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

Optimal. Leaf size=73 \[ \frac{5}{3} a^2 b^4 x^9+\frac{20}{7} a^3 b^3 x^7+3 a^4 b^2 x^5+2 a^5 b x^3+a^6 x+\frac{6}{11} a b^5 x^{11}+\frac{b^6 x^{13}}{13} \]

[Out]

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

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Rubi [A]  time = 0.0343958, antiderivative size = 73, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {28, 194} \[ \frac{5}{3} a^2 b^4 x^9+\frac{20}{7} a^3 b^3 x^7+3 a^4 b^2 x^5+2 a^5 b x^3+a^6 x+\frac{6}{11} a b^5 x^{11}+\frac{b^6 x^{13}}{13} \]

Antiderivative was successfully verified.

[In]

Int[(a^2 + 2*a*b*x^2 + b^2*x^4)^3,x]

[Out]

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

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 194

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

Rubi steps

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

Mathematica [A]  time = 0.0013389, size = 73, normalized size = 1. \[ \frac{5}{3} a^2 b^4 x^9+\frac{20}{7} a^3 b^3 x^7+3 a^4 b^2 x^5+2 a^5 b x^3+a^6 x+\frac{6}{11} a b^5 x^{11}+\frac{b^6 x^{13}}{13} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.04, size = 66, normalized size = 0.9 \begin{align*}{a}^{6}x+2\,{a}^{5}b{x}^{3}+3\,{a}^{4}{b}^{2}{x}^{5}+{\frac{20\,{a}^{3}{b}^{3}{x}^{7}}{7}}+{\frac{5\,{a}^{2}{b}^{4}{x}^{9}}{3}}+{\frac{6\,a{b}^{5}{x}^{11}}{11}}+{\frac{{b}^{6}{x}^{13}}{13}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.97231, size = 135, normalized size = 1.85 \begin{align*} \frac{1}{13} \, b^{6} x^{13} + \frac{6}{11} \, a b^{5} x^{11} + \frac{4}{3} \, a^{2} b^{4} x^{9} + \frac{8}{7} \, a^{3} b^{3} x^{7} + a^{6} x + \frac{1}{5} \,{\left (3 \, b^{2} x^{5} + 10 \, a b x^{3}\right )} a^{4} + \frac{1}{105} \,{\left (35 \, b^{4} x^{9} + 180 \, a b^{3} x^{7} + 252 \, a^{2} b^{2} x^{5}\right )} a^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/13*b^6*x^13 + 6/11*a*b^5*x^11 + 4/3*a^2*b^4*x^9 + 8/7*a^3*b^3*x^7 + a^6*x + 1/5*(3*b^2*x^5 + 10*a*b*x^3)*a^4
+ 1/105*(35*b^4*x^9 + 180*a*b^3*x^7 + 252*a^2*b^2*x^5)*a^2

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Fricas [A]  time = 1.47239, size = 146, normalized size = 2. \begin{align*} \frac{1}{13} x^{13} b^{6} + \frac{6}{11} x^{11} b^{5} a + \frac{5}{3} x^{9} b^{4} a^{2} + \frac{20}{7} x^{7} b^{3} a^{3} + 3 x^{5} b^{2} a^{4} + 2 x^{3} b a^{5} + x a^{6} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.075363, size = 73, normalized size = 1. \begin{align*} a^{6} x + 2 a^{5} b x^{3} + 3 a^{4} b^{2} x^{5} + \frac{20 a^{3} b^{3} x^{7}}{7} + \frac{5 a^{2} b^{4} x^{9}}{3} + \frac{6 a b^{5} x^{11}}{11} + \frac{b^{6} x^{13}}{13} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.15488, size = 88, normalized size = 1.21 \begin{align*} \frac{1}{13} \, b^{6} x^{13} + \frac{6}{11} \, a b^{5} x^{11} + \frac{5}{3} \, a^{2} b^{4} x^{9} + \frac{20}{7} \, a^{3} b^{3} x^{7} + 3 \, a^{4} b^{2} x^{5} + 2 \, a^{5} b x^{3} + a^{6} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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