3.4 \(\int \frac{\left (A+B x^2\right ) \left (b x^2+c x^4\right )}{x} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0880616, antiderivative size = 33, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136 \[ \frac{1}{4} x^4 (A c+b B)+\frac{1}{2} A b x^2+\frac{1}{6} B c x^6 \]

Antiderivative was successfully verified.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0125309, size = 33, normalized size = 1. \[ \frac{1}{4} x^4 (A c+b B)+\frac{1}{2} A b x^2+\frac{1}{6} B c x^6 \]

Antiderivative was successfully verified.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 1.36578, size = 36, normalized size = 1.09 \[ \frac{1}{6} \, B c x^{6} + \frac{1}{4} \,{\left (B b + A c\right )} x^{4} + \frac{1}{2} \, A b x^{2} \]

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]

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

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Fricas [A]  time = 0.214207, size = 36, normalized size = 1.09 \[ \frac{1}{6} \, B c x^{6} + \frac{1}{4} \,{\left (B b + A c\right )} x^{4} + \frac{1}{2} \, A b x^{2} \]

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]

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

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Sympy [A]  time = 0.041677, size = 29, normalized size = 0.88 \[ \frac{A b x^{2}}{2} + \frac{B c x^{6}}{6} + x^{4} \left (\frac{A c}{4} + \frac{B b}{4}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.222867, size = 39, normalized size = 1.18 \[ \frac{1}{6} \, B c x^{6} + \frac{1}{4} \, B b x^{4} + \frac{1}{4} \, A c x^{4} + \frac{1}{2} \, A b x^{2} \]

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]

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