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

Optimal. Leaf size=92 \[ -\frac{b}{2 \left (a+b x^2\right ) (b c-a d)^2}-\frac{d}{2 \left (c+d x^2\right ) (b c-a d)^2}-\frac{b d \log \left (a+b x^2\right )}{(b c-a d)^3}+\frac{b d \log \left (c+d x^2\right )}{(b c-a d)^3} \]

[Out]

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

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Rubi [A]  time = 0.176817, antiderivative size = 92, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1 \[ -\frac{b}{2 \left (a+b x^2\right ) (b c-a d)^2}-\frac{d}{2 \left (c+d x^2\right ) (b c-a d)^2}-\frac{b d \log \left (a+b x^2\right )}{(b c-a d)^3}+\frac{b d \log \left (c+d x^2\right )}{(b c-a d)^3} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 33.1255, size = 76, normalized size = 0.83 \[ \frac{b d \log{\left (a + b x^{2} \right )}}{\left (a d - b c\right )^{3}} - \frac{b d \log{\left (c + d x^{2} \right )}}{\left (a d - b c\right )^{3}} - \frac{b}{2 \left (a + b x^{2}\right ) \left (a d - b c\right )^{2}} - \frac{d}{2 \left (c + d x^{2}\right ) \left (a d - b c\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.119363, size = 77, normalized size = 0.84 \[ \frac{\frac{b (a d-b c)}{a+b x^2}+\frac{d (a d-b c)}{c+d x^2}-2 b d \log \left (a+b x^2\right )+2 b d \log \left (c+d x^2\right )}{2 (b c-a d)^3} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.024, size = 143, normalized size = 1.6 \[ -{\frac{a{d}^{2}}{2\, \left ( ad-bc \right ) ^{3} \left ( d{x}^{2}+c \right ) }}+{\frac{bdc}{2\, \left ( ad-bc \right ) ^{3} \left ( d{x}^{2}+c \right ) }}-{\frac{bd\ln \left ( d{x}^{2}+c \right ) }{ \left ( ad-bc \right ) ^{3}}}+{\frac{b\ln \left ( b{x}^{2}+a \right ) d}{ \left ( ad-bc \right ) ^{3}}}-{\frac{abd}{2\, \left ( ad-bc \right ) ^{3} \left ( b{x}^{2}+a \right ) }}+{\frac{{b}^{2}c}{2\, \left ( ad-bc \right ) ^{3} \left ( b{x}^{2}+a \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 1.36526, size = 290, normalized size = 3.15 \[ -\frac{b d \log \left (b x^{2} + a\right )}{b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}} + \frac{b d \log \left (d x^{2} + c\right )}{b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}} - \frac{2 \, b d x^{2} + b c + a d}{2 \,{\left (a b^{2} c^{3} - 2 \, a^{2} b c^{2} d + a^{3} c d^{2} +{\left (b^{3} c^{2} d - 2 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} x^{4} +{\left (b^{3} c^{3} - a b^{2} c^{2} d - a^{2} b c d^{2} + a^{3} d^{3}\right )} x^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.255273, size = 342, normalized size = 3.72 \[ -\frac{b^{2} c^{2} - a^{2} d^{2} + 2 \,{\left (b^{2} c d - a b d^{2}\right )} x^{2} + 2 \,{\left (b^{2} d^{2} x^{4} + a b c d +{\left (b^{2} c d + a b d^{2}\right )} x^{2}\right )} \log \left (b x^{2} + a\right ) - 2 \,{\left (b^{2} d^{2} x^{4} + a b c d +{\left (b^{2} c d + a b d^{2}\right )} x^{2}\right )} \log \left (d x^{2} + c\right )}{2 \,{\left (a b^{3} c^{4} - 3 \, a^{2} b^{2} c^{3} d + 3 \, a^{3} b c^{2} d^{2} - a^{4} c d^{3} +{\left (b^{4} c^{3} d - 3 \, a b^{3} c^{2} d^{2} + 3 \, a^{2} b^{2} c d^{3} - a^{3} b d^{4}\right )} x^{4} +{\left (b^{4} c^{4} - 2 \, a b^{3} c^{3} d + 2 \, a^{3} b c d^{3} - a^{4} d^{4}\right )} x^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 14.4604, size = 408, normalized size = 4.43 \[ - \frac{b d \log{\left (x^{2} + \frac{- \frac{a^{4} b d^{5}}{\left (a d - b c\right )^{3}} + \frac{4 a^{3} b^{2} c d^{4}}{\left (a d - b c\right )^{3}} - \frac{6 a^{2} b^{3} c^{2} d^{3}}{\left (a d - b c\right )^{3}} + \frac{4 a b^{4} c^{3} d^{2}}{\left (a d - b c\right )^{3}} + a b d^{2} - \frac{b^{5} c^{4} d}{\left (a d - b c\right )^{3}} + b^{2} c d}{2 b^{2} d^{2}} \right )}}{\left (a d - b c\right )^{3}} + \frac{b d \log{\left (x^{2} + \frac{\frac{a^{4} b d^{5}}{\left (a d - b c\right )^{3}} - \frac{4 a^{3} b^{2} c d^{4}}{\left (a d - b c\right )^{3}} + \frac{6 a^{2} b^{3} c^{2} d^{3}}{\left (a d - b c\right )^{3}} - \frac{4 a b^{4} c^{3} d^{2}}{\left (a d - b c\right )^{3}} + a b d^{2} + \frac{b^{5} c^{4} d}{\left (a d - b c\right )^{3}} + b^{2} c d}{2 b^{2} d^{2}} \right )}}{\left (a d - b c\right )^{3}} - \frac{a d + b c + 2 b d x^{2}}{2 a^{3} c d^{2} - 4 a^{2} b c^{2} d + 2 a b^{2} c^{3} + x^{4} \left (2 a^{2} b d^{3} - 4 a b^{2} c d^{2} + 2 b^{3} c^{2} d\right ) + x^{2} \left (2 a^{3} d^{3} - 2 a^{2} b c d^{2} - 2 a b^{2} c^{2} d + 2 b^{3} c^{3}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.251424, size = 220, normalized size = 2.39 \[ \frac{b^{2} d{\rm ln}\left ({\left | \frac{b c}{b x^{2} + a} - \frac{a d}{b x^{2} + a} + d \right |}\right )}{b^{4} c^{3} - 3 \, a b^{3} c^{2} d + 3 \, a^{2} b^{2} c d^{2} - a^{3} b d^{3}} - \frac{b^{3}}{2 \,{\left (b^{4} c^{2} - 2 \, a b^{3} c d + a^{2} b^{2} d^{2}\right )}{\left (b x^{2} + a\right )}} + \frac{b d^{2}}{2 \,{\left (b c - a d\right )}^{3}{\left (\frac{b c}{b x^{2} + a} - \frac{a d}{b x^{2} + a} + d\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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