Optimal. Leaf size=215 \[ -\frac{b^{5/4} \log \left (-\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{2 \sqrt{2} c^{9/4}}+\frac{b^{5/4} \log \left (\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{2 \sqrt{2} c^{9/4}}-\frac{b^{5/4} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )}{\sqrt{2} c^{9/4}}+\frac{b^{5/4} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}+1\right )}{\sqrt{2} c^{9/4}}-\frac{2 b \sqrt{x}}{c^2}+\frac{2 x^{5/2}}{5 c} \]
[Out]
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Rubi [A] time = 0.40537, antiderivative size = 215, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 9, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.474 \[ -\frac{b^{5/4} \log \left (-\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{2 \sqrt{2} c^{9/4}}+\frac{b^{5/4} \log \left (\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{2 \sqrt{2} c^{9/4}}-\frac{b^{5/4} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )}{\sqrt{2} c^{9/4}}+\frac{b^{5/4} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}+1\right )}{\sqrt{2} c^{9/4}}-\frac{2 b \sqrt{x}}{c^2}+\frac{2 x^{5/2}}{5 c} \]
Antiderivative was successfully verified.
[In] Int[x^(11/2)/(b*x^2 + c*x^4),x]
[Out]
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Rubi in Sympy [A] time = 72.0698, size = 204, normalized size = 0.95 \[ - \frac{\sqrt{2} b^{\frac{5}{4}} \log{\left (- \sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x} + \sqrt{b} + \sqrt{c} x \right )}}{4 c^{\frac{9}{4}}} + \frac{\sqrt{2} b^{\frac{5}{4}} \log{\left (\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x} + \sqrt{b} + \sqrt{c} x \right )}}{4 c^{\frac{9}{4}}} - \frac{\sqrt{2} b^{\frac{5}{4}} \operatorname{atan}{\left (1 - \frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}} \right )}}{2 c^{\frac{9}{4}}} + \frac{\sqrt{2} b^{\frac{5}{4}} \operatorname{atan}{\left (1 + \frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}} \right )}}{2 c^{\frac{9}{4}}} - \frac{2 b \sqrt{x}}{c^{2}} + \frac{2 x^{\frac{5}{2}}}{5 c} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**(11/2)/(c*x**4+b*x**2),x)
[Out]
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Mathematica [A] time = 0.0731065, size = 203, normalized size = 0.94 \[ \frac{-5 \sqrt{2} b^{5/4} \log \left (-\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )+5 \sqrt{2} b^{5/4} \log \left (\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )-10 \sqrt{2} b^{5/4} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )+10 \sqrt{2} b^{5/4} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}+1\right )-40 b \sqrt [4]{c} \sqrt{x}+8 c^{5/4} x^{5/2}}{20 c^{9/4}} \]
Antiderivative was successfully verified.
[In] Integrate[x^(11/2)/(b*x^2 + c*x^4),x]
[Out]
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Maple [A] time = 0.01, size = 152, normalized size = 0.7 \[{\frac{2}{5\,c}{x}^{{\frac{5}{2}}}}-2\,{\frac{b\sqrt{x}}{{c}^{2}}}+{\frac{b\sqrt{2}}{4\,{c}^{2}}\sqrt [4]{{\frac{b}{c}}}\ln \left ({1 \left ( x+\sqrt [4]{{\frac{b}{c}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{b}{c}}} \right ) \left ( x-\sqrt [4]{{\frac{b}{c}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{b}{c}}} \right ) ^{-1}} \right ) }+{\frac{b\sqrt{2}}{2\,{c}^{2}}\sqrt [4]{{\frac{b}{c}}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}}+1 \right ) }+{\frac{b\sqrt{2}}{2\,{c}^{2}}\sqrt [4]{{\frac{b}{c}}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}}-1 \right ) } \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^(11/2)/(c*x^4+b*x^2),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^(11/2)/(c*x^4 + b*x^2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.280556, size = 207, normalized size = 0.96 \[ -\frac{20 \, c^{2} \left (-\frac{b^{5}}{c^{9}}\right )^{\frac{1}{4}} \arctan \left (\frac{c^{2} \left (-\frac{b^{5}}{c^{9}}\right )^{\frac{1}{4}}}{b \sqrt{x} + \sqrt{c^{4} \sqrt{-\frac{b^{5}}{c^{9}}} + b^{2} x}}\right ) - 5 \, c^{2} \left (-\frac{b^{5}}{c^{9}}\right )^{\frac{1}{4}} \log \left (c^{2} \left (-\frac{b^{5}}{c^{9}}\right )^{\frac{1}{4}} + b \sqrt{x}\right ) + 5 \, c^{2} \left (-\frac{b^{5}}{c^{9}}\right )^{\frac{1}{4}} \log \left (-c^{2} \left (-\frac{b^{5}}{c^{9}}\right )^{\frac{1}{4}} + b \sqrt{x}\right ) - 4 \,{\left (c x^{2} - 5 \, b\right )} \sqrt{x}}{10 \, c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^(11/2)/(c*x^4 + b*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(x**(11/2)/(c*x**4+b*x**2),x)
[Out]
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GIAC/XCAS [A] time = 0.276417, size = 265, normalized size = 1.23 \[ \frac{\sqrt{2} \left (b c^{3}\right )^{\frac{1}{4}} b \arctan \left (\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{b}{c}\right )^{\frac{1}{4}} + 2 \, \sqrt{x}\right )}}{2 \, \left (\frac{b}{c}\right )^{\frac{1}{4}}}\right )}{2 \, c^{3}} + \frac{\sqrt{2} \left (b c^{3}\right )^{\frac{1}{4}} b \arctan \left (-\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{b}{c}\right )^{\frac{1}{4}} - 2 \, \sqrt{x}\right )}}{2 \, \left (\frac{b}{c}\right )^{\frac{1}{4}}}\right )}{2 \, c^{3}} + \frac{\sqrt{2} \left (b c^{3}\right )^{\frac{1}{4}} b{\rm ln}\left (\sqrt{2} \sqrt{x} \left (\frac{b}{c}\right )^{\frac{1}{4}} + x + \sqrt{\frac{b}{c}}\right )}{4 \, c^{3}} - \frac{\sqrt{2} \left (b c^{3}\right )^{\frac{1}{4}} b{\rm ln}\left (-\sqrt{2} \sqrt{x} \left (\frac{b}{c}\right )^{\frac{1}{4}} + x + \sqrt{\frac{b}{c}}\right )}{4 \, c^{3}} + \frac{2 \,{\left (c^{4} x^{\frac{5}{2}} - 5 \, b c^{3} \sqrt{x}\right )}}{5 \, c^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^(11/2)/(c*x^4 + b*x^2),x, algorithm="giac")
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