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

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

[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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Rubi [A]  time = 0.03, antiderivative size = 53, normalized size of antiderivative = 1.00, 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 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])

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]]

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*}

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Mathematica [A]  time = 0.06, 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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fricas [B]  time = 0.99, size = 115, normalized size = 2.17 \[ -\frac {a^{2} b^{2} x - 3 \, a^{4} + 2 \, {\left (b^{4} x^{2} - 2 \, a^{2} b^{2} x + a^{4}\right )} \log \left (b \sqrt {x} + a\right ) - 2 \, {\left (b^{4} x^{2} - 2 \, a^{2} b^{2} x + a^{4}\right )} \log \left (\sqrt {x}\right ) - 2 \, {\left (a b^{3} x - 2 \, a^{3} b\right )} \sqrt {x}}{a^{3} b^{4} x^{2} - 2 \, a^{5} b^{2} x + a^{7}} \]

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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giac [A]  time = 0.16, size = 48, normalized size = 0.91 \[ -\frac {2 \, \log \left ({\left | b \sqrt {x} + a \right |}\right )}{a^{3}} + \frac {\log \left ({\left | x \right |}\right )}{a^{3}} + \frac {2 \, a b \sqrt {x} + 3 \, a^{2}}{{\left (b \sqrt {x} + a\right )}^{2} a^{3}} \]

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)

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maple [A]  time = 0.01, size = 48, normalized size = 0.91 \[ \frac {1}{\left (b \sqrt {x}+a \right )^{2} a}+\frac {2}{\left (b \sqrt {x}+a \right ) a^{2}}+\frac {\ln \relax (x )}{a^{3}}-\frac {2 \ln \left (b \sqrt {x}+a \right )}{a^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.89, size = 54, normalized size = 1.02 \[ \frac {2 \, b \sqrt {x} + 3 \, a}{a^{2} b^{2} x + 2 \, a^{3} b \sqrt {x} + a^{4}} - \frac {2 \, \log \left (b \sqrt {x} + a\right )}{a^{3}} + \frac {\log \relax (x)}{a^{3}} \]

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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mupad [B]  time = 0.07, size = 52, normalized size = 0.98 \[ \frac {\frac {3}{a}+\frac {2\,b\,\sqrt {x}}{a^2}}{b^2\,x+a^2+2\,a\,b\,\sqrt {x}}-\frac {4\,\mathrm {atanh}\left (\frac {2\,b\,\sqrt {x}}{a}+1\right )}{a^3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x*(a + b*x^(1/2))^3),x)

[Out]

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

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sympy [A]  time = 2.59, size = 364, normalized size = 6.87 \[ \begin {cases} \frac {\tilde {\infty }}{x^{\frac {3}{2}}} & \text {for}\: a = 0 \wedge b = 0 \\\frac {\log {\relax (x )}}{a^{3}} & \text {for}\: b = 0 \\- \frac {2}{3 b^{3} x^{\frac {3}{2}}} & \text {for}\: a = 0 \\\frac {a^{2} \sqrt {x} \log {\relax (x )}}{a^{5} \sqrt {x} + 2 a^{4} b x + a^{3} b^{2} x^{\frac {3}{2}}} - \frac {2 a^{2} \sqrt {x} \log {\left (\frac {a}{b} + \sqrt {x} \right )}}{a^{5} \sqrt {x} + 2 a^{4} b x + a^{3} b^{2} x^{\frac {3}{2}}} + \frac {3 a^{2} \sqrt {x}}{a^{5} \sqrt {x} + 2 a^{4} b x + a^{3} b^{2} x^{\frac {3}{2}}} + \frac {2 a b x \log {\relax (x )}}{a^{5} \sqrt {x} + 2 a^{4} b x + a^{3} b^{2} x^{\frac {3}{2}}} - \frac {4 a b x \log {\left (\frac {a}{b} + \sqrt {x} \right )}}{a^{5} \sqrt {x} + 2 a^{4} b x + a^{3} b^{2} x^{\frac {3}{2}}} + \frac {2 a b x}{a^{5} \sqrt {x} + 2 a^{4} b x + a^{3} b^{2} x^{\frac {3}{2}}} + \frac {b^{2} x^{\frac {3}{2}} \log {\relax (x )}}{a^{5} \sqrt {x} + 2 a^{4} b x + a^{3} b^{2} x^{\frac {3}{2}}} - \frac {2 b^{2} x^{\frac {3}{2}} \log {\left (\frac {a}{b} + \sqrt {x} \right )}}{a^{5} \sqrt {x} + 2 a^{4} b x + a^{3} b^{2} x^{\frac {3}{2}}} & \text {otherwise} \end {cases} \]

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)), (log(x)/a**3, Eq(b, 0)), (-2/(3*b**3*x**(3/2)), Eq(a, 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)) + 3*a**2*sqrt(x)/(a**5*sqrt(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)) + 2*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)), True))

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