Optimal. Leaf size=71 \[ \frac{3 a^2 \tanh ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{a+c x^4}}\right )}{16 \sqrt{c}}+\frac{3}{16} a x^2 \sqrt{a+c x^4}+\frac{1}{8} x^2 \left (a+c x^4\right )^{3/2} \]
[Out]
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Rubi [A] time = 0.0791766, antiderivative size = 71, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.308 \[ \frac{3 a^2 \tanh ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{a+c x^4}}\right )}{16 \sqrt{c}}+\frac{3}{16} a x^2 \sqrt{a+c x^4}+\frac{1}{8} x^2 \left (a+c x^4\right )^{3/2} \]
Antiderivative was successfully verified.
[In] Int[x*(a + c*x^4)^(3/2),x]
[Out]
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Rubi in Sympy [A] time = 6.5933, size = 65, normalized size = 0.92 \[ \frac{3 a^{2} \operatorname{atanh}{\left (\frac{\sqrt{c} x^{2}}{\sqrt{a + c x^{4}}} \right )}}{16 \sqrt{c}} + \frac{3 a x^{2} \sqrt{a + c x^{4}}}{16} + \frac{x^{2} \left (a + c x^{4}\right )^{\frac{3}{2}}}{8} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x*(c*x**4+a)**(3/2),x)
[Out]
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Mathematica [A] time = 0.0656864, size = 64, normalized size = 0.9 \[ \frac{1}{16} \left (\frac{3 a^2 \log \left (\sqrt{c} \sqrt{a+c x^4}+c x^2\right )}{\sqrt{c}}+x^2 \sqrt{a+c x^4} \left (5 a+2 c x^4\right )\right ) \]
Antiderivative was successfully verified.
[In] Integrate[x*(a + c*x^4)^(3/2),x]
[Out]
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Maple [A] time = 0.014, size = 58, normalized size = 0.8 \[{\frac{3\,{a}^{2}}{16}\ln \left ({x}^{2}\sqrt{c}+\sqrt{c{x}^{4}+a} \right ){\frac{1}{\sqrt{c}}}}+{\frac{c{x}^{6}}{8}\sqrt{c{x}^{4}+a}}+{\frac{5\,a{x}^{2}}{16}\sqrt{c{x}^{4}+a}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x*(c*x^4+a)^(3/2),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((c*x^4 + a)^(3/2)*x,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.263374, size = 1, normalized size = 0.01 \[ \left [\frac{3 \, a^{2} \log \left (-2 \, \sqrt{c x^{4} + a} c x^{2} -{\left (2 \, c x^{4} + a\right )} \sqrt{c}\right ) + 2 \,{\left (2 \, c x^{6} + 5 \, a x^{2}\right )} \sqrt{c x^{4} + a} \sqrt{c}}{32 \, \sqrt{c}}, \frac{3 \, a^{2} \arctan \left (\frac{\sqrt{-c} x^{2}}{\sqrt{c x^{4} + a}}\right ) +{\left (2 \, c x^{6} + 5 \, a x^{2}\right )} \sqrt{c x^{4} + a} \sqrt{-c}}{16 \, \sqrt{-c}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^4 + a)^(3/2)*x,x, algorithm="fricas")
[Out]
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Sympy [A] time = 10.0555, size = 73, normalized size = 1.03 \[ \frac{5 a^{\frac{3}{2}} x^{2} \sqrt{1 + \frac{c x^{4}}{a}}}{16} + \frac{\sqrt{a} c x^{6} \sqrt{1 + \frac{c x^{4}}{a}}}{8} + \frac{3 a^{2} \operatorname{asinh}{\left (\frac{\sqrt{c} x^{2}}{\sqrt{a}} \right )}}{16 \sqrt{c}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x*(c*x**4+a)**(3/2),x)
[Out]
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GIAC/XCAS [A] time = 0.224441, size = 72, normalized size = 1.01 \[ \frac{1}{16} \,{\left (2 \, c x^{4} + 5 \, a\right )} \sqrt{c x^{4} + a} x^{2} - \frac{3 \, a^{2}{\rm ln}\left ({\left | -\sqrt{c} x^{2} + \sqrt{c x^{4} + a} \right |}\right )}{16 \, \sqrt{c}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^4 + a)^(3/2)*x,x, algorithm="giac")
[Out]