Optimal. Leaf size=109 \[ \frac{b^{3/2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{\sqrt{a} (b c-a d)^2}-\frac{\sqrt{d} (3 b c-a d) \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{2 c^{3/2} (b c-a d)^2}-\frac{d x}{2 c \left (c+d x^2\right ) (b c-a d)} \]
[Out]
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Rubi [A] time = 0.212676, antiderivative size = 109, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158 \[ \frac{b^{3/2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{\sqrt{a} (b c-a d)^2}-\frac{\sqrt{d} (3 b c-a d) \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{2 c^{3/2} (b c-a d)^2}-\frac{d x}{2 c \left (c+d x^2\right ) (b c-a d)} \]
Antiderivative was successfully verified.
[In] Int[1/((a + b*x^2)*(c + d*x^2)^2),x]
[Out]
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Rubi in Sympy [A] time = 41.2097, size = 94, normalized size = 0.86 \[ \frac{d x}{2 c \left (c + d x^{2}\right ) \left (a d - b c\right )} + \frac{\sqrt{d} \left (a d - 3 b c\right ) \operatorname{atan}{\left (\frac{\sqrt{d} x}{\sqrt{c}} \right )}}{2 c^{\frac{3}{2}} \left (a d - b c\right )^{2}} + \frac{b^{\frac{3}{2}} \operatorname{atan}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{\sqrt{a} \left (a d - b c\right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(b*x**2+a)/(d*x**2+c)**2,x)
[Out]
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Mathematica [A] time = 0.316056, size = 95, normalized size = 0.87 \[ \frac{\frac{2 b^{3/2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{\sqrt{a}}+\frac{\sqrt{d} (a d-3 b c) \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{c^{3/2}}+\frac{d x (a d-b c)}{c \left (c+d x^2\right )}}{2 (b c-a d)^2} \]
Antiderivative was successfully verified.
[In] Integrate[1/((a + b*x^2)*(c + d*x^2)^2),x]
[Out]
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Maple [A] time = 0.016, size = 144, normalized size = 1.3 \[{\frac{{d}^{2}xa}{2\, \left ( ad-bc \right ) ^{2}c \left ( d{x}^{2}+c \right ) }}-{\frac{dxb}{2\, \left ( ad-bc \right ) ^{2} \left ( d{x}^{2}+c \right ) }}+{\frac{a{d}^{2}}{2\, \left ( ad-bc \right ) ^{2}c}\arctan \left ({dx{\frac{1}{\sqrt{cd}}}} \right ){\frac{1}{\sqrt{cd}}}}-{\frac{3\,bd}{2\, \left ( ad-bc \right ) ^{2}}\arctan \left ({dx{\frac{1}{\sqrt{cd}}}} \right ){\frac{1}{\sqrt{cd}}}}+{\frac{{b}^{2}}{ \left ( ad-bc \right ) ^{2}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(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, algorithm="maxima")
[Out]
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Fricas [A] time = 0.317974, size = 1, normalized size = 0.01 \[ \left [\frac{2 \,{\left (b c d x^{2} + b c^{2}\right )} \sqrt{-\frac{b}{a}} \log \left (\frac{b x^{2} + 2 \, a x \sqrt{-\frac{b}{a}} - a}{b x^{2} + a}\right ) -{\left (3 \, b c^{2} - a c d +{\left (3 \, b c d - a d^{2}\right )} x^{2}\right )} \sqrt{-\frac{d}{c}} \log \left (\frac{d x^{2} + 2 \, c x \sqrt{-\frac{d}{c}} - c}{d x^{2} + c}\right ) - 2 \,{\left (b c d - a d^{2}\right )} x}{4 \,{\left (b^{2} c^{4} - 2 \, a b c^{3} d + a^{2} c^{2} d^{2} +{\left (b^{2} c^{3} d - 2 \, a b c^{2} d^{2} + a^{2} c d^{3}\right )} x^{2}\right )}}, -\frac{{\left (3 \, b c^{2} - a c d +{\left (3 \, b c d - a d^{2}\right )} x^{2}\right )} \sqrt{\frac{d}{c}} \arctan \left (\frac{d x}{c \sqrt{\frac{d}{c}}}\right ) -{\left (b c d x^{2} + b c^{2}\right )} \sqrt{-\frac{b}{a}} \log \left (\frac{b x^{2} + 2 \, a x \sqrt{-\frac{b}{a}} - a}{b x^{2} + a}\right ) +{\left (b c d - a d^{2}\right )} x}{2 \,{\left (b^{2} c^{4} - 2 \, a b c^{3} d + a^{2} c^{2} d^{2} +{\left (b^{2} c^{3} d - 2 \, a b c^{2} d^{2} + a^{2} c d^{3}\right )} x^{2}\right )}}, \frac{4 \,{\left (b c d x^{2} + b c^{2}\right )} \sqrt{\frac{b}{a}} \arctan \left (\frac{b x}{a \sqrt{\frac{b}{a}}}\right ) -{\left (3 \, b c^{2} - a c d +{\left (3 \, b c d - a d^{2}\right )} x^{2}\right )} \sqrt{-\frac{d}{c}} \log \left (\frac{d x^{2} + 2 \, c x \sqrt{-\frac{d}{c}} - c}{d x^{2} + c}\right ) - 2 \,{\left (b c d - a d^{2}\right )} x}{4 \,{\left (b^{2} c^{4} - 2 \, a b c^{3} d + a^{2} c^{2} d^{2} +{\left (b^{2} c^{3} d - 2 \, a b c^{2} d^{2} + a^{2} c d^{3}\right )} x^{2}\right )}}, \frac{2 \,{\left (b c d x^{2} + b c^{2}\right )} \sqrt{\frac{b}{a}} \arctan \left (\frac{b x}{a \sqrt{\frac{b}{a}}}\right ) -{\left (3 \, b c^{2} - a c d +{\left (3 \, b c d - a d^{2}\right )} x^{2}\right )} \sqrt{\frac{d}{c}} \arctan \left (\frac{d x}{c \sqrt{\frac{d}{c}}}\right ) -{\left (b c d - a d^{2}\right )} x}{2 \,{\left (b^{2} c^{4} - 2 \, a b c^{3} d + a^{2} c^{2} d^{2} +{\left (b^{2} c^{3} d - 2 \, a b c^{2} d^{2} + a^{2} c d^{3}\right )} x^{2}\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x^2 + a)*(d*x^2 + c)^2),x, algorithm="fricas")
[Out]
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Sympy [A] time = 46.0257, size = 2033, normalized size = 18.65 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(b*x**2+a)/(d*x**2+c)**2,x)
[Out]
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GIAC/XCAS [A] time = 0.223712, size = 165, normalized size = 1.51 \[ \frac{b^{2} \arctan \left (\frac{b x}{\sqrt{a b}}\right )}{{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \sqrt{a b}} - \frac{{\left (3 \, b c d - a d^{2}\right )} \arctan \left (\frac{d x}{\sqrt{c d}}\right )}{2 \,{\left (b^{2} c^{3} - 2 \, a b c^{2} d + a^{2} c d^{2}\right )} \sqrt{c d}} - \frac{d x}{2 \,{\left (b c^{2} - a c d\right )}{\left (d x^{2} + c\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x^2 + a)*(d*x^2 + c)^2),x, algorithm="giac")
[Out]