Optimal. Leaf size=243 \[ -\frac{9 b^{5/4} \log \left (-\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{8 \sqrt{2} c^{13/4}}+\frac{9 b^{5/4} \log \left (\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{8 \sqrt{2} c^{13/4}}-\frac{9 b^{5/4} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )}{4 \sqrt{2} c^{13/4}}+\frac{9 b^{5/4} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}+1\right )}{4 \sqrt{2} c^{13/4}}-\frac{9 b \sqrt{x}}{2 c^3}-\frac{x^{9/2}}{2 c \left (b+c x^2\right )}+\frac{9 x^{5/2}}{10 c^2} \]
[Out]
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Rubi [A] time = 0.440643, antiderivative size = 243, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 10, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.526 \[ -\frac{9 b^{5/4} \log \left (-\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{8 \sqrt{2} c^{13/4}}+\frac{9 b^{5/4} \log \left (\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{8 \sqrt{2} c^{13/4}}-\frac{9 b^{5/4} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )}{4 \sqrt{2} c^{13/4}}+\frac{9 b^{5/4} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}+1\right )}{4 \sqrt{2} c^{13/4}}-\frac{9 b \sqrt{x}}{2 c^3}-\frac{x^{9/2}}{2 c \left (b+c x^2\right )}+\frac{9 x^{5/2}}{10 c^2} \]
Antiderivative was successfully verified.
[In] Int[x^(19/2)/(b*x^2 + c*x^4)^2,x]
[Out]
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Rubi in Sympy [A] time = 79.4076, size = 230, normalized size = 0.95 \[ - \frac{9 \sqrt{2} b^{\frac{5}{4}} \log{\left (- \sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x} + \sqrt{b} + \sqrt{c} x \right )}}{16 c^{\frac{13}{4}}} + \frac{9 \sqrt{2} b^{\frac{5}{4}} \log{\left (\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x} + \sqrt{b} + \sqrt{c} x \right )}}{16 c^{\frac{13}{4}}} - \frac{9 \sqrt{2} b^{\frac{5}{4}} \operatorname{atan}{\left (1 - \frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}} \right )}}{8 c^{\frac{13}{4}}} + \frac{9 \sqrt{2} b^{\frac{5}{4}} \operatorname{atan}{\left (1 + \frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}} \right )}}{8 c^{\frac{13}{4}}} - \frac{9 b \sqrt{x}}{2 c^{3}} - \frac{x^{\frac{9}{2}}}{2 c \left (b + c x^{2}\right )} + \frac{9 x^{\frac{5}{2}}}{10 c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**(19/2)/(c*x**4+b*x**2)**2,x)
[Out]
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Mathematica [A] time = 0.376528, size = 227, normalized size = 0.93 \[ \frac{-45 \sqrt{2} b^{5/4} \log \left (-\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )+45 \sqrt{2} b^{5/4} \log \left (\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )-90 \sqrt{2} b^{5/4} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )+90 \sqrt{2} b^{5/4} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}+1\right )-\frac{40 b^2 \sqrt [4]{c} \sqrt{x}}{b+c x^2}-320 b \sqrt [4]{c} \sqrt{x}+32 c^{5/4} x^{5/2}}{80 c^{13/4}} \]
Antiderivative was successfully verified.
[In] Integrate[x^(19/2)/(b*x^2 + c*x^4)^2,x]
[Out]
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Maple [A] time = 0.018, size = 172, normalized size = 0.7 \[{\frac{2}{5\,{c}^{2}}{x}^{{\frac{5}{2}}}}-4\,{\frac{b\sqrt{x}}{{c}^{3}}}-{\frac{{b}^{2}}{2\,{c}^{3} \left ( c{x}^{2}+b \right ) }\sqrt{x}}+{\frac{9\,b\sqrt{2}}{16\,{c}^{3}}\sqrt [4]{{\frac{b}{c}}}\ln \left ({1 \left ( x+\sqrt [4]{{\frac{b}{c}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{b}{c}}} \right ) \left ( x-\sqrt [4]{{\frac{b}{c}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{b}{c}}} \right ) ^{-1}} \right ) }+{\frac{9\,b\sqrt{2}}{8\,{c}^{3}}\sqrt [4]{{\frac{b}{c}}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}}+1 \right ) }+{\frac{9\,b\sqrt{2}}{8\,{c}^{3}}\sqrt [4]{{\frac{b}{c}}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}}-1 \right ) } \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^(19/2)/(c*x^4+b*x^2)^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^(19/2)/(c*x^4 + b*x^2)^2,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.285359, size = 284, normalized size = 1.17 \[ -\frac{180 \,{\left (c^{4} x^{2} + b c^{3}\right )} \left (-\frac{b^{5}}{c^{13}}\right )^{\frac{1}{4}} \arctan \left (\frac{c^{3} \left (-\frac{b^{5}}{c^{13}}\right )^{\frac{1}{4}}}{b \sqrt{x} + \sqrt{c^{6} \sqrt{-\frac{b^{5}}{c^{13}}} + b^{2} x}}\right ) - 45 \,{\left (c^{4} x^{2} + b c^{3}\right )} \left (-\frac{b^{5}}{c^{13}}\right )^{\frac{1}{4}} \log \left (9 \, c^{3} \left (-\frac{b^{5}}{c^{13}}\right )^{\frac{1}{4}} + 9 \, b \sqrt{x}\right ) + 45 \,{\left (c^{4} x^{2} + b c^{3}\right )} \left (-\frac{b^{5}}{c^{13}}\right )^{\frac{1}{4}} \log \left (-9 \, c^{3} \left (-\frac{b^{5}}{c^{13}}\right )^{\frac{1}{4}} + 9 \, b \sqrt{x}\right ) - 4 \,{\left (4 \, c^{2} x^{4} - 36 \, b c x^{2} - 45 \, b^{2}\right )} \sqrt{x}}{40 \,{\left (c^{4} x^{2} + b c^{3}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^(19/2)/(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**(19/2)/(c*x**4+b*x**2)**2,x)
[Out]
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GIAC/XCAS [A] time = 0.278981, size = 292, normalized size = 1.2 \[ \frac{9 \, \sqrt{2} \left (b c^{3}\right )^{\frac{1}{4}} b \arctan \left (\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{b}{c}\right )^{\frac{1}{4}} + 2 \, \sqrt{x}\right )}}{2 \, \left (\frac{b}{c}\right )^{\frac{1}{4}}}\right )}{8 \, c^{4}} + \frac{9 \, \sqrt{2} \left (b c^{3}\right )^{\frac{1}{4}} b \arctan \left (-\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{b}{c}\right )^{\frac{1}{4}} - 2 \, \sqrt{x}\right )}}{2 \, \left (\frac{b}{c}\right )^{\frac{1}{4}}}\right )}{8 \, c^{4}} + \frac{9 \, \sqrt{2} \left (b c^{3}\right )^{\frac{1}{4}} b{\rm ln}\left (\sqrt{2} \sqrt{x} \left (\frac{b}{c}\right )^{\frac{1}{4}} + x + \sqrt{\frac{b}{c}}\right )}{16 \, c^{4}} - \frac{9 \, \sqrt{2} \left (b c^{3}\right )^{\frac{1}{4}} b{\rm ln}\left (-\sqrt{2} \sqrt{x} \left (\frac{b}{c}\right )^{\frac{1}{4}} + x + \sqrt{\frac{b}{c}}\right )}{16 \, c^{4}} - \frac{b^{2} \sqrt{x}}{2 \,{\left (c x^{2} + b\right )} c^{3}} + \frac{2 \,{\left (c^{8} x^{\frac{5}{2}} - 10 \, b c^{7} \sqrt{x}\right )}}{5 \, c^{10}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^(19/2)/(c*x^4 + b*x^2)^2,x, algorithm="giac")
[Out]