Optimal. Leaf size=108 \[ -\frac{\left (3 b^2-4 a c\right ) \tanh ^{-1}\left (\frac{2 a+b x^3}{2 \sqrt{a} \sqrt{a+b x^3+c x^6}}\right )}{24 a^{5/2}}+\frac{b \sqrt{a+b x^3+c x^6}}{4 a^2 x^3}-\frac{\sqrt{a+b x^3+c x^6}}{6 a x^6} \]
[Out]
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Rubi [A] time = 0.24279, antiderivative size = 108, 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 \[ -\frac{\left (3 b^2-4 a c\right ) \tanh ^{-1}\left (\frac{2 a+b x^3}{2 \sqrt{a} \sqrt{a+b x^3+c x^6}}\right )}{24 a^{5/2}}+\frac{b \sqrt{a+b x^3+c x^6}}{4 a^2 x^3}-\frac{\sqrt{a+b x^3+c x^6}}{6 a x^6} \]
Antiderivative was successfully verified.
[In] Int[1/(x^7*Sqrt[a + b*x^3 + c*x^6]),x]
[Out]
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Rubi in Sympy [A] time = 27.3505, size = 95, normalized size = 0.88 \[ - \frac{\sqrt{a + b x^{3} + c x^{6}}}{6 a x^{6}} + \frac{b \sqrt{a + b x^{3} + c x^{6}}}{4 a^{2} x^{3}} - \frac{\left (- 4 a c + 3 b^{2}\right ) \operatorname{atanh}{\left (\frac{2 a + b x^{3}}{2 \sqrt{a} \sqrt{a + b x^{3} + c x^{6}}} \right )}}{24 a^{\frac{5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/x**7/(c*x**6+b*x**3+a)**(1/2),x)
[Out]
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Mathematica [A] time = 0.305705, size = 97, normalized size = 0.9 \[ \frac{\left (3 b^2-4 a c\right ) \left (\log \left (x^3\right )-\log \left (2 \sqrt{a} \sqrt{a+b x^3+c x^6}+2 a+b x^3\right )\right )-\frac{2 \sqrt{a} \left (2 a-3 b x^3\right ) \sqrt{a+b x^3+c x^6}}{x^6}}{24 a^{5/2}} \]
Antiderivative was successfully verified.
[In] Integrate[1/(x^7*Sqrt[a + b*x^3 + c*x^6]),x]
[Out]
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Maple [F] time = 0.041, size = 0, normalized size = 0. \[ \int{\frac{1}{{x}^{7}}{\frac{1}{\sqrt{c{x}^{6}+b{x}^{3}+a}}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/x^7/(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(1/(sqrt(c*x^6 + b*x^3 + a)*x^7),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.280626, size = 1, normalized size = 0.01 \[ \left [-\frac{{\left (3 \, b^{2} - 4 \, a c\right )} x^{6} \log \left (-\frac{4 \, \sqrt{c x^{6} + b x^{3} + a}{\left (a b x^{3} + 2 \, a^{2}\right )} +{\left ({\left (b^{2} + 4 \, a c\right )} x^{6} + 8 \, a b x^{3} + 8 \, a^{2}\right )} \sqrt{a}}{x^{6}}\right ) - 4 \, \sqrt{c x^{6} + b x^{3} + a}{\left (3 \, b x^{3} - 2 \, a\right )} \sqrt{a}}{48 \, a^{\frac{5}{2}} x^{6}}, -\frac{{\left (3 \, b^{2} - 4 \, a c\right )} x^{6} \arctan \left (\frac{{\left (b x^{3} + 2 \, a\right )} \sqrt{-a}}{2 \, \sqrt{c x^{6} + b x^{3} + a} a}\right ) - 2 \, \sqrt{c x^{6} + b x^{3} + a}{\left (3 \, b x^{3} - 2 \, a\right )} \sqrt{-a}}{24 \, \sqrt{-a} a^{2} x^{6}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(sqrt(c*x^6 + b*x^3 + a)*x^7),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{x^{7} \sqrt{a + b x^{3} + c x^{6}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/x**7/(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. \[ \int \frac{1}{\sqrt{c x^{6} + b x^{3} + a} x^{7}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(sqrt(c*x^6 + b*x^3 + a)*x^7),x, algorithm="giac")
[Out]