3.130 \(\int x^5 \sqrt{d+e x^2} (a+b \text{sech}^{-1}(c x)) \, dx\)

Optimal. Leaf size=447 \[ \frac{d^2 \left (d+e x^2\right )^{3/2} \left (a+b \text{sech}^{-1}(c x)\right )}{3 e^3}-\frac{2 d \left (d+e x^2\right )^{5/2} \left (a+b \text{sech}^{-1}(c x)\right )}{5 e^3}+\frac{\left (d+e x^2\right )^{7/2} \left (a+b \text{sech}^{-1}(c x)\right )}{7 e^3}+\frac{b \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \sqrt{1-c^2 x^2} \left (23 c^4 d^2+12 c^2 d e-75 e^2\right ) \sqrt{d+e x^2}}{1680 c^6 e^2}-\frac{b \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \left (-35 c^4 d^2 e+105 c^6 d^3+63 c^2 d e^2+75 e^3\right ) \tan ^{-1}\left (\frac{\sqrt{e} \sqrt{1-c^2 x^2}}{c \sqrt{d+e x^2}}\right )}{1680 c^7 e^{5/2}}-\frac{8 b d^{7/2} \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \tanh ^{-1}\left (\frac{\sqrt{d+e x^2}}{\sqrt{d} \sqrt{1-c^2 x^2}}\right )}{105 e^3}-\frac{b \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \sqrt{1-c^2 x^2} \left (d+e x^2\right )^{5/2}}{42 c^2 e^2}+\frac{b \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \sqrt{1-c^2 x^2} \left (29 c^2 d-25 e\right ) \left (d+e x^2\right )^{3/2}}{840 c^4 e^2} \]

[Out]

(b*(23*c^4*d^2 + 12*c^2*d*e - 75*e^2)*Sqrt[(1 + c*x)^(-1)]*Sqrt[1 + c*x]*Sqrt[1 - c^2*x^2]*Sqrt[d + e*x^2])/(1
680*c^6*e^2) + (b*(29*c^2*d - 25*e)*Sqrt[(1 + c*x)^(-1)]*Sqrt[1 + c*x]*Sqrt[1 - c^2*x^2]*(d + e*x^2)^(3/2))/(8
40*c^4*e^2) - (b*Sqrt[(1 + c*x)^(-1)]*Sqrt[1 + c*x]*Sqrt[1 - c^2*x^2]*(d + e*x^2)^(5/2))/(42*c^2*e^2) + (d^2*(
d + e*x^2)^(3/2)*(a + b*ArcSech[c*x]))/(3*e^3) - (2*d*(d + e*x^2)^(5/2)*(a + b*ArcSech[c*x]))/(5*e^3) + ((d +
e*x^2)^(7/2)*(a + b*ArcSech[c*x]))/(7*e^3) - (b*(105*c^6*d^3 - 35*c^4*d^2*e + 63*c^2*d*e^2 + 75*e^3)*Sqrt[(1 +
c*x)^(-1)]*Sqrt[1 + c*x]*ArcTan[(Sqrt[e]*Sqrt[1 - c^2*x^2])/(c*Sqrt[d + e*x^2])])/(1680*c^7*e^(5/2)) - (8*b*d
^(7/2)*Sqrt[(1 + c*x)^(-1)]*Sqrt[1 + c*x]*ArcTanh[Sqrt[d + e*x^2]/(Sqrt[d]*Sqrt[1 - c^2*x^2])])/(105*e^3)

________________________________________________________________________________________

Rubi [A]  time = 1.39547, antiderivative size = 447, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 12, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.522, Rules used = {266, 43, 6301, 12, 1615, 154, 157, 63, 217, 203, 93, 207} \[ \frac{d^2 \left (d+e x^2\right )^{3/2} \left (a+b \text{sech}^{-1}(c x)\right )}{3 e^3}-\frac{2 d \left (d+e x^2\right )^{5/2} \left (a+b \text{sech}^{-1}(c x)\right )}{5 e^3}+\frac{\left (d+e x^2\right )^{7/2} \left (a+b \text{sech}^{-1}(c x)\right )}{7 e^3}+\frac{b \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \sqrt{1-c^2 x^2} \left (23 c^4 d^2+12 c^2 d e-75 e^2\right ) \sqrt{d+e x^2}}{1680 c^6 e^2}-\frac{b \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \left (-35 c^4 d^2 e+105 c^6 d^3+63 c^2 d e^2+75 e^3\right ) \tan ^{-1}\left (\frac{\sqrt{e} \sqrt{1-c^2 x^2}}{c \sqrt{d+e x^2}}\right )}{1680 c^7 e^{5/2}}-\frac{8 b d^{7/2} \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \tanh ^{-1}\left (\frac{\sqrt{d+e x^2}}{\sqrt{d} \sqrt{1-c^2 x^2}}\right )}{105 e^3}-\frac{b \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \sqrt{1-c^2 x^2} \left (d+e x^2\right )^{5/2}}{42 c^2 e^2}+\frac{b \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \sqrt{1-c^2 x^2} \left (29 c^2 d-25 e\right ) \left (d+e x^2\right )^{3/2}}{840 c^4 e^2} \]

Antiderivative was successfully verified.

[In]

Int[x^5*Sqrt[d + e*x^2]*(a + b*ArcSech[c*x]),x]

[Out]

(b*(23*c^4*d^2 + 12*c^2*d*e - 75*e^2)*Sqrt[(1 + c*x)^(-1)]*Sqrt[1 + c*x]*Sqrt[1 - c^2*x^2]*Sqrt[d + e*x^2])/(1
680*c^6*e^2) + (b*(29*c^2*d - 25*e)*Sqrt[(1 + c*x)^(-1)]*Sqrt[1 + c*x]*Sqrt[1 - c^2*x^2]*(d + e*x^2)^(3/2))/(8
40*c^4*e^2) - (b*Sqrt[(1 + c*x)^(-1)]*Sqrt[1 + c*x]*Sqrt[1 - c^2*x^2]*(d + e*x^2)^(5/2))/(42*c^2*e^2) + (d^2*(
d + e*x^2)^(3/2)*(a + b*ArcSech[c*x]))/(3*e^3) - (2*d*(d + e*x^2)^(5/2)*(a + b*ArcSech[c*x]))/(5*e^3) + ((d +
e*x^2)^(7/2)*(a + b*ArcSech[c*x]))/(7*e^3) - (b*(105*c^6*d^3 - 35*c^4*d^2*e + 63*c^2*d*e^2 + 75*e^3)*Sqrt[(1 +
c*x)^(-1)]*Sqrt[1 + c*x]*ArcTan[(Sqrt[e]*Sqrt[1 - c^2*x^2])/(c*Sqrt[d + e*x^2])])/(1680*c^7*e^(5/2)) - (8*b*d
^(7/2)*Sqrt[(1 + c*x)^(-1)]*Sqrt[1 + c*x]*ArcTanh[Sqrt[d + e*x^2]/(Sqrt[d]*Sqrt[1 - c^2*x^2])])/(105*e^3)

Rule 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 6301

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]}, Dist[a + b*ArcSech[c*x], u, x] + Dist[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, 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]))

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 1615

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, Expon[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] + Dist[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]] /; F
reeQ[{a, b, c, d, e, f, m, n, p}, x] && PolyQ[Px, x] && IntegersQ[2*m, 2*n, 2*p]

Rule 154

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)), x_Symb
ol] :> Simp[(h*(a + b*x)^m*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/(d*f*(m + n + p + 2)), x] + Dist[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] /; FreeQ[{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]

Rule 157

Int[(((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)))/((a_.) + (b_.)*(x_)), x_Symbol]
:> Dist[h/b, Int[(c + d*x)^n*(e + f*x)^p, x], x] + Dist[(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]

Rule 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[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] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 217

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]

Rule 203

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTan[(Rt[b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rule 93

Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x_)), x_Symbol] :> With[{q = Denomin
ator[m]}, Dist[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] && LtQ[-1, m, 0] && SimplerQ[
a + b*x, c + d*x]

Rule 207

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTanh[(Rt[b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && NegQ[a/b] && (LtQ[a, 0] || GtQ[b, 0])

Rubi steps

\begin{align*} \int x^5 \sqrt{d+e x^2} \left (a+b \text{sech}^{-1}(c x)\right ) \, dx &=\frac{d^2 \left (d+e x^2\right )^{3/2} \left (a+b \text{sech}^{-1}(c x)\right )}{3 e^3}-\frac{2 d \left (d+e x^2\right )^{5/2} \left (a+b \text{sech}^{-1}(c x)\right )}{5 e^3}+\frac{\left (d+e x^2\right )^{7/2} \left (a+b \text{sech}^{-1}(c x)\right )}{7 e^3}+\left (b \sqrt{\frac{1}{1+c x}} \sqrt{1+c x}\right ) \int \frac{\left (d+e x^2\right )^{3/2} \left (8 d^2-12 d e x^2+15 e^2 x^4\right )}{105 e^3 x \sqrt{1-c^2 x^2}} \, dx\\ &=\frac{d^2 \left (d+e x^2\right )^{3/2} \left (a+b \text{sech}^{-1}(c x)\right )}{3 e^3}-\frac{2 d \left (d+e x^2\right )^{5/2} \left (a+b \text{sech}^{-1}(c x)\right )}{5 e^3}+\frac{\left (d+e x^2\right )^{7/2} \left (a+b \text{sech}^{-1}(c x)\right )}{7 e^3}+\frac{\left (b \sqrt{\frac{1}{1+c x}} \sqrt{1+c x}\right ) \int \frac{\left (d+e x^2\right )^{3/2} \left (8 d^2-12 d e x^2+15 e^2 x^4\right )}{x \sqrt{1-c^2 x^2}} \, dx}{105 e^3}\\ &=\frac{d^2 \left (d+e x^2\right )^{3/2} \left (a+b \text{sech}^{-1}(c x)\right )}{3 e^3}-\frac{2 d \left (d+e x^2\right )^{5/2} \left (a+b \text{sech}^{-1}(c x)\right )}{5 e^3}+\frac{\left (d+e x^2\right )^{7/2} \left (a+b \text{sech}^{-1}(c x)\right )}{7 e^3}+\frac{\left (b \sqrt{\frac{1}{1+c x}} \sqrt{1+c x}\right ) \operatorname{Subst}\left (\int \frac{(d+e x)^{3/2} \left (8 d^2-12 d e x+15 e^2 x^2\right )}{x \sqrt{1-c^2 x}} \, dx,x,x^2\right )}{210 e^3}\\ &=-\frac{b \sqrt{\frac{1}{1+c x}} \sqrt{1+c x} \sqrt{1-c^2 x^2} \left (d+e x^2\right )^{5/2}}{42 c^2 e^2}+\frac{d^2 \left (d+e x^2\right )^{3/2} \left (a+b \text{sech}^{-1}(c x)\right )}{3 e^3}-\frac{2 d \left (d+e x^2\right )^{5/2} \left (a+b \text{sech}^{-1}(c x)\right )}{5 e^3}+\frac{\left (d+e x^2\right )^{7/2} \left (a+b \text{sech}^{-1}(c x)\right )}{7 e^3}-\frac{\left (b \sqrt{\frac{1}{1+c x}} \sqrt{1+c x}\right ) \operatorname{Subst}\left (\int \frac{(d+e x)^{3/2} \left (-24 c^2 d^2 e+\frac{3}{2} \left (29 c^2 d-25 e\right ) e^2 x\right )}{x \sqrt{1-c^2 x}} \, dx,x,x^2\right )}{630 c^2 e^4}\\ &=\frac{b \left (29 c^2 d-25 e\right ) \sqrt{\frac{1}{1+c x}} \sqrt{1+c x} \sqrt{1-c^2 x^2} \left (d+e x^2\right )^{3/2}}{840 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 )^{5/2}}{42 c^2 e^2}+\frac{d^2 \left (d+e x^2\right )^{3/2} \left (a+b \text{sech}^{-1}(c x)\right )}{3 e^3}-\frac{2 d \left (d+e x^2\right )^{5/2} \left (a+b \text{sech}^{-1}(c x)\right )}{5 e^3}+\frac{\left (d+e x^2\right )^{7/2} \left (a+b \text{sech}^{-1}(c x)\right )}{7 e^3}+\frac{\left (b \sqrt{\frac{1}{1+c x}} \sqrt{1+c x}\right ) \operatorname{Subst}\left (\int \frac{\sqrt{d+e x} \left (48 c^4 d^3 e-\frac{3}{4} e^2 \left (23 c^4 d^2+12 c^2 d e-75 e^2\right ) x\right )}{x \sqrt{1-c^2 x}} \, dx,x,x^2\right )}{1260 c^4 e^4}\\ &=\frac{b \left (23 c^4 d^2+12 c^2 d e-75 e^2\right ) \sqrt{\frac{1}{1+c x}} \sqrt{1+c x} \sqrt{1-c^2 x^2} \sqrt{d+e x^2}}{1680 c^6 e^2}+\frac{b \left (29 c^2 d-25 e\right ) \sqrt{\frac{1}{1+c x}} \sqrt{1+c x} \sqrt{1-c^2 x^2} \left (d+e x^2\right )^{3/2}}{840 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 )^{5/2}}{42 c^2 e^2}+\frac{d^2 \left (d+e x^2\right )^{3/2} \left (a+b \text{sech}^{-1}(c x)\right )}{3 e^3}-\frac{2 d \left (d+e x^2\right )^{5/2} \left (a+b \text{sech}^{-1}(c x)\right )}{5 e^3}+\frac{\left (d+e x^2\right )^{7/2} \left (a+b \text{sech}^{-1}(c x)\right )}{7 e^3}-\frac{\left (b \sqrt{\frac{1}{1+c x}} \sqrt{1+c x}\right ) \operatorname{Subst}\left (\int \frac{-48 c^6 d^4 e-\frac{3}{8} e^2 \left (105 c^6 d^3-35 c^4 d^2 e+63 c^2 d e^2+75 e^3\right ) x}{x \sqrt{1-c^2 x} \sqrt{d+e x}} \, dx,x,x^2\right )}{1260 c^6 e^4}\\ &=\frac{b \left (23 c^4 d^2+12 c^2 d e-75 e^2\right ) \sqrt{\frac{1}{1+c x}} \sqrt{1+c x} \sqrt{1-c^2 x^2} \sqrt{d+e x^2}}{1680 c^6 e^2}+\frac{b \left (29 c^2 d-25 e\right ) \sqrt{\frac{1}{1+c x}} \sqrt{1+c x} \sqrt{1-c^2 x^2} \left (d+e x^2\right )^{3/2}}{840 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 )^{5/2}}{42 c^2 e^2}+\frac{d^2 \left (d+e x^2\right )^{3/2} \left (a+b \text{sech}^{-1}(c x)\right )}{3 e^3}-\frac{2 d \left (d+e x^2\right )^{5/2} \left (a+b \text{sech}^{-1}(c x)\right )}{5 e^3}+\frac{\left (d+e x^2\right )^{7/2} \left (a+b \text{sech}^{-1}(c x)\right )}{7 e^3}+\frac{\left (4 b d^4 \sqrt{\frac{1}{1+c x}} \sqrt{1+c x}\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{1-c^2 x} \sqrt{d+e x}} \, dx,x,x^2\right )}{105 e^3}+\frac{\left (b \left (105 c^6 d^3-35 c^4 d^2 e+63 c^2 d e^2+75 e^3\right ) \sqrt{\frac{1}{1+c x}} \sqrt{1+c x}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-c^2 x} \sqrt{d+e x}} \, dx,x,x^2\right )}{3360 c^6 e^2}\\ &=\frac{b \left (23 c^4 d^2+12 c^2 d e-75 e^2\right ) \sqrt{\frac{1}{1+c x}} \sqrt{1+c x} \sqrt{1-c^2 x^2} \sqrt{d+e x^2}}{1680 c^6 e^2}+\frac{b \left (29 c^2 d-25 e\right ) \sqrt{\frac{1}{1+c x}} \sqrt{1+c x} \sqrt{1-c^2 x^2} \left (d+e x^2\right )^{3/2}}{840 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 )^{5/2}}{42 c^2 e^2}+\frac{d^2 \left (d+e x^2\right )^{3/2} \left (a+b \text{sech}^{-1}(c x)\right )}{3 e^3}-\frac{2 d \left (d+e x^2\right )^{5/2} \left (a+b \text{sech}^{-1}(c x)\right )}{5 e^3}+\frac{\left (d+e x^2\right )^{7/2} \left (a+b \text{sech}^{-1}(c x)\right )}{7 e^3}+\frac{\left (8 b d^4 \sqrt{\frac{1}{1+c x}} \sqrt{1+c x}\right ) \operatorname{Subst}\left (\int \frac{1}{-d+x^2} \, dx,x,\frac{\sqrt{d+e x^2}}{\sqrt{1-c^2 x^2}}\right )}{105 e^3}-\frac{\left (b \left (105 c^6 d^3-35 c^4 d^2 e+63 c^2 d e^2+75 e^3\right ) \sqrt{\frac{1}{1+c x}} \sqrt{1+c x}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{d+\frac{e}{c^2}-\frac{e x^2}{c^2}}} \, dx,x,\sqrt{1-c^2 x^2}\right )}{1680 c^8 e^2}\\ &=\frac{b \left (23 c^4 d^2+12 c^2 d e-75 e^2\right ) \sqrt{\frac{1}{1+c x}} \sqrt{1+c x} \sqrt{1-c^2 x^2} \sqrt{d+e x^2}}{1680 c^6 e^2}+\frac{b \left (29 c^2 d-25 e\right ) \sqrt{\frac{1}{1+c x}} \sqrt{1+c x} \sqrt{1-c^2 x^2} \left (d+e x^2\right )^{3/2}}{840 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 )^{5/2}}{42 c^2 e^2}+\frac{d^2 \left (d+e x^2\right )^{3/2} \left (a+b \text{sech}^{-1}(c x)\right )}{3 e^3}-\frac{2 d \left (d+e x^2\right )^{5/2} \left (a+b \text{sech}^{-1}(c x)\right )}{5 e^3}+\frac{\left (d+e x^2\right )^{7/2} \left (a+b \text{sech}^{-1}(c x)\right )}{7 e^3}-\frac{8 b d^{7/2} \sqrt{\frac{1}{1+c x}} \sqrt{1+c x} \tanh ^{-1}\left (\frac{\sqrt{d+e x^2}}{\sqrt{d} \sqrt{1-c^2 x^2}}\right )}{105 e^3}-\frac{\left (b \left (105 c^6 d^3-35 c^4 d^2 e+63 c^2 d e^2+75 e^3\right ) \sqrt{\frac{1}{1+c x}} \sqrt{1+c x}\right ) \operatorname{Subst}\left (\int \frac{1}{1+\frac{e x^2}{c^2}} \, dx,x,\frac{\sqrt{1-c^2 x^2}}{\sqrt{d+e x^2}}\right )}{1680 c^8 e^2}\\ &=\frac{b \left (23 c^4 d^2+12 c^2 d e-75 e^2\right ) \sqrt{\frac{1}{1+c x}} \sqrt{1+c x} \sqrt{1-c^2 x^2} \sqrt{d+e x^2}}{1680 c^6 e^2}+\frac{b \left (29 c^2 d-25 e\right ) \sqrt{\frac{1}{1+c x}} \sqrt{1+c x} \sqrt{1-c^2 x^2} \left (d+e x^2\right )^{3/2}}{840 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 )^{5/2}}{42 c^2 e^2}+\frac{d^2 \left (d+e x^2\right )^{3/2} \left (a+b \text{sech}^{-1}(c x)\right )}{3 e^3}-\frac{2 d \left (d+e x^2\right )^{5/2} \left (a+b \text{sech}^{-1}(c x)\right )}{5 e^3}+\frac{\left (d+e x^2\right )^{7/2} \left (a+b \text{sech}^{-1}(c x)\right )}{7 e^3}-\frac{b \left (105 c^6 d^3-35 c^4 d^2 e+63 c^2 d e^2+75 e^3\right ) \sqrt{\frac{1}{1+c x}} \sqrt{1+c x} \tan ^{-1}\left (\frac{\sqrt{e} \sqrt{1-c^2 x^2}}{c \sqrt{d+e x^2}}\right )}{1680 c^7 e^{5/2}}-\frac{8 b d^{7/2} \sqrt{\frac{1}{1+c x}} \sqrt{1+c x} \tanh ^{-1}\left (\frac{\sqrt{d+e x^2}}{\sqrt{d} \sqrt{1-c^2 x^2}}\right )}{105 e^3}\\ \end{align*}

Mathematica [A]  time = 3.01421, size = 409, normalized size = 0.91 \[ \frac{\sqrt{d+e x^2} \left (16 a c^6 \left (-4 d^2 e x^2+8 d^3+3 d e^2 x^4+15 e^3 x^6\right )-b e \sqrt{\frac{1-c x}{c x+1}} (c x+1) \left (c^4 \left (-41 d^2+22 d e x^2+40 e^2 x^4\right )+2 c^2 e \left (19 d+25 e x^2\right )+75 e^2\right )+16 b c^6 \text{sech}^{-1}(c x) \left (-4 d^2 e x^2+8 d^3+3 d e^2 x^4+15 e^3 x^6\right )\right )}{1680 c^6 e^3}-\frac{b \sqrt{\frac{1-c x}{c x+1}} \sqrt{c^2 x^2-1} \left (\sqrt{c^2} \sqrt{e} \sqrt{c^2 d+e} \left (-35 c^4 d^2 e+105 c^6 d^3+63 c^2 d e^2+75 e^3\right ) \sqrt{\frac{c^2 \left (d+e x^2\right )}{c^2 d+e}} \sinh ^{-1}\left (\frac{c \sqrt{e} \sqrt{c^2 x^2-1}}{\sqrt{c^2} \sqrt{c^2 d+e}}\right )+128 c^9 d^{7/2} \sqrt{d+e x^2} \tan ^{-1}\left (\frac{\sqrt{d} \sqrt{c^2 x^2-1}}{\sqrt{d+e x^2}}\right )\right )}{1680 c^9 e^3 (c x-1) \sqrt{d+e x^2}} \]

Antiderivative was successfully verified.

[In]

Integrate[x^5*Sqrt[d + e*x^2]*(a + b*ArcSech[c*x]),x]

[Out]

(Sqrt[d + e*x^2]*(16*a*c^6*(8*d^3 - 4*d^2*e*x^2 + 3*d*e^2*x^4 + 15*e^3*x^6) - b*e*Sqrt[(1 - c*x)/(1 + c*x)]*(1
+ c*x)*(75*e^2 + 2*c^2*e*(19*d + 25*e*x^2) + c^4*(-41*d^2 + 22*d*e*x^2 + 40*e^2*x^4)) + 16*b*c^6*(8*d^3 - 4*d
^2*e*x^2 + 3*d*e^2*x^4 + 15*e^3*x^6)*ArcSech[c*x]))/(1680*c^6*e^3) - (b*Sqrt[(1 - c*x)/(1 + c*x)]*Sqrt[-1 + c^
2*x^2]*(Sqrt[c^2]*Sqrt[e]*Sqrt[c^2*d + e]*(105*c^6*d^3 - 35*c^4*d^2*e + 63*c^2*d*e^2 + 75*e^3)*Sqrt[(c^2*(d +
e*x^2))/(c^2*d + e)]*ArcSinh[(c*Sqrt[e]*Sqrt[-1 + c^2*x^2])/(Sqrt[c^2]*Sqrt[c^2*d + e])] + 128*c^9*d^(7/2)*Sqr
t[d + e*x^2]*ArcTan[(Sqrt[d]*Sqrt[-1 + c^2*x^2])/Sqrt[d + e*x^2]]))/(1680*c^9*e^3*(-1 + c*x)*Sqrt[d + e*x^2])

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Maple [F]  time = 1.95, size = 0, normalized size = 0. \begin{align*} \int{x}^{5} \left ( a+b{\rm arcsech} \left (cx\right ) \right ) \sqrt{e{x}^{2}+d}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5*(a+b*arcsech(c*x))*(e*x^2+d)^(1/2),x)

[Out]

int(x^5*(a+b*arcsech(c*x))*(e*x^2+d)^(1/2),x)

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*(a+b*arcsech(c*x))*(e*x^2+d)^(1/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 15.839, size = 4373, normalized size = 9.78 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*(a+b*arcsech(c*x))*(e*x^2+d)^(1/2),x, algorithm="fricas")

[Out]

[1/6720*(128*b*c^7*d^(7/2)*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) - (105*b*c^6*d^3 - 35*b*c^4*d
^2*e + 63*b*c^2*d*e^2 + 75*b*e^3)*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) + 64*(15
*b*c^7*e^3*x^6 + 3*b*c^7*d*e^2*x^4 - 4*b*c^7*d^2*e*x^2 + 8*b*c^7*d^3)*sqrt(e*x^2 + d)*log((c*x*sqrt(-(c^2*x^2
- 1)/(c^2*x^2)) + 1)/(c*x)) + 4*(240*a*c^7*e^3*x^6 + 48*a*c^7*d*e^2*x^4 - 64*a*c^7*d^2*e*x^2 + 128*a*c^7*d^3 -
(40*b*c^6*e^3*x^5 + 2*(11*b*c^6*d*e^2 + 25*b*c^4*e^3)*x^3 - (41*b*c^6*d^2*e - 38*b*c^4*d*e^2 - 75*b*c^2*e^3)*
x)*sqrt(-(c^2*x^2 - 1)/(c^2*x^2)))*sqrt(e*x^2 + d))/(c^7*e^3), 1/3360*(64*b*c^7*d^(7/2)*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) - (105*b*c^6*d^3 - 35*b*c^4*d^2*e + 63*b*c^2*d*e^2 + 75*b*e^3)*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)) + 32*(15*b*c^7*e^3*x^6 + 3*b*c^7*d*e^2*x^4 - 4*b*c^7*d^2*e*x^2 + 8*b*c^7*d^3)*sqrt(e*x^2
+ d)*log((c*x*sqrt(-(c^2*x^2 - 1)/(c^2*x^2)) + 1)/(c*x)) + 2*(240*a*c^7*e^3*x^6 + 48*a*c^7*d*e^2*x^4 - 64*a*c
^7*d^2*e*x^2 + 128*a*c^7*d^3 - (40*b*c^6*e^3*x^5 + 2*(11*b*c^6*d*e^2 + 25*b*c^4*e^3)*x^3 - (41*b*c^6*d^2*e - 3
8*b*c^4*d*e^2 - 75*b*c^2*e^3)*x)*sqrt(-(c^2*x^2 - 1)/(c^2*x^2)))*sqrt(e*x^2 + d))/(c^7*e^3), -1/6720*(256*b*c^
7*sqrt(-d)*d^3*arctan(-1/2*((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
))/(c^2*d*e*x^4 + (c^2*d^2 - d*e)*x^2 - d^2)) + (105*b*c^6*d^3 - 35*b*c^4*d^2*e + 63*b*c^2*d*e^2 + 75*b*e^3)*s
qrt(-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) - 64*(15*b*c^7*e^3*x^6 + 3*b*c^7*d*e^2*x^4
- 4*b*c^7*d^2*e*x^2 + 8*b*c^7*d^3)*sqrt(e*x^2 + d)*log((c*x*sqrt(-(c^2*x^2 - 1)/(c^2*x^2)) + 1)/(c*x)) - 4*(24
0*a*c^7*e^3*x^6 + 48*a*c^7*d*e^2*x^4 - 64*a*c^7*d^2*e*x^2 + 128*a*c^7*d^3 - (40*b*c^6*e^3*x^5 + 2*(11*b*c^6*d*
e^2 + 25*b*c^4*e^3)*x^3 - (41*b*c^6*d^2*e - 38*b*c^4*d*e^2 - 75*b*c^2*e^3)*x)*sqrt(-(c^2*x^2 - 1)/(c^2*x^2)))*
sqrt(e*x^2 + d))/(c^7*e^3), -1/3360*(128*b*c^7*sqrt(-d)*d^3*arctan(-1/2*((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))/(c^2*d*e*x^4 + (c^2*d^2 - d*e)*x^2 - d^2)) + (105*b*c^6*d^3 -
35*b*c^4*d^2*e + 63*b*c^2*d*e^2 + 75*b*e^3)*sqrt(e)*arctan(1/2*(2*c^2*e*x^3 + (c^2*d - e)*x)*sqrt(e*x^2 + d)*s
qrt(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)) - 32*(15*b*c^7*e^3*x^6 + 3*b*
c^7*d*e^2*x^4 - 4*b*c^7*d^2*e*x^2 + 8*b*c^7*d^3)*sqrt(e*x^2 + d)*log((c*x*sqrt(-(c^2*x^2 - 1)/(c^2*x^2)) + 1)/
(c*x)) - 2*(240*a*c^7*e^3*x^6 + 48*a*c^7*d*e^2*x^4 - 64*a*c^7*d^2*e*x^2 + 128*a*c^7*d^3 - (40*b*c^6*e^3*x^5 +
2*(11*b*c^6*d*e^2 + 25*b*c^4*e^3)*x^3 - (41*b*c^6*d^2*e - 38*b*c^4*d*e^2 - 75*b*c^2*e^3)*x)*sqrt(-(c^2*x^2 - 1
)/(c^2*x^2)))*sqrt(e*x^2 + d))/(c^7*e^3)]

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**5*(a+b*asech(c*x))*(e*x**2+d)**(1/2),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{e x^{2} + d}{\left (b \operatorname{arsech}\left (c x\right ) + a\right )} x^{5}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*(a+b*arcsech(c*x))*(e*x^2+d)^(1/2),x, algorithm="giac")

[Out]

integrate(sqrt(e*x^2 + d)*(b*arcsech(c*x) + a)*x^5, x)