\(\int e^{\coth ^{-1}(a x)} (c-a c x)^{9/2} \, dx\) [226]

Optimal result

Integrand size = 18, antiderivative size = 254 \[ \int e^{\coth ^{-1}(a x)} (c-a c x)^{9/2} \, dx=-\frac {32 \left (a-\frac {1}{x}\right )^3 \left (1+\frac {1}{a x}\right )^{3/2} (c-a c x)^{9/2}}{99 a^4 \left (1-\frac {1}{a x}\right )^{9/2}}+\frac {9088 \left (1+\frac {1}{a x}\right )^{3/2} (c-a c x)^{9/2}}{3465 a^4 \left (1-\frac {1}{a x}\right )^{9/2} x^3}-\frac {768 \left (1+\frac {1}{a x}\right )^{3/2} (c-a c x)^{9/2}}{385 a^3 \left (1-\frac {1}{a x}\right )^{9/2} x^2}+\frac {128 \left (1+\frac {1}{a x}\right )^{3/2} (c-a c x)^{9/2}}{231 a^2 \left (1-\frac {1}{a x}\right )^{9/2} x}+\frac {2 \left (a-\frac {1}{x}\right )^4 \left (1+\frac {1}{a x}\right )^{3/2} x (c-a c x)^{9/2}}{11 a^4 \left (1-\frac {1}{a x}\right )^{9/2}} \]

[Out]

-32/99*(a-1/x)^3*(1+1/a/x)^(3/2)*(-a*c*x+c)^(9/2)/a^4/(1-1/a/x)^(9/2)+9088/3465*(1+1/a/x)^(3/2)*(-a*c*x+c)^(9/
2)/a^4/(1-1/a/x)^(9/2)/x^3-768/385*(1+1/a/x)^(3/2)*(-a*c*x+c)^(9/2)/a^3/(1-1/a/x)^(9/2)/x^2+128/231*(1+1/a/x)^
(3/2)*(-a*c*x+c)^(9/2)/a^2/(1-1/a/x)^(9/2)/x+2/11*(a-1/x)^4*(1+1/a/x)^(3/2)*x*(-a*c*x+c)^(9/2)/a^4/(1-1/a/x)^(
9/2)

Rubi [A] (verified)

Time = 0.15 (sec) , antiderivative size = 254, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {6311, 6316, 96, 91, 79, 37} \[ \int e^{\coth ^{-1}(a x)} (c-a c x)^{9/2} \, dx=\frac {9088 \left (\frac {1}{a x}+1\right )^{3/2} (c-a c x)^{9/2}}{3465 a^4 x^3 \left (1-\frac {1}{a x}\right )^{9/2}}-\frac {32 \left (a-\frac {1}{x}\right )^3 \left (\frac {1}{a x}+1\right )^{3/2} (c-a c x)^{9/2}}{99 a^4 \left (1-\frac {1}{a x}\right )^{9/2}}+\frac {2 x \left (a-\frac {1}{x}\right )^4 \left (\frac {1}{a x}+1\right )^{3/2} (c-a c x)^{9/2}}{11 a^4 \left (1-\frac {1}{a x}\right )^{9/2}}-\frac {768 \left (\frac {1}{a x}+1\right )^{3/2} (c-a c x)^{9/2}}{385 a^3 x^2 \left (1-\frac {1}{a x}\right )^{9/2}}+\frac {128 \left (\frac {1}{a x}+1\right )^{3/2} (c-a c x)^{9/2}}{231 a^2 x \left (1-\frac {1}{a x}\right )^{9/2}} \]

[In]

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

[Out]

(-32*(a - x^(-1))^3*(1 + 1/(a*x))^(3/2)*(c - a*c*x)^(9/2))/(99*a^4*(1 - 1/(a*x))^(9/2)) + (9088*(1 + 1/(a*x))^
(3/2)*(c - a*c*x)^(9/2))/(3465*a^4*(1 - 1/(a*x))^(9/2)*x^3) - (768*(1 + 1/(a*x))^(3/2)*(c - a*c*x)^(9/2))/(385
*a^3*(1 - 1/(a*x))^(9/2)*x^2) + (128*(1 + 1/(a*x))^(3/2)*(c - a*c*x)^(9/2))/(231*a^2*(1 - 1/(a*x))^(9/2)*x) +
(2*(a - x^(-1))^4*(1 + 1/(a*x))^(3/2)*x*(c - a*c*x)^(9/2))/(11*a^4*(1 - 1/(a*x))^(9/2))

Rule 37

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^(n +
1)/((b*c - a*d)*(m + 1))), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[m + n + 2, 0] && NeQ
[m, -1]

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 91

Int[((a_.) + (b_.)*(x_))^2*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*c - a*d
)^2*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d^2*(d*e - c*f)*(n + 1))), x] - Dist[1/(d^2*(d*e - c*f)*(n + 1)), In
t[(c + d*x)^(n + 1)*(e + f*x)^p*Simp[a^2*d^2*f*(n + p + 2) + b^2*c*(d*e*(n + 1) + c*f*(p + 1)) - 2*a*b*d*(d*e*
(n + 1) + c*f*(p + 1)) - b^2*d*(d*e - c*f)*(n + 1)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && (LtQ
[n, -1] || (EqQ[n + p + 3, 0] && NeQ[n, -1] && (SumSimplerQ[n, 1] ||  !SumSimplerQ[p, 1])))

Rule 96

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

Rule 6311

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

Rule 6316

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

Rubi steps \begin{align*} \text {integral}& = \frac {(c-a c x)^{9/2} \int e^{\coth ^{-1}(a x)} \left (1-\frac {1}{a x}\right )^{9/2} x^{9/2} \, dx}{\left (1-\frac {1}{a x}\right )^{9/2} x^{9/2}} \\ & = -\frac {\left (\left (\frac {1}{x}\right )^{9/2} (c-a c x)^{9/2}\right ) \text {Subst}\left (\int \frac {\left (1-\frac {x}{a}\right )^4 \sqrt {1+\frac {x}{a}}}{x^{13/2}} \, dx,x,\frac {1}{x}\right )}{\left (1-\frac {1}{a x}\right )^{9/2}} \\ & = \frac {2 \left (a-\frac {1}{x}\right )^4 \left (1+\frac {1}{a x}\right )^{3/2} x (c-a c x)^{9/2}}{11 a^4 \left (1-\frac {1}{a x}\right )^{9/2}}+\frac {\left (16 \left (\frac {1}{x}\right )^{9/2} (c-a c x)^{9/2}\right ) \text {Subst}\left (\int \frac {\left (1-\frac {x}{a}\right )^3 \sqrt {1+\frac {x}{a}}}{x^{11/2}} \, dx,x,\frac {1}{x}\right )}{11 a \left (1-\frac {1}{a x}\right )^{9/2}} \\ & = -\frac {32 \left (a-\frac {1}{x}\right )^3 \left (1+\frac {1}{a x}\right )^{3/2} (c-a c x)^{9/2}}{99 a^4 \left (1-\frac {1}{a x}\right )^{9/2}}+\frac {2 \left (a-\frac {1}{x}\right )^4 \left (1+\frac {1}{a x}\right )^{3/2} x (c-a c x)^{9/2}}{11 a^4 \left (1-\frac {1}{a x}\right )^{9/2}}-\frac {\left (64 \left (\frac {1}{x}\right )^{9/2} (c-a c x)^{9/2}\right ) \text {Subst}\left (\int \frac {\left (1-\frac {x}{a}\right )^2 \sqrt {1+\frac {x}{a}}}{x^{9/2}} \, dx,x,\frac {1}{x}\right )}{33 a^2 \left (1-\frac {1}{a x}\right )^{9/2}} \\ & = -\frac {32 \left (a-\frac {1}{x}\right )^3 \left (1+\frac {1}{a x}\right )^{3/2} (c-a c x)^{9/2}}{99 a^4 \left (1-\frac {1}{a x}\right )^{9/2}}+\frac {128 \left (1+\frac {1}{a x}\right )^{3/2} (c-a c x)^{9/2}}{231 a^2 \left (1-\frac {1}{a x}\right )^{9/2} x}+\frac {2 \left (a-\frac {1}{x}\right )^4 \left (1+\frac {1}{a x}\right )^{3/2} x (c-a c x)^{9/2}}{11 a^4 \left (1-\frac {1}{a x}\right )^{9/2}}-\frac {\left (128 \left (\frac {1}{x}\right )^{9/2} (c-a c x)^{9/2}\right ) \text {Subst}\left (\int \frac {\left (-\frac {9}{a}+\frac {7 x}{2 a^2}\right ) \sqrt {1+\frac {x}{a}}}{x^{7/2}} \, dx,x,\frac {1}{x}\right )}{231 a^2 \left (1-\frac {1}{a x}\right )^{9/2}} \\ & = -\frac {32 \left (a-\frac {1}{x}\right )^3 \left (1+\frac {1}{a x}\right )^{3/2} (c-a c x)^{9/2}}{99 a^4 \left (1-\frac {1}{a x}\right )^{9/2}}-\frac {768 \left (1+\frac {1}{a x}\right )^{3/2} (c-a c x)^{9/2}}{385 a^3 \left (1-\frac {1}{a x}\right )^{9/2} x^2}+\frac {128 \left (1+\frac {1}{a x}\right )^{3/2} (c-a c x)^{9/2}}{231 a^2 \left (1-\frac {1}{a x}\right )^{9/2} x}+\frac {2 \left (a-\frac {1}{x}\right )^4 \left (1+\frac {1}{a x}\right )^{3/2} x (c-a c x)^{9/2}}{11 a^4 \left (1-\frac {1}{a x}\right )^{9/2}}-\frac {\left (4544 \left (\frac {1}{x}\right )^{9/2} (c-a c x)^{9/2}\right ) \text {Subst}\left (\int \frac {\sqrt {1+\frac {x}{a}}}{x^{5/2}} \, dx,x,\frac {1}{x}\right )}{1155 a^4 \left (1-\frac {1}{a x}\right )^{9/2}} \\ & = -\frac {32 \left (a-\frac {1}{x}\right )^3 \left (1+\frac {1}{a x}\right )^{3/2} (c-a c x)^{9/2}}{99 a^4 \left (1-\frac {1}{a x}\right )^{9/2}}+\frac {9088 \left (1+\frac {1}{a x}\right )^{3/2} (c-a c x)^{9/2}}{3465 a^4 \left (1-\frac {1}{a x}\right )^{9/2} x^3}-\frac {768 \left (1+\frac {1}{a x}\right )^{3/2} (c-a c x)^{9/2}}{385 a^3 \left (1-\frac {1}{a x}\right )^{9/2} x^2}+\frac {128 \left (1+\frac {1}{a x}\right )^{3/2} (c-a c x)^{9/2}}{231 a^2 \left (1-\frac {1}{a x}\right )^{9/2} x}+\frac {2 \left (a-\frac {1}{x}\right )^4 \left (1+\frac {1}{a x}\right )^{3/2} x (c-a c x)^{9/2}}{11 a^4 \left (1-\frac {1}{a x}\right )^{9/2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.08 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.34 \[ \int e^{\coth ^{-1}(a x)} (c-a c x)^{9/2} \, dx=\frac {2 c^4 \sqrt {1+\frac {1}{a x}} \sqrt {c-a c x} \left (5419-977 a x-1866 a^2 x^2+2710 a^3 x^3-1505 a^4 x^4+315 a^5 x^5\right )}{3465 a \sqrt {1-\frac {1}{a x}}} \]

[In]

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

[Out]

(2*c^4*Sqrt[1 + 1/(a*x)]*Sqrt[c - a*c*x]*(5419 - 977*a*x - 1866*a^2*x^2 + 2710*a^3*x^3 - 1505*a^4*x^4 + 315*a^
5*x^5))/(3465*a*Sqrt[1 - 1/(a*x)])

Maple [A] (verified)

Time = 0.41 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.27

method	result	size
default	\(\frac {2 \sqrt {-c \left (a x -1\right )}\, c^{4} \left (a x +1\right ) \left (315 a^{4} x^{4}-1820 a^{3} x^{3}+4530 a^{2} x^{2}-6396 a x +5419\right )}{3465 \sqrt {\frac {a x -1}{a x +1}}\, a}\)	\(69\)
gosper	\(\frac {2 \left (a x +1\right ) \left (315 a^{4} x^{4}-1820 a^{3} x^{3}+4530 a^{2} x^{2}-6396 a x +5419\right ) \left (-a c x +c \right )^{\frac {9}{2}}}{3465 a \left (a x -1\right )^{4} \sqrt {\frac {a x -1}{a x +1}}}\)	\(72\)
risch	\(-\frac {2 c^{5} \left (a x -1\right ) \left (315 a^{5} x^{5}-1505 a^{4} x^{4}+2710 a^{3} x^{3}-1866 a^{2} x^{2}-977 a x +5419\right )}{3465 \sqrt {\frac {a x -1}{a x +1}}\, \sqrt {-c \left (a x -1\right )}\, a}\)	\(77\)

[In]

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

[Out]

2/3465/((a*x-1)/(a*x+1))^(1/2)*(-c*(a*x-1))^(1/2)*c^4*(a*x+1)*(315*a^4*x^4-1820*a^3*x^3+4530*a^2*x^2-6396*a*x+
5419)/a

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 105, normalized size of antiderivative = 0.41 \[ \int e^{\coth ^{-1}(a x)} (c-a c x)^{9/2} \, dx=\frac {2 \, {\left (315 \, a^{6} c^{4} x^{6} - 1190 \, a^{5} c^{4} x^{5} + 1205 \, a^{4} c^{4} x^{4} + 844 \, a^{3} c^{4} x^{3} - 2843 \, a^{2} c^{4} x^{2} + 4442 \, a c^{4} x + 5419 \, c^{4}\right )} \sqrt {-a c x + c} \sqrt {\frac {a x - 1}{a x + 1}}}{3465 \, {\left (a^{2} x - a\right )}} \]

[In]

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

[Out]

2/3465*(315*a^6*c^4*x^6 - 1190*a^5*c^4*x^5 + 1205*a^4*c^4*x^4 + 844*a^3*c^4*x^3 - 2843*a^2*c^4*x^2 + 4442*a*c^
4*x + 5419*c^4)*sqrt(-a*c*x + c)*sqrt((a*x - 1)/(a*x + 1))/(a^2*x - a)

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.39 \[ \int e^{\coth ^{-1}(a x)} (c-a c x)^{9/2} \, dx=\frac {2 \, {\left (315 \, a^{5} \sqrt {-c} c^{4} x^{5} - 1505 \, a^{4} \sqrt {-c} c^{4} x^{4} + 2710 \, a^{3} \sqrt {-c} c^{4} x^{3} - 1866 \, a^{2} \sqrt {-c} c^{4} x^{2} - 977 \, a \sqrt {-c} c^{4} x + 5419 \, \sqrt {-c} c^{4}\right )} \sqrt {a x + 1}}{3465 \, a} \]

[In]

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

[Out]

2/3465*(315*a^5*sqrt(-c)*c^4*x^5 - 1505*a^4*sqrt(-c)*c^4*x^4 + 2710*a^3*sqrt(-c)*c^4*x^3 - 1866*a^2*sqrt(-c)*c
^4*x^2 - 977*a*sqrt(-c)*c^4*x + 5419*sqrt(-c)*c^4)*sqrt(a*x + 1)/a

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 147, normalized size of antiderivative = 0.58 \[ \int e^{\coth ^{-1}(a x)} (c-a c x)^{9/2} \, dx=\frac {2 \, {\left (4096 \, \sqrt {2} \sqrt {-c} c^{3} - \frac {315 \, {\left (a c x + c\right )}^{5} \sqrt {-a c x - c} - 3080 \, {\left (a c x + c\right )}^{4} \sqrt {-a c x - c} c + 11880 \, {\left (a c x + c\right )}^{3} \sqrt {-a c x - c} c^{2} - 22176 \, {\left (a c x + c\right )}^{2} \sqrt {-a c x - c} c^{3} - 18480 \, {\left (-a c x - c\right )}^{\frac {3}{2}} c^{4}}{c^{2}}\right )} c^{2}}{3465 \, a {\left | c \right |} \mathrm {sgn}\left (a x + 1\right )} \]

[In]

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

[Out]

2/3465*(4096*sqrt(2)*sqrt(-c)*c^3 - (315*(a*c*x + c)^5*sqrt(-a*c*x - c) - 3080*(a*c*x + c)^4*sqrt(-a*c*x - c)*
c + 11880*(a*c*x + c)^3*sqrt(-a*c*x - c)*c^2 - 22176*(a*c*x + c)^2*sqrt(-a*c*x - c)*c^3 - 18480*(-a*c*x - c)^(
3/2)*c^4)/c^2)*c^2/(a*abs(c)*sgn(a*x + 1))

Mupad [B] (verification not implemented)

Time = 4.53 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.30 \[ \int e^{\coth ^{-1}(a x)} (c-a c x)^{9/2} \, dx=\frac {2\,c^4\,\sqrt {c-a\,c\,x}\,{\left (a\,x+1\right )}^2\,\sqrt {\frac {a\,x-1}{a\,x+1}}\,\left (315\,a^4\,x^4-1820\,a^3\,x^3+4530\,a^2\,x^2-6396\,a\,x+5419\right )}{3465\,a\,\left (a\,x-1\right )} \]

[In]

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

[Out]

(2*c^4*(c - a*c*x)^(1/2)*(a*x + 1)^2*((a*x - 1)/(a*x + 1))^(1/2)*(4530*a^2*x^2 - 6396*a*x - 1820*a^3*x^3 + 315
*a^4*x^4 + 5419))/(3465*a*(a*x - 1))