3.581 \(\int \frac {e^{2 \tanh ^{-1}(a x)} \sqrt {c-\frac {c}{a x}}}{x^5} \, dx\)

Optimal. Leaf size=121 \[ -\frac {2 a^4 \left (c-\frac {c}{a x}\right )^{9/2}}{9 c^4}+\frac {10 a^4 \left (c-\frac {c}{a x}\right )^{7/2}}{7 c^3}-\frac {18 a^4 \left (c-\frac {c}{a x}\right )^{5/2}}{5 c^2}+\frac {14 a^4 \left (c-\frac {c}{a x}\right )^{3/2}}{3 c}-4 a^4 \sqrt {c-\frac {c}{a x}} \]

[Out]

14/3*a^4*(c-c/a/x)^(3/2)/c-18/5*a^4*(c-c/a/x)^(5/2)/c^2+10/7*a^4*(c-c/a/x)^(7/2)/c^3-2/9*a^4*(c-c/a/x)^(9/2)/c
^4-4*a^4*(c-c/a/x)^(1/2)

________________________________________________________________________________________

Rubi [A]  time = 0.24, antiderivative size = 121, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {6133, 25, 514, 446, 77} \[ -\frac {2 a^4 \left (c-\frac {c}{a x}\right )^{9/2}}{9 c^4}+\frac {10 a^4 \left (c-\frac {c}{a x}\right )^{7/2}}{7 c^3}-\frac {18 a^4 \left (c-\frac {c}{a x}\right )^{5/2}}{5 c^2}+\frac {14 a^4 \left (c-\frac {c}{a x}\right )^{3/2}}{3 c}-4 a^4 \sqrt {c-\frac {c}{a x}} \]

Antiderivative was successfully verified.

[In]

Int[(E^(2*ArcTanh[a*x])*Sqrt[c - c/(a*x)])/x^5,x]

[Out]

-4*a^4*Sqrt[c - c/(a*x)] + (14*a^4*(c - c/(a*x))^(3/2))/(3*c) - (18*a^4*(c - c/(a*x))^(5/2))/(5*c^2) + (10*a^4
*(c - c/(a*x))^(7/2))/(7*c^3) - (2*a^4*(c - c/(a*x))^(9/2))/(9*c^4)

Rule 25

Int[(u_.)*((a_) + (b_.)*(x_)^(n_.))^(m_.)*((c_) + (d_.)*(x_)^(q_.))^(p_.), x_Symbol] :> Dist[(d/a)^p, Int[(u*(
a + b*x^n)^(m + p))/x^(n*p), x], x] /; FreeQ[{a, b, c, d, m, n}, x] && EqQ[q, -n] && IntegerQ[p] && EqQ[a*c -
b*d, 0] &&  !(IntegerQ[m] && NegQ[n])

Rule 77

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((ILtQ[
n, 0] && ILtQ[p, 0]) || EqQ[p, 1] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1
, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))

Rule 446

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

Rule 514

Int[(x_)^(m_.)*((c_) + (d_.)*(x_)^(mn_.))^(q_.)*((a_) + (b_.)*(x_)^(n_.))^(p_.), x_Symbol] :> Int[x^(m - n*q)*
(a + b*x^n)^p*(d + c*x^n)^q, x] /; FreeQ[{a, b, c, d, m, n, p}, x] && EqQ[mn, -n] && IntegerQ[q] && (PosQ[n] |
|  !IntegerQ[p])

Rule 6133

Int[E^(ArcTanh[(a_.)*(x_)]*(n_))*(u_.)*((c_) + (d_.)/(x_))^(p_), x_Symbol] :> Int[(u*(c + d/x)^p*(1 + a*x)^(n/
2))/(1 - a*x)^(n/2), x] /; FreeQ[{a, c, d, p}, x] && EqQ[c^2 - a^2*d^2, 0] &&  !IntegerQ[p] && IntegerQ[n/2] &
&  !GtQ[c, 0]

Rubi steps

\begin {align*} \int \frac {e^{2 \tanh ^{-1}(a x)} \sqrt {c-\frac {c}{a x}}}{x^5} \, dx &=\int \frac {\sqrt {c-\frac {c}{a x}} (1+a x)}{x^5 (1-a x)} \, dx\\ &=-\frac {c \int \frac {1+a x}{\sqrt {c-\frac {c}{a x}} x^6} \, dx}{a}\\ &=-\frac {c \int \frac {a+\frac {1}{x}}{\sqrt {c-\frac {c}{a x}} x^5} \, dx}{a}\\ &=\frac {c \operatorname {Subst}\left (\int \frac {x^3 (a+x)}{\sqrt {c-\frac {c x}{a}}} \, dx,x,\frac {1}{x}\right )}{a}\\ &=\frac {c \operatorname {Subst}\left (\int \left (\frac {2 a^4}{\sqrt {c-\frac {c x}{a}}}-\frac {7 a^4 \sqrt {c-\frac {c x}{a}}}{c}+\frac {9 a^4 \left (c-\frac {c x}{a}\right )^{3/2}}{c^2}-\frac {5 a^4 \left (c-\frac {c x}{a}\right )^{5/2}}{c^3}+\frac {a^4 \left (c-\frac {c x}{a}\right )^{7/2}}{c^4}\right ) \, dx,x,\frac {1}{x}\right )}{a}\\ &=-4 a^4 \sqrt {c-\frac {c}{a x}}+\frac {14 a^4 \left (c-\frac {c}{a x}\right )^{3/2}}{3 c}-\frac {18 a^4 \left (c-\frac {c}{a x}\right )^{5/2}}{5 c^2}+\frac {10 a^4 \left (c-\frac {c}{a x}\right )^{7/2}}{7 c^3}-\frac {2 a^4 \left (c-\frac {c}{a x}\right )^{9/2}}{9 c^4}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.06, size = 52, normalized size = 0.43 \[ -\frac {2 \left (272 a^4 x^4+136 a^3 x^3+102 a^2 x^2+85 a x+35\right ) \sqrt {c-\frac {c}{a x}}}{315 x^4} \]

Antiderivative was successfully verified.

[In]

Integrate[(E^(2*ArcTanh[a*x])*Sqrt[c - c/(a*x)])/x^5,x]

[Out]

(-2*Sqrt[c - c/(a*x)]*(35 + 85*a*x + 102*a^2*x^2 + 136*a^3*x^3 + 272*a^4*x^4))/(315*x^4)

________________________________________________________________________________________

fricas [A]  time = 0.62, size = 52, normalized size = 0.43 \[ -\frac {2 \, {\left (272 \, a^{4} x^{4} + 136 \, a^{3} x^{3} + 102 \, a^{2} x^{2} + 85 \, a x + 35\right )} \sqrt {\frac {a c x - c}{a x}}}{315 \, x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x+1)^2/(-a^2*x^2+1)*(c-c/a/x)^(1/2)/x^5,x, algorithm="fricas")

[Out]

-2/315*(272*a^4*x^4 + 136*a^3*x^3 + 102*a^2*x^2 + 85*a*x + 35)*sqrt((a*c*x - c)/(a*x))/x^4

________________________________________________________________________________________

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*x+1)^2/(-a^2*x^2+1)*(c-c/a/x)^(1/2)/x^5,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)]Unable to divide, perhaps due to rounding error%%%{%%%{%%{[-315,0]:[1,0,%%%{-1,[1]%%%}]%%},[0,9]%%%},[10]
%%%}+%%%{%%{[%%%{2835,[0,8]%%%},0]:[1,0,%%%{%%%{-1,[1]%%%},[2,2]%%%}+%%%{%%%{1,[1]%%%},[1,1]%%%}]%%},[9]%%%}+%
%%{%%%{%%{[-11340,0]:[1,0,%%%{-1,[1]%%%}]%%},[2,9]%%%}+%%%{%%{[11340,0]:[1,0,%%%{-1,[1]%%%}]%%},[1,8]%%%},[8]%
%%}+%%%{%%{[%%%{26460,[2,8]%%%}+%%%{-26460,[1,7]%%%},0]:[1,0,%%%{%%%{-1,[1]%%%},[2,2]%%%}+%%%{%%%{1,[1]%%%},[1
,1]%%%}]%%},[7]%%%}+%%%{%%%{%%{[-39690,0]:[1,0,%%%{-1,[1]%%%}]%%},[4,9]%%%}+%%%{%%{[79380,0]:[1,0,%%%{-1,[1]%%
%}]%%},[3,8]%%%}+%%%{%%{[-39690,0]:[1,0,%%%{-1,[1]%%%}]%%},[2,7]%%%},[6]%%%}+%%%{%%{[%%%{39690,[4,8]%%%}+%%%{-
79380,[3,7]%%%}+%%%{39690,[2,6]%%%},0]:[1,0,%%%{%%%{-1,[1]%%%},[2,2]%%%}+%%%{%%%{1,[1]%%%},[1,1]%%%}]%%},[5]%%
%}+%%%{%%%{%%{[-26460,0]:[1,0,%%%{-1,[1]%%%}]%%},[6,9]%%%}+%%%{%%{[79380,0]:[1,0,%%%{-1,[1]%%%}]%%},[5,8]%%%}+
%%%{%%{[-79380,0]:[1,0,%%%{-1,[1]%%%}]%%},[4,7]%%%}+%%%{%%{[26460,0]:[1,0,%%%{-1,[1]%%%}]%%},[3,6]%%%},[4]%%%}
+%%%{%%{[%%%{11340,[6,8]%%%}+%%%{-34020,[5,7]%%%}+%%%{34020,[4,6]%%%}+%%%{-11340,[3,5]%%%},0]:[1,0,%%%{%%%{-1,
[1]%%%},[2,2]%%%}+%%%{%%%{1,[1]%%%},[1,1]%%%}]%%},[3]%%%}+%%%{%%%{%%{[-2835,0]:[1,0,%%%{-1,[1]%%%}]%%},[8,9]%%
%}+%%%{%%{[11340,0]:[1,0,%%%{-1,[1]%%%}]%%},[7,8]%%%}+%%%{%%{[-17010,0]:[1,0,%%%{-1,[1]%%%}]%%},[6,7]%%%}+%%%{
%%{[11340,0]:[1,0,%%%{-1,[1]%%%}]%%},[5,6]%%%}+%%%{%%{[-2835,0]:[1,0,%%%{-1,[1]%%%}]%%},[4,5]%%%},[2]%%%}+%%%{
%%{[%%%{315,[8,8]%%%}+%%%{-1260,[7,7]%%%}+%%%{1890,[6,6]%%%}+%%%{-1260,[5,5]%%%}+%%%{315,[4,4]%%%},0]:[1,0,%%%
{%%%{-1,[1]%%%},[2,2]%%%}+%%%{%%%{1,[1]%%%},[1,1]%%%}]%%},[1]%%%} / %%%{%%%{%%{poly1[%%%{315,[10]%%%},0]:[1,0,
%%%{-1,[1]%%%}]%%},[20,20]%%%}+%%%{%%{poly1[%%%{-3150,[10]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[19,19]%%%}+%%%{%%{p
oly1[%%%{14175,[10]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[18,18]%%%}+%%%{%%{poly1[%%%{-37800,[10]%%%},0]:[1,0,%%%{-1
,[1]%%%}]%%},[17,17]%%%}+%%%{%%{poly1[%%%{66150,[10]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[16,16]%%%}+%%%{%%{poly1[%
%%{-79380,[10]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[15,15]%%%}+%%%{%%{poly1[%%%{66150,[10]%%%},0]:[1,0,%%%{-1,[1]%%
%}]%%},[14,14]%%%}+%%%{%%{poly1[%%%{-37800,[10]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[13,13]%%%}+%%%{%%{poly1[%%%{14
175,[10]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[12,12]%%%}+%%%{%%{poly1[%%%{-3150,[10]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%}
,[11,11]%%%}+%%%{%%{poly1[%%%{315,[10]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[10,10]%%%},[10]%%%}+%%%{%%{[%%%{%%%{-28
35,[10]%%%},[20,19]%%%}+%%%{%%%{28350,[10]%%%},[19,18]%%%}+%%%{%%%{-127575,[10]%%%},[18,17]%%%}+%%%{%%%{340200
,[10]%%%},[17,16]%%%}+%%%{%%%{-595350,[10]%%%},[16,15]%%%}+%%%{%%%{714420,[10]%%%},[15,14]%%%}+%%%{%%%{-595350
,[10]%%%},[14,13]%%%}+%%%{%%%{340200,[10]%%%},[13,12]%%%}+%%%{%%%{-127575,[10]%%%},[12,11]%%%}+%%%{%%%{28350,[
10]%%%},[11,10]%%%}+%%%{%%%{-2835,[10]%%%},[10,9]%%%},0]:[1,0,%%%{%%%{-1,[1]%%%},[2,2]%%%}+%%%{%%%{1,[1]%%%},[
1,1]%%%}]%%},[9]%%%}+%%%{%%%{%%{poly1[%%%{11340,[10]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[22,20]%%%}+%%%{%%{poly1[%
%%{-124740,[10]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[21,19]%%%}+%%%{%%{poly1[%%%{623700,[10]%%%},0]:[1,0,%%%{-1,[1]
%%%}]%%},[20,18]%%%}+%%%{%%{poly1[%%%{-1871100,[10]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[19,17]%%%}+%%%{%%{poly1[%%
%{3742200,[10]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[18,16]%%%}+%%%{%%{poly1[%%%{-5239080,[10]%%%},0]:[1,0,%%%{-1,[1
]%%%}]%%},[17,15]%%%}+%%%{%%{poly1[%%%{5239080,[10]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[16,14]%%%}+%%%{%%{poly1[%%
%{-3742200,[10]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[15,13]%%%}+%%%{%%{poly1[%%%{1871100,[10]%%%},0]:[1,0,%%%{-1,[1
]%%%}]%%},[14,12]%%%}+%%%{%%{poly1[%%%{-623700,[10]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[13,11]%%%}+%%%{%%{poly1[%%
%{124740,[10]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[12,10]%%%}+%%%{%%{poly1[%%%{-11340,[10]%%%},0]:[1,0,%%%{-1,[1]%%
%}]%%},[11,9]%%%},[8]%%%}+%%%{%%{[%%%{%%%{-26460,[10]%%%},[22,19]%%%}+%%%{%%%{291060,[10]%%%},[21,18]%%%}+%%%{
%%%{-1455300,[10]%%%},[20,17]%%%}+%%%{%%%{4365900,[10]%%%},[19,16]%%%}+%%%{%%%{-8731800,[10]%%%},[18,15]%%%}+%
%%{%%%{12224520,[10]%%%},[17,14]%%%}+%%%{%%%{-12224520,[10]%%%},[16,13]%%%}+%%%{%%%{8731800,[10]%%%},[15,12]%%
%}+%%%{%%%{-4365900,[10]%%%},[14,11]%%%}+%%%{%%%{1455300,[10]%%%},[13,10]%%%}+%%%{%%%{-291060,[10]%%%},[12,9]%
%%}+%%%{%%%{26460,[10]%%%},[11,8]%%%},0]:[1,0,%%%{%%%{-1,[1]%%%},[2,2]%%%}+%%%{%%%{1,[1]%%%},[1,1]%%%}]%%},[7]
%%%}+%%%{%%%{%%{poly1[%%%{39690,[10]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[24,20]%%%}+%%%{%%{poly1[%%%{-476280,[10]%
%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[23,19]%%%}+%%%{%%{poly1[%%%{2619540,[10]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[22,18
]%%%}+%%%{%%{poly1[%%%{-8731800,[10]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[21,17]%%%}+%%%{%%{poly1[%%%{19646550,[10]
%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[20,16]%%%}+%%%{%%{poly1[%%%{-31434480,[10]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[19
,15]%%%}+%%%{%%{poly1[%%%{36673560,[10]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[18,14]%%%}+%%%{%%{poly1[%%%{-31434480,
[10]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[17,13]%%%}+%%%{%%{poly1[%%%{19646550,[10]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},
[16,12]%%%}+%%%{%%{poly1[%%%{-8731800,[10]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[15,11]%%%}+%%%{%%{poly1[%%%{2619540
,[10]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[14,10]%%%}+%%%{%%{poly1[%%%{-476280,[10]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},
[13,9]%%%}+%%%{%%{poly1[%%%{39690,[10]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[12,8]%%%},[6]%%%}+%%%{%%{[%%%{%%%{-3969
0,[10]%%%},[24,19]%%%}+%%%{%%%{476280,[10]%%%},[23,18]%%%}+%%%{%%%{-2619540,[10]%%%},[22,17]%%%}+%%%{%%%{87318
00,[10]%%%},[21,16]%%%}+%%%{%%%{-19646550,[10]%%%},[20,15]%%%}+%%%{%%%{31434480,[10]%%%},[19,14]%%%}+%%%{%%%{-
36673560,[10]%%%},[18,13]%%%}+%%%{%%%{31434480,[10]%%%},[17,12]%%%}+%%%{%%%{-19646550,[10]%%%},[16,11]%%%}+%%%
{%%%{8731800,[10]%%%},[15,10]%%%}+%%%{%%%{-2619540,[10]%%%},[14,9]%%%}+%%%{%%%{476280,[10]%%%},[13,8]%%%}+%%%{
%%%{-39690,[10]%%%},[12,7]%%%},0]:[1,0,%%%{%%%{-1,[1]%%%},[2,2]%%%}+%%%{%%%{1,[1]%%%},[1,1]%%%}]%%},[5]%%%}+%%
%{%%%{%%{poly1[%%%{26460,[10]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[26,20]%%%}+%%%{%%{poly1[%%%{-343980,[10]%%%},0]:
[1,0,%%%{-1,[1]%%%}]%%},[25,19]%%%}+%%%{%%{poly1[%%%{2063880,[10]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[24,18]%%%}+%
%%{%%{poly1[%%%{-7567560,[10]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[23,17]%%%}+%%%{%%{poly1[%%%{18918900,[10]%%%},0]
:[1,0,%%%{-1,[1]%%%}]%%},[22,16]%%%}+%%%{%%{poly1[%%%{-34054020,[10]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[21,15]%%%
}+%%%{%%{poly1[%%%{45405360,[10]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[20,14]%%%}+%%%{%%{poly1[%%%{-45405360,[10]%%%
},0]:[1,0,%%%{-1,[1]%%%}]%%},[19,13]%%%}+%%%{%%{poly1[%%%{34054020,[10]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[18,12]
%%%}+%%%{%%{poly1[%%%{-18918900,[10]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[17,11]%%%}+%%%{%%{poly1[%%%{7567560,[10]%
%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[16,10]%%%}+%%%{%%{poly1[%%%{-2063880,[10]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[15,9
]%%%}+%%%{%%{poly1[%%%{343980,[10]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[14,8]%%%}+%%%{%%{poly1[%%%{-26460,[10]%%%},
0]:[1,0,%%%{-1,[1]%%%}]%%},[13,7]%%%},[4]%%%}+%%%{%%{[%%%{%%%{-11340,[10]%%%},[26,19]%%%}+%%%{%%%{147420,[10]%
%%},[25,18]%%%}+%%%{%%%{-884520,[10]%%%},[24,17]%%%}+%%%{%%%{3243240,[10]%%%},[23,16]%%%}+%%%{%%%{-8108100,[10
]%%%},[22,15]%%%}+%%%{%%%{14594580,[10]%%%},[21,14]%%%}+%%%{%%%{-19459440,[10]%%%},[20,13]%%%}+%%%{%%%{1945944
0,[10]%%%},[19,12]%%%}+%%%{%%%{-14594580,[10]%%%},[18,11]%%%}+%%%{%%%{8108100,[10]%%%},[17,10]%%%}+%%%{%%%{-32
43240,[10]%%%},[16,9]%%%}+%%%{%%%{884520,[10]%%%},[15,8]%%%}+%%%{%%%{-147420,[10]%%%},[14,7]%%%}+%%%{%%%{11340
,[10]%%%},[13,6]%%%},0]:[1,0,%%%{%%%{-1,[1]%%%},[2,2]%%%}+%%%{%%%{1,[1]%%%},[1,1]%%%}]%%},[3]%%%}+%%%{%%%{%%{p
oly1[%%%{2835,[10]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[28,20]%%%}+%%%{%%{poly1[%%%{-39690,[10]%%%},0]:[1,0,%%%{-1,
[1]%%%}]%%},[27,19]%%%}+%%%{%%{poly1[%%%{257985,[10]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[26,18]%%%}+%%%{%%{poly1[%
%%{-1031940,[10]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[25,17]%%%}+%%%{%%{poly1[%%%{2837835,[10]%%%},0]:[1,0,%%%{-1,[
1]%%%}]%%},[24,16]%%%}+%%%{%%{poly1[%%%{-5675670,[10]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[23,15]%%%}+%%%{%%{poly1[
%%%{8513505,[10]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[22,14]%%%}+%%%{%%{poly1[%%%{-9729720,[10]%%%},0]:[1,0,%%%{-1,
[1]%%%}]%%},[21,13]%%%}+%%%{%%{poly1[%%%{8513505,[10]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[20,12]%%%}+%%%{%%{poly1[
%%%{-5675670,[10]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[19,11]%%%}+%%%{%%{poly1[%%%{2837835,[10]%%%},0]:[1,0,%%%{-1,
[1]%%%}]%%},[18,10]%%%}+%%%{%%{poly1[%%%{-1031940,[10]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[17,9]%%%}+%%%{%%{poly1[
%%%{257985,[10]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[16,8]%%%}+%%%{%%{poly1[%%%{-39690,[10]%%%},0]:[1,0,%%%{-1,[1]%
%%}]%%},[15,7]%%%}+%%%{%%{poly1[%%%{2835,[10]%%%},0]:[1,0,%%%{-1,[1]%%%}]%%},[14,6]%%%},[2]%%%}+%%%{%%{[%%%{%%
%{-315,[10]%%%},[28,19]%%%}+%%%{%%%{4410,[10]%%%},[27,18]%%%}+%%%{%%%{-28665,[10]%%%},[26,17]%%%}+%%%{%%%{1146
60,[10]%%%},[25,16]%%%}+%%%{%%%{-315315,[10]%%%},[24,15]%%%}+%%%{%%%{630630,[10]%%%},[23,14]%%%}+%%%{%%%{-9459
45,[10]%%%},[22,13]%%%}+%%%{%%%{1081080,[10]%%%},[21,12]%%%}+%%%{%%%{-945945,[10]%%%},[20,11]%%%}+%%%{%%%{6306
30,[10]%%%},[19,10]%%%}+%%%{%%%{-315315,[10]%%%},[18,9]%%%}+%%%{%%%{114660,[10]%%%},[17,8]%%%}+%%%{%%%{-28665,
[10]%%%},[16,7]%%%}+%%%{%%%{4410,[10]%%%},[15,6]%%%}+%%%{%%%{-315,[10]%%%},[14,5]%%%},0]:[1,0,%%%{%%%{-1,[1]%%
%},[2,2]%%%}+%%%{%%%{1,[1]%%%},[1,1]%%%}]%%},[1]%%%} Error: Bad Argument Value

________________________________________________________________________________________

maple [A]  time = 0.03, size = 51, normalized size = 0.42 \[ -\frac {2 \sqrt {\frac {c \left (a x -1\right )}{a x}}\, \left (272 x^{4} a^{4}+136 x^{3} a^{3}+102 a^{2} x^{2}+85 a x +35\right )}{315 x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a*x+1)^2/(-a^2*x^2+1)*(c-c/a/x)^(1/2)/x^5,x)

[Out]

-2/315*(c*(a*x-1)/a/x)^(1/2)*(272*a^4*x^4+136*a^3*x^3+102*a^2*x^2+85*a*x+35)/x^4

________________________________________________________________________________________

maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ -\int \frac {{\left (a x + 1\right )}^{2} \sqrt {c - \frac {c}{a x}}}{{\left (a^{2} x^{2} - 1\right )} x^{5}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x+1)^2/(-a^2*x^2+1)*(c-c/a/x)^(1/2)/x^5,x, algorithm="maxima")

[Out]

-integrate((a*x + 1)^2*sqrt(c - c/(a*x))/((a^2*x^2 - 1)*x^5), x)

________________________________________________________________________________________

mupad [B]  time = 0.90, size = 98, normalized size = 0.81 \[ -\frac {544\,a^4\,\sqrt {c-\frac {c}{a\,x}}}{315}-\frac {2\,\sqrt {c-\frac {c}{a\,x}}}{9\,x^4}-\frac {34\,a\,\sqrt {c-\frac {c}{a\,x}}}{63\,x^3}-\frac {68\,a^2\,\sqrt {c-\frac {c}{a\,x}}}{105\,x^2}-\frac {272\,a^3\,\sqrt {c-\frac {c}{a\,x}}}{315\,x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-((c - c/(a*x))^(1/2)*(a*x + 1)^2)/(x^5*(a^2*x^2 - 1)),x)

[Out]

- (544*a^4*(c - c/(a*x))^(1/2))/315 - (2*(c - c/(a*x))^(1/2))/(9*x^4) - (34*a*(c - c/(a*x))^(1/2))/(63*x^3) -
(68*a^2*(c - c/(a*x))^(1/2))/(105*x^2) - (272*a^3*(c - c/(a*x))^(1/2))/(315*x)

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ - \int \frac {\sqrt {c - \frac {c}{a x}}}{a x^{6} - x^{5}}\, dx - \int \frac {a x \sqrt {c - \frac {c}{a x}}}{a x^{6} - x^{5}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x+1)**2/(-a**2*x**2+1)*(c-c/a/x)**(1/2)/x**5,x)

[Out]

-Integral(sqrt(c - c/(a*x))/(a*x**6 - x**5), x) - Integral(a*x*sqrt(c - c/(a*x))/(a*x**6 - x**5), x)

________________________________________________________________________________________