Optimal. Leaf size=76 \[ -\frac{15 \sqrt{c} \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b}}\right )}{8 b^{7/2}}-\frac{15}{8 b^3 x}+\frac{5}{8 b^2 x \left (b+c x^2\right )}+\frac{1}{4 b x \left (b+c x^2\right )^2} \]
[Out]
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Rubi [A] time = 0.0855407, antiderivative size = 76, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.235 \[ -\frac{15 \sqrt{c} \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b}}\right )}{8 b^{7/2}}-\frac{15}{8 b^3 x}+\frac{5}{8 b^2 x \left (b+c x^2\right )}+\frac{1}{4 b x \left (b+c x^2\right )^2} \]
Antiderivative was successfully verified.
[In] Int[x^4/(b*x^2 + c*x^4)^3,x]
[Out]
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Rubi in Sympy [A] time = 16.8851, size = 65, normalized size = 0.86 \[ \frac{1}{4 b x \left (b + c x^{2}\right )^{2}} + \frac{5}{8 b^{2} x \left (b + c x^{2}\right )} - \frac{15}{8 b^{3} x} - \frac{15 \sqrt{c} \operatorname{atan}{\left (\frac{\sqrt{c} x}{\sqrt{b}} \right )}}{8 b^{\frac{7}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**4/(c*x**4+b*x**2)**3,x)
[Out]
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Mathematica [A] time = 0.074645, size = 68, normalized size = 0.89 \[ -\frac{15 \sqrt{c} \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b}}\right )}{8 b^{7/2}}-\frac{8 b^2+25 b c x^2+15 c^2 x^4}{8 b^3 x \left (b+c x^2\right )^2} \]
Antiderivative was successfully verified.
[In] Integrate[x^4/(b*x^2 + c*x^4)^3,x]
[Out]
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Maple [A] time = 0.016, size = 66, normalized size = 0.9 \[ -{\frac{7\,{c}^{2}{x}^{3}}{8\,{b}^{3} \left ( c{x}^{2}+b \right ) ^{2}}}-{\frac{9\,cx}{8\,{b}^{2} \left ( c{x}^{2}+b \right ) ^{2}}}-{\frac{15\,c}{8\,{b}^{3}}\arctan \left ({cx{\frac{1}{\sqrt{bc}}}} \right ){\frac{1}{\sqrt{bc}}}}-{\frac{1}{{b}^{3}x}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^4/(c*x^4+b*x^2)^3,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(x^4/(c*x^4 + b*x^2)^3,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.264044, size = 1, normalized size = 0.01 \[ \left [-\frac{30 \, c^{2} x^{4} + 50 \, b c x^{2} - 15 \,{\left (c^{2} x^{5} + 2 \, b c x^{3} + b^{2} x\right )} \sqrt{-\frac{c}{b}} \log \left (\frac{c x^{2} - 2 \, b x \sqrt{-\frac{c}{b}} - b}{c x^{2} + b}\right ) + 16 \, b^{2}}{16 \,{\left (b^{3} c^{2} x^{5} + 2 \, b^{4} c x^{3} + b^{5} x\right )}}, -\frac{15 \, c^{2} x^{4} + 25 \, b c x^{2} + 15 \,{\left (c^{2} x^{5} + 2 \, b c x^{3} + b^{2} x\right )} \sqrt{\frac{c}{b}} \arctan \left (\frac{c x}{b \sqrt{\frac{c}{b}}}\right ) + 8 \, b^{2}}{8 \,{\left (b^{3} c^{2} x^{5} + 2 \, b^{4} c x^{3} + b^{5} x\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^4/(c*x^4 + b*x^2)^3,x, algorithm="fricas")
[Out]
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Sympy [A] time = 2.3972, size = 114, normalized size = 1.5 \[ \frac{15 \sqrt{- \frac{c}{b^{7}}} \log{\left (- \frac{b^{4} \sqrt{- \frac{c}{b^{7}}}}{c} + x \right )}}{16} - \frac{15 \sqrt{- \frac{c}{b^{7}}} \log{\left (\frac{b^{4} \sqrt{- \frac{c}{b^{7}}}}{c} + x \right )}}{16} - \frac{8 b^{2} + 25 b c x^{2} + 15 c^{2} x^{4}}{8 b^{5} x + 16 b^{4} c x^{3} + 8 b^{3} c^{2} x^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**4/(c*x**4+b*x**2)**3,x)
[Out]
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GIAC/XCAS [A] time = 0.271848, size = 77, normalized size = 1.01 \[ -\frac{15 \, c \arctan \left (\frac{c x}{\sqrt{b c}}\right )}{8 \, \sqrt{b c} b^{3}} - \frac{7 \, c^{2} x^{3} + 9 \, b c x}{8 \,{\left (c x^{2} + b\right )}^{2} b^{3}} - \frac{1}{b^{3} x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^4/(c*x^4 + b*x^2)^3,x, algorithm="giac")
[Out]