3.1677 \(\int \frac{1}{(a+\frac{b}{x})^2 x^{3/2}} \, dx\)

Optimal. Leaf size=46 \[ \frac{\tan ^{-1}\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{b}}\right )}{a^{3/2} \sqrt{b}}-\frac{\sqrt{x}}{a (a x+b)} \]

[Out]

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

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Rubi [A]  time = 0.0158499, antiderivative size = 46, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267, Rules used = {263, 47, 63, 205} \[ \frac{\tan ^{-1}\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{b}}\right )}{a^{3/2} \sqrt{b}}-\frac{\sqrt{x}}{a (a x+b)} \]

Antiderivative was successfully verified.

[In]

Int[1/((a + b/x)^2*x^(3/2)),x]

[Out]

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

Rule 263

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Int[x^(m + n*p)*(b + a/x^n)^p, x] /; FreeQ[{a, b, m
, n}, x] && IntegerQ[p] && NegQ[n]

Rule 47

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^n)/(b*
(m + 1)), x] - Dist[(d*n)/(b*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d},
x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && LtQ[m, -1] &&  !(IntegerQ[n] &&  !IntegerQ[m]) &&  !(ILeQ[m + n + 2, 0
] && (FractionQ[m] || GeQ[2*n + m + 1, 0])) && 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 205

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

Rubi steps

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

Mathematica [A]  time = 0.0207734, size = 46, normalized size = 1. \[ \frac{\tan ^{-1}\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{b}}\right )}{a^{3/2} \sqrt{b}}-\frac{\sqrt{x}}{a (a x+b)} \]

Antiderivative was successfully verified.

[In]

Integrate[1/((a + b/x)^2*x^(3/2)),x]

[Out]

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

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Maple [A]  time = 0.009, size = 37, normalized size = 0.8 \begin{align*} -{\frac{1}{a \left ( ax+b \right ) }\sqrt{x}}+{\frac{1}{a}\arctan \left ({a\sqrt{x}{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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)^2/x^(3/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.81852, size = 277, normalized size = 6.02 \begin{align*} \left [-\frac{2 \, a b \sqrt{x} + \sqrt{-a b}{\left (a x + b\right )} \log \left (\frac{a x - b - 2 \, \sqrt{-a b} \sqrt{x}}{a x + b}\right )}{2 \,{\left (a^{3} b x + a^{2} b^{2}\right )}}, -\frac{a b \sqrt{x} + \sqrt{a b}{\left (a x + b\right )} \arctan \left (\frac{\sqrt{a b}}{a \sqrt{x}}\right )}{a^{3} b x + a^{2} b^{2}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 21.7266, size = 337, normalized size = 7.33 \begin{align*} \begin{cases} \tilde{\infty } x^{\frac{3}{2}} & \text{for}\: a = 0 \wedge b = 0 \\\frac{2 x^{\frac{3}{2}}}{3 b^{2}} & \text{for}\: a = 0 \\- \frac{2}{a^{2} \sqrt{x}} & \text{for}\: b = 0 \\- \frac{2 i a \sqrt{b} \sqrt{x} \sqrt{\frac{1}{a}}}{2 i a^{3} \sqrt{b} x \sqrt{\frac{1}{a}} + 2 i a^{2} b^{\frac{3}{2}} \sqrt{\frac{1}{a}}} + \frac{a x \log{\left (- i \sqrt{b} \sqrt{\frac{1}{a}} + \sqrt{x} \right )}}{2 i a^{3} \sqrt{b} x \sqrt{\frac{1}{a}} + 2 i a^{2} b^{\frac{3}{2}} \sqrt{\frac{1}{a}}} - \frac{a x \log{\left (i \sqrt{b} \sqrt{\frac{1}{a}} + \sqrt{x} \right )}}{2 i a^{3} \sqrt{b} x \sqrt{\frac{1}{a}} + 2 i a^{2} b^{\frac{3}{2}} \sqrt{\frac{1}{a}}} + \frac{b \log{\left (- i \sqrt{b} \sqrt{\frac{1}{a}} + \sqrt{x} \right )}}{2 i a^{3} \sqrt{b} x \sqrt{\frac{1}{a}} + 2 i a^{2} b^{\frac{3}{2}} \sqrt{\frac{1}{a}}} - \frac{b \log{\left (i \sqrt{b} \sqrt{\frac{1}{a}} + \sqrt{x} \right )}}{2 i a^{3} \sqrt{b} x \sqrt{\frac{1}{a}} + 2 i a^{2} b^{\frac{3}{2}} \sqrt{\frac{1}{a}}} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.11157, size = 49, normalized size = 1.07 \begin{align*} \frac{\arctan \left (\frac{a \sqrt{x}}{\sqrt{a b}}\right )}{\sqrt{a b} a} - \frac{\sqrt{x}}{{\left (a x + b\right )} a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

arctan(a*sqrt(x)/sqrt(a*b))/(sqrt(a*b)*a) - sqrt(x)/((a*x + b)*a)