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

Optimal. Leaf size=43 \[ \frac{a^3 x^7}{7}+\frac{1}{3} a^2 b x^9+\frac{3}{11} a b^2 x^{11}+\frac{b^3 x^{13}}{13} \]

[Out]

(a^3*x^7)/7 + (a^2*b*x^9)/3 + (3*a*b^2*x^11)/11 + (b^3*x^13)/13

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Rubi [A]  time = 0.046116, antiderivative size = 43, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077 \[ \frac{a^3 x^7}{7}+\frac{1}{3} a^2 b x^9+\frac{3}{11} a b^2 x^{11}+\frac{b^3 x^{13}}{13} \]

Antiderivative was successfully verified.

[In]  Int[x^6*(a + b*x^2)^3,x]

[Out]

(a^3*x^7)/7 + (a^2*b*x^9)/3 + (3*a*b^2*x^11)/11 + (b^3*x^13)/13

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Rubi in Sympy [A]  time = 7.4988, size = 37, normalized size = 0.86 \[ \frac{a^{3} x^{7}}{7} + \frac{a^{2} b x^{9}}{3} + \frac{3 a b^{2} x^{11}}{11} + \frac{b^{3} x^{13}}{13} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**6*(b*x**2+a)**3,x)

[Out]

a**3*x**7/7 + a**2*b*x**9/3 + 3*a*b**2*x**11/11 + b**3*x**13/13

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Mathematica [A]  time = 0.00321871, size = 43, normalized size = 1. \[ \frac{a^3 x^7}{7}+\frac{1}{3} a^2 b x^9+\frac{3}{11} a b^2 x^{11}+\frac{b^3 x^{13}}{13} \]

Antiderivative was successfully verified.

[In]  Integrate[x^6*(a + b*x^2)^3,x]

[Out]

(a^3*x^7)/7 + (a^2*b*x^9)/3 + (3*a*b^2*x^11)/11 + (b^3*x^13)/13

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Maple [A]  time = 0.002, size = 36, normalized size = 0.8 \[{\frac{{a}^{3}{x}^{7}}{7}}+{\frac{{a}^{2}b{x}^{9}}{3}}+{\frac{3\,a{b}^{2}{x}^{11}}{11}}+{\frac{{b}^{3}{x}^{13}}{13}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^6*(b*x^2+a)^3,x)

[Out]

1/7*a^3*x^7+1/3*a^2*b*x^9+3/11*a*b^2*x^11+1/13*b^3*x^13

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Maxima [A]  time = 1.34434, size = 47, normalized size = 1.09 \[ \frac{1}{13} \, b^{3} x^{13} + \frac{3}{11} \, a b^{2} x^{11} + \frac{1}{3} \, a^{2} b x^{9} + \frac{1}{7} \, a^{3} x^{7} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/13*b^3*x^13 + 3/11*a*b^2*x^11 + 1/3*a^2*b*x^9 + 1/7*a^3*x^7

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Fricas [A]  time = 0.179628, size = 1, normalized size = 0.02 \[ \frac{1}{13} x^{13} b^{3} + \frac{3}{11} x^{11} b^{2} a + \frac{1}{3} x^{9} b a^{2} + \frac{1}{7} x^{7} a^{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/13*x^13*b^3 + 3/11*x^11*b^2*a + 1/3*x^9*b*a^2 + 1/7*x^7*a^3

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Sympy [A]  time = 0.101922, size = 37, normalized size = 0.86 \[ \frac{a^{3} x^{7}}{7} + \frac{a^{2} b x^{9}}{3} + \frac{3 a b^{2} x^{11}}{11} + \frac{b^{3} x^{13}}{13} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**6*(b*x**2+a)**3,x)

[Out]

a**3*x**7/7 + a**2*b*x**9/3 + 3*a*b**2*x**11/11 + b**3*x**13/13

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GIAC/XCAS [A]  time = 0.207793, size = 47, normalized size = 1.09 \[ \frac{1}{13} \, b^{3} x^{13} + \frac{3}{11} \, a b^{2} x^{11} + \frac{1}{3} \, a^{2} b x^{9} + \frac{1}{7} \, a^{3} x^{7} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/13*b^3*x^13 + 3/11*a*b^2*x^11 + 1/3*a^2*b*x^9 + 1/7*a^3*x^7