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

Optimal. Leaf size=34 \[ \frac{\left (a+b x^2\right )^5}{10 b^2}-\frac{a \left (a+b x^2\right )^4}{8 b^2} \]

[Out]

-(a*(a + b*x^2)^4)/(8*b^2) + (a + b*x^2)^5/(10*b^2)

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Rubi [A]  time = 0.0848432, antiderivative size = 34, 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 \[ \frac{\left (a+b x^2\right )^5}{10 b^2}-\frac{a \left (a+b x^2\right )^4}{8 b^2} \]

Antiderivative was successfully verified.

[In]  Int[x^3*(a + b*x^2)^3,x]

[Out]

-(a*(a + b*x^2)^4)/(8*b^2) + (a + b*x^2)^5/(10*b^2)

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Rubi in Sympy [A]  time = 8.96201, size = 27, normalized size = 0.79 \[ - \frac{a \left (a + b x^{2}\right )^{4}}{8 b^{2}} + \frac{\left (a + b x^{2}\right )^{5}}{10 b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-a*(a + b*x**2)**4/(8*b**2) + (a + b*x**2)**5/(10*b**2)

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Mathematica [A]  time = 0.00322607, size = 43, normalized size = 1.26 \[ \frac{a^3 x^4}{4}+\frac{1}{2} a^2 b x^6+\frac{3}{8} a b^2 x^8+\frac{b^3 x^{10}}{10} \]

Antiderivative was successfully verified.

[In]  Integrate[x^3*(a + b*x^2)^3,x]

[Out]

(a^3*x^4)/4 + (a^2*b*x^6)/2 + (3*a*b^2*x^8)/8 + (b^3*x^10)/10

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Maple [A]  time = 0., size = 36, normalized size = 1.1 \[{\frac{{b}^{3}{x}^{10}}{10}}+{\frac{3\,a{b}^{2}{x}^{8}}{8}}+{\frac{{a}^{2}b{x}^{6}}{2}}+{\frac{{a}^{3}{x}^{4}}{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^3*(b*x^2+a)^3,x)

[Out]

1/10*b^3*x^10+3/8*a*b^2*x^8+1/2*a^2*b*x^6+1/4*a^3*x^4

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Maxima [A]  time = 1.35244, size = 47, normalized size = 1.38 \[ \frac{1}{10} \, b^{3} x^{10} + \frac{3}{8} \, a b^{2} x^{8} + \frac{1}{2} \, a^{2} b x^{6} + \frac{1}{4} \, a^{3} x^{4} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/10*b^3*x^10 + 3/8*a*b^2*x^8 + 1/2*a^2*b*x^6 + 1/4*a^3*x^4

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Fricas [A]  time = 0.182997, size = 1, normalized size = 0.03 \[ \frac{1}{10} x^{10} b^{3} + \frac{3}{8} x^{8} b^{2} a + \frac{1}{2} x^{6} b a^{2} + \frac{1}{4} x^{4} a^{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/10*x^10*b^3 + 3/8*x^8*b^2*a + 1/2*x^6*b*a^2 + 1/4*x^4*a^3

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Sympy [A]  time = 0.099702, size = 37, normalized size = 1.09 \[ \frac{a^{3} x^{4}}{4} + \frac{a^{2} b x^{6}}{2} + \frac{3 a b^{2} x^{8}}{8} + \frac{b^{3} x^{10}}{10} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**3*(b*x**2+a)**3,x)

[Out]

a**3*x**4/4 + a**2*b*x**6/2 + 3*a*b**2*x**8/8 + b**3*x**10/10

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GIAC/XCAS [A]  time = 0.206832, size = 47, normalized size = 1.38 \[ \frac{1}{10} \, b^{3} x^{10} + \frac{3}{8} \, a b^{2} x^{8} + \frac{1}{2} \, a^{2} b x^{6} + \frac{1}{4} \, a^{3} x^{4} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/10*b^3*x^10 + 3/8*a*b^2*x^8 + 1/2*a^2*b*x^6 + 1/4*a^3*x^4