Optimal. Leaf size=171 \[ \frac{\left (48 a^2 c^2-120 a b^2 c+35 b^4\right ) \tanh ^{-1}\left (\frac{b+2 c x^3}{2 \sqrt{c} \sqrt{a+b x^3+c x^6}}\right )}{384 c^{9/2}}-\frac{\left (5 b \left (21 b^2-44 a c\right )-2 c x^3 \left (35 b^2-36 a c\right )\right ) \sqrt{a+b x^3+c x^6}}{576 c^4}-\frac{7 b x^6 \sqrt{a+b x^3+c x^6}}{72 c^2}+\frac{x^9 \sqrt{a+b x^3+c x^6}}{12 c} \]
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Rubi [A] time = 0.497091, antiderivative size = 171, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3 \[ \frac{\left (48 a^2 c^2-120 a b^2 c+35 b^4\right ) \tanh ^{-1}\left (\frac{b+2 c x^3}{2 \sqrt{c} \sqrt{a+b x^3+c x^6}}\right )}{384 c^{9/2}}-\frac{\left (5 b \left (21 b^2-44 a c\right )-2 c x^3 \left (35 b^2-36 a c\right )\right ) \sqrt{a+b x^3+c x^6}}{576 c^4}-\frac{7 b x^6 \sqrt{a+b x^3+c x^6}}{72 c^2}+\frac{x^9 \sqrt{a+b x^3+c x^6}}{12 c} \]
Antiderivative was successfully verified.
[In] Int[x^14/Sqrt[a + b*x^3 + c*x^6],x]
[Out]
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Rubi in Sympy [A] time = 42.0434, size = 162, normalized size = 0.95 \[ - \frac{7 b x^{6} \sqrt{a + b x^{3} + c x^{6}}}{72 c^{2}} + \frac{x^{9} \sqrt{a + b x^{3} + c x^{6}}}{12 c} - \frac{\left (\frac{5 b \left (- 44 a c + 21 b^{2}\right )}{8} - \frac{c x^{3} \left (- 36 a c + 35 b^{2}\right )}{4}\right ) \sqrt{a + b x^{3} + c x^{6}}}{72 c^{4}} + \frac{\left (48 a^{2} c^{2} - 120 a b^{2} c + 35 b^{4}\right ) \operatorname{atanh}{\left (\frac{b + 2 c x^{3}}{2 \sqrt{c} \sqrt{a + b x^{3} + c x^{6}}} \right )}}{384 c^{\frac{9}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**14/(c*x**6+b*x**3+a)**(1/2),x)
[Out]
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Mathematica [A] time = 0.185408, size = 135, normalized size = 0.79 \[ \frac{3 \left (48 a^2 c^2-120 a b^2 c+35 b^4\right ) \log \left (2 \sqrt{c} \sqrt{a+b x^3+c x^6}+b+2 c x^3\right )+2 \sqrt{c} \sqrt{a+b x^3+c x^6} \left (4 b c \left (55 a-14 c x^6\right )+24 c^2 x^3 \left (2 c x^6-3 a\right )-105 b^3+70 b^2 c x^3\right )}{1152 c^{9/2}} \]
Antiderivative was successfully verified.
[In] Integrate[x^14/Sqrt[a + b*x^3 + c*x^6],x]
[Out]
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Maple [F] time = 0.036, size = 0, normalized size = 0. \[ \int{{x}^{14}{\frac{1}{\sqrt{c{x}^{6}+b{x}^{3}+a}}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^14/(c*x^6+b*x^3+a)^(1/2),x)
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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^14/sqrt(c*x^6 + b*x^3 + a),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.290877, size = 1, normalized size = 0.01 \[ \left [\frac{4 \,{\left (48 \, c^{3} x^{9} - 56 \, b c^{2} x^{6} + 2 \,{\left (35 \, b^{2} c - 36 \, a c^{2}\right )} x^{3} - 105 \, b^{3} + 220 \, a b c\right )} \sqrt{c x^{6} + b x^{3} + a} \sqrt{c} + 3 \,{\left (35 \, b^{4} - 120 \, a b^{2} c + 48 \, a^{2} c^{2}\right )} \log \left (-4 \, \sqrt{c x^{6} + b x^{3} + a}{\left (2 \, c^{2} x^{3} + b c\right )} -{\left (8 \, c^{2} x^{6} + 8 \, b c x^{3} + b^{2} + 4 \, a c\right )} \sqrt{c}\right )}{2304 \, c^{\frac{9}{2}}}, \frac{2 \,{\left (48 \, c^{3} x^{9} - 56 \, b c^{2} x^{6} + 2 \,{\left (35 \, b^{2} c - 36 \, a c^{2}\right )} x^{3} - 105 \, b^{3} + 220 \, a b c\right )} \sqrt{c x^{6} + b x^{3} + a} \sqrt{-c} + 3 \,{\left (35 \, b^{4} - 120 \, a b^{2} c + 48 \, a^{2} c^{2}\right )} \arctan \left (\frac{{\left (2 \, c x^{3} + b\right )} \sqrt{-c}}{2 \, \sqrt{c x^{6} + b x^{3} + a} c}\right )}{1152 \, \sqrt{-c} c^{4}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^14/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. \[ \int \frac{x^{14}}{\sqrt{a + b x^{3} + c x^{6}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**14/(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. \[ \int \frac{x^{14}}{\sqrt{c x^{6} + b x^{3} + a}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^14/sqrt(c*x^6 + b*x^3 + a),x, algorithm="giac")
[Out]