Integrand size = 20, antiderivative size = 266 \[ \int \frac {a+b \text {sech}^{-1}(c x)}{\left (d+e x^2\right )^{5/2}} \, dx=\frac {b e x \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2}}{3 d^2 \left (c^2 d+e\right ) \sqrt {d+e x^2}}+\frac {x \left (a+b \text {sech}^{-1}(c x)\right )}{3 d \left (d+e x^2\right )^{3/2}}+\frac {2 x \left (a+b \text {sech}^{-1}(c x)\right )}{3 d^2 \sqrt {d+e x^2}}+\frac {b c \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {d+e x^2} E\left (\arcsin (c x)\left |-\frac {e}{c^2 d}\right .\right )}{3 d^2 \left (c^2 d+e\right ) \sqrt {1+\frac {e x^2}{d}}}+\frac {2 b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1+\frac {e x^2}{d}} \operatorname {EllipticF}\left (\arcsin (c x),-\frac {e}{c^2 d}\right )}{3 c d^2 \sqrt {d+e x^2}} \]
1/3*x*(a+b*arcsech(c*x))/d/(e*x^2+d)^(3/2)+2/3*x*(a+b*arcsech(c*x))/d^2/(e *x^2+d)^(1/2)+1/3*b*e*x*(1/(c*x+1))^(1/2)*(c*x+1)^(1/2)*(-c^2*x^2+1)^(1/2) /d^2/(c^2*d+e)/(e*x^2+d)^(1/2)+1/3*b*c*EllipticE(c*x,(-e/c^2/d)^(1/2))*(1/ (c*x+1))^(1/2)*(c*x+1)^(1/2)*(e*x^2+d)^(1/2)/d^2/(c^2*d+e)/(1+e*x^2/d)^(1/ 2)+2/3*b*EllipticF(c*x,(-e/c^2/d)^(1/2))*(1/(c*x+1))^(1/2)*(c*x+1)^(1/2)*( 1+e*x^2/d)^(1/2)/c/d^2/(e*x^2+d)^(1/2)
Result contains complex when optimal does not.
Time = 26.79 (sec) , antiderivative size = 517, normalized size of antiderivative = 1.94 \[ \int \frac {a+b \text {sech}^{-1}(c x)}{\left (d+e x^2\right )^{5/2}} \, dx=\frac {\frac {b \sqrt {\frac {1-c x}{1+c x}} (-c d+e x) \left (d+e x^2\right )}{c^2 d+e}+a x \left (3 d+2 e x^2\right )+b x \left (3 d+2 e x^2\right ) \text {sech}^{-1}(c x)-\frac {i b \sqrt {\frac {1-c x}{1+c x}} (1+c x) \sqrt {\frac {c \left (\sqrt {d}-i \sqrt {e} x\right )}{\left (c \sqrt {d}-i \sqrt {e}\right ) (1+c x)}} \sqrt {\frac {c \left (\sqrt {d}+i \sqrt {e} x\right )}{\left (c \sqrt {d}+i \sqrt {e}\right ) (1+c x)}} \left (d+e x^2\right ) \left (\left (c \sqrt {d}-i \sqrt {e}\right ) E\left (i \text {arcsinh}\left (\sqrt {\frac {\left (c^2 d+e\right ) (1-c x)}{\left (c \sqrt {d}+i \sqrt {e}\right )^2 (1+c x)}}\right )|\frac {\left (c \sqrt {d}+i \sqrt {e}\right )^2}{\left (c \sqrt {d}-i \sqrt {e}\right )^2}\right )-2 \left (3 c \sqrt {d}+2 i \sqrt {e}\right ) \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {\left (c^2 d+e\right ) (1-c x)}{\left (c \sqrt {d}+i \sqrt {e}\right )^2 (1+c x)}}\right ),\frac {\left (c \sqrt {d}+i \sqrt {e}\right )^2}{\left (c \sqrt {d}-i \sqrt {e}\right )^2}\right )\right )}{c \left (c \sqrt {d}+i \sqrt {e}\right ) \sqrt {-\frac {\left (c \sqrt {d}-i \sqrt {e}\right ) (-1+c x)}{\left (c \sqrt {d}+i \sqrt {e}\right ) (1+c x)}}}}{3 d^2 \left (d+e x^2\right )^{3/2}} \]
((b*Sqrt[(1 - c*x)/(1 + c*x)]*(-(c*d) + e*x)*(d + e*x^2))/(c^2*d + e) + a* x*(3*d + 2*e*x^2) + b*x*(3*d + 2*e*x^2)*ArcSech[c*x] - (I*b*Sqrt[(1 - c*x) /(1 + c*x)]*(1 + c*x)*Sqrt[(c*(Sqrt[d] - I*Sqrt[e]*x))/((c*Sqrt[d] - I*Sqr t[e])*(1 + c*x))]*Sqrt[(c*(Sqrt[d] + I*Sqrt[e]*x))/((c*Sqrt[d] + I*Sqrt[e] )*(1 + c*x))]*(d + e*x^2)*((c*Sqrt[d] - I*Sqrt[e])*EllipticE[I*ArcSinh[Sqr t[((c^2*d + e)*(1 - c*x))/((c*Sqrt[d] + I*Sqrt[e])^2*(1 + c*x))]], (c*Sqrt [d] + I*Sqrt[e])^2/(c*Sqrt[d] - I*Sqrt[e])^2] - 2*(3*c*Sqrt[d] + (2*I)*Sqr t[e])*EllipticF[I*ArcSinh[Sqrt[((c^2*d + e)*(1 - c*x))/((c*Sqrt[d] + I*Sqr t[e])^2*(1 + c*x))]], (c*Sqrt[d] + I*Sqrt[e])^2/(c*Sqrt[d] - I*Sqrt[e])^2] ))/(c*(c*Sqrt[d] + I*Sqrt[e])*Sqrt[-(((c*Sqrt[d] - I*Sqrt[e])*(-1 + c*x))/ ((c*Sqrt[d] + I*Sqrt[e])*(1 + c*x)))]))/(3*d^2*(d + e*x^2)^(3/2))
Time = 0.46 (sec) , antiderivative size = 224, normalized size of antiderivative = 0.84, number of steps used = 10, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6845, 27, 402, 25, 27, 399, 323, 321, 330, 327}
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.
| \(\displaystyle \int \frac {a+b \text {sech}^{-1}(c x)}{\left (d+e x^2\right )^{5/2}} \, dx\) |
| \(\Big \downarrow \) 6845 |
| \(\displaystyle b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \int \frac {2 e x^2+3 d}{3 d^2 \sqrt {1-c^2 x^2} \left (e x^2+d\right )^{3/2}}dx+\frac {2 x \left (a+b \text {sech}^{-1}(c x)\right )}{3 d^2 \sqrt {d+e x^2}}+\frac {x \left (a+b \text {sech}^{-1}(c x)\right )}{3 d \left (d+e x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \int \frac {2 e x^2+3 d}{\sqrt {1-c^2 x^2} \left (e x^2+d\right )^{3/2}}dx}{3 d^2}+\frac {2 x \left (a+b \text {sech}^{-1}(c x)\right )}{3 d^2 \sqrt {d+e x^2}}+\frac {x \left (a+b \text {sech}^{-1}(c x)\right )}{3 d \left (d+e x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 402 |
| \(\displaystyle \frac {b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \left (\frac {e x \sqrt {1-c^2 x^2}}{\left (c^2 d+e\right ) \sqrt {d+e x^2}}-\frac {\int -\frac {d \left (e x^2 c^2+3 d c^2+2 e\right )}{\sqrt {1-c^2 x^2} \sqrt {e x^2+d}}dx}{d \left (c^2 d+e\right )}\right )}{3 d^2}+\frac {2 x \left (a+b \text {sech}^{-1}(c x)\right )}{3 d^2 \sqrt {d+e x^2}}+\frac {x \left (a+b \text {sech}^{-1}(c x)\right )}{3 d \left (d+e x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \left (\frac {\int \frac {d \left (e x^2 c^2+3 d c^2+2 e\right )}{\sqrt {1-c^2 x^2} \sqrt {e x^2+d}}dx}{d \left (c^2 d+e\right )}+\frac {e x \sqrt {1-c^2 x^2}}{\left (c^2 d+e\right ) \sqrt {d+e x^2}}\right )}{3 d^2}+\frac {2 x \left (a+b \text {sech}^{-1}(c x)\right )}{3 d^2 \sqrt {d+e x^2}}+\frac {x \left (a+b \text {sech}^{-1}(c x)\right )}{3 d \left (d+e x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \left (\frac {\int \frac {e x^2 c^2+3 d c^2+2 e}{\sqrt {1-c^2 x^2} \sqrt {e x^2+d}}dx}{c^2 d+e}+\frac {e x \sqrt {1-c^2 x^2}}{\left (c^2 d+e\right ) \sqrt {d+e x^2}}\right )}{3 d^2}+\frac {2 x \left (a+b \text {sech}^{-1}(c x)\right )}{3 d^2 \sqrt {d+e x^2}}+\frac {x \left (a+b \text {sech}^{-1}(c x)\right )}{3 d \left (d+e x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 399 |
| \(\displaystyle \frac {b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \left (\frac {c^2 \int \frac {\sqrt {e x^2+d}}{\sqrt {1-c^2 x^2}}dx+2 \left (c^2 d+e\right ) \int \frac {1}{\sqrt {1-c^2 x^2} \sqrt {e x^2+d}}dx}{c^2 d+e}+\frac {e x \sqrt {1-c^2 x^2}}{\left (c^2 d+e\right ) \sqrt {d+e x^2}}\right )}{3 d^2}+\frac {2 x \left (a+b \text {sech}^{-1}(c x)\right )}{3 d^2 \sqrt {d+e x^2}}+\frac {x \left (a+b \text {sech}^{-1}(c x)\right )}{3 d \left (d+e x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 323 |
| \(\displaystyle \frac {b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \left (\frac {c^2 \int \frac {\sqrt {e x^2+d}}{\sqrt {1-c^2 x^2}}dx+\frac {2 \left (c^2 d+e\right ) \sqrt {\frac {e x^2}{d}+1} \int \frac {1}{\sqrt {1-c^2 x^2} \sqrt {\frac {e x^2}{d}+1}}dx}{\sqrt {d+e x^2}}}{c^2 d+e}+\frac {e x \sqrt {1-c^2 x^2}}{\left (c^2 d+e\right ) \sqrt {d+e x^2}}\right )}{3 d^2}+\frac {2 x \left (a+b \text {sech}^{-1}(c x)\right )}{3 d^2 \sqrt {d+e x^2}}+\frac {x \left (a+b \text {sech}^{-1}(c x)\right )}{3 d \left (d+e x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 321 |
| \(\displaystyle \frac {b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \left (\frac {c^2 \int \frac {\sqrt {e x^2+d}}{\sqrt {1-c^2 x^2}}dx+\frac {2 \left (c^2 d+e\right ) \sqrt {\frac {e x^2}{d}+1} \operatorname {EllipticF}\left (\arcsin (c x),-\frac {e}{c^2 d}\right )}{c \sqrt {d+e x^2}}}{c^2 d+e}+\frac {e x \sqrt {1-c^2 x^2}}{\left (c^2 d+e\right ) \sqrt {d+e x^2}}\right )}{3 d^2}+\frac {2 x \left (a+b \text {sech}^{-1}(c x)\right )}{3 d^2 \sqrt {d+e x^2}}+\frac {x \left (a+b \text {sech}^{-1}(c x)\right )}{3 d \left (d+e x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 330 |
| \(\displaystyle \frac {b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \left (\frac {\frac {c^2 \sqrt {d+e x^2} \int \frac {\sqrt {\frac {e x^2}{d}+1}}{\sqrt {1-c^2 x^2}}dx}{\sqrt {\frac {e x^2}{d}+1}}+\frac {2 \left (c^2 d+e\right ) \sqrt {\frac {e x^2}{d}+1} \operatorname {EllipticF}\left (\arcsin (c x),-\frac {e}{c^2 d}\right )}{c \sqrt {d+e x^2}}}{c^2 d+e}+\frac {e x \sqrt {1-c^2 x^2}}{\left (c^2 d+e\right ) \sqrt {d+e x^2}}\right )}{3 d^2}+\frac {2 x \left (a+b \text {sech}^{-1}(c x)\right )}{3 d^2 \sqrt {d+e x^2}}+\frac {x \left (a+b \text {sech}^{-1}(c x)\right )}{3 d \left (d+e x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 327 |
| \(\displaystyle \frac {2 x \left (a+b \text {sech}^{-1}(c x)\right )}{3 d^2 \sqrt {d+e x^2}}+\frac {x \left (a+b \text {sech}^{-1}(c x)\right )}{3 d \left (d+e x^2\right )^{3/2}}+\frac {b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \left (\frac {\frac {2 \left (c^2 d+e\right ) \sqrt {\frac {e x^2}{d}+1} \operatorname {EllipticF}\left (\arcsin (c x),-\frac {e}{c^2 d}\right )}{c \sqrt {d+e x^2}}+\frac {c \sqrt {d+e x^2} E\left (\arcsin (c x)\left |-\frac {e}{c^2 d}\right .\right )}{\sqrt {\frac {e x^2}{d}+1}}}{c^2 d+e}+\frac {e x \sqrt {1-c^2 x^2}}{\left (c^2 d+e\right ) \sqrt {d+e x^2}}\right )}{3 d^2}\) |
(x*(a + b*ArcSech[c*x]))/(3*d*(d + e*x^2)^(3/2)) + (2*x*(a + b*ArcSech[c*x ]))/(3*d^2*Sqrt[d + e*x^2]) + (b*Sqrt[(1 + c*x)^(-1)]*Sqrt[1 + c*x]*((e*x* Sqrt[1 - c^2*x^2])/((c^2*d + e)*Sqrt[d + e*x^2]) + ((c*Sqrt[d + e*x^2]*Ell ipticE[ArcSin[c*x], -(e/(c^2*d))])/Sqrt[1 + (e*x^2)/d] + (2*(c^2*d + e)*Sq rt[1 + (e*x^2)/d]*EllipticF[ArcSin[c*x], -(e/(c^2*d))])/(c*Sqrt[d + e*x^2] ))/(c^2*d + e)))/(3*d^2)
3.2.76.3.1 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[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> S imp[(1/(Sqrt[a]*Sqrt[c]*Rt[-d/c, 2]))*EllipticF[ArcSin[Rt[-d/c, 2]*x], b*(c /(a*d))], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 0] && !(NegQ[b/a] && SimplerSqrtQ[-b/a, -d/c])
Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> S imp[Sqrt[1 + (d/c)*x^2]/Sqrt[c + d*x^2] Int[1/(Sqrt[a + b*x^2]*Sqrt[1 + ( d/c)*x^2]), x], x] /; FreeQ[{a, b, c, d}, x] && !GtQ[c, 0]
Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[ (Sqrt[a]/(Sqrt[c]*Rt[-d/c, 2]))*EllipticE[ArcSin[Rt[-d/c, 2]*x], b*(c/(a*d) )], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 0]
Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[ Sqrt[a + b*x^2]/Sqrt[1 + (b/a)*x^2] Int[Sqrt[1 + (b/a)*x^2]/Sqrt[c + d*x^ 2], x], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && !GtQ[a, 0]
Int[((e_) + (f_.)*(x_)^2)/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_) ^2]), x_Symbol] :> Simp[f/b Int[Sqrt[a + b*x^2]/Sqrt[c + d*x^2], x], x] + Simp[(b*e - a*f)/b Int[1/(Sqrt[a + b*x^2]*Sqrt[c + d*x^2]), x], x] /; Fr eeQ[{a, b, c, d, e, f}, x] && !((PosQ[b/a] && PosQ[d/c]) || (NegQ[b/a] && (PosQ[d/c] || (GtQ[a, 0] && ( !GtQ[c, 0] || SimplerSqrtQ[-b/a, -d/c])))))
Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_.)*((e_) + (f_.)*(x _)^2), x_Symbol] :> Simp[(-(b*e - a*f))*x*(a + b*x^2)^(p + 1)*((c + d*x^2)^ (q + 1)/(a*2*(b*c - a*d)*(p + 1))), x] + Simp[1/(a*2*(b*c - a*d)*(p + 1)) Int[(a + b*x^2)^(p + 1)*(c + d*x^2)^q*Simp[c*(b*e - a*f) + e*2*(b*c - a*d) *(p + 1) + d*(b*e - a*f)*(2*(p + q + 2) + 1)*x^2, x], x], x] /; FreeQ[{a, b , c, d, e, f, q}, x] && LtQ[p, -1]
Int[((a_.) + ArcSech[(c_.)*(x_)]*(b_.))*((d_.) + (e_.)*(x_)^2)^(p_.), x_Sym bol] :> With[{u = IntHide[(d + e*x^2)^p, x]}, Simp[(a + b*ArcSech[c*x]) u , x] + Simp[b*Sqrt[1 + c*x]*Sqrt[1/(1 + c*x)] Int[SimplifyIntegrand[u/(x* Sqrt[1 - c*x]*Sqrt[1 + c*x]), x], x], x]] /; FreeQ[{a, b, c, d, e}, x] && ( IGtQ[p, 0] || ILtQ[p + 1/2, 0])
\[\int \frac {a +b \,\operatorname {arcsech}\left (c x \right )}{\left (e \,x^{2}+d \right )^{\frac {5}{2}}}d x\]
Leaf count of result is larger than twice the leaf count of optimal. 397 vs. \(2 (182) = 364\).
Time = 0.11 (sec) , antiderivative size = 397, normalized size of antiderivative = 1.49 \[ \int \frac {a+b \text {sech}^{-1}(c x)}{\left (d+e x^2\right )^{5/2}} \, dx=\frac {{\left (2 \, {\left (b c^{3} d^{2} e + b c d e^{2}\right )} x^{3} + 3 \, {\left (b c^{3} d^{3} + b c d^{2} e\right )} x\right )} \sqrt {e x^{2} + d} \log \left (\frac {c x \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}} + 1}{c x}\right ) + {\left (2 \, {\left (a c^{3} d^{2} e + a c d e^{2}\right )} x^{3} + 3 \, {\left (a c^{3} d^{3} + a c d^{2} e\right )} x + {\left (b c^{2} d e^{2} x^{4} + b c^{2} d^{2} e x^{2}\right )} \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}}\right )} \sqrt {e x^{2} + d} + {\left ({\left (b c^{4} d e^{2} x^{4} + 2 \, b c^{4} d^{2} e x^{2} + b c^{4} d^{3}\right )} E(\arcsin \left (c x\right )\,|\,-\frac {e}{c^{2} d}) - {\left ({\left ({\left (b c^{4} - 3 \, b c^{2}\right )} d e^{2} - 2 \, b e^{3}\right )} x^{4} + {\left (b c^{4} - 3 \, b c^{2}\right )} d^{3} - 2 \, b d^{2} e + 2 \, {\left ({\left (b c^{4} - 3 \, b c^{2}\right )} d^{2} e - 2 \, b d e^{2}\right )} x^{2}\right )} F(\arcsin \left (c x\right )\,|\,-\frac {e}{c^{2} d})\right )} \sqrt {d}}{3 \, {\left (c^{3} d^{6} + c d^{5} e + {\left (c^{3} d^{4} e^{2} + c d^{3} e^{3}\right )} x^{4} + 2 \, {\left (c^{3} d^{5} e + c d^{4} e^{2}\right )} x^{2}\right )}} \]
1/3*((2*(b*c^3*d^2*e + b*c*d*e^2)*x^3 + 3*(b*c^3*d^3 + b*c*d^2*e)*x)*sqrt( e*x^2 + d)*log((c*x*sqrt(-(c^2*x^2 - 1)/(c^2*x^2)) + 1)/(c*x)) + (2*(a*c^3 *d^2*e + a*c*d*e^2)*x^3 + 3*(a*c^3*d^3 + a*c*d^2*e)*x + (b*c^2*d*e^2*x^4 + b*c^2*d^2*e*x^2)*sqrt(-(c^2*x^2 - 1)/(c^2*x^2)))*sqrt(e*x^2 + d) + ((b*c^ 4*d*e^2*x^4 + 2*b*c^4*d^2*e*x^2 + b*c^4*d^3)*elliptic_e(arcsin(c*x), -e/(c ^2*d)) - (((b*c^4 - 3*b*c^2)*d*e^2 - 2*b*e^3)*x^4 + (b*c^4 - 3*b*c^2)*d^3 - 2*b*d^2*e + 2*((b*c^4 - 3*b*c^2)*d^2*e - 2*b*d*e^2)*x^2)*elliptic_f(arcs in(c*x), -e/(c^2*d)))*sqrt(d))/(c^3*d^6 + c*d^5*e + (c^3*d^4*e^2 + c*d^3*e ^3)*x^4 + 2*(c^3*d^5*e + c*d^4*e^2)*x^2)
Timed out. \[ \int \frac {a+b \text {sech}^{-1}(c x)}{\left (d+e x^2\right )^{5/2}} \, dx=\text {Timed out} \]
\[ \int \frac {a+b \text {sech}^{-1}(c x)}{\left (d+e x^2\right )^{5/2}} \, dx=\int { \frac {b \operatorname {arsech}\left (c x\right ) + a}{{\left (e x^{2} + d\right )}^{\frac {5}{2}}} \,d x } \]
1/3*a*(2*x/(sqrt(e*x^2 + d)*d^2) + x/((e*x^2 + d)^(3/2)*d)) + b*integrate( log(sqrt(1/(c*x) + 1)*sqrt(1/(c*x) - 1) + 1/(c*x))/(e*x^2 + d)^(5/2), x)
\[ \int \frac {a+b \text {sech}^{-1}(c x)}{\left (d+e x^2\right )^{5/2}} \, dx=\int { \frac {b \operatorname {arsech}\left (c x\right ) + a}{{\left (e x^{2} + d\right )}^{\frac {5}{2}}} \,d x } \]
Timed out. \[ \int \frac {a+b \text {sech}^{-1}(c x)}{\left (d+e x^2\right )^{5/2}} \, dx=\int \frac {a+b\,\mathrm {acosh}\left (\frac {1}{c\,x}\right )}{{\left (e\,x^2+d\right )}^{5/2}} \,d x \]