Optimal. Leaf size=218 \[ \frac{\log \left (-\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{8 \sqrt{2} b^{5/4} c^{3/4}}-\frac{\log \left (\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{8 \sqrt{2} b^{5/4} c^{3/4}}-\frac{\tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )}{4 \sqrt{2} b^{5/4} c^{3/4}}+\frac{\tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}+1\right )}{4 \sqrt{2} b^{5/4} c^{3/4}}+\frac{x^{3/2}}{2 b \left (b+c x^2\right )} \]
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Rubi [A] time = 0.341862, antiderivative size = 218, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 9, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.474 \[ \frac{\log \left (-\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{8 \sqrt{2} b^{5/4} c^{3/4}}-\frac{\log \left (\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{8 \sqrt{2} b^{5/4} c^{3/4}}-\frac{\tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )}{4 \sqrt{2} b^{5/4} c^{3/4}}+\frac{\tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}+1\right )}{4 \sqrt{2} b^{5/4} c^{3/4}}+\frac{x^{3/2}}{2 b \left (b+c x^2\right )} \]
Antiderivative was successfully verified.
[In] Int[x^(9/2)/(b*x^2 + c*x^4)^2,x]
[Out]
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Rubi in Sympy [A] time = 65.8037, size = 197, normalized size = 0.9 \[ \frac{x^{\frac{3}{2}}}{2 b \left (b + c x^{2}\right )} + \frac{\sqrt{2} \log{\left (- \sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x} + \sqrt{b} + \sqrt{c} x \right )}}{16 b^{\frac{5}{4}} c^{\frac{3}{4}}} - \frac{\sqrt{2} \log{\left (\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x} + \sqrt{b} + \sqrt{c} x \right )}}{16 b^{\frac{5}{4}} c^{\frac{3}{4}}} - \frac{\sqrt{2} \operatorname{atan}{\left (1 - \frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}} \right )}}{8 b^{\frac{5}{4}} c^{\frac{3}{4}}} + \frac{\sqrt{2} \operatorname{atan}{\left (1 + \frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}} \right )}}{8 b^{\frac{5}{4}} c^{\frac{3}{4}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**(9/2)/(c*x**4+b*x**2)**2,x)
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Mathematica [A] time = 0.264392, size = 198, normalized size = 0.91 \[ \frac{\frac{\sqrt{2} \log \left (-\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{c^{3/4}}-\frac{\sqrt{2} \log \left (\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{c^{3/4}}-\frac{2 \sqrt{2} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )}{c^{3/4}}+\frac{2 \sqrt{2} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}+1\right )}{c^{3/4}}+\frac{8 \sqrt [4]{b} x^{3/2}}{b+c x^2}}{16 b^{5/4}} \]
Antiderivative was successfully verified.
[In] Integrate[x^(9/2)/(b*x^2 + c*x^4)^2,x]
[Out]
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Maple [A] time = 0.012, size = 158, normalized size = 0.7 \[{\frac{1}{2\,b \left ( c{x}^{2}+b \right ) }{x}^{{\frac{3}{2}}}}+{\frac{\sqrt{2}}{16\,bc}\ln \left ({1 \left ( x-\sqrt [4]{{\frac{b}{c}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{b}{c}}} \right ) \left ( x+\sqrt [4]{{\frac{b}{c}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{b}{c}}} \right ) ^{-1}} \right ){\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}}+{\frac{\sqrt{2}}{8\,bc}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}}+1 \right ){\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}}+{\frac{\sqrt{2}}{8\,bc}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}}-1 \right ){\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^(9/2)/(c*x^4+b*x^2)^2,x)
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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^(9/2)/(c*x^4 + b*x^2)^2,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.284939, size = 235, normalized size = 1.08 \[ \frac{4 \,{\left (b c x^{2} + b^{2}\right )} \left (-\frac{1}{b^{5} c^{3}}\right )^{\frac{1}{4}} \arctan \left (\frac{b^{4} c^{2} \left (-\frac{1}{b^{5} c^{3}}\right )^{\frac{3}{4}}}{\sqrt{-b^{3} c \sqrt{-\frac{1}{b^{5} c^{3}}} + x} + \sqrt{x}}\right ) +{\left (b c x^{2} + b^{2}\right )} \left (-\frac{1}{b^{5} c^{3}}\right )^{\frac{1}{4}} \log \left (b^{4} c^{2} \left (-\frac{1}{b^{5} c^{3}}\right )^{\frac{3}{4}} + \sqrt{x}\right ) -{\left (b c x^{2} + b^{2}\right )} \left (-\frac{1}{b^{5} c^{3}}\right )^{\frac{1}{4}} \log \left (-b^{4} c^{2} \left (-\frac{1}{b^{5} c^{3}}\right )^{\frac{3}{4}} + \sqrt{x}\right ) + 4 \, x^{\frac{3}{2}}}{8 \,{\left (b c x^{2} + b^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^(9/2)/(c*x^4 + b*x^2)^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(x**(9/2)/(c*x**4+b*x**2)**2,x)
[Out]
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GIAC/XCAS [A] time = 0.280132, size = 269, normalized size = 1.23 \[ \frac{x^{\frac{3}{2}}}{2 \,{\left (c x^{2} + b\right )} b} + \frac{\sqrt{2} \left (b c^{3}\right )^{\frac{3}{4}} \arctan \left (\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{b}{c}\right )^{\frac{1}{4}} + 2 \, \sqrt{x}\right )}}{2 \, \left (\frac{b}{c}\right )^{\frac{1}{4}}}\right )}{8 \, b^{2} c^{3}} + \frac{\sqrt{2} \left (b c^{3}\right )^{\frac{3}{4}} \arctan \left (-\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{b}{c}\right )^{\frac{1}{4}} - 2 \, \sqrt{x}\right )}}{2 \, \left (\frac{b}{c}\right )^{\frac{1}{4}}}\right )}{8 \, b^{2} c^{3}} - \frac{\sqrt{2} \left (b c^{3}\right )^{\frac{3}{4}}{\rm ln}\left (\sqrt{2} \sqrt{x} \left (\frac{b}{c}\right )^{\frac{1}{4}} + x + \sqrt{\frac{b}{c}}\right )}{16 \, b^{2} c^{3}} + \frac{\sqrt{2} \left (b c^{3}\right )^{\frac{3}{4}}{\rm ln}\left (-\sqrt{2} \sqrt{x} \left (\frac{b}{c}\right )^{\frac{1}{4}} + x + \sqrt{\frac{b}{c}}\right )}{16 \, b^{2} c^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^(9/2)/(c*x^4 + b*x^2)^2,x, algorithm="giac")
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