\(\int x^5 \sqrt {d+e x^2} (a+b \sec ^{-1}(c x)) \, dx\) [111]

Optimal result

Integrand size = 23, antiderivative size = 403 \[ \int x^5 \sqrt {d+e x^2} \left (a+b \sec ^{-1}(c x)\right ) \, dx=\frac {b \left (23 c^4 d^2+12 c^2 d e-75 e^2\right ) x \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}{1680 c^5 e^2 \sqrt {c^2 x^2}}+\frac {b \left (29 c^2 d-25 e\right ) x \sqrt {-1+c^2 x^2} \left (d+e x^2\right )^{3/2}}{840 c^3 e^2 \sqrt {c^2 x^2}}-\frac {b x \sqrt {-1+c^2 x^2} \left (d+e x^2\right )^{5/2}}{42 c e^2 \sqrt {c^2 x^2}}+\frac {d^2 \left (d+e x^2\right )^{3/2} \left (a+b \sec ^{-1}(c x)\right )}{3 e^3}-\frac {2 d \left (d+e x^2\right )^{5/2} \left (a+b \sec ^{-1}(c x)\right )}{5 e^3}+\frac {\left (d+e x^2\right )^{7/2} \left (a+b \sec ^{-1}(c x)\right )}{7 e^3}+\frac {8 b c d^{7/2} x \arctan \left (\frac {\sqrt {d+e x^2}}{\sqrt {d} \sqrt {-1+c^2 x^2}}\right )}{105 e^3 \sqrt {c^2 x^2}}-\frac {b \left (105 c^6 d^3-35 c^4 d^2 e+63 c^2 d e^2+75 e^3\right ) x \text {arctanh}\left (\frac {\sqrt {e} \sqrt {-1+c^2 x^2}}{c \sqrt {d+e x^2}}\right )}{1680 c^6 e^{5/2} \sqrt {c^2 x^2}} \]

[Out]

1/3*d^2*(e*x^2+d)^(3/2)*(a+b*arcsec(c*x))/e^3-2/5*d*(e*x^2+d)^(5/2)*(a+b*arcsec(c*x))/e^3+1/7*(e*x^2+d)^(7/2)*
(a+b*arcsec(c*x))/e^3+8/105*b*c*d^(7/2)*x*arctan((e*x^2+d)^(1/2)/d^(1/2)/(c^2*x^2-1)^(1/2))/e^3/(c^2*x^2)^(1/2
)-1/1680*b*(105*c^6*d^3-35*c^4*d^2*e+63*c^2*d*e^2+75*e^3)*x*arctanh(e^(1/2)*(c^2*x^2-1)^(1/2)/c/(e*x^2+d)^(1/2
))/c^6/e^(5/2)/(c^2*x^2)^(1/2)+1/840*b*(29*c^2*d-25*e)*x*(e*x^2+d)^(3/2)*(c^2*x^2-1)^(1/2)/c^3/e^2/(c^2*x^2)^(
1/2)-1/42*b*x*(e*x^2+d)^(5/2)*(c^2*x^2-1)^(1/2)/c/e^2/(c^2*x^2)^(1/2)+1/1680*b*(23*c^4*d^2+12*c^2*d*e-75*e^2)*
x*(c^2*x^2-1)^(1/2)*(e*x^2+d)^(1/2)/c^5/e^2/(c^2*x^2)^(1/2)

Rubi [A] (verified)

Time = 0.88 (sec) , antiderivative size = 403, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.522, Rules used = {272, 45, 5346, 12, 1629, 159, 163, 65, 223, 212, 95, 210} \[ \int x^5 \sqrt {d+e x^2} \left (a+b \sec ^{-1}(c x)\right ) \, dx=\frac {d^2 \left (d+e x^2\right )^{3/2} \left (a+b \sec ^{-1}(c x)\right )}{3 e^3}-\frac {2 d \left (d+e x^2\right )^{5/2} \left (a+b \sec ^{-1}(c x)\right )}{5 e^3}+\frac {\left (d+e x^2\right )^{7/2} \left (a+b \sec ^{-1}(c x)\right )}{7 e^3}+\frac {8 b c d^{7/2} x \arctan \left (\frac {\sqrt {d+e x^2}}{\sqrt {d} \sqrt {c^2 x^2-1}}\right )}{105 e^3 \sqrt {c^2 x^2}}-\frac {b x \left (105 c^6 d^3-35 c^4 d^2 e+63 c^2 d e^2+75 e^3\right ) \text {arctanh}\left (\frac {\sqrt {e} \sqrt {c^2 x^2-1}}{c \sqrt {d+e x^2}}\right )}{1680 c^6 e^{5/2} \sqrt {c^2 x^2}}-\frac {b x \sqrt {c^2 x^2-1} \left (d+e x^2\right )^{5/2}}{42 c e^2 \sqrt {c^2 x^2}}+\frac {b x \sqrt {c^2 x^2-1} \left (29 c^2 d-25 e\right ) \left (d+e x^2\right )^{3/2}}{840 c^3 e^2 \sqrt {c^2 x^2}}+\frac {b x \sqrt {c^2 x^2-1} \left (23 c^4 d^2+12 c^2 d e-75 e^2\right ) \sqrt {d+e x^2}}{1680 c^5 e^2 \sqrt {c^2 x^2}} \]

[In]

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

[Out]

(b*(23*c^4*d^2 + 12*c^2*d*e - 75*e^2)*x*Sqrt[-1 + c^2*x^2]*Sqrt[d + e*x^2])/(1680*c^5*e^2*Sqrt[c^2*x^2]) + (b*
(29*c^2*d - 25*e)*x*Sqrt[-1 + c^2*x^2]*(d + e*x^2)^(3/2))/(840*c^3*e^2*Sqrt[c^2*x^2]) - (b*x*Sqrt[-1 + c^2*x^2
]*(d + e*x^2)^(5/2))/(42*c*e^2*Sqrt[c^2*x^2]) + (d^2*(d + e*x^2)^(3/2)*(a + b*ArcSec[c*x]))/(3*e^3) - (2*d*(d
+ e*x^2)^(5/2)*(a + b*ArcSec[c*x]))/(5*e^3) + ((d + e*x^2)^(7/2)*(a + b*ArcSec[c*x]))/(7*e^3) + (8*b*c*d^(7/2)
*x*ArcTan[Sqrt[d + e*x^2]/(Sqrt[d]*Sqrt[-1 + c^2*x^2])])/(105*e^3*Sqrt[c^2*x^2]) - (b*(105*c^6*d^3 - 35*c^4*d^
2*e + 63*c^2*d*e^2 + 75*e^3)*x*ArcTanh[(Sqrt[e]*Sqrt[-1 + c^2*x^2])/(c*Sqrt[d + e*x^2])])/(1680*c^6*e^(5/2)*Sq
rt[c^2*x^2])

Rule 12

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

Rule 45

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 65

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 95

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 159

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 163

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 210

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])

Rule 212

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

Rule 223

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 272

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 1629

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]

Rule 5346

Int[((a_.) + ArcSec[(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*ArcSec[c*x], u, x] - Dist[b*c*(x/Sqrt[c^2*x^2]), Int[SimplifyI
ntegrand[u/(x*Sqrt[c^2*x^2 - 1]), 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])) || (I
LtQ[(m + 2*p + 1)/2, 0] &&  !ILtQ[(m - 1)/2, 0]))

Rubi steps \begin{align*} \text {integral}& = \frac {d^2 \left (d+e x^2\right )^{3/2} \left (a+b \sec ^{-1}(c x)\right )}{3 e^3}-\frac {2 d \left (d+e x^2\right )^{5/2} \left (a+b \sec ^{-1}(c x)\right )}{5 e^3}+\frac {\left (d+e x^2\right )^{7/2} \left (a+b \sec ^{-1}(c x)\right )}{7 e^3}-\frac {(b c x) \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}{\sqrt {c^2 x^2}} \\ & = \frac {d^2 \left (d+e x^2\right )^{3/2} \left (a+b \sec ^{-1}(c x)\right )}{3 e^3}-\frac {2 d \left (d+e x^2\right )^{5/2} \left (a+b \sec ^{-1}(c x)\right )}{5 e^3}+\frac {\left (d+e x^2\right )^{7/2} \left (a+b \sec ^{-1}(c x)\right )}{7 e^3}-\frac {(b c x) \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 \sqrt {c^2 x^2}} \\ & = \frac {d^2 \left (d+e x^2\right )^{3/2} \left (a+b \sec ^{-1}(c x)\right )}{3 e^3}-\frac {2 d \left (d+e x^2\right )^{5/2} \left (a+b \sec ^{-1}(c x)\right )}{5 e^3}+\frac {\left (d+e x^2\right )^{7/2} \left (a+b \sec ^{-1}(c x)\right )}{7 e^3}-\frac {(b c x) \text {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 \sqrt {c^2 x^2}} \\ & = -\frac {b x \sqrt {-1+c^2 x^2} \left (d+e x^2\right )^{5/2}}{42 c e^2 \sqrt {c^2 x^2}}+\frac {d^2 \left (d+e x^2\right )^{3/2} \left (a+b \sec ^{-1}(c x)\right )}{3 e^3}-\frac {2 d \left (d+e x^2\right )^{5/2} \left (a+b \sec ^{-1}(c x)\right )}{5 e^3}+\frac {\left (d+e x^2\right )^{7/2} \left (a+b \sec ^{-1}(c x)\right )}{7 e^3}-\frac {(b x) \text {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 e^4 \sqrt {c^2 x^2}} \\ & = \frac {b \left (29 c^2 d-25 e\right ) x \sqrt {-1+c^2 x^2} \left (d+e x^2\right )^{3/2}}{840 c^3 e^2 \sqrt {c^2 x^2}}-\frac {b x \sqrt {-1+c^2 x^2} \left (d+e x^2\right )^{5/2}}{42 c e^2 \sqrt {c^2 x^2}}+\frac {d^2 \left (d+e x^2\right )^{3/2} \left (a+b \sec ^{-1}(c x)\right )}{3 e^3}-\frac {2 d \left (d+e x^2\right )^{5/2} \left (a+b \sec ^{-1}(c x)\right )}{5 e^3}+\frac {\left (d+e x^2\right )^{7/2} \left (a+b \sec ^{-1}(c x)\right )}{7 e^3}-\frac {(b x) \text {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^3 e^4 \sqrt {c^2 x^2}} \\ & = \frac {b \left (23 c^4 d^2+12 c^2 d e-75 e^2\right ) x \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}{1680 c^5 e^2 \sqrt {c^2 x^2}}+\frac {b \left (29 c^2 d-25 e\right ) x \sqrt {-1+c^2 x^2} \left (d+e x^2\right )^{3/2}}{840 c^3 e^2 \sqrt {c^2 x^2}}-\frac {b x \sqrt {-1+c^2 x^2} \left (d+e x^2\right )^{5/2}}{42 c e^2 \sqrt {c^2 x^2}}+\frac {d^2 \left (d+e x^2\right )^{3/2} \left (a+b \sec ^{-1}(c x)\right )}{3 e^3}-\frac {2 d \left (d+e x^2\right )^{5/2} \left (a+b \sec ^{-1}(c x)\right )}{5 e^3}+\frac {\left (d+e x^2\right )^{7/2} \left (a+b \sec ^{-1}(c x)\right )}{7 e^3}-\frac {(b x) \text {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^5 e^4 \sqrt {c^2 x^2}} \\ & = \frac {b \left (23 c^4 d^2+12 c^2 d e-75 e^2\right ) x \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}{1680 c^5 e^2 \sqrt {c^2 x^2}}+\frac {b \left (29 c^2 d-25 e\right ) x \sqrt {-1+c^2 x^2} \left (d+e x^2\right )^{3/2}}{840 c^3 e^2 \sqrt {c^2 x^2}}-\frac {b x \sqrt {-1+c^2 x^2} \left (d+e x^2\right )^{5/2}}{42 c e^2 \sqrt {c^2 x^2}}+\frac {d^2 \left (d+e x^2\right )^{3/2} \left (a+b \sec ^{-1}(c x)\right )}{3 e^3}-\frac {2 d \left (d+e x^2\right )^{5/2} \left (a+b \sec ^{-1}(c x)\right )}{5 e^3}+\frac {\left (d+e x^2\right )^{7/2} \left (a+b \sec ^{-1}(c x)\right )}{7 e^3}-\frac {\left (4 b c d^4 x\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {-1+c^2 x} \sqrt {d+e x}} \, dx,x,x^2\right )}{105 e^3 \sqrt {c^2 x^2}}-\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 ) x\right ) \text {Subst}\left (\int \frac {1}{\sqrt {-1+c^2 x} \sqrt {d+e x}} \, dx,x,x^2\right )}{3360 c^5 e^2 \sqrt {c^2 x^2}} \\ & = \frac {b \left (23 c^4 d^2+12 c^2 d e-75 e^2\right ) x \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}{1680 c^5 e^2 \sqrt {c^2 x^2}}+\frac {b \left (29 c^2 d-25 e\right ) x \sqrt {-1+c^2 x^2} \left (d+e x^2\right )^{3/2}}{840 c^3 e^2 \sqrt {c^2 x^2}}-\frac {b x \sqrt {-1+c^2 x^2} \left (d+e x^2\right )^{5/2}}{42 c e^2 \sqrt {c^2 x^2}}+\frac {d^2 \left (d+e x^2\right )^{3/2} \left (a+b \sec ^{-1}(c x)\right )}{3 e^3}-\frac {2 d \left (d+e x^2\right )^{5/2} \left (a+b \sec ^{-1}(c x)\right )}{5 e^3}+\frac {\left (d+e x^2\right )^{7/2} \left (a+b \sec ^{-1}(c x)\right )}{7 e^3}-\frac {\left (8 b c d^4 x\right ) \text {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 \sqrt {c^2 x^2}}-\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 ) x\right ) \text {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^7 e^2 \sqrt {c^2 x^2}} \\ & = \frac {b \left (23 c^4 d^2+12 c^2 d e-75 e^2\right ) x \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}{1680 c^5 e^2 \sqrt {c^2 x^2}}+\frac {b \left (29 c^2 d-25 e\right ) x \sqrt {-1+c^2 x^2} \left (d+e x^2\right )^{3/2}}{840 c^3 e^2 \sqrt {c^2 x^2}}-\frac {b x \sqrt {-1+c^2 x^2} \left (d+e x^2\right )^{5/2}}{42 c e^2 \sqrt {c^2 x^2}}+\frac {d^2 \left (d+e x^2\right )^{3/2} \left (a+b \sec ^{-1}(c x)\right )}{3 e^3}-\frac {2 d \left (d+e x^2\right )^{5/2} \left (a+b \sec ^{-1}(c x)\right )}{5 e^3}+\frac {\left (d+e x^2\right )^{7/2} \left (a+b \sec ^{-1}(c x)\right )}{7 e^3}+\frac {8 b c d^{7/2} x \arctan \left (\frac {\sqrt {d+e x^2}}{\sqrt {d} \sqrt {-1+c^2 x^2}}\right )}{105 e^3 \sqrt {c^2 x^2}}-\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 ) x\right ) \text {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^7 e^2 \sqrt {c^2 x^2}} \\ & = \frac {b \left (23 c^4 d^2+12 c^2 d e-75 e^2\right ) x \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}{1680 c^5 e^2 \sqrt {c^2 x^2}}+\frac {b \left (29 c^2 d-25 e\right ) x \sqrt {-1+c^2 x^2} \left (d+e x^2\right )^{3/2}}{840 c^3 e^2 \sqrt {c^2 x^2}}-\frac {b x \sqrt {-1+c^2 x^2} \left (d+e x^2\right )^{5/2}}{42 c e^2 \sqrt {c^2 x^2}}+\frac {d^2 \left (d+e x^2\right )^{3/2} \left (a+b \sec ^{-1}(c x)\right )}{3 e^3}-\frac {2 d \left (d+e x^2\right )^{5/2} \left (a+b \sec ^{-1}(c x)\right )}{5 e^3}+\frac {\left (d+e x^2\right )^{7/2} \left (a+b \sec ^{-1}(c x)\right )}{7 e^3}+\frac {8 b c d^{7/2} x \arctan \left (\frac {\sqrt {d+e x^2}}{\sqrt {d} \sqrt {-1+c^2 x^2}}\right )}{105 e^3 \sqrt {c^2 x^2}}-\frac {b \left (105 c^6 d^3-35 c^4 d^2 e+63 c^2 d e^2+75 e^3\right ) x \text {arctanh}\left (\frac {\sqrt {e} \sqrt {-1+c^2 x^2}}{c \sqrt {d+e x^2}}\right )}{1680 c^6 e^{5/2} \sqrt {c^2 x^2}} \\ \end{align*}

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 6 vs. order 3 in optimal.

Time = 1.62 (sec) , antiderivative size = 342, normalized size of antiderivative = 0.85 \[ \int x^5 \sqrt {d+e x^2} \left (a+b \sec ^{-1}(c x)\right ) \, dx=\frac {32 a \left (d+e x^2\right ) \left (8 d^3-4 d^2 e x^2+3 d e^2 x^4+15 e^3 x^6\right )-\frac {2 b e \sqrt {1-\frac {1}{c^2 x^2}} x \left (d+e x^2\right ) \left (75 e^2+2 c^2 e \left (19 d+25 e x^2\right )+c^4 \left (-41 d^2+22 d e x^2+40 e^2 x^4\right )\right )}{c^5}+\frac {128 b d^4 \sqrt {1+\frac {d}{e x^2}} \operatorname {AppellF1}\left (1,\frac {1}{2},\frac {1}{2},2,\frac {1}{c^2 x^2},-\frac {d}{e x^2}\right )}{c x}+\frac {b e \left (105 c^6 d^3-35 c^4 d^2 e+63 c^2 d e^2+75 e^3\right ) \sqrt {1-\frac {1}{c^2 x^2}} x^3 \sqrt {1+\frac {e x^2}{d}} \operatorname {AppellF1}\left (1,\frac {1}{2},\frac {1}{2},2,c^2 x^2,-\frac {e x^2}{d}\right )}{c^5 \sqrt {1-c^2 x^2}}+32 b \left (d+e x^2\right ) \left (8 d^3-4 d^2 e x^2+3 d e^2 x^4+15 e^3 x^6\right ) \sec ^{-1}(c x)}{3360 e^3 \sqrt {d+e x^2}} \]

[In]

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

[Out]

(32*a*(d + e*x^2)*(8*d^3 - 4*d^2*e*x^2 + 3*d*e^2*x^4 + 15*e^3*x^6) - (2*b*e*Sqrt[1 - 1/(c^2*x^2)]*x*(d + e*x^2
)*(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)))/c^5 + (128*b*d^4*Sqrt[1 + d/
(e*x^2)]*AppellF1[1, 1/2, 1/2, 2, 1/(c^2*x^2), -(d/(e*x^2))])/(c*x) + (b*e*(105*c^6*d^3 - 35*c^4*d^2*e + 63*c^
2*d*e^2 + 75*e^3)*Sqrt[1 - 1/(c^2*x^2)]*x^3*Sqrt[1 + (e*x^2)/d]*AppellF1[1, 1/2, 1/2, 2, c^2*x^2, -((e*x^2)/d)
])/(c^5*Sqrt[1 - c^2*x^2]) + 32*b*(d + e*x^2)*(8*d^3 - 4*d^2*e*x^2 + 3*d*e^2*x^4 + 15*e^3*x^6)*ArcSec[c*x])/(3
360*e^3*Sqrt[d + e*x^2])

Maple [F]

\[\int x^{5} \left (a +b \,\operatorname {arcsec}\left (c x \right )\right ) \sqrt {e \,x^{2}+d}d x\]

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 2.37 (sec) , antiderivative size = 1701, normalized size of antiderivative = 4.22 \[ \int x^5 \sqrt {d+e x^2} \left (a+b \sec ^{-1}(c x)\right ) \, dx=\text {Too large to display} \]

[In]

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

[Out]

[1/6720*(128*b*c^7*sqrt(-d)*d^3*log(((c^4*d^2 - 6*c^2*d*e + e^2)*x^4 - 8*(c^2*d^2 - d*e)*x^2 - 4*sqrt(c^2*x^2
- 1)*((c^2*d - e)*x^2 - 2*d)*sqrt(e*x^2 + d)*sqrt(-d) + 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^3*e*
x^2 + c^3*d - c*e)*sqrt(c^2*x^2 - 1)*sqrt(e*x^2 + d)*sqrt(e) + e^2) + 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 + 16*(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)*arcsec(c*x) - (40*b*c^5*e^3*x^4 - 41*b*c^5*d^2*e + 38*b*c^3*d*e^2 + 75*b*c*e^3 + 2*(11*b*c^5*d*e^2 + 25
*b*c^3*e^3)*x^2)*sqrt(c^2*x^2 - 1))*sqrt(e*x^2 + d))/(c^7*e^3), 1/6720*(256*b*c^7*d^(7/2)*arctan(-1/2*sqrt(c^2
*x^2 - 1)*((c^2*d - e)*x^2 - 2*d)*sqrt(e*x^2 + d)*sqrt(d)/(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)*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^3*e*x^2 + c^3*d - c*e)*sqrt(c^2*x^2 - 1)*sqrt(e*x^2 + d)*sqrt(e) + e^2) + 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 + 16*(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)*arcsec(c*x) - (40*b*c^5*e^3*x^4 - 41*b*c^5*d^2*e + 38*b*c^3*d*e^2 + 75
*b*c*e^3 + 2*(11*b*c^5*d*e^2 + 25*b*c^3*e^3)*x^2)*sqrt(c^2*x^2 - 1))*sqrt(e*x^2 + d))/(c^7*e^3), 1/3360*(64*b*
c^7*sqrt(-d)*d^3*log(((c^4*d^2 - 6*c^2*d*e + e^2)*x^4 - 8*(c^2*d^2 - d*e)*x^2 - 4*sqrt(c^2*x^2 - 1)*((c^2*d -
e)*x^2 - 2*d)*sqrt(e*x^2 + d)*sqrt(-d) + 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^2 + c^2*d - e)*sqrt(c^2*x^2 - 1)*sqrt(e*x^2 + d)*sqrt(-e)/(c^3*e^2*x^4 -
c*d*e + (c^3*d*e - c*e^2)*x^2)) + 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 + 16*(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)*arcsec(c*x) - (40*b*c^5*e^3*x
^4 - 41*b*c^5*d^2*e + 38*b*c^3*d*e^2 + 75*b*c*e^3 + 2*(11*b*c^5*d*e^2 + 25*b*c^3*e^3)*x^2)*sqrt(c^2*x^2 - 1))*
sqrt(e*x^2 + d))/(c^7*e^3), 1/3360*(128*b*c^7*d^(7/2)*arctan(-1/2*sqrt(c^2*x^2 - 1)*((c^2*d - e)*x^2 - 2*d)*sq
rt(e*x^2 + d)*sqrt(d)/(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^2 + c^2*d - e)*sqrt(c^2*x^2 - 1)*sqrt(e*x^2 + d)*sqrt(-e)/(c^
3*e^2*x^4 - c*d*e + (c^3*d*e - c*e^2)*x^2)) + 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 + 16*(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)*arcsec(c*x) - (40
*b*c^5*e^3*x^4 - 41*b*c^5*d^2*e + 38*b*c^3*d*e^2 + 75*b*c*e^3 + 2*(11*b*c^5*d*e^2 + 25*b*c^3*e^3)*x^2)*sqrt(c^
2*x^2 - 1))*sqrt(e*x^2 + d))/(c^7*e^3)]

Sympy [F]

\[ \int x^5 \sqrt {d+e x^2} \left (a+b \sec ^{-1}(c x)\right ) \, dx=\int x^{5} \left (a + b \operatorname {asec}{\left (c x \right )}\right ) \sqrt {d + e x^{2}}\, dx \]

[In]

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

[Out]

Integral(x**5*(a + b*asec(c*x))*sqrt(d + e*x**2), x)

Maxima [F(-2)]

Exception generated. \[ \int x^5 \sqrt {d+e x^2} \left (a+b \sec ^{-1}(c x)\right ) \, dx=\text {Exception raised: ValueError} \]

[In]

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

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(e>0)', see `assume?` for more
details)Is e

Giac [F]

\[ \int x^5 \sqrt {d+e x^2} \left (a+b \sec ^{-1}(c x)\right ) \, dx=\int { \sqrt {e x^{2} + d} {\left (b \operatorname {arcsec}\left (c x\right ) + a\right )} x^{5} \,d x } \]

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int x^5 \sqrt {d+e x^2} \left (a+b \sec ^{-1}(c x)\right ) \, dx=\int x^5\,\sqrt {e\,x^2+d}\,\left (a+b\,\mathrm {acos}\left (\frac {1}{c\,x}\right )\right ) \,d x \]

[In]

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

[Out]

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