Optimal. Leaf size=80 \[ -\frac{3 b^2 \tanh ^{-1}\left (\frac{\sqrt{a+b \sqrt{x}}}{\sqrt{a}}\right )}{2 a^{5/2}}+\frac{3 b \sqrt{a+b \sqrt{x}}}{2 a^2 \sqrt{x}}-\frac{\sqrt{a+b \sqrt{x}}}{a x} \]
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Rubi [A] time = 0.104909, antiderivative size = 80, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.235 \[ -\frac{3 b^2 \tanh ^{-1}\left (\frac{\sqrt{a+b \sqrt{x}}}{\sqrt{a}}\right )}{2 a^{5/2}}+\frac{3 b \sqrt{a+b \sqrt{x}}}{2 a^2 \sqrt{x}}-\frac{\sqrt{a+b \sqrt{x}}}{a x} \]
Antiderivative was successfully verified.
[In] Int[1/(Sqrt[a + b*Sqrt[x]]*x^2),x]
[Out]
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Rubi in Sympy [A] time = 10.3309, size = 70, normalized size = 0.88 \[ - \frac{\sqrt{a + b \sqrt{x}}}{a x} + \frac{3 b \sqrt{a + b \sqrt{x}}}{2 a^{2} \sqrt{x}} - \frac{3 b^{2} \operatorname{atanh}{\left (\frac{\sqrt{a + b \sqrt{x}}}{\sqrt{a}} \right )}}{2 a^{\frac{5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/x**2/(a+b*x**(1/2))**(1/2),x)
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Mathematica [A] time = 0.0653856, size = 69, normalized size = 0.86 \[ \left (\frac{3 b}{2 a^2 \sqrt{x}}-\frac{1}{a x}\right ) \sqrt{a+b \sqrt{x}}-\frac{3 b^2 \tanh ^{-1}\left (\frac{\sqrt{a+b \sqrt{x}}}{\sqrt{a}}\right )}{2 a^{5/2}} \]
Antiderivative was successfully verified.
[In] Integrate[1/(Sqrt[a + b*Sqrt[x]]*x^2),x]
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Maple [A] time = 0.007, size = 72, normalized size = 0.9 \[ 4\,{b}^{2} \left ( -1/4\,{\frac{\sqrt{a+b\sqrt{x}}}{a{b}^{2}x}}-3/4\,{\frac{1}{a} \left ( -1/2\,{\frac{\sqrt{a+b\sqrt{x}}}{ab\sqrt{x}}}+1/2\,{\frac{1}{{a}^{3/2}}{\it Artanh} \left ({\frac{\sqrt{a+b\sqrt{x}}}{\sqrt{a}}} \right ) } \right ) } \right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/x^2/(a+b*x^(1/2))^(1/2),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(1/(sqrt(b*sqrt(x) + a)*x^2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.25914, size = 1, normalized size = 0.01 \[ \left [\frac{3 \, b^{2} x \log \left (\frac{\sqrt{a} b \sqrt{x} - 2 \, \sqrt{b \sqrt{x} + a} a + 2 \, a^{\frac{3}{2}}}{\sqrt{x}}\right ) + 2 \,{\left (3 \, \sqrt{a} b \sqrt{x} - 2 \, a^{\frac{3}{2}}\right )} \sqrt{b \sqrt{x} + a}}{4 \, a^{\frac{5}{2}} x}, \frac{3 \, b^{2} x \arctan \left (\frac{a}{\sqrt{b \sqrt{x} + a} \sqrt{-a}}\right ) +{\left (3 \, \sqrt{-a} b \sqrt{x} - 2 \, \sqrt{-a} a\right )} \sqrt{b \sqrt{x} + a}}{2 \, \sqrt{-a} a^{2} x}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(sqrt(b*sqrt(x) + a)*x^2),x, algorithm="fricas")
[Out]
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Sympy [A] time = 16.7029, size = 110, normalized size = 1.38 \[ - \frac{1}{\sqrt{b} x^{\frac{5}{4}} \sqrt{\frac{a}{b \sqrt{x}} + 1}} + \frac{\sqrt{b}}{2 a x^{\frac{3}{4}} \sqrt{\frac{a}{b \sqrt{x}} + 1}} + \frac{3 b^{\frac{3}{2}}}{2 a^{2} \sqrt [4]{x} \sqrt{\frac{a}{b \sqrt{x}} + 1}} - \frac{3 b^{2} \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{b} \sqrt [4]{x}} \right )}}{2 a^{\frac{5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/x**2/(a+b*x**(1/2))**(1/2),x)
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GIAC/XCAS [A] time = 0.256131, size = 89, normalized size = 1.11 \[ \frac{1}{2} \, b^{2}{\left (\frac{3 \, \arctan \left (\frac{\sqrt{b \sqrt{x} + a}}{\sqrt{-a}}\right )}{\sqrt{-a} a^{2}} + \frac{3 \,{\left (b \sqrt{x} + a\right )}^{\frac{3}{2}} - 5 \, \sqrt{b \sqrt{x} + a} a}{a^{2} b^{2} x}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(sqrt(b*sqrt(x) + a)*x^2),x, algorithm="giac")
[Out]