3.1742 \(\int \frac{x^2}{(a+\frac{b}{x})^{5/2}} \, dx\)

Optimal. Leaf size=138 \[ \frac{105 b^3}{8 a^5 \sqrt{a+\frac{b}{x}}}+\frac{35 b^3}{8 a^4 \left (a+\frac{b}{x}\right )^{3/2}}+\frac{21 b^2 x}{8 a^3 \left (a+\frac{b}{x}\right )^{3/2}}-\frac{105 b^3 \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right )}{8 a^{11/2}}-\frac{3 b x^2}{4 a^2 \left (a+\frac{b}{x}\right )^{3/2}}+\frac{x^3}{3 a \left (a+\frac{b}{x}\right )^{3/2}} \]

[Out]

(35*b^3)/(8*a^4*(a + b/x)^(3/2)) + (105*b^3)/(8*a^5*Sqrt[a + b/x]) + (21*b^2*x)/(8*a^3*(a + b/x)^(3/2)) - (3*b
*x^2)/(4*a^2*(a + b/x)^(3/2)) + x^3/(3*a*(a + b/x)^(3/2)) - (105*b^3*ArcTanh[Sqrt[a + b/x]/Sqrt[a]])/(8*a^(11/
2))

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Rubi [A]  time = 0.0663102, antiderivative size = 134, normalized size of antiderivative = 0.97, number of steps used = 8, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267, Rules used = {266, 51, 63, 208} \[ \frac{105 b^2 x \sqrt{a+\frac{b}{x}}}{8 a^5}-\frac{105 b^3 \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right )}{8 a^{11/2}}-\frac{35 b x^2 \sqrt{a+\frac{b}{x}}}{4 a^4}+\frac{7 x^3 \sqrt{a+\frac{b}{x}}}{a^3}-\frac{6 x^3}{a^2 \sqrt{a+\frac{b}{x}}}-\frac{2 x^3}{3 a \left (a+\frac{b}{x}\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

Int[x^2/(a + b/x)^(5/2),x]

[Out]

(105*b^2*Sqrt[a + b/x]*x)/(8*a^5) - (35*b*Sqrt[a + b/x]*x^2)/(4*a^4) - (2*x^3)/(3*a*(a + b/x)^(3/2)) - (6*x^3)
/(a^2*Sqrt[a + b/x]) + (7*Sqrt[a + b/x]*x^3)/a^3 - (105*b^3*ArcTanh[Sqrt[a + b/x]/Sqrt[a]])/(8*a^(11/2))

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{x^2}{\left (a+\frac{b}{x}\right )^{5/2}} \, dx &=-\operatorname{Subst}\left (\int \frac{1}{x^4 (a+b x)^{5/2}} \, dx,x,\frac{1}{x}\right )\\ &=-\frac{2 x^3}{3 a \left (a+\frac{b}{x}\right )^{3/2}}-\frac{3 \operatorname{Subst}\left (\int \frac{1}{x^4 (a+b x)^{3/2}} \, dx,x,\frac{1}{x}\right )}{a}\\ &=-\frac{2 x^3}{3 a \left (a+\frac{b}{x}\right )^{3/2}}-\frac{6 x^3}{a^2 \sqrt{a+\frac{b}{x}}}-\frac{21 \operatorname{Subst}\left (\int \frac{1}{x^4 \sqrt{a+b x}} \, dx,x,\frac{1}{x}\right )}{a^2}\\ &=-\frac{2 x^3}{3 a \left (a+\frac{b}{x}\right )^{3/2}}-\frac{6 x^3}{a^2 \sqrt{a+\frac{b}{x}}}+\frac{7 \sqrt{a+\frac{b}{x}} x^3}{a^3}+\frac{(35 b) \operatorname{Subst}\left (\int \frac{1}{x^3 \sqrt{a+b x}} \, dx,x,\frac{1}{x}\right )}{2 a^3}\\ &=-\frac{35 b \sqrt{a+\frac{b}{x}} x^2}{4 a^4}-\frac{2 x^3}{3 a \left (a+\frac{b}{x}\right )^{3/2}}-\frac{6 x^3}{a^2 \sqrt{a+\frac{b}{x}}}+\frac{7 \sqrt{a+\frac{b}{x}} x^3}{a^3}-\frac{\left (105 b^2\right ) \operatorname{Subst}\left (\int \frac{1}{x^2 \sqrt{a+b x}} \, dx,x,\frac{1}{x}\right )}{8 a^4}\\ &=\frac{105 b^2 \sqrt{a+\frac{b}{x}} x}{8 a^5}-\frac{35 b \sqrt{a+\frac{b}{x}} x^2}{4 a^4}-\frac{2 x^3}{3 a \left (a+\frac{b}{x}\right )^{3/2}}-\frac{6 x^3}{a^2 \sqrt{a+\frac{b}{x}}}+\frac{7 \sqrt{a+\frac{b}{x}} x^3}{a^3}+\frac{\left (105 b^3\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+b x}} \, dx,x,\frac{1}{x}\right )}{16 a^5}\\ &=\frac{105 b^2 \sqrt{a+\frac{b}{x}} x}{8 a^5}-\frac{35 b \sqrt{a+\frac{b}{x}} x^2}{4 a^4}-\frac{2 x^3}{3 a \left (a+\frac{b}{x}\right )^{3/2}}-\frac{6 x^3}{a^2 \sqrt{a+\frac{b}{x}}}+\frac{7 \sqrt{a+\frac{b}{x}} x^3}{a^3}+\frac{\left (105 b^2\right ) \operatorname{Subst}\left (\int \frac{1}{-\frac{a}{b}+\frac{x^2}{b}} \, dx,x,\sqrt{a+\frac{b}{x}}\right )}{8 a^5}\\ &=\frac{105 b^2 \sqrt{a+\frac{b}{x}} x}{8 a^5}-\frac{35 b \sqrt{a+\frac{b}{x}} x^2}{4 a^4}-\frac{2 x^3}{3 a \left (a+\frac{b}{x}\right )^{3/2}}-\frac{6 x^3}{a^2 \sqrt{a+\frac{b}{x}}}+\frac{7 \sqrt{a+\frac{b}{x}} x^3}{a^3}-\frac{105 b^3 \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right )}{8 a^{11/2}}\\ \end{align*}

Mathematica [C]  time = 0.0146011, size = 39, normalized size = 0.28 \[ \frac{2 b^3 \, _2F_1\left (-\frac{3}{2},4;-\frac{1}{2};\frac{b}{a x}+1\right )}{3 a^4 \left (a+\frac{b}{x}\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[x^2/(a + b/x)^(5/2),x]

[Out]

(2*b^3*Hypergeometric2F1[-3/2, 4, -1/2, 1 + b/(a*x)])/(3*a^4*(a + b/x)^(3/2))

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Maple [B]  time = 0.015, size = 616, normalized size = 4.5 \begin{align*}{\frac{x}{48\, \left ( ax+b \right ) ^{3}}\sqrt{{\frac{ax+b}{x}}} \left ( 16\, \left ( a{x}^{2}+bx \right ) ^{3/2}{a}^{11/2}{x}^{3}-84\,\sqrt{a{x}^{2}+bx}{a}^{11/2}{x}^{4}b+48\, \left ( a{x}^{2}+bx \right ) ^{3/2}{a}^{9/2}{x}^{2}b-294\,\sqrt{a{x}^{2}+bx}{a}^{9/2}{x}^{3}{b}^{2}+672\,{a}^{9/2}\sqrt{ \left ( ax+b \right ) x}{x}^{3}{b}^{2}-336\,{a}^{4}\ln \left ( 1/2\,{\frac{2\,\sqrt{ \left ( ax+b \right ) x}\sqrt{a}+2\,ax+b}{\sqrt{a}}} \right ){x}^{3}{b}^{3}+48\, \left ( a{x}^{2}+bx \right ) ^{3/2}{a}^{7/2}x{b}^{2}-378\,\sqrt{a{x}^{2}+bx}{a}^{7/2}{x}^{2}{b}^{3}-384\,{a}^{7/2} \left ( \left ( ax+b \right ) x \right ) ^{3/2}x{b}^{2}+2016\,{a}^{7/2}\sqrt{ \left ( ax+b \right ) x}{x}^{2}{b}^{3}-1008\,{a}^{3}\ln \left ( 1/2\,{\frac{2\,\sqrt{ \left ( ax+b \right ) x}\sqrt{a}+2\,ax+b}{\sqrt{a}}} \right ){x}^{2}{b}^{4}+21\,\ln \left ( 1/2\,{\frac{2\,\sqrt{a{x}^{2}+bx}\sqrt{a}+2\,ax+b}{\sqrt{a}}} \right ){x}^{3}{a}^{4}{b}^{3}+16\, \left ( a{x}^{2}+bx \right ) ^{3/2}{a}^{5/2}{b}^{3}-210\,\sqrt{a{x}^{2}+bx}{a}^{5/2}x{b}^{4}-352\,{b}^{3}{a}^{5/2} \left ( \left ( ax+b \right ) x \right ) ^{3/2}+2016\,{a}^{5/2}\sqrt{ \left ( ax+b \right ) x}x{b}^{4}-1008\,{a}^{2}\ln \left ( 1/2\,{\frac{2\,\sqrt{ \left ( ax+b \right ) x}\sqrt{a}+2\,ax+b}{\sqrt{a}}} \right ) x{b}^{5}+63\,\ln \left ( 1/2\,{\frac{2\,\sqrt{a{x}^{2}+bx}\sqrt{a}+2\,ax+b}{\sqrt{a}}} \right ){x}^{2}{a}^{3}{b}^{4}-42\,\sqrt{a{x}^{2}+bx}{a}^{3/2}{b}^{5}+672\,{a}^{3/2}\sqrt{ \left ( ax+b \right ) x}{b}^{5}-336\,a\ln \left ( 1/2\,{\frac{2\,\sqrt{ \left ( ax+b \right ) x}\sqrt{a}+2\,ax+b}{\sqrt{a}}} \right ){b}^{6}+63\,\ln \left ( 1/2\,{\frac{2\,\sqrt{a{x}^{2}+bx}\sqrt{a}+2\,ax+b}{\sqrt{a}}} \right ) x{a}^{2}{b}^{5}+21\,\ln \left ( 1/2\,{\frac{2\,\sqrt{a{x}^{2}+bx}\sqrt{a}+2\,ax+b}{\sqrt{a}}} \right ) a{b}^{6} \right ){a}^{-{\frac{13}{2}}}{\frac{1}{\sqrt{ \left ( ax+b \right ) x}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/48*((a*x+b)/x)^(1/2)*x/a^(13/2)*(16*(a*x^2+b*x)^(3/2)*a^(11/2)*x^3-84*(a*x^2+b*x)^(1/2)*a^(11/2)*x^4*b+48*(a
*x^2+b*x)^(3/2)*a^(9/2)*x^2*b-294*(a*x^2+b*x)^(1/2)*a^(9/2)*x^3*b^2+672*a^(9/2)*((a*x+b)*x)^(1/2)*x^3*b^2-336*
a^4*ln(1/2*(2*((a*x+b)*x)^(1/2)*a^(1/2)+2*a*x+b)/a^(1/2))*x^3*b^3+48*(a*x^2+b*x)^(3/2)*a^(7/2)*x*b^2-378*(a*x^
2+b*x)^(1/2)*a^(7/2)*x^2*b^3-384*a^(7/2)*((a*x+b)*x)^(3/2)*x*b^2+2016*a^(7/2)*((a*x+b)*x)^(1/2)*x^2*b^3-1008*a
^3*ln(1/2*(2*((a*x+b)*x)^(1/2)*a^(1/2)+2*a*x+b)/a^(1/2))*x^2*b^4+21*ln(1/2*(2*(a*x^2+b*x)^(1/2)*a^(1/2)+2*a*x+
b)/a^(1/2))*x^3*a^4*b^3+16*(a*x^2+b*x)^(3/2)*a^(5/2)*b^3-210*(a*x^2+b*x)^(1/2)*a^(5/2)*x*b^4-352*b^3*a^(5/2)*(
(a*x+b)*x)^(3/2)+2016*a^(5/2)*((a*x+b)*x)^(1/2)*x*b^4-1008*a^2*ln(1/2*(2*((a*x+b)*x)^(1/2)*a^(1/2)+2*a*x+b)/a^
(1/2))*x*b^5+63*ln(1/2*(2*(a*x^2+b*x)^(1/2)*a^(1/2)+2*a*x+b)/a^(1/2))*x^2*a^3*b^4-42*(a*x^2+b*x)^(1/2)*a^(3/2)
*b^5+672*a^(3/2)*((a*x+b)*x)^(1/2)*b^5-336*a*ln(1/2*(2*((a*x+b)*x)^(1/2)*a^(1/2)+2*a*x+b)/a^(1/2))*b^6+63*ln(1
/2*(2*(a*x^2+b*x)^(1/2)*a^(1/2)+2*a*x+b)/a^(1/2))*x*a^2*b^5+21*ln(1/2*(2*(a*x^2+b*x)^(1/2)*a^(1/2)+2*a*x+b)/a^
(1/2))*a*b^6)/((a*x+b)*x)^(1/2)/(a*x+b)^3

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.88782, size = 609, normalized size = 4.41 \begin{align*} \left [\frac{315 \,{\left (a^{2} b^{3} x^{2} + 2 \, a b^{4} x + b^{5}\right )} \sqrt{a} \log \left (2 \, a x - 2 \, \sqrt{a} x \sqrt{\frac{a x + b}{x}} + b\right ) + 2 \,{\left (8 \, a^{5} x^{5} - 18 \, a^{4} b x^{4} + 63 \, a^{3} b^{2} x^{3} + 420 \, a^{2} b^{3} x^{2} + 315 \, a b^{4} x\right )} \sqrt{\frac{a x + b}{x}}}{48 \,{\left (a^{8} x^{2} + 2 \, a^{7} b x + a^{6} b^{2}\right )}}, \frac{315 \,{\left (a^{2} b^{3} x^{2} + 2 \, a b^{4} x + b^{5}\right )} \sqrt{-a} \arctan \left (\frac{\sqrt{-a} \sqrt{\frac{a x + b}{x}}}{a}\right ) +{\left (8 \, a^{5} x^{5} - 18 \, a^{4} b x^{4} + 63 \, a^{3} b^{2} x^{3} + 420 \, a^{2} b^{3} x^{2} + 315 \, a b^{4} x\right )} \sqrt{\frac{a x + b}{x}}}{24 \,{\left (a^{8} x^{2} + 2 \, a^{7} b x + a^{6} b^{2}\right )}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/48*(315*(a^2*b^3*x^2 + 2*a*b^4*x + b^5)*sqrt(a)*log(2*a*x - 2*sqrt(a)*x*sqrt((a*x + b)/x) + b) + 2*(8*a^5*x
^5 - 18*a^4*b*x^4 + 63*a^3*b^2*x^3 + 420*a^2*b^3*x^2 + 315*a*b^4*x)*sqrt((a*x + b)/x))/(a^8*x^2 + 2*a^7*b*x +
a^6*b^2), 1/24*(315*(a^2*b^3*x^2 + 2*a*b^4*x + b^5)*sqrt(-a)*arctan(sqrt(-a)*sqrt((a*x + b)/x)/a) + (8*a^5*x^5
- 18*a^4*b*x^4 + 63*a^3*b^2*x^3 + 420*a^2*b^3*x^2 + 315*a*b^4*x)*sqrt((a*x + b)/x))/(a^8*x^2 + 2*a^7*b*x + a^
6*b^2)]

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Sympy [B]  time = 12.3749, size = 532, normalized size = 3.86 \begin{align*} \frac{8 a^{\frac{133}{2}} b^{128} x^{72}}{24 a^{\frac{137}{2}} b^{\frac{257}{2}} x^{\frac{137}{2}} \sqrt{\frac{a x}{b} + 1} + 24 a^{\frac{135}{2}} b^{\frac{259}{2}} x^{\frac{135}{2}} \sqrt{\frac{a x}{b} + 1}} - \frac{18 a^{\frac{131}{2}} b^{129} x^{71}}{24 a^{\frac{137}{2}} b^{\frac{257}{2}} x^{\frac{137}{2}} \sqrt{\frac{a x}{b} + 1} + 24 a^{\frac{135}{2}} b^{\frac{259}{2}} x^{\frac{135}{2}} \sqrt{\frac{a x}{b} + 1}} + \frac{63 a^{\frac{129}{2}} b^{130} x^{70}}{24 a^{\frac{137}{2}} b^{\frac{257}{2}} x^{\frac{137}{2}} \sqrt{\frac{a x}{b} + 1} + 24 a^{\frac{135}{2}} b^{\frac{259}{2}} x^{\frac{135}{2}} \sqrt{\frac{a x}{b} + 1}} + \frac{420 a^{\frac{127}{2}} b^{131} x^{69}}{24 a^{\frac{137}{2}} b^{\frac{257}{2}} x^{\frac{137}{2}} \sqrt{\frac{a x}{b} + 1} + 24 a^{\frac{135}{2}} b^{\frac{259}{2}} x^{\frac{135}{2}} \sqrt{\frac{a x}{b} + 1}} + \frac{315 a^{\frac{125}{2}} b^{132} x^{68}}{24 a^{\frac{137}{2}} b^{\frac{257}{2}} x^{\frac{137}{2}} \sqrt{\frac{a x}{b} + 1} + 24 a^{\frac{135}{2}} b^{\frac{259}{2}} x^{\frac{135}{2}} \sqrt{\frac{a x}{b} + 1}} - \frac{315 a^{63} b^{\frac{263}{2}} x^{\frac{137}{2}} \sqrt{\frac{a x}{b} + 1} \operatorname{asinh}{\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{b}} \right )}}{24 a^{\frac{137}{2}} b^{\frac{257}{2}} x^{\frac{137}{2}} \sqrt{\frac{a x}{b} + 1} + 24 a^{\frac{135}{2}} b^{\frac{259}{2}} x^{\frac{135}{2}} \sqrt{\frac{a x}{b} + 1}} - \frac{315 a^{62} b^{\frac{265}{2}} x^{\frac{135}{2}} \sqrt{\frac{a x}{b} + 1} \operatorname{asinh}{\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{b}} \right )}}{24 a^{\frac{137}{2}} b^{\frac{257}{2}} x^{\frac{137}{2}} \sqrt{\frac{a x}{b} + 1} + 24 a^{\frac{135}{2}} b^{\frac{259}{2}} x^{\frac{135}{2}} \sqrt{\frac{a x}{b} + 1}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

8*a**(133/2)*b**128*x**72/(24*a**(137/2)*b**(257/2)*x**(137/2)*sqrt(a*x/b + 1) + 24*a**(135/2)*b**(259/2)*x**(
135/2)*sqrt(a*x/b + 1)) - 18*a**(131/2)*b**129*x**71/(24*a**(137/2)*b**(257/2)*x**(137/2)*sqrt(a*x/b + 1) + 24
*a**(135/2)*b**(259/2)*x**(135/2)*sqrt(a*x/b + 1)) + 63*a**(129/2)*b**130*x**70/(24*a**(137/2)*b**(257/2)*x**(
137/2)*sqrt(a*x/b + 1) + 24*a**(135/2)*b**(259/2)*x**(135/2)*sqrt(a*x/b + 1)) + 420*a**(127/2)*b**131*x**69/(2
4*a**(137/2)*b**(257/2)*x**(137/2)*sqrt(a*x/b + 1) + 24*a**(135/2)*b**(259/2)*x**(135/2)*sqrt(a*x/b + 1)) + 31
5*a**(125/2)*b**132*x**68/(24*a**(137/2)*b**(257/2)*x**(137/2)*sqrt(a*x/b + 1) + 24*a**(135/2)*b**(259/2)*x**(
135/2)*sqrt(a*x/b + 1)) - 315*a**63*b**(263/2)*x**(137/2)*sqrt(a*x/b + 1)*asinh(sqrt(a)*sqrt(x)/sqrt(b))/(24*a
**(137/2)*b**(257/2)*x**(137/2)*sqrt(a*x/b + 1) + 24*a**(135/2)*b**(259/2)*x**(135/2)*sqrt(a*x/b + 1)) - 315*a
**62*b**(265/2)*x**(135/2)*sqrt(a*x/b + 1)*asinh(sqrt(a)*sqrt(x)/sqrt(b))/(24*a**(137/2)*b**(257/2)*x**(137/2)
*sqrt(a*x/b + 1) + 24*a**(135/2)*b**(259/2)*x**(135/2)*sqrt(a*x/b + 1))

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Giac [A]  time = 1.26409, size = 203, normalized size = 1.47 \begin{align*} \frac{1}{24} \, b{\left (\frac{315 \, b^{2} \arctan \left (\frac{\sqrt{\frac{a x + b}{x}}}{\sqrt{-a}}\right )}{\sqrt{-a} a^{5}} + \frac{16 \, a^{4} b^{2} + \frac{144 \,{\left (a x + b\right )} a^{3} b^{2}}{x} - \frac{693 \,{\left (a x + b\right )}^{2} a^{2} b^{2}}{x^{2}} + \frac{840 \,{\left (a x + b\right )}^{3} a b^{2}}{x^{3}} - \frac{315 \,{\left (a x + b\right )}^{4} b^{2}}{x^{4}}}{{\left (a \sqrt{\frac{a x + b}{x}} - \frac{{\left (a x + b\right )} \sqrt{\frac{a x + b}{x}}}{x}\right )}^{3} a^{5}}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/24*b*(315*b^2*arctan(sqrt((a*x + b)/x)/sqrt(-a))/(sqrt(-a)*a^5) + (16*a^4*b^2 + 144*(a*x + b)*a^3*b^2/x - 69
3*(a*x + b)^2*a^2*b^2/x^2 + 840*(a*x + b)^3*a*b^2/x^3 - 315*(a*x + b)^4*b^2/x^4)/((a*sqrt((a*x + b)/x) - (a*x
+ b)*sqrt((a*x + b)/x)/x)^3*a^5))