Integrand size = 23, antiderivative size = 356 \[ \int \frac {x^5 \left (a+b \text {sech}^{-1}(c x)\right )}{\sqrt {d+e x^2}} \, dx=\frac {b \left (19 c^2 d-9 e\right ) \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2} \sqrt {d+e x^2}}{120 c^4 e^2}-\frac {b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2} \left (d+e x^2\right )^{3/2}}{20 c^2 e^2}+\frac {d^2 \sqrt {d+e x^2} \left (a+b \text {sech}^{-1}(c x)\right )}{e^3}-\frac {2 d \left (d+e x^2\right )^{3/2} \left (a+b \text {sech}^{-1}(c x)\right )}{3 e^3}+\frac {\left (d+e x^2\right )^{5/2} \left (a+b \text {sech}^{-1}(c x)\right )}{5 e^3}-\frac {b \left (45 c^4 d^2-10 c^2 d e+9 e^2\right ) \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \arctan \left (\frac {\sqrt {e} \sqrt {1-c^2 x^2}}{c \sqrt {d+e x^2}}\right )}{120 c^5 e^{5/2}}-\frac {8 b d^{5/2} \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \text {arctanh}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d} \sqrt {1-c^2 x^2}}\right )}{15 e^3} \]
-2/3*d*(e*x^2+d)^(3/2)*(a+b*arcsech(c*x))/e^3+1/5*(e*x^2+d)^(5/2)*(a+b*arc sech(c*x))/e^3-1/120*b*(45*c^4*d^2-10*c^2*d*e+9*e^2)*arctan(e^(1/2)*(-c^2* x^2+1)^(1/2)/c/(e*x^2+d)^(1/2))*(1/(c*x+1))^(1/2)*(c*x+1)^(1/2)/c^5/e^(5/2 )-8/15*b*d^(5/2)*arctanh((e*x^2+d)^(1/2)/d^(1/2)/(-c^2*x^2+1)^(1/2))*(1/(c *x+1))^(1/2)*(c*x+1)^(1/2)/e^3-1/20*b*(e*x^2+d)^(3/2)*(1/(c*x+1))^(1/2)*(c *x+1)^(1/2)*(-c^2*x^2+1)^(1/2)/c^2/e^2+d^2*(a+b*arcsech(c*x))*(e*x^2+d)^(1 /2)/e^3+1/120*b*(19*c^2*d-9*e)*(1/(c*x+1))^(1/2)*(c*x+1)^(1/2)*(-c^2*x^2+1 )^(1/2)*(e*x^2+d)^(1/2)/c^4/e^2
Time = 23.56 (sec) , antiderivative size = 366, normalized size of antiderivative = 1.03 \[ \int \frac {x^5 \left (a+b \text {sech}^{-1}(c x)\right )}{\sqrt {d+e x^2}} \, dx=\frac {\sqrt {d+e x^2} \left (8 a c^4 \left (8 d^2-4 d e x^2+3 e^2 x^4\right )-b e \sqrt {\frac {1-c x}{1+c x}} (1+c x) \left (9 e+c^2 \left (-13 d+6 e x^2\right )\right )+8 b c^4 \left (8 d^2-4 d e x^2+3 e^2 x^4\right ) \text {sech}^{-1}(c x)\right )}{120 c^4 e^3}+\frac {b \sqrt {\frac {1-c x}{1+c x}} \sqrt {1-c^2 x^2} \left (\sqrt {-c^2} \sqrt {-c^2 d-e} \sqrt {e} \left (45 c^4 d^2-10 c^2 d e+9 e^2\right ) \sqrt {\frac {c^2 \left (d+e x^2\right )}{c^2 d+e}} \arcsin \left (\frac {c \sqrt {e} \sqrt {1-c^2 x^2}}{\sqrt {-c^2} \sqrt {-c^2 d-e}}\right )+64 c^7 d^{5/2} \sqrt {-d-e x^2} \arctan \left (\frac {\sqrt {d} \sqrt {1-c^2 x^2}}{\sqrt {-d-e x^2}}\right )\right )}{120 c^7 e^3 (-1+c x) \sqrt {d+e x^2}} \]
(Sqrt[d + e*x^2]*(8*a*c^4*(8*d^2 - 4*d*e*x^2 + 3*e^2*x^4) - b*e*Sqrt[(1 - c*x)/(1 + c*x)]*(1 + c*x)*(9*e + c^2*(-13*d + 6*e*x^2)) + 8*b*c^4*(8*d^2 - 4*d*e*x^2 + 3*e^2*x^4)*ArcSech[c*x]))/(120*c^4*e^3) + (b*Sqrt[(1 - c*x)/( 1 + c*x)]*Sqrt[1 - c^2*x^2]*(Sqrt[-c^2]*Sqrt[-(c^2*d) - e]*Sqrt[e]*(45*c^4 *d^2 - 10*c^2*d*e + 9*e^2)*Sqrt[(c^2*(d + e*x^2))/(c^2*d + e)]*ArcSin[(c*S qrt[e]*Sqrt[1 - c^2*x^2])/(Sqrt[-c^2]*Sqrt[-(c^2*d) - e])] + 64*c^7*d^(5/2 )*Sqrt[-d - e*x^2]*ArcTan[(Sqrt[d]*Sqrt[1 - c^2*x^2])/Sqrt[-d - e*x^2]]))/ (120*c^7*e^3*(-1 + c*x)*Sqrt[d + e*x^2])
Time = 1.36 (sec) , antiderivative size = 306, normalized size of antiderivative = 0.86, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.522, Rules used = {6855, 27, 7282, 2118, 27, 171, 27, 175, 66, 104, 218, 220}
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.
| \(\displaystyle \int \frac {x^5 \left (a+b \text {sech}^{-1}(c x)\right )}{\sqrt {d+e x^2}} \, dx\) |
| \(\Big \downarrow \) 6855 |
| \(\displaystyle b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \int \frac {\sqrt {e x^2+d} \left (3 e^2 x^4-4 d e x^2+8 d^2\right )}{15 e^3 x \sqrt {1-c^2 x^2}}dx+\frac {d^2 \sqrt {d+e x^2} \left (a+b \text {sech}^{-1}(c x)\right )}{e^3}+\frac {\left (d+e x^2\right )^{5/2} \left (a+b \text {sech}^{-1}(c x)\right )}{5 e^3}-\frac {2 d \left (d+e x^2\right )^{3/2} \left (a+b \text {sech}^{-1}(c x)\right )}{3 e^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \int \frac {\sqrt {e x^2+d} \left (3 e^2 x^4-4 d e x^2+8 d^2\right )}{x \sqrt {1-c^2 x^2}}dx}{15 e^3}+\frac {d^2 \sqrt {d+e x^2} \left (a+b \text {sech}^{-1}(c x)\right )}{e^3}+\frac {\left (d+e x^2\right )^{5/2} \left (a+b \text {sech}^{-1}(c x)\right )}{5 e^3}-\frac {2 d \left (d+e x^2\right )^{3/2} \left (a+b \text {sech}^{-1}(c x)\right )}{3 e^3}\) |
| \(\Big \downarrow \) 7282 |
| \(\displaystyle \frac {b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \int \frac {\sqrt {e x^2+d} \left (3 e^2 x^4-4 d e x^2+8 d^2\right )}{x^2 \sqrt {1-c^2 x^2}}dx^2}{30 e^3}+\frac {d^2 \sqrt {d+e x^2} \left (a+b \text {sech}^{-1}(c x)\right )}{e^3}+\frac {\left (d+e x^2\right )^{5/2} \left (a+b \text {sech}^{-1}(c x)\right )}{5 e^3}-\frac {2 d \left (d+e x^2\right )^{3/2} \left (a+b \text {sech}^{-1}(c x)\right )}{3 e^3}\) |
| \(\Big \downarrow \) 2118 |
| \(\displaystyle \frac {b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \left (-\frac {\int -\frac {e \sqrt {e x^2+d} \left (32 c^2 d^2-\left (19 c^2 d-9 e\right ) e x^2\right )}{2 x^2 \sqrt {1-c^2 x^2}}dx^2}{2 c^2 e}-\frac {3 e \sqrt {1-c^2 x^2} \left (d+e x^2\right )^{3/2}}{2 c^2}\right )}{30 e^3}+\frac {d^2 \sqrt {d+e x^2} \left (a+b \text {sech}^{-1}(c x)\right )}{e^3}+\frac {\left (d+e x^2\right )^{5/2} \left (a+b \text {sech}^{-1}(c x)\right )}{5 e^3}-\frac {2 d \left (d+e x^2\right )^{3/2} \left (a+b \text {sech}^{-1}(c x)\right )}{3 e^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \left (\frac {\int \frac {\sqrt {e x^2+d} \left (32 c^2 d^2-\left (19 c^2 d-9 e\right ) e x^2\right )}{x^2 \sqrt {1-c^2 x^2}}dx^2}{4 c^2}-\frac {3 e \sqrt {1-c^2 x^2} \left (d+e x^2\right )^{3/2}}{2 c^2}\right )}{30 e^3}+\frac {d^2 \sqrt {d+e x^2} \left (a+b \text {sech}^{-1}(c x)\right )}{e^3}+\frac {\left (d+e x^2\right )^{5/2} \left (a+b \text {sech}^{-1}(c x)\right )}{5 e^3}-\frac {2 d \left (d+e x^2\right )^{3/2} \left (a+b \text {sech}^{-1}(c x)\right )}{3 e^3}\) |
| \(\Big \downarrow \) 171 |
| \(\displaystyle \frac {b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \left (\frac {\frac {e \sqrt {1-c^2 x^2} \left (19 c^2 d-9 e\right ) \sqrt {d+e x^2}}{c^2}-\frac {\int -\frac {64 d^3 c^4+e \left (45 d^2 c^4-10 d e c^2+9 e^2\right ) x^2}{2 x^2 \sqrt {1-c^2 x^2} \sqrt {e x^2+d}}dx^2}{c^2}}{4 c^2}-\frac {3 e \sqrt {1-c^2 x^2} \left (d+e x^2\right )^{3/2}}{2 c^2}\right )}{30 e^3}+\frac {d^2 \sqrt {d+e x^2} \left (a+b \text {sech}^{-1}(c x)\right )}{e^3}+\frac {\left (d+e x^2\right )^{5/2} \left (a+b \text {sech}^{-1}(c x)\right )}{5 e^3}-\frac {2 d \left (d+e x^2\right )^{3/2} \left (a+b \text {sech}^{-1}(c x)\right )}{3 e^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \left (\frac {\frac {\int \frac {64 d^3 c^4+e \left (45 d^2 c^4-10 d e c^2+9 e^2\right ) x^2}{x^2 \sqrt {1-c^2 x^2} \sqrt {e x^2+d}}dx^2}{2 c^2}+\frac {e \sqrt {1-c^2 x^2} \left (19 c^2 d-9 e\right ) \sqrt {d+e x^2}}{c^2}}{4 c^2}-\frac {3 e \sqrt {1-c^2 x^2} \left (d+e x^2\right )^{3/2}}{2 c^2}\right )}{30 e^3}+\frac {d^2 \sqrt {d+e x^2} \left (a+b \text {sech}^{-1}(c x)\right )}{e^3}+\frac {\left (d+e x^2\right )^{5/2} \left (a+b \text {sech}^{-1}(c x)\right )}{5 e^3}-\frac {2 d \left (d+e x^2\right )^{3/2} \left (a+b \text {sech}^{-1}(c x)\right )}{3 e^3}\) |
| \(\Big \downarrow \) 175 |
| \(\displaystyle \frac {b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \left (\frac {\frac {64 c^4 d^3 \int \frac {1}{x^2 \sqrt {1-c^2 x^2} \sqrt {e x^2+d}}dx^2+e \left (45 c^4 d^2-10 c^2 d e+9 e^2\right ) \int \frac {1}{\sqrt {1-c^2 x^2} \sqrt {e x^2+d}}dx^2}{2 c^2}+\frac {e \sqrt {1-c^2 x^2} \left (19 c^2 d-9 e\right ) \sqrt {d+e x^2}}{c^2}}{4 c^2}-\frac {3 e \sqrt {1-c^2 x^2} \left (d+e x^2\right )^{3/2}}{2 c^2}\right )}{30 e^3}+\frac {d^2 \sqrt {d+e x^2} \left (a+b \text {sech}^{-1}(c x)\right )}{e^3}+\frac {\left (d+e x^2\right )^{5/2} \left (a+b \text {sech}^{-1}(c x)\right )}{5 e^3}-\frac {2 d \left (d+e x^2\right )^{3/2} \left (a+b \text {sech}^{-1}(c x)\right )}{3 e^3}\) |
| \(\Big \downarrow \) 66 |
| \(\displaystyle \frac {b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \left (\frac {\frac {64 c^4 d^3 \int \frac {1}{x^2 \sqrt {1-c^2 x^2} \sqrt {e x^2+d}}dx^2+2 e \left (45 c^4 d^2-10 c^2 d e+9 e^2\right ) \int \frac {1}{-e x^4-c^2}d\frac {\sqrt {1-c^2 x^2}}{\sqrt {e x^2+d}}}{2 c^2}+\frac {e \sqrt {1-c^2 x^2} \left (19 c^2 d-9 e\right ) \sqrt {d+e x^2}}{c^2}}{4 c^2}-\frac {3 e \sqrt {1-c^2 x^2} \left (d+e x^2\right )^{3/2}}{2 c^2}\right )}{30 e^3}+\frac {d^2 \sqrt {d+e x^2} \left (a+b \text {sech}^{-1}(c x)\right )}{e^3}+\frac {\left (d+e x^2\right )^{5/2} \left (a+b \text {sech}^{-1}(c x)\right )}{5 e^3}-\frac {2 d \left (d+e x^2\right )^{3/2} \left (a+b \text {sech}^{-1}(c x)\right )}{3 e^3}\) |
| \(\Big \downarrow \) 104 |
| \(\displaystyle \frac {b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \left (\frac {\frac {128 c^4 d^3 \int \frac {1}{x^4-d}d\frac {\sqrt {e x^2+d}}{\sqrt {1-c^2 x^2}}+2 e \left (45 c^4 d^2-10 c^2 d e+9 e^2\right ) \int \frac {1}{-e x^4-c^2}d\frac {\sqrt {1-c^2 x^2}}{\sqrt {e x^2+d}}}{2 c^2}+\frac {e \sqrt {1-c^2 x^2} \left (19 c^2 d-9 e\right ) \sqrt {d+e x^2}}{c^2}}{4 c^2}-\frac {3 e \sqrt {1-c^2 x^2} \left (d+e x^2\right )^{3/2}}{2 c^2}\right )}{30 e^3}+\frac {d^2 \sqrt {d+e x^2} \left (a+b \text {sech}^{-1}(c x)\right )}{e^3}+\frac {\left (d+e x^2\right )^{5/2} \left (a+b \text {sech}^{-1}(c x)\right )}{5 e^3}-\frac {2 d \left (d+e x^2\right )^{3/2} \left (a+b \text {sech}^{-1}(c x)\right )}{3 e^3}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle \frac {b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \left (\frac {\frac {128 c^4 d^3 \int \frac {1}{x^4-d}d\frac {\sqrt {e x^2+d}}{\sqrt {1-c^2 x^2}}-\frac {2 \sqrt {e} \left (45 c^4 d^2-10 c^2 d e+9 e^2\right ) \arctan \left (\frac {\sqrt {e} \sqrt {1-c^2 x^2}}{c \sqrt {d+e x^2}}\right )}{c}}{2 c^2}+\frac {e \sqrt {1-c^2 x^2} \left (19 c^2 d-9 e\right ) \sqrt {d+e x^2}}{c^2}}{4 c^2}-\frac {3 e \sqrt {1-c^2 x^2} \left (d+e x^2\right )^{3/2}}{2 c^2}\right )}{30 e^3}+\frac {d^2 \sqrt {d+e x^2} \left (a+b \text {sech}^{-1}(c x)\right )}{e^3}+\frac {\left (d+e x^2\right )^{5/2} \left (a+b \text {sech}^{-1}(c x)\right )}{5 e^3}-\frac {2 d \left (d+e x^2\right )^{3/2} \left (a+b \text {sech}^{-1}(c x)\right )}{3 e^3}\) |
| \(\Big \downarrow \) 220 |
| \(\displaystyle \frac {d^2 \sqrt {d+e x^2} \left (a+b \text {sech}^{-1}(c x)\right )}{e^3}+\frac {\left (d+e x^2\right )^{5/2} \left (a+b \text {sech}^{-1}(c x)\right )}{5 e^3}-\frac {2 d \left (d+e x^2\right )^{3/2} \left (a+b \text {sech}^{-1}(c x)\right )}{3 e^3}+\frac {b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \left (\frac {\frac {-\frac {2 \sqrt {e} \left (45 c^4 d^2-10 c^2 d e+9 e^2\right ) \arctan \left (\frac {\sqrt {e} \sqrt {1-c^2 x^2}}{c \sqrt {d+e x^2}}\right )}{c}-128 c^4 d^{5/2} \text {arctanh}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d} \sqrt {1-c^2 x^2}}\right )}{2 c^2}+\frac {e \sqrt {1-c^2 x^2} \left (19 c^2 d-9 e\right ) \sqrt {d+e x^2}}{c^2}}{4 c^2}-\frac {3 e \sqrt {1-c^2 x^2} \left (d+e x^2\right )^{3/2}}{2 c^2}\right )}{30 e^3}\) |
(d^2*Sqrt[d + e*x^2]*(a + b*ArcSech[c*x]))/e^3 - (2*d*(d + e*x^2)^(3/2)*(a + b*ArcSech[c*x]))/(3*e^3) + ((d + e*x^2)^(5/2)*(a + b*ArcSech[c*x]))/(5* e^3) + (b*Sqrt[(1 + c*x)^(-1)]*Sqrt[1 + c*x]*((-3*e*Sqrt[1 - c^2*x^2]*(d + e*x^2)^(3/2))/(2*c^2) + (((19*c^2*d - 9*e)*e*Sqrt[1 - c^2*x^2]*Sqrt[d + e *x^2])/c^2 + ((-2*Sqrt[e]*(45*c^4*d^2 - 10*c^2*d*e + 9*e^2)*ArcTan[(Sqrt[e ]*Sqrt[1 - c^2*x^2])/(c*Sqrt[d + e*x^2])])/c - 128*c^4*d^(5/2)*ArcTanh[Sqr t[d + e*x^2]/(Sqrt[d]*Sqrt[1 - c^2*x^2])])/(2*c^2))/(4*c^2)))/(30*e^3)
3.2.50.3.1 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_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Simp[ 2 Subst[Int[1/(b - d*x^2), x], x, Sqrt[a + b*x]/Sqrt[c + d*x]], x] /; Fre eQ[{a, b, c, d}, x] && !GtQ[c - a*(d/b), 0]
Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x _)), x_] :> With[{q = Denominator[m]}, Simp[q Subst[Int[x^(q*(m + 1) - 1) /(b*e - a*f - (d*e - c*f)*x^q), x], x, (a + b*x)^(1/q)/(c + d*x)^(1/q)], x] ] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[m + n + 1, 0] && RationalQ[n] && L tQ[-1, m, 0] && SimplerQ[a + b*x, c + d*x]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_)*((g_.) + (h_.)*(x_)), x_] :> Simp[h*(a + b*x)^m*(c + d*x)^(n + 1)*(( e + f*x)^(p + 1)/(d*f*(m + n + p + 2))), x] + Simp[1/(d*f*(m + n + p + 2)) Int[(a + b*x)^(m - 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*g*(m + n + p + 2 ) - h*(b*c*e*m + a*(d*e*(n + 1) + c*f*(p + 1))) + (b*d*f*g*(m + n + p + 2) + h*(a*d*f*m - b*(d*e*(m + n + 1) + c*f*(m + p + 1))))*x, x], x], x] /; Fre eQ[{a, b, c, d, e, f, g, h, n, p}, x] && GtQ[m, 0] && NeQ[m + n + p + 2, 0] && IntegersQ[2*m, 2*n, 2*p]
Int[(((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_ )))/((a_.) + (b_.)*(x_)), x_] :> Simp[h/b Int[(c + d*x)^n*(e + f*x)^p, x] , x] + Simp[(b*g - a*h)/b Int[(c + d*x)^n*((e + f*x)^p/(a + b*x)), x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/R t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[a/b]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[b, 2])^(- 1))*ArcTanh[Rt[b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (LtQ[a, 0] || GtQ[b, 0])
Int[(Px_)*((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f _.)*(x_))^(p_.), x_Symbol] :> With[{q = Expon[Px, x], k = Coeff[Px, x, Expo n[Px, x]]}, Simp[k*(a + b*x)^(m + q - 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d*f*b^(q - 1)*(m + n + p + q + 1))), x] + Simp[1/(d*f*b^q*(m + n + p + q + 1)) Int[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p*ExpandToSum[d*f*b^q*(m + n + p + q + 1)*Px - d*f*k*(m + n + p + q + 1)*(a + b*x)^q + k*(a + b*x)^(q - 2)*(a^2*d*f*(m + n + p + q + 1) - b*(b*c*e*(m + q - 1) + a*(d*e*(n + 1) + c*f*(p + 1))) + b*(a*d*f*(2*(m + q) + n + p) - b*(d*e*(m + q + n) + c*f*(m + q + p)))*x), x], x], x] /; NeQ[m + n + p + q + 1, 0]] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && PolyQ[Px, x]
Int[((a_.) + ArcSech[(c_.)*(x_)]*(b_.))*((f_.)*(x_))^(m_.)*((d_.) + (e_.)*( x_)^2)^(p_.), x_Symbol] :> With[{u = IntHide[(f*x)^m*(d + e*x^2)^p, x]}, Si mp[(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]] /; Fre eQ[{a, b, c, d, e, f, m, p}, x] && ((IGtQ[p, 0] && !(ILtQ[(m - 1)/2, 0] && GtQ[m + 2*p + 3, 0])) || (IGtQ[(m + 1)/2, 0] && !(ILtQ[p, 0] && GtQ[m + 2 *p + 3, 0])) || (ILtQ[(m + 2*p + 1)/2, 0] && !ILtQ[(m - 1)/2, 0]))
Int[(u_)/(x_), x_Symbol] :> With[{lst = PowerVariableExpn[u, 0, x]}, Simp[1 /lst[[2]] Subst[Int[NormalizeIntegrand[Simplify[lst[[1]]/x], x], x], x, ( lst[[3]]*x)^lst[[2]]], x] /; !FalseQ[lst] && NeQ[lst[[2]], 0]] /; NonsumQ[ u] && !RationalFunctionQ[u, x]
\[\int \frac {x^{5} \left (a +b \,\operatorname {arcsech}\left (c x \right )\right )}{\sqrt {e \,x^{2}+d}}d x\]
Time = 0.75 (sec) , antiderivative size = 1679, normalized size of antiderivative = 4.72 \[ \int \frac {x^5 \left (a+b \text {sech}^{-1}(c x)\right )}{\sqrt {d+e x^2}} \, dx=\text {Too large to display} \]
[1/480*(64*b*c^5*d^(5/2)*log(((c^4*d^2 - 6*c^2*d*e + e^2)*x^4 - 8*(c^2*d^2 - d*e)*x^2 + 4*((c^3*d - c*e)*x^3 - 2*c*d*x)*sqrt(e*x^2 + d)*sqrt(d)*sqrt (-(c^2*x^2 - 1)/(c^2*x^2)) + 8*d^2)/x^4) - (45*b*c^4*d^2 - 10*b*c^2*d*e + 9*b*e^2)*sqrt(-e)*log(8*c^4*e^2*x^4 + c^4*d^2 - 6*c^2*d*e + 8*(c^4*d*e - c ^2*e^2)*x^2 - 4*(2*c^4*e*x^3 + (c^4*d - c^2*e)*x)*sqrt(e*x^2 + d)*sqrt(-e) *sqrt(-(c^2*x^2 - 1)/(c^2*x^2)) + e^2) + 32*(3*b*c^5*e^2*x^4 - 4*b*c^5*d*e *x^2 + 8*b*c^5*d^2)*sqrt(e*x^2 + d)*log((c*x*sqrt(-(c^2*x^2 - 1)/(c^2*x^2) ) + 1)/(c*x)) + 4*(24*a*c^5*e^2*x^4 - 32*a*c^5*d*e*x^2 + 64*a*c^5*d^2 - (6 *b*c^4*e^2*x^3 - (13*b*c^4*d*e - 9*b*c^2*e^2)*x)*sqrt(-(c^2*x^2 - 1)/(c^2* x^2)))*sqrt(e*x^2 + d))/(c^5*e^3), 1/240*(32*b*c^5*d^(5/2)*log(((c^4*d^2 - 6*c^2*d*e + e^2)*x^4 - 8*(c^2*d^2 - d*e)*x^2 + 4*((c^3*d - c*e)*x^3 - 2*c *d*x)*sqrt(e*x^2 + d)*sqrt(d)*sqrt(-(c^2*x^2 - 1)/(c^2*x^2)) + 8*d^2)/x^4) - (45*b*c^4*d^2 - 10*b*c^2*d*e + 9*b*e^2)*sqrt(e)*arctan(1/2*(2*c^2*e*x^3 + (c^2*d - e)*x)*sqrt(e*x^2 + d)*sqrt(e)*sqrt(-(c^2*x^2 - 1)/(c^2*x^2))/( c^2*e^2*x^4 + (c^2*d*e - e^2)*x^2 - d*e)) + 16*(3*b*c^5*e^2*x^4 - 4*b*c^5* d*e*x^2 + 8*b*c^5*d^2)*sqrt(e*x^2 + d)*log((c*x*sqrt(-(c^2*x^2 - 1)/(c^2*x ^2)) + 1)/(c*x)) + 2*(24*a*c^5*e^2*x^4 - 32*a*c^5*d*e*x^2 + 64*a*c^5*d^2 - (6*b*c^4*e^2*x^3 - (13*b*c^4*d*e - 9*b*c^2*e^2)*x)*sqrt(-(c^2*x^2 - 1)/(c ^2*x^2)))*sqrt(e*x^2 + d))/(c^5*e^3), -1/480*(128*b*c^5*sqrt(-d)*d^2*arcta n(-1/2*((c^3*d - c*e)*x^3 - 2*c*d*x)*sqrt(e*x^2 + d)*sqrt(-d)*sqrt(-(c^...
\[ \int \frac {x^5 \left (a+b \text {sech}^{-1}(c x)\right )}{\sqrt {d+e x^2}} \, dx=\int \frac {x^{5} \left (a + b \operatorname {asech}{\left (c x \right )}\right )}{\sqrt {d + e x^{2}}}\, dx \]
Exception generated. \[ \int \frac {x^5 \left (a+b \text {sech}^{-1}(c x)\right )}{\sqrt {d+e x^2}} \, dx=\text {Exception raised: ValueError} \]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(e>0)', see `assume?` for more de tails)Is e
\[ \int \frac {x^5 \left (a+b \text {sech}^{-1}(c x)\right )}{\sqrt {d+e x^2}} \, dx=\int { \frac {{\left (b \operatorname {arsech}\left (c x\right ) + a\right )} x^{5}}{\sqrt {e x^{2} + d}} \,d x } \]
Timed out. \[ \int \frac {x^5 \left (a+b \text {sech}^{-1}(c x)\right )}{\sqrt {d+e x^2}} \, dx=\int \frac {x^5\,\left (a+b\,\mathrm {acosh}\left (\frac {1}{c\,x}\right )\right )}{\sqrt {e\,x^2+d}} \,d x \]