3.1 \(\int x^2 \left (a+b x^3\right ) \left (A+B x^3\right ) \, dx\)

Optimal. Leaf size=33 \[ \frac{1}{6} x^6 (a B+A b)+\frac{1}{3} a A x^3+\frac{1}{9} b B x^9 \]

[Out]

(a*A*x^3)/3 + ((A*b + a*B)*x^6)/6 + (b*B*x^9)/9

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Rubi [A]  time = 0.104397, antiderivative size = 33, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111 \[ \frac{1}{6} x^6 (a B+A b)+\frac{1}{3} a A x^3+\frac{1}{9} b B x^9 \]

Antiderivative was successfully verified.

[In]  Int[x^2*(a + b*x^3)*(A + B*x^3),x]

[Out]

(a*A*x^3)/3 + ((A*b + a*B)*x^6)/6 + (b*B*x^9)/9

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{B b x^{9}}{9} + \frac{a \int ^{x^{3}} A\, dx}{3} + \left (\frac{A b}{3} + \frac{B a}{3}\right ) \int ^{x^{3}} x\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

B*b*x**9/9 + a*Integral(A, (x, x**3))/3 + (A*b/3 + B*a/3)*Integral(x, (x, x**3))

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Mathematica [A]  time = 0.0153083, size = 33, normalized size = 1. \[ \frac{1}{6} x^6 (a B+A b)+\frac{1}{3} a A x^3+\frac{1}{9} b B x^9 \]

Antiderivative was successfully verified.

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

[Out]

(a*A*x^3)/3 + ((A*b + a*B)*x^6)/6 + (b*B*x^9)/9

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Maple [A]  time = 0.002, size = 28, normalized size = 0.9 \[{\frac{aA{x}^{3}}{3}}+{\frac{ \left ( Ab+Ba \right ){x}^{6}}{6}}+{\frac{bB{x}^{9}}{9}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^2*(b*x^3+a)*(B*x^3+A),x)

[Out]

1/3*a*A*x^3+1/6*(A*b+B*a)*x^6+1/9*b*B*x^9

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Maxima [A]  time = 1.40642, size = 36, normalized size = 1.09 \[ \frac{1}{9} \, B b x^{9} + \frac{1}{6} \,{\left (B a + A b\right )} x^{6} + \frac{1}{3} \, A a x^{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/9*B*b*x^9 + 1/6*(B*a + A*b)*x^6 + 1/3*A*a*x^3

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Fricas [A]  time = 0.20057, size = 1, normalized size = 0.03 \[ \frac{1}{9} x^{9} b B + \frac{1}{6} x^{6} a B + \frac{1}{6} x^{6} b A + \frac{1}{3} x^{3} a A \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/9*x^9*b*B + 1/6*x^6*a*B + 1/6*x^6*b*A + 1/3*x^3*a*A

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Sympy [A]  time = 0.078523, size = 29, normalized size = 0.88 \[ \frac{A a x^{3}}{3} + \frac{B b x^{9}}{9} + x^{6} \left (\frac{A b}{6} + \frac{B a}{6}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

A*a*x**3/3 + B*b*x**9/9 + x**6*(A*b/6 + B*a/6)

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GIAC/XCAS [A]  time = 0.235945, size = 39, normalized size = 1.18 \[ \frac{1}{9} \, B b x^{9} + \frac{1}{6} \, B a x^{6} + \frac{1}{6} \, A b x^{6} + \frac{1}{3} \, A a x^{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/9*B*b*x^9 + 1/6*B*a*x^6 + 1/6*A*b*x^6 + 1/3*A*a*x^3