Optimal. Leaf size=621 \[ \frac{x \left (c x^2 \left (20 a^2 c f+a b^2 f-24 a b c d+3 b^3 d\right )+8 a^2 b c f+28 a^2 c^2 d+a b^3 f-25 a b^2 c d+3 b^4 d\right )}{8 a^2 \left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )}+\frac{\sqrt{c} \left (-\frac{-52 a^2 b c f+168 a^2 c^2 d+a b^3 f-30 a b^2 c d+3 b^4 d}{\sqrt{b^2-4 a c}}+20 a^2 c f+a b^2 f-24 a b c d+3 b^3 d\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right )}{8 \sqrt{2} a^2 \left (b^2-4 a c\right )^2 \sqrt{\sqrt{b^2-4 a c}+b}}+\frac{\sqrt{c} \left (4 a^2 c \left (5 f \sqrt{b^2-4 a c}+42 c d\right )-a b^2 \left (30 c d-f \sqrt{b^2-4 a c}\right )-4 a b c \left (6 d \sqrt{b^2-4 a c}+13 a f\right )+b^3 \left (3 d \sqrt{b^2-4 a c}+a f\right )+3 b^4 d\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right )}{8 \sqrt{2} a^2 \left (b^2-4 a c\right )^{5/2} \sqrt{b-\sqrt{b^2-4 a c}}}-\frac{6 c^2 e \tanh ^{-1}\left (\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{5/2}}+\frac{x \left (c x^2 (b d-2 a f)-a b f-2 a c d+b^2 d\right )}{4 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}+\frac{3 c e \left (b+2 c x^2\right )}{2 \left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )}-\frac{e \left (b+2 c x^2\right )}{4 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2} \]
[Out]
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Rubi [A] time = 10.4248, antiderivative size = 621, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 9, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.36 \[ \frac{x \left (c x^2 \left (20 a^2 c f+a b^2 f-24 a b c d+3 b^3 d\right )+8 a^2 b c f+28 a^2 c^2 d+a b^3 f-25 a b^2 c d+3 b^4 d\right )}{8 a^2 \left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )}+\frac{\sqrt{c} \left (-\frac{-52 a^2 b c f+168 a^2 c^2 d+a b^3 f-30 a b^2 c d+3 b^4 d}{\sqrt{b^2-4 a c}}+20 a^2 c f+a b^2 f-24 a b c d+3 b^3 d\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right )}{8 \sqrt{2} a^2 \left (b^2-4 a c\right )^2 \sqrt{\sqrt{b^2-4 a c}+b}}+\frac{\sqrt{c} \left (4 a^2 c \left (5 f \sqrt{b^2-4 a c}+42 c d\right )-a b^2 \left (30 c d-f \sqrt{b^2-4 a c}\right )-4 a b c \left (6 d \sqrt{b^2-4 a c}+13 a f\right )+b^3 \left (3 d \sqrt{b^2-4 a c}+a f\right )+3 b^4 d\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right )}{8 \sqrt{2} a^2 \left (b^2-4 a c\right )^{5/2} \sqrt{b-\sqrt{b^2-4 a c}}}-\frac{6 c^2 e \tanh ^{-1}\left (\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{5/2}}+\frac{x \left (c x^2 (b d-2 a f)-a b f-2 a c d+b^2 d\right )}{4 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}+\frac{3 c e \left (b+2 c x^2\right )}{2 \left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )}-\frac{e \left (b+2 c x^2\right )}{4 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2} \]
Antiderivative was successfully verified.
[In] Int[(d + e*x + f*x^2)/(a + b*x^2 + c*x^4)^3,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((f*x**2+e*x+d)/(c*x**4+b*x**2+a)**3,x)
[Out]
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Mathematica [A] time = 6.6145, size = 683, normalized size = 1.1 \[ \frac{12 a^2 b c e+8 a^2 b c f x+28 a^2 c^2 d x+24 a^2 c^2 e x^2+20 a^2 c^2 f x^3+a b^3 f x-25 a b^2 c d x+a b^2 c f x^3-24 a b c^2 d x^3+3 b^4 d x+3 b^3 c d x^3}{8 a^2 \left (4 a c-b^2\right )^2 \left (a+b x^2+c x^4\right )}+\frac{\sqrt{c} \left (20 a^2 c f \sqrt{b^2-4 a c}-52 a^2 b c f+168 a^2 c^2 d+a b^3 f-30 a b^2 c d-24 a b c d \sqrt{b^2-4 a c}+a b^2 f \sqrt{b^2-4 a c}+3 b^3 d \sqrt{b^2-4 a c}+3 b^4 d\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right )}{8 \sqrt{2} a^2 \left (b^2-4 a c\right )^{5/2} \sqrt{b-\sqrt{b^2-4 a c}}}+\frac{\sqrt{c} \left (20 a^2 c f \sqrt{b^2-4 a c}+52 a^2 b c f-168 a^2 c^2 d-a b^3 f+30 a b^2 c d-24 a b c d \sqrt{b^2-4 a c}+a b^2 f \sqrt{b^2-4 a c}+3 b^3 d \sqrt{b^2-4 a c}-3 b^4 d\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right )}{8 \sqrt{2} a^2 \left (b^2-4 a c\right )^{5/2} \sqrt{\sqrt{b^2-4 a c}+b}}+\frac{3 c^2 e \log \left (\sqrt{b^2-4 a c}-b-2 c x^2\right )}{\left (b^2-4 a c\right )^{5/2}}-\frac{3 c^2 e \log \left (\sqrt{b^2-4 a c}+b+2 c x^2\right )}{\left (b^2-4 a c\right )^{5/2}}+\frac{a b e+a b f x+2 a c d x+2 a c e x^2+2 a c f x^3-b^2 d x-b c d x^3}{4 a \left (4 a c-b^2\right ) \left (a+b x^2+c x^4\right )^2} \]
Antiderivative was successfully verified.
[In] Integrate[(d + e*x + f*x^2)/(a + b*x^2 + c*x^4)^3,x]
[Out]
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Maple [B] time = 0.398, size = 10809, normalized size = 17.4 \[ \text{output too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((f*x^2+e*x+d)/(c*x^4+b*x^2+a)^3,x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \frac{24 \, a^{2} c^{3} e x^{6} + 36 \, a^{2} b c^{2} e x^{4} +{\left (3 \,{\left (b^{3} c^{2} - 8 \, a b c^{3}\right )} d +{\left (a b^{2} c^{2} + 20 \, a^{2} c^{3}\right )} f\right )} x^{7} +{\left ({\left (6 \, b^{4} c - 49 \, a b^{2} c^{2} + 28 \, a^{2} c^{3}\right )} d + 2 \,{\left (a b^{3} c + 14 \, a^{2} b c^{2}\right )} f\right )} x^{5} + 8 \,{\left (a^{2} b^{2} c + 5 \, a^{3} c^{2}\right )} e x^{2} +{\left ({\left (3 \, b^{5} - 20 \, a b^{3} c - 4 \, a^{2} b c^{2}\right )} d +{\left (a b^{4} + 5 \, a^{2} b^{2} c + 36 \, a^{3} c^{2}\right )} f\right )} x^{3} - 2 \,{\left (a^{2} b^{3} - 10 \, a^{3} b c\right )} e +{\left ({\left (5 \, a b^{4} - 37 \, a^{2} b^{2} c + 44 \, a^{3} c^{2}\right )} d -{\left (a^{2} b^{3} - 16 \, a^{3} b c\right )} f\right )} x}{8 \,{\left ({\left (a^{2} b^{4} c^{2} - 8 \, a^{3} b^{2} c^{3} + 16 \, a^{4} c^{4}\right )} x^{8} + a^{4} b^{4} - 8 \, a^{5} b^{2} c + 16 \, a^{6} c^{2} + 2 \,{\left (a^{2} b^{5} c - 8 \, a^{3} b^{3} c^{2} + 16 \, a^{4} b c^{3}\right )} x^{6} +{\left (a^{2} b^{6} - 6 \, a^{3} b^{4} c + 32 \, a^{5} c^{3}\right )} x^{4} + 2 \,{\left (a^{3} b^{5} - 8 \, a^{4} b^{3} c + 16 \, a^{5} b c^{2}\right )} x^{2}\right )}} + \frac{\int \frac{48 \, a^{2} c^{2} e x +{\left (3 \,{\left (b^{3} c - 8 \, a b c^{2}\right )} d +{\left (a b^{2} c + 20 \, a^{2} c^{2}\right )} f\right )} x^{2} + 3 \,{\left (b^{4} - 9 \, a b^{2} c + 28 \, a^{2} c^{2}\right )} d +{\left (a b^{3} - 16 \, a^{2} b c\right )} f}{c x^{4} + b x^{2} + a}\,{d x}}{8 \,{\left (a^{2} b^{4} - 8 \, a^{3} b^{2} c + 16 \, a^{4} c^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((f*x^2 + e*x + d)/(c*x^4 + b*x^2 + a)^3,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((f*x^2 + e*x + d)/(c*x^4 + b*x^2 + a)^3,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((f*x**2+e*x+d)/(c*x**4+b*x**2+a)**3,x)
[Out]
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GIAC/XCAS [A] time = 50.9725, size = 1, normalized size = 0. \[ \mathit{Done} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((f*x^2 + e*x + d)/(c*x^4 + b*x^2 + a)^3,x, algorithm="giac")
[Out]