Optimal. Leaf size=45 \[ \frac{\tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{a^{3/2} \sqrt{b}}+\frac{\sqrt{x}}{a (a+b x)} \]
[Out]
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Rubi [A] time = 0.0348669, antiderivative size = 45, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231 \[ \frac{\tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{a^{3/2} \sqrt{b}}+\frac{\sqrt{x}}{a (a+b x)} \]
Antiderivative was successfully verified.
[In] Int[1/(Sqrt[x]*(a + b*x)^2),x]
[Out]
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Rubi in Sympy [A] time = 6.6163, size = 37, normalized size = 0.82 \[ \frac{\sqrt{x}}{a \left (a + b x\right )} + \frac{\operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}} \right )}}{a^{\frac{3}{2}} \sqrt{b}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(b*x+a)**2/x**(1/2),x)
[Out]
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Mathematica [A] time = 0.030017, size = 45, normalized size = 1. \[ \frac{\tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{a^{3/2} \sqrt{b}}+\frac{\sqrt{x}}{a (a+b x)} \]
Antiderivative was successfully verified.
[In] Integrate[1/(Sqrt[x]*(a + b*x)^2),x]
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Maple [A] time = 0.012, size = 36, normalized size = 0.8 \[{\frac{1}{a \left ( bx+a \right ) }\sqrt{x}}+{\frac{1}{a}\arctan \left ({b\sqrt{x}{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(b*x+a)^2/x^(1/2),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/((b*x + a)^2*sqrt(x)),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.222124, size = 1, normalized size = 0.02 \[ \left [\frac{{\left (b x + a\right )} \log \left (\frac{2 \, a b \sqrt{x} + \sqrt{-a b}{\left (b x - a\right )}}{b x + a}\right ) + 2 \, \sqrt{-a b} \sqrt{x}}{2 \,{\left (a b x + a^{2}\right )} \sqrt{-a b}}, -\frac{{\left (b x + a\right )} \arctan \left (\frac{a}{\sqrt{a b} \sqrt{x}}\right ) - \sqrt{a b} \sqrt{x}}{{\left (a b x + a^{2}\right )} \sqrt{a b}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x + a)^2*sqrt(x)),x, algorithm="fricas")
[Out]
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Sympy [A] time = 3.45395, size = 144, normalized size = 3.2 \[ \frac{a^{\frac{3}{2}} \sqrt{x} \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}} \right )}}{a^{3} \sqrt{b} \sqrt{x} + a^{2} b^{\frac{3}{2}} x^{\frac{3}{2}}} + \frac{\sqrt{a} b x^{\frac{3}{2}} \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}} \right )}}{a^{3} \sqrt{b} \sqrt{x} + a^{2} b^{\frac{3}{2}} x^{\frac{3}{2}}} + \frac{a \sqrt{b} x}{a^{3} \sqrt{b} \sqrt{x} + a^{2} b^{\frac{3}{2}} x^{\frac{3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(b*x+a)**2/x**(1/2),x)
[Out]
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GIAC/XCAS [A] time = 0.203456, size = 47, normalized size = 1.04 \[ \frac{\arctan \left (\frac{b \sqrt{x}}{\sqrt{a b}}\right )}{\sqrt{a b} a} + \frac{\sqrt{x}}{{\left (b x + a\right )} a} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x + a)^2*sqrt(x)),x, algorithm="giac")
[Out]