Integrand size = 14, antiderivative size = 305 \[ \int e^{\frac {5}{2} \text {arctanh}(a x)} x^2 \, dx=\frac {55 (1-a x)^{3/4} \sqrt [4]{1+a x}}{8 a^3}+\frac {11 (1-a x)^{3/4} (1+a x)^{5/4}}{4 a^3}+\frac {2 (1+a x)^{9/4}}{a^3 \sqrt [4]{1-a x}}+\frac {(1-a x)^{3/4} (1+a x)^{9/4}}{3 a^3}-\frac {55 \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}\right )}{8 \sqrt {2} a^3}+\frac {55 \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}\right )}{8 \sqrt {2} a^3}+\frac {55 \log \left (1+\frac {\sqrt {1-a x}}{\sqrt {1+a x}}-\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}\right )}{16 \sqrt {2} a^3}-\frac {55 \log \left (1+\frac {\sqrt {1-a x}}{\sqrt {1+a x}}+\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}\right )}{16 \sqrt {2} a^3} \]
55/8*(-a*x+1)^(3/4)*(a*x+1)^(1/4)/a^3+11/4*(-a*x+1)^(3/4)*(a*x+1)^(5/4)/a^ 3+2*(a*x+1)^(9/4)/a^3/(-a*x+1)^(1/4)+1/3*(-a*x+1)^(3/4)*(a*x+1)^(9/4)/a^3+ 55/16*arctan(-1+(-a*x+1)^(1/4)*2^(1/2)/(a*x+1)^(1/4))/a^3*2^(1/2)+55/16*ar ctan(1+(-a*x+1)^(1/4)*2^(1/2)/(a*x+1)^(1/4))/a^3*2^(1/2)+55/32*ln(1-(-a*x+ 1)^(1/4)*2^(1/2)/(a*x+1)^(1/4)+(-a*x+1)^(1/2)/(a*x+1)^(1/2))/a^3*2^(1/2)-5 5/32*ln(1+(-a*x+1)^(1/4)*2^(1/2)/(a*x+1)^(1/4)+(-a*x+1)^(1/2)/(a*x+1)^(1/2 ))/a^3*2^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 0.03 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.22 \[ \int e^{\frac {5}{2} \text {arctanh}(a x)} x^2 \, dx=\frac {(7-a x) (1+a x)^{9/4}-44 \sqrt [4]{2} (-1+a x) \operatorname {Hypergeometric2F1}\left (-\frac {5}{4},\frac {3}{4},\frac {7}{4},\frac {1}{2} (1-a x)\right )}{3 a^3 \sqrt [4]{1-a x}} \]
((7 - a*x)*(1 + a*x)^(9/4) - 44*2^(1/4)*(-1 + a*x)*Hypergeometric2F1[-5/4, 3/4, 7/4, (1 - a*x)/2])/(3*a^3*(1 - a*x)^(1/4))
Time = 0.49 (sec) , antiderivative size = 298, normalized size of antiderivative = 0.98, number of steps used = 17, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.143, Rules used = {6676, 100, 27, 90, 60, 60, 73, 854, 826, 1476, 1082, 217, 1479, 25, 27, 1103}
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.
| \(\displaystyle \int x^2 e^{\frac {5}{2} \text {arctanh}(a x)} \, dx\) |
| \(\Big \downarrow \) 6676 |
| \(\displaystyle \int \frac {x^2 (a x+1)^{5/4}}{(1-a x)^{5/4}}dx\) |
| \(\Big \downarrow \) 100 |
| \(\displaystyle \frac {2 (a x+1)^{9/4}}{a^3 \sqrt [4]{1-a x}}-\frac {2 \int \frac {a (a x+1)^{5/4} (a x+5)}{2 \sqrt [4]{1-a x}}dx}{a^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 (a x+1)^{9/4}}{a^3 \sqrt [4]{1-a x}}-\frac {\int \frac {(a x+1)^{5/4} (a x+5)}{\sqrt [4]{1-a x}}dx}{a^2}\) |
| \(\Big \downarrow \) 90 |
| \(\displaystyle \frac {2 (a x+1)^{9/4}}{a^3 \sqrt [4]{1-a x}}-\frac {\frac {11}{2} \int \frac {(a x+1)^{5/4}}{\sqrt [4]{1-a x}}dx-\frac {(1-a x)^{3/4} (a x+1)^{9/4}}{3 a}}{a^2}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {2 (a x+1)^{9/4}}{a^3 \sqrt [4]{1-a x}}-\frac {\frac {11}{2} \left (\frac {5}{4} \int \frac {\sqrt [4]{a x+1}}{\sqrt [4]{1-a x}}dx-\frac {(1-a x)^{3/4} (a x+1)^{5/4}}{2 a}\right )-\frac {(1-a x)^{3/4} (a x+1)^{9/4}}{3 a}}{a^2}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {2 (a x+1)^{9/4}}{a^3 \sqrt [4]{1-a x}}-\frac {\frac {11}{2} \left (\frac {5}{4} \left (\frac {1}{2} \int \frac {1}{\sqrt [4]{1-a x} (a x+1)^{3/4}}dx-\frac {(1-a x)^{3/4} \sqrt [4]{a x+1}}{a}\right )-\frac {(1-a x)^{3/4} (a x+1)^{5/4}}{2 a}\right )-\frac {(1-a x)^{3/4} (a x+1)^{9/4}}{3 a}}{a^2}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {2 (a x+1)^{9/4}}{a^3 \sqrt [4]{1-a x}}-\frac {\frac {11}{2} \left (\frac {5}{4} \left (-\frac {2 \int \frac {\sqrt {1-a x}}{(a x+1)^{3/4}}d\sqrt [4]{1-a x}}{a}-\frac {(1-a x)^{3/4} \sqrt [4]{a x+1}}{a}\right )-\frac {(1-a x)^{3/4} (a x+1)^{5/4}}{2 a}\right )-\frac {(1-a x)^{3/4} (a x+1)^{9/4}}{3 a}}{a^2}\) |
| \(\Big \downarrow \) 854 |
| \(\displaystyle \frac {2 (a x+1)^{9/4}}{a^3 \sqrt [4]{1-a x}}-\frac {\frac {11}{2} \left (\frac {5}{4} \left (-\frac {2 \int \frac {\sqrt {1-a x}}{2-a x}d\frac {\sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}}{a}-\frac {(1-a x)^{3/4} \sqrt [4]{a x+1}}{a}\right )-\frac {(1-a x)^{3/4} (a x+1)^{5/4}}{2 a}\right )-\frac {(1-a x)^{3/4} (a x+1)^{9/4}}{3 a}}{a^2}\) |
| \(\Big \downarrow \) 826 |
| \(\displaystyle \frac {2 (a x+1)^{9/4}}{a^3 \sqrt [4]{1-a x}}-\frac {\frac {11}{2} \left (\frac {5}{4} \left (-\frac {2 \left (\frac {1}{2} \int \frac {\sqrt {1-a x}+1}{2-a x}d\frac {\sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}-\frac {1}{2} \int \frac {1-\sqrt {1-a x}}{2-a x}d\frac {\sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}\right )}{a}-\frac {(1-a x)^{3/4} \sqrt [4]{a x+1}}{a}\right )-\frac {(1-a x)^{3/4} (a x+1)^{5/4}}{2 a}\right )-\frac {(1-a x)^{3/4} (a x+1)^{9/4}}{3 a}}{a^2}\) |
| \(\Big \downarrow \) 1476 |
| \(\displaystyle \frac {2 (a x+1)^{9/4}}{a^3 \sqrt [4]{1-a x}}-\frac {\frac {11}{2} \left (\frac {5}{4} \left (-\frac {2 \left (\frac {1}{2} \left (\frac {1}{2} \int \frac {1}{\sqrt {1-a x}-\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+1}d\frac {\sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+\frac {1}{2} \int \frac {1}{\sqrt {1-a x}+\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+1}d\frac {\sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}\right )-\frac {1}{2} \int \frac {1-\sqrt {1-a x}}{2-a x}d\frac {\sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}\right )}{a}-\frac {(1-a x)^{3/4} \sqrt [4]{a x+1}}{a}\right )-\frac {(1-a x)^{3/4} (a x+1)^{5/4}}{2 a}\right )-\frac {(1-a x)^{3/4} (a x+1)^{9/4}}{3 a}}{a^2}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {2 (a x+1)^{9/4}}{a^3 \sqrt [4]{1-a x}}-\frac {\frac {11}{2} \left (\frac {5}{4} \left (-\frac {2 \left (\frac {1}{2} \left (\frac {\int \frac {1}{-\sqrt {1-a x}-1}d\left (1-\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}\right )}{\sqrt {2}}-\frac {\int \frac {1}{-\sqrt {1-a x}-1}d\left (\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+1\right )}{\sqrt {2}}\right )-\frac {1}{2} \int \frac {1-\sqrt {1-a x}}{2-a x}d\frac {\sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}\right )}{a}-\frac {(1-a x)^{3/4} \sqrt [4]{a x+1}}{a}\right )-\frac {(1-a x)^{3/4} (a x+1)^{5/4}}{2 a}\right )-\frac {(1-a x)^{3/4} (a x+1)^{9/4}}{3 a}}{a^2}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {2 (a x+1)^{9/4}}{a^3 \sqrt [4]{1-a x}}-\frac {\frac {11}{2} \left (\frac {5}{4} \left (-\frac {2 \left (\frac {1}{2} \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}\right )}{\sqrt {2}}\right )-\frac {1}{2} \int \frac {1-\sqrt {1-a x}}{2-a x}d\frac {\sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}\right )}{a}-\frac {(1-a x)^{3/4} \sqrt [4]{a x+1}}{a}\right )-\frac {(1-a x)^{3/4} (a x+1)^{5/4}}{2 a}\right )-\frac {(1-a x)^{3/4} (a x+1)^{9/4}}{3 a}}{a^2}\) |
| \(\Big \downarrow \) 1479 |
| \(\displaystyle \frac {2 (a x+1)^{9/4}}{a^3 \sqrt [4]{1-a x}}-\frac {\frac {11}{2} \left (\frac {5}{4} \left (-\frac {2 \left (\frac {1}{2} \left (\frac {\int -\frac {\sqrt {2}-\frac {2 \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}}{\sqrt {1-a x}-\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+1}d\frac {\sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}}{2 \sqrt {2}}+\frac {\int -\frac {\sqrt {2} \left (\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+1\right )}{\sqrt {1-a x}+\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+1}d\frac {\sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}}{2 \sqrt {2}}\right )+\frac {1}{2} \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}\right )}{\sqrt {2}}\right )\right )}{a}-\frac {(1-a x)^{3/4} \sqrt [4]{a x+1}}{a}\right )-\frac {(1-a x)^{3/4} (a x+1)^{5/4}}{2 a}\right )-\frac {(1-a x)^{3/4} (a x+1)^{9/4}}{3 a}}{a^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {2 (a x+1)^{9/4}}{a^3 \sqrt [4]{1-a x}}-\frac {\frac {11}{2} \left (\frac {5}{4} \left (-\frac {2 \left (\frac {1}{2} \left (-\frac {\int \frac {\sqrt {2}-\frac {2 \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}}{\sqrt {1-a x}-\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+1}d\frac {\sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}}{2 \sqrt {2}}-\frac {\int \frac {\sqrt {2} \left (\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+1\right )}{\sqrt {1-a x}+\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+1}d\frac {\sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}}{2 \sqrt {2}}\right )+\frac {1}{2} \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}\right )}{\sqrt {2}}\right )\right )}{a}-\frac {(1-a x)^{3/4} \sqrt [4]{a x+1}}{a}\right )-\frac {(1-a x)^{3/4} (a x+1)^{5/4}}{2 a}\right )-\frac {(1-a x)^{3/4} (a x+1)^{9/4}}{3 a}}{a^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 (a x+1)^{9/4}}{a^3 \sqrt [4]{1-a x}}-\frac {\frac {11}{2} \left (\frac {5}{4} \left (-\frac {2 \left (\frac {1}{2} \left (-\frac {\int \frac {\sqrt {2}-\frac {2 \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}}{\sqrt {1-a x}-\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+1}d\frac {\sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}}{2 \sqrt {2}}-\frac {1}{2} \int \frac {\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+1}{\sqrt {1-a x}+\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+1}d\frac {\sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}\right )+\frac {1}{2} \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}\right )}{\sqrt {2}}\right )\right )}{a}-\frac {(1-a x)^{3/4} \sqrt [4]{a x+1}}{a}\right )-\frac {(1-a x)^{3/4} (a x+1)^{5/4}}{2 a}\right )-\frac {(1-a x)^{3/4} (a x+1)^{9/4}}{3 a}}{a^2}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {2 (a x+1)^{9/4}}{a^3 \sqrt [4]{1-a x}}-\frac {\frac {11}{2} \left (\frac {5}{4} \left (-\frac {2 \left (\frac {1}{2} \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}\right )}{\sqrt {2}}\right )+\frac {1}{2} \left (\frac {\log \left (\sqrt {1-a x}-\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+1\right )}{2 \sqrt {2}}-\frac {\log \left (\sqrt {1-a x}+\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+1\right )}{2 \sqrt {2}}\right )\right )}{a}-\frac {(1-a x)^{3/4} \sqrt [4]{a x+1}}{a}\right )-\frac {(1-a x)^{3/4} (a x+1)^{5/4}}{2 a}\right )-\frac {(1-a x)^{3/4} (a x+1)^{9/4}}{3 a}}{a^2}\) |
(2*(1 + a*x)^(9/4))/(a^3*(1 - a*x)^(1/4)) - (-1/3*((1 - a*x)^(3/4)*(1 + a* x)^(9/4))/a + (11*(-1/2*((1 - a*x)^(3/4)*(1 + a*x)^(5/4))/a + (5*(-(((1 - a*x)^(3/4)*(1 + a*x)^(1/4))/a) - (2*((-(ArcTan[1 - (Sqrt[2]*(1 - a*x)^(1/4 ))/(1 + a*x)^(1/4)]/Sqrt[2]) + ArcTan[1 + (Sqrt[2]*(1 - a*x)^(1/4))/(1 + a *x)^(1/4)]/Sqrt[2])/2 + (Log[1 + Sqrt[1 - a*x] - (Sqrt[2]*(1 - a*x)^(1/4)) /(1 + a*x)^(1/4)]/(2*Sqrt[2]) - Log[1 + Sqrt[1 - a*x] + (Sqrt[2]*(1 - a*x) ^(1/4))/(1 + a*x)^(1/4)]/(2*Sqrt[2]))/2))/a))/4))/2)/a^2
3.1.82.3.1 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_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^n/(b*(m + n + 1))), x] + Simp[n*((b*c - a*d)/( b*(m + n + 1))) Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d}, x] && GtQ[n, 0] && NeQ[m + n + 1, 0] && !(IGtQ[m, 0] && ( !Integer Q[n] || (GtQ[m, 0] && LtQ[m - n, 0]))) && !ILtQ[m + n + 2, 0] && IntLinear Q[a, b, c, d, m, n, x]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ {p = Denominator[m]}, Simp[p/b Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL inearQ[a, b, c, d, m, n, x]
Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p _.), x_] :> Simp[b*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d*f*(n + p + 2))), x] + Simp[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(d*f*(n + p + 2)) Int[(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 2, 0]
Int[((a_.) + (b_.)*(x_))^2*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^( p_), x_] :> Simp[(b*c - a*d)^2*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d^2*(d *e - c*f)*(n + 1))), x] - Simp[1/(d^2*(d*e - c*f)*(n + 1)) Int[(c + d*x)^ (n + 1)*(e + f*x)^p*Simp[a^2*d^2*f*(n + p + 2) + b^2*c*(d*e*(n + 1) + c*f*( p + 1)) - 2*a*b*d*(d*e*(n + 1) + c*f*(p + 1)) - b^2*d*(d*e - c*f)*(n + 1)*x , x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && (LtQ[n, -1] || (EqQ[n + p + 3, 0] && NeQ[n, -1] && (SumSimplerQ[n, 1] || !SumSimplerQ[p, 1])))
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^( -1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & & (LtQ[a, 0] || LtQ[b, 0])
Int[(x_)^2/((a_) + (b_.)*(x_)^4), x_Symbol] :> With[{r = Numerator[Rt[a/b, 2]], s = Denominator[Rt[a/b, 2]]}, Simp[1/(2*s) Int[(r + s*x^2)/(a + b*x^ 4), x], x] - Simp[1/(2*s) Int[(r - s*x^2)/(a + b*x^4), x], x]] /; FreeQ[{ a, b}, x] && (GtQ[a/b, 0] || (PosQ[a/b] && AtomQ[SplitProduct[SumBaseQ, a]] && AtomQ[SplitProduct[SumBaseQ, b]]))
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^(p + (m + 1)/n) Subst[Int[x^m/(1 - b*x^n)^(p + (m + 1)/n + 1), x], x, x/(a + b*x^n )^(1/n)], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && LtQ[-1, p, 0] && NeQ[p, - 2^(-1)] && IntegersQ[m, p + (m + 1)/n]
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*S implify[a*(c/b^2)]}, Simp[-2/b Subst[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b )], x] /; RationalQ[q] && (EqQ[q^2, 1] || !RationalQ[b^2 - 4*a*c])] /; Fre eQ[{a, b, c}, x]
Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[d*(Log[RemoveContent[a + b*x + c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]
Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ 2*(d/e), 2]}, Simp[e/(2*c) Int[1/Simp[d/e + q*x + x^2, x], x], x] + Simp[ e/(2*c) Int[1/Simp[d/e - q*x + x^2, x], x], x]] /; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && PosQ[d*e]
Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ -2*(d/e), 2]}, Simp[e/(2*c*q) Int[(q - 2*x)/Simp[d/e + q*x - x^2, x], x], x] + Simp[e/(2*c*q) Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /; F reeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]
Int[E^(ArcTanh[(a_.)*(x_)]*(n_))*((c_.)*(x_))^(m_.), x_Symbol] :> Int[(c*x) ^m*((1 + a*x)^(n/2)/(1 - a*x)^(n/2)), x] /; FreeQ[{a, c, m, n}, x] && !Int egerQ[(n - 1)/2]
\[\int {\left (\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )}^{\frac {5}{2}} x^{2}d x\]
Result contains complex when optimal does not.
Time = 0.28 (sec) , antiderivative size = 248, normalized size of antiderivative = 0.81 \[ \int e^{\frac {5}{2} \text {arctanh}(a x)} x^2 \, dx=-\frac {165 \, a^{3} \left (-\frac {1}{a^{12}}\right )^{\frac {1}{4}} \log \left (a^{3} \left (-\frac {1}{a^{12}}\right )^{\frac {1}{4}} + \sqrt {-\frac {\sqrt {-a^{2} x^{2} + 1}}{a x - 1}}\right ) + 165 i \, a^{3} \left (-\frac {1}{a^{12}}\right )^{\frac {1}{4}} \log \left (i \, a^{3} \left (-\frac {1}{a^{12}}\right )^{\frac {1}{4}} + \sqrt {-\frac {\sqrt {-a^{2} x^{2} + 1}}{a x - 1}}\right ) - 165 i \, a^{3} \left (-\frac {1}{a^{12}}\right )^{\frac {1}{4}} \log \left (-i \, a^{3} \left (-\frac {1}{a^{12}}\right )^{\frac {1}{4}} + \sqrt {-\frac {\sqrt {-a^{2} x^{2} + 1}}{a x - 1}}\right ) - 165 \, a^{3} \left (-\frac {1}{a^{12}}\right )^{\frac {1}{4}} \log \left (-a^{3} \left (-\frac {1}{a^{12}}\right )^{\frac {1}{4}} + \sqrt {-\frac {\sqrt {-a^{2} x^{2} + 1}}{a x - 1}}\right ) + 2 \, {\left (8 \, a^{3} x^{3} + 26 \, a^{2} x^{2} + 61 \, a x - 287\right )} \sqrt {-\frac {\sqrt {-a^{2} x^{2} + 1}}{a x - 1}}}{48 \, a^{3}} \]
-1/48*(165*a^3*(-1/a^12)^(1/4)*log(a^3*(-1/a^12)^(1/4) + sqrt(-sqrt(-a^2*x ^2 + 1)/(a*x - 1))) + 165*I*a^3*(-1/a^12)^(1/4)*log(I*a^3*(-1/a^12)^(1/4) + sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1))) - 165*I*a^3*(-1/a^12)^(1/4)*log(-I* a^3*(-1/a^12)^(1/4) + sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1))) - 165*a^3*(-1/a ^12)^(1/4)*log(-a^3*(-1/a^12)^(1/4) + sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1))) + 2*(8*a^3*x^3 + 26*a^2*x^2 + 61*a*x - 287)*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1)))/a^3
Timed out. \[ \int e^{\frac {5}{2} \text {arctanh}(a x)} x^2 \, dx=\text {Timed out} \]
\[ \int e^{\frac {5}{2} \text {arctanh}(a x)} x^2 \, dx=\int { x^{2} \left (\frac {a x + 1}{\sqrt {-a^{2} x^{2} + 1}}\right )^{\frac {5}{2}} \,d x } \]
Exception generated. \[ \int e^{\frac {5}{2} \text {arctanh}(a x)} x^2 \, dx=\text {Exception raised: TypeError} \]
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. \[ \int e^{\frac {5}{2} \text {arctanh}(a x)} x^2 \, dx=\int x^2\,{\left (\frac {a\,x+1}{\sqrt {1-a^2\,x^2}}\right )}^{5/2} \,d x \]