3.2 \(\int x \left (A+B x^2\right ) \left (b x^2+c x^4\right ) \, dx\)

Optimal. Leaf size=33 \[ \frac{1}{6} x^6 (A c+b B)+\frac{1}{4} A b x^4+\frac{1}{8} B c x^8 \]

[Out]

(A*b*x^4)/4 + ((b*B + A*c)*x^6)/6 + (B*c*x^8)/8

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Rubi [A]  time = 0.121096, antiderivative size = 33, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15 \[ \frac{1}{6} x^6 (A c+b B)+\frac{1}{4} A b x^4+\frac{1}{8} B c x^8 \]

Antiderivative was successfully verified.

[In]  Int[x*(A + B*x^2)*(b*x^2 + c*x^4),x]

[Out]

(A*b*x^4)/4 + ((b*B + A*c)*x^6)/6 + (B*c*x^8)/8

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{A b \int ^{x^{2}} x\, dx}{2} + \frac{B c x^{8}}{8} + x^{6} \left (\frac{A c}{6} + \frac{B b}{6}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x*(B*x**2+A)*(c*x**4+b*x**2),x)

[Out]

A*b*Integral(x, (x, x**2))/2 + B*c*x**8/8 + x**6*(A*c/6 + B*b/6)

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Mathematica [A]  time = 0.0119226, size = 33, normalized size = 1. \[ \frac{1}{6} x^6 (A c+b B)+\frac{1}{4} A b x^4+\frac{1}{8} B c x^8 \]

Antiderivative was successfully verified.

[In]  Integrate[x*(A + B*x^2)*(b*x^2 + c*x^4),x]

[Out]

(A*b*x^4)/4 + ((b*B + A*c)*x^6)/6 + (B*c*x^8)/8

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Maple [A]  time = 0., size = 28, normalized size = 0.9 \[{\frac{Ab{x}^{4}}{4}}+{\frac{ \left ( Ac+Bb \right ){x}^{6}}{6}}+{\frac{Bc{x}^{8}}{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x*(B*x^2+A)*(c*x^4+b*x^2),x)

[Out]

1/4*A*b*x^4+1/6*(A*c+B*b)*x^6+1/8*B*c*x^8

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Maxima [A]  time = 1.37073, size = 36, normalized size = 1.09 \[ \frac{1}{8} \, B c x^{8} + \frac{1}{6} \,{\left (B b + A c\right )} x^{6} + \frac{1}{4} \, A b x^{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^4 + b*x^2)*(B*x^2 + A)*x,x, algorithm="maxima")

[Out]

1/8*B*c*x^8 + 1/6*(B*b + A*c)*x^6 + 1/4*A*b*x^4

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Fricas [A]  time = 0.19886, size = 1, normalized size = 0.03 \[ \frac{1}{8} x^{8} c B + \frac{1}{6} x^{6} b B + \frac{1}{6} x^{6} c A + \frac{1}{4} x^{4} b A \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^4 + b*x^2)*(B*x^2 + A)*x,x, algorithm="fricas")

[Out]

1/8*x^8*c*B + 1/6*x^6*b*B + 1/6*x^6*c*A + 1/4*x^4*b*A

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Sympy [A]  time = 0.042262, size = 29, normalized size = 0.88 \[ \frac{A b x^{4}}{4} + \frac{B c x^{8}}{8} + x^{6} \left (\frac{A c}{6} + \frac{B b}{6}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x*(B*x**2+A)*(c*x**4+b*x**2),x)

[Out]

A*b*x**4/4 + B*c*x**8/8 + x**6*(A*c/6 + B*b/6)

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GIAC/XCAS [A]  time = 0.218945, size = 39, normalized size = 1.18 \[ \frac{1}{8} \, B c x^{8} + \frac{1}{6} \, B b x^{6} + \frac{1}{6} \, A c x^{6} + \frac{1}{4} \, A b x^{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^4 + b*x^2)*(B*x^2 + A)*x,x, algorithm="giac")

[Out]

1/8*B*c*x^8 + 1/6*B*b*x^6 + 1/6*A*c*x^6 + 1/4*A*b*x^4