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

Optimal. Leaf size=141 \[ -\frac{b^2 (b c-3 a d) \log \left (a+b x^2\right )}{2 a^2 (b c-a d)^3}+\frac{\log (x)}{a^2 c^2}+\frac{b^2}{2 a \left (a+b x^2\right ) (b c-a d)^2}-\frac{d^2 (3 b c-a d) \log \left (c+d x^2\right )}{2 c^2 (b c-a d)^3}+\frac{d^2}{2 c \left (c+d x^2\right ) (b c-a d)^2} \]

[Out]

b^2/(2*a*(b*c - a*d)^2*(a + b*x^2)) + d^2/(2*c*(b*c - a*d)^2*(c + d*x^2)) + Log[
x]/(a^2*c^2) - (b^2*(b*c - 3*a*d)*Log[a + b*x^2])/(2*a^2*(b*c - a*d)^3) - (d^2*(
3*b*c - a*d)*Log[c + d*x^2])/(2*c^2*(b*c - a*d)^3)

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Rubi [A]  time = 0.35751, antiderivative size = 141, 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 \[ -\frac{b^2 (b c-3 a d) \log \left (a+b x^2\right )}{2 a^2 (b c-a d)^3}+\frac{\log (x)}{a^2 c^2}+\frac{b^2}{2 a \left (a+b x^2\right ) (b c-a d)^2}-\frac{d^2 (3 b c-a d) \log \left (c+d x^2\right )}{2 c^2 (b c-a d)^3}+\frac{d^2}{2 c \left (c+d x^2\right ) (b c-a d)^2} \]

Antiderivative was successfully verified.

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

[Out]

b^2/(2*a*(b*c - a*d)^2*(a + b*x^2)) + d^2/(2*c*(b*c - a*d)^2*(c + d*x^2)) + Log[
x]/(a^2*c^2) - (b^2*(b*c - 3*a*d)*Log[a + b*x^2])/(2*a^2*(b*c - a*d)^3) - (d^2*(
3*b*c - a*d)*Log[c + d*x^2])/(2*c^2*(b*c - a*d)^3)

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Rubi in Sympy [A]  time = 62.5929, size = 124, normalized size = 0.88 \[ \frac{d^{2}}{2 c \left (c + d x^{2}\right ) \left (a d - b c\right )^{2}} - \frac{d^{2} \left (a d - 3 b c\right ) \log{\left (c + d x^{2} \right )}}{2 c^{2} \left (a d - b c\right )^{3}} + \frac{b^{2}}{2 a \left (a + b x^{2}\right ) \left (a d - b c\right )^{2}} - \frac{b^{2} \left (3 a d - b c\right ) \log{\left (a + b x^{2} \right )}}{2 a^{2} \left (a d - b c\right )^{3}} + \frac{\log{\left (x^{2} \right )}}{2 a^{2} c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.409594, size = 133, normalized size = 0.94 \[ \frac{1}{2} \left (\frac{b^2 (b c-3 a d) \log \left (a+b x^2\right )}{a^2 (a d-b c)^3}+\frac{2 \log (x)}{a^2 c^2}+\frac{b^2}{a \left (a+b x^2\right ) (b c-a d)^2}+\frac{d^2 (a d-3 b c) \log \left (c+d x^2\right )}{c^2 (b c-a d)^3}+\frac{d^2}{c \left (c+d x^2\right ) (b c-a d)^2}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(b^2/(a*(b*c - a*d)^2*(a + b*x^2)) + d^2/(c*(b*c - a*d)^2*(c + d*x^2)) + (2*Log[
x])/(a^2*c^2) + (b^2*(b*c - 3*a*d)*Log[a + b*x^2])/(a^2*(-(b*c) + a*d)^3) + (d^2
*(-3*b*c + a*d)*Log[c + d*x^2])/(c^2*(b*c - a*d)^3))/2

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Maple [A]  time = 0.033, size = 225, normalized size = 1.6 \[{\frac{\ln \left ( x \right ) }{{a}^{2}{c}^{2}}}+{\frac{{d}^{3}a}{2\,c \left ( ad-bc \right ) ^{3} \left ( d{x}^{2}+c \right ) }}-{\frac{{d}^{2}b}{2\, \left ( ad-bc \right ) ^{3} \left ( d{x}^{2}+c \right ) }}-{\frac{{d}^{3}\ln \left ( d{x}^{2}+c \right ) a}{2\,{c}^{2} \left ( ad-bc \right ) ^{3}}}+{\frac{3\,{d}^{2}\ln \left ( d{x}^{2}+c \right ) b}{2\,c \left ( ad-bc \right ) ^{3}}}-{\frac{3\,{b}^{2}\ln \left ( b{x}^{2}+a \right ) d}{2\,a \left ( ad-bc \right ) ^{3}}}+{\frac{{b}^{3}\ln \left ( b{x}^{2}+a \right ) c}{2\,{a}^{2} \left ( ad-bc \right ) ^{3}}}+{\frac{d{b}^{2}}{2\, \left ( ad-bc \right ) ^{3} \left ( b{x}^{2}+a \right ) }}-{\frac{{b}^{3}c}{2\,a \left ( ad-bc \right ) ^{3} \left ( b{x}^{2}+a \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 1.3684, size = 398, normalized size = 2.82 \[ -\frac{{\left (b^{3} c - 3 \, a b^{2} d\right )} \log \left (b x^{2} + a\right )}{2 \,{\left (a^{2} b^{3} c^{3} - 3 \, a^{3} b^{2} c^{2} d + 3 \, a^{4} b c d^{2} - a^{5} d^{3}\right )}} - \frac{{\left (3 \, b c d^{2} - a d^{3}\right )} \log \left (d x^{2} + c\right )}{2 \,{\left (b^{3} c^{5} - 3 \, a b^{2} c^{4} d + 3 \, a^{2} b c^{3} d^{2} - a^{3} c^{2} d^{3}\right )}} + \frac{b^{2} c^{2} + a^{2} d^{2} +{\left (b^{2} c d + a b d^{2}\right )} x^{2}}{2 \,{\left (a^{2} b^{2} c^{4} - 2 \, a^{3} b c^{3} d + a^{4} c^{2} d^{2} +{\left (a b^{3} c^{3} d - 2 \, a^{2} b^{2} c^{2} d^{2} + a^{3} b c d^{3}\right )} x^{4} +{\left (a b^{3} c^{4} - a^{2} b^{2} c^{3} d - a^{3} b c^{2} d^{2} + a^{4} c d^{3}\right )} x^{2}\right )}} + \frac{\log \left (x^{2}\right )}{2 \, a^{2} c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 4.12246, size = 729, normalized size = 5.17 \[ \frac{a b^{3} c^{4} - a^{2} b^{2} c^{3} d + a^{3} b c^{2} d^{2} - a^{4} c d^{3} +{\left (a b^{3} c^{3} d - a^{3} b c d^{3}\right )} x^{2} -{\left (a b^{3} c^{4} - 3 \, a^{2} b^{2} c^{3} d +{\left (b^{4} c^{3} d - 3 \, a b^{3} c^{2} d^{2}\right )} x^{4} +{\left (b^{4} c^{4} - 2 \, a b^{3} c^{3} d - 3 \, a^{2} b^{2} c^{2} d^{2}\right )} x^{2}\right )} \log \left (b x^{2} + a\right ) -{\left (3 \, a^{3} b c^{2} d^{2} - a^{4} c d^{3} +{\left (3 \, a^{2} b^{2} c d^{3} - a^{3} b d^{4}\right )} x^{4} +{\left (3 \, a^{2} b^{2} c^{2} d^{2} + 2 \, a^{3} b c d^{3} - a^{4} d^{4}\right )} x^{2}\right )} \log \left (d x^{2} + c\right ) + 2 \,{\left (a b^{3} c^{4} - 3 \, a^{2} b^{2} c^{3} d + 3 \, a^{3} b c^{2} d^{2} - a^{4} c d^{3} +{\left (b^{4} c^{3} d - 3 \, a b^{3} c^{2} d^{2} + 3 \, a^{2} b^{2} c d^{3} - a^{3} b d^{4}\right )} x^{4} +{\left (b^{4} c^{4} - 2 \, a b^{3} c^{3} d + 2 \, a^{3} b c d^{3} - a^{4} d^{4}\right )} x^{2}\right )} \log \left (x\right )}{2 \,{\left (a^{3} b^{3} c^{6} - 3 \, a^{4} b^{2} c^{5} d + 3 \, a^{5} b c^{4} d^{2} - a^{6} c^{3} d^{3} +{\left (a^{2} b^{4} c^{5} d - 3 \, a^{3} b^{3} c^{4} d^{2} + 3 \, a^{4} b^{2} c^{3} d^{3} - a^{5} b c^{2} d^{4}\right )} x^{4} +{\left (a^{2} b^{4} c^{6} - 2 \, a^{3} b^{3} c^{5} d + 2 \, a^{5} b c^{3} d^{3} - a^{6} c^{2} d^{4}\right )} x^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*(a*b^3*c^4 - a^2*b^2*c^3*d + a^3*b*c^2*d^2 - a^4*c*d^3 + (a*b^3*c^3*d - a^3*
b*c*d^3)*x^2 - (a*b^3*c^4 - 3*a^2*b^2*c^3*d + (b^4*c^3*d - 3*a*b^3*c^2*d^2)*x^4
+ (b^4*c^4 - 2*a*b^3*c^3*d - 3*a^2*b^2*c^2*d^2)*x^2)*log(b*x^2 + a) - (3*a^3*b*c
^2*d^2 - a^4*c*d^3 + (3*a^2*b^2*c*d^3 - a^3*b*d^4)*x^4 + (3*a^2*b^2*c^2*d^2 + 2*
a^3*b*c*d^3 - a^4*d^4)*x^2)*log(d*x^2 + c) + 2*(a*b^3*c^4 - 3*a^2*b^2*c^3*d + 3*
a^3*b*c^2*d^2 - a^4*c*d^3 + (b^4*c^3*d - 3*a*b^3*c^2*d^2 + 3*a^2*b^2*c*d^3 - a^3
*b*d^4)*x^4 + (b^4*c^4 - 2*a*b^3*c^3*d + 2*a^3*b*c*d^3 - a^4*d^4)*x^2)*log(x))/(
a^3*b^3*c^6 - 3*a^4*b^2*c^5*d + 3*a^5*b*c^4*d^2 - a^6*c^3*d^3 + (a^2*b^4*c^5*d -
3*a^3*b^3*c^4*d^2 + 3*a^4*b^2*c^3*d^3 - a^5*b*c^2*d^4)*x^4 + (a^2*b^4*c^6 - 2*a
^3*b^3*c^5*d + 2*a^5*b*c^3*d^3 - a^6*c^2*d^4)*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)**2/(d*x**2+c)**2,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)^2*(d*x^2 + c)^2*x),x, algorithm="giac")

[Out]

Exception raised: NotImplementedError