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

Optimal. Leaf size=45 \[ \frac{\log \left (a+b x^2\right )}{2 (b c-a d)}-\frac{\log \left (c+d x^2\right )}{2 (b c-a d)} \]

[Out]

Log[a + b*x^2]/(2*(b*c - a*d)) - Log[c + d*x^2]/(2*(b*c - a*d))

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Rubi [A]  time = 0.0725536, antiderivative size = 45, 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 \[ \frac{\log \left (a+b x^2\right )}{2 (b c-a d)}-\frac{\log \left (c+d x^2\right )}{2 (b c-a d)} \]

Antiderivative was successfully verified.

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

[Out]

Log[a + b*x^2]/(2*(b*c - a*d)) - Log[c + d*x^2]/(2*(b*c - a*d))

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Rubi in Sympy [A]  time = 12.8105, size = 36, normalized size = 0.8 \[ - \frac{\log{\left (a + b x^{2} \right )}}{2 \left (a d - b c\right )} + \frac{\log{\left (c + d x^{2} \right )}}{2 \left (a d - b c\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-log(a + b*x**2)/(2*(a*d - b*c)) + log(c + d*x**2)/(2*(a*d - b*c))

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Mathematica [A]  time = 0.0266405, size = 31, normalized size = 0.69 \[ \frac{\log \left (a+b x^2\right )-\log \left (c+d x^2\right )}{2 b c-2 a d} \]

Antiderivative was successfully verified.

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

[Out]

(Log[a + b*x^2] - Log[c + d*x^2])/(2*b*c - 2*a*d)

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Maple [A]  time = 0.009, size = 42, normalized size = 0.9 \[{\frac{\ln \left ( d{x}^{2}+c \right ) }{2\,ad-2\,bc}}-{\frac{\ln \left ( b{x}^{2}+a \right ) }{2\,ad-2\,bc}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x/(b*x^2+a)/(d*x^2+c),x)

[Out]

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

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Maxima [A]  time = 1.33834, size = 55, normalized size = 1.22 \[ \frac{\log \left (b x^{2} + a\right )}{2 \,{\left (b c - a d\right )}} - \frac{\log \left (d x^{2} + c\right )}{2 \,{\left (b c - a d\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*log(b*x^2 + a)/(b*c - a*d) - 1/2*log(d*x^2 + c)/(b*c - a*d)

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Fricas [A]  time = 0.232263, size = 42, normalized size = 0.93 \[ \frac{\log \left (b x^{2} + a\right ) - \log \left (d x^{2} + c\right )}{2 \,{\left (b c - a d\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*(log(b*x^2 + a) - log(d*x^2 + c))/(b*c - a*d)

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Sympy [A]  time = 3.24171, size = 138, normalized size = 3.07 \[ \frac{\log{\left (x^{2} + \frac{- \frac{a^{2} d^{2}}{a d - b c} + \frac{2 a b c d}{a d - b c} + a d - \frac{b^{2} c^{2}}{a d - b c} + b c}{2 b d} \right )}}{2 \left (a d - b c\right )} - \frac{\log{\left (x^{2} + \frac{\frac{a^{2} d^{2}}{a d - b c} - \frac{2 a b c d}{a d - b c} + a d + \frac{b^{2} c^{2}}{a d - b c} + b c}{2 b d} \right )}}{2 \left (a d - b c\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

log(x**2 + (-a**2*d**2/(a*d - b*c) + 2*a*b*c*d/(a*d - b*c) + a*d - b**2*c**2/(a*
d - b*c) + b*c)/(2*b*d))/(2*(a*d - b*c)) - log(x**2 + (a**2*d**2/(a*d - b*c) - 2
*a*b*c*d/(a*d - b*c) + a*d + b**2*c**2/(a*d - b*c) + b*c)/(2*b*d))/(2*(a*d - b*c
))

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GIAC/XCAS [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: NotImplementedError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: NotImplementedError