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

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

[Out]

35/8*b^3/a^4/(a+b/x)^(3/2)+21/8*b^2*x/a^3/(a+b/x)^(3/2)-3/4*b*x^2/a^2/(a+b/x)^(3/2)+1/3*x^3/a/(a+b/x)^(3/2)-10
5/8*b^3*arctanh((a+b/x)^(1/2)/a^(1/2))/a^(11/2)+105/8*b^3/a^5/(a+b/x)^(1/2)

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Rubi [A]  time = 0.07, 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 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]

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]]

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*}

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Mathematica [C]  time = 0.02, 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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fricas [A]  time = 1.10, size = 273, normalized size = 1.98 \[ \left [\frac {315 \, {\left (a^{2} b^{3} x^{2} + 2 \, a b^{4} x + b^{5}\right )} \sqrt {a} \log \left (2 \, a x - 2 \, \sqrt {a} x \sqrt {\frac {a x + b}{x}} + b\right ) + 2 \, {\left (8 \, a^{5} x^{5} - 18 \, a^{4} b x^{4} + 63 \, a^{3} b^{2} x^{3} + 420 \, a^{2} b^{3} x^{2} + 315 \, a b^{4} x\right )} \sqrt {\frac {a x + b}{x}}}{48 \, {\left (a^{8} x^{2} + 2 \, a^{7} b x + a^{6} b^{2}\right )}}, \frac {315 \, {\left (a^{2} b^{3} x^{2} + 2 \, a b^{4} x + b^{5}\right )} \sqrt {-a} \arctan \left (\frac {\sqrt {-a} \sqrt {\frac {a x + b}{x}}}{a}\right ) + {\left (8 \, a^{5} x^{5} - 18 \, a^{4} b x^{4} + 63 \, a^{3} b^{2} x^{3} + 420 \, a^{2} b^{3} x^{2} + 315 \, a b^{4} x\right )} \sqrt {\frac {a x + b}{x}}}{24 \, {\left (a^{8} x^{2} + 2 \, a^{7} b x + a^{6} b^{2}\right )}}\right ] \]

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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giac [A]  time = 0.18, size = 134, normalized size = 0.97 \[ \frac {1}{24} \, b^{3} {\left (\frac {315 \, \arctan \left (\frac {\sqrt {\frac {a x + b}{x}}}{\sqrt {-a}}\right )}{\sqrt {-a} a^{5}} + \frac {16 \, a^{4} + \frac {144 \, {\left (a x + b\right )} a^{3}}{x} - \frac {693 \, {\left (a x + b\right )}^{2} a^{2}}{x^{2}} + \frac {840 \, {\left (a x + b\right )}^{3} a}{x^{3}} - \frac {315 \, {\left (a x + b\right )}^{4}}{x^{4}}}{{\left (a \sqrt {\frac {a x + b}{x}} - \frac {{\left (a x + b\right )} \sqrt {\frac {a x + b}{x}}}{x}\right )}^{3} a^{5}}\right )} \]

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^3*(315*arctan(sqrt((a*x + b)/x)/sqrt(-a))/(sqrt(-a)*a^5) + (16*a^4 + 144*(a*x + b)*a^3/x - 693*(a*x + b
)^2*a^2/x^2 + 840*(a*x + b)^3*a/x^3 - 315*(a*x + b)^4/x^4)/((a*sqrt((a*x + b)/x) - (a*x + b)*sqrt((a*x + b)/x)
/x)^3*a^5))

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maple [B]  time = 0.02, size = 616, normalized size = 4.46 \[ \frac {\sqrt {\frac {a x +b}{x}}\, \left (-336 a^{4} b^{3} x^{3} \ln \left (\frac {2 a x +b +2 \sqrt {\left (a x +b \right ) x}\, \sqrt {a}}{2 \sqrt {a}}\right )+21 a^{4} b^{3} x^{3} \ln \left (\frac {2 a x +b +2 \sqrt {a \,x^{2}+b x}\, \sqrt {a}}{2 \sqrt {a}}\right )-84 \sqrt {a \,x^{2}+b x}\, a^{\frac {11}{2}} b \,x^{4}-1008 a^{3} b^{4} x^{2} \ln \left (\frac {2 a x +b +2 \sqrt {\left (a x +b \right ) x}\, \sqrt {a}}{2 \sqrt {a}}\right )+63 a^{3} b^{4} x^{2} \ln \left (\frac {2 a x +b +2 \sqrt {a \,x^{2}+b x}\, \sqrt {a}}{2 \sqrt {a}}\right )-294 \sqrt {a \,x^{2}+b x}\, a^{\frac {9}{2}} b^{2} x^{3}+672 \sqrt {\left (a x +b \right ) x}\, a^{\frac {9}{2}} b^{2} x^{3}-1008 a^{2} b^{5} x \ln \left (\frac {2 a x +b +2 \sqrt {\left (a x +b \right ) x}\, \sqrt {a}}{2 \sqrt {a}}\right )+63 a^{2} b^{5} x \ln \left (\frac {2 a x +b +2 \sqrt {a \,x^{2}+b x}\, \sqrt {a}}{2 \sqrt {a}}\right )+16 \left (a \,x^{2}+b x \right )^{\frac {3}{2}} a^{\frac {11}{2}} x^{3}-378 \sqrt {a \,x^{2}+b x}\, a^{\frac {7}{2}} b^{3} x^{2}+2016 \sqrt {\left (a x +b \right ) x}\, a^{\frac {7}{2}} b^{3} x^{2}-336 a \,b^{6} \ln \left (\frac {2 a x +b +2 \sqrt {\left (a x +b \right ) x}\, \sqrt {a}}{2 \sqrt {a}}\right )+21 a \,b^{6} \ln \left (\frac {2 a x +b +2 \sqrt {a \,x^{2}+b x}\, \sqrt {a}}{2 \sqrt {a}}\right )+48 \left (a \,x^{2}+b x \right )^{\frac {3}{2}} a^{\frac {9}{2}} b \,x^{2}-210 \sqrt {a \,x^{2}+b x}\, a^{\frac {5}{2}} b^{4} x +2016 \sqrt {\left (a x +b \right ) x}\, a^{\frac {5}{2}} b^{4} x +48 \left (a \,x^{2}+b x \right )^{\frac {3}{2}} a^{\frac {7}{2}} b^{2} x -384 \left (\left (a x +b \right ) x \right )^{\frac {3}{2}} a^{\frac {7}{2}} b^{2} x -42 \sqrt {a \,x^{2}+b x}\, a^{\frac {3}{2}} b^{5}+672 \sqrt {\left (a x +b \right ) x}\, a^{\frac {3}{2}} b^{5}+16 \left (a \,x^{2}+b x \right )^{\frac {3}{2}} a^{\frac {5}{2}} b^{3}-352 \left (\left (a x +b \right ) x \right )^{\frac {3}{2}} a^{\frac {5}{2}} b^{3}\right ) x}{48 \sqrt {\left (a x +b \right ) x}\, \left (a x +b \right )^{3} a^{\frac {13}{2}}} \]

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+2*((a*x+b)*x)^(1/2)*a^(1/2))/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+2*((a*x+b)*x)^(1/2)*a^(1/2))/a^(1/2))*x^2*b^4+21*ln(1/2*(2*a*x+b+2*(a*x^2+b*x)^(1/2)*a^(1/2
))/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+2*((a*x+b)*x)^(1/2)*a^(1/2))/a^
(1/2))*x*b^5+63*ln(1/2*(2*a*x+b+2*(a*x^2+b*x)^(1/2)*a^(1/2))/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+2*((a*x+b)*x)^(1/2)*a^(1/2))/a^(1/2))*b^6+63*ln(1
/2*(2*a*x+b+2*(a*x^2+b*x)^(1/2)*a^(1/2))/a^(1/2))*x*a^2*b^5+21*ln(1/2*(2*a*x+b+2*(a*x^2+b*x)^(1/2)*a^(1/2))/a^
(1/2))*a*b^6)/((a*x+b)*x)^(1/2)/(a*x+b)^3

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maxima [A]  time = 2.33, size = 171, normalized size = 1.24 \[ \frac {315 \, {\left (a + \frac {b}{x}\right )}^{4} b^{3} - 840 \, {\left (a + \frac {b}{x}\right )}^{3} a b^{3} + 693 \, {\left (a + \frac {b}{x}\right )}^{2} a^{2} b^{3} - 144 \, {\left (a + \frac {b}{x}\right )} a^{3} b^{3} - 16 \, a^{4} b^{3}}{24 \, {\left ({\left (a + \frac {b}{x}\right )}^{\frac {9}{2}} a^{5} - 3 \, {\left (a + \frac {b}{x}\right )}^{\frac {7}{2}} a^{6} + 3 \, {\left (a + \frac {b}{x}\right )}^{\frac {5}{2}} a^{7} - {\left (a + \frac {b}{x}\right )}^{\frac {3}{2}} a^{8}\right )}} + \frac {105 \, b^{3} \log \left (\frac {\sqrt {a + \frac {b}{x}} - \sqrt {a}}{\sqrt {a + \frac {b}{x}} + \sqrt {a}}\right )}{16 \, a^{\frac {11}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/24*(315*(a + b/x)^4*b^3 - 840*(a + b/x)^3*a*b^3 + 693*(a + b/x)^2*a^2*b^3 - 144*(a + b/x)*a^3*b^3 - 16*a^4*b
^3)/((a + b/x)^(9/2)*a^5 - 3*(a + b/x)^(7/2)*a^6 + 3*(a + b/x)^(5/2)*a^7 - (a + b/x)^(3/2)*a^8) + 105/16*b^3*l
og((sqrt(a + b/x) - sqrt(a))/(sqrt(a + b/x) + sqrt(a)))/a^(11/2)

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mupad [B]  time = 1.79, size = 113, normalized size = 0.82 \[ \frac {35\,b^3}{2\,a^4\,{\left (a+\frac {b}{x}\right )}^{3/2}}-\frac {105\,b^3\,\mathrm {atanh}\left (\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {a}}\right )}{8\,a^{11/2}}+\frac {x^3}{3\,a\,{\left (a+\frac {b}{x}\right )}^{3/2}}-\frac {3\,b\,x^2}{4\,a^2\,{\left (a+\frac {b}{x}\right )}^{3/2}}+\frac {21\,b^2\,x}{8\,a^3\,{\left (a+\frac {b}{x}\right )}^{3/2}}+\frac {105\,b^4}{8\,a^5\,x\,{\left (a+\frac {b}{x}\right )}^{3/2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2/(a + b/x)^(5/2),x)

[Out]

(35*b^3)/(2*a^4*(a + b/x)^(3/2)) - (105*b^3*atanh((a + b/x)^(1/2)/a^(1/2)))/(8*a^(11/2)) + x^3/(3*a*(a + b/x)^
(3/2)) - (3*b*x^2)/(4*a^2*(a + b/x)^(3/2)) + (21*b^2*x)/(8*a^3*(a + b/x)^(3/2)) + (105*b^4)/(8*a^5*x*(a + b/x)
^(3/2))

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sympy [B]  time = 17.64, size = 532, normalized size = 3.86 \[ \frac {8 a^{\frac {133}{2}} b^{128} x^{72}}{24 a^{\frac {137}{2}} b^{\frac {257}{2}} x^{\frac {137}{2}} \sqrt {\frac {a x}{b} + 1} + 24 a^{\frac {135}{2}} b^{\frac {259}{2}} x^{\frac {135}{2}} \sqrt {\frac {a x}{b} + 1}} - \frac {18 a^{\frac {131}{2}} b^{129} x^{71}}{24 a^{\frac {137}{2}} b^{\frac {257}{2}} x^{\frac {137}{2}} \sqrt {\frac {a x}{b} + 1} + 24 a^{\frac {135}{2}} b^{\frac {259}{2}} x^{\frac {135}{2}} \sqrt {\frac {a x}{b} + 1}} + \frac {63 a^{\frac {129}{2}} b^{130} x^{70}}{24 a^{\frac {137}{2}} b^{\frac {257}{2}} x^{\frac {137}{2}} \sqrt {\frac {a x}{b} + 1} + 24 a^{\frac {135}{2}} b^{\frac {259}{2}} x^{\frac {135}{2}} \sqrt {\frac {a x}{b} + 1}} + \frac {420 a^{\frac {127}{2}} b^{131} x^{69}}{24 a^{\frac {137}{2}} b^{\frac {257}{2}} x^{\frac {137}{2}} \sqrt {\frac {a x}{b} + 1} + 24 a^{\frac {135}{2}} b^{\frac {259}{2}} x^{\frac {135}{2}} \sqrt {\frac {a x}{b} + 1}} + \frac {315 a^{\frac {125}{2}} b^{132} x^{68}}{24 a^{\frac {137}{2}} b^{\frac {257}{2}} x^{\frac {137}{2}} \sqrt {\frac {a x}{b} + 1} + 24 a^{\frac {135}{2}} b^{\frac {259}{2}} x^{\frac {135}{2}} \sqrt {\frac {a x}{b} + 1}} - \frac {315 a^{63} b^{\frac {263}{2}} x^{\frac {137}{2}} \sqrt {\frac {a x}{b} + 1} \operatorname {asinh}{\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}} \right )}}{24 a^{\frac {137}{2}} b^{\frac {257}{2}} x^{\frac {137}{2}} \sqrt {\frac {a x}{b} + 1} + 24 a^{\frac {135}{2}} b^{\frac {259}{2}} x^{\frac {135}{2}} \sqrt {\frac {a x}{b} + 1}} - \frac {315 a^{62} b^{\frac {265}{2}} x^{\frac {135}{2}} \sqrt {\frac {a x}{b} + 1} \operatorname {asinh}{\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}} \right )}}{24 a^{\frac {137}{2}} b^{\frac {257}{2}} x^{\frac {137}{2}} \sqrt {\frac {a x}{b} + 1} + 24 a^{\frac {135}{2}} b^{\frac {259}{2}} x^{\frac {135}{2}} \sqrt {\frac {a x}{b} + 1}} \]

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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