3.2191 \(\int \frac{x}{a+b \sqrt{x}} \, dx\)

Optimal. Leaf size=51 \[ -\frac{2 a^3 \log \left (a+b \sqrt{x}\right )}{b^4}+\frac{2 a^2 \sqrt{x}}{b^3}-\frac{a x}{b^2}+\frac{2 x^{3/2}}{3 b} \]

[Out]

(2*a^2*Sqrt[x])/b^3 - (a*x)/b^2 + (2*x^(3/2))/(3*b) - (2*a^3*Log[a + b*Sqrt[x]])
/b^4

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Rubi [A]  time = 0.0762548, antiderivative size = 51, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ -\frac{2 a^3 \log \left (a+b \sqrt{x}\right )}{b^4}+\frac{2 a^2 \sqrt{x}}{b^3}-\frac{a x}{b^2}+\frac{2 x^{3/2}}{3 b} \]

Antiderivative was successfully verified.

[In]  Int[x/(a + b*Sqrt[x]),x]

[Out]

(2*a^2*Sqrt[x])/b^3 - (a*x)/b^2 + (2*x^(3/2))/(3*b) - (2*a^3*Log[a + b*Sqrt[x]])
/b^4

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \frac{2 a^{3} \log{\left (a + b \sqrt{x} \right )}}{b^{4}} - \frac{2 a \int ^{\sqrt{x}} x\, dx}{b^{2}} + \frac{2 x^{\frac{3}{2}}}{3 b} + \frac{2 \int ^{\sqrt{x}} a^{2}\, dx}{b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x/(a+b*x**(1/2)),x)

[Out]

-2*a**3*log(a + b*sqrt(x))/b**4 - 2*a*Integral(x, (x, sqrt(x)))/b**2 + 2*x**(3/2
)/(3*b) + 2*Integral(a**2, (x, sqrt(x)))/b**3

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Mathematica [A]  time = 0.0162078, size = 51, normalized size = 1. \[ -\frac{2 a^3 \log \left (a+b \sqrt{x}\right )}{b^4}+\frac{2 a^2 \sqrt{x}}{b^3}-\frac{a x}{b^2}+\frac{2 x^{3/2}}{3 b} \]

Antiderivative was successfully verified.

[In]  Integrate[x/(a + b*Sqrt[x]),x]

[Out]

(2*a^2*Sqrt[x])/b^3 - (a*x)/b^2 + (2*x^(3/2))/(3*b) - (2*a^3*Log[a + b*Sqrt[x]])
/b^4

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Maple [A]  time = 0.004, size = 44, normalized size = 0.9 \[ -{\frac{ax}{{b}^{2}}}+{\frac{2}{3\,b}{x}^{{\frac{3}{2}}}}-2\,{\frac{{a}^{3}\ln \left ( a+b\sqrt{x} \right ) }{{b}^{4}}}+2\,{\frac{{a}^{2}\sqrt{x}}{{b}^{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x/(a+b*x^(1/2)),x)

[Out]

-a*x/b^2+2/3*x^(3/2)/b-2*a^3*ln(a+b*x^(1/2))/b^4+2*a^2*x^(1/2)/b^3

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Maxima [A]  time = 1.44189, size = 82, normalized size = 1.61 \[ -\frac{2 \, a^{3} \log \left (b \sqrt{x} + a\right )}{b^{4}} + \frac{2 \,{\left (b \sqrt{x} + a\right )}^{3}}{3 \, b^{4}} - \frac{3 \,{\left (b \sqrt{x} + a\right )}^{2} a}{b^{4}} + \frac{6 \,{\left (b \sqrt{x} + a\right )} a^{2}}{b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x/(b*sqrt(x) + a),x, algorithm="maxima")

[Out]

-2*a^3*log(b*sqrt(x) + a)/b^4 + 2/3*(b*sqrt(x) + a)^3/b^4 - 3*(b*sqrt(x) + a)^2*
a/b^4 + 6*(b*sqrt(x) + a)*a^2/b^4

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Fricas [A]  time = 0.237705, size = 58, normalized size = 1.14 \[ -\frac{3 \, a b^{2} x + 6 \, a^{3} \log \left (b \sqrt{x} + a\right ) - 2 \,{\left (b^{3} x + 3 \, a^{2} b\right )} \sqrt{x}}{3 \, b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x/(b*sqrt(x) + a),x, algorithm="fricas")

[Out]

-1/3*(3*a*b^2*x + 6*a^3*log(b*sqrt(x) + a) - 2*(b^3*x + 3*a^2*b)*sqrt(x))/b^4

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Sympy [A]  time = 12.6424, size = 49, normalized size = 0.96 \[ - \frac{2 a^{3} \log{\left (1 + \frac{b \sqrt{x}}{a} \right )}}{b^{4}} + \frac{2 a^{2} \sqrt{x}}{b^{3}} - \frac{a x}{b^{2}} + \frac{2 x^{\frac{3}{2}}}{3 b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x/(a+b*x**(1/2)),x)

[Out]

-2*a**3*log(1 + b*sqrt(x)/a)/b**4 + 2*a**2*sqrt(x)/b**3 - a*x/b**2 + 2*x**(3/2)/
(3*b)

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GIAC/XCAS [A]  time = 0.220052, size = 61, normalized size = 1.2 \[ -\frac{2 \, a^{3}{\rm ln}\left ({\left | b \sqrt{x} + a \right |}\right )}{b^{4}} + \frac{2 \, b^{2} x^{\frac{3}{2}} - 3 \, a b x + 6 \, a^{2} \sqrt{x}}{3 \, b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x/(b*sqrt(x) + a),x, algorithm="giac")

[Out]

-2*a^3*ln(abs(b*sqrt(x) + a))/b^4 + 1/3*(2*b^2*x^(3/2) - 3*a*b*x + 6*a^2*sqrt(x)
)/b^3