Optimal. Leaf size=129 \[ -\frac{35 a^2 \tanh ^{-1}\left (\frac{\sqrt{b}}{\sqrt{x} \sqrt{a+\frac{b}{x}}}\right )}{4 b^{9/2}}+\frac{35 a \sqrt{a+\frac{b}{x}}}{4 b^4 \sqrt{x}}-\frac{35 \sqrt{a+\frac{b}{x}}}{6 b^3 x^{3/2}}+\frac{14}{3 b^2 x^{5/2} \sqrt{a+\frac{b}{x}}}+\frac{2}{3 b x^{7/2} \left (a+\frac{b}{x}\right )^{3/2}} \]
[Out]
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Rubi [A] time = 0.198109, antiderivative size = 129, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.294 \[ -\frac{35 a^2 \tanh ^{-1}\left (\frac{\sqrt{b}}{\sqrt{x} \sqrt{a+\frac{b}{x}}}\right )}{4 b^{9/2}}+\frac{35 a \sqrt{a+\frac{b}{x}}}{4 b^4 \sqrt{x}}-\frac{35 \sqrt{a+\frac{b}{x}}}{6 b^3 x^{3/2}}+\frac{14}{3 b^2 x^{5/2} \sqrt{a+\frac{b}{x}}}+\frac{2}{3 b x^{7/2} \left (a+\frac{b}{x}\right )^{3/2}} \]
Antiderivative was successfully verified.
[In] Int[1/((a + b/x)^(5/2)*x^(11/2)),x]
[Out]
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Rubi in Sympy [A] time = 20.9956, size = 112, normalized size = 0.87 \[ - \frac{35 a^{2} \operatorname{atanh}{\left (\frac{\sqrt{b}}{\sqrt{x} \sqrt{a + \frac{b}{x}}} \right )}}{4 b^{\frac{9}{2}}} + \frac{35 a \sqrt{a + \frac{b}{x}}}{4 b^{4} \sqrt{x}} + \frac{2}{3 b x^{\frac{7}{2}} \left (a + \frac{b}{x}\right )^{\frac{3}{2}}} + \frac{14}{3 b^{2} x^{\frac{5}{2}} \sqrt{a + \frac{b}{x}}} - \frac{35 \sqrt{a + \frac{b}{x}}}{6 b^{3} x^{\frac{3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(a+b/x)**(5/2)/x**(11/2),x)
[Out]
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Mathematica [A] time = 0.32784, size = 107, normalized size = 0.83 \[ \frac{-210 a^2 \log \left (\sqrt{b} \sqrt{x} \sqrt{a+\frac{b}{x}}+b\right )+105 a^2 \log (x)+\frac{2 \sqrt{b} \sqrt{a+\frac{b}{x}} \left (105 a^3 x^3+140 a^2 b x^2+21 a b^2 x-6 b^3\right )}{x^{3/2} (a x+b)^2}}{24 b^{9/2}} \]
Antiderivative was successfully verified.
[In] Integrate[1/((a + b/x)^(5/2)*x^(11/2)),x]
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Maple [A] time = 0.031, size = 117, normalized size = 0.9 \[ -{\frac{1}{12\, \left ( ax+b \right ) ^{2}}\sqrt{{\frac{ax+b}{x}}} \left ( 105\,{\it Artanh} \left ({\frac{\sqrt{ax+b}}{\sqrt{b}}} \right ) \sqrt{ax+b}{x}^{3}{a}^{3}+105\,{\it Artanh} \left ({\frac{\sqrt{ax+b}}{\sqrt{b}}} \right ){x}^{2}{a}^{2}b\sqrt{ax+b}-105\,{x}^{3}{a}^{3}\sqrt{b}-140\,{b}^{3/2}{x}^{2}{a}^{2}-21\,{b}^{5/2}xa+6\,{b}^{7/2} \right ){x}^{-{\frac{3}{2}}}{b}^{-{\frac{9}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(a+b/x)^(5/2)/x^(11/2),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(1/((a + b/x)^(5/2)*x^(11/2)),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.254303, size = 1, normalized size = 0.01 \[ \left [\frac{105 \,{\left (a^{3} x^{3} + a^{2} b x^{2}\right )} \sqrt{x} \sqrt{\frac{a x + b}{x}} \log \left (-\frac{2 \, b \sqrt{x} \sqrt{\frac{a x + b}{x}} -{\left (a x + 2 \, b\right )} \sqrt{b}}{x}\right ) + 2 \,{\left (105 \, a^{3} x^{3} + 140 \, a^{2} b x^{2} + 21 \, a b^{2} x - 6 \, b^{3}\right )} \sqrt{b}}{24 \,{\left (a b^{4} x^{3} + b^{5} x^{2}\right )} \sqrt{b} \sqrt{x} \sqrt{\frac{a x + b}{x}}}, \frac{105 \,{\left (a^{3} x^{3} + a^{2} b x^{2}\right )} \sqrt{x} \sqrt{\frac{a x + b}{x}} \arctan \left (\frac{b}{\sqrt{-b} \sqrt{x} \sqrt{\frac{a x + b}{x}}}\right ) +{\left (105 \, a^{3} x^{3} + 140 \, a^{2} b x^{2} + 21 \, a b^{2} x - 6 \, b^{3}\right )} \sqrt{-b}}{12 \,{\left (a b^{4} x^{3} + b^{5} x^{2}\right )} \sqrt{-b} \sqrt{x} \sqrt{\frac{a x + b}{x}}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((a + b/x)^(5/2)*x^(11/2)),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(1/(a+b/x)**(5/2)/x**(11/2),x)
[Out]
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GIAC/XCAS [A] time = 0.268367, size = 109, normalized size = 0.84 \[ \frac{1}{12} \, a^{2}{\left (\frac{105 \, \arctan \left (\frac{\sqrt{a x + b}}{\sqrt{-b}}\right )}{\sqrt{-b} b^{4}} + \frac{8 \,{\left (9 \, a x + 10 \, b\right )}}{{\left (a x + b\right )}^{\frac{3}{2}} b^{4}} + \frac{3 \,{\left (11 \,{\left (a x + b\right )}^{\frac{3}{2}} - 13 \, \sqrt{a x + b} b\right )}}{a^{2} b^{4} x^{2}}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((a + b/x)^(5/2)*x^(11/2)),x, algorithm="giac")
[Out]