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

Optimal. Leaf size=152 \[ a^{3/2} c (4 a d+5 b c) \tanh ^{-1}\left (\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {a}}\right )+\frac {c^2 x \left (a+\frac {b}{x}\right )^{7/2}}{a}-\frac {c \left (a+\frac {b}{x}\right )^{5/2} (4 a d+5 b c)}{5 a}-\frac {1}{3} c \left (a+\frac {b}{x}\right )^{3/2} (4 a d+5 b c)-a c \sqrt {a+\frac {b}{x}} (4 a d+5 b c)-\frac {2 d^2 \left (a+\frac {b}{x}\right )^{7/2}}{7 b} \]

[Out]

-1/3*c*(4*a*d+5*b*c)*(a+b/x)^(3/2)-1/5*c*(4*a*d+5*b*c)*(a+b/x)^(5/2)/a-2/7*d^2*(a+b/x)^(7/2)/b+c^2*(a+b/x)^(7/
2)*x/a+a^(3/2)*c*(4*a*d+5*b*c)*arctanh((a+b/x)^(1/2)/a^(1/2))-a*c*(4*a*d+5*b*c)*(a+b/x)^(1/2)

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Rubi [A]  time = 0.10, antiderivative size = 152, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {375, 89, 80, 50, 63, 208} \[ a^{3/2} c (4 a d+5 b c) \tanh ^{-1}\left (\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {a}}\right )+\frac {c^2 x \left (a+\frac {b}{x}\right )^{7/2}}{a}-\frac {c \left (a+\frac {b}{x}\right )^{5/2} (4 a d+5 b c)}{5 a}-\frac {1}{3} c \left (a+\frac {b}{x}\right )^{3/2} (4 a d+5 b c)-a c \sqrt {a+\frac {b}{x}} (4 a d+5 b c)-\frac {2 d^2 \left (a+\frac {b}{x}\right )^{7/2}}{7 b} \]

Antiderivative was successfully verified.

[In]

Int[(a + b/x)^(5/2)*(c + d/x)^2,x]

[Out]

-(a*c*(5*b*c + 4*a*d)*Sqrt[a + b/x]) - (c*(5*b*c + 4*a*d)*(a + b/x)^(3/2))/3 - (c*(5*b*c + 4*a*d)*(a + b/x)^(5
/2))/(5*a) - (2*d^2*(a + b/x)^(7/2))/(7*b) + (c^2*(a + b/x)^(7/2)*x)/a + a^(3/2)*c*(5*b*c + 4*a*d)*ArcTanh[Sqr
t[a + b/x]/Sqrt[a]]

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 80

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*(c + d*x)
^(n + 1)*(e + f*x)^(p + 1))/(d*f*(n + p + 2)), x] + Dist[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(
d*f*(n + p + 2)), Int[(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 2,
0]

Rule 89

Int[((a_.) + (b_.)*(x_))^2*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[((b*c - a*
d)^2*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/(d^2*(d*e - c*f)*(n + 1)), x] - Dist[1/(d^2*(d*e - c*f)*(n + 1)), In
t[(c + d*x)^(n + 1)*(e + f*x)^p*Simp[a^2*d^2*f*(n + p + 2) + b^2*c*(d*e*(n + 1) + c*f*(p + 1)) - 2*a*b*d*(d*e*
(n + 1) + c*f*(p + 1)) - b^2*d*(d*e - c*f)*(n + 1)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && (LtQ
[n, -1] || (EqQ[n + p + 3, 0] && NeQ[n, -1] && (SumSimplerQ[n, 1] ||  !SumSimplerQ[p, 1])))

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 375

Int[((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> -Subst[Int[((a + b/x^n)^p*(c +
d/x^n)^q)/x^2, x], x, 1/x] /; FreeQ[{a, b, c, d, p, q}, x] && NeQ[b*c - a*d, 0] && ILtQ[n, 0]

Rubi steps

\begin {align*} \int \left (a+\frac {b}{x}\right )^{5/2} \left (c+\frac {d}{x}\right )^2 \, dx &=-\operatorname {Subst}\left (\int \frac {(a+b x)^{5/2} (c+d x)^2}{x^2} \, dx,x,\frac {1}{x}\right )\\ &=\frac {c^2 \left (a+\frac {b}{x}\right )^{7/2} x}{a}-\frac {\operatorname {Subst}\left (\int \frac {(a+b x)^{5/2} \left (\frac {1}{2} c (5 b c+4 a d)+a d^2 x\right )}{x} \, dx,x,\frac {1}{x}\right )}{a}\\ &=-\frac {2 d^2 \left (a+\frac {b}{x}\right )^{7/2}}{7 b}+\frac {c^2 \left (a+\frac {b}{x}\right )^{7/2} x}{a}-\frac {(c (5 b c+4 a d)) \operatorname {Subst}\left (\int \frac {(a+b x)^{5/2}}{x} \, dx,x,\frac {1}{x}\right )}{2 a}\\ &=-\frac {c (5 b c+4 a d) \left (a+\frac {b}{x}\right )^{5/2}}{5 a}-\frac {2 d^2 \left (a+\frac {b}{x}\right )^{7/2}}{7 b}+\frac {c^2 \left (a+\frac {b}{x}\right )^{7/2} x}{a}-\frac {1}{2} (c (5 b c+4 a d)) \operatorname {Subst}\left (\int \frac {(a+b x)^{3/2}}{x} \, dx,x,\frac {1}{x}\right )\\ &=-\frac {1}{3} c (5 b c+4 a d) \left (a+\frac {b}{x}\right )^{3/2}-\frac {c (5 b c+4 a d) \left (a+\frac {b}{x}\right )^{5/2}}{5 a}-\frac {2 d^2 \left (a+\frac {b}{x}\right )^{7/2}}{7 b}+\frac {c^2 \left (a+\frac {b}{x}\right )^{7/2} x}{a}-\frac {1}{2} (a c (5 b c+4 a d)) \operatorname {Subst}\left (\int \frac {\sqrt {a+b x}}{x} \, dx,x,\frac {1}{x}\right )\\ &=-a c (5 b c+4 a d) \sqrt {a+\frac {b}{x}}-\frac {1}{3} c (5 b c+4 a d) \left (a+\frac {b}{x}\right )^{3/2}-\frac {c (5 b c+4 a d) \left (a+\frac {b}{x}\right )^{5/2}}{5 a}-\frac {2 d^2 \left (a+\frac {b}{x}\right )^{7/2}}{7 b}+\frac {c^2 \left (a+\frac {b}{x}\right )^{7/2} x}{a}-\frac {1}{2} \left (a^2 c (5 b c+4 a d)\right ) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,\frac {1}{x}\right )\\ &=-a c (5 b c+4 a d) \sqrt {a+\frac {b}{x}}-\frac {1}{3} c (5 b c+4 a d) \left (a+\frac {b}{x}\right )^{3/2}-\frac {c (5 b c+4 a d) \left (a+\frac {b}{x}\right )^{5/2}}{5 a}-\frac {2 d^2 \left (a+\frac {b}{x}\right )^{7/2}}{7 b}+\frac {c^2 \left (a+\frac {b}{x}\right )^{7/2} x}{a}-\frac {\left (a^2 c (5 b c+4 a d)\right ) \operatorname {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+\frac {b}{x}}\right )}{b}\\ &=-a c (5 b c+4 a d) \sqrt {a+\frac {b}{x}}-\frac {1}{3} c (5 b c+4 a d) \left (a+\frac {b}{x}\right )^{3/2}-\frac {c (5 b c+4 a d) \left (a+\frac {b}{x}\right )^{5/2}}{5 a}-\frac {2 d^2 \left (a+\frac {b}{x}\right )^{7/2}}{7 b}+\frac {c^2 \left (a+\frac {b}{x}\right )^{7/2} x}{a}+a^{3/2} c (5 b c+4 a d) \tanh ^{-1}\left (\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {a}}\right )\\ \end {align*}

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Mathematica [A]  time = 0.16, size = 121, normalized size = 0.80 \[ -\frac {c (4 a d+5 b c) \left (\sqrt {a+\frac {b}{x}} \left (23 a^2 x^2+11 a b x+3 b^2\right )-15 a^{5/2} x^2 \tanh ^{-1}\left (\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {a}}\right )\right )}{15 a x^2}+\frac {c^2 x \left (a+\frac {b}{x}\right )^{7/2}}{a}-\frac {2 d^2 \left (a+\frac {b}{x}\right )^{7/2}}{7 b} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b/x)^(5/2)*(c + d/x)^2,x]

[Out]

(-2*d^2*(a + b/x)^(7/2))/(7*b) + (c^2*(a + b/x)^(7/2)*x)/a - (c*(5*b*c + 4*a*d)*(Sqrt[a + b/x]*(3*b^2 + 11*a*b
*x + 23*a^2*x^2) - 15*a^(5/2)*x^2*ArcTanh[Sqrt[a + b/x]/Sqrt[a]]))/(15*a*x^2)

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fricas [A]  time = 1.02, size = 350, normalized size = 2.30 \[ \left [\frac {105 \, {\left (5 \, a b^{2} c^{2} + 4 \, a^{2} b c d\right )} \sqrt {a} x^{3} \log \left (2 \, a x + 2 \, \sqrt {a} x \sqrt {\frac {a x + b}{x}} + b\right ) + 2 \, {\left (105 \, a^{2} b c^{2} x^{4} - 30 \, b^{3} d^{2} - 2 \, {\left (245 \, a b^{2} c^{2} + 322 \, a^{2} b c d + 15 \, a^{3} d^{2}\right )} x^{3} - 2 \, {\left (35 \, b^{3} c^{2} + 154 \, a b^{2} c d + 45 \, a^{2} b d^{2}\right )} x^{2} - 6 \, {\left (14 \, b^{3} c d + 15 \, a b^{2} d^{2}\right )} x\right )} \sqrt {\frac {a x + b}{x}}}{210 \, b x^{3}}, -\frac {105 \, {\left (5 \, a b^{2} c^{2} + 4 \, a^{2} b c d\right )} \sqrt {-a} x^{3} \arctan \left (\frac {\sqrt {-a} \sqrt {\frac {a x + b}{x}}}{a}\right ) - {\left (105 \, a^{2} b c^{2} x^{4} - 30 \, b^{3} d^{2} - 2 \, {\left (245 \, a b^{2} c^{2} + 322 \, a^{2} b c d + 15 \, a^{3} d^{2}\right )} x^{3} - 2 \, {\left (35 \, b^{3} c^{2} + 154 \, a b^{2} c d + 45 \, a^{2} b d^{2}\right )} x^{2} - 6 \, {\left (14 \, b^{3} c d + 15 \, a b^{2} d^{2}\right )} x\right )} \sqrt {\frac {a x + b}{x}}}{105 \, b x^{3}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/210*(105*(5*a*b^2*c^2 + 4*a^2*b*c*d)*sqrt(a)*x^3*log(2*a*x + 2*sqrt(a)*x*sqrt((a*x + b)/x) + b) + 2*(105*a^
2*b*c^2*x^4 - 30*b^3*d^2 - 2*(245*a*b^2*c^2 + 322*a^2*b*c*d + 15*a^3*d^2)*x^3 - 2*(35*b^3*c^2 + 154*a*b^2*c*d
+ 45*a^2*b*d^2)*x^2 - 6*(14*b^3*c*d + 15*a*b^2*d^2)*x)*sqrt((a*x + b)/x))/(b*x^3), -1/105*(105*(5*a*b^2*c^2 +
4*a^2*b*c*d)*sqrt(-a)*x^3*arctan(sqrt(-a)*sqrt((a*x + b)/x)/a) - (105*a^2*b*c^2*x^4 - 30*b^3*d^2 - 2*(245*a*b^
2*c^2 + 322*a^2*b*c*d + 15*a^3*d^2)*x^3 - 2*(35*b^3*c^2 + 154*a*b^2*c*d + 45*a^2*b*d^2)*x^2 - 6*(14*b^3*c*d +
15*a*b^2*d^2)*x)*sqrt((a*x + b)/x))/(b*x^3)]

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giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,x):;OUTPUT:Warn
ing, integration of abs or sign assumes constant sign by intervals (correct if the argument is real):Check [ab
s(x)]Warning, choosing root of [1,0,%%%{-2,[1,2,0]%%%}+%%%{-2,[1,0,0]%%%}+%%%{-2,[0,1,1]%%%},0,%%%{1,[2,4,0]%%
%}+%%%{-2,[2,2,0]%%%}+%%%{1,[2,0,0]%%%}+%%%{2,[1,3,1]%%%}+%%%{-2,[1,1,1]%%%}+%%%{1,[0,2,2]%%%}] at parameters
values [86,-97,-82]Warning, choosing root of [1,0,%%%{-2,[1,2,0]%%%}+%%%{-2,[1,0,0]%%%}+%%%{-2,[0,1,1]%%%},0,%
%%{1,[2,4,0]%%%}+%%%{-2,[2,2,0]%%%}+%%%{1,[2,0,0]%%%}+%%%{2,[1,3,1]%%%}+%%%{-2,[1,1,1]%%%}+%%%{1,[0,2,2]%%%}]
at parameters values [7,-27,26]Warning, choosing root of [1,0,%%%{-4,[1,0,0]%%%}+%%%{-2,[0,1,1]%%%},0,%%%{1,[0
,2,2]%%%}] at parameters values [18.6420984049,-49,-86]Warning, choosing root of [1,0,%%%{-4,[1,0,0]%%%}+%%%{-
2,[0,1,1]%%%},0,%%%{1,[0,2,2]%%%}] at parameters values [78.6493344628,22,42]Warning, choosing root of [1,0,%%
%{-2,[1,0,1]%%%}+%%%{-4,[0,1,0]%%%},0,%%%{1,[2,0,2]%%%}] at parameters values [-13,74.7709350525,24]Sign error
(%%%{-b,0%%%}+%%%{2*sqrt(a)*sqrt(b),1/2%%%}+%%%{-2*a,1%%%}+%%%{a*sqrt(a)*sqrt(b)/b,3/2%%%}+%%%{-a^2*sqrt(a)*s
qrt(b)/(4*b^2),5/2%%%}+%%%{undef,7/2%%%})Evaluation time: 0.81Limit: Max order reached or unable to make serie
s expansion Error: Bad Argument Value

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maple [B]  time = 0.06, size = 336, normalized size = 2.21 \[ -\frac {\sqrt {\frac {a x +b}{x}}\, \left (-420 a^{3} b c d \,x^{5} \ln \left (\frac {2 a x +b +2 \sqrt {a \,x^{2}+b x}\, \sqrt {a}}{2 \sqrt {a}}\right )-525 a^{2} b^{2} c^{2} x^{5} \ln \left (\frac {2 a x +b +2 \sqrt {a \,x^{2}+b x}\, \sqrt {a}}{2 \sqrt {a}}\right )-840 \sqrt {a \,x^{2}+b x}\, a^{\frac {7}{2}} c d \,x^{5}-1050 \sqrt {a \,x^{2}+b x}\, a^{\frac {5}{2}} b \,c^{2} x^{5}+840 \left (a \,x^{2}+b x \right )^{\frac {3}{2}} a^{\frac {5}{2}} c d \,x^{3}+840 \left (a \,x^{2}+b x \right )^{\frac {3}{2}} a^{\frac {3}{2}} b \,c^{2} x^{3}+60 \left (a \,x^{2}+b x \right )^{\frac {3}{2}} a^{\frac {5}{2}} d^{2} x^{2}+448 \left (a \,x^{2}+b x \right )^{\frac {3}{2}} a^{\frac {3}{2}} b c d \,x^{2}+140 \left (a \,x^{2}+b x \right )^{\frac {3}{2}} \sqrt {a}\, b^{2} c^{2} x^{2}+120 \left (a \,x^{2}+b x \right )^{\frac {3}{2}} a^{\frac {3}{2}} b \,d^{2} x +168 \left (a \,x^{2}+b x \right )^{\frac {3}{2}} \sqrt {a}\, b^{2} c d x +60 \left (a \,x^{2}+b x \right )^{\frac {3}{2}} \sqrt {a}\, b^{2} d^{2}\right )}{210 \sqrt {\left (a x +b \right ) x}\, \sqrt {a}\, b \,x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/210*((a*x+b)/x)^(1/2)/x^4/b*(-840*(a*x^2+b*x)^(1/2)*a^(7/2)*x^5*c*d-1050*(a*x^2+b*x)^(1/2)*a^(5/2)*x^5*b*c^
2-420*ln(1/2*(2*a*x+b+2*(a*x^2+b*x)^(1/2)*a^(1/2))/a^(1/2))*x^5*a^3*b*c*d-525*ln(1/2*(2*a*x+b+2*(a*x^2+b*x)^(1
/2)*a^(1/2))/a^(1/2))*x^5*a^2*b^2*c^2+840*(a*x^2+b*x)^(3/2)*a^(5/2)*x^3*c*d+840*(a*x^2+b*x)^(3/2)*a^(3/2)*x^3*
b*c^2+60*(a*x^2+b*x)^(3/2)*a^(5/2)*x^2*d^2+448*(a*x^2+b*x)^(3/2)*a^(3/2)*x^2*b*c*d+140*(a*x^2+b*x)^(3/2)*a^(1/
2)*x^2*b^2*c^2+120*(a*x^2+b*x)^(3/2)*a^(3/2)*x*b*d^2+168*(a*x^2+b*x)^(3/2)*a^(1/2)*x*b^2*c*d+60*(a*x^2+b*x)^(3
/2)*a^(1/2)*b^2*d^2)/((a*x+b)*x)^(1/2)/a^(1/2)

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maxima [A]  time = 1.25, size = 181, normalized size = 1.19 \[ -\frac {2 \, {\left (a + \frac {b}{x}\right )}^{\frac {7}{2}} d^{2}}{7 \, b} + \frac {1}{6} \, {\left (6 \, \sqrt {a + \frac {b}{x}} a^{2} x - 15 \, a^{\frac {3}{2}} b \log \left (\frac {\sqrt {a + \frac {b}{x}} - \sqrt {a}}{\sqrt {a + \frac {b}{x}} + \sqrt {a}}\right ) - 4 \, {\left (a + \frac {b}{x}\right )}^{\frac {3}{2}} b - 24 \, \sqrt {a + \frac {b}{x}} a b\right )} c^{2} - \frac {2}{15} \, {\left (15 \, a^{\frac {5}{2}} \log \left (\frac {\sqrt {a + \frac {b}{x}} - \sqrt {a}}{\sqrt {a + \frac {b}{x}} + \sqrt {a}}\right ) + 6 \, {\left (a + \frac {b}{x}\right )}^{\frac {5}{2}} + 10 \, {\left (a + \frac {b}{x}\right )}^{\frac {3}{2}} a + 30 \, \sqrt {a + \frac {b}{x}} a^{2}\right )} c d \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/7*(a + b/x)^(7/2)*d^2/b + 1/6*(6*sqrt(a + b/x)*a^2*x - 15*a^(3/2)*b*log((sqrt(a + b/x) - sqrt(a))/(sqrt(a +
b/x) + sqrt(a))) - 4*(a + b/x)^(3/2)*b - 24*sqrt(a + b/x)*a*b)*c^2 - 2/15*(15*a^(5/2)*log((sqrt(a + b/x) - sq
rt(a))/(sqrt(a + b/x) + sqrt(a))) + 6*(a + b/x)^(5/2) + 10*(a + b/x)^(3/2)*a + 30*sqrt(a + b/x)*a^2)*c*d

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mupad [B]  time = 3.79, size = 271, normalized size = 1.78 \[ {\left (a+\frac {b}{x}\right )}^{3/2}\,\left (\frac {2\,a\,\left (\frac {4\,a\,d^2-4\,b\,c\,d}{b}-\frac {4\,a\,d^2}{b}\right )}{3}-\frac {2\,{\left (a\,d-b\,c\right )}^2}{3\,b}+\frac {2\,a^2\,d^2}{3\,b}\right )+\left (\frac {4\,a\,d^2-4\,b\,c\,d}{5\,b}-\frac {4\,a\,d^2}{5\,b}\right )\,{\left (a+\frac {b}{x}\right )}^{5/2}-\sqrt {a+\frac {b}{x}}\,\left (a^2\,\left (\frac {4\,a\,d^2-4\,b\,c\,d}{b}-\frac {4\,a\,d^2}{b}\right )-2\,a\,\left (2\,a\,\left (\frac {4\,a\,d^2-4\,b\,c\,d}{b}-\frac {4\,a\,d^2}{b}\right )-\frac {2\,{\left (a\,d-b\,c\right )}^2}{b}+\frac {2\,a^2\,d^2}{b}\right )\right )-\frac {2\,d^2\,{\left (a+\frac {b}{x}\right )}^{7/2}}{7\,b}+a^2\,c^2\,x\,\sqrt {a+\frac {b}{x}}-a^{3/2}\,c\,\mathrm {atan}\left (\frac {\sqrt {a+\frac {b}{x}}\,1{}\mathrm {i}}{\sqrt {a}}\right )\,\left (4\,a\,d+5\,b\,c\right )\,1{}\mathrm {i} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b/x)^(5/2)*(c + d/x)^2,x)

[Out]

(a + b/x)^(3/2)*((2*a*((4*a*d^2 - 4*b*c*d)/b - (4*a*d^2)/b))/3 - (2*(a*d - b*c)^2)/(3*b) + (2*a^2*d^2)/(3*b))
+ ((4*a*d^2 - 4*b*c*d)/(5*b) - (4*a*d^2)/(5*b))*(a + b/x)^(5/2) - (a + b/x)^(1/2)*(a^2*((4*a*d^2 - 4*b*c*d)/b
- (4*a*d^2)/b) - 2*a*(2*a*((4*a*d^2 - 4*b*c*d)/b - (4*a*d^2)/b) - (2*(a*d - b*c)^2)/b + (2*a^2*d^2)/b)) - (2*d
^2*(a + b/x)^(7/2))/(7*b) + a^2*c^2*x*(a + b/x)^(1/2) - a^(3/2)*c*atan(((a + b/x)^(1/2)*1i)/a^(1/2))*(4*a*d +
5*b*c)*1i

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sympy [A]  time = 112.03, size = 1841, normalized size = 12.11 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-16*a**(19/2)*b**(13/2)*d**2*x**6*sqrt(a*x/b + 1)/(105*a**(13/2)*b**7*x**(13/2) + 315*a**(11/2)*b**8*x**(11/2)
+ 315*a**(9/2)*b**9*x**(9/2) + 105*a**(7/2)*b**10*x**(7/2)) - 40*a**(17/2)*b**(15/2)*d**2*x**5*sqrt(a*x/b + 1
)/(105*a**(13/2)*b**7*x**(13/2) + 315*a**(11/2)*b**8*x**(11/2) + 315*a**(9/2)*b**9*x**(9/2) + 105*a**(7/2)*b**
10*x**(7/2)) - 30*a**(15/2)*b**(17/2)*d**2*x**4*sqrt(a*x/b + 1)/(105*a**(13/2)*b**7*x**(13/2) + 315*a**(11/2)*
b**8*x**(11/2) + 315*a**(9/2)*b**9*x**(9/2) + 105*a**(7/2)*b**10*x**(7/2)) - 40*a**(13/2)*b**(19/2)*d**2*x**3*
sqrt(a*x/b + 1)/(105*a**(13/2)*b**7*x**(13/2) + 315*a**(11/2)*b**8*x**(11/2) + 315*a**(9/2)*b**9*x**(9/2) + 10
5*a**(7/2)*b**10*x**(7/2)) + 8*a**(13/2)*b**(5/2)*d**2*x**3*sqrt(a*x/b + 1)/(15*a**(7/2)*b**3*x**(7/2) + 15*a*
*(5/2)*b**4*x**(5/2)) - 100*a**(11/2)*b**(21/2)*d**2*x**2*sqrt(a*x/b + 1)/(105*a**(13/2)*b**7*x**(13/2) + 315*
a**(11/2)*b**8*x**(11/2) + 315*a**(9/2)*b**9*x**(9/2) + 105*a**(7/2)*b**10*x**(7/2)) + 8*a**(11/2)*b**(7/2)*c*
d*x**3*sqrt(a*x/b + 1)/(15*a**(7/2)*b**3*x**(7/2) + 15*a**(5/2)*b**4*x**(5/2)) + 4*a**(11/2)*b**(7/2)*d**2*x**
2*sqrt(a*x/b + 1)/(15*a**(7/2)*b**3*x**(7/2) + 15*a**(5/2)*b**4*x**(5/2)) - 96*a**(9/2)*b**(23/2)*d**2*x*sqrt(
a*x/b + 1)/(105*a**(13/2)*b**7*x**(13/2) + 315*a**(11/2)*b**8*x**(11/2) + 315*a**(9/2)*b**9*x**(9/2) + 105*a**
(7/2)*b**10*x**(7/2)) + 4*a**(9/2)*b**(9/2)*c*d*x**2*sqrt(a*x/b + 1)/(15*a**(7/2)*b**3*x**(7/2) + 15*a**(5/2)*
b**4*x**(5/2)) - 16*a**(9/2)*b**(9/2)*d**2*x*sqrt(a*x/b + 1)/(15*a**(7/2)*b**3*x**(7/2) + 15*a**(5/2)*b**4*x**
(5/2)) - 30*a**(7/2)*b**(25/2)*d**2*sqrt(a*x/b + 1)/(105*a**(13/2)*b**7*x**(13/2) + 315*a**(11/2)*b**8*x**(11/
2) + 315*a**(9/2)*b**9*x**(9/2) + 105*a**(7/2)*b**10*x**(7/2)) - 16*a**(7/2)*b**(11/2)*c*d*x*sqrt(a*x/b + 1)/(
15*a**(7/2)*b**3*x**(7/2) + 15*a**(5/2)*b**4*x**(5/2)) - 12*a**(7/2)*b**(11/2)*d**2*sqrt(a*x/b + 1)/(15*a**(7/
2)*b**3*x**(7/2) + 15*a**(5/2)*b**4*x**(5/2)) - 12*a**(5/2)*b**(13/2)*c*d*sqrt(a*x/b + 1)/(15*a**(7/2)*b**3*x*
*(7/2) + 15*a**(5/2)*b**4*x**(5/2)) + a**(3/2)*b*c**2*asinh(sqrt(a)*sqrt(x)/sqrt(b)) + 16*a**10*b**6*d**2*x**(
13/2)/(105*a**(13/2)*b**7*x**(13/2) + 315*a**(11/2)*b**8*x**(11/2) + 315*a**(9/2)*b**9*x**(9/2) + 105*a**(7/2)
*b**10*x**(7/2)) + 48*a**9*b**7*d**2*x**(11/2)/(105*a**(13/2)*b**7*x**(13/2) + 315*a**(11/2)*b**8*x**(11/2) +
315*a**(9/2)*b**9*x**(9/2) + 105*a**(7/2)*b**10*x**(7/2)) + 48*a**8*b**8*d**2*x**(9/2)/(105*a**(13/2)*b**7*x**
(13/2) + 315*a**(11/2)*b**8*x**(11/2) + 315*a**(9/2)*b**9*x**(9/2) + 105*a**(7/2)*b**10*x**(7/2)) + 16*a**7*b*
*9*d**2*x**(7/2)/(105*a**(13/2)*b**7*x**(13/2) + 315*a**(11/2)*b**8*x**(11/2) + 315*a**(9/2)*b**9*x**(9/2) + 1
05*a**(7/2)*b**10*x**(7/2)) - 8*a**7*b**2*d**2*x**(7/2)/(15*a**(7/2)*b**3*x**(7/2) + 15*a**(5/2)*b**4*x**(5/2)
) - 8*a**6*b**3*c*d*x**(7/2)/(15*a**(7/2)*b**3*x**(7/2) + 15*a**(5/2)*b**4*x**(5/2)) - 8*a**6*b**3*d**2*x**(5/
2)/(15*a**(7/2)*b**3*x**(7/2) + 15*a**(5/2)*b**4*x**(5/2)) - 8*a**5*b**4*c*d*x**(5/2)/(15*a**(7/2)*b**3*x**(7/
2) + 15*a**(5/2)*b**4*x**(5/2)) - 4*a**3*c*d*atan(sqrt(a + b/x)/sqrt(-a))/sqrt(-a) + a**2*sqrt(b)*c**2*sqrt(x)
*sqrt(a*x/b + 1) - 4*a**2*b*c**2*atan(sqrt(a + b/x)/sqrt(-a))/sqrt(-a) - 4*a**2*c*d*sqrt(a + b/x) + a**2*d**2*
Piecewise((-sqrt(a)/x, Eq(b, 0)), (-2*(a + b/x)**(3/2)/(3*b), True)) - 4*a*b*c**2*sqrt(a + b/x) + 4*a*b*c*d*Pi
ecewise((-sqrt(a)/x, Eq(b, 0)), (-2*(a + b/x)**(3/2)/(3*b), True)) + b**2*c**2*Piecewise((-sqrt(a)/x, Eq(b, 0)
), (-2*(a + b/x)**(3/2)/(3*b), True))

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