3.234 \(\int \frac{1}{x \left (a+b x^2\right ) \left (c+d x^2\right )} \, dx\)

Optimal. Leaf size=62 \[ -\frac{b \log \left (a+b x^2\right )}{2 a (b c-a d)}+\frac{d \log \left (c+d x^2\right )}{2 c (b c-a d)}+\frac{\log (x)}{a c} \]

[Out]

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

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Rubi [A]  time = 0.155338, antiderivative size = 62, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091 \[ -\frac{b \log \left (a+b x^2\right )}{2 a (b c-a d)}+\frac{d \log \left (c+d x^2\right )}{2 c (b c-a d)}+\frac{\log (x)}{a c} \]

Antiderivative was successfully verified.

[In]  Int[1/(x*(a + b*x^2)*(c + d*x^2)),x]

[Out]

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

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Rubi in Sympy [A]  time = 25.999, size = 49, normalized size = 0.79 \[ - \frac{d \log{\left (c + d x^{2} \right )}}{2 c \left (a d - b c\right )} + \frac{b \log{\left (a + b x^{2} \right )}}{2 a \left (a d - b c\right )} + \frac{\log{\left (x^{2} \right )}}{2 a c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x/(b*x**2+a)/(d*x**2+c),x)

[Out]

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

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Mathematica [A]  time = 0.0441077, size = 54, normalized size = 0.87 \[ \frac{-b c \log \left (a+b x^2\right )+a d \log \left (c+d x^2\right )-2 a d \log (x)+2 b c \log (x)}{2 a b c^2-2 a^2 c d} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x*(a + b*x^2)*(c + d*x^2)),x]

[Out]

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

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Maple [A]  time = 0.013, size = 59, normalized size = 1. \[{\frac{\ln \left ( x \right ) }{ac}}-{\frac{d\ln \left ( d{x}^{2}+c \right ) }{2\,c \left ( ad-bc \right ) }}+{\frac{b\ln \left ( b{x}^{2}+a \right ) }{2\,a \left ( ad-bc \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x/(b*x^2+a)/(d*x^2+c),x)

[Out]

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

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Maxima [A]  time = 1.34522, size = 82, normalized size = 1.32 \[ -\frac{b \log \left (b x^{2} + a\right )}{2 \,{\left (a b c - a^{2} d\right )}} + \frac{d \log \left (d x^{2} + c\right )}{2 \,{\left (b c^{2} - a c d\right )}} + \frac{\log \left (x^{2}\right )}{2 \, a c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^2 + a)*(d*x^2 + c)*x),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.283252, size = 73, normalized size = 1.18 \[ -\frac{b c \log \left (b x^{2} + a\right ) - a d \log \left (d x^{2} + c\right ) - 2 \,{\left (b c - a d\right )} \log \left (x\right )}{2 \,{\left (a b c^{2} - a^{2} c d\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^2 + a)*(d*x^2 + c)*x),x, algorithm="fricas")

[Out]

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

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x/(b*x**2+a)/(d*x**2+c),x)

[Out]

Timed out

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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(1/((b*x^2 + a)*(d*x^2 + c)*x),x, algorithm="giac")

[Out]

Exception raised: NotImplementedError