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

Optimal. Leaf size=28 \[ \frac{1}{3} x^3 (a B+A b)+a A x+\frac{1}{5} b B x^5 \]

[Out]

a*A*x + ((A*b + a*B)*x^3)/3 + (b*B*x^5)/5

_______________________________________________________________________________________

Rubi [A]  time = 0.0364509, antiderivative size = 28, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067 \[ \frac{1}{3} x^3 (a B+A b)+a A x+\frac{1}{5} b B x^5 \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^2)*(A + B*x^2),x]

[Out]

a*A*x + ((A*b + a*B)*x^3)/3 + (b*B*x^5)/5

_______________________________________________________________________________________

Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{B b x^{5}}{5} + a \int A\, dx + x^{3} \left (\frac{A b}{3} + \frac{B a}{3}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

B*b*x**5/5 + a*Integral(A, x) + x**3*(A*b/3 + B*a/3)

_______________________________________________________________________________________

Mathematica [A]  time = 0.00839187, size = 28, normalized size = 1. \[ \frac{1}{3} x^3 (a B+A b)+a A x+\frac{1}{5} b B x^5 \]

Antiderivative was successfully verified.

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

[Out]

a*A*x + ((A*b + a*B)*x^3)/3 + (b*B*x^5)/5

_______________________________________________________________________________________

Maple [A]  time = 0.002, size = 25, normalized size = 0.9 \[ aAx+{\frac{ \left ( Ab+Ba \right ){x}^{3}}{3}}+{\frac{bB{x}^{5}}{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^2+a)*(B*x^2+A),x)

[Out]

a*A*x+1/3*(A*b+B*a)*x^3+1/5*b*B*x^5

_______________________________________________________________________________________

Maxima [A]  time = 1.33378, size = 32, normalized size = 1.14 \[ \frac{1}{5} \, B b x^{5} + \frac{1}{3} \,{\left (B a + A b\right )} x^{3} + A a x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/5*B*b*x^5 + 1/3*(B*a + A*b)*x^3 + A*a*x

_______________________________________________________________________________________

Fricas [A]  time = 0.21031, size = 1, normalized size = 0.04 \[ \frac{1}{5} x^{5} b B + \frac{1}{3} x^{3} a B + \frac{1}{3} x^{3} b A + x a A \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/5*x^5*b*B + 1/3*x^3*a*B + 1/3*x^3*b*A + x*a*A

_______________________________________________________________________________________

Sympy [A]  time = 0.083425, size = 26, normalized size = 0.93 \[ A a x + \frac{B b x^{5}}{5} + x^{3} \left (\frac{A b}{3} + \frac{B a}{3}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

A*a*x + B*b*x**5/5 + x**3*(A*b/3 + B*a/3)

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.210535, size = 35, normalized size = 1.25 \[ \frac{1}{5} \, B b x^{5} + \frac{1}{3} \, B a x^{3} + \frac{1}{3} \, A b x^{3} + A a x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/5*B*b*x^5 + 1/3*B*a*x^3 + 1/3*A*b*x^3 + A*a*x