3.2197 \(\int \frac{1}{(a+b \sqrt{x}) x^3} \, dx\)

Optimal. Leaf size=75 \[ \frac{2 b^3}{a^4 \sqrt{x}}-\frac{b^2}{a^3 x}-\frac{2 b^4 \log \left (a+b \sqrt{x}\right )}{a^5}+\frac{b^4 \log (x)}{a^5}+\frac{2 b}{3 a^2 x^{3/2}}-\frac{1}{2 a x^2} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0383596, antiderivative size = 75, 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} \[ \frac{2 b^3}{a^4 \sqrt{x}}-\frac{b^2}{a^3 x}-\frac{2 b^4 \log \left (a+b \sqrt{x}\right )}{a^5}+\frac{b^4 \log (x)}{a^5}+\frac{2 b}{3 a^2 x^{3/2}}-\frac{1}{2 a x^2} \]

Antiderivative was successfully verified.

[In]

Int[1/((a + b*Sqrt[x])*x^3),x]

[Out]

-1/(2*a*x^2) + (2*b)/(3*a^2*x^(3/2)) - b^2/(a^3*x) + (2*b^3)/(a^4*Sqrt[x]) - (2*b^4*Log[a + b*Sqrt[x]])/a^5 +
(b^4*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 ) x^3} \, dx &=2 \operatorname{Subst}\left (\int \frac{1}{x^5 (a+b x)} \, dx,x,\sqrt{x}\right )\\ &=2 \operatorname{Subst}\left (\int \left (\frac{1}{a x^5}-\frac{b}{a^2 x^4}+\frac{b^2}{a^3 x^3}-\frac{b^3}{a^4 x^2}+\frac{b^4}{a^5 x}-\frac{b^5}{a^5 (a+b x)}\right ) \, dx,x,\sqrt{x}\right )\\ &=-\frac{1}{2 a x^2}+\frac{2 b}{3 a^2 x^{3/2}}-\frac{b^2}{a^3 x}+\frac{2 b^3}{a^4 \sqrt{x}}-\frac{2 b^4 \log \left (a+b \sqrt{x}\right )}{a^5}+\frac{b^4 \log (x)}{a^5}\\ \end{align*}

Mathematica [A]  time = 0.0433211, size = 69, normalized size = 0.92 \[ \frac{\frac{a \left (4 a^2 b \sqrt{x}-3 a^3-6 a b^2 x+12 b^3 x^{3/2}\right )}{x^2}-12 b^4 \log \left (a+b \sqrt{x}\right )+6 b^4 \log (x)}{6 a^5} \]

Antiderivative was successfully verified.

[In]

Integrate[1/((a + b*Sqrt[x])*x^3),x]

[Out]

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

________________________________________________________________________________________

Maple [A]  time = 0.008, size = 66, normalized size = 0.9 \begin{align*} -{\frac{1}{2\,a{x}^{2}}}+{\frac{2\,b}{3\,{a}^{2}}{x}^{-{\frac{3}{2}}}}-{\frac{{b}^{2}}{{a}^{3}x}}+{\frac{{b}^{4}\ln \left ( x \right ) }{{a}^{5}}}-2\,{\frac{{b}^{4}\ln \left ( a+b\sqrt{x} \right ) }{{a}^{5}}}+2\,{\frac{{b}^{3}}{{a}^{4}\sqrt{x}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 0.953942, size = 86, normalized size = 1.15 \begin{align*} -\frac{2 \, b^{4} \log \left (b \sqrt{x} + a\right )}{a^{5}} + \frac{b^{4} \log \left (x\right )}{a^{5}} + \frac{12 \, b^{3} x^{\frac{3}{2}} - 6 \, a b^{2} x + 4 \, a^{2} b \sqrt{x} - 3 \, a^{3}}{6 \, a^{4} x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [A]  time = 1.33795, size = 173, normalized size = 2.31 \begin{align*} -\frac{12 \, b^{4} x^{2} \log \left (b \sqrt{x} + a\right ) - 12 \, b^{4} x^{2} \log \left (\sqrt{x}\right ) + 6 \, a^{2} b^{2} x + 3 \, a^{4} - 4 \,{\left (3 \, a b^{3} x + a^{3} b\right )} \sqrt{x}}{6 \, a^{5} x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/6*(12*b^4*x^2*log(b*sqrt(x) + a) - 12*b^4*x^2*log(sqrt(x)) + 6*a^2*b^2*x + 3*a^4 - 4*(3*a*b^3*x + a^3*b)*sq
rt(x))/(a^5*x^2)

________________________________________________________________________________________

Sympy [A]  time = 3.0342, size = 99, normalized size = 1.32 \begin{align*} \begin{cases} \frac{\tilde{\infty }}{x^{\frac{5}{2}}} & \text{for}\: a = 0 \wedge b = 0 \\- \frac{2}{5 b x^{\frac{5}{2}}} & \text{for}\: a = 0 \\- \frac{1}{2 a x^{2}} & \text{for}\: b = 0 \\- \frac{1}{2 a x^{2}} + \frac{2 b}{3 a^{2} x^{\frac{3}{2}}} - \frac{b^{2}}{a^{3} x} + \frac{2 b^{3}}{a^{4} \sqrt{x}} + \frac{b^{4} \log{\left (x \right )}}{a^{5}} - \frac{2 b^{4} \log{\left (\frac{a}{b} + \sqrt{x} \right )}}{a^{5}} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Giac [A]  time = 1.11152, size = 93, normalized size = 1.24 \begin{align*} -\frac{2 \, b^{4} \log \left ({\left | b \sqrt{x} + a \right |}\right )}{a^{5}} + \frac{b^{4} \log \left ({\left | x \right |}\right )}{a^{5}} + \frac{12 \, a b^{3} x^{\frac{3}{2}} - 6 \, a^{2} b^{2} x + 4 \, a^{3} b \sqrt{x} - 3 \, a^{4}}{6 \, a^{5} x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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