3.1675 \(\int \frac{\sqrt{x}}{(a+\frac{b}{x})^2} \, dx\)

Optimal. Leaf size=70 \[ \frac{5 b^{3/2} \tan ^{-1}\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{b}}\right )}{a^{7/2}}-\frac{5 b \sqrt{x}}{a^3}+\frac{5 x^{3/2}}{3 a^2}-\frac{x^{5/2}}{a (a x+b)} \]

[Out]

(-5*b*Sqrt[x])/a^3 + (5*x^(3/2))/(3*a^2) - x^(5/2)/(a*(b + a*x)) + (5*b^(3/2)*ArcTan[(Sqrt[a]*Sqrt[x])/Sqrt[b]
])/a^(7/2)

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Rubi [A]  time = 0.023811, antiderivative size = 70, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {263, 47, 50, 63, 205} \[ \frac{5 b^{3/2} \tan ^{-1}\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{b}}\right )}{a^{7/2}}-\frac{5 b \sqrt{x}}{a^3}+\frac{5 x^{3/2}}{3 a^2}-\frac{x^{5/2}}{a (a x+b)} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[x]/(a + b/x)^2,x]

[Out]

(-5*b*Sqrt[x])/a^3 + (5*x^(3/2))/(3*a^2) - x^(5/2)/(a*(b + a*x)) + (5*b^(3/2)*ArcTan[(Sqrt[a]*Sqrt[x])/Sqrt[b]
])/a^(7/2)

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 50

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^n)/(b*
(m + n + 1)), x] + Dist[(n*(b*c - a*d))/(b*(m + n + 1)), Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a
, b, c, d}, x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 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{\sqrt{x}}{\left (a+\frac{b}{x}\right )^2} \, dx &=\int \frac{x^{5/2}}{(b+a x)^2} \, dx\\ &=-\frac{x^{5/2}}{a (b+a x)}+\frac{5 \int \frac{x^{3/2}}{b+a x} \, dx}{2 a}\\ &=\frac{5 x^{3/2}}{3 a^2}-\frac{x^{5/2}}{a (b+a x)}-\frac{(5 b) \int \frac{\sqrt{x}}{b+a x} \, dx}{2 a^2}\\ &=-\frac{5 b \sqrt{x}}{a^3}+\frac{5 x^{3/2}}{3 a^2}-\frac{x^{5/2}}{a (b+a x)}+\frac{\left (5 b^2\right ) \int \frac{1}{\sqrt{x} (b+a x)} \, dx}{2 a^3}\\ &=-\frac{5 b \sqrt{x}}{a^3}+\frac{5 x^{3/2}}{3 a^2}-\frac{x^{5/2}}{a (b+a x)}+\frac{\left (5 b^2\right ) \operatorname{Subst}\left (\int \frac{1}{b+a x^2} \, dx,x,\sqrt{x}\right )}{a^3}\\ &=-\frac{5 b \sqrt{x}}{a^3}+\frac{5 x^{3/2}}{3 a^2}-\frac{x^{5/2}}{a (b+a x)}+\frac{5 b^{3/2} \tan ^{-1}\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{b}}\right )}{a^{7/2}}\\ \end{align*}

Mathematica [C]  time = 0.0042537, size = 27, normalized size = 0.39 \[ \frac{2 x^{7/2} \, _2F_1\left (2,\frac{7}{2};\frac{9}{2};-\frac{a x}{b}\right )}{7 b^2} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[x]/(a + b/x)^2,x]

[Out]

(2*x^(7/2)*Hypergeometric2F1[2, 7/2, 9/2, -((a*x)/b)])/(7*b^2)

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Maple [A]  time = 0.01, size = 61, normalized size = 0.9 \begin{align*}{\frac{2}{3\,{a}^{2}}{x}^{{\frac{3}{2}}}}-4\,{\frac{b\sqrt{x}}{{a}^{3}}}-{\frac{{b}^{2}}{{a}^{3} \left ( ax+b \right ) }\sqrt{x}}+5\,{\frac{{b}^{2}}{{a}^{3}\sqrt{ab}}\arctan \left ({\frac{a\sqrt{x}}{\sqrt{ab}}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3*x^(3/2)/a^2-4*b*x^(1/2)/a^3-b^2/a^3*x^(1/2)/(a*x+b)+5*b^2/a^3/(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(x^(1/2)/(a+b/x)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.81551, size = 366, normalized size = 5.23 \begin{align*} \left [\frac{15 \,{\left (a b x + b^{2}\right )} \sqrt{-\frac{b}{a}} \log \left (\frac{a x + 2 \, a \sqrt{x} \sqrt{-\frac{b}{a}} - b}{a x + b}\right ) + 2 \,{\left (2 \, a^{2} x^{2} - 10 \, a b x - 15 \, b^{2}\right )} \sqrt{x}}{6 \,{\left (a^{4} x + a^{3} b\right )}}, \frac{15 \,{\left (a b x + b^{2}\right )} \sqrt{\frac{b}{a}} \arctan \left (\frac{a \sqrt{x} \sqrt{\frac{b}{a}}}{b}\right ) +{\left (2 \, a^{2} x^{2} - 10 \, a b x - 15 \, b^{2}\right )} \sqrt{x}}{3 \,{\left (a^{4} x + a^{3} b\right )}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/6*(15*(a*b*x + b^2)*sqrt(-b/a)*log((a*x + 2*a*sqrt(x)*sqrt(-b/a) - b)/(a*x + b)) + 2*(2*a^2*x^2 - 10*a*b*x
- 15*b^2)*sqrt(x))/(a^4*x + a^3*b), 1/3*(15*(a*b*x + b^2)*sqrt(b/a)*arctan(a*sqrt(x)*sqrt(b/a)/b) + (2*a^2*x^2
- 10*a*b*x - 15*b^2)*sqrt(x))/(a^4*x + a^3*b)]

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Sympy [A]  time = 10.167, size = 479, normalized size = 6.84 \begin{align*} \begin{cases} \tilde{\infty } x^{\frac{7}{2}} & \text{for}\: a = 0 \wedge b = 0 \\\frac{2 x^{\frac{3}{2}}}{3 a^{2}} & \text{for}\: b = 0 \\\frac{2 x^{\frac{7}{2}}}{7 b^{2}} & \text{for}\: a = 0 \\\frac{4 i a^{3} \sqrt{b} x^{\frac{5}{2}} \sqrt{\frac{1}{a}}}{6 i a^{5} \sqrt{b} x \sqrt{\frac{1}{a}} + 6 i a^{4} b^{\frac{3}{2}} \sqrt{\frac{1}{a}}} - \frac{20 i a^{2} b^{\frac{3}{2}} x^{\frac{3}{2}} \sqrt{\frac{1}{a}}}{6 i a^{5} \sqrt{b} x \sqrt{\frac{1}{a}} + 6 i a^{4} b^{\frac{3}{2}} \sqrt{\frac{1}{a}}} - \frac{30 i a b^{\frac{5}{2}} \sqrt{x} \sqrt{\frac{1}{a}}}{6 i a^{5} \sqrt{b} x \sqrt{\frac{1}{a}} + 6 i a^{4} b^{\frac{3}{2}} \sqrt{\frac{1}{a}}} + \frac{15 a b^{2} x \log{\left (- i \sqrt{b} \sqrt{\frac{1}{a}} + \sqrt{x} \right )}}{6 i a^{5} \sqrt{b} x \sqrt{\frac{1}{a}} + 6 i a^{4} b^{\frac{3}{2}} \sqrt{\frac{1}{a}}} - \frac{15 a b^{2} x \log{\left (i \sqrt{b} \sqrt{\frac{1}{a}} + \sqrt{x} \right )}}{6 i a^{5} \sqrt{b} x \sqrt{\frac{1}{a}} + 6 i a^{4} b^{\frac{3}{2}} \sqrt{\frac{1}{a}}} + \frac{15 b^{3} \log{\left (- i \sqrt{b} \sqrt{\frac{1}{a}} + \sqrt{x} \right )}}{6 i a^{5} \sqrt{b} x \sqrt{\frac{1}{a}} + 6 i a^{4} b^{\frac{3}{2}} \sqrt{\frac{1}{a}}} - \frac{15 b^{3} \log{\left (i \sqrt{b} \sqrt{\frac{1}{a}} + \sqrt{x} \right )}}{6 i a^{5} \sqrt{b} x \sqrt{\frac{1}{a}} + 6 i a^{4} 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(x**(1/2)/(a+b/x)**2,x)

[Out]

Piecewise((zoo*x**(7/2), Eq(a, 0) & Eq(b, 0)), (2*x**(3/2)/(3*a**2), Eq(b, 0)), (2*x**(7/2)/(7*b**2), Eq(a, 0)
), (4*I*a**3*sqrt(b)*x**(5/2)*sqrt(1/a)/(6*I*a**5*sqrt(b)*x*sqrt(1/a) + 6*I*a**4*b**(3/2)*sqrt(1/a)) - 20*I*a*
*2*b**(3/2)*x**(3/2)*sqrt(1/a)/(6*I*a**5*sqrt(b)*x*sqrt(1/a) + 6*I*a**4*b**(3/2)*sqrt(1/a)) - 30*I*a*b**(5/2)*
sqrt(x)*sqrt(1/a)/(6*I*a**5*sqrt(b)*x*sqrt(1/a) + 6*I*a**4*b**(3/2)*sqrt(1/a)) + 15*a*b**2*x*log(-I*sqrt(b)*sq
rt(1/a) + sqrt(x))/(6*I*a**5*sqrt(b)*x*sqrt(1/a) + 6*I*a**4*b**(3/2)*sqrt(1/a)) - 15*a*b**2*x*log(I*sqrt(b)*sq
rt(1/a) + sqrt(x))/(6*I*a**5*sqrt(b)*x*sqrt(1/a) + 6*I*a**4*b**(3/2)*sqrt(1/a)) + 15*b**3*log(-I*sqrt(b)*sqrt(
1/a) + sqrt(x))/(6*I*a**5*sqrt(b)*x*sqrt(1/a) + 6*I*a**4*b**(3/2)*sqrt(1/a)) - 15*b**3*log(I*sqrt(b)*sqrt(1/a)
+ sqrt(x))/(6*I*a**5*sqrt(b)*x*sqrt(1/a) + 6*I*a**4*b**(3/2)*sqrt(1/a)), True))

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Giac [A]  time = 1.09972, size = 88, normalized size = 1.26 \begin{align*} \frac{5 \, b^{2} \arctan \left (\frac{a \sqrt{x}}{\sqrt{a b}}\right )}{\sqrt{a b} a^{3}} - \frac{b^{2} \sqrt{x}}{{\left (a x + b\right )} a^{3}} + \frac{2 \,{\left (a^{4} x^{\frac{3}{2}} - 6 \, a^{3} b \sqrt{x}\right )}}{3 \, a^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

5*b^2*arctan(a*sqrt(x)/sqrt(a*b))/(sqrt(a*b)*a^3) - b^2*sqrt(x)/((a*x + b)*a^3) + 2/3*(a^4*x^(3/2) - 6*a^3*b*s
qrt(x))/a^6