3.2211 \(\int \frac{1}{(a+b \sqrt{x})^3 x} \, dx\)

Optimal. Leaf size=53 \[ \frac{2}{a^2 \left (a+b \sqrt{x}\right )}-\frac{2 \log \left (a+b \sqrt{x}\right )}{a^3}+\frac{\log (x)}{a^3}+\frac{1}{a \left (a+b \sqrt{x}\right )^2} \]

[Out]

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

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Rubi [A]  time = 0.0307791, antiderivative size = 53, 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} \[ \frac{2}{a^2 \left (a+b \sqrt{x}\right )}-\frac{2 \log \left (a+b \sqrt{x}\right )}{a^3}+\frac{\log (x)}{a^3}+\frac{1}{a \left (a+b \sqrt{x}\right )^2} \]

Antiderivative was successfully verified.

[In]

Int[1/((a + b*Sqrt[x])^3*x),x]

[Out]

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

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

Mathematica [A]  time = 0.0508998, size = 44, normalized size = 0.83 \[ \frac{\frac{a \left (3 a+2 b \sqrt{x}\right )}{\left (a+b \sqrt{x}\right )^2}-2 \log \left (a+b \sqrt{x}\right )+\log (x)}{a^3} \]

Antiderivative was successfully verified.

[In]

Integrate[1/((a + b*Sqrt[x])^3*x),x]

[Out]

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

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Maple [A]  time = 0.008, size = 48, normalized size = 0.9 \begin{align*}{\frac{\ln \left ( x \right ) }{{a}^{3}}}-2\,{\frac{\ln \left ( a+b\sqrt{x} \right ) }{{a}^{3}}}+{\frac{1}{a} \left ( a+b\sqrt{x} \right ) ^{-2}}+2\,{\frac{1}{{a}^{2} \left ( a+b\sqrt{x} \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.984636, size = 73, normalized size = 1.38 \begin{align*} \frac{2 \, b \sqrt{x} + 3 \, a}{a^{2} b^{2} x + 2 \, a^{3} b \sqrt{x} + a^{4}} - \frac{2 \, \log \left (b \sqrt{x} + a\right )}{a^{3}} + \frac{\log \left (x\right )}{a^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B]  time = 1.35832, size = 254, normalized size = 4.79 \begin{align*} -\frac{a^{2} b^{2} x - 3 \, a^{4} + 2 \,{\left (b^{4} x^{2} - 2 \, a^{2} b^{2} x + a^{4}\right )} \log \left (b \sqrt{x} + a\right ) - 2 \,{\left (b^{4} x^{2} - 2 \, a^{2} b^{2} x + a^{4}\right )} \log \left (\sqrt{x}\right ) - 2 \,{\left (a b^{3} x - 2 \, a^{3} b\right )} \sqrt{x}}{a^{3} b^{4} x^{2} - 2 \, a^{5} b^{2} x + a^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 2.36452, size = 364, normalized size = 6.87 \begin{align*} \begin{cases} \frac{\tilde{\infty }}{x^{\frac{3}{2}}} & \text{for}\: a = 0 \wedge b = 0 \\- \frac{2}{3 b^{3} x^{\frac{3}{2}}} & \text{for}\: a = 0 \\\frac{\log{\left (x \right )}}{a^{3}} & \text{for}\: b = 0 \\\frac{a^{2} \sqrt{x} \log{\left (x \right )}}{a^{5} \sqrt{x} + 2 a^{4} b x + a^{3} b^{2} x^{\frac{3}{2}}} - \frac{2 a^{2} \sqrt{x} \log{\left (\frac{a}{b} + \sqrt{x} \right )}}{a^{5} \sqrt{x} + 2 a^{4} b x + a^{3} b^{2} x^{\frac{3}{2}}} + \frac{2 a b x \log{\left (x \right )}}{a^{5} \sqrt{x} + 2 a^{4} b x + a^{3} b^{2} x^{\frac{3}{2}}} - \frac{4 a b x \log{\left (\frac{a}{b} + \sqrt{x} \right )}}{a^{5} \sqrt{x} + 2 a^{4} b x + a^{3} b^{2} x^{\frac{3}{2}}} - \frac{4 a b x}{a^{5} \sqrt{x} + 2 a^{4} b x + a^{3} b^{2} x^{\frac{3}{2}}} + \frac{b^{2} x^{\frac{3}{2}} \log{\left (x \right )}}{a^{5} \sqrt{x} + 2 a^{4} b x + a^{3} b^{2} x^{\frac{3}{2}}} - \frac{2 b^{2} x^{\frac{3}{2}} \log{\left (\frac{a}{b} + \sqrt{x} \right )}}{a^{5} \sqrt{x} + 2 a^{4} b x + a^{3} b^{2} x^{\frac{3}{2}}} - \frac{3 b^{2} x^{\frac{3}{2}}}{a^{5} \sqrt{x} + 2 a^{4} b x + a^{3} b^{2} x^{\frac{3}{2}}} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((zoo/x**(3/2), Eq(a, 0) & Eq(b, 0)), (-2/(3*b**3*x**(3/2)), Eq(a, 0)), (log(x)/a**3, Eq(b, 0)), (a**
2*sqrt(x)*log(x)/(a**5*sqrt(x) + 2*a**4*b*x + a**3*b**2*x**(3/2)) - 2*a**2*sqrt(x)*log(a/b + sqrt(x))/(a**5*sq
rt(x) + 2*a**4*b*x + a**3*b**2*x**(3/2)) + 2*a*b*x*log(x)/(a**5*sqrt(x) + 2*a**4*b*x + a**3*b**2*x**(3/2)) - 4
*a*b*x*log(a/b + sqrt(x))/(a**5*sqrt(x) + 2*a**4*b*x + a**3*b**2*x**(3/2)) - 4*a*b*x/(a**5*sqrt(x) + 2*a**4*b*
x + a**3*b**2*x**(3/2)) + b**2*x**(3/2)*log(x)/(a**5*sqrt(x) + 2*a**4*b*x + a**3*b**2*x**(3/2)) - 2*b**2*x**(3
/2)*log(a/b + sqrt(x))/(a**5*sqrt(x) + 2*a**4*b*x + a**3*b**2*x**(3/2)) - 3*b**2*x**(3/2)/(a**5*sqrt(x) + 2*a*
*4*b*x + a**3*b**2*x**(3/2)), True))

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Giac [A]  time = 1.10212, size = 65, normalized size = 1.23 \begin{align*} -\frac{2 \, \log \left ({\left | b \sqrt{x} + a \right |}\right )}{a^{3}} + \frac{\log \left ({\left | x \right |}\right )}{a^{3}} + \frac{2 \, a b \sqrt{x} + 3 \, a^{2}}{{\left (b \sqrt{x} + a\right )}^{2} a^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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