3.580 \(\int \frac {e^{2 \tanh ^{-1}(a x)} \sqrt {c-\frac {c}{a x}}}{x^4} \, dx\)
Optimal. Leaf size=96 \[ \frac {2 a^3 \left (c-\frac {c}{a x}\right )^{7/2}}{7 c^3}-\frac {8 a^3 \left (c-\frac {c}{a x}\right )^{5/2}}{5 c^2}+\frac {10 a^3 \left (c-\frac {c}{a x}\right )^{3/2}}{3 c}-4 a^3 \sqrt {c-\frac {c}{a x}} \]
[Out]
10/3*a^3*(c-c/a/x)^(3/2)/c-8/5*a^3*(c-c/a/x)^(5/2)/c^2+2/7*a^3*(c-c/a/x)^(7/2)/c^3-4*a^3*(c-c/a/x)^(1/2)
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Rubi [A] time = 0.22, antiderivative size = 96, 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} \[ \frac {2 a^3 \left (c-\frac {c}{a x}\right )^{7/2}}{7 c^3}-\frac {8 a^3 \left (c-\frac {c}{a x}\right )^{5/2}}{5 c^2}+\frac {10 a^3 \left (c-\frac {c}{a x}\right )^{3/2}}{3 c}-4 a^3 \sqrt {c-\frac {c}{a x}} \]
Antiderivative was successfully verified.
[In]
Int[(E^(2*ArcTanh[a*x])*Sqrt[c - c/(a*x)])/x^4,x]
[Out]
-4*a^3*Sqrt[c - c/(a*x)] + (10*a^3*(c - c/(a*x))^(3/2))/(3*c) - (8*a^3*(c - c/(a*x))^(5/2))/(5*c^2) + (2*a^3*(
c - c/(a*x))^(7/2))/(7*c^3)
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^4} \, dx &=\int \frac {\sqrt {c-\frac {c}{a x}} (1+a x)}{x^4 (1-a x)} \, dx\\ &=-\frac {c \int \frac {1+a x}{\sqrt {c-\frac {c}{a x}} x^5} \, dx}{a}\\ &=-\frac {c \int \frac {a+\frac {1}{x}}{\sqrt {c-\frac {c}{a x}} x^4} \, dx}{a}\\ &=\frac {c \operatorname {Subst}\left (\int \frac {x^2 (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^3}{\sqrt {c-\frac {c x}{a}}}-\frac {5 a^3 \sqrt {c-\frac {c x}{a}}}{c}+\frac {4 a^3 \left (c-\frac {c x}{a}\right )^{3/2}}{c^2}-\frac {a^3 \left (c-\frac {c x}{a}\right )^{5/2}}{c^3}\right ) \, dx,x,\frac {1}{x}\right )}{a}\\ &=-4 a^3 \sqrt {c-\frac {c}{a x}}+\frac {10 a^3 \left (c-\frac {c}{a x}\right )^{3/2}}{3 c}-\frac {8 a^3 \left (c-\frac {c}{a x}\right )^{5/2}}{5 c^2}+\frac {2 a^3 \left (c-\frac {c}{a x}\right )^{7/2}}{7 c^3}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 44, normalized size = 0.46 \[ -\frac {2 \left (104 a^3 x^3+52 a^2 x^2+39 a x+15\right ) \sqrt {c-\frac {c}{a x}}}{105 x^3} \]
Antiderivative was successfully verified.
[In]
Integrate[(E^(2*ArcTanh[a*x])*Sqrt[c - c/(a*x)])/x^4,x]
[Out]
(-2*Sqrt[c - c/(a*x)]*(15 + 39*a*x + 52*a^2*x^2 + 104*a^3*x^3))/(105*x^3)
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fricas [A] time = 0.50, size = 44, normalized size = 0.46 \[ -\frac {2 \, {\left (104 \, a^{3} x^{3} + 52 \, a^{2} x^{2} + 39 \, a x + 15\right )} \sqrt {\frac {a c x - c}{a x}}}{105 \, x^{3}} \]
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^4,x, algorithm="fricas")
[Out]
-2/105*(104*a^3*x^3 + 52*a^2*x^2 + 39*a*x + 15)*sqrt((a*c*x - c)/(a*x))/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*x+1)^2/(-a^2*x^2+1)*(c-c/a/x)^(1/2)/x^4,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)]Evaluation time: 0.45Unable to divide, perhaps due to rounding error%%%{%%%{210,[2,1,9]%%%}+%%%{-210,[1,1
,8]%%%}+%%%{-210,[0,1,7]%%%},[8]%%%}+%%%{%%{[%%%{-735,[2,0,8]%%%}+%%%{735,[1,0,7]%%%}+%%%{735,[0,0,6]%%%},0,%%
%{735,[4,1,10]%%%}+%%%{-1470,[3,1,9]%%%}+%%%{735,[2,1,8]%%%}+%%%{-735,[0,1,6]%%%}]:[1,0,%%%{-2,[2,1,2]%%%}+%%%
{2,[1,1,1]%%%}+%%%{-2,[0,1,0]%%%},0,%%%{1,[4,2,4]%%%}+%%%{-2,[3,2,3]%%%}+%%%{-1,[2,2,2]%%%}+%%%{2,[1,2,1]%%%}+
%%%{1,[0,2,0]%%%}]%%},[7]%%%}+%%%{%%%{4410,[4,1,9]%%%}+%%%{-8820,[3,1,8]%%%}+%%%{4410,[1,1,6]%%%},[6]%%%}+%%%{
%%{[%%%{-3675,[4,0,8]%%%}+%%%{7350,[3,0,7]%%%}+%%%{-3675,[1,0,5]%%%},0,%%%{3675,[6,1,10]%%%}+%%%{-11025,[5,1,9
]%%%}+%%%{11025,[4,1,8]%%%}+%%%{-3675,[3,1,7]%%%}+%%%{-3675,[2,1,6]%%%}+%%%{3675,[1,1,5]%%%}]:[1,0,%%%{-2,[2,1
,2]%%%}+%%%{2,[1,1,1]%%%}+%%%{-2,[0,1,0]%%%},0,%%%{1,[4,2,4]%%%}+%%%{-2,[3,2,3]%%%}+%%%{-1,[2,2,2]%%%}+%%%{2,[
1,2,1]%%%}+%%%{1,[0,2,0]%%%}]%%},[5]%%%}+%%%{%%%{7350,[6,1,9]%%%}+%%%{-22050,[5,1,8]%%%}+%%%{14700,[4,1,7]%%%}
+%%%{7350,[3,1,6]%%%}+%%%{-7350,[2,1,5]%%%},[4]%%%}+%%%{%%{[%%%{-2205,[6,0,8]%%%}+%%%{6615,[5,0,7]%%%}+%%%{-44
10,[4,0,6]%%%}+%%%{-2205,[3,0,5]%%%}+%%%{2205,[2,0,4]%%%},0,%%%{2205,[8,1,10]%%%}+%%%{-8820,[7,1,9]%%%}+%%%{13
230,[6,1,8]%%%}+%%%{-8820,[5,1,7]%%%}+%%%{4410,[3,1,5]%%%}+%%%{-2205,[2,1,4]%%%}]:[1,0,%%%{-2,[2,1,2]%%%}+%%%{
2,[1,1,1]%%%}+%%%{-2,[0,1,0]%%%},0,%%%{1,[4,2,4]%%%}+%%%{-2,[3,2,3]%%%}+%%%{-1,[2,2,2]%%%}+%%%{2,[1,2,1]%%%}+%
%%{1,[0,2,0]%%%}]%%},[3]%%%}+%%%{%%%{1470,[8,1,9]%%%}+%%%{-5880,[7,1,8]%%%}+%%%{7350,[6,1,7]%%%}+%%%{-1470,[5,
1,6]%%%}+%%%{-2940,[4,1,5]%%%}+%%%{1470,[3,1,4]%%%},[2]%%%}+%%%{%%{[%%%{-105,[8,0,8]%%%}+%%%{420,[7,0,7]%%%}+%
%%{-525,[6,0,6]%%%}+%%%{105,[5,0,5]%%%}+%%%{210,[4,0,4]%%%}+%%%{-105,[3,0,3]%%%},0,%%%{105,[10,1,10]%%%}+%%%{-
525,[9,1,9]%%%}+%%%{1050,[8,1,8]%%%}+%%%{-1050,[7,1,7]%%%}+%%%{420,[6,1,6]%%%}+%%%{210,[5,1,5]%%%}+%%%{-315,[4
,1,4]%%%}+%%%{105,[3,1,3]%%%}]:[1,0,%%%{-2,[2,1,2]%%%}+%%%{2,[1,1,1]%%%}+%%%{-2,[0,1,0]%%%},0,%%%{1,[4,2,4]%%%
}+%%%{-2,[3,2,3]%%%}+%%%{-1,[2,2,2]%%%}+%%%{2,[1,2,1]%%%}+%%%{1,[0,2,0]%%%}]%%},[1]%%%} / %%%{%%%{65536,[16,16
,16]%%%}+%%%{-524288,[15,16,15]%%%}+%%%{1835008,[14,16,14]%%%}+%%%{-3670016,[13,16,13]%%%}+%%%{4587520,[12,16,
12]%%%}+%%%{-3670016,[11,16,11]%%%}+%%%{1835008,[10,16,10]%%%}+%%%{-524288,[9,16,9]%%%}+%%%{65536,[8,16,8]%%%}
,[8]%%%}+%%%{%%{[%%%{-229376,[16,15,15]%%%}+%%%{1835008,[15,15,14]%%%}+%%%{-6422528,[14,15,13]%%%}+%%%{1284505
6,[13,15,12]%%%}+%%%{-16056320,[12,15,11]%%%}+%%%{12845056,[11,15,10]%%%}+%%%{-6422528,[10,15,9]%%%}+%%%{18350
08,[9,15,8]%%%}+%%%{-229376,[8,15,7]%%%},0,%%%{229376,[18,16,17]%%%}+%%%{-2064384,[17,16,16]%%%}+%%%{8486912,[
16,16,15]%%%}+%%%{-21102592,[15,16,14]%%%}+%%%{35323904,[14,16,13]%%%}+%%%{-41746432,[13,16,12]%%%}+%%%{353239
04,[12,16,11]%%%}+%%%{-21102592,[11,16,10]%%%}+%%%{8486912,[10,16,9]%%%}+%%%{-2064384,[9,16,8]%%%}+%%%{229376,
[8,16,7]%%%}]:[1,0,%%%{-2,[2,1,2]%%%}+%%%{2,[1,1,1]%%%}+%%%{-2,[0,1,0]%%%},0,%%%{1,[4,2,4]%%%}+%%%{-2,[3,2,3]%
%%}+%%%{-1,[2,2,2]%%%}+%%%{2,[1,2,1]%%%}+%%%{1,[0,2,0]%%%}]%%},[7]%%%}+%%%{%%%{1376256,[18,16,16]%%%}+%%%{-123
86304,[17,16,15]%%%}+%%%{49545216,[16,16,14]%%%}+%%%{-115605504,[15,16,13]%%%}+%%%{173408256,[14,16,12]%%%}+%%
%{-173408256,[13,16,11]%%%}+%%%{115605504,[12,16,10]%%%}+%%%{-49545216,[11,16,9]%%%}+%%%{12386304,[10,16,8]%%%
}+%%%{-1376256,[9,16,7]%%%},[6]%%%}+%%%{%%{[%%%{-1146880,[18,15,15]%%%}+%%%{10321920,[17,15,14]%%%}+%%%{-41287
680,[16,15,13]%%%}+%%%{96337920,[15,15,12]%%%}+%%%{-144506880,[14,15,11]%%%}+%%%{144506880,[13,15,10]%%%}+%%%{
-96337920,[12,15,9]%%%}+%%%{41287680,[11,15,8]%%%}+%%%{-10321920,[10,15,7]%%%}+%%%{1146880,[9,15,6]%%%},0,%%%{
1146880,[20,16,17]%%%}+%%%{-11468800,[19,16,16]%%%}+%%%{52756480,[18,16,15]%%%}+%%%{-147947520,[17,16,14]%%%}+
%%%{282132480,[16,16,13]%%%}+%%%{-385351680,[15,16,12]%%%}+%%%{385351680,[14,16,11]%%%}+%%%{-282132480,[13,16,
10]%%%}+%%%{147947520,[12,16,9]%%%}+%%%{-52756480,[11,16,8]%%%}+%%%{11468800,[10,16,7]%%%}+%%%{-1146880,[9,16,
6]%%%}]:[1,0,%%%{-2,[2,1,2]%%%}+%%%{2,[1,1,1]%%%}+%%%{-2,[0,1,0]%%%},0,%%%{1,[4,2,4]%%%}+%%%{-2,[3,2,3]%%%}+%%
%{-1,[2,2,2]%%%}+%%%{2,[1,2,1]%%%}+%%%{1,[0,2,0]%%%}]%%},[5]%%%}+%%%{%%%{2293760,[20,16,16]%%%}+%%%{-22937600,
[19,16,15]%%%}+%%%{103219200,[18,16,14]%%%}+%%%{-275251200,[17,16,13]%%%}+%%%{481689600,[16,16,12]%%%}+%%%{-57
8027520,[15,16,11]%%%}+%%%{481689600,[14,16,10]%%%}+%%%{-275251200,[13,16,9]%%%}+%%%{103219200,[12,16,8]%%%}+%
%%{-22937600,[11,16,7]%%%}+%%%{2293760,[10,16,6]%%%},[4]%%%}+%%%{%%{[%%%{-688128,[20,15,15]%%%}+%%%{6881280,[1
9,15,14]%%%}+%%%{-30965760,[18,15,13]%%%}+%%%{82575360,[17,15,12]%%%}+%%%{-144506880,[16,15,11]%%%}+%%%{173408
256,[15,15,10]%%%}+%%%{-144506880,[14,15,9]%%%}+%%%{82575360,[13,15,8]%%%}+%%%{-30965760,[12,15,7]%%%}+%%%{688
1280,[11,15,6]%%%}+%%%{-688128,[10,15,5]%%%},0,%%%{688128,[22,16,17]%%%}+%%%{-7569408,[21,16,16]%%%}+%%%{38535
168,[20,16,15]%%%}+%%%{-120422400,[19,16,14]%%%}+%%%{258048000,[18,16,13]%%%}+%%%{-400490496,[17,16,12]%%%}+%%
%{462422016,[16,16,11]%%%}+%%%{-400490496,[15,16,10]%%%}+%%%{258048000,[14,16,9]%%%}+%%%{-120422400,[13,16,8]%
%%}+%%%{38535168,[12,16,7]%%%}+%%%{-7569408,[11,16,6]%%%}+%%%{688128,[10,16,5]%%%}]:[1,0,%%%{-2,[2,1,2]%%%}+%%
%{2,[1,1,1]%%%}+%%%{-2,[0,1,0]%%%},0,%%%{1,[4,2,4]%%%}+%%%{-2,[3,2,3]%%%}+%%%{-1,[2,2,2]%%%}+%%%{2,[1,2,1]%%%}
+%%%{1,[0,2,0]%%%}]%%},[3]%%%}+%%%{%%%{458752,[22,16,16]%%%}+%%%{-5046272,[21,16,15]%%%}+%%%{25231360,[20,16,1
4]%%%}+%%%{-75694080,[19,16,13]%%%}+%%%{151388160,[18,16,12]%%%}+%%%{-211943424,[17,16,11]%%%}+%%%{211943424,[
16,16,10]%%%}+%%%{-151388160,[15,16,9]%%%}+%%%{75694080,[14,16,8]%%%}+%%%{-25231360,[13,16,7]%%%}+%%%{5046272,
[12,16,6]%%%}+%%%{-458752,[11,16,5]%%%},[2]%%%}+%%%{%%{[%%%{-32768,[22,15,15]%%%}+%%%{360448,[21,15,14]%%%}+%%
%{-1802240,[20,15,13]%%%}+%%%{5406720,[19,15,12]%%%}+%%%{-10813440,[18,15,11]%%%}+%%%{15138816,[17,15,10]%%%}+
%%%{-15138816,[16,15,9]%%%}+%%%{10813440,[15,15,8]%%%}+%%%{-5406720,[14,15,7]%%%}+%%%{1802240,[13,15,6]%%%}+%%
%{-360448,[12,15,5]%%%}+%%%{32768,[11,15,4]%%%},0,%%%{32768,[24,16,17]%%%}+%%%{-393216,[23,16,16]%%%}+%%%{2195
456,[22,16,15]%%%}+%%%{-7569408,[21,16,14]%%%}+%%%{18022400,[20,16,13]%%%}+%%%{-31358976,[19,16,12]%%%}+%%%{41
091072,[18,16,11]%%%}+%%%{-41091072,[17,16,10]%%%}+%%%{31358976,[16,16,9]%%%}+%%%{-18022400,[15,16,8]%%%}+%%%{
7569408,[14,16,7]%%%}+%%%{-2195456,[13,16,6]%%%}+%%%{393216,[12,16,5]%%%}+%%%{-32768,[11,16,4]%%%}]:[1,0,%%%{-
2,[2,1,2]%%%}+%%%{2,[1,1,1]%%%}+%%%{-2,[0,1,0]%%%},0,%%%{1,[4,2,4]%%%}+%%%{-2,[3,2,3]%%%}+%%%{-1,[2,2,2]%%%}+%
%%{2,[1,2,1]%%%}+%%%{1,[0,2,0]%%%}]%%},[1]%%%} Error: Bad Argument Value
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maple [A] time = 0.03, size = 43, normalized size = 0.45 \[ -\frac {2 \sqrt {\frac {c \left (a x -1\right )}{a x}}\, \left (104 x^{3} a^{3}+52 a^{2} x^{2}+39 a x +15\right )}{105 x^{3}} \]
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^4,x)
[Out]
-2/105*(c*(a*x-1)/a/x)^(1/2)*(104*a^3*x^3+52*a^2*x^2+39*a*x+15)/x^3
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ -\int \frac {{\left (a x + 1\right )}^{2} \sqrt {c - \frac {c}{a x}}}{{\left (a^{2} x^{2} - 1\right )} x^{4}}\,{d x} \]
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^4,x, algorithm="maxima")
[Out]
-integrate((a*x + 1)^2*sqrt(c - c/(a*x))/((a^2*x^2 - 1)*x^4), x)
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mupad [B] time = 0.91, size = 77, normalized size = 0.80 \[ -\frac {208\,a^3\,\sqrt {c-\frac {c}{a\,x}}}{105}-\frac {2\,\sqrt {c-\frac {c}{a\,x}}}{7\,x^3}-\frac {26\,a\,\sqrt {c-\frac {c}{a\,x}}}{35\,x^2}-\frac {104\,a^2\,\sqrt {c-\frac {c}{a\,x}}}{105\,x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(-((c - c/(a*x))^(1/2)*(a*x + 1)^2)/(x^4*(a^2*x^2 - 1)),x)
[Out]
- (208*a^3*(c - c/(a*x))^(1/2))/105 - (2*(c - c/(a*x))^(1/2))/(7*x^3) - (26*a*(c - c/(a*x))^(1/2))/(35*x^2) -
(104*a^2*(c - c/(a*x))^(1/2))/(105*x)
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ - \int \frac {\sqrt {c - \frac {c}{a x}}}{a x^{5} - x^{4}}\, dx - \int \frac {a x \sqrt {c - \frac {c}{a x}}}{a x^{5} - x^{4}}\, dx \]
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**4,x)
[Out]
-Integral(sqrt(c - c/(a*x))/(a*x**5 - x**4), x) - Integral(a*x*sqrt(c - c/(a*x))/(a*x**5 - x**4), x)
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