Integrand size = 27, antiderivative size = 136 \[ \int e^{2 \text {arctanh}(a x)} x^2 \left (c-a^2 c x^2\right )^{3/2} \, dx=\frac {3 c x \sqrt {c-a^2 c x^2}}{16 a^2}-\frac {2 x^2 \left (c-a^2 c x^2\right )^{3/2}}{5 a}-\frac {1}{6} x^3 \left (c-a^2 c x^2\right )^{3/2}-\frac {(32+45 a x) \left (c-a^2 c x^2\right )^{3/2}}{120 a^3}+\frac {3 c^{3/2} \arctan \left (\frac {a \sqrt {c} x}{\sqrt {c-a^2 c x^2}}\right )}{16 a^3} \]
-2/5*x^2*(-a^2*c*x^2+c)^(3/2)/a-1/6*x^3*(-a^2*c*x^2+c)^(3/2)-1/120*(45*a*x +32)*(-a^2*c*x^2+c)^(3/2)/a^3+3/16*c^(3/2)*arctan(a*x*c^(1/2)/(-a^2*c*x^2+ c)^(1/2))/a^3+3/16*c*x*(-a^2*c*x^2+c)^(1/2)/a^2
Time = 0.13 (sec) , antiderivative size = 105, normalized size of antiderivative = 0.77 \[ \int e^{2 \text {arctanh}(a x)} x^2 \left (c-a^2 c x^2\right )^{3/2} \, dx=\frac {c \sqrt {c-a^2 c x^2} \left (-64-45 a x-32 a^2 x^2+50 a^3 x^3+96 a^4 x^4+40 a^5 x^5\right )-45 c^{3/2} \arctan \left (\frac {a x \sqrt {c-a^2 c x^2}}{\sqrt {c} \left (-1+a^2 x^2\right )}\right )}{240 a^3} \]
(c*Sqrt[c - a^2*c*x^2]*(-64 - 45*a*x - 32*a^2*x^2 + 50*a^3*x^3 + 96*a^4*x^ 4 + 40*a^5*x^5) - 45*c^(3/2)*ArcTan[(a*x*Sqrt[c - a^2*c*x^2])/(Sqrt[c]*(-1 + a^2*x^2))])/(240*a^3)
Time = 0.47 (sec) , antiderivative size = 187, normalized size of antiderivative = 1.38, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.407, Rules used = {6701, 541, 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.
| \(\displaystyle \int x^2 e^{2 \text {arctanh}(a x)} \left (c-a^2 c x^2\right )^{3/2} \, dx\) |
| \(\Big \downarrow \) 6701 |
| \(\displaystyle c \int x^2 (a x+1)^2 \sqrt {c-a^2 c x^2}dx\) |
| \(\Big \downarrow \) 541 |
| \(\displaystyle c \left (-\frac {\int -3 a^2 c x^2 (4 a x+3) \sqrt {c-a^2 c x^2}dx}{6 a^2 c}-\frac {x^3 \left (c-a^2 c x^2\right )^{3/2}}{6 c}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle c \left (\frac {1}{2} \int x^2 (4 a x+3) \sqrt {c-a^2 c x^2}dx-\frac {x^3 \left (c-a^2 c x^2\right )^{3/2}}{6 c}\right )\) |
| \(\Big \downarrow \) 533 |
| \(\displaystyle c \left (\frac {1}{2} \left (\frac {\int a c x (15 a x+8) \sqrt {c-a^2 c x^2}dx}{5 a^2 c}-\frac {4 x^2 \left (c-a^2 c x^2\right )^{3/2}}{5 a c}\right )-\frac {x^3 \left (c-a^2 c x^2\right )^{3/2}}{6 c}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle c \left (\frac {1}{2} \left (\frac {\int x (15 a x+8) \sqrt {c-a^2 c x^2}dx}{5 a}-\frac {4 x^2 \left (c-a^2 c x^2\right )^{3/2}}{5 a c}\right )-\frac {x^3 \left (c-a^2 c x^2\right )^{3/2}}{6 c}\right )\) |
| \(\Big \downarrow \) 533 |
| \(\displaystyle c \left (\frac {1}{2} \left (\frac {\frac {\int a c (32 a x+15) \sqrt {c-a^2 c x^2}dx}{4 a^2 c}-\frac {15 x \left (c-a^2 c x^2\right )^{3/2}}{4 a c}}{5 a}-\frac {4 x^2 \left (c-a^2 c x^2\right )^{3/2}}{5 a c}\right )-\frac {x^3 \left (c-a^2 c x^2\right )^{3/2}}{6 c}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle c \left (\frac {1}{2} \left (\frac {\frac {\int (32 a x+15) \sqrt {c-a^2 c x^2}dx}{4 a}-\frac {15 x \left (c-a^2 c x^2\right )^{3/2}}{4 a c}}{5 a}-\frac {4 x^2 \left (c-a^2 c x^2\right )^{3/2}}{5 a c}\right )-\frac {x^3 \left (c-a^2 c x^2\right )^{3/2}}{6 c}\right )\) |
| \(\Big \downarrow \) 455 |
| \(\displaystyle c \left (\frac {1}{2} \left (\frac {\frac {15 \int \sqrt {c-a^2 c x^2}dx-\frac {32 \left (c-a^2 c x^2\right )^{3/2}}{3 a c}}{4 a}-\frac {15 x \left (c-a^2 c x^2\right )^{3/2}}{4 a c}}{5 a}-\frac {4 x^2 \left (c-a^2 c x^2\right )^{3/2}}{5 a c}\right )-\frac {x^3 \left (c-a^2 c x^2\right )^{3/2}}{6 c}\right )\) |
| \(\Big \downarrow \) 211 |
| \(\displaystyle c \left (\frac {1}{2} \left (\frac {\frac {15 \left (\frac {1}{2} c \int \frac {1}{\sqrt {c-a^2 c x^2}}dx+\frac {1}{2} x \sqrt {c-a^2 c x^2}\right )-\frac {32 \left (c-a^2 c x^2\right )^{3/2}}{3 a c}}{4 a}-\frac {15 x \left (c-a^2 c x^2\right )^{3/2}}{4 a c}}{5 a}-\frac {4 x^2 \left (c-a^2 c x^2\right )^{3/2}}{5 a c}\right )-\frac {x^3 \left (c-a^2 c x^2\right )^{3/2}}{6 c}\right )\) |
| \(\Big \downarrow \) 224 |
| \(\displaystyle c \left (\frac {1}{2} \left (\frac {\frac {15 \left (\frac {1}{2} c \int \frac {1}{\frac {a^2 c x^2}{c-a^2 c x^2}+1}d\frac {x}{\sqrt {c-a^2 c x^2}}+\frac {1}{2} x \sqrt {c-a^2 c x^2}\right )-\frac {32 \left (c-a^2 c x^2\right )^{3/2}}{3 a c}}{4 a}-\frac {15 x \left (c-a^2 c x^2\right )^{3/2}}{4 a c}}{5 a}-\frac {4 x^2 \left (c-a^2 c x^2\right )^{3/2}}{5 a c}\right )-\frac {x^3 \left (c-a^2 c x^2\right )^{3/2}}{6 c}\right )\) |
| \(\Big \downarrow \) 216 |
| \(\displaystyle c \left (\frac {1}{2} \left (\frac {\frac {15 \left (\frac {\sqrt {c} \arctan \left (\frac {a \sqrt {c} x}{\sqrt {c-a^2 c x^2}}\right )}{2 a}+\frac {1}{2} x \sqrt {c-a^2 c x^2}\right )-\frac {32 \left (c-a^2 c x^2\right )^{3/2}}{3 a c}}{4 a}-\frac {15 x \left (c-a^2 c x^2\right )^{3/2}}{4 a c}}{5 a}-\frac {4 x^2 \left (c-a^2 c x^2\right )^{3/2}}{5 a c}\right )-\frac {x^3 \left (c-a^2 c x^2\right )^{3/2}}{6 c}\right )\) |
c*(-1/6*(x^3*(c - a^2*c*x^2)^(3/2))/c + ((-4*x^2*(c - a^2*c*x^2)^(3/2))/(5 *a*c) + ((-15*x*(c - a^2*c*x^2)^(3/2))/(4*a*c) + ((-32*(c - a^2*c*x^2)^(3/ 2))/(3*a*c) + 15*((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))/2)
3.11.87.3.1 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 = 0.24 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.80
| method | result | size |
| risch | \(-\frac {\left (40 a^{5} x^{5}+96 a^{4} x^{4}+50 a^{3} x^{3}-32 a^{2} x^{2}-45 a x -64\right ) \left (a^{2} x^{2}-1\right ) c^{2}}{240 a^{3} \sqrt {-c \left (a^{2} x^{2}-1\right )}}+\frac {3 \arctan \left (\frac {\sqrt {a^{2} c}\, x}{\sqrt {-a^{2} c \,x^{2}+c}}\right ) c^{2}}{16 a^{2} \sqrt {a^{2} c}}\) | \(109\) |
| default | \(\frac {x \left (-a^{2} c \,x^{2}+c \right )^{\frac {5}{2}}}{6 a^{2} c}-\frac {13 \left (\frac {x \left (-a^{2} c \,x^{2}+c \right )^{\frac {3}{2}}}{4}+\frac {3 c \left (\frac {x \sqrt {-a^{2} c \,x^{2}+c}}{2}+\frac {c \arctan \left (\frac {\sqrt {a^{2} c}\, x}{\sqrt {-a^{2} c \,x^{2}+c}}\right )}{2 \sqrt {a^{2} c}}\right )}{4}\right )}{6 a^{2}}+\frac {2 \left (-a^{2} c \,x^{2}+c \right )^{\frac {5}{2}}}{5 a^{3} c}-\frac {2 \left (\frac {\left (-a^{2} c \left (x -\frac {1}{a}\right )^{2}-2 \left (x -\frac {1}{a}\right ) a c \right )^{\frac {3}{2}}}{3}-a c \left (-\frac {\left (-2 a^{2} c \left (x -\frac {1}{a}\right )-2 a c \right ) \sqrt {-a^{2} c \left (x -\frac {1}{a}\right )^{2}-2 \left (x -\frac {1}{a}\right ) a c}}{4 a^{2} c}+\frac {c \arctan \left (\frac {\sqrt {a^{2} c}\, x}{\sqrt {-a^{2} c \left (x -\frac {1}{a}\right )^{2}-2 \left (x -\frac {1}{a}\right ) a c}}\right )}{2 \sqrt {a^{2} c}}\right )\right )}{a^{3}}\) | \(266\) |
-1/240*(40*a^5*x^5+96*a^4*x^4+50*a^3*x^3-32*a^2*x^2-45*a*x-64)*(a^2*x^2-1) /a^3/(-c*(a^2*x^2-1))^(1/2)*c^2+3/16/a^2/(a^2*c)^(1/2)*arctan((a^2*c)^(1/2 )*x/(-a^2*c*x^2+c)^(1/2))*c^2
Time = 0.29 (sec) , antiderivative size = 216, normalized size of antiderivative = 1.59 \[ \int e^{2 \text {arctanh}(a x)} x^2 \left (c-a^2 c x^2\right )^{3/2} \, dx=\left [\frac {45 \, \sqrt {-c} c \log \left (2 \, a^{2} c x^{2} + 2 \, \sqrt {-a^{2} c x^{2} + c} a \sqrt {-c} x - c\right ) + 2 \, {\left (40 \, a^{5} c x^{5} + 96 \, a^{4} c x^{4} + 50 \, a^{3} c x^{3} - 32 \, a^{2} c x^{2} - 45 \, a c x - 64 \, c\right )} \sqrt {-a^{2} c x^{2} + c}}{480 \, a^{3}}, -\frac {45 \, c^{\frac {3}{2}} \arctan \left (\frac {\sqrt {-a^{2} c x^{2} + c} a \sqrt {c} x}{a^{2} c x^{2} - c}\right ) - {\left (40 \, a^{5} c x^{5} + 96 \, a^{4} c x^{4} + 50 \, a^{3} c x^{3} - 32 \, a^{2} c x^{2} - 45 \, a c x - 64 \, c\right )} \sqrt {-a^{2} c x^{2} + c}}{240 \, a^{3}}\right ] \]
[1/480*(45*sqrt(-c)*c*log(2*a^2*c*x^2 + 2*sqrt(-a^2*c*x^2 + c)*a*sqrt(-c)* x - c) + 2*(40*a^5*c*x^5 + 96*a^4*c*x^4 + 50*a^3*c*x^3 - 32*a^2*c*x^2 - 45 *a*c*x - 64*c)*sqrt(-a^2*c*x^2 + c))/a^3, -1/240*(45*c^(3/2)*arctan(sqrt(- a^2*c*x^2 + c)*a*sqrt(c)*x/(a^2*c*x^2 - c)) - (40*a^5*c*x^5 + 96*a^4*c*x^4 + 50*a^3*c*x^3 - 32*a^2*c*x^2 - 45*a*c*x - 64*c)*sqrt(-a^2*c*x^2 + c))/a^ 3]
Time = 3.38 (sec) , antiderivative size = 287, normalized size of antiderivative = 2.11 \[ \int e^{2 \text {arctanh}(a x)} x^2 \left (c-a^2 c x^2\right )^{3/2} \, dx=a^{2} c \left (\begin {cases} \sqrt {- a^{2} c x^{2} + c} \left (\frac {x^{5}}{6} - \frac {x^{3}}{24 a^{2}} - \frac {x}{16 a^{4}}\right ) + \frac {c \left (\begin {cases} \frac {\log {\left (- 2 a^{2} c x + 2 \sqrt {- a^{2} c} \sqrt {- a^{2} c x^{2} + c} \right )}}{\sqrt {- a^{2} c}} & \text {for}\: c \neq 0 \\\frac {x \log {\left (x \right )}}{\sqrt {- a^{2} c x^{2}}} & \text {otherwise} \end {cases}\right )}{16 a^{4}} & \text {for}\: a^{2} c \neq 0 \\\frac {\sqrt {c} x^{5}}{5} & \text {otherwise} \end {cases}\right ) + 2 a c \left (\begin {cases} \sqrt {- a^{2} c x^{2} + c} \left (\frac {x^{4}}{5} - \frac {x^{2}}{15 a^{2}} - \frac {2}{15 a^{4}}\right ) & \text {for}\: a^{2} c \neq 0 \\\frac {\sqrt {c} x^{4}}{4} & \text {otherwise} \end {cases}\right ) + c \left (\begin {cases} \left (\frac {x^{3}}{4} - \frac {x}{8 a^{2}}\right ) \sqrt {- a^{2} c x^{2} + c} + \frac {c \left (\begin {cases} \frac {\log {\left (- 2 a^{2} c x + 2 \sqrt {- a^{2} c} \sqrt {- a^{2} c x^{2} + c} \right )}}{\sqrt {- a^{2} c}} & \text {for}\: c \neq 0 \\\frac {x \log {\left (x \right )}}{\sqrt {- a^{2} c x^{2}}} & \text {otherwise} \end {cases}\right )}{8 a^{2}} & \text {for}\: a^{2} c \neq 0 \\\frac {\sqrt {c} x^{3}}{3} & \text {otherwise} \end {cases}\right ) \]
a**2*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)) + 2*a*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)), (sq rt(c)*x**4/4, True)) + c*Piecewise(((x**3/4 - x/(8*a**2))*sqrt(-a**2*c*x** 2 + c) + 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))/(8*a* *2), Ne(a**2*c, 0)), (sqrt(c)*x**3/3, True))
Time = 0.40 (sec) , antiderivative size = 189, normalized size of antiderivative = 1.39 \[ \int e^{2 \text {arctanh}(a x)} x^2 \left (c-a^2 c x^2\right )^{3/2} \, dx=-\frac {1}{240} \, a {\left (\frac {130 \, {\left (-a^{2} c x^{2} + c\right )}^{\frac {3}{2}} x}{a^{3}} - \frac {40 \, {\left (-a^{2} c x^{2} + c\right )}^{\frac {5}{2}} x}{a^{3} c} - \frac {240 \, \sqrt {a^{2} c x^{2} - 4 \, a c x + 3 \, c} c x}{a^{3}} + \frac {195 \, \sqrt {-a^{2} c x^{2} + c} c x}{a^{3}} + \frac {195 \, c^{\frac {3}{2}} \arcsin \left (a x\right )}{a^{4}} + \frac {160 \, {\left (-a^{2} c x^{2} + c\right )}^{\frac {3}{2}}}{a^{4}} - \frac {96 \, {\left (-a^{2} c x^{2} + c\right )}^{\frac {5}{2}}}{a^{4} c} + \frac {480 \, \sqrt {a^{2} c x^{2} - 4 \, a c x + 3 \, c} c}{a^{4}} - \frac {240 \, c^{3} \arcsin \left (a x - 2\right )}{a^{7} \left (-\frac {c}{a^{2}}\right )^{\frac {3}{2}}}\right )} \]
-1/240*a*(130*(-a^2*c*x^2 + c)^(3/2)*x/a^3 - 40*(-a^2*c*x^2 + c)^(5/2)*x/( a^3*c) - 240*sqrt(a^2*c*x^2 - 4*a*c*x + 3*c)*c*x/a^3 + 195*sqrt(-a^2*c*x^2 + c)*c*x/a^3 + 195*c^(3/2)*arcsin(a*x)/a^4 + 160*(-a^2*c*x^2 + c)^(3/2)/a ^4 - 96*(-a^2*c*x^2 + c)^(5/2)/(a^4*c) + 480*sqrt(a^2*c*x^2 - 4*a*c*x + 3* c)*c/a^4 - 240*c^3*arcsin(a*x - 2)/(a^7*(-c/a^2)^(3/2)))
Time = 0.29 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.79 \[ \int e^{2 \text {arctanh}(a x)} x^2 \left (c-a^2 c x^2\right )^{3/2} \, dx=\frac {1}{240} \, \sqrt {-a^{2} c x^{2} + c} {\left ({\left (2 \, {\left ({\left (4 \, {\left (5 \, a^{2} c x + 12 \, a c\right )} x + 25 \, c\right )} x - \frac {16 \, c}{a}\right )} x - \frac {45 \, c}{a^{2}}\right )} x - \frac {64 \, c}{a^{3}}\right )} - \frac {3 \, c^{2} \log \left ({\left | -\sqrt {-a^{2} c} x + \sqrt {-a^{2} c x^{2} + c} \right |}\right )}{16 \, a^{2} \sqrt {-c} {\left | a \right |}} \]
1/240*sqrt(-a^2*c*x^2 + c)*((2*((4*(5*a^2*c*x + 12*a*c)*x + 25*c)*x - 16*c /a)*x - 45*c/a^2)*x - 64*c/a^3) - 3/16*c^2*log(abs(-sqrt(-a^2*c)*x + sqrt( -a^2*c*x^2 + c)))/(a^2*sqrt(-c)*abs(a))
Timed out. \[ \int e^{2 \text {arctanh}(a x)} x^2 \left (c-a^2 c x^2\right )^{3/2} \, dx=\int -\frac {x^2\,{\left (c-a^2\,c\,x^2\right )}^{3/2}\,{\left (a\,x+1\right )}^2}{a^2\,x^2-1} \,d x \]