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

Optimal. Leaf size=30 \[ \frac{a^2 x^6}{6}+\frac{2}{7} a b x^7+\frac{b^2 x^8}{8} \]

[Out]

(a^2*x^6)/6 + (2*a*b*x^7)/7 + (b^2*x^8)/8

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Rubi [A]  time = 0.0470887, antiderivative size = 30, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133 \[ \frac{a^2 x^6}{6}+\frac{2}{7} a b x^7+\frac{b^2 x^8}{8} \]

Antiderivative was successfully verified.

[In]  Int[x*(a*x^2 + b*x^3)^2,x]

[Out]

(a^2*x^6)/6 + (2*a*b*x^7)/7 + (b^2*x^8)/8

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Rubi in Sympy [A]  time = 7.52156, size = 26, normalized size = 0.87 \[ \frac{a^{2} x^{6}}{6} + \frac{2 a b x^{7}}{7} + \frac{b^{2} x^{8}}{8} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a**2*x**6/6 + 2*a*b*x**7/7 + b**2*x**8/8

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Mathematica [A]  time = 0.00316111, size = 30, normalized size = 1. \[ \frac{a^2 x^6}{6}+\frac{2}{7} a b x^7+\frac{b^2 x^8}{8} \]

Antiderivative was successfully verified.

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

[Out]

(a^2*x^6)/6 + (2*a*b*x^7)/7 + (b^2*x^8)/8

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Maple [A]  time = 0., size = 25, normalized size = 0.8 \[{\frac{{a}^{2}{x}^{6}}{6}}+{\frac{2\,ab{x}^{7}}{7}}+{\frac{{b}^{2}{x}^{8}}{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x*(b*x^3+a*x^2)^2,x)

[Out]

1/6*a^2*x^6+2/7*a*b*x^7+1/8*b^2*x^8

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Maxima [A]  time = 1.42729, size = 32, normalized size = 1.07 \[ \frac{1}{8} \, b^{2} x^{8} + \frac{2}{7} \, a b x^{7} + \frac{1}{6} \, a^{2} x^{6} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/8*b^2*x^8 + 2/7*a*b*x^7 + 1/6*a^2*x^6

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Fricas [A]  time = 0.193738, size = 1, normalized size = 0.03 \[ \frac{1}{8} x^{8} b^{2} + \frac{2}{7} x^{7} b a + \frac{1}{6} x^{6} a^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/8*x^8*b^2 + 2/7*x^7*b*a + 1/6*x^6*a^2

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Sympy [A]  time = 0.091962, size = 26, normalized size = 0.87 \[ \frac{a^{2} x^{6}}{6} + \frac{2 a b x^{7}}{7} + \frac{b^{2} x^{8}}{8} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a**2*x**6/6 + 2*a*b*x**7/7 + b**2*x**8/8

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GIAC/XCAS [A]  time = 0.217163, size = 32, normalized size = 1.07 \[ \frac{1}{8} \, b^{2} x^{8} + \frac{2}{7} \, a b x^{7} + \frac{1}{6} \, a^{2} x^{6} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/8*b^2*x^8 + 2/7*a*b*x^7 + 1/6*a^2*x^6