Optimal. Leaf size=144 \[ -\frac{b^{5/2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{a^{3/2} (b c-a d)^2}+\frac{d^{3/2} (5 b c-3 a d) \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{2 c^{5/2} (b c-a d)^2}-\frac{2 b c-3 a d}{2 a c^2 x (b c-a d)}-\frac{d}{2 c x \left (c+d x^2\right ) (b c-a d)} \]
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Rubi [A] time = 0.530002, antiderivative size = 144, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ -\frac{b^{5/2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{a^{3/2} (b c-a d)^2}+\frac{d^{3/2} (5 b c-3 a d) \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{2 c^{5/2} (b c-a d)^2}-\frac{2 b c-3 a d}{2 a c^2 x (b c-a d)}-\frac{d}{2 c x \left (c+d x^2\right ) (b c-a d)} \]
Antiderivative was successfully verified.
[In] Int[1/(x^2*(a + b*x^2)*(c + d*x^2)^2),x]
[Out]
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Rubi in Sympy [A] time = 102.365, size = 121, normalized size = 0.84 \[ \frac{d}{2 c x \left (c + d x^{2}\right ) \left (a d - b c\right )} - \frac{d^{\frac{3}{2}} \left (3 a d - 5 b c\right ) \operatorname{atan}{\left (\frac{\sqrt{d} x}{\sqrt{c}} \right )}}{2 c^{\frac{5}{2}} \left (a d - b c\right )^{2}} - \frac{3 a d - 2 b c}{2 a c^{2} x \left (a d - b c\right )} - \frac{b^{\frac{5}{2}} \operatorname{atan}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{a^{\frac{3}{2}} \left (a d - b c\right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/x**2/(b*x**2+a)/(d*x**2+c)**2,x)
[Out]
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Mathematica [A] time = 0.480483, size = 123, normalized size = 0.85 \[ -\frac{b^{5/2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{a^{3/2} (a d-b c)^2}+\frac{d^{3/2} (5 b c-3 a d) \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{2 c^{5/2} (b c-a d)^2}+\frac{d^2 x}{2 c^2 \left (c+d x^2\right ) (b c-a d)}-\frac{1}{a c^2 x} \]
Antiderivative was successfully verified.
[In] Integrate[1/(x^2*(a + b*x^2)*(c + d*x^2)^2),x]
[Out]
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Maple [A] time = 0.02, size = 169, normalized size = 1.2 \[ -{\frac{1}{a{c}^{2}x}}-{\frac{{d}^{3}xa}{2\,{c}^{2} \left ( ad-bc \right ) ^{2} \left ( d{x}^{2}+c \right ) }}+{\frac{{d}^{2}xb}{2\,c \left ( ad-bc \right ) ^{2} \left ( d{x}^{2}+c \right ) }}-{\frac{3\,a{d}^{3}}{2\,{c}^{2} \left ( ad-bc \right ) ^{2}}\arctan \left ({dx{\frac{1}{\sqrt{cd}}}} \right ){\frac{1}{\sqrt{cd}}}}+{\frac{5\,b{d}^{2}}{2\,c \left ( ad-bc \right ) ^{2}}\arctan \left ({dx{\frac{1}{\sqrt{cd}}}} \right ){\frac{1}{\sqrt{cd}}}}-{\frac{{b}^{3}}{a \left ( ad-bc \right ) ^{2}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/x^2/(b*x^2+a)/(d*x^2+c)^2,x)
[Out]
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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)*(d*x^2 + c)^2*x^2),x, algorithm="maxima")
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Fricas [A] time = 0.614686, 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)*(d*x^2 + c)^2*x^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/x**2/(b*x**2+a)/(d*x**2+c)**2,x)
[Out]
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GIAC/XCAS [A] time = 0.283047, size = 221, normalized size = 1.53 \[ -\frac{b^{3} \arctan \left (\frac{b x}{\sqrt{a b}}\right )}{{\left (a b^{2} c^{2} - 2 \, a^{2} b c d + a^{3} d^{2}\right )} \sqrt{a b}} + \frac{{\left (5 \, b c d^{2} - 3 \, a d^{3}\right )} \arctan \left (\frac{d x}{\sqrt{c d}}\right )}{2 \,{\left (b^{2} c^{4} - 2 \, a b c^{3} d + a^{2} c^{2} d^{2}\right )} \sqrt{c d}} - \frac{2 \, b c d x^{2} - 3 \, a d^{2} x^{2} + 2 \, b c^{2} - 2 \, a c d}{2 \,{\left (a b c^{3} - a^{2} c^{2} d\right )}{\left (d x^{3} + c x\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x^2 + a)*(d*x^2 + c)^2*x^2),x, algorithm="giac")
[Out]