{b}^{6}{x}^{19}}{19}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^6*(b^2*x^4+2*a*b*x^2+a^2)^3,x)

[Out]

1/7*a^6*x^7+2/3*a^5*b*x^9+15/11*a^4*b^2*x^11+20/13*a^3*b^3*x^13+a^2*b^4*x^15+6/17*a*b^5*x^17+1/19*b^6*x^19

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Maxima [A]  time = 1.00599, size = 90, normalized size = 1.14 \begin{align*} \frac{1}{19} \, b^{6} x^{19} + \frac{6}{17} \, a b^{5} x^{17} + a^{2} b^{4} x^{15} + \frac{20}{13} \, a^{3} b^{3} x^{13} + \frac{15}{11} \, a^{4} b^{2} x^{11} + \frac{2}{3} \, a^{5} b x^{9} + \frac{1}{7} \, a^{6} x^{7} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^6*(b^2*x^4+2*a*b*x^2+a^2)^3,x, algorithm="maxima")

[Out]

1/19*b^6*x^19 + 6/17*a*b^5*x^17 + a^2*b^4*x^15 + 20/13*a^3*b^3*x^13 + 15/11*a^4*b^2*x^11 + 2/3*a^5*b*x^9 + 1/7
*a^6*x^7

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Fricas [A]  time = 1.45639, size = 162, normalized size = 2.05 \begin{align*} \frac{1}{19} x^{19} b^{6} + \frac{6}{17} x^{17} b^{5} a + x^{15} b^{4} a^{2} + \frac{20}{13} x^{13} b^{3} a^{3} + \frac{15}{11} x^{11} b^{2} a^{4} + \frac{2}{3} x^{9} b a^{5} + \frac{1}{7} x^{7} a^{6} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^6*(b^2*x^4+2*a*b*x^2+a^2)^3,x, algorithm="fricas")

[Out]

1/19*x^19*b^6 + 6/17*x^17*b^5*a + x^15*b^4*a^2 + 20/13*x^13*b^3*a^3 + 15/11*x^11*b^2*a^4 + 2/3*x^9*b*a^5 + 1/7
*x^7*a^6

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Sympy [A]  time = 0.078663, size = 76, normalized size = 0.96 \begin{align*} \frac{a^{6} x^{7}}{7} + \frac{2 a^{5} b x^{9}}{3} + \frac{15 a^{4} b^{2} x^{11}}{11} + \frac{20 a^{3} b^{3} x^{13}}{13} + a^{2} b^{4} x^{15} + \frac{6 a b^{5} x^{17}}{17} + \frac{b^{6} x^{19}}{19} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**6*(b**2*x**4+2*a*b*x**2+a**2)**3,x)

[Out]

a**6*x**7/7 + 2*a**5*b*x**9/3 + 15*a**4*b**2*x**11/11 + 20*a**3*b**3*x**13/13 + a**2*b**4*x**15 + 6*a*b**5*x**
17/17 + b**6*x**19/19

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Giac [A]  time = 1.14545, size = 90, normalized size = 1.14 \begin{align*} \frac{1}{19} \, b^{6} x^{19} + \frac{6}{17} \, a b^{5} x^{17} + a^{2} b^{4} x^{15} + \frac{20}{13} \, a^{3} b^{3} x^{13} + \frac{15}{11} \, a^{4} b^{2} x^{11} + \frac{2}{3} \, a^{5} b x^{9} + \frac{1}{7} \, a^{6} x^{7} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^6*(b^2*x^4+2*a*b*x^2+a^2)^3,x, algorithm="giac")

[Out]

1/19*b^6*x^19 + 6/17*a*b^5*x^17 + a^2*b^4*x^15 + 20/13*a^3*b^3*x^13 + 15/11*a^4*b^2*x^11 + 2/3*a^5*b*x^9 + 1/7
*a^6*x^7