Integrand size = 27, antiderivative size = 180 \[ \int e^{2 \text {arctanh}(a x)} x^3 \left (c-a^2 c x^2\right )^{3/2} \, dx=-\frac {c x \sqrt {c-a^2 c x^2}}{8 a^3}-\frac {c x^3 \sqrt {c-a^2 c x^2}}{12 a}+\frac {1}{3} a c x^5 \sqrt {c-a^2 c x^2}-\frac {11 \left (c-a^2 c x^2\right )^{3/2}}{21 a^4}-\frac {1}{7} x^4 \left (c-a^2 c x^2\right )^{3/2}+\frac {11 \left (c-a^2 c x^2\right )^{5/2}}{35 a^4 c}+\frac {c^{3/2} \arctan \left (\frac {a \sqrt {c} x}{\sqrt {c-a^2 c x^2}}\right )}{8 a^4} \] Output:
-1/8*c*x*(-a^2*c*x^2+c)^(1/2)/a^3-1/12*c*x^3*(-a^2*c*x^2+c)^(1/2)/a+1/3*a* c*x^5*(-a^2*c*x^2+c)^(1/2)-11/21*(-a^2*c*x^2+c)^(3/2)/a^4-1/7*x^4*(-a^2*c* x^2+c)^(3/2)+11/35*(-a^2*c*x^2+c)^(5/2)/a^4/c+1/8*c^(3/2)*arctan(a*c^(1/2) *x/(-a^2*c*x^2+c)^(1/2))/a^4
Time = 0.24 (sec) , antiderivative size = 113, normalized size of antiderivative = 0.63 \[ \int e^{2 \text {arctanh}(a x)} x^3 \left (c-a^2 c x^2\right )^{3/2} \, dx=\frac {c \sqrt {c-a^2 c x^2} \left (-176-105 a x-88 a^2 x^2-70 a^3 x^3+144 a^4 x^4+280 a^5 x^5+120 a^6 x^6\right )-105 c^{3/2} \arctan \left (\frac {a x \sqrt {c-a^2 c x^2}}{\sqrt {c} \left (-1+a^2 x^2\right )}\right )}{840 a^4} \] Input:
Integrate[E^(2*ArcTanh[a*x])*x^3*(c - a^2*c*x^2)^(3/2),x]
Output:
(c*Sqrt[c - a^2*c*x^2]*(-176 - 105*a*x - 88*a^2*x^2 - 70*a^3*x^3 + 144*a^4 *x^4 + 280*a^5*x^5 + 120*a^6*x^6) - 105*c^(3/2)*ArcTan[(a*x*Sqrt[c - a^2*c *x^2])/(Sqrt[c]*(-1 + a^2*x^2))])/(840*a^4)
Time = 0.84 (sec) , antiderivative size = 220, normalized size of antiderivative = 1.22, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.519, Rules used = {6701, 541, 25, 27, 533, 27, 533, 27, 533, 27, 455, 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.
| \(\displaystyle \int x^3 e^{2 \text {arctanh}(a x)} \left (c-a^2 c x^2\right )^{3/2} \, dx\) |
| \(\Big \downarrow \) 6701 |
| \(\displaystyle c \int x^3 (a x+1)^2 \sqrt {c-a^2 c x^2}dx\) |
| \(\Big \downarrow \) 541 |
| \(\displaystyle c \left (-\frac {\int -a^2 c x^3 (14 a x+11) \sqrt {c-a^2 c x^2}dx}{7 a^2 c}-\frac {x^4 \left (c-a^2 c x^2\right )^{3/2}}{7 c}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle c \left (\frac {\int a^2 c x^3 (14 a x+11) \sqrt {c-a^2 c x^2}dx}{7 a^2 c}-\frac {x^4 \left (c-a^2 c x^2\right )^{3/2}}{7 c}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle c \left (\frac {1}{7} \int x^3 (14 a x+11) \sqrt {c-a^2 c x^2}dx-\frac {x^4 \left (c-a^2 c x^2\right )^{3/2}}{7 c}\right )\) |
| \(\Big \downarrow \) 533 |
| \(\displaystyle c \left (\frac {1}{7} \left (\frac {\int 6 a c x^2 (11 a x+7) \sqrt {c-a^2 c x^2}dx}{6 a^2 c}-\frac {7 x^3 \left (c-a^2 c x^2\right )^{3/2}}{3 a c}\right )-\frac {x^4 \left (c-a^2 c x^2\right )^{3/2}}{7 c}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle c \left (\frac {1}{7} \left (\frac {\int x^2 (11 a x+7) \sqrt {c-a^2 c x^2}dx}{a}-\frac {7 x^3 \left (c-a^2 c x^2\right )^{3/2}}{3 a c}\right )-\frac {x^4 \left (c-a^2 c x^2\right )^{3/2}}{7 c}\right )\) |
| \(\Big \downarrow \) 533 |
| \(\displaystyle c \left (\frac {1}{7} \left (\frac {\frac {\int a c x (35 a x+22) \sqrt {c-a^2 c x^2}dx}{5 a^2 c}-\frac {11 x^2 \left (c-a^2 c x^2\right )^{3/2}}{5 a c}}{a}-\frac {7 x^3 \left (c-a^2 c x^2\right )^{3/2}}{3 a c}\right )-\frac {x^4 \left (c-a^2 c x^2\right )^{3/2}}{7 c}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle c \left (\frac {1}{7} \left (\frac {\frac {\int x (35 a x+22) \sqrt {c-a^2 c x^2}dx}{5 a}-\frac {11 x^2 \left (c-a^2 c x^2\right )^{3/2}}{5 a c}}{a}-\frac {7 x^3 \left (c-a^2 c x^2\right )^{3/2}}{3 a c}\right )-\frac {x^4 \left (c-a^2 c x^2\right )^{3/2}}{7 c}\right )\) |
| \(\Big \downarrow \) 533 |
| \(\displaystyle c \left (\frac {1}{7} \left (\frac {\frac {\frac {\int a c (88 a x+35) \sqrt {c-a^2 c x^2}dx}{4 a^2 c}-\frac {35 x \left (c-a^2 c x^2\right )^{3/2}}{4 a c}}{5 a}-\frac {11 x^2 \left (c-a^2 c x^2\right )^{3/2}}{5 a c}}{a}-\frac {7 x^3 \left (c-a^2 c x^2\right )^{3/2}}{3 a c}\right )-\frac {x^4 \left (c-a^2 c x^2\right )^{3/2}}{7 c}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle c \left (\frac {1}{7} \left (\frac {\frac {\frac {\int (88 a x+35) \sqrt {c-a^2 c x^2}dx}{4 a}-\frac {35 x \left (c-a^2 c x^2\right )^{3/2}}{4 a c}}{5 a}-\frac {11 x^2 \left (c-a^2 c x^2\right )^{3/2}}{5 a c}}{a}-\frac {7 x^3 \left (c-a^2 c x^2\right )^{3/2}}{3 a c}\right )-\frac {x^4 \left (c-a^2 c x^2\right )^{3/2}}{7 c}\right )\) |
| \(\Big \downarrow \) 455 |
| \(\displaystyle c \left (\frac {1}{7} \left (\frac {\frac {\frac {35 \int \sqrt {c-a^2 c x^2}dx-\frac {88 \left (c-a^2 c x^2\right )^{3/2}}{3 a c}}{4 a}-\frac {35 x \left (c-a^2 c x^2\right )^{3/2}}{4 a c}}{5 a}-\frac {11 x^2 \left (c-a^2 c x^2\right )^{3/2}}{5 a c}}{a}-\frac {7 x^3 \left (c-a^2 c x^2\right )^{3/2}}{3 a c}\right )-\frac {x^4 \left (c-a^2 c x^2\right )^{3/2}}{7 c}\right )\) |
| \(\Big \downarrow \) 211 |
| \(\displaystyle c \left (\frac {1}{7} \left (\frac {\frac {\frac {35 \left (\frac {1}{2} c \int \frac {1}{\sqrt {c-a^2 c x^2}}dx+\frac {1}{2} x \sqrt {c-a^2 c x^2}\right )-\frac {88 \left (c-a^2 c x^2\right )^{3/2}}{3 a c}}{4 a}-\frac {35 x \left (c-a^2 c x^2\right )^{3/2}}{4 a c}}{5 a}-\frac {11 x^2 \left (c-a^2 c x^2\right )^{3/2}}{5 a c}}{a}-\frac {7 x^3 \left (c-a^2 c x^2\right )^{3/2}}{3 a c}\right )-\frac {x^4 \left (c-a^2 c x^2\right )^{3/2}}{7 c}\right )\) |
| \(\Big \downarrow \) 224 |
| \(\displaystyle c \left (\frac {1}{7} \left (\frac {\frac {\frac {35 \left (\frac {1}{2} c \int \frac {1}{\frac {a^2 c x^2}{c-a^2 c x^2}+1}d\frac {x}{\sqrt {c-a^2 c x^2}}+\frac {1}{2} x \sqrt {c-a^2 c x^2}\right )-\frac {88 \left (c-a^2 c x^2\right )^{3/2}}{3 a c}}{4 a}-\frac {35 x \left (c-a^2 c x^2\right )^{3/2}}{4 a c}}{5 a}-\frac {11 x^2 \left (c-a^2 c x^2\right )^{3/2}}{5 a c}}{a}-\frac {7 x^3 \left (c-a^2 c x^2\right )^{3/2}}{3 a c}\right )-\frac {x^4 \left (c-a^2 c x^2\right )^{3/2}}{7 c}\right )\) |
| \(\Big \downarrow \) 216 |
| \(\displaystyle c \left (\frac {1}{7} \left (\frac {\frac {\frac {35 \left (\frac {\sqrt {c} \arctan \left (\frac {a \sqrt {c} x}{\sqrt {c-a^2 c x^2}}\right )}{2 a}+\frac {1}{2} x \sqrt {c-a^2 c x^2}\right )-\frac {88 \left (c-a^2 c x^2\right )^{3/2}}{3 a c}}{4 a}-\frac {35 x \left (c-a^2 c x^2\right )^{3/2}}{4 a c}}{5 a}-\frac {11 x^2 \left (c-a^2 c x^2\right )^{3/2}}{5 a c}}{a}-\frac {7 x^3 \left (c-a^2 c x^2\right )^{3/2}}{3 a c}\right )-\frac {x^4 \left (c-a^2 c x^2\right )^{3/2}}{7 c}\right )\) |
Input:
Int[E^(2*ArcTanh[a*x])*x^3*(c - a^2*c*x^2)^(3/2),x]
Output:
c*(-1/7*(x^4*(c - a^2*c*x^2)^(3/2))/c + ((-7*x^3*(c - a^2*c*x^2)^(3/2))/(3 *a*c) + ((-11*x^2*(c - a^2*c*x^2)^(3/2))/(5*a*c) + ((-35*x*(c - a^2*c*x^2) ^(3/2))/(4*a*c) + ((-88*(c - a^2*c*x^2)^(3/2))/(3*a*c) + 35*((x*Sqrt[c - a ^2*c*x^2])/2 + (Sqrt[c]*ArcTan[(a*Sqrt[c]*x)/Sqrt[c - a^2*c*x^2]])/(2*a))) /(4*a))/(5*a))/a)/7)
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 = 0.18 (sec) , antiderivative size = 117, normalized size of antiderivative = 0.65
| method | result | size |
| risch | \(-\frac {\left (120 x^{6} a^{6}+280 a^{5} x^{5}+144 a^{4} x^{4}-70 a^{3} x^{3}-88 a^{2} x^{2}-105 a x -176\right ) \left (a^{2} x^{2}-1\right ) c^{2}}{840 a^{4} \sqrt {-c \left (a^{2} x^{2}-1\right )}}+\frac {\arctan \left (\frac {\sqrt {a^{2} c}\, x}{\sqrt {-a^{2} c \,x^{2}+c}}\right ) c^{2}}{8 a^{3} \sqrt {a^{2} c}}\) | \(117\) |
| default | \(\frac {x^{2} \left (-a^{2} c \,x^{2}+c \right )^{\frac {5}{2}}}{7 a^{2} c}+\frac {16 \left (-a^{2} c \,x^{2}+c \right )^{\frac {5}{2}}}{35 a^{4} c}-\frac {2 \left (\frac {x \left (-a^{2} c \,x^{2}+c \right )^{\frac {3}{2}}}{4}+\frac {3 c \left (\frac {x \sqrt {-a^{2} c \,x^{2}+c}}{2}+\frac {c \arctan \left (\frac {\sqrt {a^{2} c}\, x}{\sqrt {-a^{2} c \,x^{2}+c}}\right )}{2 \sqrt {a^{2} c}}\right )}{4}\right )}{a^{3}}-\frac {2 \left (-\frac {x \left (-a^{2} c \,x^{2}+c \right )^{\frac {5}{2}}}{6 a^{2} c}+\frac {\frac {x \left (-a^{2} c \,x^{2}+c \right )^{\frac {3}{2}}}{4}+\frac {3 c \left (\frac {x \sqrt {-a^{2} c \,x^{2}+c}}{2}+\frac {c \arctan \left (\frac {\sqrt {a^{2} c}\, x}{\sqrt {-a^{2} c \,x^{2}+c}}\right )}{2 \sqrt {a^{2} c}}\right )}{4}}{6 a^{2}}\right )}{a}-\frac {2 \left (\frac {\left (-\left (x -\frac {1}{a}\right )^{2} a^{2} c -2 \left (x -\frac {1}{a}\right ) a c \right )^{\frac {3}{2}}}{3}-a c \left (-\frac {\left (-2 \left (x -\frac {1}{a}\right ) a^{2} c -2 a c \right ) \sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2} c -2 \left (x -\frac {1}{a}\right ) a c}}{4 a^{2} c}+\frac {c \arctan \left (\frac {\sqrt {a^{2} c}\, x}{\sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2} c -2 \left (x -\frac {1}{a}\right ) a c}}\right )}{2 \sqrt {a^{2} c}}\right )\right )}{a^{4}}\) | \(371\) |
Input:
int((a*x+1)^2/(-a^2*x^2+1)*x^3*(-a^2*c*x^2+c)^(3/2),x,method=_RETURNVERBOS E)
Output:
-1/840*(120*a^6*x^6+280*a^5*x^5+144*a^4*x^4-70*a^3*x^3-88*a^2*x^2-105*a*x- 176)*(a^2*x^2-1)/a^4/(-c*(a^2*x^2-1))^(1/2)*c^2+1/8/a^3/(a^2*c)^(1/2)*arct an((a^2*c)^(1/2)*x/(-a^2*c*x^2+c)^(1/2))*c^2
Time = 0.14 (sec) , antiderivative size = 234, normalized size of antiderivative = 1.30 \[ \int e^{2 \text {arctanh}(a x)} x^3 \left (c-a^2 c x^2\right )^{3/2} \, dx=\left [\frac {105 \, \sqrt {-c} c \log \left (2 \, a^{2} c x^{2} + 2 \, \sqrt {-a^{2} c x^{2} + c} a \sqrt {-c} x - c\right ) + 2 \, {\left (120 \, a^{6} c x^{6} + 280 \, a^{5} c x^{5} + 144 \, a^{4} c x^{4} - 70 \, a^{3} c x^{3} - 88 \, a^{2} c x^{2} - 105 \, a c x - 176 \, c\right )} \sqrt {-a^{2} c x^{2} + c}}{1680 \, a^{4}}, -\frac {105 \, c^{\frac {3}{2}} \arctan \left (\frac {\sqrt {-a^{2} c x^{2} + c} a \sqrt {c} x}{a^{2} c x^{2} - c}\right ) - {\left (120 \, a^{6} c x^{6} + 280 \, a^{5} c x^{5} + 144 \, a^{4} c x^{4} - 70 \, a^{3} c x^{3} - 88 \, a^{2} c x^{2} - 105 \, a c x - 176 \, c\right )} \sqrt {-a^{2} c x^{2} + c}}{840 \, a^{4}}\right ] \] Input:
integrate((a*x+1)^2/(-a^2*x^2+1)*x^3*(-a^2*c*x^2+c)^(3/2),x, algorithm="fr icas")
Output:
[1/1680*(105*sqrt(-c)*c*log(2*a^2*c*x^2 + 2*sqrt(-a^2*c*x^2 + c)*a*sqrt(-c )*x - c) + 2*(120*a^6*c*x^6 + 280*a^5*c*x^5 + 144*a^4*c*x^4 - 70*a^3*c*x^3 - 88*a^2*c*x^2 - 105*a*c*x - 176*c)*sqrt(-a^2*c*x^2 + c))/a^4, -1/840*(10 5*c^(3/2)*arctan(sqrt(-a^2*c*x^2 + c)*a*sqrt(c)*x/(a^2*c*x^2 - c)) - (120* a^6*c*x^6 + 280*a^5*c*x^5 + 144*a^4*c*x^4 - 70*a^3*c*x^3 - 88*a^2*c*x^2 - 105*a*c*x - 176*c)*sqrt(-a^2*c*x^2 + c))/a^4]
Time = 3.53 (sec) , antiderivative size = 235, normalized size of antiderivative = 1.31 \[ \int e^{2 \text {arctanh}(a x)} x^3 \left (c-a^2 c x^2\right )^{3/2} \, dx=a^{2} c \left (\begin {cases} \sqrt {- a^{2} c x^{2} + c} \left (\frac {x^{6}}{7} - \frac {x^{4}}{35 a^{2}} - \frac {4 x^{2}}{105 a^{4}} - \frac {8}{105 a^{6}}\right ) & \text {for}\: a^{2} c \neq 0 \\\frac {\sqrt {c} x^{6}}{6} & \text {otherwise} \end {cases}\right ) + 2 a c \left (\begin {cases} \sqrt {- a^{2} c x^{2} + c} \left (\frac {x^{5}}{6} - \frac {x^{3}}{24 a^{2}} - \frac {x}{16 a^{4}}\right ) + \frac {c \left (\begin {cases} \frac {\log {\left (- 2 a^{2} c x + 2 \sqrt {- a^{2} c} \sqrt {- a^{2} c x^{2} + c} \right )}}{\sqrt {- a^{2} c}} & \text {for}\: c \neq 0 \\\frac {x \log {\left (x \right )}}{\sqrt {- a^{2} c x^{2}}} & \text {otherwise} \end {cases}\right )}{16 a^{4}} & \text {for}\: a^{2} c \neq 0 \\\frac {\sqrt {c} x^{5}}{5} & \text {otherwise} \end {cases}\right ) + c \left (\begin {cases} \sqrt {- a^{2} c x^{2} + c} \left (\frac {x^{4}}{5} - \frac {x^{2}}{15 a^{2}} - \frac {2}{15 a^{4}}\right ) & \text {for}\: a^{2} c \neq 0 \\\frac {\sqrt {c} x^{4}}{4} & \text {otherwise} \end {cases}\right ) \] Input:
integrate((a*x+1)**2/(-a**2*x**2+1)*x**3*(-a**2*c*x**2+c)**(3/2),x)
Output:
a**2*c*Piecewise((sqrt(-a**2*c*x**2 + c)*(x**6/7 - x**4/(35*a**2) - 4*x**2 /(105*a**4) - 8/(105*a**6)), Ne(a**2*c, 0)), (sqrt(c)*x**6/6, True)) + 2*a *c*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*sqrt(-a**2*c)*sqrt(-a**2*c*x**2 + c) )/sqrt(-a**2*c), Ne(c, 0)), (x*log(x)/sqrt(-a**2*c*x**2), True))/(16*a**4) , Ne(a**2*c, 0)), (sqrt(c)*x**5/5, True)) + c*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 = 0.29 (sec) , antiderivative size = 213, normalized size of antiderivative = 1.18 \[ \int e^{2 \text {arctanh}(a x)} x^3 \left (c-a^2 c x^2\right )^{3/2} \, dx=\frac {1}{840} \, a {\left (\frac {120 \, {\left (-a^{2} c x^{2} + c\right )}^{\frac {5}{2}} x^{2}}{a^{3} c} - \frac {490 \, {\left (-a^{2} c x^{2} + c\right )}^{\frac {3}{2}} x}{a^{4}} + \frac {280 \, {\left (-a^{2} c x^{2} + c\right )}^{\frac {5}{2}} x}{a^{4} c} + \frac {840 \, \sqrt {a^{2} c x^{2} - 4 \, a c x + 3 \, c} c x}{a^{4}} - \frac {735 \, \sqrt {-a^{2} c x^{2} + c} c x}{a^{4}} - \frac {735 \, c^{\frac {3}{2}} \arcsin \left (a x\right )}{a^{5}} - \frac {560 \, {\left (-a^{2} c x^{2} + c\right )}^{\frac {3}{2}}}{a^{5}} + \frac {384 \, {\left (-a^{2} c x^{2} + c\right )}^{\frac {5}{2}}}{a^{5} c} - \frac {1680 \, \sqrt {a^{2} c x^{2} - 4 \, a c x + 3 \, c} c}{a^{5}} + \frac {840 \, c^{3} \arcsin \left (a x - 2\right )}{a^{8} \left (-\frac {c}{a^{2}}\right )^{\frac {3}{2}}}\right )} \] Input:
integrate((a*x+1)^2/(-a^2*x^2+1)*x^3*(-a^2*c*x^2+c)^(3/2),x, algorithm="ma xima")
Output:
1/840*a*(120*(-a^2*c*x^2 + c)^(5/2)*x^2/(a^3*c) - 490*(-a^2*c*x^2 + c)^(3/ 2)*x/a^4 + 280*(-a^2*c*x^2 + c)^(5/2)*x/(a^4*c) + 840*sqrt(a^2*c*x^2 - 4*a *c*x + 3*c)*c*x/a^4 - 735*sqrt(-a^2*c*x^2 + c)*c*x/a^4 - 735*c^(3/2)*arcsi n(a*x)/a^5 - 560*(-a^2*c*x^2 + c)^(3/2)/a^5 + 384*(-a^2*c*x^2 + c)^(5/2)/( a^5*c) - 1680*sqrt(a^2*c*x^2 - 4*a*c*x + 3*c)*c/a^5 + 840*c^3*arcsin(a*x - 2)/(a^8*(-c/a^2)^(3/2)))
Time = 0.17 (sec) , antiderivative size = 117, normalized size of antiderivative = 0.65 \[ \int e^{2 \text {arctanh}(a x)} x^3 \left (c-a^2 c x^2\right )^{3/2} \, dx=\frac {1}{840} \, \sqrt {-a^{2} c x^{2} + c} {\left ({\left (2 \, {\left ({\left (4 \, {\left (5 \, {\left (3 \, a^{2} c x + 7 \, a c\right )} x + 18 \, c\right )} x - \frac {35 \, c}{a}\right )} x - \frac {44 \, c}{a^{2}}\right )} x - \frac {105 \, c}{a^{3}}\right )} x - \frac {176 \, c}{a^{4}}\right )} - \frac {c^{2} \log \left ({\left | -\sqrt {-a^{2} c} x + \sqrt {-a^{2} c x^{2} + c} \right |}\right )}{8 \, a^{3} \sqrt {-c} {\left | a \right |}} \] Input:
integrate((a*x+1)^2/(-a^2*x^2+1)*x^3*(-a^2*c*x^2+c)^(3/2),x, algorithm="gi ac")
Output:
1/840*sqrt(-a^2*c*x^2 + c)*((2*((4*(5*(3*a^2*c*x + 7*a*c)*x + 18*c)*x - 35 *c/a)*x - 44*c/a^2)*x - 105*c/a^3)*x - 176*c/a^4) - 1/8*c^2*log(abs(-sqrt( -a^2*c)*x + sqrt(-a^2*c*x^2 + c)))/(a^3*sqrt(-c)*abs(a))
Timed out. \[ \int e^{2 \text {arctanh}(a x)} x^3 \left (c-a^2 c x^2\right )^{3/2} \, dx=\int -\frac {x^3\,{\left (c-a^2\,c\,x^2\right )}^{3/2}\,{\left (a\,x+1\right )}^2}{a^2\,x^2-1} \,d x \] Input:
int(-(x^3*(c - a^2*c*x^2)^(3/2)*(a*x + 1)^2)/(a^2*x^2 - 1),x)
Output:
int(-(x^3*(c - a^2*c*x^2)^(3/2)*(a*x + 1)^2)/(a^2*x^2 - 1), x)
Time = 0.16 (sec) , antiderivative size = 139, normalized size of antiderivative = 0.77 \[ \int e^{2 \text {arctanh}(a x)} x^3 \left (c-a^2 c x^2\right )^{3/2} \, dx=\frac {\sqrt {c}\, c \left (105 \mathit {asin} \left (a x \right )+120 \sqrt {-a^{2} x^{2}+1}\, a^{6} x^{6}+280 \sqrt {-a^{2} x^{2}+1}\, a^{5} x^{5}+144 \sqrt {-a^{2} x^{2}+1}\, a^{4} x^{4}-70 \sqrt {-a^{2} x^{2}+1}\, a^{3} x^{3}-88 \sqrt {-a^{2} x^{2}+1}\, a^{2} x^{2}-105 \sqrt {-a^{2} x^{2}+1}\, a x -176 \sqrt {-a^{2} x^{2}+1}+176\right )}{840 a^{4}} \] Input:
int((a*x+1)^2/(-a^2*x^2+1)*x^3*(-a^2*c*x^2+c)^(3/2),x)
Output:
(sqrt(c)*c*(105*asin(a*x) + 120*sqrt( - a**2*x**2 + 1)*a**6*x**6 + 280*sqr t( - a**2*x**2 + 1)*a**5*x**5 + 144*sqrt( - a**2*x**2 + 1)*a**4*x**4 - 70* sqrt( - a**2*x**2 + 1)*a**3*x**3 - 88*sqrt( - a**2*x**2 + 1)*a**2*x**2 - 1 05*sqrt( - a**2*x**2 + 1)*a*x - 176*sqrt( - a**2*x**2 + 1) + 176))/(840*a* *4)