Optimal. Leaf size=68 \[ \frac{\sqrt{a+b x^3+c x^6}}{3 c}-\frac{b \tanh ^{-1}\left (\frac{b+2 c x^3}{2 \sqrt{c} \sqrt{a+b x^3+c x^6}}\right )}{6 c^{3/2}} \]
[Out]
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Rubi [A] time = 0.114341, antiderivative size = 68, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ \frac{\sqrt{a+b x^3+c x^6}}{3 c}-\frac{b \tanh ^{-1}\left (\frac{b+2 c x^3}{2 \sqrt{c} \sqrt{a+b x^3+c x^6}}\right )}{6 c^{3/2}} \]
Antiderivative was successfully verified.
[In] Int[x^5/Sqrt[a + b*x^3 + c*x^6],x]
[Out]
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Rubi in Sympy [A] time = 12.9748, size = 58, normalized size = 0.85 \[ - \frac{b \operatorname{atanh}{\left (\frac{b + 2 c x^{3}}{2 \sqrt{c} \sqrt{a + b x^{3} + c x^{6}}} \right )}}{6 c^{\frac{3}{2}}} + \frac{\sqrt{a + b x^{3} + c x^{6}}}{3 c} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**5/(c*x**6+b*x**3+a)**(1/2),x)
[Out]
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Mathematica [A] time = 0.0381865, size = 66, normalized size = 0.97 \[ \frac{\sqrt{a+b x^3+c x^6}}{3 c}-\frac{b \log \left (2 \sqrt{c} \sqrt{a+b x^3+c x^6}+b+2 c x^3\right )}{6 c^{3/2}} \]
Antiderivative was successfully verified.
[In] Integrate[x^5/Sqrt[a + b*x^3 + c*x^6],x]
[Out]
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Maple [F] time = 0.032, size = 0, normalized size = 0. \[ \int{{x}^{5}{\frac{1}{\sqrt{c{x}^{6}+b{x}^{3}+a}}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^5/(c*x^6+b*x^3+a)^(1/2),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(x^5/sqrt(c*x^6 + b*x^3 + a),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.290517, size = 1, normalized size = 0.01 \[ \left [\frac{b \log \left (4 \, \sqrt{c x^{6} + b x^{3} + a}{\left (2 \, c^{2} x^{3} + b c\right )} -{\left (8 \, c^{2} x^{6} + 8 \, b c x^{3} + b^{2} + 4 \, a c\right )} \sqrt{c}\right ) + 4 \, \sqrt{c x^{6} + b x^{3} + a} \sqrt{c}}{12 \, c^{\frac{3}{2}}}, -\frac{b \arctan \left (\frac{{\left (2 \, c x^{3} + b\right )} \sqrt{-c}}{2 \, \sqrt{c x^{6} + b x^{3} + a} c}\right ) - 2 \, \sqrt{c x^{6} + b x^{3} + a} \sqrt{-c}}{6 \, \sqrt{-c} c}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^5/sqrt(c*x^6 + b*x^3 + a),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{5}}{\sqrt{a + b x^{3} + c x^{6}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**5/(c*x**6+b*x**3+a)**(1/2),x)
[Out]
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GIAC/XCAS [A] time = 0.291017, size = 82, normalized size = 1.21 \[ \frac{b{\rm ln}\left ({\left | -2 \,{\left (\sqrt{c} x^{3} - \sqrt{c x^{6} + b x^{3} + a}\right )} \sqrt{c} - b \right |}\right )}{6 \, c^{\frac{3}{2}}} + \frac{\sqrt{c x^{6} + b x^{3} + a}}{3 \, c} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^5/sqrt(c*x^6 + b*x^3 + a),x, algorithm="giac")
[Out]