Optimal. Leaf size=74 \[ \frac{3 a \tanh ^{-1}\left (\frac{\sqrt{b}}{\sqrt{x} \sqrt{a+\frac{b}{x}}}\right )}{b^{5/2}}-\frac{3 \sqrt{a+\frac{b}{x}}}{b^2 \sqrt{x}}+\frac{2}{b x^{3/2} \sqrt{a+\frac{b}{x}}} \]
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Rubi [A] time = 0.116558, antiderivative size = 74, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.294 \[ \frac{3 a \tanh ^{-1}\left (\frac{\sqrt{b}}{\sqrt{x} \sqrt{a+\frac{b}{x}}}\right )}{b^{5/2}}-\frac{3 \sqrt{a+\frac{b}{x}}}{b^2 \sqrt{x}}+\frac{2}{b x^{3/2} \sqrt{a+\frac{b}{x}}} \]
Antiderivative was successfully verified.
[In] Int[1/((a + b/x)^(3/2)*x^(7/2)),x]
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Rubi in Sympy [A] time = 11.9811, size = 63, normalized size = 0.85 \[ \frac{3 a \operatorname{atanh}{\left (\frac{\sqrt{b}}{\sqrt{x} \sqrt{a + \frac{b}{x}}} \right )}}{b^{\frac{5}{2}}} + \frac{2}{b x^{\frac{3}{2}} \sqrt{a + \frac{b}{x}}} - \frac{3 \sqrt{a + \frac{b}{x}}}{b^{2} \sqrt{x}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(a+b/x)**(3/2)/x**(7/2),x)
[Out]
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Mathematica [A] time = 0.395971, size = 79, normalized size = 1.07 \[ \frac{-\frac{2 \sqrt{b} \sqrt{a+\frac{b}{x}} (3 a x+b)}{\sqrt{x} (a x+b)}+6 a \log \left (\sqrt{b} \sqrt{x} \sqrt{a+\frac{b}{x}}+b\right )-3 a \log (x)}{2 b^{5/2}} \]
Antiderivative was successfully verified.
[In] Integrate[1/((a + b/x)^(3/2)*x^(7/2)),x]
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Maple [A] time = 0.027, size = 61, normalized size = 0.8 \[ -{\frac{1}{ax+b}\sqrt{{\frac{ax+b}{x}}} \left ( -3\,{\it Artanh} \left ({\frac{\sqrt{ax+b}}{\sqrt{b}}} \right ) \sqrt{ax+b}xa+3\,ax\sqrt{b}+{b}^{{\frac{3}{2}}} \right ){\frac{1}{\sqrt{x}}}{b}^{-{\frac{5}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(a+b/x)^(3/2)/x^(7/2),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)^(3/2)*x^(7/2)),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.249103, size = 1, normalized size = 0.01 \[ \left [\frac{3 \, a x^{\frac{3}{2}} \sqrt{\frac{a x + b}{x}} \log \left (\frac{2 \, b \sqrt{x} \sqrt{\frac{a x + b}{x}} +{\left (a x + 2 \, b\right )} \sqrt{b}}{x}\right ) - 2 \,{\left (3 \, a x + b\right )} \sqrt{b}}{2 \, b^{\frac{5}{2}} x^{\frac{3}{2}} \sqrt{\frac{a x + b}{x}}}, -\frac{3 \, a x^{\frac{3}{2}} \sqrt{\frac{a x + b}{x}} \arctan \left (\frac{b}{\sqrt{-b} \sqrt{x} \sqrt{\frac{a x + b}{x}}}\right ) +{\left (3 \, a x + b\right )} \sqrt{-b}}{\sqrt{-b} b^{2} x^{\frac{3}{2}} \sqrt{\frac{a x + b}{x}}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((a + b/x)^(3/2)*x^(7/2)),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)**(3/2)/x**(7/2),x)
[Out]
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GIAC/XCAS [A] time = 0.25126, size = 78, normalized size = 1.05 \[ -a{\left (\frac{3 \, \arctan \left (\frac{\sqrt{a x + b}}{\sqrt{-b}}\right )}{\sqrt{-b} b^{2}} + \frac{3 \, a x + b}{{\left ({\left (a x + b\right )}^{\frac{3}{2}} - \sqrt{a x + b} b\right )} b^{2}}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((a + b/x)^(3/2)*x^(7/2)),x, algorithm="giac")
[Out]