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

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

[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

_______________________________________________________________________________________

Rubi [A]  time = 0.104694, 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 \[ \frac{a^6 x^7}{7}+\frac{2}{3} a^5 b x^9+\frac{15}{11} a^4 b^2 x^{11}+\frac{20}{13} a^3 b^3 x^{13}+a^2 b^4 x^{15}+\frac{6}{17} a b^5 x^{17}+\frac{b^6 x^{19}}{19} \]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 23.1571, size = 76, normalized size = 0.96 \[ \frac{a^{6} x^{7}}{7} + \frac{2 a^{5} b x^{9}}{3} + \frac{15 a^{4} b^{2} x^{11}}{11} + \frac{20 a^{3} b^{3} x^{13}}{13} + a^{2} b^{4} x^{15} + \frac{6 a b^{5} x^{17}}{17} + \frac{b^{6} x^{19}}{19} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_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

_______________________________________________________________________________________

Mathematica [A]  time = 0.00407754, size = 79, normalized size = 1. \[ \frac{a^6 x^7}{7}+\frac{2}{3} a^5 b x^9+\frac{15}{11} a^4 b^2 x^{11}+\frac{20}{13} a^3 b^3 x^{13}+a^2 b^4 x^{15}+\frac{6}{17} a b^5 x^{17}+\frac{b^6 x^{19}}{19} \]

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

_______________________________________________________________________________________

Maple [A]  time = 0.001, size = 68, normalized size = 0.9 \[{\frac{{a}^{6}{x}^{7}}{7}}+{\frac{2\,{a}^{5}b{x}^{9}}{3}}+{\frac{15\,{a}^{4}{b}^{2}{x}^{11}}{11}}+{\frac{20\,{a}^{3}{b}^{3}{x}^{13}}{13}}+{a}^{2}{b}^{4}{x}^{15}+{\frac{6\,a{b}^{5}{x}^{17}}{17}}+{\frac{{b}^{6}{x}^{19}}{19}} \]

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+6/1
7*a*b^5*x^17+1/19*b^6*x^19

_______________________________________________________________________________________

Maxima [A]  time = 0.695062, size = 90, normalized size = 1.14 \[ \frac{1}{19} \, b^{6} x^{19} + \frac{6}{17} \, a b^{5} x^{17} + a^{2} b^{4} x^{15} + \frac{20}{13} \, a^{3} b^{3} x^{13} + \frac{15}{11} \, a^{4} b^{2} x^{11} + \frac{2}{3} \, a^{5} b x^{9} + \frac{1}{7} \, a^{6} x^{7} \]

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^6,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

_______________________________________________________________________________________

Fricas [A]  time = 0.236235, size = 1, normalized size = 0.01 \[ \frac{1}{19} x^{19} b^{6} + \frac{6}{17} x^{17} b^{5} a + x^{15} b^{4} a^{2} + \frac{20}{13} x^{13} b^{3} a^{3} + \frac{15}{11} x^{11} b^{2} a^{4} + \frac{2}{3} x^{9} b a^{5} + \frac{1}{7} x^{7} a^{6} \]

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^6,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

_______________________________________________________________________________________

Sympy [A]  time = 0.134051, size = 76, normalized size = 0.96 \[ \frac{a^{6} x^{7}}{7} + \frac{2 a^{5} b x^{9}}{3} + \frac{15 a^{4} b^{2} x^{11}}{11} + \frac{20 a^{3} b^{3} x^{13}}{13} + a^{2} b^{4} x^{15} + \frac{6 a b^{5} x^{17}}{17} + \frac{b^{6} x^{19}}{19} \]

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

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.267808, size = 90, normalized size = 1.14 \[ \frac{1}{19} \, b^{6} x^{19} + \frac{6}{17} \, a b^{5} x^{17} + a^{2} b^{4} x^{15} + \frac{20}{13} \, a^{3} b^{3} x^{13} + \frac{15}{11} \, a^{4} b^{2} x^{11} + \frac{2}{3} \, a^{5} b x^{9} + \frac{1}{7} \, a^{6} x^{7} \]

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^6,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