Integrand size = 27, antiderivative size = 281 \[ \int \frac {e^{3 \text {arctanh}(a x)} \sqrt {c-\frac {c}{a x}}}{x^5} \, dx=-\frac {1576 a^4 \sqrt {c-\frac {c}{a x}} \sqrt {1+a x}}{315 \sqrt {1-a x}}-\frac {2 \sqrt {c-\frac {c}{a x}} \sqrt {1+a x}}{9 x^4 \sqrt {1-a x}}-\frac {38 a \sqrt {c-\frac {c}{a x}} \sqrt {1+a x}}{63 x^3 \sqrt {1-a x}}-\frac {92 a^2 \sqrt {c-\frac {c}{a x}} \sqrt {1+a x}}{105 x^2 \sqrt {1-a x}}-\frac {472 a^3 \sqrt {c-\frac {c}{a x}} \sqrt {1+a x}}{315 x \sqrt {1-a x}}+\frac {4 \sqrt {2} a^{9/2} \sqrt {c-\frac {c}{a x}} \sqrt {x} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {a} \sqrt {x}}{\sqrt {1+a x}}\right )}{\sqrt {1-a x}} \] Output:
-1576/315*a^4*(c-c/a/x)^(1/2)*(a*x+1)^(1/2)/(-a*x+1)^(1/2)-2/9*(c-c/a/x)^( 1/2)*(a*x+1)^(1/2)/x^4/(-a*x+1)^(1/2)-38/63*a*(c-c/a/x)^(1/2)*(a*x+1)^(1/2 )/x^3/(-a*x+1)^(1/2)-92/105*a^2*(c-c/a/x)^(1/2)*(a*x+1)^(1/2)/x^2/(-a*x+1) ^(1/2)-472/315*a^3*(c-c/a/x)^(1/2)*(a*x+1)^(1/2)/x/(-a*x+1)^(1/2)+4*2^(1/2 )*a^(9/2)*(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*x+1)^(1/2)
Time = 0.06 (sec) , antiderivative size = 117, normalized size of antiderivative = 0.42 \[ \int \frac {e^{3 \text {arctanh}(a x)} \sqrt {c-\frac {c}{a x}}}{x^5} \, dx=\frac {2 \sqrt {c-\frac {c}{a x}} \left (-\sqrt {1+a x} \left (35+95 a x+138 a^2 x^2+236 a^3 x^3+788 a^4 x^4\right )+630 \sqrt {2} a^{9/2} x^{9/2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {a} \sqrt {x}}{\sqrt {1+a x}}\right )\right )}{315 x^4 \sqrt {1-a x}} \] Input:
Integrate[(E^(3*ArcTanh[a*x])*Sqrt[c - c/(a*x)])/x^5,x]
Output:
(2*Sqrt[c - c/(a*x)]*(-(Sqrt[1 + a*x]*(35 + 95*a*x + 138*a^2*x^2 + 236*a^3 *x^3 + 788*a^4*x^4)) + 630*Sqrt[2]*a^(9/2)*x^(9/2)*ArcTanh[(Sqrt[2]*Sqrt[a ]*Sqrt[x])/Sqrt[1 + a*x]]))/(315*x^4*Sqrt[1 - a*x])
Time = 0.60 (sec) , antiderivative size = 182, normalized size of antiderivative = 0.65, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.556, Rules used = {6684, 6679, 109, 27, 169, 27, 169, 25, 27, 169, 27, 169, 27, 104, 219}
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.
| \(\displaystyle \int \frac {e^{3 \text {arctanh}(a x)} \sqrt {c-\frac {c}{a x}}}{x^5} \, dx\) |
| \(\Big \downarrow \) 6684 |
| \(\displaystyle \frac {\sqrt {x} \sqrt {c-\frac {c}{a x}} \int \frac {e^{3 \text {arctanh}(a x)} \sqrt {1-a x}}{x^{11/2}}dx}{\sqrt {1-a x}}\) |
| \(\Big \downarrow \) 6679 |
| \(\displaystyle \frac {\sqrt {x} \sqrt {c-\frac {c}{a x}} \int \frac {(a x+1)^{3/2}}{x^{11/2} (1-a x)}dx}{\sqrt {1-a x}}\) |
| \(\Big \downarrow \) 109 |
| \(\displaystyle \frac {\sqrt {x} \sqrt {c-\frac {c}{a x}} \left (-\frac {2}{9} \int -\frac {a (17 a x+19)}{2 x^{9/2} (1-a x) \sqrt {a x+1}}dx-\frac {2 \sqrt {a x+1}}{9 x^{9/2}}\right )}{\sqrt {1-a x}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\sqrt {x} \sqrt {c-\frac {c}{a x}} \left (\frac {1}{9} a \int \frac {17 a x+19}{x^{9/2} (1-a x) \sqrt {a x+1}}dx-\frac {2 \sqrt {a x+1}}{9 x^{9/2}}\right )}{\sqrt {1-a x}}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle \frac {\sqrt {x} \sqrt {c-\frac {c}{a x}} \left (\frac {1}{9} a \left (-\frac {2}{7} \int -\frac {3 a (19 a x+23)}{x^{7/2} (1-a x) \sqrt {a x+1}}dx-\frac {38 \sqrt {a x+1}}{7 x^{7/2}}\right )-\frac {2 \sqrt {a x+1}}{9 x^{9/2}}\right )}{\sqrt {1-a x}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\sqrt {x} \sqrt {c-\frac {c}{a x}} \left (\frac {1}{9} a \left (\frac {6}{7} a \int \frac {19 a x+23}{x^{7/2} (1-a x) \sqrt {a x+1}}dx-\frac {38 \sqrt {a x+1}}{7 x^{7/2}}\right )-\frac {2 \sqrt {a x+1}}{9 x^{9/2}}\right )}{\sqrt {1-a x}}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle \frac {\sqrt {x} \sqrt {c-\frac {c}{a x}} \left (\frac {1}{9} a \left (\frac {6}{7} a \left (-\frac {2}{5} \int -\frac {a (46 a x+59)}{x^{5/2} (1-a x) \sqrt {a x+1}}dx-\frac {46 \sqrt {a x+1}}{5 x^{5/2}}\right )-\frac {38 \sqrt {a x+1}}{7 x^{7/2}}\right )-\frac {2 \sqrt {a x+1}}{9 x^{9/2}}\right )}{\sqrt {1-a x}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\sqrt {x} \sqrt {c-\frac {c}{a x}} \left (\frac {1}{9} a \left (\frac {6}{7} a \left (\frac {2}{5} \int \frac {a (46 a x+59)}{x^{5/2} (1-a x) \sqrt {a x+1}}dx-\frac {46 \sqrt {a x+1}}{5 x^{5/2}}\right )-\frac {38 \sqrt {a x+1}}{7 x^{7/2}}\right )-\frac {2 \sqrt {a x+1}}{9 x^{9/2}}\right )}{\sqrt {1-a x}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\sqrt {x} \sqrt {c-\frac {c}{a x}} \left (\frac {1}{9} a \left (\frac {6}{7} a \left (\frac {2}{5} a \int \frac {46 a x+59}{x^{5/2} (1-a x) \sqrt {a x+1}}dx-\frac {46 \sqrt {a x+1}}{5 x^{5/2}}\right )-\frac {38 \sqrt {a x+1}}{7 x^{7/2}}\right )-\frac {2 \sqrt {a x+1}}{9 x^{9/2}}\right )}{\sqrt {1-a x}}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle \frac {\sqrt {x} \sqrt {c-\frac {c}{a x}} \left (\frac {1}{9} a \left (\frac {6}{7} a \left (\frac {2}{5} a \left (-\frac {2}{3} \int -\frac {a (118 a x+197)}{2 x^{3/2} (1-a x) \sqrt {a x+1}}dx-\frac {118 \sqrt {a x+1}}{3 x^{3/2}}\right )-\frac {46 \sqrt {a x+1}}{5 x^{5/2}}\right )-\frac {38 \sqrt {a x+1}}{7 x^{7/2}}\right )-\frac {2 \sqrt {a x+1}}{9 x^{9/2}}\right )}{\sqrt {1-a x}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\sqrt {x} \sqrt {c-\frac {c}{a x}} \left (\frac {1}{9} a \left (\frac {6}{7} a \left (\frac {2}{5} a \left (\frac {1}{3} a \int \frac {118 a x+197}{x^{3/2} (1-a x) \sqrt {a x+1}}dx-\frac {118 \sqrt {a x+1}}{3 x^{3/2}}\right )-\frac {46 \sqrt {a x+1}}{5 x^{5/2}}\right )-\frac {38 \sqrt {a x+1}}{7 x^{7/2}}\right )-\frac {2 \sqrt {a x+1}}{9 x^{9/2}}\right )}{\sqrt {1-a x}}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle \frac {\sqrt {x} \sqrt {c-\frac {c}{a x}} \left (\frac {1}{9} a \left (\frac {6}{7} a \left (\frac {2}{5} a \left (\frac {1}{3} a \left (-2 \int -\frac {315 a}{2 \sqrt {x} (1-a x) \sqrt {a x+1}}dx-\frac {394 \sqrt {a x+1}}{\sqrt {x}}\right )-\frac {118 \sqrt {a x+1}}{3 x^{3/2}}\right )-\frac {46 \sqrt {a x+1}}{5 x^{5/2}}\right )-\frac {38 \sqrt {a x+1}}{7 x^{7/2}}\right )-\frac {2 \sqrt {a x+1}}{9 x^{9/2}}\right )}{\sqrt {1-a x}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\sqrt {x} \sqrt {c-\frac {c}{a x}} \left (\frac {1}{9} a \left (\frac {6}{7} a \left (\frac {2}{5} a \left (\frac {1}{3} a \left (315 a \int \frac {1}{\sqrt {x} (1-a x) \sqrt {a x+1}}dx-\frac {394 \sqrt {a x+1}}{\sqrt {x}}\right )-\frac {118 \sqrt {a x+1}}{3 x^{3/2}}\right )-\frac {46 \sqrt {a x+1}}{5 x^{5/2}}\right )-\frac {38 \sqrt {a x+1}}{7 x^{7/2}}\right )-\frac {2 \sqrt {a x+1}}{9 x^{9/2}}\right )}{\sqrt {1-a x}}\) |
| \(\Big \downarrow \) 104 |
| \(\displaystyle \frac {\sqrt {x} \sqrt {c-\frac {c}{a x}} \left (\frac {1}{9} a \left (\frac {6}{7} a \left (\frac {2}{5} a \left (\frac {1}{3} a \left (630 a \int \frac {1}{1-\frac {2 a x}{a x+1}}d\frac {\sqrt {x}}{\sqrt {a x+1}}-\frac {394 \sqrt {a x+1}}{\sqrt {x}}\right )-\frac {118 \sqrt {a x+1}}{3 x^{3/2}}\right )-\frac {46 \sqrt {a x+1}}{5 x^{5/2}}\right )-\frac {38 \sqrt {a x+1}}{7 x^{7/2}}\right )-\frac {2 \sqrt {a x+1}}{9 x^{9/2}}\right )}{\sqrt {1-a x}}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {\sqrt {x} \left (\frac {1}{9} a \left (\frac {6}{7} a \left (\frac {2}{5} a \left (\frac {1}{3} a \left (315 \sqrt {2} \sqrt {a} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {a} \sqrt {x}}{\sqrt {a x+1}}\right )-\frac {394 \sqrt {a x+1}}{\sqrt {x}}\right )-\frac {118 \sqrt {a x+1}}{3 x^{3/2}}\right )-\frac {46 \sqrt {a x+1}}{5 x^{5/2}}\right )-\frac {38 \sqrt {a x+1}}{7 x^{7/2}}\right )-\frac {2 \sqrt {a x+1}}{9 x^{9/2}}\right ) \sqrt {c-\frac {c}{a x}}}{\sqrt {1-a x}}\) |
Input:
Int[(E^(3*ArcTanh[a*x])*Sqrt[c - c/(a*x)])/x^5,x]
Output:
(Sqrt[c - c/(a*x)]*Sqrt[x]*((-2*Sqrt[1 + a*x])/(9*x^(9/2)) + (a*((-38*Sqrt [1 + a*x])/(7*x^(7/2)) + (6*a*((-46*Sqrt[1 + a*x])/(5*x^(5/2)) + (2*a*((-1 18*Sqrt[1 + a*x])/(3*x^(3/2)) + (a*((-394*Sqrt[1 + a*x])/Sqrt[x] + 315*Sqr t[2]*Sqrt[a]*ArcTanh[(Sqrt[2]*Sqrt[a]*Sqrt[x])/Sqrt[1 + a*x]]))/3))/5))/7) )/9))/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[(((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[(b*c - a*d)*(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*((e + f *x)^(p + 1)/(b*(b*e - a*f)*(m + 1))), x] + Simp[1/(b*(b*e - a*f)*(m + 1)) Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 2)*(e + f*x)^p*Simp[a*d*(d*e*(n - 1) + c*f*(p + 1)) + b*c*(d*e*(m - n + 2) - c*f*(m + p + 2)) + d*(a*d*f*(n + p) + b*(d*e*(m + 1) - c*f*(m + n + p + 1)))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && LtQ[m, -1] && GtQ[n, 1] && (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[(b*g - a*h)*(a + b*x)^(m + 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/((m + 1)*(b*c - a*d)*(b*e - a*f))), x] + S imp[1/((m + 1)*(b*c - a*d)*(b*e - a*f)) Int[(a + b*x)^(m + 1)*(c + d*x)^n *(e + f*x)^p*Simp[(a*d*f*g - b*(d*e + c*f)*g + b*c*e*h)*(m + 1) - (b*g - a* h)*(d*e*(n + 1) + c*f*(p + 1)) - d*f*(b*g - a*h)*(m + n + p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && LtQ[m, -1] && IntegersQ[ 2*m, 2*n, 2*p]
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[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 = 232, normalized size of antiderivative = 0.83
| method | result | size |
| risch | \(-\frac {2 \left (788 a^{5} x^{5}+1024 a^{4} x^{4}+374 a^{3} x^{3}+233 a^{2} x^{2}+130 a x +35\right ) \sqrt {\frac {c \left (a x -1\right )}{a x}}\, \sqrt {\frac {c a x \left (-a^{2} x^{2}+1\right )}{a x -1}}}{315 x^{4} \sqrt {-\left (a x +1\right ) a c x}\, \sqrt {-a^{2} x^{2}+1}}+\frac {4 a^{4} \ln \left (\frac {-4 c -3 \left (x -\frac {1}{a}\right ) a c +2 \sqrt {-2 c}\, \sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2} c -3 \left (x -\frac {1}{a}\right ) a c -2 c}}{x -\frac {1}{a}}\right ) \sqrt {\frac {c \left (a x -1\right )}{a x}}\, \sqrt {\frac {c a x \left (-a^{2} x^{2}+1\right )}{a x -1}}}{\sqrt {-2 c}\, \sqrt {-a^{2} x^{2}+1}}\) | \(232\) |
| default | \(\frac {\sqrt {\frac {c \left (a x -1\right )}{a x}}\, \sqrt {-a^{2} x^{2}+1}\, \left (788 x^{4} \sqrt {-x \left (a x +1\right )}\, a^{4} \sqrt {2}\, \sqrt {-\frac {1}{a}}+630 a^{4} \ln \left (\frac {2 \sqrt {2}\, \sqrt {-\frac {1}{a}}\, \sqrt {-x \left (a x +1\right )}\, a -3 a x -1}{a x -1}\right ) x^{5}+236 x^{3} \sqrt {-x \left (a x +1\right )}\, a^{3} \sqrt {2}\, \sqrt {-\frac {1}{a}}+138 x^{2} \sqrt {-x \left (a x +1\right )}\, a^{2} \sqrt {2}\, \sqrt {-\frac {1}{a}}+95 x \sqrt {-x \left (a x +1\right )}\, a \sqrt {2}\, \sqrt {-\frac {1}{a}}+35 \sqrt {-x \left (a x +1\right )}\, \sqrt {2}\, \sqrt {-\frac {1}{a}}\right ) \sqrt {2}}{315 x^{4} \left (a x -1\right ) \sqrt {-x \left (a x +1\right )}\, \sqrt {-\frac {1}{a}}}\) | \(237\) |
Input:
int((a*x+1)^3/(-a^2*x^2+1)^(3/2)*(c-c/a/x)^(1/2)/x^5,x,method=_RETURNVERBO SE)
Output:
-2/315*(788*a^5*x^5+1024*a^4*x^4+374*a^3*x^3+233*a^2*x^2+130*a*x+35)/x^4/( -(a*x+1)*a*c*x)^(1/2)*(c*(a*x-1)/a/x)^(1/2)*(c/(a*x-1)*a*x*(-a^2*x^2+1))^( 1/2)/(-a^2*x^2+1)^(1/2)+4*a^4/(-2*c)^(1/2)*ln((-4*c-3*(x-1/a)*a*c+2*(-2*c) ^(1/2)*(-(x-1/a)^2*a^2*c-3*(x-1/a)*a*c-2*c)^(1/2))/(x-1/a))*(c*(a*x-1)/a/x )^(1/2)*(c/(a*x-1)*a*x*(-a^2*x^2+1))^(1/2)/(-a^2*x^2+1)^(1/2)
Time = 0.13 (sec) , antiderivative size = 369, normalized size of antiderivative = 1.31 \[ \int \frac {e^{3 \text {arctanh}(a x)} \sqrt {c-\frac {c}{a x}}}{x^5} \, dx=\left [\frac {315 \, \sqrt {2} {\left (a^{5} x^{5} - a^{4} x^{4}\right )} \sqrt {-c} \log \left (-\frac {17 \, a^{3} c x^{3} - 3 \, a^{2} c x^{2} - 13 \, a c x + 4 \, \sqrt {2} {\left (3 \, a^{2} x^{2} + a x\right )} \sqrt {-a^{2} x^{2} + 1} \sqrt {-c} \sqrt {\frac {a c x - c}{a x}} - c}{a^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, a x - 1}\right ) + 2 \, {\left (788 \, a^{4} x^{4} + 236 \, a^{3} x^{3} + 138 \, a^{2} x^{2} + 95 \, a x + 35\right )} \sqrt {-a^{2} x^{2} + 1} \sqrt {\frac {a c x - c}{a x}}}{315 \, {\left (a x^{5} - x^{4}\right )}}, -\frac {2 \, {\left (315 \, \sqrt {2} {\left (a^{5} x^{5} - a^{4} x^{4}\right )} \sqrt {c} \arctan \left (\frac {2 \, \sqrt {2} \sqrt {-a^{2} x^{2} + 1} a \sqrt {c} x \sqrt {\frac {a c x - c}{a x}}}{3 \, a^{2} c x^{2} - 2 \, a c x - c}\right ) - {\left (788 \, a^{4} x^{4} + 236 \, a^{3} x^{3} + 138 \, a^{2} x^{2} + 95 \, a x + 35\right )} \sqrt {-a^{2} x^{2} + 1} \sqrt {\frac {a c x - c}{a x}}\right )}}{315 \, {\left (a x^{5} - x^{4}\right )}}\right ] \] Input:
integrate((a*x+1)^3/(-a^2*x^2+1)^(3/2)*(c-c/a/x)^(1/2)/x^5,x, algorithm="f ricas")
Output:
[1/315*(315*sqrt(2)*(a^5*x^5 - a^4*x^4)*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)) + 2*(78 8*a^4*x^4 + 236*a^3*x^3 + 138*a^2*x^2 + 95*a*x + 35)*sqrt(-a^2*x^2 + 1)*sq rt((a*c*x - c)/(a*x)))/(a*x^5 - x^4), -2/315*(315*sqrt(2)*(a^5*x^5 - a^4*x ^4)*sqrt(c)*arctan(2*sqrt(2)*sqrt(-a^2*x^2 + 1)*a*sqrt(c)*x*sqrt((a*c*x - c)/(a*x))/(3*a^2*c*x^2 - 2*a*c*x - c)) - (788*a^4*x^4 + 236*a^3*x^3 + 138* a^2*x^2 + 95*a*x + 35)*sqrt(-a^2*x^2 + 1)*sqrt((a*c*x - c)/(a*x)))/(a*x^5 - x^4)]
\[ \int \frac {e^{3 \text {arctanh}(a x)} \sqrt {c-\frac {c}{a x}}}{x^5} \, dx=\int \frac {\sqrt {- c \left (-1 + \frac {1}{a x}\right )} \left (a x + 1\right )^{3}}{x^{5} \left (- \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {3}{2}}}\, dx \] Input:
integrate((a*x+1)**3/(-a**2*x**2+1)**(3/2)*(c-c/a/x)**(1/2)/x**5,x)
Output:
Integral(sqrt(-c*(-1 + 1/(a*x)))*(a*x + 1)**3/(x**5*(-(a*x - 1)*(a*x + 1)) **(3/2)), x)
\[ \int \frac {e^{3 \text {arctanh}(a x)} \sqrt {c-\frac {c}{a x}}}{x^5} \, dx=\int { \frac {{\left (a x + 1\right )}^{3} \sqrt {c - \frac {c}{a x}}}{{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} x^{5}} \,d x } \] Input:
integrate((a*x+1)^3/(-a^2*x^2+1)^(3/2)*(c-c/a/x)^(1/2)/x^5,x, algorithm="m axima")
Output:
integrate((a*x + 1)^3*sqrt(c - c/(a*x))/((-a^2*x^2 + 1)^(3/2)*x^5), x)
Exception generated. \[ \int \frac {e^{3 \text {arctanh}(a x)} \sqrt {c-\frac {c}{a x}}}{x^5} \, dx=\text {Exception raised: TypeError} \] Input:
integrate((a*x+1)^3/(-a^2*x^2+1)^(3/2)*(c-c/a/x)^(1/2)/x^5,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. \[ \int \frac {e^{3 \text {arctanh}(a x)} \sqrt {c-\frac {c}{a x}}}{x^5} \, dx=\int \frac {\sqrt {c-\frac {c}{a\,x}}\,{\left (a\,x+1\right )}^3}{x^5\,{\left (1-a^2\,x^2\right )}^{3/2}} \,d x \] Input:
int(((c - c/(a*x))^(1/2)*(a*x + 1)^3)/(x^5*(1 - a^2*x^2)^(3/2)),x)
Output:
int(((c - c/(a*x))^(1/2)*(a*x + 1)^3)/(x^5*(1 - a^2*x^2)^(3/2)), x)
Time = 0.22 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.46 \[ \int \frac {e^{3 \text {arctanh}(a x)} \sqrt {c-\frac {c}{a x}}}{x^5} \, dx=\frac {2 \sqrt {c}\, \left (-630 \sqrt {2}\, \mathit {atan} \left (\frac {\sqrt {x}\, \sqrt {a}\, \sqrt {a x +1}\, \sqrt {2}\, i}{a x +1}\right ) a^{5} x^{5}+788 \sqrt {x}\, \sqrt {a}\, \sqrt {a x +1}\, a^{4} i \,x^{4}+236 \sqrt {x}\, \sqrt {a}\, \sqrt {a x +1}\, a^{3} i \,x^{3}+138 \sqrt {x}\, \sqrt {a}\, \sqrt {a x +1}\, a^{2} i \,x^{2}+95 \sqrt {x}\, \sqrt {a}\, \sqrt {a x +1}\, a i x +35 \sqrt {x}\, \sqrt {a}\, \sqrt {a x +1}\, i \right )}{315 a \,x^{5}} \] Input:
int((a*x+1)^3/(-a^2*x^2+1)^(3/2)*(c-c/a/x)^(1/2)/x^5,x)
Output:
(2*sqrt(c)*( - 630*sqrt(2)*atan((sqrt(x)*sqrt(a)*sqrt(a*x + 1)*sqrt(2)*i)/ (a*x + 1))*a**5*x**5 + 788*sqrt(x)*sqrt(a)*sqrt(a*x + 1)*a**4*i*x**4 + 236 *sqrt(x)*sqrt(a)*sqrt(a*x + 1)*a**3*i*x**3 + 138*sqrt(x)*sqrt(a)*sqrt(a*x + 1)*a**2*i*x**2 + 95*sqrt(x)*sqrt(a)*sqrt(a*x + 1)*a*i*x + 35*sqrt(x)*sqr t(a)*sqrt(a*x + 1)*i))/(315*a*x**5)