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

Optimal. Leaf size=34 \[ \frac{(b B-A c) \log \left (b+c x^2\right )}{2 b c}+\frac{A \log (x)}{b} \]

[Out]

(A*Log[x])/b + ((b*B - A*c)*Log[b + c*x^2])/(2*b*c)

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Rubi [A]  time = 0.104874, antiderivative size = 34, 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 \[ \frac{(b B-A c) \log \left (b+c x^2\right )}{2 b c}+\frac{A \log (x)}{b} \]

Antiderivative was successfully verified.

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

[Out]

(A*Log[x])/b + ((b*B - A*c)*Log[b + c*x^2])/(2*b*c)

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Rubi in Sympy [A]  time = 14.5494, size = 29, normalized size = 0.85 \[ \frac{A \log{\left (x^{2} \right )}}{2 b} - \frac{\left (A c - B b\right ) \log{\left (b + c x^{2} \right )}}{2 b c} \]

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*log(x**2)/(2*b) - (A*c - B*b)*log(b + c*x**2)/(2*b*c)

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Mathematica [A]  time = 0.0205189, size = 34, normalized size = 1. \[ \frac{(b B-A c) \log \left (b+c x^2\right )}{2 b c}+\frac{A \log (x)}{b} \]

Antiderivative was successfully verified.

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

[Out]

(A*Log[x])/b + ((b*B - A*c)*Log[b + c*x^2])/(2*b*c)

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Maple [A]  time = 0.007, size = 37, normalized size = 1.1 \[{\frac{A\ln \left ( x \right ) }{b}}-{\frac{\ln \left ( c{x}^{2}+b \right ) A}{2\,b}}+{\frac{\ln \left ( c{x}^{2}+b \right ) B}{2\,c}} \]

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]

A*ln(x)/b-1/2/b*ln(c*x^2+b)*A+1/2/c*ln(c*x^2+b)*B

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Maxima [A]  time = 1.37236, size = 47, normalized size = 1.38 \[ \frac{A \log \left (x^{2}\right )}{2 \, b} + \frac{{\left (B b - A c\right )} \log \left (c x^{2} + b\right )}{2 \, b c} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*A*log(x^2)/b + 1/2*(B*b - A*c)*log(c*x^2 + b)/(b*c)

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Fricas [A]  time = 0.210755, size = 43, normalized size = 1.26 \[ \frac{2 \, A c \log \left (x\right ) +{\left (B b - A c\right )} \log \left (c x^{2} + b\right )}{2 \, b c} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*(2*A*c*log(x) + (B*b - A*c)*log(c*x^2 + b))/(b*c)

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Sympy [A]  time = 1.10029, size = 26, normalized size = 0.76 \[ \frac{A \log{\left (x \right )}}{b} + \frac{\left (- A c + B b\right ) \log{\left (\frac{b}{c} + x^{2} \right )}}{2 b c} \]

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*log(x)/b + (-A*c + B*b)*log(b/c + x**2)/(2*b*c)

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GIAC/XCAS [A]  time = 0.211408, size = 46, normalized size = 1.35 \[ \frac{A{\rm ln}\left ({\left | x \right |}\right )}{b} + \frac{{\left (B b - A c\right )}{\rm ln}\left ({\left | c x^{2} + b \right |}\right )}{2 \, b c} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

A*ln(abs(x))/b + 1/2*(B*b - A*c)*ln(abs(c*x^2 + b))/(b*c)