3.2193 \(\int \frac{x}{a+b \sqrt{x}} \, dx\)

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

[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 = 0.0279938, 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, Rules used = {266, 43} \[ \frac{2 a^2 \sqrt{x}}{b^3}-\frac{2 a^3 \log \left (a+b \sqrt{x}\right )}{b^4}-\frac{a x}{b^2}+\frac{2 x^{3/2}}{3 b} \]

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

Rule 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int \frac{x}{a+b \sqrt{x}} \, dx &=2 \operatorname{Subst}\left (\int \frac{x^3}{a+b x} \, dx,x,\sqrt{x}\right )\\ &=2 \operatorname{Subst}\left (\int \left (\frac{a^2}{b^3}-\frac{a x}{b^2}+\frac{x^2}{b}-\frac{a^3}{b^3 (a+b x)}\right ) \, dx,x,\sqrt{x}\right )\\ &=\frac{2 a^2 \sqrt{x}}{b^3}-\frac{a x}{b^2}+\frac{2 x^{3/2}}{3 b}-\frac{2 a^3 \log \left (a+b \sqrt{x}\right )}{b^4}\\ \end{align*}

Mathematica [A]  time = 0.0222261, size = 51, normalized size = 1. \[ \frac{2 a^2 \sqrt{x}}{b^3}-\frac{2 a^3 \log \left (a+b \sqrt{x}\right )}{b^4}-\frac{a x}{b^2}+\frac{2 x^{3/2}}{3 b} \]

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 = 0.003, size = 44, normalized size = 0.9 \begin{align*} -{\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}}} \end{align*}

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 = 0.981197, size = 82, normalized size = 1.61 \begin{align*} -\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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(a+b*x^(1/2)),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 = 1.33225, size = 107, normalized size = 2.1 \begin{align*} -\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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(a+b*x^(1/2)),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 = 0.224569, size = 54, normalized size = 1.06 \begin{align*} \begin{cases} - \frac{2 a^{3} \log{\left (\frac{a}{b} + \sqrt{x} \right )}}{b^{4}} + \frac{2 a^{2} \sqrt{x}}{b^{3}} - \frac{a x}{b^{2}} + \frac{2 x^{\frac{3}{2}}}{3 b} & \text{for}\: b \neq 0 \\\frac{x^{2}}{2 a} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(a+b*x**(1/2)),x)

[Out]

Piecewise((-2*a**3*log(a/b + sqrt(x))/b**4 + 2*a**2*sqrt(x)/b**3 - a*x/b**2 + 2*x**(3/2)/(3*b), Ne(b, 0)), (x*
*2/(2*a), True))

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Giac [A]  time = 1.11229, size = 61, normalized size = 1.2 \begin{align*} -\frac{2 \, a^{3} \log \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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(a+b*x^(1/2)),x, algorithm="giac")

[Out]

-2*a^3*log(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