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

Optimal. Leaf size=33 \[ \frac{1}{5} x^5 (a B+A b)+\frac{1}{2} a A x^2+\frac{1}{8} b B x^8 \]

[Out]

(a*A*x^2)/2 + ((A*b + a*B)*x^5)/5 + (b*B*x^8)/8

_______________________________________________________________________________________

Rubi [A]  time = 0.0623301, antiderivative size = 33, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.062 \[ \frac{1}{5} x^5 (a B+A b)+\frac{1}{2} a A x^2+\frac{1}{8} b B x^8 \]

Antiderivative was successfully verified.

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

[Out]

(a*A*x^2)/2 + ((A*b + a*B)*x^5)/5 + (b*B*x^8)/8

_______________________________________________________________________________________

Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ A a \int x\, dx + \frac{B b x^{8}}{8} + x^{5} \left (\frac{A b}{5} + \frac{B a}{5}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

A*a*Integral(x, x) + B*b*x**8/8 + x**5*(A*b/5 + B*a/5)

_______________________________________________________________________________________

Mathematica [A]  time = 0.010034, size = 33, normalized size = 1. \[ \frac{1}{5} x^5 (a B+A b)+\frac{1}{2} a A x^2+\frac{1}{8} b B x^8 \]

Antiderivative was successfully verified.

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

[Out]

(a*A*x^2)/2 + ((A*b + a*B)*x^5)/5 + (b*B*x^8)/8

_______________________________________________________________________________________

Maple [A]  time = 0., size = 28, normalized size = 0.9 \[{\frac{aA{x}^{2}}{2}}+{\frac{ \left ( Ab+Ba \right ){x}^{5}}{5}}+{\frac{bB{x}^{8}}{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x*(b*x^3+a)*(B*x^3+A),x)

[Out]

1/2*a*A*x^2+1/5*(A*b+B*a)*x^5+1/8*b*B*x^8

_______________________________________________________________________________________

Maxima [A]  time = 1.37498, size = 36, normalized size = 1.09 \[ \frac{1}{8} \, B b x^{8} + \frac{1}{5} \,{\left (B a + A b\right )} x^{5} + \frac{1}{2} \, A a x^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/8*B*b*x^8 + 1/5*(B*a + A*b)*x^5 + 1/2*A*a*x^2

_______________________________________________________________________________________

Fricas [A]  time = 0.203789, size = 1, normalized size = 0.03 \[ \frac{1}{8} x^{8} b B + \frac{1}{5} x^{5} a B + \frac{1}{5} x^{5} b A + \frac{1}{2} x^{2} a A \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/8*x^8*b*B + 1/5*x^5*a*B + 1/5*x^5*b*A + 1/2*x^2*a*A

_______________________________________________________________________________________

Sympy [A]  time = 0.076719, size = 29, normalized size = 0.88 \[ \frac{A a x^{2}}{2} + \frac{B b x^{8}}{8} + x^{5} \left (\frac{A b}{5} + \frac{B a}{5}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

A*a*x**2/2 + B*b*x**8/8 + x**5*(A*b/5 + B*a/5)

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.220171, size = 39, normalized size = 1.18 \[ \frac{1}{8} \, B b x^{8} + \frac{1}{5} \, B a x^{5} + \frac{1}{5} \, A b x^{5} + \frac{1}{2} \, A a x^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/8*B*b*x^8 + 1/5*B*a*x^5 + 1/5*A*b*x^5 + 1/2*A*a*x^2