Optimal. Leaf size=121 \[ -\frac{b \left (5 b^2-12 a c\right ) \tanh ^{-1}\left (\frac{b+2 c x^3}{2 \sqrt{c} \sqrt{a+b x^3+c x^6}}\right )}{48 c^{7/2}}+\frac{\left (-16 a c+15 b^2-10 b c x^3\right ) \sqrt{a+b x^3+c x^6}}{72 c^3}+\frac{x^6 \sqrt{a+b x^3+c x^6}}{9 c} \]
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Rubi [A] time = 0.253063, antiderivative size = 121, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ -\frac{b \left (5 b^2-12 a c\right ) \tanh ^{-1}\left (\frac{b+2 c x^3}{2 \sqrt{c} \sqrt{a+b x^3+c x^6}}\right )}{48 c^{7/2}}+\frac{\left (-16 a c+15 b^2-10 b c x^3\right ) \sqrt{a+b x^3+c x^6}}{72 c^3}+\frac{x^6 \sqrt{a+b x^3+c x^6}}{9 c} \]
Antiderivative was successfully verified.
[In] Int[x^11/Sqrt[a + b*x^3 + c*x^6],x]
[Out]
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Rubi in Sympy [A] time = 26.0097, size = 114, normalized size = 0.94 \[ - \frac{b \left (- 12 a c + 5 b^{2}\right ) \operatorname{atanh}{\left (\frac{b + 2 c x^{3}}{2 \sqrt{c} \sqrt{a + b x^{3} + c x^{6}}} \right )}}{48 c^{\frac{7}{2}}} + \frac{x^{6} \sqrt{a + b x^{3} + c x^{6}}}{9 c} + \frac{\sqrt{a + b x^{3} + c x^{6}} \left (- 4 a c + \frac{15 b^{2}}{4} - \frac{5 b c x^{3}}{2}\right )}{18 c^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**11/(c*x**6+b*x**3+a)**(1/2),x)
[Out]
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Mathematica [A] time = 0.100495, size = 102, normalized size = 0.84 \[ \frac{\left (36 a b c-15 b^3\right ) \log \left (2 \sqrt{c} \sqrt{a+b x^3+c x^6}+b+2 c x^3\right )+2 \sqrt{c} \sqrt{a+b x^3+c x^6} \left (8 c \left (c x^6-2 a\right )+15 b^2-10 b c x^3\right )}{144 c^{7/2}} \]
Antiderivative was successfully verified.
[In] Integrate[x^11/Sqrt[a + b*x^3 + c*x^6],x]
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Maple [F] time = 0.034, size = 0, normalized size = 0. \[ \int{{x}^{11}{\frac{1}{\sqrt{c{x}^{6}+b{x}^{3}+a}}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^11/(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^11/sqrt(c*x^6 + b*x^3 + a),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.28028, size = 1, normalized size = 0.01 \[ \left [\frac{4 \,{\left (8 \, c^{2} x^{6} - 10 \, b c x^{3} + 15 \, b^{2} - 16 \, a c\right )} \sqrt{c x^{6} + b x^{3} + a} \sqrt{c} - 3 \,{\left (5 \, b^{3} - 12 \, a b c\right )} \log \left (-4 \, \sqrt{c x^{6} + b x^{3} + a}{\left (2 \, c^{2} x^{3} + b c\right )} -{\left (8 \, c^{2} x^{6} + 8 \, b c x^{3} + b^{2} + 4 \, a c\right )} \sqrt{c}\right )}{288 \, c^{\frac{7}{2}}}, \frac{2 \,{\left (8 \, c^{2} x^{6} - 10 \, b c x^{3} + 15 \, b^{2} - 16 \, a c\right )} \sqrt{c x^{6} + b x^{3} + a} \sqrt{-c} - 3 \,{\left (5 \, b^{3} - 12 \, a b c\right )} \arctan \left (\frac{{\left (2 \, c x^{3} + b\right )} \sqrt{-c}}{2 \, \sqrt{c x^{6} + b x^{3} + a} c}\right )}{144 \, \sqrt{-c} c^{3}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^11/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. \[ \int \frac{x^{11}}{\sqrt{a + b x^{3} + c x^{6}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**11/(c*x**6+b*x**3+a)**(1/2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{11}}{\sqrt{c x^{6} + b x^{3} + a}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^11/sqrt(c*x^6 + b*x^3 + a),x, algorithm="giac")
[Out]