3.128 \(\int x^3 (d+e x^2)^{3/2} (a+b \text{csch}^{-1}(c x)) \, dx\)

Optimal. Leaf size=384 \[ -\frac{d \left (d+e x^2\right )^{5/2} \left (a+b \text{csch}^{-1}(c x)\right )}{5 e^2}+\frac{\left (d+e x^2\right )^{7/2} \left (a+b \text{csch}^{-1}(c x)\right )}{7 e^2}-\frac{b x \sqrt{-c^2 x^2-1} \left (3 c^4 d^2+38 c^2 d e-25 e^2\right ) \sqrt{d+e x^2}}{560 c^5 e \sqrt{-c^2 x^2}}-\frac{2 b c d^{7/2} x \tan ^{-1}\left (\frac{\sqrt{d+e x^2}}{\sqrt{d} \sqrt{-c^2 x^2-1}}\right )}{35 e^2 \sqrt{-c^2 x^2}}-\frac{b x \left (35 c^4 d^2 e+35 c^6 d^3-63 c^2 d e^2+25 e^3\right ) \tan ^{-1}\left (\frac{\sqrt{e} \sqrt{-c^2 x^2-1}}{c \sqrt{d+e x^2}}\right )}{560 c^6 e^{3/2} \sqrt{-c^2 x^2}}+\frac{b x \sqrt{-c^2 x^2-1} \left (d+e x^2\right )^{5/2}}{42 c e \sqrt{-c^2 x^2}}+\frac{b x \sqrt{-c^2 x^2-1} \left (13 c^2 d-25 e\right ) \left (d+e x^2\right )^{3/2}}{840 c^3 e \sqrt{-c^2 x^2}} \]

[Out]

-(b*(3*c^4*d^2 + 38*c^2*d*e - 25*e^2)*x*Sqrt[-1 - c^2*x^2]*Sqrt[d + e*x^2])/(560*c^5*e*Sqrt[-(c^2*x^2)]) + (b*
(13*c^2*d - 25*e)*x*Sqrt[-1 - c^2*x^2]*(d + e*x^2)^(3/2))/(840*c^3*e*Sqrt[-(c^2*x^2)]) + (b*x*Sqrt[-1 - c^2*x^
2]*(d + e*x^2)^(5/2))/(42*c*e*Sqrt[-(c^2*x^2)]) - (d*(d + e*x^2)^(5/2)*(a + b*ArcCsch[c*x]))/(5*e^2) + ((d + e
*x^2)^(7/2)*(a + b*ArcCsch[c*x]))/(7*e^2) - (b*(35*c^6*d^3 + 35*c^4*d^2*e - 63*c^2*d*e^2 + 25*e^3)*x*ArcTan[(S
qrt[e]*Sqrt[-1 - c^2*x^2])/(c*Sqrt[d + e*x^2])])/(560*c^6*e^(3/2)*Sqrt[-(c^2*x^2)]) - (2*b*c*d^(7/2)*x*ArcTan[
Sqrt[d + e*x^2]/(Sqrt[d]*Sqrt[-1 - c^2*x^2])])/(35*e^2*Sqrt[-(c^2*x^2)])

________________________________________________________________________________________

Rubi [A]  time = 0.51424, antiderivative size = 384, 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, 6302, 12, 573, 154, 157, 63, 217, 203, 93, 204} \[ -\frac{d \left (d+e x^2\right )^{5/2} \left (a+b \text{csch}^{-1}(c x)\right )}{5 e^2}+\frac{\left (d+e x^2\right )^{7/2} \left (a+b \text{csch}^{-1}(c x)\right )}{7 e^2}-\frac{b x \sqrt{-c^2 x^2-1} \left (3 c^4 d^2+38 c^2 d e-25 e^2\right ) \sqrt{d+e x^2}}{560 c^5 e \sqrt{-c^2 x^2}}-\frac{2 b c d^{7/2} x \tan ^{-1}\left (\frac{\sqrt{d+e x^2}}{\sqrt{d} \sqrt{-c^2 x^2-1}}\right )}{35 e^2 \sqrt{-c^2 x^2}}-\frac{b x \left (35 c^4 d^2 e+35 c^6 d^3-63 c^2 d e^2+25 e^3\right ) \tan ^{-1}\left (\frac{\sqrt{e} \sqrt{-c^2 x^2-1}}{c \sqrt{d+e x^2}}\right )}{560 c^6 e^{3/2} \sqrt{-c^2 x^2}}+\frac{b x \sqrt{-c^2 x^2-1} \left (d+e x^2\right )^{5/2}}{42 c e \sqrt{-c^2 x^2}}+\frac{b x \sqrt{-c^2 x^2-1} \left (13 c^2 d-25 e\right ) \left (d+e x^2\right )^{3/2}}{840 c^3 e \sqrt{-c^2 x^2}} \]

Antiderivative was successfully verified.

[In]

Int[x^3*(d + e*x^2)^(3/2)*(a + b*ArcCsch[c*x]),x]

[Out]

-(b*(3*c^4*d^2 + 38*c^2*d*e - 25*e^2)*x*Sqrt[-1 - c^2*x^2]*Sqrt[d + e*x^2])/(560*c^5*e*Sqrt[-(c^2*x^2)]) + (b*
(13*c^2*d - 25*e)*x*Sqrt[-1 - c^2*x^2]*(d + e*x^2)^(3/2))/(840*c^3*e*Sqrt[-(c^2*x^2)]) + (b*x*Sqrt[-1 - c^2*x^
2]*(d + e*x^2)^(5/2))/(42*c*e*Sqrt[-(c^2*x^2)]) - (d*(d + e*x^2)^(5/2)*(a + b*ArcCsch[c*x]))/(5*e^2) + ((d + e
*x^2)^(7/2)*(a + b*ArcCsch[c*x]))/(7*e^2) - (b*(35*c^6*d^3 + 35*c^4*d^2*e - 63*c^2*d*e^2 + 25*e^3)*x*ArcTan[(S
qrt[e]*Sqrt[-1 - c^2*x^2])/(c*Sqrt[d + e*x^2])])/(560*c^6*e^(3/2)*Sqrt[-(c^2*x^2)]) - (2*b*c*d^(7/2)*x*ArcTan[
Sqrt[d + e*x^2]/(Sqrt[d]*Sqrt[-1 - c^2*x^2])])/(35*e^2*Sqrt[-(c^2*x^2)])

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 6302

Int[((a_.) + ArcCsch[(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*ArcCsch[c*x], u, x] - Dist[(b*c*x)/Sqrt[-(c^2*x^2)], Int[Simp
lifyIntegrand[u/(x*Sqrt[-1 - c^2*x^2]), 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 573

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.)*((e_) + (f_.)*(x_)^(n_))^(r_.), x
_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a + b*x)^p*(c + d*x)^q*(e + f*x)^r, x], x, x^n],
x] /; FreeQ[{a, b, c, d, e, f, m, n, p, q, r}, x] && IntegerQ[Simplify[(m + 1)/n]]

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 204

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

Rubi steps

\begin{align*} \int x^3 \left (d+e x^2\right )^{3/2} \left (a+b \text{csch}^{-1}(c x)\right ) \, dx &=-\frac{d \left (d+e x^2\right )^{5/2} \left (a+b \text{csch}^{-1}(c x)\right )}{5 e^2}+\frac{\left (d+e x^2\right )^{7/2} \left (a+b \text{csch}^{-1}(c x)\right )}{7 e^2}-\frac{(b c x) \int \frac{\left (d+e x^2\right )^{5/2} \left (-2 d+5 e x^2\right )}{35 e^2 x \sqrt{-1-c^2 x^2}} \, dx}{\sqrt{-c^2 x^2}}\\ &=-\frac{d \left (d+e x^2\right )^{5/2} \left (a+b \text{csch}^{-1}(c x)\right )}{5 e^2}+\frac{\left (d+e x^2\right )^{7/2} \left (a+b \text{csch}^{-1}(c x)\right )}{7 e^2}-\frac{(b c x) \int \frac{\left (d+e x^2\right )^{5/2} \left (-2 d+5 e x^2\right )}{x \sqrt{-1-c^2 x^2}} \, dx}{35 e^2 \sqrt{-c^2 x^2}}\\ &=-\frac{d \left (d+e x^2\right )^{5/2} \left (a+b \text{csch}^{-1}(c x)\right )}{5 e^2}+\frac{\left (d+e x^2\right )^{7/2} \left (a+b \text{csch}^{-1}(c x)\right )}{7 e^2}-\frac{(b c x) \operatorname{Subst}\left (\int \frac{(d+e x)^{5/2} (-2 d+5 e x)}{x \sqrt{-1-c^2 x}} \, dx,x,x^2\right )}{70 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 \sqrt{-c^2 x^2}}-\frac{d \left (d+e x^2\right )^{5/2} \left (a+b \text{csch}^{-1}(c x)\right )}{5 e^2}+\frac{\left (d+e x^2\right )^{7/2} \left (a+b \text{csch}^{-1}(c x)\right )}{7 e^2}+\frac{(b x) \operatorname{Subst}\left (\int \frac{(d+e x)^{3/2} \left (6 c^2 d^2-\frac{1}{2} \left (13 c^2 d-25 e\right ) e x\right )}{x \sqrt{-1-c^2 x}} \, dx,x,x^2\right )}{210 c e^2 \sqrt{-c^2 x^2}}\\ &=\frac{b \left (13 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 \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 \sqrt{-c^2 x^2}}-\frac{d \left (d+e x^2\right )^{5/2} \left (a+b \text{csch}^{-1}(c x)\right )}{5 e^2}+\frac{\left (d+e x^2\right )^{7/2} \left (a+b \text{csch}^{-1}(c x)\right )}{7 e^2}-\frac{(b x) \operatorname{Subst}\left (\int \frac{\sqrt{d+e x} \left (-12 c^4 d^3-\frac{3}{4} e \left (3 c^4 d^2+38 c^2 d e-25 e^2\right ) x\right )}{x \sqrt{-1-c^2 x}} \, dx,x,x^2\right )}{420 c^3 e^2 \sqrt{-c^2 x^2}}\\ &=-\frac{b \left (3 c^4 d^2+38 c^2 d e-25 e^2\right ) x \sqrt{-1-c^2 x^2} \sqrt{d+e x^2}}{560 c^5 e \sqrt{-c^2 x^2}}+\frac{b \left (13 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 \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 \sqrt{-c^2 x^2}}-\frac{d \left (d+e x^2\right )^{5/2} \left (a+b \text{csch}^{-1}(c x)\right )}{5 e^2}+\frac{\left (d+e x^2\right )^{7/2} \left (a+b \text{csch}^{-1}(c x)\right )}{7 e^2}+\frac{(b x) \operatorname{Subst}\left (\int \frac{12 c^6 d^4+\frac{3}{8} e \left (35 c^6 d^3+35 c^4 d^2 e-63 c^2 d e^2+25 e^3\right ) x}{x \sqrt{-1-c^2 x} \sqrt{d+e x}} \, dx,x,x^2\right )}{420 c^5 e^2 \sqrt{-c^2 x^2}}\\ &=-\frac{b \left (3 c^4 d^2+38 c^2 d e-25 e^2\right ) x \sqrt{-1-c^2 x^2} \sqrt{d+e x^2}}{560 c^5 e \sqrt{-c^2 x^2}}+\frac{b \left (13 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 \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 \sqrt{-c^2 x^2}}-\frac{d \left (d+e x^2\right )^{5/2} \left (a+b \text{csch}^{-1}(c x)\right )}{5 e^2}+\frac{\left (d+e x^2\right )^{7/2} \left (a+b \text{csch}^{-1}(c x)\right )}{7 e^2}+\frac{\left (b c d^4 x\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{-1-c^2 x} \sqrt{d+e x}} \, dx,x,x^2\right )}{35 e^2 \sqrt{-c^2 x^2}}+\frac{\left (b \left (35 c^6 d^3+35 c^4 d^2 e-63 c^2 d e^2+25 e^3\right ) x\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{-1-c^2 x} \sqrt{d+e x}} \, dx,x,x^2\right )}{1120 c^5 e \sqrt{-c^2 x^2}}\\ &=-\frac{b \left (3 c^4 d^2+38 c^2 d e-25 e^2\right ) x \sqrt{-1-c^2 x^2} \sqrt{d+e x^2}}{560 c^5 e \sqrt{-c^2 x^2}}+\frac{b \left (13 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 \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 \sqrt{-c^2 x^2}}-\frac{d \left (d+e x^2\right )^{5/2} \left (a+b \text{csch}^{-1}(c x)\right )}{5 e^2}+\frac{\left (d+e x^2\right )^{7/2} \left (a+b \text{csch}^{-1}(c x)\right )}{7 e^2}+\frac{\left (2 b c d^4 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 )}{35 e^2 \sqrt{-c^2 x^2}}-\frac{\left (b \left (35 c^6 d^3+35 c^4 d^2 e-63 c^2 d e^2+25 e^3\right ) 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 )}{560 c^7 e \sqrt{-c^2 x^2}}\\ &=-\frac{b \left (3 c^4 d^2+38 c^2 d e-25 e^2\right ) x \sqrt{-1-c^2 x^2} \sqrt{d+e x^2}}{560 c^5 e \sqrt{-c^2 x^2}}+\frac{b \left (13 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 \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 \sqrt{-c^2 x^2}}-\frac{d \left (d+e x^2\right )^{5/2} \left (a+b \text{csch}^{-1}(c x)\right )}{5 e^2}+\frac{\left (d+e x^2\right )^{7/2} \left (a+b \text{csch}^{-1}(c x)\right )}{7 e^2}-\frac{2 b c d^{7/2} x \tan ^{-1}\left (\frac{\sqrt{d+e x^2}}{\sqrt{d} \sqrt{-1-c^2 x^2}}\right )}{35 e^2 \sqrt{-c^2 x^2}}-\frac{\left (b \left (35 c^6 d^3+35 c^4 d^2 e-63 c^2 d e^2+25 e^3\right ) 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 )}{560 c^7 e \sqrt{-c^2 x^2}}\\ &=-\frac{b \left (3 c^4 d^2+38 c^2 d e-25 e^2\right ) x \sqrt{-1-c^2 x^2} \sqrt{d+e x^2}}{560 c^5 e \sqrt{-c^2 x^2}}+\frac{b \left (13 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 \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 \sqrt{-c^2 x^2}}-\frac{d \left (d+e x^2\right )^{5/2} \left (a+b \text{csch}^{-1}(c x)\right )}{5 e^2}+\frac{\left (d+e x^2\right )^{7/2} \left (a+b \text{csch}^{-1}(c x)\right )}{7 e^2}-\frac{b \left (35 c^6 d^3+35 c^4 d^2 e-63 c^2 d e^2+25 e^3\right ) x \tan ^{-1}\left (\frac{\sqrt{e} \sqrt{-1-c^2 x^2}}{c \sqrt{d+e x^2}}\right )}{560 c^6 e^{3/2} \sqrt{-c^2 x^2}}-\frac{2 b c d^{7/2} x \tan ^{-1}\left (\frac{\sqrt{d+e x^2}}{\sqrt{d} \sqrt{-1-c^2 x^2}}\right )}{35 e^2 \sqrt{-c^2 x^2}}\\ \end{align*}

Mathematica [C]  time = 0.734967, size = 318, normalized size = 0.83 \[ \frac{\sqrt{d+e x^2} \left (-48 a c^5 \left (2 d-5 e x^2\right ) \left (d+e x^2\right )^2+b e x \sqrt{\frac{1}{c^2 x^2}+1} \left (c^4 \left (57 d^2+106 d e x^2+40 e^2 x^4\right )-2 c^2 e \left (82 d+25 e x^2\right )+75 e^2\right )-48 b c^5 \text{csch}^{-1}(c x) \left (2 d-5 e x^2\right ) \left (d+e x^2\right )^2\right )}{1680 c^5 e^2}-\frac{b \left (\frac{e x^4 \sqrt{\frac{1}{c^2 x^2}+1} \left (35 c^4 d^2 e+35 c^6 d^3-63 c^2 d e^2+25 e^3\right ) \sqrt{\frac{e x^2}{d}+1} F_1\left (1;\frac{1}{2},\frac{1}{2};2;-c^2 x^2,-\frac{e x^2}{d}\right )}{\sqrt{c^2 x^2+1}}-32 c^4 d^4 \sqrt{\frac{d}{e x^2}+1} F_1\left (1;\frac{1}{2},\frac{1}{2};2;-\frac{1}{c^2 x^2},-\frac{d}{e x^2}\right )\right )}{1120 c^5 e^2 x \sqrt{d+e x^2}} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[x^3*(d + e*x^2)^(3/2)*(a + b*ArcCsch[c*x]),x]

[Out]

-(b*(-32*c^4*d^4*Sqrt[1 + d/(e*x^2)]*AppellF1[1, 1/2, 1/2, 2, -(1/(c^2*x^2)), -(d/(e*x^2))] + (e*(35*c^6*d^3 +
35*c^4*d^2*e - 63*c^2*d*e^2 + 25*e^3)*Sqrt[1 + 1/(c^2*x^2)]*x^4*Sqrt[1 + (e*x^2)/d]*AppellF1[1, 1/2, 1/2, 2,
-(c^2*x^2), -((e*x^2)/d)])/Sqrt[1 + c^2*x^2]))/(1120*c^5*e^2*x*Sqrt[d + e*x^2]) + (Sqrt[d + e*x^2]*(-48*a*c^5*
(2*d - 5*e*x^2)*(d + e*x^2)^2 + b*e*Sqrt[1 + 1/(c^2*x^2)]*x*(75*e^2 - 2*c^2*e*(82*d + 25*e*x^2) + c^4*(57*d^2
+ 106*d*e*x^2 + 40*e^2*x^4)) - 48*b*c^5*(2*d - 5*e*x^2)*(d + e*x^2)^2*ArcCsch[c*x]))/(1680*c^5*e^2)

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Maple [F]  time = 0.459, size = 0, normalized size = 0. \begin{align*} \int{x}^{3} \left ( e{x}^{2}+d \right ) ^{{\frac{3}{2}}} \left ( a+b{\rm arccsch} \left (cx\right ) \right ) \, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*(e*x^2+d)^(3/2)*(a+b*arccsch(c*x)),x)

[Out]

int(x^3*(e*x^2+d)^(3/2)*(a+b*arccsch(c*x)),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^3*(e*x^2+d)^(3/2)*(a+b*arccsch(c*x)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 19.9241, size = 4343, normalized size = 11.31 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(e*x^2+d)^(3/2)*(a+b*arccsch(c*x)),x, algorithm="fricas")

[Out]

[1/6720*(96*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) + 3*(35*b*c^6*d^3 + 35*b*c^4*d^
2*e - 63*b*c^2*d*e^2 + 25*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) + 192*(5*b*c
^7*e^3*x^6 + 8*b*c^7*d*e^2*x^4 + b*c^7*d^2*e*x^2 - 2*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 + 384*a*c^7*d*e^2*x^4 + 48*a*c^7*d^2*e*x^2 - 96*a*c^7*d^3 + (40*b*
c^6*e^3*x^5 + 2*(53*b*c^6*d*e^2 - 25*b*c^4*e^3)*x^3 + (57*b*c^6*d^2*e - 164*b*c^4*d*e^2 + 75*b*c^2*e^3)*x)*sqr
t((c^2*x^2 + 1)/(c^2*x^2)))*sqrt(e*x^2 + d))/(c^7*e^2), 1/3360*(48*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) + 3*(35*b*c^6*d^3 + 35*b*c^4*d^2*e - 63*b*c^2*d*e^2 + 25*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)) + 96*(5*b*c^7*e^3*x^6 + 8*b*c^7*d*e^2*x^4 + b*c^7*d^2*e*x^2 - 2*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 + 384*a*c^7*d*e^2*x^4 + 48*a*c^7*d^2*e*
x^2 - 96*a*c^7*d^3 + (40*b*c^6*e^3*x^5 + 2*(53*b*c^6*d*e^2 - 25*b*c^4*e^3)*x^3 + (57*b*c^6*d^2*e - 164*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^2), -1/6720*(192*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)) - 3*(35*b*c^6*d^3 + 35*b*c^4*d^2*e - 63*b*c^2*d*e^2 + 25*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) - 192*(5*b*c^7*e^3*x^6 + 8*b*c^7*d*e^2*x^4 + b*c^7*d^2*e*
x^2 - 2*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 +
384*a*c^7*d*e^2*x^4 + 48*a*c^7*d^2*e*x^2 - 96*a*c^7*d^3 + (40*b*c^6*e^3*x^5 + 2*(53*b*c^6*d*e^2 - 25*b*c^4*e^
3)*x^3 + (57*b*c^6*d^2*e - 164*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^2), -1/3360*(96*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)*sq
rt((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)) - 3*(35*b*c^6*d^3 + 35*b*c^4*d^2*e - 63
*b*c^2*d*e^2 + 25*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)) - 96*(5*b*c^7*e^3*x^6 + 8*b*c^7*d*e^2*x^4 + b*c
^7*d^2*e*x^2 - 2*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 + 384*a*c^7*d*e^2*x^4 + 48*a*c^7*d^2*e*x^2 - 96*a*c^7*d^3 + (40*b*c^6*e^3*x^5 + 2*(53*b*c^6*d*e^2 - 25
*b*c^4*e^3)*x^3 + (57*b*c^6*d^2*e - 164*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^2)]

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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**3*(e*x**2+d)**(3/2)*(a+b*acsch(c*x)),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (e x^{2} + d\right )}^{\frac{3}{2}}{\left (b \operatorname{arcsch}\left (c x\right ) + a\right )} x^{3}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(e*x^2+d)^(3/2)*(a+b*arccsch(c*x)),x, algorithm="giac")

[Out]

integrate((e*x^2 + d)^(3/2)*(b*arccsch(c*x) + a)*x^3, x)