Optimal. Leaf size=217 \[ -\frac{c^{7/4} \log \left (-\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{2 \sqrt{2} b^{11/4}}+\frac{c^{7/4} \log \left (\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{2 \sqrt{2} b^{11/4}}-\frac{c^{7/4} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )}{\sqrt{2} b^{11/4}}+\frac{c^{7/4} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}+1\right )}{\sqrt{2} b^{11/4}}+\frac{2 c}{3 b^2 x^{3/2}}-\frac{2}{7 b x^{7/2}} \]
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Rubi [A] time = 0.387007, antiderivative size = 217, 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 \[ -\frac{c^{7/4} \log \left (-\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{2 \sqrt{2} b^{11/4}}+\frac{c^{7/4} \log \left (\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{2 \sqrt{2} b^{11/4}}-\frac{c^{7/4} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )}{\sqrt{2} b^{11/4}}+\frac{c^{7/4} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}+1\right )}{\sqrt{2} b^{11/4}}+\frac{2 c}{3 b^2 x^{3/2}}-\frac{2}{7 b x^{7/2}} \]
Antiderivative was successfully verified.
[In] Int[1/(x^(5/2)*(b*x^2 + c*x^4)),x]
[Out]
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Rubi in Sympy [A] time = 71.9281, size = 206, normalized size = 0.95 \[ - \frac{2}{7 b x^{\frac{7}{2}}} + \frac{2 c}{3 b^{2} x^{\frac{3}{2}}} - \frac{\sqrt{2} c^{\frac{7}{4}} \log{\left (- \sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x} + \sqrt{b} + \sqrt{c} x \right )}}{4 b^{\frac{11}{4}}} + \frac{\sqrt{2} c^{\frac{7}{4}} \log{\left (\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x} + \sqrt{b} + \sqrt{c} x \right )}}{4 b^{\frac{11}{4}}} - \frac{\sqrt{2} c^{\frac{7}{4}} \operatorname{atan}{\left (1 - \frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}} \right )}}{2 b^{\frac{11}{4}}} + \frac{\sqrt{2} c^{\frac{7}{4}} \operatorname{atan}{\left (1 + \frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}} \right )}}{2 b^{\frac{11}{4}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/x**(5/2)/(c*x**4+b*x**2),x)
[Out]
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Mathematica [A] time = 0.0965798, size = 221, normalized size = 1.02 \[ \frac{56 b^{3/4} c x^2-24 b^{7/4}-21 \sqrt{2} c^{7/4} x^{7/2} \log \left (-\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )+21 \sqrt{2} c^{7/4} x^{7/2} \log \left (\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )-42 \sqrt{2} c^{7/4} x^{7/2} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )+42 \sqrt{2} c^{7/4} x^{7/2} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}+1\right )}{84 b^{11/4} x^{7/2}} \]
Antiderivative was successfully verified.
[In] Integrate[1/(x^(5/2)*(b*x^2 + c*x^4)),x]
[Out]
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Maple [A] time = 0.014, size = 158, normalized size = 0.7 \[{\frac{{c}^{2}\sqrt{2}}{4\,{b}^{3}}\sqrt [4]{{\frac{b}{c}}}\ln \left ({1 \left ( x+\sqrt [4]{{\frac{b}{c}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{b}{c}}} \right ) \left ( x-\sqrt [4]{{\frac{b}{c}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{b}{c}}} \right ) ^{-1}} \right ) }+{\frac{{c}^{2}\sqrt{2}}{2\,{b}^{3}}\sqrt [4]{{\frac{b}{c}}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}}+1 \right ) }+{\frac{{c}^{2}\sqrt{2}}{2\,{b}^{3}}\sqrt [4]{{\frac{b}{c}}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}}-1 \right ) }-{\frac{2}{7\,b}{x}^{-{\frac{7}{2}}}}+{\frac{2\,c}{3\,{b}^{2}}{x}^{-{\frac{3}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/x^(5/2)/(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(1/((c*x^4 + b*x^2)*x^(5/2)),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.293846, size = 224, normalized size = 1.03 \[ -\frac{84 \, b^{2} x^{\frac{7}{2}} \left (-\frac{c^{7}}{b^{11}}\right )^{\frac{1}{4}} \arctan \left (\frac{b^{3} \left (-\frac{c^{7}}{b^{11}}\right )^{\frac{1}{4}}}{c^{2} \sqrt{x} + \sqrt{b^{6} \sqrt{-\frac{c^{7}}{b^{11}}} + c^{4} x}}\right ) - 21 \, b^{2} x^{\frac{7}{2}} \left (-\frac{c^{7}}{b^{11}}\right )^{\frac{1}{4}} \log \left (b^{3} \left (-\frac{c^{7}}{b^{11}}\right )^{\frac{1}{4}} + c^{2} \sqrt{x}\right ) + 21 \, b^{2} x^{\frac{7}{2}} \left (-\frac{c^{7}}{b^{11}}\right )^{\frac{1}{4}} \log \left (-b^{3} \left (-\frac{c^{7}}{b^{11}}\right )^{\frac{1}{4}} + c^{2} \sqrt{x}\right ) - 28 \, c x^{2} + 12 \, b}{42 \, b^{2} x^{\frac{7}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((c*x^4 + b*x^2)*x^(5/2)),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(1/x**(5/2)/(c*x**4+b*x**2),x)
[Out]
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GIAC/XCAS [A] time = 0.274811, size = 259, normalized size = 1.19 \[ \frac{\sqrt{2} \left (b c^{3}\right )^{\frac{1}{4}} c \arctan \left (\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{b}{c}\right )^{\frac{1}{4}} + 2 \, \sqrt{x}\right )}}{2 \, \left (\frac{b}{c}\right )^{\frac{1}{4}}}\right )}{2 \, b^{3}} + \frac{\sqrt{2} \left (b c^{3}\right )^{\frac{1}{4}} c \arctan \left (-\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{b}{c}\right )^{\frac{1}{4}} - 2 \, \sqrt{x}\right )}}{2 \, \left (\frac{b}{c}\right )^{\frac{1}{4}}}\right )}{2 \, b^{3}} + \frac{\sqrt{2} \left (b c^{3}\right )^{\frac{1}{4}} c{\rm ln}\left (\sqrt{2} \sqrt{x} \left (\frac{b}{c}\right )^{\frac{1}{4}} + x + \sqrt{\frac{b}{c}}\right )}{4 \, b^{3}} - \frac{\sqrt{2} \left (b c^{3}\right )^{\frac{1}{4}} c{\rm ln}\left (-\sqrt{2} \sqrt{x} \left (\frac{b}{c}\right )^{\frac{1}{4}} + x + \sqrt{\frac{b}{c}}\right )}{4 \, b^{3}} + \frac{2 \,{\left (7 \, c x^{2} - 3 \, b\right )}}{21 \, b^{2} x^{\frac{7}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((c*x^4 + b*x^2)*x^(5/2)),x, algorithm="giac")
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