Optimal. Leaf size=72 \[ a^{5/2} \left (-\tanh ^{-1}\left (\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right )\right )+a^2 \sqrt{a+b x^2}+\frac{1}{3} a \left (a+b x^2\right )^{3/2}+\frac{1}{5} \left (a+b x^2\right )^{5/2} \]
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Rubi [A] time = 0.126076, antiderivative size = 72, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267 \[ a^{5/2} \left (-\tanh ^{-1}\left (\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right )\right )+a^2 \sqrt{a+b x^2}+\frac{1}{3} a \left (a+b x^2\right )^{3/2}+\frac{1}{5} \left (a+b x^2\right )^{5/2} \]
Antiderivative was successfully verified.
[In] Int[(a + b*x^2)^(5/2)/x,x]
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Rubi in Sympy [A] time = 11.9501, size = 60, normalized size = 0.83 \[ - a^{\frac{5}{2}} \operatorname{atanh}{\left (\frac{\sqrt{a + b x^{2}}}{\sqrt{a}} \right )} + a^{2} \sqrt{a + b x^{2}} + \frac{a \left (a + b x^{2}\right )^{\frac{3}{2}}}{3} + \frac{\left (a + b x^{2}\right )^{\frac{5}{2}}}{5} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b*x**2+a)**(5/2)/x,x)
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Mathematica [A] time = 0.074612, size = 72, normalized size = 1. \[ -a^{5/2} \log \left (\sqrt{a} \sqrt{a+b x^2}+a\right )+a^{5/2} \log (x)+\frac{1}{15} \sqrt{a+b x^2} \left (23 a^2+11 a b x^2+3 b^2 x^4\right ) \]
Antiderivative was successfully verified.
[In] Integrate[(a + b*x^2)^(5/2)/x,x]
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Maple [A] time = 0.007, size = 66, normalized size = 0.9 \[{\frac{1}{5} \left ( b{x}^{2}+a \right ) ^{{\frac{5}{2}}}}+{\frac{a}{3} \left ( b{x}^{2}+a \right ) ^{{\frac{3}{2}}}}-{a}^{{\frac{5}{2}}}\ln \left ({\frac{1}{x} \left ( 2\,a+2\,\sqrt{a}\sqrt{b{x}^{2}+a} \right ) } \right ) +{a}^{2}\sqrt{b{x}^{2}+a} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b*x^2+a)^(5/2)/x,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((b*x^2 + a)^(5/2)/x,x, algorithm="maxima")
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Fricas [A] time = 0.24829, size = 1, normalized size = 0.01 \[ \left [\frac{1}{2} \, a^{\frac{5}{2}} \log \left (-\frac{b x^{2} - 2 \, \sqrt{b x^{2} + a} \sqrt{a} + 2 \, a}{x^{2}}\right ) + \frac{1}{15} \,{\left (3 \, b^{2} x^{4} + 11 \, a b x^{2} + 23 \, a^{2}\right )} \sqrt{b x^{2} + a}, -\sqrt{-a} a^{2} \arctan \left (\frac{a}{\sqrt{b x^{2} + a} \sqrt{-a}}\right ) + \frac{1}{15} \,{\left (3 \, b^{2} x^{4} + 11 \, a b x^{2} + 23 \, a^{2}\right )} \sqrt{b x^{2} + a}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^2 + a)^(5/2)/x,x, algorithm="fricas")
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Sympy [A] time = 11.8058, size = 105, normalized size = 1.46 \[ \frac{23 a^{\frac{5}{2}} \sqrt{1 + \frac{b x^{2}}{a}}}{15} + \frac{a^{\frac{5}{2}} \log{\left (\frac{b x^{2}}{a} \right )}}{2} - a^{\frac{5}{2}} \log{\left (\sqrt{1 + \frac{b x^{2}}{a}} + 1 \right )} + \frac{11 a^{\frac{3}{2}} b x^{2} \sqrt{1 + \frac{b x^{2}}{a}}}{15} + \frac{\sqrt{a} b^{2} x^{4} \sqrt{1 + \frac{b x^{2}}{a}}}{5} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x**2+a)**(5/2)/x,x)
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GIAC/XCAS [A] time = 0.210943, size = 84, normalized size = 1.17 \[ \frac{a^{3} \arctan \left (\frac{\sqrt{b x^{2} + a}}{\sqrt{-a}}\right )}{\sqrt{-a}} + \frac{1}{5} \,{\left (b x^{2} + a\right )}^{\frac{5}{2}} + \frac{1}{3} \,{\left (b x^{2} + a\right )}^{\frac{3}{2}} a + \sqrt{b x^{2} + a} a^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^2 + a)^(5/2)/x,x, algorithm="giac")
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