3.1745 \(\int \frac{1}{(a+\frac{b}{x})^{5/2} x} \, dx\)

Optimal. Leaf size=60 \[ -\frac{2}{a^2 \sqrt{a+\frac{b}{x}}}+\frac{2 \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right )}{a^{5/2}}-\frac{2}{3 a \left (a+\frac{b}{x}\right )^{3/2}} \]

[Out]

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

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Rubi [A]  time = 0.0291079, antiderivative size = 60, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267, Rules used = {266, 51, 63, 208} \[ -\frac{2}{a^2 \sqrt{a+\frac{b}{x}}}+\frac{2 \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right )}{a^{5/2}}-\frac{2}{3 a \left (a+\frac{b}{x}\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

Int[1/((a + b/x)^(5/2)*x),x]

[Out]

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

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 51

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n + 1
))/((b*c - a*d)*(m + 1)), x] - Dist[(d*(m + n + 2))/((b*c - a*d)*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^n,
x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && LtQ[m, -1] &&  !(LtQ[n, -1] && (EqQ[a, 0] || (NeQ[
c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && IntLinearQ[a, b, c, d, m, n, x]

Rule 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 208

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-(a/b), 2]*ArcTanh[x/Rt[-(a/b), 2]])/a, x] /; FreeQ[{a,
b}, x] && NegQ[a/b]

Rubi steps

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

Mathematica [C]  time = 0.0146855, size = 36, normalized size = 0.6 \[ -\frac{2 \, _2F_1\left (-\frac{3}{2},1;-\frac{1}{2};\frac{b}{a x}+1\right )}{3 a \left (a+\frac{b}{x}\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/((a + b/x)^(5/2)*x),x]

[Out]

(-2*Hypergeometric2F1[-3/2, 1, -1/2, 1 + b/(a*x)])/(3*a*(a + b/x)^(3/2))

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Maple [B]  time = 0.01, size = 274, normalized size = 4.6 \begin{align*}{\frac{x}{3\, \left ( ax+b \right ) ^{3}b}\sqrt{{\frac{ax+b}{x}}} \left ( -6\,{a}^{7/2}\sqrt{ \left ( ax+b \right ) x}{x}^{3}+3\,\ln \left ( 1/2\,{\frac{2\,\sqrt{ \left ( ax+b \right ) x}\sqrt{a}+2\,ax+b}{\sqrt{a}}} \right ){x}^{3}{a}^{3}b+6\,{a}^{5/2} \left ( \left ( ax+b \right ) x \right ) ^{3/2}x-18\,{a}^{5/2}\sqrt{ \left ( ax+b \right ) x}{x}^{2}b+9\,\ln \left ( 1/2\,{\frac{2\,\sqrt{ \left ( ax+b \right ) x}\sqrt{a}+2\,ax+b}{\sqrt{a}}} \right ){x}^{2}{a}^{2}{b}^{2}+4\,b{a}^{3/2} \left ( \left ( ax+b \right ) x \right ) ^{3/2}-18\,{a}^{3/2}\sqrt{ \left ( ax+b \right ) x}x{b}^{2}+9\,\ln \left ( 1/2\,{\frac{2\,\sqrt{ \left ( ax+b \right ) x}\sqrt{a}+2\,ax+b}{\sqrt{a}}} \right ) xa{b}^{3}-6\,\sqrt{a}\sqrt{ \left ( ax+b \right ) x}{b}^{3}+3\,\ln \left ( 1/2\,{\frac{2\,\sqrt{ \left ( ax+b \right ) x}\sqrt{a}+2\,ax+b}{\sqrt{a}}} \right ){b}^{4} \right ){a}^{-{\frac{5}{2}}}{\frac{1}{\sqrt{ \left ( ax+b \right ) x}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a+b/x)^(5/2)/x,x)

[Out]

1/3*((a*x+b)/x)^(1/2)*x/a^(5/2)*(-6*a^(7/2)*((a*x+b)*x)^(1/2)*x^3+3*ln(1/2*(2*((a*x+b)*x)^(1/2)*a^(1/2)+2*a*x+
b)/a^(1/2))*x^3*a^3*b+6*a^(5/2)*((a*x+b)*x)^(3/2)*x-18*a^(5/2)*((a*x+b)*x)^(1/2)*x^2*b+9*ln(1/2*(2*((a*x+b)*x)
^(1/2)*a^(1/2)+2*a*x+b)/a^(1/2))*x^2*a^2*b^2+4*b*a^(3/2)*((a*x+b)*x)^(3/2)-18*a^(3/2)*((a*x+b)*x)^(1/2)*x*b^2+
9*ln(1/2*(2*((a*x+b)*x)^(1/2)*a^(1/2)+2*a*x+b)/a^(1/2))*x*a*b^3-6*a^(1/2)*((a*x+b)*x)^(1/2)*b^3+3*ln(1/2*(2*((
a*x+b)*x)^(1/2)*a^(1/2)+2*a*x+b)/a^(1/2))*b^4)/((a*x+b)*x)^(1/2)/(a*x+b)^3/b

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b/x)^(5/2)/x,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 1.51183, size = 440, normalized size = 7.33 \begin{align*} \left [\frac{3 \,{\left (a^{2} x^{2} + 2 \, a b x + b^{2}\right )} \sqrt{a} \log \left (2 \, a x + 2 \, \sqrt{a} x \sqrt{\frac{a x + b}{x}} + b\right ) - 2 \,{\left (4 \, a^{2} x^{2} + 3 \, a b x\right )} \sqrt{\frac{a x + b}{x}}}{3 \,{\left (a^{5} x^{2} + 2 \, a^{4} b x + a^{3} b^{2}\right )}}, -\frac{2 \,{\left (3 \,{\left (a^{2} x^{2} + 2 \, a b x + b^{2}\right )} \sqrt{-a} \arctan \left (\frac{\sqrt{-a} \sqrt{\frac{a x + b}{x}}}{a}\right ) +{\left (4 \, a^{2} x^{2} + 3 \, a b x\right )} \sqrt{\frac{a x + b}{x}}\right )}}{3 \,{\left (a^{5} x^{2} + 2 \, a^{4} b x + a^{3} b^{2}\right )}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b/x)^(5/2)/x,x, algorithm="fricas")

[Out]

[1/3*(3*(a^2*x^2 + 2*a*b*x + b^2)*sqrt(a)*log(2*a*x + 2*sqrt(a)*x*sqrt((a*x + b)/x) + b) - 2*(4*a^2*x^2 + 3*a*
b*x)*sqrt((a*x + b)/x))/(a^5*x^2 + 2*a^4*b*x + a^3*b^2), -2/3*(3*(a^2*x^2 + 2*a*b*x + b^2)*sqrt(-a)*arctan(sqr
t(-a)*sqrt((a*x + b)/x)/a) + (4*a^2*x^2 + 3*a*b*x)*sqrt((a*x + b)/x))/(a^5*x^2 + 2*a^4*b*x + a^3*b^2)]

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Sympy [B]  time = 2.99058, size = 700, normalized size = 11.67 \begin{align*} - \frac{8 a^{7} x^{3} \sqrt{1 + \frac{b}{a x}}}{3 a^{\frac{19}{2}} x^{3} + 9 a^{\frac{17}{2}} b x^{2} + 9 a^{\frac{15}{2}} b^{2} x + 3 a^{\frac{13}{2}} b^{3}} - \frac{3 a^{7} x^{3} \log{\left (\frac{b}{a x} \right )}}{3 a^{\frac{19}{2}} x^{3} + 9 a^{\frac{17}{2}} b x^{2} + 9 a^{\frac{15}{2}} b^{2} x + 3 a^{\frac{13}{2}} b^{3}} + \frac{6 a^{7} x^{3} \log{\left (\sqrt{1 + \frac{b}{a x}} + 1 \right )}}{3 a^{\frac{19}{2}} x^{3} + 9 a^{\frac{17}{2}} b x^{2} + 9 a^{\frac{15}{2}} b^{2} x + 3 a^{\frac{13}{2}} b^{3}} - \frac{14 a^{6} b x^{2} \sqrt{1 + \frac{b}{a x}}}{3 a^{\frac{19}{2}} x^{3} + 9 a^{\frac{17}{2}} b x^{2} + 9 a^{\frac{15}{2}} b^{2} x + 3 a^{\frac{13}{2}} b^{3}} - \frac{9 a^{6} b x^{2} \log{\left (\frac{b}{a x} \right )}}{3 a^{\frac{19}{2}} x^{3} + 9 a^{\frac{17}{2}} b x^{2} + 9 a^{\frac{15}{2}} b^{2} x + 3 a^{\frac{13}{2}} b^{3}} + \frac{18 a^{6} b x^{2} \log{\left (\sqrt{1 + \frac{b}{a x}} + 1 \right )}}{3 a^{\frac{19}{2}} x^{3} + 9 a^{\frac{17}{2}} b x^{2} + 9 a^{\frac{15}{2}} b^{2} x + 3 a^{\frac{13}{2}} b^{3}} - \frac{6 a^{5} b^{2} x \sqrt{1 + \frac{b}{a x}}}{3 a^{\frac{19}{2}} x^{3} + 9 a^{\frac{17}{2}} b x^{2} + 9 a^{\frac{15}{2}} b^{2} x + 3 a^{\frac{13}{2}} b^{3}} - \frac{9 a^{5} b^{2} x \log{\left (\frac{b}{a x} \right )}}{3 a^{\frac{19}{2}} x^{3} + 9 a^{\frac{17}{2}} b x^{2} + 9 a^{\frac{15}{2}} b^{2} x + 3 a^{\frac{13}{2}} b^{3}} + \frac{18 a^{5} b^{2} x \log{\left (\sqrt{1 + \frac{b}{a x}} + 1 \right )}}{3 a^{\frac{19}{2}} x^{3} + 9 a^{\frac{17}{2}} b x^{2} + 9 a^{\frac{15}{2}} b^{2} x + 3 a^{\frac{13}{2}} b^{3}} - \frac{3 a^{4} b^{3} \log{\left (\frac{b}{a x} \right )}}{3 a^{\frac{19}{2}} x^{3} + 9 a^{\frac{17}{2}} b x^{2} + 9 a^{\frac{15}{2}} b^{2} x + 3 a^{\frac{13}{2}} b^{3}} + \frac{6 a^{4} b^{3} \log{\left (\sqrt{1 + \frac{b}{a x}} + 1 \right )}}{3 a^{\frac{19}{2}} x^{3} + 9 a^{\frac{17}{2}} b x^{2} + 9 a^{\frac{15}{2}} b^{2} x + 3 a^{\frac{13}{2}} b^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b/x)**(5/2)/x,x)

[Out]

-8*a**7*x**3*sqrt(1 + b/(a*x))/(3*a**(19/2)*x**3 + 9*a**(17/2)*b*x**2 + 9*a**(15/2)*b**2*x + 3*a**(13/2)*b**3)
- 3*a**7*x**3*log(b/(a*x))/(3*a**(19/2)*x**3 + 9*a**(17/2)*b*x**2 + 9*a**(15/2)*b**2*x + 3*a**(13/2)*b**3) +
6*a**7*x**3*log(sqrt(1 + b/(a*x)) + 1)/(3*a**(19/2)*x**3 + 9*a**(17/2)*b*x**2 + 9*a**(15/2)*b**2*x + 3*a**(13/
2)*b**3) - 14*a**6*b*x**2*sqrt(1 + b/(a*x))/(3*a**(19/2)*x**3 + 9*a**(17/2)*b*x**2 + 9*a**(15/2)*b**2*x + 3*a*
*(13/2)*b**3) - 9*a**6*b*x**2*log(b/(a*x))/(3*a**(19/2)*x**3 + 9*a**(17/2)*b*x**2 + 9*a**(15/2)*b**2*x + 3*a**
(13/2)*b**3) + 18*a**6*b*x**2*log(sqrt(1 + b/(a*x)) + 1)/(3*a**(19/2)*x**3 + 9*a**(17/2)*b*x**2 + 9*a**(15/2)*
b**2*x + 3*a**(13/2)*b**3) - 6*a**5*b**2*x*sqrt(1 + b/(a*x))/(3*a**(19/2)*x**3 + 9*a**(17/2)*b*x**2 + 9*a**(15
/2)*b**2*x + 3*a**(13/2)*b**3) - 9*a**5*b**2*x*log(b/(a*x))/(3*a**(19/2)*x**3 + 9*a**(17/2)*b*x**2 + 9*a**(15/
2)*b**2*x + 3*a**(13/2)*b**3) + 18*a**5*b**2*x*log(sqrt(1 + b/(a*x)) + 1)/(3*a**(19/2)*x**3 + 9*a**(17/2)*b*x*
*2 + 9*a**(15/2)*b**2*x + 3*a**(13/2)*b**3) - 3*a**4*b**3*log(b/(a*x))/(3*a**(19/2)*x**3 + 9*a**(17/2)*b*x**2
+ 9*a**(15/2)*b**2*x + 3*a**(13/2)*b**3) + 6*a**4*b**3*log(sqrt(1 + b/(a*x)) + 1)/(3*a**(19/2)*x**3 + 9*a**(17
/2)*b*x**2 + 9*a**(15/2)*b**2*x + 3*a**(13/2)*b**3)

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Giac [A]  time = 1.29793, size = 99, normalized size = 1.65 \begin{align*} -\frac{2}{3} \, b{\left (\frac{{\left (a + \frac{3 \,{\left (a x + b\right )}}{x}\right )} x}{{\left (a x + b\right )} a^{2} b \sqrt{\frac{a x + b}{x}}} + \frac{3 \, \arctan \left (\frac{\sqrt{\frac{a x + b}{x}}}{\sqrt{-a}}\right )}{\sqrt{-a} a^{2} b}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b/x)^(5/2)/x,x, algorithm="giac")

[Out]

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