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

Optimal. Leaf size=100 \[ -\frac{b^2 \log \left (a+b x^2\right )}{2 a (b c-a d)^2}+\frac{d (2 b c-a d) \log \left (c+d x^2\right )}{2 c^2 (b c-a d)^2}-\frac{d}{2 c \left (c+d x^2\right ) (b c-a d)}+\frac{\log (x)}{a c^2} \]

[Out]

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

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Rubi [A]  time = 0.229838, antiderivative size = 100, 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 \[ -\frac{b^2 \log \left (a+b x^2\right )}{2 a (b c-a d)^2}+\frac{d (2 b c-a d) \log \left (c+d x^2\right )}{2 c^2 (b c-a d)^2}-\frac{d}{2 c \left (c+d x^2\right ) (b c-a d)}+\frac{\log (x)}{a c^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 38.8441, size = 85, normalized size = 0.85 \[ \frac{d}{2 c \left (c + d x^{2}\right ) \left (a d - b c\right )} - \frac{d \left (a d - 2 b c\right ) \log{\left (c + d x^{2} \right )}}{2 c^{2} \left (a d - b c\right )^{2}} - \frac{b^{2} \log{\left (a + b x^{2} \right )}}{2 a \left (a d - b c\right )^{2}} + \frac{\log{\left (x^{2} \right )}}{2 a c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.16313, size = 98, normalized size = 0.98 \[ \frac{1}{2} \left (-\frac{b^2 \log \left (a+b x^2\right )}{a (b c-a d)^2}+\frac{d (2 b c-a d) \log \left (c+d x^2\right )}{c^2 (b c-a d)^2}-\frac{d}{c \left (c+d x^2\right ) (b c-a d)}+\frac{2 \log (x)}{a c^2}\right ) \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.023, size = 139, normalized size = 1.4 \[{\frac{\ln \left ( x \right ) }{a{c}^{2}}}+{\frac{a{d}^{2}}{2\,c \left ( ad-bc \right ) ^{2} \left ( d{x}^{2}+c \right ) }}-{\frac{bd}{2\, \left ( ad-bc \right ) ^{2} \left ( d{x}^{2}+c \right ) }}-{\frac{{d}^{2}\ln \left ( d{x}^{2}+c \right ) a}{2\,{c}^{2} \left ( ad-bc \right ) ^{2}}}+{\frac{d\ln \left ( d{x}^{2}+c \right ) b}{c \left ( ad-bc \right ) ^{2}}}-{\frac{{b}^{2}\ln \left ( b{x}^{2}+a \right ) }{2\,a \left ( ad-bc \right ) ^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 1.35437, size = 186, normalized size = 1.86 \[ -\frac{b^{2} \log \left (b x^{2} + a\right )}{2 \,{\left (a b^{2} c^{2} - 2 \, a^{2} b c d + a^{3} d^{2}\right )}} + \frac{{\left (2 \, b c d - a d^{2}\right )} \log \left (d x^{2} + c\right )}{2 \,{\left (b^{2} c^{4} - 2 \, a b c^{3} d + a^{2} c^{2} d^{2}\right )}} - \frac{d}{2 \,{\left (b c^{3} - a c^{2} d +{\left (b c^{2} d - a c d^{2}\right )} x^{2}\right )}} + \frac{\log \left (x^{2}\right )}{2 \, a c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2*b^2*log(b*x^2 + a)/(a*b^2*c^2 - 2*a^2*b*c*d + a^3*d^2) + 1/2*(2*b*c*d - a*d
^2)*log(d*x^2 + c)/(b^2*c^4 - 2*a*b*c^3*d + a^2*c^2*d^2) - 1/2*d/(b*c^3 - a*c^2*
d + (b*c^2*d - a*c*d^2)*x^2) + 1/2*log(x^2)/(a*c^2)

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Fricas [A]  time = 1.00962, size = 296, normalized size = 2.96 \[ -\frac{a b c^{2} d - a^{2} c d^{2} +{\left (b^{2} c^{2} d x^{2} + b^{2} c^{3}\right )} \log \left (b x^{2} + a\right ) -{\left (2 \, a b c^{2} d - a^{2} c d^{2} +{\left (2 \, a b c d^{2} - a^{2} d^{3}\right )} x^{2}\right )} \log \left (d x^{2} + c\right ) - 2 \,{\left (b^{2} c^{3} - 2 \, a b c^{2} d + a^{2} c d^{2} +{\left (b^{2} c^{2} d - 2 \, a b c d^{2} + a^{2} d^{3}\right )} x^{2}\right )} \log \left (x\right )}{2 \,{\left (a b^{2} c^{5} - 2 \, a^{2} b c^{4} d + a^{3} c^{3} d^{2} +{\left (a b^{2} c^{4} d - 2 \, a^{2} b c^{3} d^{2} + a^{3} c^{2} d^{3}\right )} x^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2*(a*b*c^2*d - a^2*c*d^2 + (b^2*c^2*d*x^2 + b^2*c^3)*log(b*x^2 + a) - (2*a*b*
c^2*d - a^2*c*d^2 + (2*a*b*c*d^2 - a^2*d^3)*x^2)*log(d*x^2 + c) - 2*(b^2*c^3 - 2
*a*b*c^2*d + a^2*c*d^2 + (b^2*c^2*d - 2*a*b*c*d^2 + a^2*d^3)*x^2)*log(x))/(a*b^2
*c^5 - 2*a^2*b*c^4*d + a^3*c^3*d^2 + (a*b^2*c^4*d - 2*a^2*b*c^3*d^2 + a^3*c^2*d^
3)*x^2)

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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)**2,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.282932, size = 250, normalized size = 2.5 \[ -\frac{b^{3}{\rm ln}\left ({\left | b x^{2} + a \right |}\right )}{2 \,{\left (a b^{3} c^{2} - 2 \, a^{2} b^{2} c d + a^{3} b d^{2}\right )}} + \frac{{\left (2 \, b c d^{2} - a d^{3}\right )}{\rm ln}\left ({\left | d x^{2} + c \right |}\right )}{2 \,{\left (b^{2} c^{4} d - 2 \, a b c^{3} d^{2} + a^{2} c^{2} d^{3}\right )}} - \frac{2 \, b c d^{2} x^{2} - a d^{3} x^{2} + 3 \, b c^{2} d - 2 \, a c d^{2}}{2 \,{\left (b^{2} c^{4} - 2 \, a b c^{3} d + a^{2} c^{2} d^{2}\right )}{\left (d x^{2} + c\right )}} + \frac{{\rm ln}\left (x^{2}\right )}{2 \, a c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2*b^3*ln(abs(b*x^2 + a))/(a*b^3*c^2 - 2*a^2*b^2*c*d + a^3*b*d^2) + 1/2*(2*b*c
*d^2 - a*d^3)*ln(abs(d*x^2 + c))/(b^2*c^4*d - 2*a*b*c^3*d^2 + a^2*c^2*d^3) - 1/2
*(2*b*c*d^2*x^2 - a*d^3*x^2 + 3*b*c^2*d - 2*a*c*d^2)/((b^2*c^4 - 2*a*b*c^3*d + a
^2*c^2*d^2)*(d*x^2 + c)) + 1/2*ln(x^2)/(a*c^2)