3.106 \(\int \frac{x \left (b+2 c x^2\right )}{a+b x^2+c x^4} \, dx\)

Optimal. Leaf size=17 \[ \frac{1}{2} \log \left (a+b x^2+c x^4\right ) \]

[Out]

Log[a + b*x^2 + c*x^4]/2

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Rubi [A]  time = 0.0105358, antiderivative size = 17, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.042 \[ \frac{1}{2} \log \left (a+b x^2+c x^4\right ) \]

Antiderivative was successfully verified.

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

[Out]

Log[a + b*x^2 + c*x^4]/2

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Rubi in Sympy [A]  time = 4.94338, size = 14, normalized size = 0.82 \[ \frac{\log{\left (a + b x^{2} + c x^{4} \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

log(a + b*x**2 + c*x**4)/2

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Mathematica [A]  time = 0.00970252, size = 17, normalized size = 1. \[ \frac{1}{2} \log \left (a+b x^2+c x^4\right ) \]

Antiderivative was successfully verified.

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

[Out]

Log[a + b*x^2 + c*x^4]/2

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Maple [A]  time = 0.002, size = 16, normalized size = 0.9 \[{\frac{\ln \left ( c{x}^{4}+b{x}^{2}+a \right ) }{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*ln(c*x^4+b*x^2+a)

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Maxima [A]  time = 0.768463, size = 20, normalized size = 1.18 \[ \frac{1}{2} \, \log \left (c x^{4} + b x^{2} + a\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*log(c*x^4 + b*x^2 + a)

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Fricas [A]  time = 0.270551, size = 20, normalized size = 1.18 \[ \frac{1}{2} \, \log \left (c x^{4} + b x^{2} + a\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*log(c*x^4 + b*x^2 + a)

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Sympy [A]  time = 1.59331, size = 14, normalized size = 0.82 \[ \frac{\log{\left (a + b x^{2} + c x^{4} \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

log(a + b*x**2 + c*x**4)/2

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GIAC/XCAS [A]  time = 0.290826, size = 20, normalized size = 1.18 \[ \frac{1}{2} \,{\rm ln}\left (c x^{4} + b x^{2} + a\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*ln(c*x^4 + b*x^2 + a)