Optimal. Leaf size=202 \[ -\frac{\sqrt [4]{c} \log \left (-\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{2 \sqrt{2} b^{5/4}}+\frac{\sqrt [4]{c} \log \left (\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{2 \sqrt{2} b^{5/4}}+\frac{\sqrt [4]{c} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )}{\sqrt{2} b^{5/4}}-\frac{\sqrt [4]{c} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}+1\right )}{\sqrt{2} b^{5/4}}-\frac{2}{b \sqrt{x}} \]
[Out]
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Rubi [A] time = 0.338436, antiderivative size = 202, 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 \[ -\frac{\sqrt [4]{c} \log \left (-\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{2 \sqrt{2} b^{5/4}}+\frac{\sqrt [4]{c} \log \left (\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{2 \sqrt{2} b^{5/4}}+\frac{\sqrt [4]{c} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )}{\sqrt{2} b^{5/4}}-\frac{\sqrt [4]{c} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}+1\right )}{\sqrt{2} b^{5/4}}-\frac{2}{b \sqrt{x}} \]
Antiderivative was successfully verified.
[In] Int[Sqrt[x]/(b*x^2 + c*x^4),x]
[Out]
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Rubi in Sympy [A] time = 65.8307, size = 190, normalized size = 0.94 \[ - \frac{2}{b \sqrt{x}} - \frac{\sqrt{2} \sqrt [4]{c} \log{\left (- \sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x} + \sqrt{b} + \sqrt{c} x \right )}}{4 b^{\frac{5}{4}}} + \frac{\sqrt{2} \sqrt [4]{c} \log{\left (\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x} + \sqrt{b} + \sqrt{c} x \right )}}{4 b^{\frac{5}{4}}} + \frac{\sqrt{2} \sqrt [4]{c} \operatorname{atan}{\left (1 - \frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}} \right )}}{2 b^{\frac{5}{4}}} - \frac{\sqrt{2} \sqrt [4]{c} \operatorname{atan}{\left (1 + \frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}} \right )}}{2 b^{\frac{5}{4}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**(1/2)/(c*x**4+b*x**2),x)
[Out]
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Mathematica [A] time = 0.118246, size = 189, normalized size = 0.94 \[ \frac{-\sqrt{2} \sqrt [4]{c} \log \left (-\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )+\sqrt{2} \sqrt [4]{c} \log \left (\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )+2 \sqrt{2} \sqrt [4]{c} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )-2 \sqrt{2} \sqrt [4]{c} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}+1\right )-\frac{8 \sqrt [4]{b}}{\sqrt{x}}}{4 b^{5/4}} \]
Antiderivative was successfully verified.
[In] Integrate[Sqrt[x]/(b*x^2 + c*x^4),x]
[Out]
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Maple [A] time = 0.012, size = 140, normalized size = 0.7 \[ -{\frac{\sqrt{2}}{4\,b}\ln \left ({1 \left ( x-\sqrt [4]{{\frac{b}{c}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{b}{c}}} \right ) \left ( x+\sqrt [4]{{\frac{b}{c}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{b}{c}}} \right ) ^{-1}} \right ){\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}}-{\frac{\sqrt{2}}{2\,b}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}}+1 \right ){\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}}-{\frac{\sqrt{2}}{2\,b}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}}-1 \right ){\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}}-2\,{\frac{1}{b\sqrt{x}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^(1/2)/(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(sqrt(x)/(c*x^4 + b*x^2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.283527, size = 180, normalized size = 0.89 \[ -\frac{4 \, b \sqrt{x} \left (-\frac{c}{b^{5}}\right )^{\frac{1}{4}} \arctan \left (\frac{b^{4} \left (-\frac{c}{b^{5}}\right )^{\frac{3}{4}}}{c \sqrt{x} + \sqrt{-b^{3} c \sqrt{-\frac{c}{b^{5}}} + c^{2} x}}\right ) + b \sqrt{x} \left (-\frac{c}{b^{5}}\right )^{\frac{1}{4}} \log \left (b^{4} \left (-\frac{c}{b^{5}}\right )^{\frac{3}{4}} + c \sqrt{x}\right ) - b \sqrt{x} \left (-\frac{c}{b^{5}}\right )^{\frac{1}{4}} \log \left (-b^{4} \left (-\frac{c}{b^{5}}\right )^{\frac{3}{4}} + c \sqrt{x}\right ) + 4}{2 \, b \sqrt{x}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(x)/(c*x^4 + b*x^2),x, algorithm="fricas")
[Out]
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Sympy [A] time = 86.7168, size = 175, normalized size = 0.87 \[ \begin{cases} \frac{\tilde{\infty }}{x^{\frac{5}{2}}} & \text{for}\: b = 0 \wedge c = 0 \\- \frac{2}{5 c x^{\frac{5}{2}}} & \text{for}\: b = 0 \\- \frac{2}{b \sqrt{x}} & \text{for}\: c = 0 \\- \frac{2}{b \sqrt{x}} + \frac{\left (-1\right )^{\frac{3}{4}} \log{\left (- \sqrt [4]{-1} \sqrt [4]{b} \sqrt [4]{\frac{1}{c}} + \sqrt{x} \right )}}{2 b^{\frac{5}{4}} c \left (\frac{1}{c}\right )^{\frac{5}{4}}} - \frac{\left (-1\right )^{\frac{3}{4}} \log{\left (\sqrt [4]{-1} \sqrt [4]{b} \sqrt [4]{\frac{1}{c}} + \sqrt{x} \right )}}{2 b^{\frac{5}{4}} c \left (\frac{1}{c}\right )^{\frac{5}{4}}} - \frac{\left (-1\right )^{\frac{3}{4}} \operatorname{atan}{\left (\frac{\left (-1\right )^{\frac{3}{4}} \sqrt{x}}{\sqrt [4]{b} \sqrt [4]{\frac{1}{c}}} \right )}}{b^{\frac{5}{4}} c \left (\frac{1}{c}\right )^{\frac{5}{4}}} & \text{otherwise} \end{cases} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**(1/2)/(c*x**4+b*x**2),x)
[Out]
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GIAC/XCAS [A] time = 0.277707, size = 257, normalized size = 1.27 \[ -\frac{2}{b \sqrt{x}} - \frac{\sqrt{2} \left (b c^{3}\right )^{\frac{3}{4}} \arctan \left (\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{b}{c}\right )^{\frac{1}{4}} + 2 \, \sqrt{x}\right )}}{2 \, \left (\frac{b}{c}\right )^{\frac{1}{4}}}\right )}{2 \, b^{2} c^{2}} - \frac{\sqrt{2} \left (b c^{3}\right )^{\frac{3}{4}} \arctan \left (-\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{b}{c}\right )^{\frac{1}{4}} - 2 \, \sqrt{x}\right )}}{2 \, \left (\frac{b}{c}\right )^{\frac{1}{4}}}\right )}{2 \, b^{2} c^{2}} + \frac{\sqrt{2} \left (b c^{3}\right )^{\frac{3}{4}}{\rm ln}\left (\sqrt{2} \sqrt{x} \left (\frac{b}{c}\right )^{\frac{1}{4}} + x + \sqrt{\frac{b}{c}}\right )}{4 \, b^{2} c^{2}} - \frac{\sqrt{2} \left (b c^{3}\right )^{\frac{3}{4}}{\rm ln}\left (-\sqrt{2} \sqrt{x} \left (\frac{b}{c}\right )^{\frac{1}{4}} + x + \sqrt{\frac{b}{c}}\right )}{4 \, b^{2} c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(x)/(c*x^4 + b*x^2),x, algorithm="giac")
[Out]