Optimal. Leaf size=287 \[ -\frac{\log \left (-\sqrt{2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt{x}+\sqrt [4]{-a}+\sqrt [4]{c} x\right )}{4 \sqrt{2} (-a)^{3/8} c^{5/8}}+\frac{\log \left (\sqrt{2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt{x}+\sqrt [4]{-a}+\sqrt [4]{c} x\right )}{4 \sqrt{2} (-a)^{3/8} c^{5/8}}-\frac{\tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}\right )}{2 \sqrt{2} (-a)^{3/8} c^{5/8}}+\frac{\tan ^{-1}\left (\frac{\sqrt{2} \sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}+1\right )}{2 \sqrt{2} (-a)^{3/8} c^{5/8}}-\frac{\tan ^{-1}\left (\frac{\sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}\right )}{2 (-a)^{3/8} c^{5/8}}-\frac{\tanh ^{-1}\left (\frac{\sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}\right )}{2 (-a)^{3/8} c^{5/8}} \]
[Out]
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Rubi [A] time = 0.480318, antiderivative size = 287, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 11, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.733 \[ -\frac{\log \left (-\sqrt{2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt{x}+\sqrt [4]{-a}+\sqrt [4]{c} x\right )}{4 \sqrt{2} (-a)^{3/8} c^{5/8}}+\frac{\log \left (\sqrt{2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt{x}+\sqrt [4]{-a}+\sqrt [4]{c} x\right )}{4 \sqrt{2} (-a)^{3/8} c^{5/8}}-\frac{\tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}\right )}{2 \sqrt{2} (-a)^{3/8} c^{5/8}}+\frac{\tan ^{-1}\left (\frac{\sqrt{2} \sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}+1\right )}{2 \sqrt{2} (-a)^{3/8} c^{5/8}}-\frac{\tan ^{-1}\left (\frac{\sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}\right )}{2 (-a)^{3/8} c^{5/8}}-\frac{\tanh ^{-1}\left (\frac{\sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}\right )}{2 (-a)^{3/8} c^{5/8}} \]
Antiderivative was successfully verified.
[In] Int[x^(3/2)/(a + c*x^4),x]
[Out]
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Rubi in Sympy [A] time = 99.5278, size = 264, normalized size = 0.92 \[ - \frac{\sqrt{2} \log{\left (- \sqrt{2} \sqrt [8]{c} \sqrt{x} \sqrt [8]{- a} + \sqrt [4]{c} x + \sqrt [4]{- a} \right )}}{8 c^{\frac{5}{8}} \left (- a\right )^{\frac{3}{8}}} + \frac{\sqrt{2} \log{\left (\sqrt{2} \sqrt [8]{c} \sqrt{x} \sqrt [8]{- a} + \sqrt [4]{c} x + \sqrt [4]{- a} \right )}}{8 c^{\frac{5}{8}} \left (- a\right )^{\frac{3}{8}}} - \frac{\operatorname{atan}{\left (\frac{\sqrt [8]{c} \sqrt{x}}{\sqrt [8]{- a}} \right )}}{2 c^{\frac{5}{8}} \left (- a\right )^{\frac{3}{8}}} + \frac{\sqrt{2} \operatorname{atan}{\left (\frac{\sqrt{2} \sqrt [8]{c} \sqrt{x}}{\sqrt [8]{- a}} - 1 \right )}}{4 c^{\frac{5}{8}} \left (- a\right )^{\frac{3}{8}}} + \frac{\sqrt{2} \operatorname{atan}{\left (\frac{\sqrt{2} \sqrt [8]{c} \sqrt{x}}{\sqrt [8]{- a}} + 1 \right )}}{4 c^{\frac{5}{8}} \left (- a\right )^{\frac{3}{8}}} - \frac{\operatorname{atanh}{\left (\frac{\sqrt [8]{c} \sqrt{x}}{\sqrt [8]{- a}} \right )}}{2 c^{\frac{5}{8}} \left (- a\right )^{\frac{3}{8}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**(3/2)/(c*x**4+a),x)
[Out]
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Mathematica [A] time = 0.410355, size = 348, normalized size = 1.21 \[ -\frac{\cos \left (\frac{\pi }{8}\right ) \log \left (-2 \sqrt [8]{a} \sqrt [8]{c} \sqrt{x} \sin \left (\frac{\pi }{8}\right )+\sqrt [4]{a}+\sqrt [4]{c} x\right )-\cos \left (\frac{\pi }{8}\right ) \log \left (2 \sqrt [8]{a} \sqrt [8]{c} \sqrt{x} \sin \left (\frac{\pi }{8}\right )+\sqrt [4]{a}+\sqrt [4]{c} x\right )-\sin \left (\frac{\pi }{8}\right ) \log \left (-2 \sqrt [8]{a} \sqrt [8]{c} \sqrt{x} \cos \left (\frac{\pi }{8}\right )+\sqrt [4]{a}+\sqrt [4]{c} x\right )+\sin \left (\frac{\pi }{8}\right ) \log \left (2 \sqrt [8]{a} \sqrt [8]{c} \sqrt{x} \cos \left (\frac{\pi }{8}\right )+\sqrt [4]{a}+\sqrt [4]{c} x\right )+2 \sin \left (\frac{\pi }{8}\right ) \tan ^{-1}\left (\frac{\sqrt [8]{c} \sqrt{x} \sec \left (\frac{\pi }{8}\right )}{\sqrt [8]{a}}-\tan \left (\frac{\pi }{8}\right )\right )+2 \sin \left (\frac{\pi }{8}\right ) \tan ^{-1}\left (\frac{\sqrt [8]{c} \sqrt{x} \sec \left (\frac{\pi }{8}\right )}{\sqrt [8]{a}}+\tan \left (\frac{\pi }{8}\right )\right )+2 \cos \left (\frac{\pi }{8}\right ) \tan ^{-1}\left (\cot \left (\frac{\pi }{8}\right )-\frac{\sqrt [8]{c} \sqrt{x} \csc \left (\frac{\pi }{8}\right )}{\sqrt [8]{a}}\right )-2 \cos \left (\frac{\pi }{8}\right ) \tan ^{-1}\left (\frac{\sqrt [8]{c} \sqrt{x} \csc \left (\frac{\pi }{8}\right )}{\sqrt [8]{a}}+\cot \left (\frac{\pi }{8}\right )\right )}{4 a^{3/8} c^{5/8}} \]
Antiderivative was successfully verified.
[In] Integrate[x^(3/2)/(a + c*x^4),x]
[Out]
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Maple [C] time = 0.023, size = 29, normalized size = 0.1 \[{\frac{1}{4\,c}\sum _{{\it \_R}={\it RootOf} \left ({{\it \_Z}}^{8}c+a \right ) }{\frac{1}{{{\it \_R}}^{3}}\ln \left ( \sqrt{x}-{\it \_R} \right ) }} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^(3/2)/(c*x^4+a),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{\frac{3}{2}}}{c x^{4} + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^(3/2)/(c*x^4 + a),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.265412, size = 610, normalized size = 2.13 \[ \frac{1}{8} \, \sqrt{2}{\left (4 \, \sqrt{2} \left (-\frac{1}{a^{3} c^{5}}\right )^{\frac{1}{8}} \arctan \left (\frac{a^{2} c^{3} \left (-\frac{1}{a^{3} c^{5}}\right )^{\frac{5}{8}}}{\sqrt{-a c \left (-\frac{1}{a^{3} c^{5}}\right )^{\frac{1}{4}} + x} + \sqrt{x}}\right ) - \sqrt{2} \left (-\frac{1}{a^{3} c^{5}}\right )^{\frac{1}{8}} \log \left (a^{2} c^{3} \left (-\frac{1}{a^{3} c^{5}}\right )^{\frac{5}{8}} + \sqrt{x}\right ) + \sqrt{2} \left (-\frac{1}{a^{3} c^{5}}\right )^{\frac{1}{8}} \log \left (-a^{2} c^{3} \left (-\frac{1}{a^{3} c^{5}}\right )^{\frac{5}{8}} + \sqrt{x}\right ) - 4 \, \left (-\frac{1}{a^{3} c^{5}}\right )^{\frac{1}{8}} \arctan \left (\frac{a^{2} c^{3} \left (-\frac{1}{a^{3} c^{5}}\right )^{\frac{5}{8}}}{a^{2} c^{3} \left (-\frac{1}{a^{3} c^{5}}\right )^{\frac{5}{8}} + \sqrt{2} \sqrt{x} + \sqrt{2 \, \sqrt{2} a^{2} c^{3} \sqrt{x} \left (-\frac{1}{a^{3} c^{5}}\right )^{\frac{5}{8}} - 2 \, a c \left (-\frac{1}{a^{3} c^{5}}\right )^{\frac{1}{4}} + 2 \, x}}\right ) - 4 \, \left (-\frac{1}{a^{3} c^{5}}\right )^{\frac{1}{8}} \arctan \left (-\frac{a^{2} c^{3} \left (-\frac{1}{a^{3} c^{5}}\right )^{\frac{5}{8}}}{a^{2} c^{3} \left (-\frac{1}{a^{3} c^{5}}\right )^{\frac{5}{8}} - \sqrt{2} \sqrt{x} - \sqrt{-2 \, \sqrt{2} a^{2} c^{3} \sqrt{x} \left (-\frac{1}{a^{3} c^{5}}\right )^{\frac{5}{8}} - 2 \, a c \left (-\frac{1}{a^{3} c^{5}}\right )^{\frac{1}{4}} + 2 \, x}}\right ) + \left (-\frac{1}{a^{3} c^{5}}\right )^{\frac{1}{8}} \log \left (2 \, \sqrt{2} a^{2} c^{3} \sqrt{x} \left (-\frac{1}{a^{3} c^{5}}\right )^{\frac{5}{8}} - 2 \, a c \left (-\frac{1}{a^{3} c^{5}}\right )^{\frac{1}{4}} + 2 \, x\right ) - \left (-\frac{1}{a^{3} c^{5}}\right )^{\frac{1}{8}} \log \left (-2 \, \sqrt{2} a^{2} c^{3} \sqrt{x} \left (-\frac{1}{a^{3} c^{5}}\right )^{\frac{5}{8}} - 2 \, a c \left (-\frac{1}{a^{3} c^{5}}\right )^{\frac{1}{4}} + 2 \, x\right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^(3/2)/(c*x^4 + a),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**(3/2)/(c*x**4+a),x)
[Out]
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GIAC/XCAS [A] time = 0.294627, size = 590, normalized size = 2.06 \[ -\frac{\sqrt{-\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{5}{8}} \arctan \left (\frac{\sqrt{-\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{1}{8}} + 2 \, \sqrt{x}}{\sqrt{\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{1}{8}}}\right )}{4 \, a} - \frac{\sqrt{-\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{5}{8}} \arctan \left (-\frac{\sqrt{-\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{1}{8}} - 2 \, \sqrt{x}}{\sqrt{\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{1}{8}}}\right )}{4 \, a} + \frac{\sqrt{\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{5}{8}} \arctan \left (\frac{\sqrt{\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{1}{8}} + 2 \, \sqrt{x}}{\sqrt{-\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{1}{8}}}\right )}{4 \, a} + \frac{\sqrt{\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{5}{8}} \arctan \left (-\frac{\sqrt{\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{1}{8}} - 2 \, \sqrt{x}}{\sqrt{-\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{1}{8}}}\right )}{4 \, a} - \frac{\sqrt{-\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{5}{8}}{\rm ln}\left (\sqrt{x} \sqrt{\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{1}{8}} + x + \left (\frac{a}{c}\right )^{\frac{1}{4}}\right )}{8 \, a} + \frac{\sqrt{-\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{5}{8}}{\rm ln}\left (-\sqrt{x} \sqrt{\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{1}{8}} + x + \left (\frac{a}{c}\right )^{\frac{1}{4}}\right )}{8 \, a} + \frac{\sqrt{\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{5}{8}}{\rm ln}\left (\sqrt{x} \sqrt{-\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{1}{8}} + x + \left (\frac{a}{c}\right )^{\frac{1}{4}}\right )}{8 \, a} - \frac{\sqrt{\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{5}{8}}{\rm ln}\left (-\sqrt{x} \sqrt{-\sqrt{2} + 2} \left (\frac{a}{c}\right )^{\frac{1}{8}} + x + \left (\frac{a}{c}\right )^{\frac{1}{4}}\right )}{8 \, a} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^(3/2)/(c*x^4 + a),x, algorithm="giac")
[Out]