Optimal. Leaf size=168 \[ \frac{x \left (a-b x^2\right )}{8 a^2 \left (a+b x^2\right )^{3/2} \sqrt{a^2-b^2 x^4}}+\frac{9 x \left (a-b x^2\right )}{32 a^3 \sqrt{a+b x^2} \sqrt{a^2-b^2 x^4}}+\frac{19 \sqrt{a+b x^2} \sqrt{a-b x^2} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{b} x}{\sqrt{a-b x^2}}\right )}{32 \sqrt{2} a^3 \sqrt{b} \sqrt{a^2-b^2 x^4}} \]
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Rubi [A] time = 0.247323, antiderivative size = 168, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214 \[ \frac{x \left (a-b x^2\right )}{8 a^2 \left (a+b x^2\right )^{3/2} \sqrt{a^2-b^2 x^4}}+\frac{9 x \left (a-b x^2\right )}{32 a^3 \sqrt{a+b x^2} \sqrt{a^2-b^2 x^4}}+\frac{19 \sqrt{a+b x^2} \sqrt{a-b x^2} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{b} x}{\sqrt{a-b x^2}}\right )}{32 \sqrt{2} a^3 \sqrt{b} \sqrt{a^2-b^2 x^4}} \]
Antiderivative was successfully verified.
[In] Int[1/((a + b*x^2)^(5/2)*Sqrt[a^2 - b^2*x^4]),x]
[Out]
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Rubi in Sympy [A] time = 43.8884, size = 136, normalized size = 0.81 \[ \frac{x \sqrt{a^{2} - b^{2} x^{4}}}{8 a^{2} \left (a + b x^{2}\right )^{\frac{5}{2}}} + \frac{9 x \sqrt{a^{2} - b^{2} x^{4}}}{32 a^{3} \left (a + b x^{2}\right )^{\frac{3}{2}}} + \frac{19 \sqrt{2} \sqrt{a^{2} - b^{2} x^{4}} \operatorname{atan}{\left (\frac{\sqrt{2} \sqrt{b} x}{\sqrt{a - b x^{2}}} \right )}}{64 a^{3} \sqrt{b} \sqrt{a - b x^{2}} \sqrt{a + b x^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(b*x**2+a)**(5/2)/(-b**2*x**4+a**2)**(1/2),x)
[Out]
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Mathematica [A] time = 0.123945, size = 123, normalized size = 0.73 \[ \frac{\sqrt{a^2-b^2 x^4} \left (2 \sqrt{b} x \sqrt{a-b x^2} \left (13 a+9 b x^2\right )+19 \sqrt{2} \left (a+b x^2\right )^2 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{b} x}{\sqrt{a-b x^2}}\right )\right )}{64 a^3 \sqrt{b} \sqrt{a-b x^2} \left (a+b x^2\right )^{5/2}} \]
Antiderivative was successfully verified.
[In] Integrate[1/((a + b*x^2)^(5/2)*Sqrt[a^2 - b^2*x^4]),x]
[Out]
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Maple [B] time = 0.072, size = 729, normalized size = 4.3 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(b*x^2+a)^(5/2)/(-b^2*x^4+a^2)^(1/2),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{-b^{2} x^{4} + a^{2}}{\left (b x^{2} + a\right )}^{\frac{5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(sqrt(-b^2*x^4 + a^2)*(b*x^2 + a)^(5/2)),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.279335, size = 1, normalized size = 0.01 \[ \left [\frac{\sqrt{2}{\left (2 \, \sqrt{2} \sqrt{-b^{2} x^{4} + a^{2}}{\left (9 \, b x^{3} + 13 \, a x\right )} \sqrt{b x^{2} + a} \sqrt{-b} + 19 \,{\left (b^{3} x^{6} + 3 \, a b^{2} x^{4} + 3 \, a^{2} b x^{2} + a^{3}\right )} \log \left (-\frac{4 \, \sqrt{-b^{2} x^{4} + a^{2}} \sqrt{b x^{2} + a} b x + \sqrt{2}{\left (3 \, b^{2} x^{4} + 2 \, a b x^{2} - a^{2}\right )} \sqrt{-b}}{b^{2} x^{4} + 2 \, a b x^{2} + a^{2}}\right )\right )}}{128 \,{\left (a^{3} b^{3} x^{6} + 3 \, a^{4} b^{2} x^{4} + 3 \, a^{5} b x^{2} + a^{6}\right )} \sqrt{-b}}, \frac{\sqrt{2}{\left (\sqrt{2} \sqrt{-b^{2} x^{4} + a^{2}}{\left (9 \, b x^{3} + 13 \, a x\right )} \sqrt{b x^{2} + a} \sqrt{b} - 19 \,{\left (b^{3} x^{6} + 3 \, a b^{2} x^{4} + 3 \, a^{2} b x^{2} + a^{3}\right )} \arctan \left (\frac{\sqrt{2} \sqrt{-b^{2} x^{4} + a^{2}} \sqrt{b x^{2} + a} \sqrt{b}}{2 \,{\left (b^{2} x^{3} + a b x\right )}}\right )\right )}}{64 \,{\left (a^{3} b^{3} x^{6} + 3 \, a^{4} b^{2} x^{4} + 3 \, a^{5} b x^{2} + a^{6}\right )} \sqrt{b}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(sqrt(-b^2*x^4 + a^2)*(b*x^2 + a)^(5/2)),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{- \left (- a + b x^{2}\right ) \left (a + b x^{2}\right )} \left (a + b x^{2}\right )^{\frac{5}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(b*x**2+a)**(5/2)/(-b**2*x**4+a**2)**(1/2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{-b^{2} x^{4} + a^{2}}{\left (b x^{2} + a\right )}^{\frac{5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(sqrt(-b^2*x^4 + a^2)*(b*x^2 + a)^(5/2)),x, algorithm="giac")
[Out]