Optimal. Leaf size=329 \[ \frac{11 x^{5/2}}{64 a^2 \left (a+c x^4\right )}-\frac{33 \log \left (-\sqrt{2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt{x}+\sqrt [4]{-a}+\sqrt [4]{c} x\right )}{512 \sqrt{2} (-a)^{19/8} c^{5/8}}+\frac{33 \log \left (\sqrt{2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt{x}+\sqrt [4]{-a}+\sqrt [4]{c} x\right )}{512 \sqrt{2} (-a)^{19/8} c^{5/8}}-\frac{33 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}\right )}{256 \sqrt{2} (-a)^{19/8} c^{5/8}}+\frac{33 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}+1\right )}{256 \sqrt{2} (-a)^{19/8} c^{5/8}}-\frac{33 \tan ^{-1}\left (\frac{\sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}\right )}{256 (-a)^{19/8} c^{5/8}}-\frac{33 \tanh ^{-1}\left (\frac{\sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}\right )}{256 (-a)^{19/8} c^{5/8}}+\frac{x^{5/2}}{8 a \left (a+c x^4\right )^2} \]
[Out]
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Rubi [A] time = 0.625741, antiderivative size = 329, normalized size of antiderivative = 1., number of steps used = 16, number of rules used = 12, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.8 \[ \frac{11 x^{5/2}}{64 a^2 \left (a+c x^4\right )}-\frac{33 \log \left (-\sqrt{2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt{x}+\sqrt [4]{-a}+\sqrt [4]{c} x\right )}{512 \sqrt{2} (-a)^{19/8} c^{5/8}}+\frac{33 \log \left (\sqrt{2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt{x}+\sqrt [4]{-a}+\sqrt [4]{c} x\right )}{512 \sqrt{2} (-a)^{19/8} c^{5/8}}-\frac{33 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}\right )}{256 \sqrt{2} (-a)^{19/8} c^{5/8}}+\frac{33 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}+1\right )}{256 \sqrt{2} (-a)^{19/8} c^{5/8}}-\frac{33 \tan ^{-1}\left (\frac{\sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}\right )}{256 (-a)^{19/8} c^{5/8}}-\frac{33 \tanh ^{-1}\left (\frac{\sqrt [8]{c} \sqrt{x}}{\sqrt [8]{-a}}\right )}{256 (-a)^{19/8} c^{5/8}}+\frac{x^{5/2}}{8 a \left (a+c x^4\right )^2} \]
Antiderivative was successfully verified.
[In] Int[x^(3/2)/(a + c*x^4)^3,x]
[Out]
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Rubi in Sympy [A] time = 127.908, size = 309, normalized size = 0.94 \[ - \frac{33 \sqrt{2} \log{\left (- \sqrt{2} \sqrt [8]{c} \sqrt{x} \sqrt [8]{- a} + \sqrt [4]{c} x + \sqrt [4]{- a} \right )}}{1024 c^{\frac{5}{8}} \left (- a\right )^{\frac{19}{8}}} + \frac{33 \sqrt{2} \log{\left (\sqrt{2} \sqrt [8]{c} \sqrt{x} \sqrt [8]{- a} + \sqrt [4]{c} x + \sqrt [4]{- a} \right )}}{1024 c^{\frac{5}{8}} \left (- a\right )^{\frac{19}{8}}} - \frac{33 \operatorname{atan}{\left (\frac{\sqrt [8]{c} \sqrt{x}}{\sqrt [8]{- a}} \right )}}{256 c^{\frac{5}{8}} \left (- a\right )^{\frac{19}{8}}} + \frac{33 \sqrt{2} \operatorname{atan}{\left (\frac{\sqrt{2} \sqrt [8]{c} \sqrt{x}}{\sqrt [8]{- a}} - 1 \right )}}{512 c^{\frac{5}{8}} \left (- a\right )^{\frac{19}{8}}} + \frac{33 \sqrt{2} \operatorname{atan}{\left (\frac{\sqrt{2} \sqrt [8]{c} \sqrt{x}}{\sqrt [8]{- a}} + 1 \right )}}{512 c^{\frac{5}{8}} \left (- a\right )^{\frac{19}{8}}} - \frac{33 \operatorname{atanh}{\left (\frac{\sqrt [8]{c} \sqrt{x}}{\sqrt [8]{- a}} \right )}}{256 c^{\frac{5}{8}} \left (- a\right )^{\frac{19}{8}}} + \frac{x^{\frac{5}{2}}}{8 a \left (a + c x^{4}\right )^{2}} + \frac{11 x^{\frac{5}{2}}}{64 a^{2} \left (a + c x^{4}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**(3/2)/(c*x**4+a)**3,x)
[Out]
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Mathematica [A] time = 1.02643, size = 427, normalized size = 1.3 \[ \frac{\frac{88 a^{3/8} x^{5/2}}{a+c x^4}+\frac{64 a^{11/8} x^{5/2}}{\left (a+c x^4\right )^2}-\frac{33 \cos \left (\frac{\pi }{8}\right ) \log \left (-2 \sqrt [8]{a} \sqrt [8]{c} \sqrt{x} \sin \left (\frac{\pi }{8}\right )+\sqrt [4]{a}+\sqrt [4]{c} x\right )}{c^{5/8}}+\frac{33 \cos \left (\frac{\pi }{8}\right ) \log \left (2 \sqrt [8]{a} \sqrt [8]{c} \sqrt{x} \sin \left (\frac{\pi }{8}\right )+\sqrt [4]{a}+\sqrt [4]{c} x\right )}{c^{5/8}}+\frac{33 \sin \left (\frac{\pi }{8}\right ) \log \left (-2 \sqrt [8]{a} \sqrt [8]{c} \sqrt{x} \cos \left (\frac{\pi }{8}\right )+\sqrt [4]{a}+\sqrt [4]{c} x\right )}{c^{5/8}}-\frac{33 \sin \left (\frac{\pi }{8}\right ) \log \left (2 \sqrt [8]{a} \sqrt [8]{c} \sqrt{x} \cos \left (\frac{\pi }{8}\right )+\sqrt [4]{a}+\sqrt [4]{c} x\right )}{c^{5/8}}-\frac{66 \sin \left (\frac{\pi }{8}\right ) \tan ^{-1}\left (\frac{\sqrt [8]{c} \sqrt{x} \sec \left (\frac{\pi }{8}\right )}{\sqrt [8]{a}}-\tan \left (\frac{\pi }{8}\right )\right )}{c^{5/8}}-\frac{66 \sin \left (\frac{\pi }{8}\right ) \tan ^{-1}\left (\frac{\sqrt [8]{c} \sqrt{x} \sec \left (\frac{\pi }{8}\right )}{\sqrt [8]{a}}+\tan \left (\frac{\pi }{8}\right )\right )}{c^{5/8}}-\frac{66 \cos \left (\frac{\pi }{8}\right ) \tan ^{-1}\left (\cot \left (\frac{\pi }{8}\right )-\frac{\sqrt [8]{c} \sqrt{x} \csc \left (\frac{\pi }{8}\right )}{\sqrt [8]{a}}\right )}{c^{5/8}}+\frac{66 \cos \left (\frac{\pi }{8}\right ) \tan ^{-1}\left (\frac{\sqrt [8]{c} \sqrt{x} \csc \left (\frac{\pi }{8}\right )}{\sqrt [8]{a}}+\cot \left (\frac{\pi }{8}\right )\right )}{c^{5/8}}}{512 a^{19/8}} \]
Antiderivative was successfully verified.
[In] Integrate[x^(3/2)/(a + c*x^4)^3,x]
[Out]
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Maple [C] time = 0.027, size = 62, normalized size = 0.2 \[ 2\,{\frac{1}{ \left ( c{x}^{4}+a \right ) ^{2}} \left ({\frac{19\,{x}^{5/2}}{128\,a}}+{\frac{11\,c{x}^{13/2}}{128\,{a}^{2}}} \right ) }+{\frac{33}{512\,{a}^{2}c}\sum _{{\it \_R}={\it RootOf} \left ({{\it \_Z}}^{8}c+a \right ) }{\frac{1}{{{\it \_R}}^{3}}\ln \left ( \sqrt{x}-{\it \_R} \right ) }} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^(3/2)/(c*x^4+a)^3,x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \frac{11 \, c x^{\frac{13}{2}} + 19 \, a x^{\frac{5}{2}}}{64 \,{\left (a^{2} c^{2} x^{8} + 2 \, a^{3} c x^{4} + a^{4}\right )}} + 33 \, \int \frac{x^{\frac{3}{2}}}{128 \,{\left (a^{2} c x^{4} + a^{3}\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^(3/2)/(c*x^4 + a)^3,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.264942, size = 906, normalized size = 2.75 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^(3/2)/(c*x^4 + a)^3,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)**3,x)
[Out]
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GIAC/XCAS [A] time = 0.355334, size = 626, normalized size = 1.9 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^(3/2)/(c*x^4 + a)^3,x, algorithm="giac")
[Out]