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

Optimal. Leaf size=19 \[ \frac{1}{2} \log \left (a-b x^2-c x^4\right ) \]

[Out]

Log[a - b*x^2 - c*x^4]/2

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Rubi [A]  time = 0.0103489, antiderivative size = 19, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.038 \[ \frac{1}{2} \log \left (a-b x^2-c x^4\right ) \]

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 = 5.40042, size = 14, normalized size = 0.74 \[ \frac{\log{\left (- a + b x^{2} + c x^{4} \right )}}{2} \]

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 = 0.0104468, size = 19, 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 = 0.001, size = 18, normalized size = 1. \[{\frac{\ln \left ( c{x}^{4}+b{x}^{2}-a \right ) }{2}} \]

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 = 0.74384, size = 23, normalized size = 1.21 \[ \frac{1}{2} \, \log \left (c x^{4} + b x^{2} - a\right ) \]

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 = 0.2538, size = 23, normalized size = 1.21 \[ \frac{1}{2} \, \log \left (c x^{4} + b x^{2} - a\right ) \]

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 = 1.59674, size = 14, normalized size = 0.74 \[ \frac{\log{\left (- a + b x^{2} + c x^{4} \right )}}{2} \]

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 = 0.290812, size = 23, normalized size = 1.21 \[ \frac{1}{2} \,{\rm ln}\left (c x^{4} + b x^{2} - a\right ) \]

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)