3.2200 \(\int \frac{x^2}{(a+b \sqrt{x})^2} \, dx\)

Optimal. Leaf size=83 \[ \frac{2 a^5}{b^6 \left (a+b \sqrt{x}\right )}-\frac{8 a^3 \sqrt{x}}{b^5}+\frac{3 a^2 x}{b^4}+\frac{10 a^4 \log \left (a+b \sqrt{x}\right )}{b^6}-\frac{4 a x^{3/2}}{3 b^3}+\frac{x^2}{2 b^2} \]

[Out]

(2*a^5)/(b^6*(a + b*Sqrt[x])) - (8*a^3*Sqrt[x])/b^5 + (3*a^2*x)/b^4 - (4*a*x^(3/2))/(3*b^3) + x^2/(2*b^2) + (1
0*a^4*Log[a + b*Sqrt[x]])/b^6

________________________________________________________________________________________

Rubi [A]  time = 0.0596308, antiderivative size = 83, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {266, 43} \[ \frac{2 a^5}{b^6 \left (a+b \sqrt{x}\right )}-\frac{8 a^3 \sqrt{x}}{b^5}+\frac{3 a^2 x}{b^4}+\frac{10 a^4 \log \left (a+b \sqrt{x}\right )}{b^6}-\frac{4 a x^{3/2}}{3 b^3}+\frac{x^2}{2 b^2} \]

Antiderivative was successfully verified.

[In]

Int[x^2/(a + b*Sqrt[x])^2,x]

[Out]

(2*a^5)/(b^6*(a + b*Sqrt[x])) - (8*a^3*Sqrt[x])/b^5 + (3*a^2*x)/b^4 - (4*a*x^(3/2))/(3*b^3) + x^2/(2*b^2) + (1
0*a^4*Log[a + b*Sqrt[x]])/b^6

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^2}{\left (a+b \sqrt{x}\right )^2} \, dx &=2 \operatorname{Subst}\left (\int \frac{x^5}{(a+b x)^2} \, dx,x,\sqrt{x}\right )\\ &=2 \operatorname{Subst}\left (\int \left (-\frac{4 a^3}{b^5}+\frac{3 a^2 x}{b^4}-\frac{2 a x^2}{b^3}+\frac{x^3}{b^2}-\frac{a^5}{b^5 (a+b x)^2}+\frac{5 a^4}{b^5 (a+b x)}\right ) \, dx,x,\sqrt{x}\right )\\ &=\frac{2 a^5}{b^6 \left (a+b \sqrt{x}\right )}-\frac{8 a^3 \sqrt{x}}{b^5}+\frac{3 a^2 x}{b^4}-\frac{4 a x^{3/2}}{3 b^3}+\frac{x^2}{2 b^2}+\frac{10 a^4 \log \left (a+b \sqrt{x}\right )}{b^6}\\ \end{align*}

Mathematica [A]  time = 0.0569048, size = 78, normalized size = 0.94 \[ \frac{18 a^2 b^2 x+\frac{12 a^5}{a+b \sqrt{x}}-48 a^3 b \sqrt{x}+60 a^4 \log \left (a+b \sqrt{x}\right )-8 a b^3 x^{3/2}+3 b^4 x^2}{6 b^6} \]

Antiderivative was successfully verified.

[In]

Integrate[x^2/(a + b*Sqrt[x])^2,x]

[Out]

((12*a^5)/(a + b*Sqrt[x]) - 48*a^3*b*Sqrt[x] + 18*a^2*b^2*x - 8*a*b^3*x^(3/2) + 3*b^4*x^2 + 60*a^4*Log[a + b*S
qrt[x]])/(6*b^6)

________________________________________________________________________________________

Maple [A]  time = 0.007, size = 72, normalized size = 0.9 \begin{align*} 3\,{\frac{{a}^{2}x}{{b}^{4}}}-{\frac{4\,a}{3\,{b}^{3}}{x}^{{\frac{3}{2}}}}+{\frac{{x}^{2}}{2\,{b}^{2}}}+10\,{\frac{{a}^{4}\ln \left ( a+b\sqrt{x} \right ) }{{b}^{6}}}-8\,{\frac{{a}^{3}\sqrt{x}}{{b}^{5}}}+2\,{\frac{{a}^{5}}{{b}^{6} \left ( a+b\sqrt{x} \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3*a^2*x/b^4-4/3*a*x^(3/2)/b^3+1/2*x^2/b^2+10*a^4*ln(a+b*x^(1/2))/b^6-8*a^3*x^(1/2)/b^5+2*a^5/b^6/(a+b*x^(1/2))

________________________________________________________________________________________

Maxima [A]  time = 0.971959, size = 128, normalized size = 1.54 \begin{align*} \frac{10 \, a^{4} \log \left (b \sqrt{x} + a\right )}{b^{6}} + \frac{{\left (b \sqrt{x} + a\right )}^{4}}{2 \, b^{6}} - \frac{10 \,{\left (b \sqrt{x} + a\right )}^{3} a}{3 \, b^{6}} + \frac{10 \,{\left (b \sqrt{x} + a\right )}^{2} a^{2}}{b^{6}} - \frac{20 \,{\left (b \sqrt{x} + a\right )} a^{3}}{b^{6}} + \frac{2 \, a^{5}}{{\left (b \sqrt{x} + a\right )} b^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

10*a^4*log(b*sqrt(x) + a)/b^6 + 1/2*(b*sqrt(x) + a)^4/b^6 - 10/3*(b*sqrt(x) + a)^3*a/b^6 + 10*(b*sqrt(x) + a)^
2*a^2/b^6 - 20*(b*sqrt(x) + a)*a^3/b^6 + 2*a^5/((b*sqrt(x) + a)*b^6)

________________________________________________________________________________________

Fricas [A]  time = 1.28098, size = 230, normalized size = 2.77 \begin{align*} \frac{3 \, b^{6} x^{3} + 15 \, a^{2} b^{4} x^{2} - 18 \, a^{4} b^{2} x - 12 \, a^{6} + 60 \,{\left (a^{4} b^{2} x - a^{6}\right )} \log \left (b \sqrt{x} + a\right ) - 4 \,{\left (2 \, a b^{5} x^{2} + 10 \, a^{3} b^{3} x - 15 \, a^{5} b\right )} \sqrt{x}}{6 \,{\left (b^{8} x - a^{2} b^{6}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*(3*b^6*x^3 + 15*a^2*b^4*x^2 - 18*a^4*b^2*x - 12*a^6 + 60*(a^4*b^2*x - a^6)*log(b*sqrt(x) + a) - 4*(2*a*b^5
*x^2 + 10*a^3*b^3*x - 15*a^5*b)*sqrt(x))/(b^8*x - a^2*b^6)

________________________________________________________________________________________

Sympy [A]  time = 1.24107, size = 212, normalized size = 2.55 \begin{align*} \begin{cases} \frac{60 a^{5} \log{\left (\frac{a}{b} + \sqrt{x} \right )}}{6 a b^{6} + 6 b^{7} \sqrt{x}} + \frac{60 a^{5}}{6 a b^{6} + 6 b^{7} \sqrt{x}} + \frac{60 a^{4} b \sqrt{x} \log{\left (\frac{a}{b} + \sqrt{x} \right )}}{6 a b^{6} + 6 b^{7} \sqrt{x}} - \frac{30 a^{3} b^{2} x}{6 a b^{6} + 6 b^{7} \sqrt{x}} + \frac{10 a^{2} b^{3} x^{\frac{3}{2}}}{6 a b^{6} + 6 b^{7} \sqrt{x}} - \frac{5 a b^{4} x^{2}}{6 a b^{6} + 6 b^{7} \sqrt{x}} + \frac{3 b^{5} x^{\frac{5}{2}}}{6 a b^{6} + 6 b^{7} \sqrt{x}} & \text{for}\: b \neq 0 \\\frac{x^{3}}{3 a^{2}} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((60*a**5*log(a/b + sqrt(x))/(6*a*b**6 + 6*b**7*sqrt(x)) + 60*a**5/(6*a*b**6 + 6*b**7*sqrt(x)) + 60*a
**4*b*sqrt(x)*log(a/b + sqrt(x))/(6*a*b**6 + 6*b**7*sqrt(x)) - 30*a**3*b**2*x/(6*a*b**6 + 6*b**7*sqrt(x)) + 10
*a**2*b**3*x**(3/2)/(6*a*b**6 + 6*b**7*sqrt(x)) - 5*a*b**4*x**2/(6*a*b**6 + 6*b**7*sqrt(x)) + 3*b**5*x**(5/2)/
(6*a*b**6 + 6*b**7*sqrt(x)), Ne(b, 0)), (x**3/(3*a**2), True))

________________________________________________________________________________________

Giac [A]  time = 1.11074, size = 105, normalized size = 1.27 \begin{align*} \frac{10 \, a^{4} \log \left ({\left | b \sqrt{x} + a \right |}\right )}{b^{6}} + \frac{2 \, a^{5}}{{\left (b \sqrt{x} + a\right )} b^{6}} + \frac{3 \, b^{6} x^{2} - 8 \, a b^{5} x^{\frac{3}{2}} + 18 \, a^{2} b^{4} x - 48 \, a^{3} b^{3} \sqrt{x}}{6 \, b^{8}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

10*a^4*log(abs(b*sqrt(x) + a))/b^6 + 2*a^5/((b*sqrt(x) + a)*b^6) + 1/6*(3*b^6*x^2 - 8*a*b^5*x^(3/2) + 18*a^2*b
^4*x - 48*a^3*b^3*sqrt(x))/b^8