Optimal. Leaf size=167 \[ \frac{b^{3/2} (b c-5 a d) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 a^{3/2} (b c-a d)^3}+\frac{d^{3/2} (5 b c-a d) \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{2 c^{3/2} (b c-a d)^3}+\frac{b x}{2 a \left (a+b x^2\right ) \left (c+d x^2\right ) (b c-a d)}+\frac{d x (a d+b c)}{2 a c \left (c+d x^2\right ) (b c-a d)^2} \]
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Rubi [A] time = 0.456696, antiderivative size = 167, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21 \[ \frac{b^{3/2} (b c-5 a d) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 a^{3/2} (b c-a d)^3}+\frac{d^{3/2} (5 b c-a d) \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{2 c^{3/2} (b c-a d)^3}+\frac{b x}{2 a \left (a+b x^2\right ) \left (c+d x^2\right ) (b c-a d)}+\frac{d x (a d+b c)}{2 a c \left (c+d x^2\right ) (b c-a d)^2} \]
Antiderivative was successfully verified.
[In] Int[1/((a + b*x^2)^2*(c + d*x^2)^2),x]
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Rubi in Sympy [A] time = 99.6727, size = 141, normalized size = 0.84 \[ \frac{d x}{2 c \left (a + b x^{2}\right ) \left (c + d x^{2}\right ) \left (a d - b c\right )} + \frac{d^{\frac{3}{2}} \left (a d - 5 b c\right ) \operatorname{atan}{\left (\frac{\sqrt{d} x}{\sqrt{c}} \right )}}{2 c^{\frac{3}{2}} \left (a d - b c\right )^{3}} + \frac{b x \left (a d + b c\right )}{2 a c \left (a + b x^{2}\right ) \left (a d - b c\right )^{2}} + \frac{b^{\frac{3}{2}} \left (5 a d - b c\right ) \operatorname{atan}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{2 a^{\frac{3}{2}} \left (a d - b c\right )^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(b*x**2+a)**2/(d*x**2+c)**2,x)
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Mathematica [A] time = 0.555684, size = 136, normalized size = 0.81 \[ \frac{1}{2} \left (\frac{b^{3/2} (5 a d-b c) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{a^{3/2} (a d-b c)^3}+\frac{x (b c-a d) \left (\frac{b^2}{a^2+a b x^2}+\frac{d^2}{c^2+c d x^2}\right )+\frac{d^{3/2} (5 b c-a d) \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{c^{3/2}}}{(b c-a d)^3}\right ) \]
Antiderivative was successfully verified.
[In] Integrate[1/((a + b*x^2)^2*(c + d*x^2)^2),x]
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Maple [A] time = 0.002, size = 238, normalized size = 1.4 \[{\frac{{d}^{3}xa}{2\, \left ( ad-bc \right ) ^{3}c \left ( d{x}^{2}+c \right ) }}-{\frac{{d}^{2}xb}{2\, \left ( ad-bc \right ) ^{3} \left ( d{x}^{2}+c \right ) }}+{\frac{{d}^{3}a}{2\, \left ( ad-bc \right ) ^{3}c}\arctan \left ({dx{\frac{1}{\sqrt{cd}}}} \right ){\frac{1}{\sqrt{cd}}}}-{\frac{5\,{d}^{2}b}{2\, \left ( ad-bc \right ) ^{3}}\arctan \left ({dx{\frac{1}{\sqrt{cd}}}} \right ){\frac{1}{\sqrt{cd}}}}+{\frac{{b}^{2}xd}{2\, \left ( ad-bc \right ) ^{3} \left ( b{x}^{2}+a \right ) }}-{\frac{{b}^{3}xc}{2\, \left ( ad-bc \right ) ^{3}a \left ( b{x}^{2}+a \right ) }}+{\frac{5\,{b}^{2}d}{2\, \left ( ad-bc \right ) ^{3}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}-{\frac{{b}^{3}c}{2\, \left ( ad-bc \right ) ^{3}a}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(b*x^2+a)^2/(d*x^2+c)^2,x)
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x^2 + a)^2*(d*x^2 + c)^2),x, algorithm="maxima")
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Fricas [A] time = 0.994173, size = 1, normalized size = 0.01 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x^2 + a)^2*(d*x^2 + c)^2),x, algorithm="fricas")
[Out]
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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/(b*x**2+a)**2/(d*x**2+c)**2,x)
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GIAC/XCAS [A] time = 0.394462, size = 1, normalized size = 0.01 \[ \mathit{Done} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x^2 + a)^2*(d*x^2 + c)^2),x, algorithm="giac")
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