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

Optimal. Leaf size=163 \[ -\frac {2 a^4 \left (c-\frac {c}{a x}\right )^{9/2}}{9 c^4}+\frac {2 a^4 \left (c-\frac {c}{a x}\right )^{7/2}}{7 c^3}-\frac {2 a^4 \left (c-\frac {c}{a x}\right )^{5/2}}{5 c^2}-\frac {2 a^4 \left (c-\frac {c}{a x}\right )^{3/2}}{3 c}-4 a^4 \sqrt {c-\frac {c}{a x}}+4 \sqrt {2} a^4 \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {2} \sqrt {c}}\right ) \]

[Out]

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

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Rubi [A]  time = 0.30, antiderivative size = 163, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 8, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.296, Rules used = {6133, 25, 514, 446, 88, 50, 63, 208} \[ -\frac {2 a^4 \left (c-\frac {c}{a x}\right )^{9/2}}{9 c^4}+\frac {2 a^4 \left (c-\frac {c}{a x}\right )^{7/2}}{7 c^3}-\frac {2 a^4 \left (c-\frac {c}{a x}\right )^{5/2}}{5 c^2}-\frac {2 a^4 \left (c-\frac {c}{a x}\right )^{3/2}}{3 c}-4 a^4 \sqrt {c-\frac {c}{a x}}+4 \sqrt {2} a^4 \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {2} \sqrt {c}}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 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 88

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandI
ntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && IntegersQ[m, n] &&
(IntegerQ[p] || (GtQ[m, 0] && GeQ[n, -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 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 {a \int \frac {\left (c-\frac {c}{a x}\right )^{3/2}}{x^4 (1+a x)} \, dx}{c}\\ &=-\frac {a \int \frac {\left (c-\frac {c}{a x}\right )^{3/2}}{\left (a+\frac {1}{x}\right ) x^5} \, dx}{c}\\ &=\frac {a \operatorname {Subst}\left (\int \frac {x^3 \left (c-\frac {c x}{a}\right )^{3/2}}{a+x} \, dx,x,\frac {1}{x}\right )}{c}\\ &=\frac {a \operatorname {Subst}\left (\int \left (a^2 \left (c-\frac {c x}{a}\right )^{3/2}-\frac {a^3 \left (c-\frac {c x}{a}\right )^{3/2}}{a+x}-\frac {a^2 \left (c-\frac {c x}{a}\right )^{5/2}}{c}+\frac {a^2 \left (c-\frac {c x}{a}\right )^{7/2}}{c^2}\right ) \, dx,x,\frac {1}{x}\right )}{c}\\ &=-\frac {2 a^4 \left (c-\frac {c}{a x}\right )^{5/2}}{5 c^2}+\frac {2 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}-\frac {a^4 \operatorname {Subst}\left (\int \frac {\left (c-\frac {c x}{a}\right )^{3/2}}{a+x} \, dx,x,\frac {1}{x}\right )}{c}\\ &=-\frac {2 a^4 \left (c-\frac {c}{a x}\right )^{3/2}}{3 c}-\frac {2 a^4 \left (c-\frac {c}{a x}\right )^{5/2}}{5 c^2}+\frac {2 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}-\left (2 a^4\right ) \operatorname {Subst}\left (\int \frac {\sqrt {c-\frac {c x}{a}}}{a+x} \, dx,x,\frac {1}{x}\right )\\ &=-4 a^4 \sqrt {c-\frac {c}{a x}}-\frac {2 a^4 \left (c-\frac {c}{a x}\right )^{3/2}}{3 c}-\frac {2 a^4 \left (c-\frac {c}{a x}\right )^{5/2}}{5 c^2}+\frac {2 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}-\left (4 a^4 c\right ) \operatorname {Subst}\left (\int \frac {1}{(a+x) \sqrt {c-\frac {c x}{a}}} \, dx,x,\frac {1}{x}\right )\\ &=-4 a^4 \sqrt {c-\frac {c}{a x}}-\frac {2 a^4 \left (c-\frac {c}{a x}\right )^{3/2}}{3 c}-\frac {2 a^4 \left (c-\frac {c}{a x}\right )^{5/2}}{5 c^2}+\frac {2 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}+\left (8 a^5\right ) \operatorname {Subst}\left (\int \frac {1}{2 a-\frac {a x^2}{c}} \, dx,x,\sqrt {c-\frac {c}{a x}}\right )\\ &=-4 a^4 \sqrt {c-\frac {c}{a x}}-\frac {2 a^4 \left (c-\frac {c}{a x}\right )^{3/2}}{3 c}-\frac {2 a^4 \left (c-\frac {c}{a x}\right )^{5/2}}{5 c^2}+\frac {2 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}+4 \sqrt {2} a^4 \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {2} \sqrt {c}}\right )\\ \end {align*}

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Mathematica [A]  time = 0.14, size = 95, normalized size = 0.58 \[ 4 \sqrt {2} a^4 \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {2} \sqrt {c}}\right )-\frac {2 \left (788 a^4 x^4-236 a^3 x^3+138 a^2 x^2-95 a x+35\right ) \sqrt {c-\frac {c}{a x}}}{315 x^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*Sqrt[c - c/(a*x)]*(35 - 95*a*x + 138*a^2*x^2 - 236*a^3*x^3 + 788*a^4*x^4))/(315*x^4) + 4*Sqrt[2]*a^4*Sqrt[
c]*ArcTanh[Sqrt[c - c/(a*x)]/(Sqrt[2]*Sqrt[c])]

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fricas [A]  time = 0.51, size = 217, normalized size = 1.33 \[ \left [\frac {2 \, {\left (315 \, \sqrt {2} a^{4} \sqrt {c} x^{4} \log \left (-\frac {2 \, \sqrt {2} a \sqrt {c} x \sqrt {\frac {a c x - c}{a x}} + 3 \, a c x - c}{a x + 1}\right ) - {\left (788 \, a^{4} x^{4} - 236 \, a^{3} x^{3} + 138 \, a^{2} x^{2} - 95 \, a x + 35\right )} \sqrt {\frac {a c x - c}{a x}}\right )}}{315 \, x^{4}}, -\frac {2 \, {\left (630 \, \sqrt {2} a^{4} \sqrt {-c} x^{4} \arctan \left (\frac {\sqrt {2} \sqrt {-c} \sqrt {\frac {a c x - c}{a x}}}{2 \, c}\right ) + {\left (788 \, a^{4} x^{4} - 236 \, a^{3} x^{3} + 138 \, a^{2} x^{2} - 95 \, a x + 35\right )} \sqrt {\frac {a c x - c}{a x}}\right )}}{315 \, x^{4}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[2/315*(315*sqrt(2)*a^4*sqrt(c)*x^4*log(-(2*sqrt(2)*a*sqrt(c)*x*sqrt((a*c*x - c)/(a*x)) + 3*a*c*x - c)/(a*x +
1)) - (788*a^4*x^4 - 236*a^3*x^3 + 138*a^2*x^2 - 95*a*x + 35)*sqrt((a*c*x - c)/(a*x)))/x^4, -2/315*(630*sqrt(2
)*a^4*sqrt(-c)*x^4*arctan(1/2*sqrt(2)*sqrt(-c)*sqrt((a*c*x - c)/(a*x))/c) + (788*a^4*x^4 - 236*a^3*x^3 + 138*a
^2*x^2 - 95*a*x + 35)*sqrt((a*c*x - c)/(a*x)))/x^4]

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giac [B]  time = 2.35, size = 434, normalized size = 2.66 \[ -\frac {4 \, \sqrt {2} a^{5} c \arctan \left (\frac {\sqrt {2} {\left ({\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - a c x}\right )} a + \sqrt {c} {\left | a \right |}\right )}}{2 \, a \sqrt {-c}}\right )}{\sqrt {-c} {\left | a \right |} \mathrm {sgn}\relax (x)} - \frac {2 \, {\left (1260 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - a c x}\right )}^{8} a^{9} c - 1260 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - a c x}\right )}^{7} a^{8} c^{\frac {3}{2}} {\left | a \right |} + 2100 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - a c x}\right )}^{6} a^{9} c^{2} - 3150 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - a c x}\right )}^{5} a^{8} c^{\frac {5}{2}} {\left | a \right |} + 3528 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - a c x}\right )}^{4} a^{9} c^{3} - 2625 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - a c x}\right )}^{3} a^{8} c^{\frac {7}{2}} {\left | a \right |} + 1215 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - a c x}\right )}^{2} a^{9} c^{4} - 315 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - a c x}\right )} a^{8} c^{\frac {9}{2}} {\left | a \right |} + 35 \, a^{9} c^{5}\right )}}{315 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - a c x}\right )}^{9} a^{4} {\left | a \right |} \mathrm {sgn}\relax (x)} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-4*sqrt(2)*a^5*c*arctan(1/2*sqrt(2)*((sqrt(a^2*c)*x - sqrt(a^2*c*x^2 - a*c*x))*a + sqrt(c)*abs(a))/(a*sqrt(-c)
))/(sqrt(-c)*abs(a)*sgn(x)) - 2/315*(1260*(sqrt(a^2*c)*x - sqrt(a^2*c*x^2 - a*c*x))^8*a^9*c - 1260*(sqrt(a^2*c
)*x - sqrt(a^2*c*x^2 - a*c*x))^7*a^8*c^(3/2)*abs(a) + 2100*(sqrt(a^2*c)*x - sqrt(a^2*c*x^2 - a*c*x))^6*a^9*c^2
- 3150*(sqrt(a^2*c)*x - sqrt(a^2*c*x^2 - a*c*x))^5*a^8*c^(5/2)*abs(a) + 3528*(sqrt(a^2*c)*x - sqrt(a^2*c*x^2
- a*c*x))^4*a^9*c^3 - 2625*(sqrt(a^2*c)*x - sqrt(a^2*c*x^2 - a*c*x))^3*a^8*c^(7/2)*abs(a) + 1215*(sqrt(a^2*c)*
x - sqrt(a^2*c*x^2 - a*c*x))^2*a^9*c^4 - 315*(sqrt(a^2*c)*x - sqrt(a^2*c*x^2 - a*c*x))*a^8*c^(9/2)*abs(a) + 35
*a^9*c^5)/((sqrt(a^2*c)*x - sqrt(a^2*c*x^2 - a*c*x))^9*a^4*abs(a)*sgn(x))

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maple [B]  time = 0.05, size = 326, normalized size = 2.00 \[ \frac {\sqrt {\frac {c \left (a x -1\right )}{a x}}\, \left (630 \sqrt {\left (a x -1\right ) x}\, a^{\frac {11}{2}} \sqrt {\frac {1}{a}}\, x^{6}-1890 \sqrt {a \,x^{2}-x}\, a^{\frac {11}{2}} \sqrt {\frac {1}{a}}\, x^{6}+1260 \left (a \,x^{2}-x \right )^{\frac {3}{2}} a^{\frac {9}{2}} \sqrt {\frac {1}{a}}\, x^{4}+945 \ln \left (\frac {2 \sqrt {a \,x^{2}-x}\, \sqrt {a}+2 a x -1}{2 \sqrt {a}}\right ) \sqrt {\frac {1}{a}}\, x^{6} a^{5}-630 a^{\frac {9}{2}} \sqrt {2}\, \ln \left (\frac {2 \sqrt {2}\, \sqrt {\frac {1}{a}}\, \sqrt {\left (a x -1\right ) x}\, a -3 a x +1}{a x +1}\right ) x^{6}-945 \ln \left (\frac {2 \sqrt {\left (a x -1\right ) x}\, \sqrt {a}+2 a x -1}{2 \sqrt {a}}\right ) \sqrt {\frac {1}{a}}\, x^{6} a^{5}-316 \left (a \,x^{2}-x \right )^{\frac {3}{2}} \sqrt {\frac {1}{a}}\, a^{\frac {7}{2}} x^{3}+156 a^{\frac {5}{2}} \left (a \,x^{2}-x \right )^{\frac {3}{2}} x^{2} \sqrt {\frac {1}{a}}-120 a^{\frac {3}{2}} \left (a \,x^{2}-x \right )^{\frac {3}{2}} x \sqrt {\frac {1}{a}}+70 \left (a \,x^{2}-x \right )^{\frac {3}{2}} \sqrt {a}\, \sqrt {\frac {1}{a}}\right )}{315 x^{5} \sqrt {\left (a x -1\right ) x}\, \sqrt {a}\, \sqrt {\frac {1}{a}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/315*(c*(a*x-1)/a/x)^(1/2)/x^5*(630*((a*x-1)*x)^(1/2)*a^(11/2)*(1/a)^(1/2)*x^6-1890*(a*x^2-x)^(1/2)*a^(11/2)*
(1/a)^(1/2)*x^6+1260*(a*x^2-x)^(3/2)*a^(9/2)*(1/a)^(1/2)*x^4+945*ln(1/2*(2*(a*x^2-x)^(1/2)*a^(1/2)+2*a*x-1)/a^
(1/2))*(1/a)^(1/2)*x^6*a^5-630*a^(9/2)*2^(1/2)*ln((2*2^(1/2)*(1/a)^(1/2)*((a*x-1)*x)^(1/2)*a-3*a*x+1)/(a*x+1))
*x^6-945*ln(1/2*(2*((a*x-1)*x)^(1/2)*a^(1/2)+2*a*x-1)/a^(1/2))*(1/a)^(1/2)*x^6*a^5-316*(a*x^2-x)^(3/2)*(1/a)^(
1/2)*a^(7/2)*x^3+156*a^(5/2)*(a*x^2-x)^(3/2)*x^2*(1/a)^(1/2)-120*a^(3/2)*(a*x^2-x)^(3/2)*x*(1/a)^(1/2)+70*(a*x
^2-x)^(3/2)*a^(1/2)*(1/a)^(1/2))/((a*x-1)*x)^(1/2)/a^(1/2)/(1/a)^(1/2)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 2.62, size = 138, normalized size = 0.85 \[ \frac {2\,a^4\,{\left (c-\frac {c}{a\,x}\right )}^{7/2}}{7\,c^3}-\frac {2\,a^4\,{\left (c-\frac {c}{a\,x}\right )}^{3/2}}{3\,c}-\frac {2\,a^4\,{\left (c-\frac {c}{a\,x}\right )}^{5/2}}{5\,c^2}-4\,a^4\,\sqrt {c-\frac {c}{a\,x}}-\frac {2\,a^4\,{\left (c-\frac {c}{a\,x}\right )}^{9/2}}{9\,c^4}-\sqrt {2}\,a^4\,\sqrt {c}\,\mathrm {atan}\left (\frac {\sqrt {2}\,\sqrt {c-\frac {c}{a\,x}}\,1{}\mathrm {i}}{2\,\sqrt {c}}\right )\,4{}\mathrm {i} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(2*a^4*(c - c/(a*x))^(7/2))/(7*c^3) - (2*a^4*(c - c/(a*x))^(3/2))/(3*c) - (2*a^4*(c - c/(a*x))^(5/2))/(5*c^2)
- 4*a^4*(c - c/(a*x))^(1/2) - (2*a^4*(c - c/(a*x))^(9/2))/(9*c^4) - 2^(1/2)*a^4*c^(1/2)*atan((2^(1/2)*(c - c/(
a*x))^(1/2)*1i)/(2*c^(1/2)))*4i

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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