Optimal. Leaf size=95 \[ -\frac{a^3 \tanh ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{a+c x^4}}\right )}{32 c^{3/2}}+\frac{a^2 x^2 \sqrt{a+c x^4}}{32 c}+\frac{1}{12} x^6 \left (a+c x^4\right )^{3/2}+\frac{1}{16} a x^6 \sqrt{a+c x^4} \]
[Out]
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Rubi [A] time = 0.14832, antiderivative size = 95, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333 \[ -\frac{a^3 \tanh ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{a+c x^4}}\right )}{32 c^{3/2}}+\frac{a^2 x^2 \sqrt{a+c x^4}}{32 c}+\frac{1}{12} x^6 \left (a+c x^4\right )^{3/2}+\frac{1}{16} a x^6 \sqrt{a+c x^4} \]
Antiderivative was successfully verified.
[In] Int[x^5*(a + c*x^4)^(3/2),x]
[Out]
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Rubi in Sympy [A] time = 15.3602, size = 82, normalized size = 0.86 \[ - \frac{a^{3} \operatorname{atanh}{\left (\frac{\sqrt{c} x^{2}}{\sqrt{a + c x^{4}}} \right )}}{32 c^{\frac{3}{2}}} + \frac{a^{2} x^{2} \sqrt{a + c x^{4}}}{32 c} + \frac{a x^{6} \sqrt{a + c x^{4}}}{16} + \frac{x^{6} \left (a + c x^{4}\right )^{\frac{3}{2}}}{12} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**5*(c*x**4+a)**(3/2),x)
[Out]
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Mathematica [A] time = 0.0670835, size = 82, normalized size = 0.86 \[ \frac{1}{2} \sqrt{a+c x^4} \left (\frac{a^2 x^2}{16 c}+\frac{7 a x^6}{24}+\frac{c x^{10}}{6}\right )-\frac{a^3 \log \left (\sqrt{c} \sqrt{a+c x^4}+c x^2\right )}{32 c^{3/2}} \]
Antiderivative was successfully verified.
[In] Integrate[x^5*(a + c*x^4)^(3/2),x]
[Out]
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Maple [A] time = 0.025, size = 78, normalized size = 0.8 \[{\frac{{a}^{2}{x}^{2}}{32\,c}\sqrt{c{x}^{4}+a}}-{\frac{{a}^{3}}{32}\ln \left ({x}^{2}\sqrt{c}+\sqrt{c{x}^{4}+a} \right ){c}^{-{\frac{3}{2}}}}+{\frac{c{x}^{10}}{12}\sqrt{c{x}^{4}+a}}+{\frac{7\,{x}^{6}a}{48}\sqrt{c{x}^{4}+a}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^5*(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^5,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.29038, size = 1, normalized size = 0.01 \[ \left [\frac{3 \, a^{3} \log \left (2 \, \sqrt{c x^{4} + a} c x^{2} -{\left (2 \, c x^{4} + a\right )} \sqrt{c}\right ) + 2 \,{\left (8 \, c^{2} x^{10} + 14 \, a c x^{6} + 3 \, a^{2} x^{2}\right )} \sqrt{c x^{4} + a} \sqrt{c}}{192 \, c^{\frac{3}{2}}}, -\frac{3 \, a^{3} \arctan \left (\frac{\sqrt{-c} x^{2}}{\sqrt{c x^{4} + a}}\right ) -{\left (8 \, c^{2} x^{10} + 14 \, a c x^{6} + 3 \, a^{2} x^{2}\right )} \sqrt{c x^{4} + a} \sqrt{-c}}{96 \, \sqrt{-c} c}\right ] \]
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 = 18.403, size = 122, normalized size = 1.28 \[ \frac{a^{\frac{5}{2}} x^{2}}{32 c \sqrt{1 + \frac{c x^{4}}{a}}} + \frac{17 a^{\frac{3}{2}} x^{6}}{96 \sqrt{1 + \frac{c x^{4}}{a}}} + \frac{11 \sqrt{a} c x^{10}}{48 \sqrt{1 + \frac{c x^{4}}{a}}} - \frac{a^{3} \operatorname{asinh}{\left (\frac{\sqrt{c} x^{2}}{\sqrt{a}} \right )}}{32 c^{\frac{3}{2}}} + \frac{c^{2} x^{14}}{12 \sqrt{a} \sqrt{1 + \frac{c x^{4}}{a}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**5*(c*x**4+a)**(3/2),x)
[Out]
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GIAC/XCAS [A] time = 0.228599, size = 90, normalized size = 0.95 \[ \frac{1}{96} \,{\left (2 \,{\left (4 \, c x^{4} + 7 \, a\right )} x^{4} + \frac{3 \, a^{2}}{c}\right )} \sqrt{c x^{4} + a} x^{2} + \frac{a^{3}{\rm ln}\left ({\left | -\sqrt{c} x^{2} + \sqrt{c x^{4} + a} \right |}\right )}{32 \, c^{\frac{3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^4 + a)^(3/2)*x^5,x, algorithm="giac")
[Out]