3.270 \(\int e^{-3 \coth ^{-1}(a x)} (c-a c x)^{9/2} \, dx\)

Optimal. Leaf size=368 \[ -\frac{1024 \left (a-\frac{1}{x}\right )^3 (c-a c x)^{9/2}}{231 a^6 x^2 \left (1-\frac{1}{a x}\right )^{9/2} \sqrt{\frac{1}{a x}+1}}+\frac{4096 (c-a c x)^{9/2}}{231 a^4 x^3 \left (1-\frac{1}{a x}\right )^{9/2} \sqrt{\frac{1}{a x}+1}}-\frac{40960 (c-a c x)^{9/2}}{231 a^5 x^4 \left (1-\frac{1}{a x}\right )^{9/2} \sqrt{\frac{1}{a x}+1}}-\frac{94208 (c-a c x)^{9/2}}{231 a^6 x^5 \left (1-\frac{1}{a x}\right )^{9/2} \sqrt{\frac{1}{a x}+1}}+\frac{2 x \left (a-\frac{1}{x}\right )^6 (c-a c x)^{9/2}}{11 a^6 \left (1-\frac{1}{a x}\right )^{9/2} \sqrt{\frac{1}{a x}+1}}-\frac{16 \left (a-\frac{1}{x}\right )^5 (c-a c x)^{9/2}}{33 a^6 \left (1-\frac{1}{a x}\right )^{9/2} \sqrt{\frac{1}{a x}+1}}+\frac{320 \left (a-\frac{1}{x}\right )^4 (c-a c x)^{9/2}}{231 a^6 x \left (1-\frac{1}{a x}\right )^{9/2} \sqrt{\frac{1}{a x}+1}} \]

[Out]

(-16*(a - x^(-1))^5*(c - a*c*x)^(9/2))/(33*a^6*(1 - 1/(a*x))^(9/2)*Sqrt[1 + 1/(a*x)]) - (94208*(c - a*c*x)^(9/
2))/(231*a^6*(1 - 1/(a*x))^(9/2)*Sqrt[1 + 1/(a*x)]*x^5) - (40960*(c - a*c*x)^(9/2))/(231*a^5*(1 - 1/(a*x))^(9/
2)*Sqrt[1 + 1/(a*x)]*x^4) + (4096*(c - a*c*x)^(9/2))/(231*a^4*(1 - 1/(a*x))^(9/2)*Sqrt[1 + 1/(a*x)]*x^3) - (10
24*(a - x^(-1))^3*(c - a*c*x)^(9/2))/(231*a^6*(1 - 1/(a*x))^(9/2)*Sqrt[1 + 1/(a*x)]*x^2) + (320*(a - x^(-1))^4
*(c - a*c*x)^(9/2))/(231*a^6*(1 - 1/(a*x))^(9/2)*Sqrt[1 + 1/(a*x)]*x) + (2*(a - x^(-1))^6*x*(c - a*c*x)^(9/2))
/(11*a^6*(1 - 1/(a*x))^(9/2)*Sqrt[1 + 1/(a*x)])

________________________________________________________________________________________

Rubi [A]  time = 0.266833, antiderivative size = 368, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 6, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {6176, 6181, 94, 89, 78, 37} \[ -\frac{1024 \left (a-\frac{1}{x}\right )^3 (c-a c x)^{9/2}}{231 a^6 x^2 \left (1-\frac{1}{a x}\right )^{9/2} \sqrt{\frac{1}{a x}+1}}+\frac{4096 (c-a c x)^{9/2}}{231 a^4 x^3 \left (1-\frac{1}{a x}\right )^{9/2} \sqrt{\frac{1}{a x}+1}}-\frac{40960 (c-a c x)^{9/2}}{231 a^5 x^4 \left (1-\frac{1}{a x}\right )^{9/2} \sqrt{\frac{1}{a x}+1}}-\frac{94208 (c-a c x)^{9/2}}{231 a^6 x^5 \left (1-\frac{1}{a x}\right )^{9/2} \sqrt{\frac{1}{a x}+1}}+\frac{2 x \left (a-\frac{1}{x}\right )^6 (c-a c x)^{9/2}}{11 a^6 \left (1-\frac{1}{a x}\right )^{9/2} \sqrt{\frac{1}{a x}+1}}-\frac{16 \left (a-\frac{1}{x}\right )^5 (c-a c x)^{9/2}}{33 a^6 \left (1-\frac{1}{a x}\right )^{9/2} \sqrt{\frac{1}{a x}+1}}+\frac{320 \left (a-\frac{1}{x}\right )^4 (c-a c x)^{9/2}}{231 a^6 x \left (1-\frac{1}{a x}\right )^{9/2} \sqrt{\frac{1}{a x}+1}} \]

Antiderivative was successfully verified.

[In]

Int[(c - a*c*x)^(9/2)/E^(3*ArcCoth[a*x]),x]

[Out]

(-16*(a - x^(-1))^5*(c - a*c*x)^(9/2))/(33*a^6*(1 - 1/(a*x))^(9/2)*Sqrt[1 + 1/(a*x)]) - (94208*(c - a*c*x)^(9/
2))/(231*a^6*(1 - 1/(a*x))^(9/2)*Sqrt[1 + 1/(a*x)]*x^5) - (40960*(c - a*c*x)^(9/2))/(231*a^5*(1 - 1/(a*x))^(9/
2)*Sqrt[1 + 1/(a*x)]*x^4) + (4096*(c - a*c*x)^(9/2))/(231*a^4*(1 - 1/(a*x))^(9/2)*Sqrt[1 + 1/(a*x)]*x^3) - (10
24*(a - x^(-1))^3*(c - a*c*x)^(9/2))/(231*a^6*(1 - 1/(a*x))^(9/2)*Sqrt[1 + 1/(a*x)]*x^2) + (320*(a - x^(-1))^4
*(c - a*c*x)^(9/2))/(231*a^6*(1 - 1/(a*x))^(9/2)*Sqrt[1 + 1/(a*x)]*x) + (2*(a - x^(-1))^6*x*(c - a*c*x)^(9/2))
/(11*a^6*(1 - 1/(a*x))^(9/2)*Sqrt[1 + 1/(a*x)])

Rule 6176

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 6181

Int[E^(ArcCoth[(a_.)*(x_)]*(n_.))*((c_) + (d_.)/(x_))^(p_.)*(x_)^(m_), x_Symbol] :> -Dist[c^p*x^m*(1/x)^m, Sub
st[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]

Rule 94

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[p, 1] &&  !SumSimplerQ[m, 1])

Rule 89

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 78

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] || LtQ
[p, n]))))

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]

Rubi steps

\begin{align*} \int e^{-3 \coth ^{-1}(a x)} (c-a c x)^{9/2} \, dx &=\frac{(c-a c x)^{9/2} \int e^{-3 \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 ) \operatorname{Subst}\left (\int \frac{\left (1-\frac{x}{a}\right )^6}{x^{13/2} \left (1+\frac{x}{a}\right )^{3/2}} \, dx,x,\frac{1}{x}\right )}{\left (1-\frac{1}{a x}\right )^{9/2}}\\ &=\frac{2 \left (a-\frac{1}{x}\right )^6 x (c-a c x)^{9/2}}{11 a^6 \left (1-\frac{1}{a x}\right )^{9/2} \sqrt{1+\frac{1}{a x}}}+\frac{\left (24 \left (\frac{1}{x}\right )^{9/2} (c-a c x)^{9/2}\right ) \operatorname{Subst}\left (\int \frac{\left (1-\frac{x}{a}\right )^5}{x^{11/2} \left (1+\frac{x}{a}\right )^{3/2}} \, dx,x,\frac{1}{x}\right )}{11 a \left (1-\frac{1}{a x}\right )^{9/2}}\\ &=-\frac{16 \left (a-\frac{1}{x}\right )^5 (c-a c x)^{9/2}}{33 a^6 \left (1-\frac{1}{a x}\right )^{9/2} \sqrt{1+\frac{1}{a x}}}+\frac{2 \left (a-\frac{1}{x}\right )^6 x (c-a c x)^{9/2}}{11 a^6 \left (1-\frac{1}{a x}\right )^{9/2} \sqrt{1+\frac{1}{a x}}}-\frac{\left (160 \left (\frac{1}{x}\right )^{9/2} (c-a c x)^{9/2}\right ) \operatorname{Subst}\left (\int \frac{\left (1-\frac{x}{a}\right )^4}{x^{9/2} \left (1+\frac{x}{a}\right )^{3/2}} \, dx,x,\frac{1}{x}\right )}{33 a^2 \left (1-\frac{1}{a x}\right )^{9/2}}\\ &=-\frac{16 \left (a-\frac{1}{x}\right )^5 (c-a c x)^{9/2}}{33 a^6 \left (1-\frac{1}{a x}\right )^{9/2} \sqrt{1+\frac{1}{a x}}}+\frac{320 \left (a-\frac{1}{x}\right )^4 (c-a c x)^{9/2}}{231 a^6 \left (1-\frac{1}{a x}\right )^{9/2} \sqrt{1+\frac{1}{a x}} x}+\frac{2 \left (a-\frac{1}{x}\right )^6 x (c-a c x)^{9/2}}{11 a^6 \left (1-\frac{1}{a x}\right )^{9/2} \sqrt{1+\frac{1}{a x}}}+\frac{\left (2560 \left (\frac{1}{x}\right )^{9/2} (c-a c x)^{9/2}\right ) \operatorname{Subst}\left (\int \frac{\left (1-\frac{x}{a}\right )^3}{x^{7/2} \left (1+\frac{x}{a}\right )^{3/2}} \, dx,x,\frac{1}{x}\right )}{231 a^3 \left (1-\frac{1}{a x}\right )^{9/2}}\\ &=-\frac{16 \left (a-\frac{1}{x}\right )^5 (c-a c x)^{9/2}}{33 a^6 \left (1-\frac{1}{a x}\right )^{9/2} \sqrt{1+\frac{1}{a x}}}-\frac{1024 \left (a-\frac{1}{x}\right )^3 (c-a c x)^{9/2}}{231 a^6 \left (1-\frac{1}{a x}\right )^{9/2} \sqrt{1+\frac{1}{a x}} x^2}+\frac{320 \left (a-\frac{1}{x}\right )^4 (c-a c x)^{9/2}}{231 a^6 \left (1-\frac{1}{a x}\right )^{9/2} \sqrt{1+\frac{1}{a x}} x}+\frac{2 \left (a-\frac{1}{x}\right )^6 x (c-a c x)^{9/2}}{11 a^6 \left (1-\frac{1}{a x}\right )^{9/2} \sqrt{1+\frac{1}{a x}}}-\frac{\left (2048 \left (\frac{1}{x}\right )^{9/2} (c-a c x)^{9/2}\right ) \operatorname{Subst}\left (\int \frac{\left (1-\frac{x}{a}\right )^2}{x^{5/2} \left (1+\frac{x}{a}\right )^{3/2}} \, dx,x,\frac{1}{x}\right )}{77 a^4 \left (1-\frac{1}{a x}\right )^{9/2}}\\ &=-\frac{16 \left (a-\frac{1}{x}\right )^5 (c-a c x)^{9/2}}{33 a^6 \left (1-\frac{1}{a x}\right )^{9/2} \sqrt{1+\frac{1}{a x}}}+\frac{4096 (c-a c x)^{9/2}}{231 a^4 \left (1-\frac{1}{a x}\right )^{9/2} \sqrt{1+\frac{1}{a x}} x^3}-\frac{1024 \left (a-\frac{1}{x}\right )^3 (c-a c x)^{9/2}}{231 a^6 \left (1-\frac{1}{a x}\right )^{9/2} \sqrt{1+\frac{1}{a x}} x^2}+\frac{320 \left (a-\frac{1}{x}\right )^4 (c-a c x)^{9/2}}{231 a^6 \left (1-\frac{1}{a x}\right )^{9/2} \sqrt{1+\frac{1}{a x}} x}+\frac{2 \left (a-\frac{1}{x}\right )^6 x (c-a c x)^{9/2}}{11 a^6 \left (1-\frac{1}{a x}\right )^{9/2} \sqrt{1+\frac{1}{a x}}}-\frac{\left (4096 \left (\frac{1}{x}\right )^{9/2} (c-a c x)^{9/2}\right ) \operatorname{Subst}\left (\int \frac{-\frac{5}{a}+\frac{3 x}{2 a^2}}{x^{3/2} \left (1+\frac{x}{a}\right )^{3/2}} \, dx,x,\frac{1}{x}\right )}{231 a^4 \left (1-\frac{1}{a x}\right )^{9/2}}\\ &=-\frac{16 \left (a-\frac{1}{x}\right )^5 (c-a c x)^{9/2}}{33 a^6 \left (1-\frac{1}{a x}\right )^{9/2} \sqrt{1+\frac{1}{a x}}}-\frac{40960 (c-a c x)^{9/2}}{231 a^5 \left (1-\frac{1}{a x}\right )^{9/2} \sqrt{1+\frac{1}{a x}} x^4}+\frac{4096 (c-a c x)^{9/2}}{231 a^4 \left (1-\frac{1}{a x}\right )^{9/2} \sqrt{1+\frac{1}{a x}} x^3}-\frac{1024 \left (a-\frac{1}{x}\right )^3 (c-a c x)^{9/2}}{231 a^6 \left (1-\frac{1}{a x}\right )^{9/2} \sqrt{1+\frac{1}{a x}} x^2}+\frac{320 \left (a-\frac{1}{x}\right )^4 (c-a c x)^{9/2}}{231 a^6 \left (1-\frac{1}{a x}\right )^{9/2} \sqrt{1+\frac{1}{a x}} x}+\frac{2 \left (a-\frac{1}{x}\right )^6 x (c-a c x)^{9/2}}{11 a^6 \left (1-\frac{1}{a x}\right )^{9/2} \sqrt{1+\frac{1}{a x}}}-\frac{\left (47104 \left (\frac{1}{x}\right )^{9/2} (c-a c x)^{9/2}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{x} \left (1+\frac{x}{a}\right )^{3/2}} \, dx,x,\frac{1}{x}\right )}{231 a^6 \left (1-\frac{1}{a x}\right )^{9/2}}\\ &=-\frac{16 \left (a-\frac{1}{x}\right )^5 (c-a c x)^{9/2}}{33 a^6 \left (1-\frac{1}{a x}\right )^{9/2} \sqrt{1+\frac{1}{a x}}}-\frac{94208 (c-a c x)^{9/2}}{231 a^6 \left (1-\frac{1}{a x}\right )^{9/2} \sqrt{1+\frac{1}{a x}} x^5}-\frac{40960 (c-a c x)^{9/2}}{231 a^5 \left (1-\frac{1}{a x}\right )^{9/2} \sqrt{1+\frac{1}{a x}} x^4}+\frac{4096 (c-a c x)^{9/2}}{231 a^4 \left (1-\frac{1}{a x}\right )^{9/2} \sqrt{1+\frac{1}{a x}} x^3}-\frac{1024 \left (a-\frac{1}{x}\right )^3 (c-a c x)^{9/2}}{231 a^6 \left (1-\frac{1}{a x}\right )^{9/2} \sqrt{1+\frac{1}{a x}} x^2}+\frac{320 \left (a-\frac{1}{x}\right )^4 (c-a c x)^{9/2}}{231 a^6 \left (1-\frac{1}{a x}\right )^{9/2} \sqrt{1+\frac{1}{a x}} x}+\frac{2 \left (a-\frac{1}{x}\right )^6 x (c-a c x)^{9/2}}{11 a^6 \left (1-\frac{1}{a x}\right )^{9/2} \sqrt{1+\frac{1}{a x}}}\\ \end{align*}

Mathematica [A]  time = 0.0511153, size = 84, normalized size = 0.23 \[ \frac{2 c^4 \left (21 a^6 x^6-182 a^5 x^5+755 a^4 x^4-2132 a^3 x^3+5419 a^2 x^2-23062 a x-46355\right ) \sqrt{c-a c x}}{231 a^2 x \sqrt{1-\frac{1}{a^2 x^2}}} \]

Antiderivative was successfully verified.

[In]

Integrate[(c - a*c*x)^(9/2)/E^(3*ArcCoth[a*x]),x]

[Out]

(2*c^4*Sqrt[c - a*c*x]*(-46355 - 23062*a*x + 5419*a^2*x^2 - 2132*a^3*x^3 + 755*a^4*x^4 - 182*a^5*x^5 + 21*a^6*
x^6))/(231*a^2*Sqrt[1 - 1/(a^2*x^2)]*x)

________________________________________________________________________________________

Maple [A]  time = 0.051, size = 88, normalized size = 0.2 \begin{align*}{\frac{ \left ( 2\,ax+2 \right ) \left ( 21\,{x}^{6}{a}^{6}-182\,{x}^{5}{a}^{5}+755\,{x}^{4}{a}^{4}-2132\,{x}^{3}{a}^{3}+5419\,{a}^{2}{x}^{2}-23062\,ax-46355 \right ) }{231\,a \left ( ax-1 \right ) ^{6}} \left ( -acx+c \right ) ^{{\frac{9}{2}}} \left ({\frac{ax-1}{ax+1}} \right ) ^{{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-a*c*x+c)^(9/2)*((a*x-1)/(a*x+1))^(3/2),x)

[Out]

2/231*(a*x+1)*(21*a^6*x^6-182*a^5*x^5+755*a^4*x^4-2132*a^3*x^3+5419*a^2*x^2-23062*a*x-46355)*(-a*c*x+c)^(9/2)*
((a*x-1)/(a*x+1))^(3/2)/a/(a*x-1)^6

________________________________________________________________________________________

Maxima [A]  time = 1.11471, size = 205, normalized size = 0.56 \begin{align*} \frac{2 \,{\left (21 \, a^{7} \sqrt{-c} c^{4} x^{7} - 161 \, a^{6} \sqrt{-c} c^{4} x^{6} + 573 \, a^{5} \sqrt{-c} c^{4} x^{5} - 1377 \, a^{4} \sqrt{-c} c^{4} x^{4} + 3287 \, a^{3} \sqrt{-c} c^{4} x^{3} - 17643 \, a^{2} \sqrt{-c} c^{4} x^{2} - 69417 \, a \sqrt{-c} c^{4} x - 46355 \, \sqrt{-c} c^{4}\right )}{\left (a x - 1\right )}^{2}}{231 \,{\left (a^{3} x^{2} - 2 \, a^{2} x + a\right )}{\left (a x + 1\right )}^{\frac{3}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-a*c*x+c)^(9/2)*((a*x-1)/(a*x+1))^(3/2),x, algorithm="maxima")

[Out]

2/231*(21*a^7*sqrt(-c)*c^4*x^7 - 161*a^6*sqrt(-c)*c^4*x^6 + 573*a^5*sqrt(-c)*c^4*x^5 - 1377*a^4*sqrt(-c)*c^4*x
^4 + 3287*a^3*sqrt(-c)*c^4*x^3 - 17643*a^2*sqrt(-c)*c^4*x^2 - 69417*a*sqrt(-c)*c^4*x - 46355*sqrt(-c)*c^4)*(a*
x - 1)^2/((a^3*x^2 - 2*a^2*x + a)*(a*x + 1)^(3/2))

________________________________________________________________________________________

Fricas [A]  time = 1.51568, size = 244, normalized size = 0.66 \begin{align*} \frac{2 \,{\left (21 \, a^{6} c^{4} x^{6} - 182 \, a^{5} c^{4} x^{5} + 755 \, a^{4} c^{4} x^{4} - 2132 \, a^{3} c^{4} x^{3} + 5419 \, a^{2} c^{4} x^{2} - 23062 \, a c^{4} x - 46355 \, c^{4}\right )} \sqrt{-a c x + c} \sqrt{\frac{a x - 1}{a x + 1}}}{231 \,{\left (a^{2} x - a\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-a*c*x+c)^(9/2)*((a*x-1)/(a*x+1))^(3/2),x, algorithm="fricas")

[Out]

2/231*(21*a^6*c^4*x^6 - 182*a^5*c^4*x^5 + 755*a^4*c^4*x^4 - 2132*a^3*c^4*x^3 + 5419*a^2*c^4*x^2 - 23062*a*c^4*
x - 46355*c^4)*sqrt(-a*c*x + c)*sqrt((a*x - 1)/(a*x + 1))/(a^2*x - a)

________________________________________________________________________________________

Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-a*c*x+c)**(9/2)*((a*x-1)/(a*x+1))**(3/2),x)

[Out]

Timed out

________________________________________________________________________________________

Giac [A]  time = 1.3907, size = 203, normalized size = 0.55 \begin{align*} -\frac{2 \,{\left (21 \,{\left (a c x + c\right )}^{5} \sqrt{-a c x - c} - 308 \,{\left (a c x + c\right )}^{4} \sqrt{-a c x - c} c + 1980 \,{\left (a c x + c\right )}^{3} \sqrt{-a c x - c} c^{2} - 7392 \,{\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} - 44352 \, \sqrt{-a c x - c} c^{5} + \frac{14784 \, c^{6}}{\sqrt{-a c x - c}}\right )}{\left | c \right |}}{231 \, a c^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-a*c*x+c)^(9/2)*((a*x-1)/(a*x+1))^(3/2),x, algorithm="giac")

[Out]

-2/231*(21*(a*c*x + c)^5*sqrt(-a*c*x - c) - 308*(a*c*x + c)^4*sqrt(-a*c*x - c)*c + 1980*(a*c*x + c)^3*sqrt(-a*
c*x - c)*c^2 - 7392*(a*c*x + c)^2*sqrt(-a*c*x - c)*c^3 - 18480*(-a*c*x - c)^(3/2)*c^4 - 44352*sqrt(-a*c*x - c)
*c^5 + 14784*c^6/sqrt(-a*c*x - c))*abs(c)/(a*c^2)