3.50 \(\int x^3 (a+b x) (A+B x) \, dx\)

Optimal. Leaf size=33 \[ \frac{1}{5} x^5 (a B+A b)+\frac{1}{4} a A x^4+\frac{1}{6} b B x^6 \]

[Out]

(a*A*x^4)/4 + ((A*b + a*B)*x^5)/5 + (b*B*x^6)/6

_______________________________________________________________________________________

Rubi [A]  time = 0.0746319, antiderivative size = 33, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.071 \[ \frac{1}{5} x^5 (a B+A b)+\frac{1}{4} a A x^4+\frac{1}{6} b B x^6 \]

Antiderivative was successfully verified.

[In]  Int[x^3*(a + b*x)*(A + B*x),x]

[Out]

(a*A*x^4)/4 + ((A*b + a*B)*x^5)/5 + (b*B*x^6)/6

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 10.1892, size = 29, normalized size = 0.88 \[ \frac{A a x^{4}}{4} + \frac{B b x^{6}}{6} + x^{5} \left (\frac{A b}{5} + \frac{B a}{5}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

A*a*x**4/4 + B*b*x**6/6 + x**5*(A*b/5 + B*a/5)

_______________________________________________________________________________________

Mathematica [A]  time = 0.00733753, size = 33, normalized size = 1. \[ \frac{1}{5} x^5 (a B+A b)+\frac{1}{4} a A x^4+\frac{1}{6} b B x^6 \]

Antiderivative was successfully verified.

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

[Out]

(a*A*x^4)/4 + ((A*b + a*B)*x^5)/5 + (b*B*x^6)/6

_______________________________________________________________________________________

Maple [A]  time = 0.002, size = 28, normalized size = 0.9 \[{\frac{aA{x}^{4}}{4}}+{\frac{ \left ( Ab+Ba \right ){x}^{5}}{5}}+{\frac{bB{x}^{6}}{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^3*(b*x+a)*(B*x+A),x)

[Out]

1/4*a*A*x^4+1/5*(A*b+B*a)*x^5+1/6*b*B*x^6

_______________________________________________________________________________________

Maxima [A]  time = 1.35277, size = 36, normalized size = 1.09 \[ \frac{1}{6} \, B b x^{6} + \frac{1}{4} \, A a x^{4} + \frac{1}{5} \,{\left (B a + A b\right )} x^{5} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/6*B*b*x^6 + 1/4*A*a*x^4 + 1/5*(B*a + A*b)*x^5

_______________________________________________________________________________________

Fricas [A]  time = 0.18226, size = 1, normalized size = 0.03 \[ \frac{1}{6} x^{6} b B + \frac{1}{5} x^{5} a B + \frac{1}{5} x^{5} b A + \frac{1}{4} x^{4} a A \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/6*x^6*b*B + 1/5*x^5*a*B + 1/5*x^5*b*A + 1/4*x^4*a*A

_______________________________________________________________________________________

Sympy [A]  time = 0.084536, size = 29, normalized size = 0.88 \[ \frac{A a x^{4}}{4} + \frac{B b x^{6}}{6} + x^{5} \left (\frac{A b}{5} + \frac{B a}{5}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

A*a*x**4/4 + B*b*x**6/6 + x**5*(A*b/5 + B*a/5)

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.229772, size = 39, normalized size = 1.18 \[ \frac{1}{6} \, B b x^{6} + \frac{1}{5} \, B a x^{5} + \frac{1}{5} \, A b x^{5} + \frac{1}{4} \, A a x^{4} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/6*B*b*x^6 + 1/5*B*a*x^5 + 1/5*A*b*x^5 + 1/4*A*a*x^4