Integrand size = 27, antiderivative size = 187 \[ \int e^{2 \text {arctanh}(a x)} x^3 \left (c-a^2 c x^2\right )^{5/2} \, dx=\frac {3 c^2 x \sqrt {c-a^2 c x^2}}{64 a^3}+\frac {c x \left (c-a^2 c x^2\right )^{3/2}}{32 a^3}-\frac {13 x^2 \left (c-a^2 c x^2\right )^{5/2}}{63 a^2}-\frac {x^3 \left (c-a^2 c x^2\right )^{5/2}}{4 a}-\frac {1}{9} x^4 \left (c-a^2 c x^2\right )^{5/2}-\frac {(208+315 a x) \left (c-a^2 c x^2\right )^{5/2}}{2520 a^4}+\frac {3 c^{5/2} \arctan \left (\frac {a \sqrt {c} x}{\sqrt {c-a^2 c x^2}}\right )}{64 a^4} \]
1/32*c*x*(-a^2*c*x^2+c)^(3/2)/a^3-13/63*x^2*(-a^2*c*x^2+c)^(5/2)/a^2-1/4*x ^3*(-a^2*c*x^2+c)^(5/2)/a-1/9*x^4*(-a^2*c*x^2+c)^(5/2)-1/2520*(315*a*x+208 )*(-a^2*c*x^2+c)^(5/2)/a^4+3/64*c^(5/2)*arctan(a*x*c^(1/2)/(-a^2*c*x^2+c)^ (1/2))/a^4+3/64*c^2*x*(-a^2*c*x^2+c)^(1/2)/a^3
Time = 0.20 (sec) , antiderivative size = 131, normalized size of antiderivative = 0.70 \[ \int e^{2 \text {arctanh}(a x)} x^3 \left (c-a^2 c x^2\right )^{5/2} \, dx=-\frac {c^2 \left (\sqrt {c-a^2 c x^2} \left (1664+945 a x+832 a^2 x^2+630 a^3 x^3-4416 a^4 x^4-7560 a^5 x^5-320 a^6 x^6+5040 a^7 x^7+2240 a^8 x^8\right )+945 \sqrt {c} \arctan \left (\frac {a x \sqrt {c-a^2 c x^2}}{\sqrt {c} \left (-1+a^2 x^2\right )}\right )\right )}{20160 a^4} \]
-1/20160*(c^2*(Sqrt[c - a^2*c*x^2]*(1664 + 945*a*x + 832*a^2*x^2 + 630*a^3 *x^3 - 4416*a^4*x^4 - 7560*a^5*x^5 - 320*a^6*x^6 + 5040*a^7*x^7 + 2240*a^8 *x^8) + 945*Sqrt[c]*ArcTan[(a*x*Sqrt[c - a^2*c*x^2])/(Sqrt[c]*(-1 + a^2*x^ 2))]))/a^4
Time = 0.53 (sec) , antiderivative size = 249, normalized size of antiderivative = 1.33, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.556, Rules used = {6701, 541, 25, 27, 533, 27, 533, 27, 533, 27, 455, 211, 211, 224, 216}
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.
| \(\displaystyle \int x^3 e^{2 \text {arctanh}(a x)} \left (c-a^2 c x^2\right )^{5/2} \, dx\) |
| \(\Big \downarrow \) 6701 |
| \(\displaystyle c \int x^3 (a x+1)^2 \left (c-a^2 c x^2\right )^{3/2}dx\) |
| \(\Big \downarrow \) 541 |
| \(\displaystyle c \left (-\frac {\int -a^2 c x^3 (18 a x+13) \left (c-a^2 c x^2\right )^{3/2}dx}{9 a^2 c}-\frac {x^4 \left (c-a^2 c x^2\right )^{5/2}}{9 c}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle c \left (\frac {\int a^2 c x^3 (18 a x+13) \left (c-a^2 c x^2\right )^{3/2}dx}{9 a^2 c}-\frac {x^4 \left (c-a^2 c x^2\right )^{5/2}}{9 c}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle c \left (\frac {1}{9} \int x^3 (18 a x+13) \left (c-a^2 c x^2\right )^{3/2}dx-\frac {x^4 \left (c-a^2 c x^2\right )^{5/2}}{9 c}\right )\) |
| \(\Big \downarrow \) 533 |
| \(\displaystyle c \left (\frac {1}{9} \left (\frac {\int 2 a c x^2 (52 a x+27) \left (c-a^2 c x^2\right )^{3/2}dx}{8 a^2 c}-\frac {9 x^3 \left (c-a^2 c x^2\right )^{5/2}}{4 a c}\right )-\frac {x^4 \left (c-a^2 c x^2\right )^{5/2}}{9 c}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle c \left (\frac {1}{9} \left (\frac {\int x^2 (52 a x+27) \left (c-a^2 c x^2\right )^{3/2}dx}{4 a}-\frac {9 x^3 \left (c-a^2 c x^2\right )^{5/2}}{4 a c}\right )-\frac {x^4 \left (c-a^2 c x^2\right )^{5/2}}{9 c}\right )\) |
| \(\Big \downarrow \) 533 |
| \(\displaystyle c \left (\frac {1}{9} \left (\frac {\frac {\int a c x (189 a x+104) \left (c-a^2 c x^2\right )^{3/2}dx}{7 a^2 c}-\frac {52 x^2 \left (c-a^2 c x^2\right )^{5/2}}{7 a c}}{4 a}-\frac {9 x^3 \left (c-a^2 c x^2\right )^{5/2}}{4 a c}\right )-\frac {x^4 \left (c-a^2 c x^2\right )^{5/2}}{9 c}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle c \left (\frac {1}{9} \left (\frac {\frac {\int x (189 a x+104) \left (c-a^2 c x^2\right )^{3/2}dx}{7 a}-\frac {52 x^2 \left (c-a^2 c x^2\right )^{5/2}}{7 a c}}{4 a}-\frac {9 x^3 \left (c-a^2 c x^2\right )^{5/2}}{4 a c}\right )-\frac {x^4 \left (c-a^2 c x^2\right )^{5/2}}{9 c}\right )\) |
| \(\Big \downarrow \) 533 |
| \(\displaystyle c \left (\frac {1}{9} \left (\frac {\frac {\frac {\int 3 a c (208 a x+63) \left (c-a^2 c x^2\right )^{3/2}dx}{6 a^2 c}-\frac {63 x \left (c-a^2 c x^2\right )^{5/2}}{2 a c}}{7 a}-\frac {52 x^2 \left (c-a^2 c x^2\right )^{5/2}}{7 a c}}{4 a}-\frac {9 x^3 \left (c-a^2 c x^2\right )^{5/2}}{4 a c}\right )-\frac {x^4 \left (c-a^2 c x^2\right )^{5/2}}{9 c}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle c \left (\frac {1}{9} \left (\frac {\frac {\frac {\int (208 a x+63) \left (c-a^2 c x^2\right )^{3/2}dx}{2 a}-\frac {63 x \left (c-a^2 c x^2\right )^{5/2}}{2 a c}}{7 a}-\frac {52 x^2 \left (c-a^2 c x^2\right )^{5/2}}{7 a c}}{4 a}-\frac {9 x^3 \left (c-a^2 c x^2\right )^{5/2}}{4 a c}\right )-\frac {x^4 \left (c-a^2 c x^2\right )^{5/2}}{9 c}\right )\) |
| \(\Big \downarrow \) 455 |
| \(\displaystyle c \left (\frac {1}{9} \left (\frac {\frac {\frac {63 \int \left (c-a^2 c x^2\right )^{3/2}dx-\frac {208 \left (c-a^2 c x^2\right )^{5/2}}{5 a c}}{2 a}-\frac {63 x \left (c-a^2 c x^2\right )^{5/2}}{2 a c}}{7 a}-\frac {52 x^2 \left (c-a^2 c x^2\right )^{5/2}}{7 a c}}{4 a}-\frac {9 x^3 \left (c-a^2 c x^2\right )^{5/2}}{4 a c}\right )-\frac {x^4 \left (c-a^2 c x^2\right )^{5/2}}{9 c}\right )\) |
| \(\Big \downarrow \) 211 |
| \(\displaystyle c \left (\frac {1}{9} \left (\frac {\frac {\frac {63 \left (\frac {3}{4} c \int \sqrt {c-a^2 c x^2}dx+\frac {1}{4} x \left (c-a^2 c x^2\right )^{3/2}\right )-\frac {208 \left (c-a^2 c x^2\right )^{5/2}}{5 a c}}{2 a}-\frac {63 x \left (c-a^2 c x^2\right )^{5/2}}{2 a c}}{7 a}-\frac {52 x^2 \left (c-a^2 c x^2\right )^{5/2}}{7 a c}}{4 a}-\frac {9 x^3 \left (c-a^2 c x^2\right )^{5/2}}{4 a c}\right )-\frac {x^4 \left (c-a^2 c x^2\right )^{5/2}}{9 c}\right )\) |
| \(\Big \downarrow \) 211 |
| \(\displaystyle c \left (\frac {1}{9} \left (\frac {\frac {\frac {63 \left (\frac {3}{4} c \left (\frac {1}{2} c \int \frac {1}{\sqrt {c-a^2 c x^2}}dx+\frac {1}{2} x \sqrt {c-a^2 c x^2}\right )+\frac {1}{4} x \left (c-a^2 c x^2\right )^{3/2}\right )-\frac {208 \left (c-a^2 c x^2\right )^{5/2}}{5 a c}}{2 a}-\frac {63 x \left (c-a^2 c x^2\right )^{5/2}}{2 a c}}{7 a}-\frac {52 x^2 \left (c-a^2 c x^2\right )^{5/2}}{7 a c}}{4 a}-\frac {9 x^3 \left (c-a^2 c x^2\right )^{5/2}}{4 a c}\right )-\frac {x^4 \left (c-a^2 c x^2\right )^{5/2}}{9 c}\right )\) |
| \(\Big \downarrow \) 224 |
| \(\displaystyle c \left (\frac {1}{9} \left (\frac {\frac {\frac {63 \left (\frac {3}{4} c \left (\frac {1}{2} c \int \frac {1}{\frac {a^2 c x^2}{c-a^2 c x^2}+1}d\frac {x}{\sqrt {c-a^2 c x^2}}+\frac {1}{2} x \sqrt {c-a^2 c x^2}\right )+\frac {1}{4} x \left (c-a^2 c x^2\right )^{3/2}\right )-\frac {208 \left (c-a^2 c x^2\right )^{5/2}}{5 a c}}{2 a}-\frac {63 x \left (c-a^2 c x^2\right )^{5/2}}{2 a c}}{7 a}-\frac {52 x^2 \left (c-a^2 c x^2\right )^{5/2}}{7 a c}}{4 a}-\frac {9 x^3 \left (c-a^2 c x^2\right )^{5/2}}{4 a c}\right )-\frac {x^4 \left (c-a^2 c x^2\right )^{5/2}}{9 c}\right )\) |
| \(\Big \downarrow \) 216 |
| \(\displaystyle c \left (\frac {1}{9} \left (\frac {\frac {\frac {63 \left (\frac {3}{4} c \left (\frac {\sqrt {c} \arctan \left (\frac {a \sqrt {c} x}{\sqrt {c-a^2 c x^2}}\right )}{2 a}+\frac {1}{2} x \sqrt {c-a^2 c x^2}\right )+\frac {1}{4} x \left (c-a^2 c x^2\right )^{3/2}\right )-\frac {208 \left (c-a^2 c x^2\right )^{5/2}}{5 a c}}{2 a}-\frac {63 x \left (c-a^2 c x^2\right )^{5/2}}{2 a c}}{7 a}-\frac {52 x^2 \left (c-a^2 c x^2\right )^{5/2}}{7 a c}}{4 a}-\frac {9 x^3 \left (c-a^2 c x^2\right )^{5/2}}{4 a c}\right )-\frac {x^4 \left (c-a^2 c x^2\right )^{5/2}}{9 c}\right )\) |
c*(-1/9*(x^4*(c - a^2*c*x^2)^(5/2))/c + ((-9*x^3*(c - a^2*c*x^2)^(5/2))/(4 *a*c) + ((-52*x^2*(c - a^2*c*x^2)^(5/2))/(7*a*c) + ((-63*x*(c - a^2*c*x^2) ^(5/2))/(2*a*c) + ((-208*(c - a^2*c*x^2)^(5/2))/(5*a*c) + 63*((x*(c - a^2* c*x^2)^(3/2))/4 + (3*c*((x*Sqrt[c - a^2*c*x^2])/2 + (Sqrt[c]*ArcTan[(a*Sqr t[c]*x)/Sqrt[c - a^2*c*x^2]])/(2*a)))/4))/(2*a))/(7*a))/(4*a))/9)
3.11.98.3.1 Defintions of rubi rules used
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_)^2)^(p_), x_Symbol] :> Simp[x*((a + b*x^2)^p/(2*p + 1 )), x] + Simp[2*a*(p/(2*p + 1)) Int[(a + b*x^2)^(p - 1), x], x] /; FreeQ[ {a, b}, x] && GtQ[p, 0] && (IntegerQ[4*p] || IntegerQ[6*p])
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*A rcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a , 0] || GtQ[b, 0])
Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a, b}, x] && !GtQ[a, 0]
Int[((c_) + (d_.)*(x_))*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[d*(( a + b*x^2)^(p + 1)/(2*b*(p + 1))), x] + Simp[c Int[(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, p}, x] && !LeQ[p, -1]
Int[(x_)^(m_.)*((c_) + (d_.)*(x_))*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[d*x^m*((a + b*x^2)^(p + 1)/(b*(m + 2*p + 2))), x] - Simp[1/(b*(m + 2* p + 2)) Int[x^(m - 1)*(a + b*x^2)^p*Simp[a*d*m - b*c*(m + 2*p + 2)*x, x], x], x] /; FreeQ[{a, b, c, d, p}, x] && IGtQ[m, 0] && GtQ[p, -1] && Integer Q[2*p]
Int[(x_)^(m_.)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbo l] :> Simp[d^n*x^(m + n - 1)*((a + b*x^2)^(p + 1)/(b*(m + n + 2*p + 1))), x ] + Simp[1/(b*(m + n + 2*p + 1)) Int[x^m*(a + b*x^2)^p*ExpandToSum[b*(m + n + 2*p + 1)*(c + d*x)^n - b*d^n*(m + n + 2*p + 1)*x^n - a*d^n*(m + n - 1) *x^(n - 2), x], x], x] /; FreeQ[{a, b, c, d, m, p}, x] && IGtQ[n, 1] && IGt Q[m, -2] && GtQ[p, -1] && IntegerQ[2*p]
Int[E^(ArcTanh[(a_.)*(x_)]*(n_))*(x_)^(m_.)*((c_) + (d_.)*(x_)^2)^(p_.), x_ Symbol] :> Simp[c^(n/2) Int[x^m*(c + d*x^2)^(p - n/2)*(1 + a*x)^n, x], x] /; FreeQ[{a, c, d, m, p}, x] && EqQ[a^2*c + d, 0] && !(IntegerQ[p] || GtQ [c, 0]) && IGtQ[n/2, 0]
Time = 0.25 (sec) , antiderivative size = 133, normalized size of antiderivative = 0.71
| method | result | size |
| risch | \(\frac {\left (2240 a^{8} x^{8}+5040 a^{7} x^{7}-320 a^{6} x^{6}-7560 a^{5} x^{5}-4416 a^{4} x^{4}+630 a^{3} x^{3}+832 a^{2} x^{2}+945 a x +1664\right ) \left (a^{2} x^{2}-1\right ) c^{3}}{20160 a^{4} \sqrt {-c \left (a^{2} x^{2}-1\right )}}+\frac {3 \arctan \left (\frac {\sqrt {a^{2} c}\, x}{\sqrt {-a^{2} c \,x^{2}+c}}\right ) c^{3}}{64 a^{3} \sqrt {a^{2} c}}\) | \(133\) |
| default | \(\frac {x^{2} \left (-a^{2} c \,x^{2}+c \right )^{\frac {7}{2}}}{9 a^{2} c}+\frac {20 \left (-a^{2} c \,x^{2}+c \right )^{\frac {7}{2}}}{63 c \,a^{4}}-\frac {2 \left (\frac {x \left (-a^{2} c \,x^{2}+c \right )^{\frac {5}{2}}}{6}+\frac {5 c \left (\frac {x \left (-a^{2} c \,x^{2}+c \right )^{\frac {3}{2}}}{4}+\frac {3 c \left (\frac {x \sqrt {-a^{2} c \,x^{2}+c}}{2}+\frac {c \arctan \left (\frac {\sqrt {a^{2} c}\, x}{\sqrt {-a^{2} c \,x^{2}+c}}\right )}{2 \sqrt {a^{2} c}}\right )}{4}\right )}{6}\right )}{a^{3}}-\frac {2 \left (-\frac {x \left (-a^{2} c \,x^{2}+c \right )^{\frac {7}{2}}}{8 a^{2} c}+\frac {\frac {x \left (-a^{2} c \,x^{2}+c \right )^{\frac {5}{2}}}{6}+\frac {5 c \left (\frac {x \left (-a^{2} c \,x^{2}+c \right )^{\frac {3}{2}}}{4}+\frac {3 c \left (\frac {x \sqrt {-a^{2} c \,x^{2}+c}}{2}+\frac {c \arctan \left (\frac {\sqrt {a^{2} c}\, x}{\sqrt {-a^{2} c \,x^{2}+c}}\right )}{2 \sqrt {a^{2} c}}\right )}{4}\right )}{6}}{8 a^{2}}\right )}{a}-\frac {2 \left (\frac {\left (-a^{2} c \left (x -\frac {1}{a}\right )^{2}-2 \left (x -\frac {1}{a}\right ) a c \right )^{\frac {5}{2}}}{5}-a c \left (-\frac {\left (-2 a^{2} c \left (x -\frac {1}{a}\right )-2 a c \right ) \left (-a^{2} c \left (x -\frac {1}{a}\right )^{2}-2 \left (x -\frac {1}{a}\right ) a c \right )^{\frac {3}{2}}}{8 a^{2} c}+\frac {3 c \left (-\frac {\left (-2 a^{2} c \left (x -\frac {1}{a}\right )-2 a c \right ) \sqrt {-a^{2} c \left (x -\frac {1}{a}\right )^{2}-2 \left (x -\frac {1}{a}\right ) a c}}{4 a^{2} c}+\frac {c \arctan \left (\frac {\sqrt {a^{2} c}\, x}{\sqrt {-a^{2} c \left (x -\frac {1}{a}\right )^{2}-2 \left (x -\frac {1}{a}\right ) a c}}\right )}{2 \sqrt {a^{2} c}}\right )}{4}\right )\right )}{a^{4}}\) | \(470\) |
1/20160*(2240*a^8*x^8+5040*a^7*x^7-320*a^6*x^6-7560*a^5*x^5-4416*a^4*x^4+6 30*a^3*x^3+832*a^2*x^2+945*a*x+1664)*(a^2*x^2-1)/a^4/(-c*(a^2*x^2-1))^(1/2 )*c^3+3/64/a^3/(a^2*c)^(1/2)*arctan((a^2*c)^(1/2)*x/(-a^2*c*x^2+c)^(1/2))* c^3
Time = 0.27 (sec) , antiderivative size = 307, normalized size of antiderivative = 1.64 \[ \int e^{2 \text {arctanh}(a x)} x^3 \left (c-a^2 c x^2\right )^{5/2} \, dx=\left [\frac {945 \, \sqrt {-c} c^{2} \log \left (2 \, a^{2} c x^{2} + 2 \, \sqrt {-a^{2} c x^{2} + c} a \sqrt {-c} x - c\right ) - 2 \, {\left (2240 \, a^{8} c^{2} x^{8} + 5040 \, a^{7} c^{2} x^{7} - 320 \, a^{6} c^{2} x^{6} - 7560 \, a^{5} c^{2} x^{5} - 4416 \, a^{4} c^{2} x^{4} + 630 \, a^{3} c^{2} x^{3} + 832 \, a^{2} c^{2} x^{2} + 945 \, a c^{2} x + 1664 \, c^{2}\right )} \sqrt {-a^{2} c x^{2} + c}}{40320 \, a^{4}}, -\frac {945 \, c^{\frac {5}{2}} \arctan \left (\frac {\sqrt {-a^{2} c x^{2} + c} a \sqrt {c} x}{a^{2} c x^{2} - c}\right ) + {\left (2240 \, a^{8} c^{2} x^{8} + 5040 \, a^{7} c^{2} x^{7} - 320 \, a^{6} c^{2} x^{6} - 7560 \, a^{5} c^{2} x^{5} - 4416 \, a^{4} c^{2} x^{4} + 630 \, a^{3} c^{2} x^{3} + 832 \, a^{2} c^{2} x^{2} + 945 \, a c^{2} x + 1664 \, c^{2}\right )} \sqrt {-a^{2} c x^{2} + c}}{20160 \, a^{4}}\right ] \]
[1/40320*(945*sqrt(-c)*c^2*log(2*a^2*c*x^2 + 2*sqrt(-a^2*c*x^2 + c)*a*sqrt (-c)*x - c) - 2*(2240*a^8*c^2*x^8 + 5040*a^7*c^2*x^7 - 320*a^6*c^2*x^6 - 7 560*a^5*c^2*x^5 - 4416*a^4*c^2*x^4 + 630*a^3*c^2*x^3 + 832*a^2*c^2*x^2 + 9 45*a*c^2*x + 1664*c^2)*sqrt(-a^2*c*x^2 + c))/a^4, -1/20160*(945*c^(5/2)*ar ctan(sqrt(-a^2*c*x^2 + c)*a*sqrt(c)*x/(a^2*c*x^2 - c)) + (2240*a^8*c^2*x^8 + 5040*a^7*c^2*x^7 - 320*a^6*c^2*x^6 - 7560*a^5*c^2*x^5 - 4416*a^4*c^2*x^ 4 + 630*a^3*c^2*x^3 + 832*a^2*c^2*x^2 + 945*a*c^2*x + 1664*c^2)*sqrt(-a^2* c*x^2 + c))/a^4]
Time = 5.79 (sec) , antiderivative size = 391, normalized size of antiderivative = 2.09 \[ \int e^{2 \text {arctanh}(a x)} x^3 \left (c-a^2 c x^2\right )^{5/2} \, dx=- a^{4} c^{2} \left (\begin {cases} \sqrt {- a^{2} c x^{2} + c} \left (\frac {x^{8}}{9} - \frac {x^{6}}{63 a^{2}} - \frac {2 x^{4}}{105 a^{4}} - \frac {8 x^{2}}{315 a^{6}} - \frac {16}{315 a^{8}}\right ) & \text {for}\: a^{2} c \neq 0 \\\frac {\sqrt {c} x^{8}}{8} & \text {otherwise} \end {cases}\right ) - 2 a^{3} c^{2} \left (\begin {cases} \sqrt {- a^{2} c x^{2} + c} \left (\frac {x^{7}}{8} - \frac {x^{5}}{48 a^{2}} - \frac {5 x^{3}}{192 a^{4}} - \frac {5 x}{128 a^{6}}\right ) + \frac {5 c \left (\begin {cases} \frac {\log {\left (- 2 a^{2} c x + 2 \sqrt {- a^{2} c} \sqrt {- a^{2} c x^{2} + c} \right )}}{\sqrt {- a^{2} c}} & \text {for}\: c \neq 0 \\\frac {x \log {\left (x \right )}}{\sqrt {- a^{2} c x^{2}}} & \text {otherwise} \end {cases}\right )}{128 a^{6}} & \text {for}\: a^{2} c \neq 0 \\\frac {\sqrt {c} x^{7}}{7} & \text {otherwise} \end {cases}\right ) + 2 a c^{2} \left (\begin {cases} \sqrt {- a^{2} c x^{2} + c} \left (\frac {x^{5}}{6} - \frac {x^{3}}{24 a^{2}} - \frac {x}{16 a^{4}}\right ) + \frac {c \left (\begin {cases} \frac {\log {\left (- 2 a^{2} c x + 2 \sqrt {- a^{2} c} \sqrt {- a^{2} c x^{2} + c} \right )}}{\sqrt {- a^{2} c}} & \text {for}\: c \neq 0 \\\frac {x \log {\left (x \right )}}{\sqrt {- a^{2} c x^{2}}} & \text {otherwise} \end {cases}\right )}{16 a^{4}} & \text {for}\: a^{2} c \neq 0 \\\frac {\sqrt {c} x^{5}}{5} & \text {otherwise} \end {cases}\right ) + c^{2} \left (\begin {cases} \sqrt {- a^{2} c x^{2} + c} \left (\frac {x^{4}}{5} - \frac {x^{2}}{15 a^{2}} - \frac {2}{15 a^{4}}\right ) & \text {for}\: a^{2} c \neq 0 \\\frac {\sqrt {c} x^{4}}{4} & \text {otherwise} \end {cases}\right ) \]
-a**4*c**2*Piecewise((sqrt(-a**2*c*x**2 + c)*(x**8/9 - x**6/(63*a**2) - 2* x**4/(105*a**4) - 8*x**2/(315*a**6) - 16/(315*a**8)), Ne(a**2*c, 0)), (sqr t(c)*x**8/8, True)) - 2*a**3*c**2*Piecewise((sqrt(-a**2*c*x**2 + c)*(x**7/ 8 - x**5/(48*a**2) - 5*x**3/(192*a**4) - 5*x/(128*a**6)) + 5*c*Piecewise(( log(-2*a**2*c*x + 2*sqrt(-a**2*c)*sqrt(-a**2*c*x**2 + c))/sqrt(-a**2*c), N e(c, 0)), (x*log(x)/sqrt(-a**2*c*x**2), True))/(128*a**6), Ne(a**2*c, 0)), (sqrt(c)*x**7/7, True)) + 2*a*c**2*Piecewise((sqrt(-a**2*c*x**2 + c)*(x** 5/6 - x**3/(24*a**2) - x/(16*a**4)) + c*Piecewise((log(-2*a**2*c*x + 2*sqr t(-a**2*c)*sqrt(-a**2*c*x**2 + c))/sqrt(-a**2*c), Ne(c, 0)), (x*log(x)/sqr t(-a**2*c*x**2), True))/(16*a**4), Ne(a**2*c, 0)), (sqrt(c)*x**5/5, True)) + c**2*Piecewise((sqrt(-a**2*c*x**2 + c)*(x**4/5 - x**2/(15*a**2) - 2/(15 *a**4)), Ne(a**2*c, 0)), (sqrt(c)*x**4/4, True))
Time = 0.51 (sec) , antiderivative size = 239, normalized size of antiderivative = 1.28 \[ \int e^{2 \text {arctanh}(a x)} x^3 \left (c-a^2 c x^2\right )^{5/2} \, dx=\frac {1}{20160} \, {\left (\frac {2240 \, {\left (-a^{2} c x^{2} + c\right )}^{\frac {7}{2}} x^{2}}{a^{3} c} - \frac {7560 \, {\left (-a^{2} c x^{2} + c\right )}^{\frac {5}{2}} x}{a^{4}} + \frac {5040 \, {\left (-a^{2} c x^{2} + c\right )}^{\frac {7}{2}} x}{a^{4} c} + \frac {630 \, {\left (-a^{2} c x^{2} + c\right )}^{\frac {3}{2}} c x}{a^{4}} + \frac {15120 \, \sqrt {a^{2} c x^{2} - 4 \, a c x + 3 \, c} c^{2} x}{a^{4}} - \frac {14175 \, \sqrt {-a^{2} c x^{2} + c} c^{2} x}{a^{4}} - \frac {14175 \, c^{\frac {5}{2}} \arcsin \left (a x\right )}{a^{5}} - \frac {8064 \, {\left (-a^{2} c x^{2} + c\right )}^{\frac {5}{2}}}{a^{5}} + \frac {6400 \, {\left (-a^{2} c x^{2} + c\right )}^{\frac {7}{2}}}{a^{5} c} - \frac {30240 \, \sqrt {a^{2} c x^{2} - 4 \, a c x + 3 \, c} c^{2}}{a^{5}} + \frac {15120 \, c^{4} \arcsin \left (a x - 2\right )}{a^{8} \left (-\frac {c}{a^{2}}\right )^{\frac {3}{2}}}\right )} a \]
1/20160*(2240*(-a^2*c*x^2 + c)^(7/2)*x^2/(a^3*c) - 7560*(-a^2*c*x^2 + c)^( 5/2)*x/a^4 + 5040*(-a^2*c*x^2 + c)^(7/2)*x/(a^4*c) + 630*(-a^2*c*x^2 + c)^ (3/2)*c*x/a^4 + 15120*sqrt(a^2*c*x^2 - 4*a*c*x + 3*c)*c^2*x/a^4 - 14175*sq rt(-a^2*c*x^2 + c)*c^2*x/a^4 - 14175*c^(5/2)*arcsin(a*x)/a^5 - 8064*(-a^2* c*x^2 + c)^(5/2)/a^5 + 6400*(-a^2*c*x^2 + c)^(7/2)/(a^5*c) - 30240*sqrt(a^ 2*c*x^2 - 4*a*c*x + 3*c)*c^2/a^5 + 15120*c^4*arcsin(a*x - 2)/(a^8*(-c/a^2) ^(3/2)))*a
Time = 0.30 (sec) , antiderivative size = 155, normalized size of antiderivative = 0.83 \[ \int e^{2 \text {arctanh}(a x)} x^3 \left (c-a^2 c x^2\right )^{5/2} \, dx=\frac {1}{20160} \, \sqrt {-a^{2} c x^{2} + c} {\left ({\left (2 \, {\left ({\left (4 \, {\left (552 \, c^{2} + 5 \, {\left (189 \, a c^{2} + 2 \, {\left (4 \, a^{2} c^{2} - 7 \, {\left (4 \, a^{4} c^{2} x + 9 \, a^{3} c^{2}\right )} x\right )} x\right )} x\right )} x - \frac {315 \, c^{2}}{a}\right )} x - \frac {416 \, c^{2}}{a^{2}}\right )} x - \frac {945 \, c^{2}}{a^{3}}\right )} x - \frac {1664 \, c^{2}}{a^{4}}\right )} - \frac {3 \, c^{3} \log \left ({\left | -\sqrt {-a^{2} c} x + \sqrt {-a^{2} c x^{2} + c} \right |}\right )}{64 \, a^{3} \sqrt {-c} {\left | a \right |}} \]
1/20160*sqrt(-a^2*c*x^2 + c)*((2*((4*(552*c^2 + 5*(189*a*c^2 + 2*(4*a^2*c^ 2 - 7*(4*a^4*c^2*x + 9*a^3*c^2)*x)*x)*x)*x - 315*c^2/a)*x - 416*c^2/a^2)*x - 945*c^2/a^3)*x - 1664*c^2/a^4) - 3/64*c^3*log(abs(-sqrt(-a^2*c)*x + sqr t(-a^2*c*x^2 + c)))/(a^3*sqrt(-c)*abs(a))
Timed out. \[ \int e^{2 \text {arctanh}(a x)} x^3 \left (c-a^2 c x^2\right )^{5/2} \, dx=\int -\frac {x^3\,{\left (c-a^2\,c\,x^2\right )}^{5/2}\,{\left (a\,x+1\right )}^2}{a^2\,x^2-1} \,d x \]