Optimal. Leaf size=63 \[ -\frac{\left (a+c x^4\right )^{3/2}}{4 x^4}+\frac{3}{4} c \sqrt{a+c x^4}-\frac{3}{4} \sqrt{a} c \tanh ^{-1}\left (\frac{\sqrt{a+c x^4}}{\sqrt{a}}\right ) \]
[Out]
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Rubi [A] time = 0.0938859, antiderivative size = 63, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333 \[ -\frac{\left (a+c x^4\right )^{3/2}}{4 x^4}+\frac{3}{4} c \sqrt{a+c x^4}-\frac{3}{4} \sqrt{a} c \tanh ^{-1}\left (\frac{\sqrt{a+c x^4}}{\sqrt{a}}\right ) \]
Antiderivative was successfully verified.
[In] Int[(a + c*x^4)^(3/2)/x^5,x]
[Out]
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Rubi in Sympy [A] time = 8.95702, size = 56, normalized size = 0.89 \[ - \frac{3 \sqrt{a} c \operatorname{atanh}{\left (\frac{\sqrt{a + c x^{4}}}{\sqrt{a}} \right )}}{4} + \frac{3 c \sqrt{a + c x^{4}}}{4} - \frac{\left (a + c x^{4}\right )^{\frac{3}{2}}}{4 x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((c*x**4+a)**(3/2)/x**5,x)
[Out]
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Mathematica [A] time = 0.0819188, size = 55, normalized size = 0.87 \[ \left (\frac{c}{2}-\frac{a}{4 x^4}\right ) \sqrt{a+c x^4}-\frac{3}{4} \sqrt{a} c \tanh ^{-1}\left (\frac{\sqrt{a+c x^4}}{\sqrt{a}}\right ) \]
Antiderivative was successfully verified.
[In] Integrate[(a + c*x^4)^(3/2)/x^5,x]
[Out]
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Maple [A] time = 0.021, size = 58, normalized size = 0.9 \[{\frac{c}{2}\sqrt{c{x}^{4}+a}}-{\frac{a}{4\,{x}^{4}}\sqrt{c{x}^{4}+a}}-{\frac{3\,c}{4}\sqrt{a}\ln \left ({\frac{1}{{x}^{2}} \left ( 2\,a+2\,\sqrt{a}\sqrt{c{x}^{4}+a} \right ) } \right ) } \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((c*x^4+a)^(3/2)/x^5,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^5,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.246822, size = 1, normalized size = 0.02 \[ \left [\frac{3 \, \sqrt{a} c x^{4} \log \left (\frac{c x^{4} - 2 \, \sqrt{c x^{4} + a} \sqrt{a} + 2 \, a}{x^{4}}\right ) + 2 \,{\left (2 \, c x^{4} - a\right )} \sqrt{c x^{4} + a}}{8 \, x^{4}}, -\frac{3 \, \sqrt{-a} c x^{4} \arctan \left (\frac{\sqrt{c x^{4} + a}}{\sqrt{-a}}\right ) -{\left (2 \, c x^{4} - a\right )} \sqrt{c x^{4} + a}}{4 \, x^{4}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^4 + a)^(3/2)/x^5,x, algorithm="fricas")
[Out]
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Sympy [A] time = 8.94732, size = 95, normalized size = 1.51 \[ - \frac{3 \sqrt{a} c \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{c} x^{2}} \right )}}{4} - \frac{a^{2}}{4 \sqrt{c} x^{6} \sqrt{\frac{a}{c x^{4}} + 1}} + \frac{a \sqrt{c}}{4 x^{2} \sqrt{\frac{a}{c x^{4}} + 1}} + \frac{c^{\frac{3}{2}} x^{2}}{2 \sqrt{\frac{a}{c x^{4}} + 1}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x**4+a)**(3/2)/x**5,x)
[Out]
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GIAC/XCAS [A] time = 0.214447, size = 77, normalized size = 1.22 \[ \frac{1}{4} \,{\left (\frac{3 \, a \arctan \left (\frac{\sqrt{c x^{4} + a}}{\sqrt{-a}}\right )}{\sqrt{-a}} + 2 \, \sqrt{c x^{4} + a} - \frac{\sqrt{c x^{4} + a} a}{c x^{4}}\right )} c \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^4 + a)^(3/2)/x^5,x, algorithm="giac")
[Out]