3.2201 \(\int \frac{x}{(a+b \sqrt{x})^2} \, dx\)

Optimal. Leaf size=54 \[ \frac{2 a^3}{b^4 \left (a+b \sqrt{x}\right )}+\frac{6 a^2 \log \left (a+b \sqrt{x}\right )}{b^4}-\frac{4 a \sqrt{x}}{b^3}+\frac{x}{b^2} \]

[Out]

(2*a^3)/(b^4*(a + b*Sqrt[x])) - (4*a*Sqrt[x])/b^3 + x/b^2 + (6*a^2*Log[a + b*Sqrt[x]])/b^4

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Rubi [A]  time = 0.0363245, antiderivative size = 54, 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} \[ \frac{2 a^3}{b^4 \left (a+b \sqrt{x}\right )}+\frac{6 a^2 \log \left (a+b \sqrt{x}\right )}{b^4}-\frac{4 a \sqrt{x}}{b^3}+\frac{x}{b^2} \]

Antiderivative was successfully verified.

[In]

Int[x/(a + b*Sqrt[x])^2,x]

[Out]

(2*a^3)/(b^4*(a + b*Sqrt[x])) - (4*a*Sqrt[x])/b^3 + x/b^2 + (6*a^2*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}{\left (a+b \sqrt{x}\right )^2} \, dx &=2 \operatorname{Subst}\left (\int \frac{x^3}{(a+b x)^2} \, dx,x,\sqrt{x}\right )\\ &=2 \operatorname{Subst}\left (\int \left (-\frac{2 a}{b^3}+\frac{x}{b^2}-\frac{a^3}{b^3 (a+b x)^2}+\frac{3 a^2}{b^3 (a+b x)}\right ) \, dx,x,\sqrt{x}\right )\\ &=\frac{2 a^3}{b^4 \left (a+b \sqrt{x}\right )}-\frac{4 a \sqrt{x}}{b^3}+\frac{x}{b^2}+\frac{6 a^2 \log \left (a+b \sqrt{x}\right )}{b^4}\\ \end{align*}

Mathematica [A]  time = 0.0368101, size = 50, normalized size = 0.93 \[ \frac{\frac{2 a^3}{a+b \sqrt{x}}+6 a^2 \log \left (a+b \sqrt{x}\right )-4 a b \sqrt{x}+b^2 x}{b^4} \]

Antiderivative was successfully verified.

[In]

Integrate[x/(a + b*Sqrt[x])^2,x]

[Out]

((2*a^3)/(a + b*Sqrt[x]) - 4*a*b*Sqrt[x] + b^2*x + 6*a^2*Log[a + b*Sqrt[x]])/b^4

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Maple [A]  time = 0.006, size = 49, normalized size = 0.9 \begin{align*}{\frac{x}{{b}^{2}}}+6\,{\frac{{a}^{2}\ln \left ( a+b\sqrt{x} \right ) }{{b}^{4}}}-4\,{\frac{a\sqrt{x}}{{b}^{3}}}+2\,{\frac{{a}^{3}}{{b}^{4} \left ( a+b\sqrt{x} \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x/(a+b*x^(1/2))^2,x)

[Out]

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

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Maxima [A]  time = 0.978247, size = 81, normalized size = 1.5 \begin{align*} \frac{6 \, a^{2} \log \left (b \sqrt{x} + a\right )}{b^{4}} + \frac{{\left (b \sqrt{x} + a\right )}^{2}}{b^{4}} - \frac{6 \,{\left (b \sqrt{x} + a\right )} a}{b^{4}} + \frac{2 \, a^{3}}{{\left (b \sqrt{x} + a\right )} b^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.27179, size = 167, normalized size = 3.09 \begin{align*} \frac{b^{4} x^{2} - a^{2} b^{2} x - 2 \, a^{4} + 6 \,{\left (a^{2} b^{2} x - a^{4}\right )} \log \left (b \sqrt{x} + a\right ) - 2 \,{\left (2 \, a b^{3} x - 3 \, a^{3} b\right )} \sqrt{x}}{b^{6} x - a^{2} b^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.618201, size = 134, normalized size = 2.48 \begin{align*} \begin{cases} \frac{6 a^{3} \log{\left (\frac{a}{b} + \sqrt{x} \right )}}{a b^{4} + b^{5} \sqrt{x}} + \frac{6 a^{3}}{a b^{4} + b^{5} \sqrt{x}} + \frac{6 a^{2} b \sqrt{x} \log{\left (\frac{a}{b} + \sqrt{x} \right )}}{a b^{4} + b^{5} \sqrt{x}} - \frac{3 a b^{2} x}{a b^{4} + b^{5} \sqrt{x}} + \frac{b^{3} x^{\frac{3}{2}}}{a b^{4} + b^{5} \sqrt{x}} & \text{for}\: b \neq 0 \\\frac{x^{2}}{2 a^{2}} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((6*a**3*log(a/b + sqrt(x))/(a*b**4 + b**5*sqrt(x)) + 6*a**3/(a*b**4 + b**5*sqrt(x)) + 6*a**2*b*sqrt(
x)*log(a/b + sqrt(x))/(a*b**4 + b**5*sqrt(x)) - 3*a*b**2*x/(a*b**4 + b**5*sqrt(x)) + b**3*x**(3/2)/(a*b**4 + b
**5*sqrt(x)), Ne(b, 0)), (x**2/(2*a**2), True))

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Giac [A]  time = 1.12539, size = 70, normalized size = 1.3 \begin{align*} \frac{6 \, a^{2} \log \left ({\left | b \sqrt{x} + a \right |}\right )}{b^{4}} + \frac{2 \, a^{3}}{{\left (b \sqrt{x} + a\right )} b^{4}} + \frac{b^{2} x - 4 \, a b \sqrt{x}}{b^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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