Integrand size = 27, antiderivative size = 248 \[ \int e^{3 \text {arctanh}(a x)} \sqrt {c-\frac {c}{a x}} x^2 \, dx=-\frac {13 \sqrt {c-\frac {c}{a x}} x \sqrt {1+a x}}{8 a^2 \sqrt {1-a x}}-\frac {3 \sqrt {c-\frac {c}{a x}} x (1+a x)^{3/2}}{4 a^2 \sqrt {1-a x}}-\frac {\sqrt {c-\frac {c}{a x}} x^2 (1+a x)^{3/2}}{3 a \sqrt {1-a x}}-\frac {45 \sqrt {c-\frac {c}{a x}} \sqrt {x} \text {arcsinh}\left (\sqrt {a} \sqrt {x}\right )}{8 a^{5/2} \sqrt {1-a x}}+\frac {4 \sqrt {2} \sqrt {c-\frac {c}{a x}} \sqrt {x} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {a} \sqrt {x}}{\sqrt {1+a x}}\right )}{a^{5/2} \sqrt {1-a x}} \] Output:
-13/8*(c-c/a/x)^(1/2)*x*(a*x+1)^(1/2)/a^2/(-a*x+1)^(1/2)-3/4*(c-c/a/x)^(1/ 2)*x*(a*x+1)^(3/2)/a^2/(-a*x+1)^(1/2)-1/3*(c-c/a/x)^(1/2)*x^2*(a*x+1)^(3/2 )/a/(-a*x+1)^(1/2)-45/8*(c-c/a/x)^(1/2)*x^(1/2)*arcsinh(a^(1/2)*x^(1/2))/a ^(5/2)/(-a*x+1)^(1/2)+4*2^(1/2)*(c-c/a/x)^(1/2)*x^(1/2)*arctanh(2^(1/2)*a^ (1/2)*x^(1/2)/(a*x+1)^(1/2))/a^(5/2)/(-a*x+1)^(1/2)
Time = 0.07 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.49 \[ \int e^{3 \text {arctanh}(a x)} \sqrt {c-\frac {c}{a x}} x^2 \, dx=-\frac {\sqrt {c-\frac {c}{a x}} \sqrt {x} \left (\sqrt {a} \sqrt {x} \sqrt {1+a x} \left (57+26 a x+8 a^2 x^2\right )+135 \text {arcsinh}\left (\sqrt {a} \sqrt {x}\right )-96 \sqrt {2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {a} \sqrt {x}}{\sqrt {1+a x}}\right )\right )}{24 a^{5/2} \sqrt {1-a x}} \] Input:
Integrate[E^(3*ArcTanh[a*x])*Sqrt[c - c/(a*x)]*x^2,x]
Output:
-1/24*(Sqrt[c - c/(a*x)]*Sqrt[x]*(Sqrt[a]*Sqrt[x]*Sqrt[1 + a*x]*(57 + 26*a *x + 8*a^2*x^2) + 135*ArcSinh[Sqrt[a]*Sqrt[x]] - 96*Sqrt[2]*ArcTanh[(Sqrt[ 2]*Sqrt[a]*Sqrt[x])/Sqrt[1 + a*x]]))/(a^(5/2)*Sqrt[1 - a*x])
Time = 0.59 (sec) , antiderivative size = 168, normalized size of antiderivative = 0.68, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.481, Rules used = {6684, 6679, 112, 27, 171, 27, 171, 27, 175, 63, 104, 219, 222}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int x^2 e^{3 \text {arctanh}(a x)} \sqrt {c-\frac {c}{a x}} \, dx\) |
| \(\Big \downarrow \) 6684 |
| \(\displaystyle \frac {\sqrt {x} \sqrt {c-\frac {c}{a x}} \int e^{3 \text {arctanh}(a x)} x^{3/2} \sqrt {1-a x}dx}{\sqrt {1-a x}}\) |
| \(\Big \downarrow \) 6679 |
| \(\displaystyle \frac {\sqrt {x} \sqrt {c-\frac {c}{a x}} \int \frac {x^{3/2} (a x+1)^{3/2}}{1-a x}dx}{\sqrt {1-a x}}\) |
| \(\Big \downarrow \) 112 |
| \(\displaystyle \frac {\sqrt {x} \sqrt {c-\frac {c}{a x}} \left (\frac {\int \frac {3 \sqrt {x} \sqrt {a x+1} (3 a x+1)}{2 (1-a x)}dx}{3 a}-\frac {x^{3/2} (a x+1)^{3/2}}{3 a}\right )}{\sqrt {1-a x}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\sqrt {x} \sqrt {c-\frac {c}{a x}} \left (\frac {\int \frac {\sqrt {x} \sqrt {a x+1} (3 a x+1)}{1-a x}dx}{2 a}-\frac {x^{3/2} (a x+1)^{3/2}}{3 a}\right )}{\sqrt {1-a x}}\) |
| \(\Big \downarrow \) 171 |
| \(\displaystyle \frac {\sqrt {x} \sqrt {c-\frac {c}{a x}} \left (\frac {-\frac {\int -\frac {a \sqrt {a x+1} (13 a x+3)}{2 \sqrt {x} (1-a x)}dx}{2 a^2}-\frac {3 \sqrt {x} (a x+1)^{3/2}}{2 a}}{2 a}-\frac {x^{3/2} (a x+1)^{3/2}}{3 a}\right )}{\sqrt {1-a x}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\sqrt {x} \sqrt {c-\frac {c}{a x}} \left (\frac {\frac {\int \frac {\sqrt {a x+1} (13 a x+3)}{\sqrt {x} (1-a x)}dx}{4 a}-\frac {3 \sqrt {x} (a x+1)^{3/2}}{2 a}}{2 a}-\frac {x^{3/2} (a x+1)^{3/2}}{3 a}\right )}{\sqrt {1-a x}}\) |
| \(\Big \downarrow \) 171 |
| \(\displaystyle \frac {\sqrt {x} \sqrt {c-\frac {c}{a x}} \left (\frac {\frac {-\frac {\int -\frac {a (45 a x+19)}{2 \sqrt {x} (1-a x) \sqrt {a x+1}}dx}{a}-13 \sqrt {x} \sqrt {a x+1}}{4 a}-\frac {3 \sqrt {x} (a x+1)^{3/2}}{2 a}}{2 a}-\frac {x^{3/2} (a x+1)^{3/2}}{3 a}\right )}{\sqrt {1-a x}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\sqrt {x} \sqrt {c-\frac {c}{a x}} \left (\frac {\frac {\frac {1}{2} \int \frac {45 a x+19}{\sqrt {x} (1-a x) \sqrt {a x+1}}dx-13 \sqrt {x} \sqrt {a x+1}}{4 a}-\frac {3 \sqrt {x} (a x+1)^{3/2}}{2 a}}{2 a}-\frac {x^{3/2} (a x+1)^{3/2}}{3 a}\right )}{\sqrt {1-a x}}\) |
| \(\Big \downarrow \) 175 |
| \(\displaystyle \frac {\sqrt {x} \sqrt {c-\frac {c}{a x}} \left (\frac {\frac {\frac {1}{2} \left (64 \int \frac {1}{\sqrt {x} (1-a x) \sqrt {a x+1}}dx-45 \int \frac {1}{\sqrt {x} \sqrt {a x+1}}dx\right )-13 \sqrt {x} \sqrt {a x+1}}{4 a}-\frac {3 \sqrt {x} (a x+1)^{3/2}}{2 a}}{2 a}-\frac {x^{3/2} (a x+1)^{3/2}}{3 a}\right )}{\sqrt {1-a x}}\) |
| \(\Big \downarrow \) 63 |
| \(\displaystyle \frac {\sqrt {x} \sqrt {c-\frac {c}{a x}} \left (\frac {\frac {\frac {1}{2} \left (64 \int \frac {1}{\sqrt {x} (1-a x) \sqrt {a x+1}}dx-90 \int \frac {1}{\sqrt {a x+1}}d\sqrt {x}\right )-13 \sqrt {x} \sqrt {a x+1}}{4 a}-\frac {3 \sqrt {x} (a x+1)^{3/2}}{2 a}}{2 a}-\frac {x^{3/2} (a x+1)^{3/2}}{3 a}\right )}{\sqrt {1-a x}}\) |
| \(\Big \downarrow \) 104 |
| \(\displaystyle \frac {\sqrt {x} \sqrt {c-\frac {c}{a x}} \left (\frac {\frac {\frac {1}{2} \left (128 \int \frac {1}{1-\frac {2 a x}{a x+1}}d\frac {\sqrt {x}}{\sqrt {a x+1}}-90 \int \frac {1}{\sqrt {a x+1}}d\sqrt {x}\right )-13 \sqrt {x} \sqrt {a x+1}}{4 a}-\frac {3 \sqrt {x} (a x+1)^{3/2}}{2 a}}{2 a}-\frac {x^{3/2} (a x+1)^{3/2}}{3 a}\right )}{\sqrt {1-a x}}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {\sqrt {x} \sqrt {c-\frac {c}{a x}} \left (\frac {\frac {\frac {1}{2} \left (\frac {64 \sqrt {2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {a} \sqrt {x}}{\sqrt {a x+1}}\right )}{\sqrt {a}}-90 \int \frac {1}{\sqrt {a x+1}}d\sqrt {x}\right )-13 \sqrt {x} \sqrt {a x+1}}{4 a}-\frac {3 \sqrt {x} (a x+1)^{3/2}}{2 a}}{2 a}-\frac {x^{3/2} (a x+1)^{3/2}}{3 a}\right )}{\sqrt {1-a x}}\) |
| \(\Big \downarrow \) 222 |
| \(\displaystyle \frac {\sqrt {x} \sqrt {c-\frac {c}{a x}} \left (\frac {\frac {\frac {1}{2} \left (\frac {64 \sqrt {2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {a} \sqrt {x}}{\sqrt {a x+1}}\right )}{\sqrt {a}}-\frac {90 \text {arcsinh}\left (\sqrt {a} \sqrt {x}\right )}{\sqrt {a}}\right )-13 \sqrt {x} \sqrt {a x+1}}{4 a}-\frac {3 \sqrt {x} (a x+1)^{3/2}}{2 a}}{2 a}-\frac {x^{3/2} (a x+1)^{3/2}}{3 a}\right )}{\sqrt {1-a x}}\) |
Input:
Int[E^(3*ArcTanh[a*x])*Sqrt[c - c/(a*x)]*x^2,x]
Output:
(Sqrt[c - c/(a*x)]*Sqrt[x]*(-1/3*(x^(3/2)*(1 + a*x)^(3/2))/a + ((-3*Sqrt[x ]*(1 + a*x)^(3/2))/(2*a) + (-13*Sqrt[x]*Sqrt[1 + a*x] + ((-90*ArcSinh[Sqrt [a]*Sqrt[x]])/Sqrt[a] + (64*Sqrt[2]*ArcTanh[(Sqrt[2]*Sqrt[a]*Sqrt[x])/Sqrt [1 + a*x]])/Sqrt[a])/2)/(4*a))/(2*a)))/Sqrt[1 - a*x]
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[1/(Sqrt[(b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Simp[2/b S ubst[Int[1/Sqrt[c + d*(x^2/b)], x], x, Sqrt[b*x]], x] /; FreeQ[{b, c, d}, x ] && GtQ[c, 0]
Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x _)), x_] :> With[{q = Denominator[m]}, Simp[q Subst[Int[x^(q*(m + 1) - 1) /(b*e - a*f - (d*e - c*f)*x^q), x], x, (a + b*x)^(1/q)/(c + d*x)^(1/q)], x] ] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[m + n + 1, 0] && RationalQ[n] && L tQ[-1, m, 0] && SimplerQ[a + b*x, c + d*x]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> Simp[(a + b*x)^m*(c + d*x)^n*((e + f*x)^(p + 1)/(f*(m + n + p + 1))), x] - Simp[1/(f*(m + n + p + 1)) Int[(a + b*x)^(m - 1)*(c + d*x) ^(n - 1)*(e + f*x)^p*Simp[c*m*(b*e - a*f) + a*n*(d*e - c*f) + (d*m*(b*e - a *f) + b*n*(d*e - c*f))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && GtQ[m, 0] && GtQ[n, 0] && NeQ[m + n + p + 1, 0] && (IntegersQ[2*m, 2*n, 2*p ] || (IntegersQ[m, n + p] || IntegersQ[p, m + n]))
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_)*((g_.) + (h_.)*(x_)), x_] :> Simp[h*(a + b*x)^m*(c + d*x)^(n + 1)*(( e + f*x)^(p + 1)/(d*f*(m + n + p + 2))), x] + Simp[1/(d*f*(m + n + p + 2)) Int[(a + b*x)^(m - 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*g*(m + n + p + 2 ) - h*(b*c*e*m + a*(d*e*(n + 1) + c*f*(p + 1))) + (b*d*f*g*(m + n + p + 2) + h*(a*d*f*m - b*(d*e*(m + n + 1) + c*f*(m + p + 1))))*x, x], x], x] /; Fre eQ[{a, b, c, d, e, f, g, h, n, p}, x] && GtQ[m, 0] && NeQ[m + n + p + 2, 0] && IntegersQ[2*m, 2*n, 2*p]
Int[(((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_ )))/((a_.) + (b_.)*(x_)), x_] :> Simp[h/b Int[(c + d*x)^n*(e + f*x)^p, x] , x] + Simp[(b*g - a*h)/b Int[(c + d*x)^n*((e + f*x)^p/(a + b*x)), x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSinh[Rt[b, 2]*(x/Sqrt [a])]/Rt[b, 2], x] /; FreeQ[{a, b}, x] && GtQ[a, 0] && PosQ[b]
Int[E^(ArcTanh[(a_.)*(x_)]*(n_.))*(u_.)*((c_) + (d_.)*(x_))^(p_.), x_Symbol ] :> Simp[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])
Int[E^(ArcTanh[(a_.)*(x_)]*(n_.))*(u_.)*((c_) + (d_.)/(x_))^(p_), x_Symbol] :> Simp[x^p*((c + d/x)^p/(1 + c*(x/d))^p) Int[u*(1 + c*(x/d))^p*(E^(n*Ar cTanh[a*x])/x^p), x], x] /; FreeQ[{a, c, d, n, p}, x] && EqQ[c^2 - a^2*d^2, 0] && !IntegerQ[p]
Time = 0.22 (sec) , antiderivative size = 219, normalized size of antiderivative = 0.88
| method | result | size |
| default | \(\frac {\sqrt {\frac {c \left (a x -1\right )}{a x}}\, x \sqrt {-a^{2} x^{2}+1}\, \left (16 \sqrt {-x \left (a x +1\right )}\, a^{\frac {7}{2}} \sqrt {2}\, \sqrt {-\frac {1}{a}}\, x^{2}+52 \sqrt {-x \left (a x +1\right )}\, a^{\frac {5}{2}} \sqrt {2}\, \sqrt {-\frac {1}{a}}\, x +114 \sqrt {-x \left (a x +1\right )}\, a^{\frac {3}{2}} \sqrt {2}\, \sqrt {-\frac {1}{a}}-135 \arctan \left (\frac {2 a x +1}{2 \sqrt {a}\, \sqrt {-x \left (a x +1\right )}}\right ) a \sqrt {2}\, \sqrt {-\frac {1}{a}}+192 \ln \left (\frac {2 \sqrt {2}\, \sqrt {-\frac {1}{a}}\, \sqrt {-x \left (a x +1\right )}\, a -3 a x -1}{a x -1}\right ) \sqrt {a}\right ) \sqrt {2}}{96 a^{\frac {7}{2}} \left (a x -1\right ) \sqrt {-x \left (a x +1\right )}\, \sqrt {-\frac {1}{a}}}\) | \(219\) |
| risch | \(-\frac {\left (8 a^{2} x^{2}+26 a x +57\right ) \left (a x +1\right ) x \sqrt {\frac {c \left (a x -1\right )}{a x}}\, \sqrt {\frac {c a x \left (-a^{2} x^{2}+1\right )}{a x -1}}}{24 a^{2} \sqrt {-\left (a x +1\right ) a c x}\, \sqrt {-a^{2} x^{2}+1}}-\frac {\left (\frac {45 \arctan \left (\frac {\sqrt {a^{2} c}\, \left (x +\frac {1}{2 a}\right )}{\sqrt {-a^{2} c \,x^{2}-a c x}}\right )}{16 a^{2} \sqrt {a^{2} c}}-\frac {4 \ln \left (\frac {-4 c -3 \left (x -\frac {1}{a}\right ) a c +2 \sqrt {-2 c}\, \sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2} c -3 \left (x -\frac {1}{a}\right ) a c -2 c}}{x -\frac {1}{a}}\right )}{a^{3} \sqrt {-2 c}}\right ) \sqrt {\frac {c \left (a x -1\right )}{a x}}\, \sqrt {\frac {c a x \left (-a^{2} x^{2}+1\right )}{a x -1}}}{\sqrt {-a^{2} x^{2}+1}}\) | \(262\) |
Input:
int((a*x+1)^3/(-a^2*x^2+1)^(3/2)*(c-c/a/x)^(1/2)*x^2,x,method=_RETURNVERBO SE)
Output:
1/96*(c*(a*x-1)/a/x)^(1/2)*x*(-a^2*x^2+1)^(1/2)*(16*(-x*(a*x+1))^(1/2)*a^( 7/2)*2^(1/2)*(-1/a)^(1/2)*x^2+52*(-x*(a*x+1))^(1/2)*a^(5/2)*2^(1/2)*(-1/a) ^(1/2)*x+114*(-x*(a*x+1))^(1/2)*a^(3/2)*2^(1/2)*(-1/a)^(1/2)-135*arctan(1/ 2/a^(1/2)*(2*a*x+1)/(-x*(a*x+1))^(1/2))*a*2^(1/2)*(-1/a)^(1/2)+192*ln((2*2 ^(1/2)*(-1/a)^(1/2)*(-x*(a*x+1))^(1/2)*a-3*a*x-1)/(a*x-1))*a^(1/2))*2^(1/2 )/a^(7/2)/(a*x-1)/(-x*(a*x+1))^(1/2)/(-1/a)^(1/2)
Time = 0.14 (sec) , antiderivative size = 484, normalized size of antiderivative = 1.95 \[ \int e^{3 \text {arctanh}(a x)} \sqrt {c-\frac {c}{a x}} x^2 \, dx=\left [\frac {96 \, \sqrt {2} {\left (a x - 1\right )} \sqrt {-c} \log \left (-\frac {17 \, a^{3} c x^{3} - 3 \, a^{2} c x^{2} - 13 \, a c x + 4 \, \sqrt {2} {\left (3 \, a^{2} x^{2} + a x\right )} \sqrt {-a^{2} x^{2} + 1} \sqrt {-c} \sqrt {\frac {a c x - c}{a x}} - c}{a^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, a x - 1}\right ) + 135 \, {\left (a x - 1\right )} \sqrt {-c} \log \left (-\frac {8 \, a^{3} c x^{3} - 7 \, a c x - 4 \, {\left (2 \, a^{2} x^{2} + a x\right )} \sqrt {-a^{2} x^{2} + 1} \sqrt {-c} \sqrt {\frac {a c x - c}{a x}} - c}{a x - 1}\right ) + 4 \, {\left (8 \, a^{3} x^{3} + 26 \, a^{2} x^{2} + 57 \, a x\right )} \sqrt {-a^{2} x^{2} + 1} \sqrt {\frac {a c x - c}{a x}}}{96 \, {\left (a^{4} x - a^{3}\right )}}, -\frac {96 \, \sqrt {2} {\left (a x - 1\right )} \sqrt {c} \arctan \left (\frac {2 \, \sqrt {2} \sqrt {-a^{2} x^{2} + 1} a \sqrt {c} x \sqrt {\frac {a c x - c}{a x}}}{3 \, a^{2} c x^{2} - 2 \, a c x - c}\right ) - 135 \, {\left (a x - 1\right )} \sqrt {c} \arctan \left (\frac {2 \, \sqrt {-a^{2} x^{2} + 1} a \sqrt {c} x \sqrt {\frac {a c x - c}{a x}}}{2 \, a^{2} c x^{2} - a c x - c}\right ) - 2 \, {\left (8 \, a^{3} x^{3} + 26 \, a^{2} x^{2} + 57 \, a x\right )} \sqrt {-a^{2} x^{2} + 1} \sqrt {\frac {a c x - c}{a x}}}{48 \, {\left (a^{4} x - a^{3}\right )}}\right ] \] Input:
integrate((a*x+1)^3/(-a^2*x^2+1)^(3/2)*(c-c/a/x)^(1/2)*x^2,x, algorithm="f ricas")
Output:
[1/96*(96*sqrt(2)*(a*x - 1)*sqrt(-c)*log(-(17*a^3*c*x^3 - 3*a^2*c*x^2 - 13 *a*c*x + 4*sqrt(2)*(3*a^2*x^2 + a*x)*sqrt(-a^2*x^2 + 1)*sqrt(-c)*sqrt((a*c *x - c)/(a*x)) - c)/(a^3*x^3 - 3*a^2*x^2 + 3*a*x - 1)) + 135*(a*x - 1)*sqr t(-c)*log(-(8*a^3*c*x^3 - 7*a*c*x - 4*(2*a^2*x^2 + a*x)*sqrt(-a^2*x^2 + 1) *sqrt(-c)*sqrt((a*c*x - c)/(a*x)) - c)/(a*x - 1)) + 4*(8*a^3*x^3 + 26*a^2* x^2 + 57*a*x)*sqrt(-a^2*x^2 + 1)*sqrt((a*c*x - c)/(a*x)))/(a^4*x - a^3), - 1/48*(96*sqrt(2)*(a*x - 1)*sqrt(c)*arctan(2*sqrt(2)*sqrt(-a^2*x^2 + 1)*a*s qrt(c)*x*sqrt((a*c*x - c)/(a*x))/(3*a^2*c*x^2 - 2*a*c*x - c)) - 135*(a*x - 1)*sqrt(c)*arctan(2*sqrt(-a^2*x^2 + 1)*a*sqrt(c)*x*sqrt((a*c*x - c)/(a*x) )/(2*a^2*c*x^2 - a*c*x - c)) - 2*(8*a^3*x^3 + 26*a^2*x^2 + 57*a*x)*sqrt(-a ^2*x^2 + 1)*sqrt((a*c*x - c)/(a*x)))/(a^4*x - a^3)]
\[ \int e^{3 \text {arctanh}(a x)} \sqrt {c-\frac {c}{a x}} x^2 \, dx=\int \frac {x^{2} \sqrt {- c \left (-1 + \frac {1}{a x}\right )} \left (a x + 1\right )^{3}}{\left (- \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {3}{2}}}\, dx \] Input:
integrate((a*x+1)**3/(-a**2*x**2+1)**(3/2)*(c-c/a/x)**(1/2)*x**2,x)
Output:
Integral(x**2*sqrt(-c*(-1 + 1/(a*x)))*(a*x + 1)**3/(-(a*x - 1)*(a*x + 1))* *(3/2), x)
\[ \int e^{3 \text {arctanh}(a x)} \sqrt {c-\frac {c}{a x}} x^2 \, dx=\int { \frac {{\left (a x + 1\right )}^{3} \sqrt {c - \frac {c}{a x}} x^{2}}{{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate((a*x+1)^3/(-a^2*x^2+1)^(3/2)*(c-c/a/x)^(1/2)*x^2,x, algorithm="m axima")
Output:
integrate((a*x + 1)^3*sqrt(c - c/(a*x))*x^2/(-a^2*x^2 + 1)^(3/2), x)
Exception generated. \[ \int e^{3 \text {arctanh}(a x)} \sqrt {c-\frac {c}{a x}} x^2 \, dx=\text {Exception raised: TypeError} \] Input:
integrate((a*x+1)^3/(-a^2*x^2+1)^(3/2)*(c-c/a/x)^(1/2)*x^2,x, algorithm="g iac")
Output:
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value
Timed out. \[ \int e^{3 \text {arctanh}(a x)} \sqrt {c-\frac {c}{a x}} x^2 \, dx=\int \frac {x^2\,\sqrt {c-\frac {c}{a\,x}}\,{\left (a\,x+1\right )}^3}{{\left (1-a^2\,x^2\right )}^{3/2}} \,d x \] Input:
int((x^2*(c - c/(a*x))^(1/2)*(a*x + 1)^3)/(1 - a^2*x^2)^(3/2),x)
Output:
int((x^2*(c - c/(a*x))^(1/2)*(a*x + 1)^3)/(1 - a^2*x^2)^(3/2), x)
Time = 0.19 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.42 \[ \int e^{3 \text {arctanh}(a x)} \sqrt {c-\frac {c}{a x}} x^2 \, dx=\frac {\sqrt {c}\, \left (-96 \sqrt {2}\, \mathit {atan} \left (\frac {\sqrt {x}\, \sqrt {a}\, \sqrt {a x +1}\, \sqrt {2}\, i}{a x +1}\right )+135 \mathit {atan} \left (\frac {\sqrt {x}\, \sqrt {a}\, \sqrt {a x +1}\, i}{a x +1}\right )+8 \sqrt {x}\, \sqrt {a}\, \sqrt {a x +1}\, a^{2} i \,x^{2}+26 \sqrt {x}\, \sqrt {a}\, \sqrt {a x +1}\, a i x +57 \sqrt {x}\, \sqrt {a}\, \sqrt {a x +1}\, i \right )}{24 a^{3}} \] Input:
int((a*x+1)^3/(-a^2*x^2+1)^(3/2)*(c-c/a/x)^(1/2)*x^2,x)
Output:
(sqrt(c)*( - 96*sqrt(2)*atan((sqrt(x)*sqrt(a)*sqrt(a*x + 1)*sqrt(2)*i)/(a* x + 1)) + 135*atan((sqrt(x)*sqrt(a)*sqrt(a*x + 1)*i)/(a*x + 1)) + 8*sqrt(x )*sqrt(a)*sqrt(a*x + 1)*a**2*i*x**2 + 26*sqrt(x)*sqrt(a)*sqrt(a*x + 1)*a*i *x + 57*sqrt(x)*sqrt(a)*sqrt(a*x + 1)*i))/(24*a**3)