3.459 \(\int \frac{1}{\sqrt{x} (a+b x)^2} \, dx\)

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]

Sqrt[x]/(a*(a + b*x)) + ArcTan[(Sqrt[b]*Sqrt[x])/Sqrt[a]]/(a^(3/2)*Sqrt[b])

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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]

Sqrt[x]/(a*(a + b*x)) + ArcTan[(Sqrt[b]*Sqrt[x])/Sqrt[a]]/(a^(3/2)*Sqrt[b])

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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]

sqrt(x)/(a*(a + b*x)) + atan(sqrt(b)*sqrt(x)/sqrt(a))/(a**(3/2)*sqrt(b))

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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]

[Out]

Sqrt[x]/(a*(a + b*x)) + ArcTan[(Sqrt[b]*Sqrt[x])/Sqrt[a]]/(a^(3/2)*Sqrt[b])

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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]

x^(1/2)/a/(b*x+a)+1/a/(a*b)^(1/2)*arctan(x^(1/2)*b/(a*b)^(1/2))

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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]

Exception raised: ValueError

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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]

[1/2*((b*x + a)*log((2*a*b*sqrt(x) + sqrt(-a*b)*(b*x - a))/(b*x + a)) + 2*sqrt(-
a*b)*sqrt(x))/((a*b*x + a^2)*sqrt(-a*b)), -((b*x + a)*arctan(a/(sqrt(a*b)*sqrt(x
))) - sqrt(a*b)*sqrt(x))/((a*b*x + a^2)*sqrt(a*b))]

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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]

a**(3/2)*sqrt(x)*atan(sqrt(b)*sqrt(x)/sqrt(a))/(a**3*sqrt(b)*sqrt(x) + a**2*b**(
3/2)*x**(3/2)) + sqrt(a)*b*x**(3/2)*atan(sqrt(b)*sqrt(x)/sqrt(a))/(a**3*sqrt(b)*
sqrt(x) + a**2*b**(3/2)*x**(3/2)) + a*sqrt(b)*x/(a**3*sqrt(b)*sqrt(x) + a**2*b**
(3/2)*x**(3/2))

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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]

arctan(b*sqrt(x)/sqrt(a*b))/(sqrt(a*b)*a) + sqrt(x)/((b*x + a)*a)