\(\int e^{2 \coth ^{-1}(a x)} (c-\frac {c}{a x})^{9/2} \, dx\) [446]

Optimal result

Integrand size = 24, antiderivative size = 143 \[ \int e^{2 \coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^{9/2} \, dx=\frac {5 c^4 \sqrt {c-\frac {c}{a x}}}{a}+\frac {5 c^3 \left (c-\frac {c}{a x}\right )^{3/2}}{3 a}+\frac {c^2 \left (c-\frac {c}{a x}\right )^{5/2}}{a}+\frac {5 c \left (c-\frac {c}{a x}\right )^{7/2}}{7 a}+\left (c-\frac {c}{a x}\right )^{9/2} x-\frac {5 c^{9/2} \text {arctanh}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {c}}\right )}{a} \]

[Out]

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

Rubi [A] (verified)

Time = 0.18 (sec) , antiderivative size = 143, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {6302, 6268, 25, 528, 382, 79, 52, 65, 214} \[ \int e^{2 \coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^{9/2} \, dx=-\frac {5 c^{9/2} \text {arctanh}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {c}}\right )}{a}+\frac {5 c^4 \sqrt {c-\frac {c}{a x}}}{a}+\frac {5 c^3 \left (c-\frac {c}{a x}\right )^{3/2}}{3 a}+\frac {c^2 \left (c-\frac {c}{a x}\right )^{5/2}}{a}+\frac {5 c \left (c-\frac {c}{a x}\right )^{7/2}}{7 a}+x \left (c-\frac {c}{a x}\right )^{9/2} \]

[In]

Int[E^(2*ArcCoth[a*x])*(c - c/(a*x))^(9/2),x]

[Out]

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

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 52

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 65

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 79

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

Rule 214

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x/Rt[-a/b, 2]], x] /; FreeQ[{a, b},
x] && NegQ[a/b]

Rule 382

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]

Rule 528

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 6268

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]

Rule 6302

Int[E^(ArcCoth[(a_.)*(x_)]*(n_))*(u_.), x_Symbol] :> Dist[(-1)^(n/2), Int[u*E^(n*ArcTanh[a*x]), x], x] /; Free
Q[a, x] && IntegerQ[n/2]

Rubi steps \begin{align*} \text {integral}& = -\int e^{2 \text {arctanh}(a x)} \left (c-\frac {c}{a x}\right )^{9/2} \, dx \\ & = -\int \frac {\left (c-\frac {c}{a x}\right )^{9/2} (1+a x)}{1-a x} \, dx \\ & = \frac {c \int \frac {\left (c-\frac {c}{a x}\right )^{7/2} (1+a x)}{x} \, dx}{a} \\ & = \frac {c \int \left (a+\frac {1}{x}\right ) \left (c-\frac {c}{a x}\right )^{7/2} \, dx}{a} \\ & = -\frac {c \text {Subst}\left (\int \frac {(a+x) \left (c-\frac {c x}{a}\right )^{7/2}}{x^2} \, dx,x,\frac {1}{x}\right )}{a} \\ & = \left (c-\frac {c}{a x}\right )^{9/2} x+\frac {(5 c) \text {Subst}\left (\int \frac {\left (c-\frac {c x}{a}\right )^{7/2}}{x} \, dx,x,\frac {1}{x}\right )}{2 a} \\ & = \frac {5 c \left (c-\frac {c}{a x}\right )^{7/2}}{7 a}+\left (c-\frac {c}{a x}\right )^{9/2} x+\frac {\left (5 c^2\right ) \text {Subst}\left (\int \frac {\left (c-\frac {c x}{a}\right )^{5/2}}{x} \, dx,x,\frac {1}{x}\right )}{2 a} \\ & = \frac {c^2 \left (c-\frac {c}{a x}\right )^{5/2}}{a}+\frac {5 c \left (c-\frac {c}{a x}\right )^{7/2}}{7 a}+\left (c-\frac {c}{a x}\right )^{9/2} x+\frac {\left (5 c^3\right ) \text {Subst}\left (\int \frac {\left (c-\frac {c x}{a}\right )^{3/2}}{x} \, dx,x,\frac {1}{x}\right )}{2 a} \\ & = \frac {5 c^3 \left (c-\frac {c}{a x}\right )^{3/2}}{3 a}+\frac {c^2 \left (c-\frac {c}{a x}\right )^{5/2}}{a}+\frac {5 c \left (c-\frac {c}{a x}\right )^{7/2}}{7 a}+\left (c-\frac {c}{a x}\right )^{9/2} x+\frac {\left (5 c^4\right ) \text {Subst}\left (\int \frac {\sqrt {c-\frac {c x}{a}}}{x} \, dx,x,\frac {1}{x}\right )}{2 a} \\ & = \frac {5 c^4 \sqrt {c-\frac {c}{a x}}}{a}+\frac {5 c^3 \left (c-\frac {c}{a x}\right )^{3/2}}{3 a}+\frac {c^2 \left (c-\frac {c}{a x}\right )^{5/2}}{a}+\frac {5 c \left (c-\frac {c}{a x}\right )^{7/2}}{7 a}+\left (c-\frac {c}{a x}\right )^{9/2} x+\frac {\left (5 c^5\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {c-\frac {c x}{a}}} \, dx,x,\frac {1}{x}\right )}{2 a} \\ & = \frac {5 c^4 \sqrt {c-\frac {c}{a x}}}{a}+\frac {5 c^3 \left (c-\frac {c}{a x}\right )^{3/2}}{3 a}+\frac {c^2 \left (c-\frac {c}{a x}\right )^{5/2}}{a}+\frac {5 c \left (c-\frac {c}{a x}\right )^{7/2}}{7 a}+\left (c-\frac {c}{a x}\right )^{9/2} x-\left (5 c^4\right ) \text {Subst}\left (\int \frac {1}{a-\frac {a x^2}{c}} \, dx,x,\sqrt {c-\frac {c}{a x}}\right ) \\ & = \frac {5 c^4 \sqrt {c-\frac {c}{a x}}}{a}+\frac {5 c^3 \left (c-\frac {c}{a x}\right )^{3/2}}{3 a}+\frac {c^2 \left (c-\frac {c}{a x}\right )^{5/2}}{a}+\frac {5 c \left (c-\frac {c}{a x}\right )^{7/2}}{7 a}+\left (c-\frac {c}{a x}\right )^{9/2} x-\frac {5 c^{9/2} \text {arctanh}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {c}}\right )}{a} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.16 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.64 \[ \int e^{2 \coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^{9/2} \, dx=\frac {c^4 \sqrt {c-\frac {c}{a x}} \left (6-18 a x+4 a^2 x^2+92 a^3 x^3+21 a^4 x^4\right )}{21 a^4 x^3}-\frac {5 c^{9/2} \text {arctanh}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {c}}\right )}{a} \]

[In]

Integrate[E^(2*ArcCoth[a*x])*(c - c/(a*x))^(9/2),x]

[Out]

(c^4*Sqrt[c - c/(a*x)]*(6 - 18*a*x + 4*a^2*x^2 + 92*a^3*x^3 + 21*a^4*x^4))/(21*a^4*x^3) - (5*c^(9/2)*ArcTanh[S
qrt[c - c/(a*x)]/Sqrt[c]])/a

Maple [A] (verified)

Time = 0.55 (sec) , antiderivative size = 155, normalized size of antiderivative = 1.08

method	result	size
risch	\(\frac {\left (21 a^{5} x^{5}+71 a^{4} x^{4}-88 a^{3} x^{3}-22 a^{2} x^{2}+24 a x -6\right ) c^{4} \sqrt {\frac {c \left (a x -1\right )}{a x}}}{21 x^{3} a^{4} \left (a x -1\right )}-\frac {5 \ln \left (\frac {-\frac {1}{2} a c +a^{2} c x}{\sqrt {a^{2} c}}+\sqrt {a^{2} c \,x^{2}-a c x}\right ) c^{4} \sqrt {\frac {c \left (a x -1\right )}{a x}}\, \sqrt {c \left (a x -1\right ) a x}}{2 \sqrt {a^{2} c}\, \left (a x -1\right )}\)	\(155\)
default	\(-\frac {\sqrt {\frac {c \left (a x -1\right )}{a x}}\, c^{4} \left (-210 a^{\frac {9}{2}} \sqrt {a \,x^{2}-x}\, x^{5}+105 \ln \left (\frac {2 \sqrt {a \,x^{2}-x}\, \sqrt {a}+2 a x -1}{2 \sqrt {a}}\right ) a^{4} x^{5}+168 a^{\frac {7}{2}} \left (a \,x^{2}-x \right )^{\frac {3}{2}} x^{3}-16 a^{\frac {5}{2}} \left (a \,x^{2}-x \right )^{\frac {3}{2}} x^{2}-24 a^{\frac {3}{2}} \left (a \,x^{2}-x \right )^{\frac {3}{2}} x +12 \left (a \,x^{2}-x \right )^{\frac {3}{2}} \sqrt {a}\right )}{42 x^{4} \sqrt {\left (a x -1\right ) x}\, a^{\frac {9}{2}}}\)	\(163\)

[In]

int(1/(a*x-1)*(a*x+1)*(c-c/a/x)^(9/2),x,method=_RETURNVERBOSE)

[Out]

1/21*(21*a^5*x^5+71*a^4*x^4-88*a^3*x^3-22*a^2*x^2+24*a*x-6)/x^3*c^4/a^4/(a*x-1)*(c*(a*x-1)/a/x)^(1/2)-5/2*ln((
-1/2*a*c+a^2*c*x)/(a^2*c)^(1/2)+(a^2*c*x^2-a*c*x)^(1/2))/(a^2*c)^(1/2)*c^4/(a*x-1)*(c*(a*x-1)/a/x)^(1/2)*(c*(a
*x-1)*a*x)^(1/2)

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 234, normalized size of antiderivative = 1.64 \[ \int e^{2 \coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^{9/2} \, dx=\left [\frac {105 \, a^{3} c^{\frac {9}{2}} x^{3} \log \left (-2 \, a c x + 2 \, a \sqrt {c} x \sqrt {\frac {a c x - c}{a x}} + c\right ) + 2 \, {\left (21 \, a^{4} c^{4} x^{4} + 92 \, a^{3} c^{4} x^{3} + 4 \, a^{2} c^{4} x^{2} - 18 \, a c^{4} x + 6 \, c^{4}\right )} \sqrt {\frac {a c x - c}{a x}}}{42 \, a^{4} x^{3}}, \frac {105 \, a^{3} \sqrt {-c} c^{4} x^{3} \arctan \left (\frac {\sqrt {-c} \sqrt {\frac {a c x - c}{a x}}}{c}\right ) + {\left (21 \, a^{4} c^{4} x^{4} + 92 \, a^{3} c^{4} x^{3} + 4 \, a^{2} c^{4} x^{2} - 18 \, a c^{4} x + 6 \, c^{4}\right )} \sqrt {\frac {a c x - c}{a x}}}{21 \, a^{4} x^{3}}\right ] \]

[In]

integrate(1/(a*x-1)*(a*x+1)*(c-c/a/x)^(9/2),x, algorithm="fricas")

[Out]

[1/42*(105*a^3*c^(9/2)*x^3*log(-2*a*c*x + 2*a*sqrt(c)*x*sqrt((a*c*x - c)/(a*x)) + c) + 2*(21*a^4*c^4*x^4 + 92*
a^3*c^4*x^3 + 4*a^2*c^4*x^2 - 18*a*c^4*x + 6*c^4)*sqrt((a*c*x - c)/(a*x)))/(a^4*x^3), 1/21*(105*a^3*sqrt(-c)*c
^4*x^3*arctan(sqrt(-c)*sqrt((a*c*x - c)/(a*x))/c) + (21*a^4*c^4*x^4 + 92*a^3*c^4*x^3 + 4*a^2*c^4*x^2 - 18*a*c^
4*x + 6*c^4)*sqrt((a*c*x - c)/(a*x)))/(a^4*x^3)]

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 22.54 (sec) , antiderivative size = 2222, normalized size of antiderivative = 15.54 \[ \int e^{2 \coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^{9/2} \, dx=\text {Too large to display} \]

[In]

integrate(1/(a*x-1)*(a*x+1)*(c-c/a/x)**(9/2),x)

[Out]

c**4*Piecewise((-sqrt(c)*acosh(sqrt(a)*sqrt(x))/a + sqrt(c)*sqrt(x)*sqrt(a*x - 1)/sqrt(a), Abs(a*x) > 1), (-I*
sqrt(a)*sqrt(c)*x**(3/2)/sqrt(-a*x + 1) + I*sqrt(c)*asin(sqrt(a)*sqrt(x))/a + I*sqrt(c)*sqrt(x)/(sqrt(a)*sqrt(
-a*x + 1)), True)) + 2*c**4*Piecewise((2*c*atan(sqrt(c - c/(a*x))/sqrt(-c))/sqrt(-c) + 2*sqrt(c - c/(a*x)), Ne
(c/a, 0)), (-sqrt(c)*log(x), True))/a + 2*c**4*Piecewise((-4*a**(11/2)*sqrt(c)*x**(7/2)/(15*a**(7/2)*x**(7/2)
- 15*a**(5/2)*x**(5/2)) + 4*a**(9/2)*sqrt(c)*x**(5/2)/(15*a**(7/2)*x**(7/2) - 15*a**(5/2)*x**(5/2)) + 4*a**5*s
qrt(c)*x**3*sqrt(a*x - 1)/(15*a**(7/2)*x**(7/2) - 15*a**(5/2)*x**(5/2)) - 2*a**4*sqrt(c)*x**2*sqrt(a*x - 1)/(1
5*a**(7/2)*x**(7/2) - 15*a**(5/2)*x**(5/2)) - 8*a**3*sqrt(c)*x*sqrt(a*x - 1)/(15*a**(7/2)*x**(7/2) - 15*a**(5/
2)*x**(5/2)) + 6*a**2*sqrt(c)*sqrt(a*x - 1)/(15*a**(7/2)*x**(7/2) - 15*a**(5/2)*x**(5/2)), Abs(a*x) > 1), (-4*
a**(11/2)*sqrt(c)*x**(7/2)/(15*a**(7/2)*x**(7/2) - 15*a**(5/2)*x**(5/2)) + 4*a**(9/2)*sqrt(c)*x**(5/2)/(15*a**
(7/2)*x**(7/2) - 15*a**(5/2)*x**(5/2)) + 4*I*a**5*sqrt(c)*x**3*sqrt(-a*x + 1)/(15*a**(7/2)*x**(7/2) - 15*a**(5
/2)*x**(5/2)) - 2*I*a**4*sqrt(c)*x**2*sqrt(-a*x + 1)/(15*a**(7/2)*x**(7/2) - 15*a**(5/2)*x**(5/2)) - 8*I*a**3*
sqrt(c)*x*sqrt(-a*x + 1)/(15*a**(7/2)*x**(7/2) - 15*a**(5/2)*x**(5/2)) + 6*I*a**2*sqrt(c)*sqrt(-a*x + 1)/(15*a
**(7/2)*x**(7/2) - 15*a**(5/2)*x**(5/2)), True))/a**3 - c**4*Piecewise((-16*a**(19/2)*sqrt(c)*x**(13/2)/(105*a
**(13/2)*x**(13/2) - 315*a**(11/2)*x**(11/2) + 315*a**(9/2)*x**(9/2) - 105*a**(7/2)*x**(7/2)) + 48*a**(17/2)*s
qrt(c)*x**(11/2)/(105*a**(13/2)*x**(13/2) - 315*a**(11/2)*x**(11/2) + 315*a**(9/2)*x**(9/2) - 105*a**(7/2)*x**
(7/2)) - 48*a**(15/2)*sqrt(c)*x**(9/2)/(105*a**(13/2)*x**(13/2) - 315*a**(11/2)*x**(11/2) + 315*a**(9/2)*x**(9
/2) - 105*a**(7/2)*x**(7/2)) + 16*a**(13/2)*sqrt(c)*x**(7/2)/(105*a**(13/2)*x**(13/2) - 315*a**(11/2)*x**(11/2
) + 315*a**(9/2)*x**(9/2) - 105*a**(7/2)*x**(7/2)) + 16*a**9*sqrt(c)*x**6*sqrt(a*x - 1)/(105*a**(13/2)*x**(13/
2) - 315*a**(11/2)*x**(11/2) + 315*a**(9/2)*x**(9/2) - 105*a**(7/2)*x**(7/2)) - 40*a**8*sqrt(c)*x**5*sqrt(a*x
- 1)/(105*a**(13/2)*x**(13/2) - 315*a**(11/2)*x**(11/2) + 315*a**(9/2)*x**(9/2) - 105*a**(7/2)*x**(7/2)) + 30*
a**7*sqrt(c)*x**4*sqrt(a*x - 1)/(105*a**(13/2)*x**(13/2) - 315*a**(11/2)*x**(11/2) + 315*a**(9/2)*x**(9/2) - 1
05*a**(7/2)*x**(7/2)) - 40*a**6*sqrt(c)*x**3*sqrt(a*x - 1)/(105*a**(13/2)*x**(13/2) - 315*a**(11/2)*x**(11/2)
+ 315*a**(9/2)*x**(9/2) - 105*a**(7/2)*x**(7/2)) + 100*a**5*sqrt(c)*x**2*sqrt(a*x - 1)/(105*a**(13/2)*x**(13/2
) - 315*a**(11/2)*x**(11/2) + 315*a**(9/2)*x**(9/2) - 105*a**(7/2)*x**(7/2)) - 96*a**4*sqrt(c)*x*sqrt(a*x - 1)
/(105*a**(13/2)*x**(13/2) - 315*a**(11/2)*x**(11/2) + 315*a**(9/2)*x**(9/2) - 105*a**(7/2)*x**(7/2)) + 30*a**3
*sqrt(c)*sqrt(a*x - 1)/(105*a**(13/2)*x**(13/2) - 315*a**(11/2)*x**(11/2) + 315*a**(9/2)*x**(9/2) - 105*a**(7/
2)*x**(7/2)), Abs(a*x) > 1), (-16*a**(19/2)*sqrt(c)*x**(13/2)/(105*a**(13/2)*x**(13/2) - 315*a**(11/2)*x**(11/
2) + 315*a**(9/2)*x**(9/2) - 105*a**(7/2)*x**(7/2)) + 48*a**(17/2)*sqrt(c)*x**(11/2)/(105*a**(13/2)*x**(13/2)
- 315*a**(11/2)*x**(11/2) + 315*a**(9/2)*x**(9/2) - 105*a**(7/2)*x**(7/2)) - 48*a**(15/2)*sqrt(c)*x**(9/2)/(10
5*a**(13/2)*x**(13/2) - 315*a**(11/2)*x**(11/2) + 315*a**(9/2)*x**(9/2) - 105*a**(7/2)*x**(7/2)) + 16*a**(13/2
)*sqrt(c)*x**(7/2)/(105*a**(13/2)*x**(13/2) - 315*a**(11/2)*x**(11/2) + 315*a**(9/2)*x**(9/2) - 105*a**(7/2)*x
**(7/2)) + 16*I*a**9*sqrt(c)*x**6*sqrt(-a*x + 1)/(105*a**(13/2)*x**(13/2) - 315*a**(11/2)*x**(11/2) + 315*a**(
9/2)*x**(9/2) - 105*a**(7/2)*x**(7/2)) - 40*I*a**8*sqrt(c)*x**5*sqrt(-a*x + 1)/(105*a**(13/2)*x**(13/2) - 315*
a**(11/2)*x**(11/2) + 315*a**(9/2)*x**(9/2) - 105*a**(7/2)*x**(7/2)) + 30*I*a**7*sqrt(c)*x**4*sqrt(-a*x + 1)/(
105*a**(13/2)*x**(13/2) - 315*a**(11/2)*x**(11/2) + 315*a**(9/2)*x**(9/2) - 105*a**(7/2)*x**(7/2)) - 40*I*a**6
*sqrt(c)*x**3*sqrt(-a*x + 1)/(105*a**(13/2)*x**(13/2) - 315*a**(11/2)*x**(11/2) + 315*a**(9/2)*x**(9/2) - 105*
a**(7/2)*x**(7/2)) + 100*I*a**5*sqrt(c)*x**2*sqrt(-a*x + 1)/(105*a**(13/2)*x**(13/2) - 315*a**(11/2)*x**(11/2)
+ 315*a**(9/2)*x**(9/2) - 105*a**(7/2)*x**(7/2)) - 96*I*a**4*sqrt(c)*x*sqrt(-a*x + 1)/(105*a**(13/2)*x**(13/2
) - 315*a**(11/2)*x**(11/2) + 315*a**(9/2)*x**(9/2) - 105*a**(7/2)*x**(7/2)) + 30*I*a**3*sqrt(c)*sqrt(-a*x + 1
)/(105*a**(13/2)*x**(13/2) - 315*a**(11/2)*x**(11/2) + 315*a**(9/2)*x**(9/2) - 105*a**(7/2)*x**(7/2)), True))/
a**4

Maxima [F]

\[ \int e^{2 \coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^{9/2} \, dx=\int { \frac {{\left (a x + 1\right )} {\left (c - \frac {c}{a x}\right )}^{\frac {9}{2}}}{a x - 1} \,d x } \]

[In]

integrate(1/(a*x-1)*(a*x+1)*(c-c/a/x)^(9/2),x, algorithm="maxima")

[Out]

integrate((a*x + 1)*(c - c/(a*x))^(9/2)/(a*x - 1), x)

Giac [F(-2)]

Exception generated. \[ \int e^{2 \coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^{9/2} \, dx=\text {Exception raised: TypeError} \]

[In]

integrate(1/(a*x-1)*(a*x+1)*(c-c/a/x)^(9/2),x, algorithm="giac")

[Out]

Exception raised: TypeError >> an error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:Limit: Max order reached or unable to make series expansion Error: Bad Argument Value

Mupad [F(-1)]

Timed out. \[ \int e^{2 \coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^{9/2} \, dx=\int \frac {{\left (c-\frac {c}{a\,x}\right )}^{9/2}\,\left (a\,x+1\right )}{a\,x-1} \,d x \]

[In]

int(((c - c/(a*x))^(9/2)*(a*x + 1))/(a*x - 1),x)

[Out]

int(((c - c/(a*x))^(9/2)*(a*x + 1))/(a*x - 1), x)