3.13 \(\int x^5 \left (a+b x^2\right )^2 \, dx\)

Optimal. Leaf size=30 \[ \frac{a^2 x^6}{6}+\frac{1}{4} a b x^8+\frac{b^2 x^{10}}{10} \]

[Out]

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

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Rubi [A]  time = 0.061023, antiderivative size = 30, 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 \[ \frac{a^2 x^6}{6}+\frac{1}{4} a b x^8+\frac{b^2 x^{10}}{10} \]

Antiderivative was successfully verified.

[In]  Int[x^5*(a + b*x^2)^2,x]

[Out]

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

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Rubi in Sympy [A]  time = 7.76487, size = 24, normalized size = 0.8 \[ \frac{a^{2} x^{6}}{6} + \frac{a b x^{8}}{4} + \frac{b^{2} x^{10}}{10} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**5*(b*x**2+a)**2,x)

[Out]

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

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Mathematica [A]  time = 0.00178551, size = 30, normalized size = 1. \[ \frac{a^2 x^6}{6}+\frac{1}{4} a b x^8+\frac{b^2 x^{10}}{10} \]

Antiderivative was successfully verified.

[In]  Integrate[x^5*(a + b*x^2)^2,x]

[Out]

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

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Maple [A]  time = 0.001, size = 25, normalized size = 0.8 \[{\frac{{a}^{2}{x}^{6}}{6}}+{\frac{ab{x}^{8}}{4}}+{\frac{{b}^{2}{x}^{10}}{10}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^5*(b*x^2+a)^2,x)

[Out]

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

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Maxima [A]  time = 1.34741, size = 32, normalized size = 1.07 \[ \frac{1}{10} \, b^{2} x^{10} + \frac{1}{4} \, a b x^{8} + \frac{1}{6} \, a^{2} x^{6} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^2*x^5,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.196389, size = 1, normalized size = 0.03 \[ \frac{1}{10} x^{10} b^{2} + \frac{1}{4} x^{8} b a + \frac{1}{6} x^{6} a^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^2*x^5,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.094627, size = 24, normalized size = 0.8 \[ \frac{a^{2} x^{6}}{6} + \frac{a b x^{8}}{4} + \frac{b^{2} x^{10}}{10} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**5*(b*x**2+a)**2,x)

[Out]

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

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GIAC/XCAS [A]  time = 0.207913, size = 32, normalized size = 1.07 \[ \frac{1}{10} \, b^{2} x^{10} + \frac{1}{4} \, a b x^{8} + \frac{1}{6} \, a^{2} x^{6} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^2*x^5,x, algorithm="giac")

[Out]

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