Optimal. Leaf size=92 \[ -\frac{5 a^3 \tanh ^{-1}\left (\frac{\sqrt{b}}{x \sqrt{a+\frac{b}{x^2}}}\right )}{16 \sqrt{b}}-\frac{5 a^2 \sqrt{a+\frac{b}{x^2}}}{16 x}-\frac{5 a \left (a+\frac{b}{x^2}\right )^{3/2}}{24 x}-\frac{\left (a+\frac{b}{x^2}\right )^{5/2}}{6 x} \]
[Out]
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Rubi [A] time = 0.103693, antiderivative size = 92, 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 \[ -\frac{5 a^3 \tanh ^{-1}\left (\frac{\sqrt{b}}{x \sqrt{a+\frac{b}{x^2}}}\right )}{16 \sqrt{b}}-\frac{5 a^2 \sqrt{a+\frac{b}{x^2}}}{16 x}-\frac{5 a \left (a+\frac{b}{x^2}\right )^{3/2}}{24 x}-\frac{\left (a+\frac{b}{x^2}\right )^{5/2}}{6 x} \]
Antiderivative was successfully verified.
[In] Int[(a + b/x^2)^(5/2)/x^2,x]
[Out]
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Rubi in Sympy [A] time = 7.67312, size = 80, normalized size = 0.87 \[ - \frac{5 a^{3} \operatorname{atanh}{\left (\frac{\sqrt{b}}{x \sqrt{a + \frac{b}{x^{2}}}} \right )}}{16 \sqrt{b}} - \frac{5 a^{2} \sqrt{a + \frac{b}{x^{2}}}}{16 x} - \frac{5 a \left (a + \frac{b}{x^{2}}\right )^{\frac{3}{2}}}{24 x} - \frac{\left (a + \frac{b}{x^{2}}\right )^{\frac{5}{2}}}{6 x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((a+b/x**2)**(5/2)/x**2,x)
[Out]
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Mathematica [A] time = 0.119784, size = 112, normalized size = 1.22 \[ \frac{\sqrt{a+\frac{b}{x^2}} \left (-15 a^3 x^6 \log \left (\sqrt{b} \sqrt{a x^2+b}+b\right )+15 a^3 x^6 \log (x)-\sqrt{b} \sqrt{a x^2+b} \left (33 a^2 x^4+26 a b x^2+8 b^2\right )\right )}{48 \sqrt{b} x^5 \sqrt{a x^2+b}} \]
Antiderivative was successfully verified.
[In] Integrate[(a + b/x^2)^(5/2)/x^2,x]
[Out]
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Maple [B] time = 0.018, size = 166, normalized size = 1.8 \[ -{\frac{1}{48\,{b}^{3}x} \left ({\frac{a{x}^{2}+b}{{x}^{2}}} \right ) ^{{\frac{5}{2}}} \left ( -3\, \left ( a{x}^{2}+b \right ) ^{5/2}{x}^{6}{a}^{3}+15\,{b}^{5/2}\ln \left ( 2\,{\frac{\sqrt{b}\sqrt{a{x}^{2}+b}+b}{x}} \right ){x}^{6}{a}^{3}+3\, \left ( a{x}^{2}+b \right ) ^{7/2}{x}^{4}{a}^{2}-5\, \left ( a{x}^{2}+b \right ) ^{3/2}{x}^{6}{a}^{3}b-15\,\sqrt{a{x}^{2}+b}{x}^{6}{a}^{3}{b}^{2}+2\, \left ( a{x}^{2}+b \right ) ^{7/2}{x}^{2}ab+8\, \left ( a{x}^{2}+b \right ) ^{7/2}{b}^{2} \right ) \left ( a{x}^{2}+b \right ) ^{-{\frac{5}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((a+b/x^2)^(5/2)/x^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((a + b/x^2)^(5/2)/x^2,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.263277, size = 1, normalized size = 0.01 \[ \left [\frac{15 \, a^{3} \sqrt{b} x^{5} \log \left (\frac{2 \, b x \sqrt{\frac{a x^{2} + b}{x^{2}}} -{\left (a x^{2} + 2 \, b\right )} \sqrt{b}}{x^{2}}\right ) - 2 \,{\left (33 \, a^{2} b x^{4} + 26 \, a b^{2} x^{2} + 8 \, b^{3}\right )} \sqrt{\frac{a x^{2} + b}{x^{2}}}}{96 \, b x^{5}}, \frac{15 \, a^{3} \sqrt{-b} x^{5} \arctan \left (\frac{\sqrt{-b}}{x \sqrt{\frac{a x^{2} + b}{x^{2}}}}\right ) -{\left (33 \, a^{2} b x^{4} + 26 \, a b^{2} x^{2} + 8 \, b^{3}\right )} \sqrt{\frac{a x^{2} + b}{x^{2}}}}{48 \, b x^{5}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((a + b/x^2)^(5/2)/x^2,x, algorithm="fricas")
[Out]
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Sympy [A] time = 18.1762, size = 99, normalized size = 1.08 \[ - \frac{11 a^{\frac{5}{2}} \sqrt{1 + \frac{b}{a x^{2}}}}{16 x} - \frac{13 a^{\frac{3}{2}} b \sqrt{1 + \frac{b}{a x^{2}}}}{24 x^{3}} - \frac{\sqrt{a} b^{2} \sqrt{1 + \frac{b}{a x^{2}}}}{6 x^{5}} - \frac{5 a^{3} \operatorname{asinh}{\left (\frac{\sqrt{b}}{\sqrt{a} x} \right )}}{16 \sqrt{b}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((a+b/x**2)**(5/2)/x**2,x)
[Out]
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GIAC/XCAS [A] time = 0.283501, size = 104, normalized size = 1.13 \[ \frac{1}{48} \, a^{3}{\left (\frac{15 \, \arctan \left (\frac{\sqrt{a x^{2} + b}}{\sqrt{-b}}\right )}{\sqrt{-b}} - \frac{33 \,{\left (a x^{2} + b\right )}^{\frac{5}{2}} - 40 \,{\left (a x^{2} + b\right )}^{\frac{3}{2}} b + 15 \, \sqrt{a x^{2} + b} b^{2}}{a^{3} x^{6}}\right )}{\rm sign}\left (x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((a + b/x^2)^(5/2)/x^2,x, algorithm="giac")
[Out]