3.30 \(\int x^7 \left (a+b x^2\right )^3 \, dx\)

Optimal. Leaf size=43 \[ \frac{a^3 x^8}{8}+\frac{3}{10} a^2 b x^{10}+\frac{1}{4} a b^2 x^{12}+\frac{b^3 x^{14}}{14} \]

[Out]

(a^3*x^8)/8 + (3*a^2*b*x^10)/10 + (a*b^2*x^12)/4 + (b^3*x^14)/14

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Rubi [A]  time = 0.0782016, antiderivative size = 43, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ \frac{a^3 x^8}{8}+\frac{3}{10} a^2 b x^{10}+\frac{1}{4} a b^2 x^{12}+\frac{b^3 x^{14}}{14} \]

Antiderivative was successfully verified.

[In]  Int[x^7*(a + b*x^2)^3,x]

[Out]

(a^3*x^8)/8 + (3*a^2*b*x^10)/10 + (a*b^2*x^12)/4 + (b^3*x^14)/14

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Rubi in Sympy [A]  time = 10.4321, size = 37, normalized size = 0.86 \[ \frac{a^{3} x^{8}}{8} + \frac{3 a^{2} b x^{10}}{10} + \frac{a b^{2} x^{12}}{4} + \frac{b^{3} x^{14}}{14} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**7*(b*x**2+a)**3,x)

[Out]

a**3*x**8/8 + 3*a**2*b*x**10/10 + a*b**2*x**12/4 + b**3*x**14/14

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Mathematica [A]  time = 0.00327151, size = 43, normalized size = 1. \[ \frac{a^3 x^8}{8}+\frac{3}{10} a^2 b x^{10}+\frac{1}{4} a b^2 x^{12}+\frac{b^3 x^{14}}{14} \]

Antiderivative was successfully verified.

[In]  Integrate[x^7*(a + b*x^2)^3,x]

[Out]

(a^3*x^8)/8 + (3*a^2*b*x^10)/10 + (a*b^2*x^12)/4 + (b^3*x^14)/14

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Maple [A]  time = 0., size = 36, normalized size = 0.8 \[{\frac{{a}^{3}{x}^{8}}{8}}+{\frac{3\,{a}^{2}b{x}^{10}}{10}}+{\frac{a{b}^{2}{x}^{12}}{4}}+{\frac{{b}^{3}{x}^{14}}{14}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^7*(b*x^2+a)^3,x)

[Out]

1/8*a^3*x^8+3/10*a^2*b*x^10+1/4*a*b^2*x^12+1/14*b^3*x^14

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Maxima [A]  time = 1.34176, size = 47, normalized size = 1.09 \[ \frac{1}{14} \, b^{3} x^{14} + \frac{1}{4} \, a b^{2} x^{12} + \frac{3}{10} \, a^{2} b x^{10} + \frac{1}{8} \, a^{3} x^{8} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^3*x^7,x, algorithm="maxima")

[Out]

1/14*b^3*x^14 + 1/4*a*b^2*x^12 + 3/10*a^2*b*x^10 + 1/8*a^3*x^8

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Fricas [A]  time = 0.182858, size = 1, normalized size = 0.02 \[ \frac{1}{14} x^{14} b^{3} + \frac{1}{4} x^{12} b^{2} a + \frac{3}{10} x^{10} b a^{2} + \frac{1}{8} x^{8} a^{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^3*x^7,x, algorithm="fricas")

[Out]

1/14*x^14*b^3 + 1/4*x^12*b^2*a + 3/10*x^10*b*a^2 + 1/8*x^8*a^3

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Sympy [A]  time = 0.105183, size = 37, normalized size = 0.86 \[ \frac{a^{3} x^{8}}{8} + \frac{3 a^{2} b x^{10}}{10} + \frac{a b^{2} x^{12}}{4} + \frac{b^{3} x^{14}}{14} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**7*(b*x**2+a)**3,x)

[Out]

a**3*x**8/8 + 3*a**2*b*x**10/10 + a*b**2*x**12/4 + b**3*x**14/14

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GIAC/XCAS [A]  time = 0.205491, size = 47, normalized size = 1.09 \[ \frac{1}{14} \, b^{3} x^{14} + \frac{1}{4} \, a b^{2} x^{12} + \frac{3}{10} \, a^{2} b x^{10} + \frac{1}{8} \, a^{3} x^{8} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^3*x^7,x, algorithm="giac")

[Out]

1/14*b^3*x^14 + 1/4*a*b^2*x^12 + 3/10*a^2*b*x^10 + 1/8*a^3*x^8