3.754 \(\int \frac {e^{2 \tanh ^{-1}(a x)} \sqrt {c-\frac {c}{a^2 x^2}}}{x^4} \, dx\)

Optimal. Leaf size=156 \[ -\frac {7 a^2 \sqrt {c-\frac {c}{a^2 x^2}}}{8 x}-\frac {2 a \sqrt {c-\frac {c}{a^2 x^2}}}{3 x^2}-\frac {\sqrt {c-\frac {c}{a^2 x^2}}}{4 x^3}-\frac {7 a^4 x \sqrt {c-\frac {c}{a^2 x^2}} \tanh ^{-1}\left (\sqrt {1-a x} \sqrt {a x+1}\right )}{8 \sqrt {1-a x} \sqrt {a x+1}}-\frac {4}{3} a^3 \sqrt {c-\frac {c}{a^2 x^2}} \]

[Out]

-4/3*a^3*(c-c/a^2/x^2)^(1/2)-1/4*(c-c/a^2/x^2)^(1/2)/x^3-2/3*a*(c-c/a^2/x^2)^(1/2)/x^2-7/8*a^2*(c-c/a^2/x^2)^(
1/2)/x-7/8*a^4*x*arctanh((-a*x+1)^(1/2)*(a*x+1)^(1/2))*(c-c/a^2/x^2)^(1/2)/(-a*x+1)^(1/2)/(a*x+1)^(1/2)

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Rubi [A]  time = 0.41, antiderivative size = 156, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.259, Rules used = {6159, 6129, 98, 151, 12, 92, 208} \[ -\frac {4}{3} a^3 \sqrt {c-\frac {c}{a^2 x^2}}-\frac {7 a^2 \sqrt {c-\frac {c}{a^2 x^2}}}{8 x}-\frac {2 a \sqrt {c-\frac {c}{a^2 x^2}}}{3 x^2}-\frac {\sqrt {c-\frac {c}{a^2 x^2}}}{4 x^3}-\frac {7 a^4 x \sqrt {c-\frac {c}{a^2 x^2}} \tanh ^{-1}\left (\sqrt {1-a x} \sqrt {a x+1}\right )}{8 \sqrt {1-a x} \sqrt {a x+1}} \]

Antiderivative was successfully verified.

[In]

Int[(E^(2*ArcTanh[a*x])*Sqrt[c - c/(a^2*x^2)])/x^4,x]

[Out]

(-4*a^3*Sqrt[c - c/(a^2*x^2)])/3 - Sqrt[c - c/(a^2*x^2)]/(4*x^3) - (2*a*Sqrt[c - c/(a^2*x^2)])/(3*x^2) - (7*a^
2*Sqrt[c - c/(a^2*x^2)])/(8*x) - (7*a^4*Sqrt[c - c/(a^2*x^2)]*x*ArcTanh[Sqrt[1 - a*x]*Sqrt[1 + a*x]])/(8*Sqrt[
1 - a*x]*Sqrt[1 + a*x])

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 92

Int[1/(Sqrt[(a_.) + (b_.)*(x_)]*Sqrt[(c_.) + (d_.)*(x_)]*((e_.) + (f_.)*(x_))), x_Symbol] :> Dist[b*f, Subst[I
nt[1/(d*(b*e - a*f)^2 + b*f^2*x^2), x], x, Sqrt[a + b*x]*Sqrt[c + d*x]], x] /; FreeQ[{a, b, c, d, e, f}, x] &&
EqQ[2*b*d*e - f*(b*c + a*d), 0]

Rule 98

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

Rule 151

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)), x_Symb
ol] :> Simp[((b*g - a*h)*(a + b*x)^(m + 1)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/((m + 1)*(b*c - a*d)*(b*e - a*
f)), x] + Dist[1/((m + 1)*(b*c - a*d)*(b*e - a*f)), Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*Simp[(a*d*f*
g - b*(d*e + c*f)*g + b*c*e*h)*(m + 1) - (b*g - a*h)*(d*e*(n + 1) + c*f*(p + 1)) - d*f*(b*g - a*h)*(m + n + p
+ 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && LtQ[m, -1] && IntegerQ[m]

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 6129

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

Rule 6159

Int[E^(ArcTanh[(a_.)*(x_)]*(n_))*(u_.)*((c_) + (d_.)/(x_)^2)^(p_), x_Symbol] :> Dist[(x^(2*p)*(c + d/x^2)^p)/(
(1 - a*x)^p*(1 + a*x)^p), Int[(u*(1 - a*x)^p*(1 + a*x)^p*E^(n*ArcTanh[a*x]))/x^(2*p), x], x] /; FreeQ[{a, c, d
, n, p}, x] && EqQ[c + a^2*d, 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^2 x^2}}}{x^4} \, dx &=\frac {\left (\sqrt {c-\frac {c}{a^2 x^2}} x\right ) \int \frac {e^{2 \tanh ^{-1}(a x)} \sqrt {1-a x} \sqrt {1+a x}}{x^5} \, dx}{\sqrt {1-a x} \sqrt {1+a x}}\\ &=\frac {\left (\sqrt {c-\frac {c}{a^2 x^2}} x\right ) \int \frac {(1+a x)^{3/2}}{x^5 \sqrt {1-a x}} \, dx}{\sqrt {1-a x} \sqrt {1+a x}}\\ &=-\frac {\sqrt {c-\frac {c}{a^2 x^2}}}{4 x^3}-\frac {\left (\sqrt {c-\frac {c}{a^2 x^2}} x\right ) \int \frac {-8 a-7 a^2 x}{x^4 \sqrt {1-a x} \sqrt {1+a x}} \, dx}{4 \sqrt {1-a x} \sqrt {1+a x}}\\ &=-\frac {\sqrt {c-\frac {c}{a^2 x^2}}}{4 x^3}-\frac {2 a \sqrt {c-\frac {c}{a^2 x^2}}}{3 x^2}+\frac {\left (\sqrt {c-\frac {c}{a^2 x^2}} x\right ) \int \frac {21 a^2+16 a^3 x}{x^3 \sqrt {1-a x} \sqrt {1+a x}} \, dx}{12 \sqrt {1-a x} \sqrt {1+a x}}\\ &=-\frac {\sqrt {c-\frac {c}{a^2 x^2}}}{4 x^3}-\frac {2 a \sqrt {c-\frac {c}{a^2 x^2}}}{3 x^2}-\frac {7 a^2 \sqrt {c-\frac {c}{a^2 x^2}}}{8 x}-\frac {\left (\sqrt {c-\frac {c}{a^2 x^2}} x\right ) \int \frac {-32 a^3-21 a^4 x}{x^2 \sqrt {1-a x} \sqrt {1+a x}} \, dx}{24 \sqrt {1-a x} \sqrt {1+a x}}\\ &=-\frac {4}{3} a^3 \sqrt {c-\frac {c}{a^2 x^2}}-\frac {\sqrt {c-\frac {c}{a^2 x^2}}}{4 x^3}-\frac {2 a \sqrt {c-\frac {c}{a^2 x^2}}}{3 x^2}-\frac {7 a^2 \sqrt {c-\frac {c}{a^2 x^2}}}{8 x}+\frac {\left (\sqrt {c-\frac {c}{a^2 x^2}} x\right ) \int \frac {21 a^4}{x \sqrt {1-a x} \sqrt {1+a x}} \, dx}{24 \sqrt {1-a x} \sqrt {1+a x}}\\ &=-\frac {4}{3} a^3 \sqrt {c-\frac {c}{a^2 x^2}}-\frac {\sqrt {c-\frac {c}{a^2 x^2}}}{4 x^3}-\frac {2 a \sqrt {c-\frac {c}{a^2 x^2}}}{3 x^2}-\frac {7 a^2 \sqrt {c-\frac {c}{a^2 x^2}}}{8 x}+\frac {\left (7 a^4 \sqrt {c-\frac {c}{a^2 x^2}} x\right ) \int \frac {1}{x \sqrt {1-a x} \sqrt {1+a x}} \, dx}{8 \sqrt {1-a x} \sqrt {1+a x}}\\ &=-\frac {4}{3} a^3 \sqrt {c-\frac {c}{a^2 x^2}}-\frac {\sqrt {c-\frac {c}{a^2 x^2}}}{4 x^3}-\frac {2 a \sqrt {c-\frac {c}{a^2 x^2}}}{3 x^2}-\frac {7 a^2 \sqrt {c-\frac {c}{a^2 x^2}}}{8 x}-\frac {\left (7 a^5 \sqrt {c-\frac {c}{a^2 x^2}} x\right ) \operatorname {Subst}\left (\int \frac {1}{a-a x^2} \, dx,x,\sqrt {1-a x} \sqrt {1+a x}\right )}{8 \sqrt {1-a x} \sqrt {1+a x}}\\ &=-\frac {4}{3} a^3 \sqrt {c-\frac {c}{a^2 x^2}}-\frac {\sqrt {c-\frac {c}{a^2 x^2}}}{4 x^3}-\frac {2 a \sqrt {c-\frac {c}{a^2 x^2}}}{3 x^2}-\frac {7 a^2 \sqrt {c-\frac {c}{a^2 x^2}}}{8 x}-\frac {7 a^4 \sqrt {c-\frac {c}{a^2 x^2}} x \tanh ^{-1}\left (\sqrt {1-a x} \sqrt {1+a x}\right )}{8 \sqrt {1-a x} \sqrt {1+a x}}\\ \end {align*}

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Mathematica [A]  time = 0.08, size = 95, normalized size = 0.61 \[ \frac {\sqrt {c-\frac {c}{a^2 x^2}} \left (21 a^4 x^4 \tan ^{-1}\left (\frac {1}{\sqrt {a^2 x^2-1}}\right )-\sqrt {a^2 x^2-1} \left (32 a^3 x^3+21 a^2 x^2+16 a x+6\right )\right )}{24 x^3 \sqrt {a^2 x^2-1}} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[(E^(2*ArcTanh[a*x])*Sqrt[c - c/(a^2*x^2)])/x^4,x]

[Out]

(Sqrt[c - c/(a^2*x^2)]*(-(Sqrt[-1 + a^2*x^2]*(6 + 16*a*x + 21*a^2*x^2 + 32*a^3*x^3)) + 21*a^4*x^4*ArcTan[1/Sqr
t[-1 + a^2*x^2]]))/(24*x^3*Sqrt[-1 + a^2*x^2])

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fricas [A]  time = 0.76, size = 217, normalized size = 1.39 \[ \left [\frac {21 \, a^{3} \sqrt {-c} x^{3} \log \left (-\frac {a^{2} c x^{2} - 2 \, a \sqrt {-c} x \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}} - 2 \, c}{x^{2}}\right ) - 2 \, {\left (32 \, a^{3} x^{3} + 21 \, a^{2} x^{2} + 16 \, a x + 6\right )} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}}}{48 \, x^{3}}, \frac {21 \, a^{3} \sqrt {c} x^{3} \arctan \left (\frac {a \sqrt {c} x \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}}}{a^{2} c x^{2} - c}\right ) - {\left (32 \, a^{3} x^{3} + 21 \, a^{2} x^{2} + 16 \, a x + 6\right )} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}}}{24 \, x^{3}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x+1)^2/(-a^2*x^2+1)*(c-c/a^2/x^2)^(1/2)/x^4,x, algorithm="fricas")

[Out]

[1/48*(21*a^3*sqrt(-c)*x^3*log(-(a^2*c*x^2 - 2*a*sqrt(-c)*x*sqrt((a^2*c*x^2 - c)/(a^2*x^2)) - 2*c)/x^2) - 2*(3
2*a^3*x^3 + 21*a^2*x^2 + 16*a*x + 6)*sqrt((a^2*c*x^2 - c)/(a^2*x^2)))/x^3, 1/24*(21*a^3*sqrt(c)*x^3*arctan(a*s
qrt(c)*x*sqrt((a^2*c*x^2 - c)/(a^2*x^2))/(a^2*c*x^2 - c)) - (32*a^3*x^3 + 21*a^2*x^2 + 16*a*x + 6)*sqrt((a^2*c
*x^2 - c)/(a^2*x^2)))/x^3]

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giac [B]  time = 6.42, size = 316, normalized size = 2.03 \[ -\frac {1}{12} \, {\left (21 \, a^{2} \sqrt {c} \arctan \left (-\frac {\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - c}}{\sqrt {c}}\right ) \mathrm {sgn}\relax (x) - \frac {21 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - c}\right )}^{7} a^{2} c \mathrm {sgn}\relax (x) + 45 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - c}\right )}^{5} a^{2} c^{2} \mathrm {sgn}\relax (x) - 96 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - c}\right )}^{4} a c^{\frac {5}{2}} {\left | a \right |} \mathrm {sgn}\relax (x) - 45 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - c}\right )}^{3} a^{2} c^{3} \mathrm {sgn}\relax (x) - 128 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - c}\right )}^{2} a c^{\frac {7}{2}} {\left | a \right |} \mathrm {sgn}\relax (x) - 21 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - c}\right )} a^{2} c^{4} \mathrm {sgn}\relax (x) - 32 \, a c^{\frac {9}{2}} {\left | a \right |} \mathrm {sgn}\relax (x)}{{\left ({\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - c}\right )}^{2} + c\right )}^{4}}\right )} {\left | a \right |} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x+1)^2/(-a^2*x^2+1)*(c-c/a^2/x^2)^(1/2)/x^4,x, algorithm="giac")

[Out]

-1/12*(21*a^2*sqrt(c)*arctan(-(sqrt(a^2*c)*x - sqrt(a^2*c*x^2 - c))/sqrt(c))*sgn(x) - (21*(sqrt(a^2*c)*x - sqr
t(a^2*c*x^2 - c))^7*a^2*c*sgn(x) + 45*(sqrt(a^2*c)*x - sqrt(a^2*c*x^2 - c))^5*a^2*c^2*sgn(x) - 96*(sqrt(a^2*c)
*x - sqrt(a^2*c*x^2 - c))^4*a*c^(5/2)*abs(a)*sgn(x) - 45*(sqrt(a^2*c)*x - sqrt(a^2*c*x^2 - c))^3*a^2*c^3*sgn(x
) - 128*(sqrt(a^2*c)*x - sqrt(a^2*c*x^2 - c))^2*a*c^(7/2)*abs(a)*sgn(x) - 21*(sqrt(a^2*c)*x - sqrt(a^2*c*x^2 -
c))*a^2*c^4*sgn(x) - 32*a*c^(9/2)*abs(a)*sgn(x))/((sqrt(a^2*c)*x - sqrt(a^2*c*x^2 - c))^2 + c)^4)*abs(a)

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maple [B]  time = 0.05, size = 410, normalized size = 2.63 \[ \frac {\sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2} x^{2}}}\, a^{2} \left (-48 \sqrt {-\frac {c}{a^{2}}}\, \sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}}\, x^{5} a^{3} c +48 \sqrt {-\frac {c}{a^{2}}}\, \left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )^{\frac {3}{2}} x^{3} a^{3}+48 \sqrt {-\frac {c}{a^{2}}}\, c^{\frac {3}{2}} \ln \left (x \sqrt {c}+\sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}}\right ) x^{4} a -48 \sqrt {-\frac {c}{a^{2}}}\, c^{\frac {3}{2}} \ln \left (\frac {\sqrt {c}\, \sqrt {\frac {\left (a x -1\right ) \left (a x +1\right ) c}{a^{2}}}+c x}{\sqrt {c}}\right ) x^{4} a -48 \sqrt {-\frac {c}{a^{2}}}\, \sqrt {\frac {\left (a x -1\right ) \left (a x +1\right ) c}{a^{2}}}\, x^{4} a^{2} c +21 \sqrt {-\frac {c}{a^{2}}}\, \sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}}\, x^{4} a^{2} c +27 \sqrt {-\frac {c}{a^{2}}}\, \left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )^{\frac {3}{2}} x^{2} a^{2}+21 \ln \left (\frac {2 \sqrt {-\frac {c}{a^{2}}}\, \sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}}\, a^{2}-2 c}{a^{2} x}\right ) x^{4} c^{2}+16 a \left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )^{\frac {3}{2}} \sqrt {-\frac {c}{a^{2}}}\, x +6 \left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )^{\frac {3}{2}} \sqrt {-\frac {c}{a^{2}}}\right )}{24 x^{3} \sqrt {-\frac {c}{a^{2}}}\, \sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}}\, c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a*x+1)^2/(-a^2*x^2+1)*(c-c/a^2/x^2)^(1/2)/x^4,x)

[Out]

1/24*(c*(a^2*x^2-1)/a^2/x^2)^(1/2)/x^3*a^2*(-48*(-c/a^2)^(1/2)*(c*(a^2*x^2-1)/a^2)^(1/2)*x^5*a^3*c+48*(-c/a^2)
^(1/2)*(c*(a^2*x^2-1)/a^2)^(3/2)*x^3*a^3+48*(-c/a^2)^(1/2)*c^(3/2)*ln(x*c^(1/2)+(c*(a^2*x^2-1)/a^2)^(1/2))*x^4
*a-48*(-c/a^2)^(1/2)*c^(3/2)*ln((c^(1/2)*((a*x-1)*(a*x+1)*c/a^2)^(1/2)+c*x)/c^(1/2))*x^4*a-48*(-c/a^2)^(1/2)*(
(a*x-1)*(a*x+1)*c/a^2)^(1/2)*x^4*a^2*c+21*(-c/a^2)^(1/2)*(c*(a^2*x^2-1)/a^2)^(1/2)*x^4*a^2*c+27*(-c/a^2)^(1/2)
*(c*(a^2*x^2-1)/a^2)^(3/2)*x^2*a^2+21*ln(2*((-c/a^2)^(1/2)*(c*(a^2*x^2-1)/a^2)^(1/2)*a^2-c)/a^2/x)*x^4*c^2+16*
a*(c*(a^2*x^2-1)/a^2)^(3/2)*(-c/a^2)^(1/2)*x+6*(c*(a^2*x^2-1)/a^2)^(3/2)*(-c/a^2)^(1/2))/(-c/a^2)^(1/2)/(c*(a^
2*x^2-1)/a^2)^(1/2)/c

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ -\int \frac {{\left (a x + 1\right )}^{2} \sqrt {c - \frac {c}{a^{2} x^{2}}}}{{\left (a^{2} x^{2} - 1\right )} x^{4}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x+1)^2/(-a^2*x^2+1)*(c-c/a^2/x^2)^(1/2)/x^4,x, algorithm="maxima")

[Out]

-integrate((a*x + 1)^2*sqrt(c - c/(a^2*x^2))/((a^2*x^2 - 1)*x^4), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ -\int \frac {\sqrt {c-\frac {c}{a^2\,x^2}}\,{\left (a\,x+1\right )}^2}{x^4\,\left (a^2\,x^2-1\right )} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-((c - c/(a^2*x^2))^(1/2)*(a*x + 1)^2)/(x^4*(a^2*x^2 - 1)),x)

[Out]

-int(((c - c/(a^2*x^2))^(1/2)*(a*x + 1)^2)/(x^4*(a^2*x^2 - 1)), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ - \int \frac {\sqrt {c - \frac {c}{a^{2} x^{2}}}}{a x^{5} - x^{4}}\, dx - \int \frac {a x \sqrt {c - \frac {c}{a^{2} x^{2}}}}{a x^{5} - x^{4}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x+1)**2/(-a**2*x**2+1)*(c-c/a**2/x**2)**(1/2)/x**4,x)

[Out]

-Integral(sqrt(c - c/(a**2*x**2))/(a*x**5 - x**4), x) - Integral(a*x*sqrt(c - c/(a**2*x**2))/(a*x**5 - x**4),
x)

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