3.2212 \(\int \frac{1}{(a+b \sqrt{x})^3 x^2} \, dx\)

Optimal. Leaf size=85 \[ \frac{6 b^2}{a^4 \left (a+b \sqrt{x}\right )}+\frac{b^2}{a^3 \left (a+b \sqrt{x}\right )^2}-\frac{12 b^2 \log \left (a+b \sqrt{x}\right )}{a^5}+\frac{6 b^2 \log (x)}{a^5}+\frac{6 b}{a^4 \sqrt{x}}-\frac{1}{a^3 x} \]

[Out]

b^2/(a^3*(a + b*Sqrt[x])^2) + (6*b^2)/(a^4*(a + b*Sqrt[x])) - 1/(a^3*x) + (6*b)/(a^4*Sqrt[x]) - (12*b^2*Log[a
+ b*Sqrt[x]])/a^5 + (6*b^2*Log[x])/a^5

________________________________________________________________________________________

Rubi [A]  time = 0.0565994, antiderivative size = 85, 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, 44} \[ \frac{6 b^2}{a^4 \left (a+b \sqrt{x}\right )}+\frac{b^2}{a^3 \left (a+b \sqrt{x}\right )^2}-\frac{12 b^2 \log \left (a+b \sqrt{x}\right )}{a^5}+\frac{6 b^2 \log (x)}{a^5}+\frac{6 b}{a^4 \sqrt{x}}-\frac{1}{a^3 x} \]

Antiderivative was successfully verified.

[In]

Int[1/((a + b*Sqrt[x])^3*x^2),x]

[Out]

b^2/(a^3*(a + b*Sqrt[x])^2) + (6*b^2)/(a^4*(a + b*Sqrt[x])) - 1/(a^3*x) + (6*b)/(a^4*Sqrt[x]) - (12*b^2*Log[a
+ b*Sqrt[x]])/a^5 + (6*b^2*Log[x])/a^5

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 44

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}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && L
tQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int \frac{1}{\left (a+b \sqrt{x}\right )^3 x^2} \, dx &=2 \operatorname{Subst}\left (\int \frac{1}{x^3 (a+b x)^3} \, dx,x,\sqrt{x}\right )\\ &=2 \operatorname{Subst}\left (\int \left (\frac{1}{a^3 x^3}-\frac{3 b}{a^4 x^2}+\frac{6 b^2}{a^5 x}-\frac{b^3}{a^3 (a+b x)^3}-\frac{3 b^3}{a^4 (a+b x)^2}-\frac{6 b^3}{a^5 (a+b x)}\right ) \, dx,x,\sqrt{x}\right )\\ &=\frac{b^2}{a^3 \left (a+b \sqrt{x}\right )^2}+\frac{6 b^2}{a^4 \left (a+b \sqrt{x}\right )}-\frac{1}{a^3 x}+\frac{6 b}{a^4 \sqrt{x}}-\frac{12 b^2 \log \left (a+b \sqrt{x}\right )}{a^5}+\frac{6 b^2 \log (x)}{a^5}\\ \end{align*}

Mathematica [A]  time = 0.0954464, size = 77, normalized size = 0.91 \[ \frac{\frac{a \left (4 a^2 b \sqrt{x}-a^3+18 a b^2 x+12 b^3 x^{3/2}\right )}{x \left (a+b \sqrt{x}\right )^2}-12 b^2 \log \left (a+b \sqrt{x}\right )+6 b^2 \log (x)}{a^5} \]

Antiderivative was successfully verified.

[In]

Integrate[1/((a + b*Sqrt[x])^3*x^2),x]

[Out]

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

________________________________________________________________________________________

Maple [A]  time = 0.01, size = 78, normalized size = 0.9 \begin{align*} -{\frac{1}{{a}^{3}x}}+6\,{\frac{{b}^{2}\ln \left ( x \right ) }{{a}^{5}}}-12\,{\frac{{b}^{2}\ln \left ( a+b\sqrt{x} \right ) }{{a}^{5}}}+6\,{\frac{b}{{a}^{4}\sqrt{x}}}+{\frac{{b}^{2}}{{a}^{3}} \left ( a+b\sqrt{x} \right ) ^{-2}}+6\,{\frac{{b}^{2}}{{a}^{4} \left ( a+b\sqrt{x} \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^2/(a+b*x^(1/2))^3,x)

[Out]

-1/a^3/x+6*b^2*ln(x)/a^5-12*b^2*ln(a+b*x^(1/2))/a^5+6*b/a^4/x^(1/2)+b^2/a^3/(a+b*x^(1/2))^2+6*b^2/a^4/(a+b*x^(
1/2))

________________________________________________________________________________________

Maxima [A]  time = 0.981166, size = 115, normalized size = 1.35 \begin{align*} \frac{12 \, b^{3} x^{\frac{3}{2}} + 18 \, a b^{2} x + 4 \, a^{2} b \sqrt{x} - a^{3}}{a^{4} b^{2} x^{2} + 2 \, a^{5} b x^{\frac{3}{2}} + a^{6} x} - \frac{12 \, b^{2} \log \left (b \sqrt{x} + a\right )}{a^{5}} + \frac{6 \, b^{2} \log \left (x\right )}{a^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^2/(a+b*x^(1/2))^3,x, algorithm="maxima")

[Out]

(12*b^3*x^(3/2) + 18*a*b^2*x + 4*a^2*b*sqrt(x) - a^3)/(a^4*b^2*x^2 + 2*a^5*b*x^(3/2) + a^6*x) - 12*b^2*log(b*s
qrt(x) + a)/a^5 + 6*b^2*log(x)/a^5

________________________________________________________________________________________

Fricas [B]  time = 1.27856, size = 331, normalized size = 3.89 \begin{align*} -\frac{6 \, a^{2} b^{4} x^{2} - 9 \, a^{4} b^{2} x + a^{6} + 12 \,{\left (b^{6} x^{3} - 2 \, a^{2} b^{4} x^{2} + a^{4} b^{2} x\right )} \log \left (b \sqrt{x} + a\right ) - 12 \,{\left (b^{6} x^{3} - 2 \, a^{2} b^{4} x^{2} + a^{4} b^{2} x\right )} \log \left (\sqrt{x}\right ) - 2 \,{\left (6 \, a b^{5} x^{2} - 10 \, a^{3} b^{3} x + 3 \, a^{5} b\right )} \sqrt{x}}{a^{5} b^{4} x^{3} - 2 \, a^{7} b^{2} x^{2} + a^{9} x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^2/(a+b*x^(1/2))^3,x, algorithm="fricas")

[Out]

-(6*a^2*b^4*x^2 - 9*a^4*b^2*x + a^6 + 12*(b^6*x^3 - 2*a^2*b^4*x^2 + a^4*b^2*x)*log(b*sqrt(x) + a) - 12*(b^6*x^
3 - 2*a^2*b^4*x^2 + a^4*b^2*x)*log(sqrt(x)) - 2*(6*a*b^5*x^2 - 10*a^3*b^3*x + 3*a^5*b)*sqrt(x))/(a^5*b^4*x^3 -
2*a^7*b^2*x^2 + a^9*x)

________________________________________________________________________________________

Sympy [A]  time = 5.14509, size = 478, normalized size = 5.62 \begin{align*} \begin{cases} \frac{\tilde{\infty }}{x^{\frac{5}{2}}} & \text{for}\: a = 0 \wedge b = 0 \\- \frac{2}{5 b^{3} x^{\frac{5}{2}}} & \text{for}\: a = 0 \\- \frac{1}{a^{3} x} & \text{for}\: b = 0 \\- \frac{a^{4} \sqrt{x}}{a^{7} x^{\frac{3}{2}} + 2 a^{6} b x^{2} + a^{5} b^{2} x^{\frac{5}{2}}} + \frac{4 a^{3} b x}{a^{7} x^{\frac{3}{2}} + 2 a^{6} b x^{2} + a^{5} b^{2} x^{\frac{5}{2}}} + \frac{6 a^{2} b^{2} x^{\frac{3}{2}} \log{\left (x \right )}}{a^{7} x^{\frac{3}{2}} + 2 a^{6} b x^{2} + a^{5} b^{2} x^{\frac{5}{2}}} - \frac{12 a^{2} b^{2} x^{\frac{3}{2}} \log{\left (\frac{a}{b} + \sqrt{x} \right )}}{a^{7} x^{\frac{3}{2}} + 2 a^{6} b x^{2} + a^{5} b^{2} x^{\frac{5}{2}}} + \frac{12 a b^{3} x^{2} \log{\left (x \right )}}{a^{7} x^{\frac{3}{2}} + 2 a^{6} b x^{2} + a^{5} b^{2} x^{\frac{5}{2}}} - \frac{24 a b^{3} x^{2} \log{\left (\frac{a}{b} + \sqrt{x} \right )}}{a^{7} x^{\frac{3}{2}} + 2 a^{6} b x^{2} + a^{5} b^{2} x^{\frac{5}{2}}} - \frac{24 a b^{3} x^{2}}{a^{7} x^{\frac{3}{2}} + 2 a^{6} b x^{2} + a^{5} b^{2} x^{\frac{5}{2}}} + \frac{6 b^{4} x^{\frac{5}{2}} \log{\left (x \right )}}{a^{7} x^{\frac{3}{2}} + 2 a^{6} b x^{2} + a^{5} b^{2} x^{\frac{5}{2}}} - \frac{12 b^{4} x^{\frac{5}{2}} \log{\left (\frac{a}{b} + \sqrt{x} \right )}}{a^{7} x^{\frac{3}{2}} + 2 a^{6} b x^{2} + a^{5} b^{2} x^{\frac{5}{2}}} - \frac{18 b^{4} x^{\frac{5}{2}}}{a^{7} x^{\frac{3}{2}} + 2 a^{6} b x^{2} + a^{5} b^{2} x^{\frac{5}{2}}} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**2/(a+b*x**(1/2))**3,x)

[Out]

Piecewise((zoo/x**(5/2), Eq(a, 0) & Eq(b, 0)), (-2/(5*b**3*x**(5/2)), Eq(a, 0)), (-1/(a**3*x), Eq(b, 0)), (-a*
*4*sqrt(x)/(a**7*x**(3/2) + 2*a**6*b*x**2 + a**5*b**2*x**(5/2)) + 4*a**3*b*x/(a**7*x**(3/2) + 2*a**6*b*x**2 +
a**5*b**2*x**(5/2)) + 6*a**2*b**2*x**(3/2)*log(x)/(a**7*x**(3/2) + 2*a**6*b*x**2 + a**5*b**2*x**(5/2)) - 12*a*
*2*b**2*x**(3/2)*log(a/b + sqrt(x))/(a**7*x**(3/2) + 2*a**6*b*x**2 + a**5*b**2*x**(5/2)) + 12*a*b**3*x**2*log(
x)/(a**7*x**(3/2) + 2*a**6*b*x**2 + a**5*b**2*x**(5/2)) - 24*a*b**3*x**2*log(a/b + sqrt(x))/(a**7*x**(3/2) + 2
*a**6*b*x**2 + a**5*b**2*x**(5/2)) - 24*a*b**3*x**2/(a**7*x**(3/2) + 2*a**6*b*x**2 + a**5*b**2*x**(5/2)) + 6*b
**4*x**(5/2)*log(x)/(a**7*x**(3/2) + 2*a**6*b*x**2 + a**5*b**2*x**(5/2)) - 12*b**4*x**(5/2)*log(a/b + sqrt(x))
/(a**7*x**(3/2) + 2*a**6*b*x**2 + a**5*b**2*x**(5/2)) - 18*b**4*x**(5/2)/(a**7*x**(3/2) + 2*a**6*b*x**2 + a**5
*b**2*x**(5/2)), True))

________________________________________________________________________________________

Giac [A]  time = 1.11207, size = 100, normalized size = 1.18 \begin{align*} -\frac{12 \, b^{2} \log \left ({\left | b \sqrt{x} + a \right |}\right )}{a^{5}} + \frac{6 \, b^{2} \log \left ({\left | x \right |}\right )}{a^{5}} + \frac{12 \, b^{3} x^{\frac{3}{2}} + 18 \, a b^{2} x + 4 \, a^{2} b \sqrt{x} - a^{3}}{{\left (b x + a \sqrt{x}\right )}^{2} a^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^2/(a+b*x^(1/2))^3,x, algorithm="giac")

[Out]

-12*b^2*log(abs(b*sqrt(x) + a))/a^5 + 6*b^2*log(abs(x))/a^5 + (12*b^3*x^(3/2) + 18*a*b^2*x + 4*a^2*b*sqrt(x) -
a^3)/((b*x + a*sqrt(x))^2*a^4)