Optimal. Leaf size=51 \[ -\frac{\tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^6}}{\sqrt{b c-a d}}\right )}{3 \sqrt{b} \sqrt{b c-a d}} \]
[Out]
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Rubi [A] time = 0.13127, antiderivative size = 51, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125 \[ -\frac{\tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^6}}{\sqrt{b c-a d}}\right )}{3 \sqrt{b} \sqrt{b c-a d}} \]
Antiderivative was successfully verified.
[In] Int[x^5/((a + b*x^6)*Sqrt[c + d*x^6]),x]
[Out]
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Rubi in Sympy [A] time = 15.0472, size = 42, normalized size = 0.82 \[ \frac{\operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{c + d x^{6}}}{\sqrt{a d - b c}} \right )}}{3 \sqrt{b} \sqrt{a d - b c}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**5/(b*x**6+a)/(d*x**6+c)**(1/2),x)
[Out]
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Mathematica [A] time = 0.0392107, size = 51, normalized size = 1. \[ -\frac{\tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^6}}{\sqrt{b c-a d}}\right )}{3 \sqrt{b} \sqrt{b c-a d}} \]
Antiderivative was successfully verified.
[In] Integrate[x^5/((a + b*x^6)*Sqrt[c + d*x^6]),x]
[Out]
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Maple [F] time = 0.051, size = 0, normalized size = 0. \[ \int{\frac{{x}^{5}}{b{x}^{6}+a}{\frac{1}{\sqrt{d{x}^{6}+c}}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^5/(b*x^6+a)/(d*x^6+c)^(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/((b*x^6 + a)*sqrt(d*x^6 + c)),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.218899, size = 1, normalized size = 0.02 \[ \left [\frac{\log \left (\frac{{\left (b d x^{6} + 2 \, b c - a d\right )} \sqrt{b^{2} c - a b d} - 2 \, \sqrt{d x^{6} + c}{\left (b^{2} c - a b d\right )}}{b x^{6} + a}\right )}{6 \, \sqrt{b^{2} c - a b d}}, -\frac{\arctan \left (-\frac{b c - a d}{\sqrt{d x^{6} + c} \sqrt{-b^{2} c + a b d}}\right )}{3 \, \sqrt{-b^{2} c + a b d}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^5/((b*x^6 + a)*sqrt(d*x^6 + c)),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{5}}{\left (a + b x^{6}\right ) \sqrt{c + d x^{6}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**5/(b*x**6+a)/(d*x**6+c)**(1/2),x)
[Out]
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GIAC/XCAS [A] time = 0.211612, size = 54, normalized size = 1.06 \[ \frac{\arctan \left (\frac{\sqrt{d x^{6} + c} b}{\sqrt{-b^{2} c + a b d}}\right )}{3 \, \sqrt{-b^{2} c + a b d}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^5/((b*x^6 + a)*sqrt(d*x^6 + c)),x, algorithm="giac")
[Out]