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

Optimal. Leaf size=33 \[ \frac{1}{4} x^4 (a B+A b)+\frac{1}{2} a A x^2+\frac{1}{6} b B x^6 \]

[Out]

(a*A*x^2)/2 + ((A*b + a*B)*x^4)/4 + (b*B*x^6)/6

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Rubi [A]  time = 0.0777268, antiderivative size = 33, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125 \[ \frac{1}{4} x^4 (a B+A b)+\frac{1}{2} a A x^2+\frac{1}{6} b B x^6 \]

Antiderivative was successfully verified.

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

[Out]

(a*A*x^2)/2 + ((A*b + a*B)*x^4)/4 + (b*B*x^6)/6

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{B b x^{6}}{6} + \frac{a \int ^{x^{2}} A\, dx}{2} + \left (\frac{A b}{2} + \frac{B a}{2}\right ) \int ^{x^{2}} x\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

B*b*x**6/6 + a*Integral(A, (x, x**2))/2 + (A*b/2 + B*a/2)*Integral(x, (x, x**2))

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Mathematica [A]  time = 0.0115034, size = 33, normalized size = 1. \[ \frac{1}{4} x^4 (a B+A b)+\frac{1}{2} a A x^2+\frac{1}{6} b B x^6 \]

Antiderivative was successfully verified.

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

[Out]

(a*A*x^2)/2 + ((A*b + a*B)*x^4)/4 + (b*B*x^6)/6

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Maple [A]  time = 0.001, size = 28, normalized size = 0.9 \[{\frac{aA{x}^{2}}{2}}+{\frac{ \left ( Ab+Ba \right ){x}^{4}}{4}}+{\frac{bB{x}^{6}}{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x*(b*x^2+a)*(B*x^2+A),x)

[Out]

1/2*a*A*x^2+1/4*(A*b+B*a)*x^4+1/6*b*B*x^6

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Maxima [A]  time = 1.33638, size = 36, normalized size = 1.09 \[ \frac{1}{6} \, B b x^{6} + \frac{1}{4} \,{\left (B a + A b\right )} x^{4} + \frac{1}{2} \, A a x^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/6*B*b*x^6 + 1/4*(B*a + A*b)*x^4 + 1/2*A*a*x^2

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Fricas [A]  time = 0.197312, size = 1, normalized size = 0.03 \[ \frac{1}{6} x^{6} b B + \frac{1}{4} x^{4} a B + \frac{1}{4} x^{4} b A + \frac{1}{2} x^{2} a A \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/6*x^6*b*B + 1/4*x^4*a*B + 1/4*x^4*b*A + 1/2*x^2*a*A

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Sympy [A]  time = 0.087729, size = 29, normalized size = 0.88 \[ \frac{A a x^{2}}{2} + \frac{B b x^{6}}{6} + x^{4} \left (\frac{A b}{4} + \frac{B a}{4}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

A*a*x**2/2 + B*b*x**6/6 + x**4*(A*b/4 + B*a/4)

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GIAC/XCAS [A]  time = 0.21267, size = 39, normalized size = 1.18 \[ \frac{1}{6} \, B b x^{6} + \frac{1}{4} \, B a x^{4} + \frac{1}{4} \, A b x^{4} + \frac{1}{2} \, A a x^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/6*B*b*x^6 + 1/4*B*a*x^4 + 1/4*A*b*x^4 + 1/2*A*a*x^2