Optimal. Leaf size=514 \[ -\frac{-10 a A c-a b C+3 A b^2}{2 a^2 x \left (b^2-4 a c\right )}-\frac{\sqrt{c} \left (A \left (3 b^2 \sqrt{b^2-4 a c}-10 a c \sqrt{b^2-4 a c}-16 a b c+3 b^3\right )-a C \left (b \sqrt{b^2-4 a c}-12 a c+b^2\right )\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right )}{2 \sqrt{2} a^2 \left (b^2-4 a c\right )^{3/2} \sqrt{b-\sqrt{b^2-4 a c}}}-\frac{\sqrt{c} \left (-\frac{A \left (3 b^3-16 a b c\right )-a C \left (b^2-12 a c\right )}{\sqrt{b^2-4 a c}}-10 a A c-a b C+3 A b^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right )}{2 \sqrt{2} a^2 \left (b^2-4 a c\right ) \sqrt{\sqrt{b^2-4 a c}+b}}+\frac{b B \left (b^2-6 a c\right ) \tanh ^{-1}\left (\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right )}{2 a^2 \left (b^2-4 a c\right )^{3/2}}-\frac{B \log \left (a+b x^2+c x^4\right )}{4 a^2}+\frac{B \log (x)}{a^2}+\frac{A \left (b^2-2 a c\right )+c x^2 (A b-2 a C)-a b C}{2 a x \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}+\frac{B \left (-2 a c+b^2+b c x^2\right )}{2 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )} \]
[Out]
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Rubi [A] time = 3.79182, antiderivative size = 514, normalized size of antiderivative = 1., number of steps used = 15, number of rules used = 13, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.464 \[ -\frac{-10 a A c-a b C+3 A b^2}{2 a^2 x \left (b^2-4 a c\right )}-\frac{\sqrt{c} \left (A \left (3 b^2 \sqrt{b^2-4 a c}-10 a c \sqrt{b^2-4 a c}-16 a b c+3 b^3\right )-a C \left (b \sqrt{b^2-4 a c}-12 a c+b^2\right )\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right )}{2 \sqrt{2} a^2 \left (b^2-4 a c\right )^{3/2} \sqrt{b-\sqrt{b^2-4 a c}}}-\frac{\sqrt{c} \left (-\frac{A \left (3 b^3-16 a b c\right )-a C \left (b^2-12 a c\right )}{\sqrt{b^2-4 a c}}-10 a A c-a b C+3 A b^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right )}{2 \sqrt{2} a^2 \left (b^2-4 a c\right ) \sqrt{\sqrt{b^2-4 a c}+b}}+\frac{b B \left (b^2-6 a c\right ) \tanh ^{-1}\left (\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right )}{2 a^2 \left (b^2-4 a c\right )^{3/2}}-\frac{B \log \left (a+b x^2+c x^4\right )}{4 a^2}+\frac{B \log (x)}{a^2}+\frac{A \left (b^2-2 a c\right )+c x^2 (A b-2 a C)-a b C}{2 a x \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}+\frac{B \left (-2 a c+b^2+b c x^2\right )}{2 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )} \]
Antiderivative was successfully verified.
[In] Int[(A + B*x + C*x^2)/(x^2*(a + b*x^2 + c*x^4)^2),x]
[Out]
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Rubi in Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((C*x**2+B*x+A)/x**2/(c*x**4+b*x**2+a)**2,x)
[Out]
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Mathematica [A] time = 4.33749, size = 559, normalized size = 1.09 \[ \frac{\frac{-4 a^2 c (B+C x)+2 a \left (b c x (3 A+x (B+C x))+2 A c^2 x^3+b^2 (B+C x)\right )-2 A b^2 x \left (b+c x^2\right )}{\left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}+\frac{\sqrt{2} \sqrt{c} \left (A \left (-3 b^2 \sqrt{b^2-4 a c}+10 a c \sqrt{b^2-4 a c}+16 a b c-3 b^3\right )+a C \left (b \sqrt{b^2-4 a c}-12 a c+b^2\right )\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right )}{\left (b^2-4 a c\right )^{3/2} \sqrt{b-\sqrt{b^2-4 a c}}}+\frac{\sqrt{2} \sqrt{c} \left (A \left (-3 b^2 \sqrt{b^2-4 a c}+10 a c \sqrt{b^2-4 a c}-16 a b c+3 b^3\right )+a C \left (b \sqrt{b^2-4 a c}+12 a c-b^2\right )\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right )}{\left (b^2-4 a c\right )^{3/2} \sqrt{\sqrt{b^2-4 a c}+b}}-\frac{B \left (b^2 \sqrt{b^2-4 a c}-4 a c \sqrt{b^2-4 a c}-6 a b c+b^3\right ) \log \left (\sqrt{b^2-4 a c}-b-2 c x^2\right )}{\left (b^2-4 a c\right )^{3/2}}-\frac{B \left (b^2 \sqrt{b^2-4 a c}-4 a c \sqrt{b^2-4 a c}+6 a b c-b^3\right ) \log \left (\sqrt{b^2-4 a c}+b+2 c x^2\right )}{\left (b^2-4 a c\right )^{3/2}}-\frac{4 A}{x}+4 B \log (x)}{4 a^2} \]
Antiderivative was successfully verified.
[In] Integrate[(A + B*x + C*x^2)/(x^2*(a + b*x^2 + c*x^4)^2),x]
[Out]
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Maple [B] time = 0.119, size = 6960, normalized size = 13.5 \[ \text{output too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((C*x^2+B*x+A)/x^2/(c*x^4+b*x^2+a)^2,x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \frac{B a b c x^{3} +{\left (10 \, A a c^{2} +{\left (C a b - 3 \, A b^{2}\right )} c\right )} x^{4} - 2 \, A a b^{2} + 8 \, A a^{2} c +{\left (C a b^{2} - 3 \, A b^{3} -{\left (2 \, C a^{2} - 11 \, A a b\right )} c\right )} x^{2} +{\left (B a b^{2} - 2 \, B a^{2} c\right )} x}{2 \,{\left ({\left (a^{2} b^{2} c - 4 \, a^{3} c^{2}\right )} x^{5} +{\left (a^{2} b^{3} - 4 \, a^{3} b c\right )} x^{3} +{\left (a^{3} b^{2} - 4 \, a^{4} c\right )} x\right )}} + \frac{\int \frac{C a b^{2} - 3 \, A b^{3} - 2 \,{\left (B b^{2} c - 4 \, B a c^{2}\right )} x^{3} +{\left (10 \, A a c^{2} +{\left (C a b - 3 \, A b^{2}\right )} c\right )} x^{2} -{\left (6 \, C a^{2} - 13 \, A a b\right )} c - 2 \,{\left (B b^{3} - 5 \, B a b c\right )} x}{c x^{4} + b x^{2} + a}\,{d x}}{2 \,{\left (a^{2} b^{2} - 4 \, a^{3} c\right )}} + \frac{B \log \left (x\right )}{a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((C*x^2 + B*x + A)/((c*x^4 + b*x^2 + a)^2*x^2),x, algorithm="maxima")
[Out]
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Fricas [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((C*x^2 + B*x + A)/((c*x^4 + b*x^2 + a)^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((C*x**2+B*x+A)/x**2/(c*x**4+b*x**2+a)**2,x)
[Out]
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GIAC/XCAS [F(-2)] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((C*x^2 + B*x + A)/((c*x^4 + b*x^2 + a)^2*x^2),x, algorithm="giac")
[Out]