Optimal. Leaf size=221 \[ -\frac{x \left (b+2 c x^2\right )}{2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}+\frac{\sqrt{c} \left (2 b-\sqrt{b^2-4 a c}\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right )}{\sqrt{2} \left (b^2-4 a c\right )^{3/2} \sqrt{b-\sqrt{b^2-4 a c}}}-\frac{\sqrt{c} \left (\sqrt{b^2-4 a c}+2 b\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right )}{\sqrt{2} \left (b^2-4 a c\right )^{3/2} \sqrt{\sqrt{b^2-4 a c}+b}} \]
[Out]
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Rubi [A] time = 0.52079, antiderivative size = 221, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ -\frac{x \left (b+2 c x^2\right )}{2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}+\frac{\sqrt{c} \left (2 b-\sqrt{b^2-4 a c}\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right )}{\sqrt{2} \left (b^2-4 a c\right )^{3/2} \sqrt{b-\sqrt{b^2-4 a c}}}-\frac{\sqrt{c} \left (\sqrt{b^2-4 a c}+2 b\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right )}{\sqrt{2} \left (b^2-4 a c\right )^{3/2} \sqrt{\sqrt{b^2-4 a c}+b}} \]
Antiderivative was successfully verified.
[In] Int[x^4/(a*x + b*x^3 + c*x^5)^2,x]
[Out]
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Rubi in Sympy [A] time = 47.8794, size = 201, normalized size = 0.91 \[ - \frac{\sqrt{2} \sqrt{c} \left (b + \frac{\sqrt{- 4 a c + b^{2}}}{2}\right ) \operatorname{atan}{\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b + \sqrt{- 4 a c + b^{2}}}} \right )}}{\sqrt{b + \sqrt{- 4 a c + b^{2}}} \left (- 4 a c + b^{2}\right )^{\frac{3}{2}}} + \frac{\sqrt{2} \sqrt{c} \left (b - \frac{\sqrt{- 4 a c + b^{2}}}{2}\right ) \operatorname{atan}{\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b - \sqrt{- 4 a c + b^{2}}}} \right )}}{\sqrt{b - \sqrt{- 4 a c + b^{2}}} \left (- 4 a c + b^{2}\right )^{\frac{3}{2}}} - \frac{x \left (b + 2 c x^{2}\right )}{2 \left (- 4 a c + b^{2}\right ) \left (a + b x^{2} + c x^{4}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**4/(c*x**5+b*x**3+a*x)**2,x)
[Out]
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Mathematica [A] time = 0.80278, size = 222, normalized size = 1. \[ \frac{-b x-2 c x^3}{2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}-\frac{\sqrt{c} \left (\sqrt{b^2-4 a c}-2 b\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right )}{\sqrt{2} \left (b^2-4 a c\right )^{3/2} \sqrt{b-\sqrt{b^2-4 a c}}}-\frac{\sqrt{c} \left (\sqrt{b^2-4 a c}+2 b\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right )}{\sqrt{2} \left (b^2-4 a c\right )^{3/2} \sqrt{\sqrt{b^2-4 a c}+b}} \]
Antiderivative was successfully verified.
[In] Integrate[x^4/(a*x + b*x^3 + c*x^5)^2,x]
[Out]
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Maple [A] time = 0.084, size = 342, normalized size = 1.6 \[{\frac{x}{8\,ac-2\,{b}^{2}} \left ({x}^{2}+{\frac{1}{2\,c}\sqrt{-4\,ac+{b}^{2}}}+{\frac{b}{2\,c}} \right ) ^{-1}}+{\frac{c\sqrt{2}b}{4\,ac-{b}^{2}}\arctan \left ({cx\sqrt{2}{\frac{1}{\sqrt{ \left ( b+\sqrt{-4\,ac+{b}^{2}} \right ) c}}}} \right ){\frac{1}{\sqrt{-4\,ac+{b}^{2}}}}{\frac{1}{\sqrt{ \left ( b+\sqrt{-4\,ac+{b}^{2}} \right ) c}}}}+{\frac{c\sqrt{2}}{8\,ac-2\,{b}^{2}}\arctan \left ({cx\sqrt{2}{\frac{1}{\sqrt{ \left ( b+\sqrt{-4\,ac+{b}^{2}} \right ) c}}}} \right ){\frac{1}{\sqrt{ \left ( b+\sqrt{-4\,ac+{b}^{2}} \right ) c}}}}+{\frac{x}{8\,ac-2\,{b}^{2}} \left ({x}^{2}+{\frac{b}{2\,c}}-{\frac{1}{2\,c}\sqrt{-4\,ac+{b}^{2}}} \right ) ^{-1}}+{\frac{c\sqrt{2}b}{4\,ac-{b}^{2}}{\it Artanh} \left ({cx\sqrt{2}{\frac{1}{\sqrt{ \left ( -b+\sqrt{-4\,ac+{b}^{2}} \right ) c}}}} \right ){\frac{1}{\sqrt{-4\,ac+{b}^{2}}}}{\frac{1}{\sqrt{ \left ( -b+\sqrt{-4\,ac+{b}^{2}} \right ) c}}}}-{\frac{c\sqrt{2}}{8\,ac-2\,{b}^{2}}{\it Artanh} \left ({cx\sqrt{2}{\frac{1}{\sqrt{ \left ( -b+\sqrt{-4\,ac+{b}^{2}} \right ) c}}}} \right ){\frac{1}{\sqrt{ \left ( -b+\sqrt{-4\,ac+{b}^{2}} \right ) c}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^4/(c*x^5+b*x^3+a*x)^2,x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ -\frac{2 \, c x^{3} + b x}{2 \,{\left ({\left (b^{2} c - 4 \, a c^{2}\right )} x^{4} + a b^{2} - 4 \, a^{2} c +{\left (b^{3} - 4 \, a b c\right )} x^{2}\right )}} - \frac{\int \frac{2 \, c x^{2} - b}{c x^{4} + b x^{2} + a}\,{d x}}{2 \,{\left (b^{2} - 4 \, a c\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^4/(c*x^5 + b*x^3 + a*x)^2,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.322846, size = 2268, normalized size = 10.26 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^4/(c*x^5 + b*x^3 + a*x)^2,x, algorithm="fricas")
[Out]
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Sympy [A] time = 12.4669, size = 298, normalized size = 1.35 \[ \frac{b x + 2 c x^{3}}{8 a^{2} c - 2 a b^{2} + x^{4} \left (8 a c^{2} - 2 b^{2} c\right ) + x^{2} \left (8 a b c - 2 b^{3}\right )} + \operatorname{RootSum}{\left (t^{4} \left (1048576 a^{7} c^{6} - 1572864 a^{6} b^{2} c^{5} + 983040 a^{5} b^{4} c^{4} - 327680 a^{4} b^{6} c^{3} + 61440 a^{3} b^{8} c^{2} - 6144 a^{2} b^{10} c + 256 a b^{12}\right ) + t^{2} \left (- 12288 a^{4} b c^{4} + 8192 a^{3} b^{3} c^{3} - 1536 a^{2} b^{5} c^{2} + 16 b^{9}\right ) + 16 a^{2} c^{3} + 24 a b^{2} c^{2} + 9 b^{4} c, \left ( t \mapsto t \log{\left (x + \frac{16384 t^{3} a^{5} c^{4} - 8192 t^{3} a^{4} b^{2} c^{3} + 512 t^{3} a^{2} b^{6} c - 64 t^{3} a b^{8} - 128 t a^{2} b c^{2} - 16 t a b^{3} c - 4 t b^{5}}{4 a c^{2} + 3 b^{2} c} \right )} \right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**4/(c*x**5+b*x**3+a*x)**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(x^4/(c*x^5 + b*x^3 + a*x)^2,x, algorithm="giac")
[Out]