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

Optimal. Leaf size=33 \[ \frac{1}{5} x^5 (a B+A b)+\frac{1}{3} a A x^3+\frac{1}{7} b B x^7 \]

[Out]

(a*A*x^3)/3 + ((A*b + a*B)*x^5)/5 + (b*B*x^7)/7

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Rubi [A]  time = 0.0602547, antiderivative size = 33, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.056 \[ \frac{1}{5} x^5 (a B+A b)+\frac{1}{3} a A x^3+\frac{1}{7} b B x^7 \]

Antiderivative was successfully verified.

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

[Out]

(a*A*x^3)/3 + ((A*b + a*B)*x^5)/5 + (b*B*x^7)/7

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Rubi in Sympy [A]  time = 10.0854, size = 29, normalized size = 0.88 \[ \frac{A a x^{3}}{3} + \frac{B b x^{7}}{7} + x^{5} \left (\frac{A b}{5} + \frac{B a}{5}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

A*a*x**3/3 + B*b*x**7/7 + x**5*(A*b/5 + B*a/5)

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Mathematica [A]  time = 0.0100798, size = 33, normalized size = 1. \[ \frac{1}{5} x^5 (a B+A b)+\frac{1}{3} a A x^3+\frac{1}{7} b B x^7 \]

Antiderivative was successfully verified.

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

[Out]

(a*A*x^3)/3 + ((A*b + a*B)*x^5)/5 + (b*B*x^7)/7

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Maple [A]  time = 0.001, size = 28, normalized size = 0.9 \[{\frac{aA{x}^{3}}{3}}+{\frac{ \left ( Ab+Ba \right ){x}^{5}}{5}}+{\frac{bB{x}^{7}}{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^2*(b*x^2+a)*(B*x^2+A),x)

[Out]

1/3*a*A*x^3+1/5*(A*b+B*a)*x^5+1/7*b*B*x^7

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Maxima [A]  time = 1.37163, size = 36, normalized size = 1.09 \[ \frac{1}{7} \, B b x^{7} + \frac{1}{5} \,{\left (B a + A b\right )} x^{5} + \frac{1}{3} \, A a x^{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/7*B*b*x^7 + 1/5*(B*a + A*b)*x^5 + 1/3*A*a*x^3

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Fricas [A]  time = 0.20973, size = 1, normalized size = 0.03 \[ \frac{1}{7} x^{7} b B + \frac{1}{5} x^{5} a B + \frac{1}{5} x^{5} b A + \frac{1}{3} x^{3} a A \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/7*x^7*b*B + 1/5*x^5*a*B + 1/5*x^5*b*A + 1/3*x^3*a*A

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Sympy [A]  time = 0.082555, size = 29, normalized size = 0.88 \[ \frac{A a x^{3}}{3} + \frac{B b x^{7}}{7} + x^{5} \left (\frac{A b}{5} + \frac{B a}{5}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

A*a*x**3/3 + B*b*x**7/7 + x**5*(A*b/5 + B*a/5)

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GIAC/XCAS [A]  time = 0.215866, size = 39, normalized size = 1.18 \[ \frac{1}{7} \, B b x^{7} + \frac{1}{5} \, B a x^{5} + \frac{1}{5} \, A b x^{5} + \frac{1}{3} \, A a x^{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/7*B*b*x^7 + 1/5*B*a*x^5 + 1/5*A*b*x^5 + 1/3*A*a*x^3