Optimal. Leaf size=68 \[ -\frac{3 c^2 \tanh ^{-1}\left (\frac{\sqrt{a+c x^4}}{\sqrt{a}}\right )}{16 \sqrt{a}}-\frac{3 c \sqrt{a+c x^4}}{16 x^4}-\frac{\left (a+c x^4\right )^{3/2}}{8 x^8} \]
[Out]
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Rubi [A] time = 0.0969037, antiderivative size = 68, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267 \[ -\frac{3 c^2 \tanh ^{-1}\left (\frac{\sqrt{a+c x^4}}{\sqrt{a}}\right )}{16 \sqrt{a}}-\frac{3 c \sqrt{a+c x^4}}{16 x^4}-\frac{\left (a+c x^4\right )^{3/2}}{8 x^8} \]
Antiderivative was successfully verified.
[In] Int[(a + c*x^4)^(3/2)/x^9,x]
[Out]
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Rubi in Sympy [A] time = 9.59444, size = 63, normalized size = 0.93 \[ - \frac{3 c \sqrt{a + c x^{4}}}{16 x^{4}} - \frac{\left (a + c x^{4}\right )^{\frac{3}{2}}}{8 x^{8}} - \frac{3 c^{2} \operatorname{atanh}{\left (\frac{\sqrt{a + c x^{4}}}{\sqrt{a}} \right )}}{16 \sqrt{a}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((c*x**4+a)**(3/2)/x**9,x)
[Out]
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Mathematica [A] time = 0.104716, size = 60, normalized size = 0.88 \[ \left (-\frac{a}{8 x^8}-\frac{5 c}{16 x^4}\right ) \sqrt{a+c x^4}-\frac{3 c^2 \tanh ^{-1}\left (\frac{\sqrt{a+c x^4}}{\sqrt{a}}\right )}{16 \sqrt{a}} \]
Antiderivative was successfully verified.
[In] Integrate[(a + c*x^4)^(3/2)/x^9,x]
[Out]
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Maple [A] time = 0.023, size = 63, normalized size = 0.9 \[ -{\frac{a}{8\,{x}^{8}}\sqrt{c{x}^{4}+a}}-{\frac{5\,c}{16\,{x}^{4}}\sqrt{c{x}^{4}+a}}-{\frac{3\,{c}^{2}}{16}\ln \left ({\frac{1}{{x}^{2}} \left ( 2\,a+2\,\sqrt{a}\sqrt{c{x}^{4}+a} \right ) } \right ){\frac{1}{\sqrt{a}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((c*x^4+a)^(3/2)/x^9,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^9,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.266838, size = 1, normalized size = 0.01 \[ \left [\frac{3 \, c^{2} x^{8} \log \left (\frac{{\left (c x^{4} + 2 \, a\right )} \sqrt{a} - 2 \, \sqrt{c x^{4} + a} a}{x^{4}}\right ) - 2 \,{\left (5 \, c x^{4} + 2 \, a\right )} \sqrt{c x^{4} + a} \sqrt{a}}{32 \, \sqrt{a} x^{8}}, \frac{3 \, c^{2} x^{8} \arctan \left (\frac{a}{\sqrt{c x^{4} + a} \sqrt{-a}}\right ) -{\left (5 \, c x^{4} + 2 \, a\right )} \sqrt{c x^{4} + a} \sqrt{-a}}{16 \, \sqrt{-a} x^{8}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^4 + a)^(3/2)/x^9,x, algorithm="fricas")
[Out]
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Sympy [A] time = 13.572, size = 75, normalized size = 1.1 \[ - \frac{a \sqrt{c} \sqrt{\frac{a}{c x^{4}} + 1}}{8 x^{6}} - \frac{5 c^{\frac{3}{2}} \sqrt{\frac{a}{c x^{4}} + 1}}{16 x^{2}} - \frac{3 c^{2} \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{c} x^{2}} \right )}}{16 \sqrt{a}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x**4+a)**(3/2)/x**9,x)
[Out]
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GIAC/XCAS [A] time = 0.21564, size = 82, normalized size = 1.21 \[ \frac{1}{16} \, c^{2}{\left (\frac{3 \, \arctan \left (\frac{\sqrt{c x^{4} + a}}{\sqrt{-a}}\right )}{\sqrt{-a}} - \frac{5 \,{\left (c x^{4} + a\right )}^{\frac{3}{2}} - 3 \, \sqrt{c x^{4} + a} a}{c^{2} x^{8}}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^4 + a)^(3/2)/x^9,x, algorithm="giac")
[Out]