Optimal. Leaf size=123 \[ -\frac{a^2 \sqrt{c+d x^8}}{8 b^2 \left (a+b x^8\right ) (b c-a d)}+\frac{a (4 b c-3 a d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^8}}{\sqrt{b c-a d}}\right )}{8 b^{5/2} (b c-a d)^{3/2}}+\frac{\sqrt{c+d x^8}}{4 b^2 d} \]
[Out]
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Rubi [A] time = 0.430144, antiderivative size = 123, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208 \[ -\frac{a^2 \sqrt{c+d x^8}}{8 b^2 \left (a+b x^8\right ) (b c-a d)}+\frac{a (4 b c-3 a d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^8}}{\sqrt{b c-a d}}\right )}{8 b^{5/2} (b c-a d)^{3/2}}+\frac{\sqrt{c+d x^8}}{4 b^2 d} \]
Antiderivative was successfully verified.
[In] Int[x^23/((a + b*x^8)^2*Sqrt[c + d*x^8]),x]
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Rubi in Sympy [A] time = 36.5668, size = 104, normalized size = 0.85 \[ \frac{a^{2} \sqrt{c + d x^{8}}}{8 b^{2} \left (a + b x^{8}\right ) \left (a d - b c\right )} - \frac{a \left (3 a d - 4 b c\right ) \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{c + d x^{8}}}{\sqrt{a d - b c}} \right )}}{8 b^{\frac{5}{2}} \left (a d - b c\right )^{\frac{3}{2}}} + \frac{\sqrt{c + d x^{8}}}{4 b^{2} d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**23/(b*x**8+a)**2/(d*x**8+c)**(1/2),x)
[Out]
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Mathematica [A] time = 0.393592, size = 107, normalized size = 0.87 \[ \frac{1}{8} \left (\frac{\sqrt{c+d x^8} \left (\frac{a^2}{\left (a+b x^8\right ) (a d-b c)}+\frac{2}{d}\right )}{b^2}+\frac{a (4 b c-3 a d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^8}}{\sqrt{b c-a d}}\right )}{b^{5/2} (b c-a d)^{3/2}}\right ) \]
Antiderivative was successfully verified.
[In] Integrate[x^23/((a + b*x^8)^2*Sqrt[c + d*x^8]),x]
[Out]
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Maple [F] time = 0.109, size = 0, normalized size = 0. \[ \int{\frac{{x}^{23}}{ \left ( b{x}^{8}+a \right ) ^{2}}{\frac{1}{\sqrt{d{x}^{8}+c}}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^23/(b*x^8+a)^2/(d*x^8+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^23/((b*x^8 + a)^2*sqrt(d*x^8 + c)),x, algorithm="maxima")
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Fricas [A] time = 0.240787, size = 1, normalized size = 0.01 \[ \left [\frac{2 \,{\left (2 \,{\left (b^{2} c - a b d\right )} x^{8} + 2 \, a b c - 3 \, a^{2} d\right )} \sqrt{d x^{8} + c} \sqrt{b^{2} c - a b d} +{\left ({\left (4 \, a b^{2} c d - 3 \, a^{2} b d^{2}\right )} x^{8} + 4 \, a^{2} b c d - 3 \, a^{3} d^{2}\right )} \log \left (\frac{{\left (b d x^{8} + 2 \, b c - a d\right )} \sqrt{b^{2} c - a b d} + 2 \, \sqrt{d x^{8} + c}{\left (b^{2} c - a b d\right )}}{b x^{8} + a}\right )}{16 \,{\left ({\left (b^{4} c d - a b^{3} d^{2}\right )} x^{8} + a b^{3} c d - a^{2} b^{2} d^{2}\right )} \sqrt{b^{2} c - a b d}}, \frac{{\left (2 \,{\left (b^{2} c - a b d\right )} x^{8} + 2 \, a b c - 3 \, a^{2} d\right )} \sqrt{d x^{8} + c} \sqrt{-b^{2} c + a b d} +{\left ({\left (4 \, a b^{2} c d - 3 \, a^{2} b d^{2}\right )} x^{8} + 4 \, a^{2} b c d - 3 \, a^{3} d^{2}\right )} \arctan \left (-\frac{b c - a d}{\sqrt{d x^{8} + c} \sqrt{-b^{2} c + a b d}}\right )}{8 \,{\left ({\left (b^{4} c d - a b^{3} d^{2}\right )} x^{8} + a b^{3} c d - a^{2} b^{2} d^{2}\right )} \sqrt{-b^{2} c + a b d}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^23/((b*x^8 + a)^2*sqrt(d*x^8 + c)),x, algorithm="fricas")
[Out]
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**23/(b*x**8+a)**2/(d*x**8+c)**(1/2),x)
[Out]
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GIAC/XCAS [A] time = 0.217098, size = 181, normalized size = 1.47 \[ -\frac{\sqrt{d x^{8} + c} a^{2} d}{8 \,{\left (b^{3} c - a b^{2} d\right )}{\left ({\left (d x^{8} + c\right )} b - b c + a d\right )}} - \frac{{\left (4 \, a b c - 3 \, a^{2} d\right )} \arctan \left (\frac{\sqrt{d x^{8} + c} b}{\sqrt{-b^{2} c + a b d}}\right )}{8 \,{\left (b^{3} c - a b^{2} d\right )} \sqrt{-b^{2} c + a b d}} + \frac{\sqrt{d x^{8} + c}}{4 \, b^{2} d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^23/((b*x^8 + a)^2*sqrt(d*x^8 + c)),x, algorithm="giac")
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