3.57 \(\int \frac{x^3 (a+b \text{csch}^{-1}(c x))}{\sqrt{d+e x}} \, dx\)

Optimal. Leaf size=939 \[ \frac{64 b \sqrt{\frac{\sqrt{-c^2} (d+e x)}{\sqrt{-c^2} d+e}} \sqrt{c^2 x^2+1} \Pi \left (2;\sin ^{-1}\left (\frac{\sqrt{1-\sqrt{-c^2} x}}{\sqrt{2}}\right )|\frac{2 e}{\sqrt{-c^2} d+e}\right ) d^4}{35 c e^4 \sqrt{1+\frac{1}{c^2 x^2}} x \sqrt{d+e x}}-\frac{2 \sqrt{d+e x} \left (a+b \text{csch}^{-1}(c x)\right ) d^3}{e^4}-\frac{64 b c \sqrt{\frac{c^2 (d+e x)}{c^2 d-\sqrt{-c^2} e}} \sqrt{c^2 x^2+1} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{1-\sqrt{-c^2} x}}{\sqrt{2}}\right ),-\frac{2 \sqrt{-c^2} e}{c^2 d-\sqrt{-c^2} e}\right ) d^3}{35 \left (-c^2\right )^{3/2} e^3 \sqrt{1+\frac{1}{c^2 x^2}} x \sqrt{d+e x}}+\frac{2 (d+e x)^{3/2} \left (a+b \text{csch}^{-1}(c x)\right ) d^2}{e^4}+\frac{24 b c \sqrt{d+e x} \sqrt{c^2 x^2+1} E\left (\sin ^{-1}\left (\frac{\sqrt{1-\sqrt{-c^2} x}}{\sqrt{2}}\right )|-\frac{2 \sqrt{-c^2} e}{c^2 d-\sqrt{-c^2} e}\right ) d^2}{35 \left (-c^2\right )^{3/2} e^3 \sqrt{1+\frac{1}{c^2 x^2}} x \sqrt{\frac{c^2 (d+e x)}{c^2 d-\sqrt{-c^2} e}}}-\frac{4 b \sqrt{d+e x} \left (c^2 x^2+1\right ) d}{21 c^3 e^2 \sqrt{1+\frac{1}{c^2 x^2}} x}-\frac{6 (d+e x)^{5/2} \left (a+b \text{csch}^{-1}(c x)\right ) d}{5 e^4}-\frac{32 b c \left (c^2 d^2+e^2\right ) \sqrt{\frac{c^2 (d+e x)}{c^2 d-\sqrt{-c^2} e}} \sqrt{c^2 x^2+1} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{1-\sqrt{-c^2} x}}{\sqrt{2}}\right ),-\frac{2 \sqrt{-c^2} e}{c^2 d-\sqrt{-c^2} e}\right ) d}{105 \left (-c^2\right )^{5/2} e^3 \sqrt{1+\frac{1}{c^2 x^2}} x \sqrt{d+e x}}+\frac{4 b \sqrt{d+e x} \left (c^2 x^2+1\right )}{35 c^3 e \sqrt{1+\frac{1}{c^2 x^2}}}+\frac{2 (d+e x)^{7/2} \left (a+b \text{csch}^{-1}(c x)\right )}{7 e^4}+\frac{4 b c \left (2 c^2 d^2+9 e^2\right ) \sqrt{d+e x} \sqrt{c^2 x^2+1} E\left (\sin ^{-1}\left (\frac{\sqrt{1-\sqrt{-c^2} x}}{\sqrt{2}}\right )|-\frac{2 \sqrt{-c^2} e}{c^2 d-\sqrt{-c^2} e}\right )}{105 \left (-c^2\right )^{5/2} e^3 \sqrt{1+\frac{1}{c^2 x^2}} x \sqrt{\frac{c^2 (d+e x)}{c^2 d-\sqrt{-c^2} e}}} \]

[Out]

(4*b*Sqrt[d + e*x]*(1 + c^2*x^2))/(35*c^3*e*Sqrt[1 + 1/(c^2*x^2)]) - (4*b*d*Sqrt[d + e*x]*(1 + c^2*x^2))/(21*c
^3*e^2*Sqrt[1 + 1/(c^2*x^2)]*x) - (2*d^3*Sqrt[d + e*x]*(a + b*ArcCsch[c*x]))/e^4 + (2*d^2*(d + e*x)^(3/2)*(a +
b*ArcCsch[c*x]))/e^4 - (6*d*(d + e*x)^(5/2)*(a + b*ArcCsch[c*x]))/(5*e^4) + (2*(d + e*x)^(7/2)*(a + b*ArcCsch
[c*x]))/(7*e^4) + (24*b*c*d^2*Sqrt[d + e*x]*Sqrt[1 + c^2*x^2]*EllipticE[ArcSin[Sqrt[1 - Sqrt[-c^2]*x]/Sqrt[2]]
, (-2*Sqrt[-c^2]*e)/(c^2*d - Sqrt[-c^2]*e)])/(35*(-c^2)^(3/2)*e^3*Sqrt[1 + 1/(c^2*x^2)]*x*Sqrt[(c^2*(d + e*x))
/(c^2*d - Sqrt[-c^2]*e)]) + (4*b*c*(2*c^2*d^2 + 9*e^2)*Sqrt[d + e*x]*Sqrt[1 + c^2*x^2]*EllipticE[ArcSin[Sqrt[1
- Sqrt[-c^2]*x]/Sqrt[2]], (-2*Sqrt[-c^2]*e)/(c^2*d - Sqrt[-c^2]*e)])/(105*(-c^2)^(5/2)*e^3*Sqrt[1 + 1/(c^2*x^
2)]*x*Sqrt[(c^2*(d + e*x))/(c^2*d - Sqrt[-c^2]*e)]) - (64*b*c*d^3*Sqrt[(c^2*(d + e*x))/(c^2*d - Sqrt[-c^2]*e)]
*Sqrt[1 + c^2*x^2]*EllipticF[ArcSin[Sqrt[1 - Sqrt[-c^2]*x]/Sqrt[2]], (-2*Sqrt[-c^2]*e)/(c^2*d - Sqrt[-c^2]*e)]
)/(35*(-c^2)^(3/2)*e^3*Sqrt[1 + 1/(c^2*x^2)]*x*Sqrt[d + e*x]) - (32*b*c*d*(c^2*d^2 + e^2)*Sqrt[(c^2*(d + e*x))
/(c^2*d - Sqrt[-c^2]*e)]*Sqrt[1 + c^2*x^2]*EllipticF[ArcSin[Sqrt[1 - Sqrt[-c^2]*x]/Sqrt[2]], (-2*Sqrt[-c^2]*e)
/(c^2*d - Sqrt[-c^2]*e)])/(105*(-c^2)^(5/2)*e^3*Sqrt[1 + 1/(c^2*x^2)]*x*Sqrt[d + e*x]) + (64*b*d^4*Sqrt[(Sqrt[
-c^2]*(d + e*x))/(Sqrt[-c^2]*d + e)]*Sqrt[1 + c^2*x^2]*EllipticPi[2, ArcSin[Sqrt[1 - Sqrt[-c^2]*x]/Sqrt[2]], (
2*e)/(Sqrt[-c^2]*d + e)])/(35*c*e^4*Sqrt[1 + 1/(c^2*x^2)]*x*Sqrt[d + e*x])

________________________________________________________________________________________

Rubi [A]  time = 2.87133, antiderivative size = 939, normalized size of antiderivative = 1., number of steps used = 27, number of rules used = 17, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.81, Rules used = {43, 6310, 12, 6721, 6742, 719, 424, 944, 419, 932, 168, 538, 537, 833, 844, 942, 1654} \[ \frac{64 b \sqrt{\frac{\sqrt{-c^2} (d+e x)}{\sqrt{-c^2} d+e}} \sqrt{c^2 x^2+1} \Pi \left (2;\sin ^{-1}\left (\frac{\sqrt{1-\sqrt{-c^2} x}}{\sqrt{2}}\right )|\frac{2 e}{\sqrt{-c^2} d+e}\right ) d^4}{35 c e^4 \sqrt{1+\frac{1}{c^2 x^2}} x \sqrt{d+e x}}-\frac{2 \sqrt{d+e x} \left (a+b \text{csch}^{-1}(c x)\right ) d^3}{e^4}-\frac{64 b c \sqrt{\frac{c^2 (d+e x)}{c^2 d-\sqrt{-c^2} e}} \sqrt{c^2 x^2+1} F\left (\sin ^{-1}\left (\frac{\sqrt{1-\sqrt{-c^2} x}}{\sqrt{2}}\right )|-\frac{2 \sqrt{-c^2} e}{c^2 d-\sqrt{-c^2} e}\right ) d^3}{35 \left (-c^2\right )^{3/2} e^3 \sqrt{1+\frac{1}{c^2 x^2}} x \sqrt{d+e x}}+\frac{2 (d+e x)^{3/2} \left (a+b \text{csch}^{-1}(c x)\right ) d^2}{e^4}+\frac{24 b c \sqrt{d+e x} \sqrt{c^2 x^2+1} E\left (\sin ^{-1}\left (\frac{\sqrt{1-\sqrt{-c^2} x}}{\sqrt{2}}\right )|-\frac{2 \sqrt{-c^2} e}{c^2 d-\sqrt{-c^2} e}\right ) d^2}{35 \left (-c^2\right )^{3/2} e^3 \sqrt{1+\frac{1}{c^2 x^2}} x \sqrt{\frac{c^2 (d+e x)}{c^2 d-\sqrt{-c^2} e}}}-\frac{4 b \sqrt{d+e x} \left (c^2 x^2+1\right ) d}{21 c^3 e^2 \sqrt{1+\frac{1}{c^2 x^2}} x}-\frac{6 (d+e x)^{5/2} \left (a+b \text{csch}^{-1}(c x)\right ) d}{5 e^4}-\frac{32 b c \left (c^2 d^2+e^2\right ) \sqrt{\frac{c^2 (d+e x)}{c^2 d-\sqrt{-c^2} e}} \sqrt{c^2 x^2+1} F\left (\sin ^{-1}\left (\frac{\sqrt{1-\sqrt{-c^2} x}}{\sqrt{2}}\right )|-\frac{2 \sqrt{-c^2} e}{c^2 d-\sqrt{-c^2} e}\right ) d}{105 \left (-c^2\right )^{5/2} e^3 \sqrt{1+\frac{1}{c^2 x^2}} x \sqrt{d+e x}}+\frac{4 b \sqrt{d+e x} \left (c^2 x^2+1\right )}{35 c^3 e \sqrt{1+\frac{1}{c^2 x^2}}}+\frac{2 (d+e x)^{7/2} \left (a+b \text{csch}^{-1}(c x)\right )}{7 e^4}+\frac{4 b c \left (2 c^2 d^2+9 e^2\right ) \sqrt{d+e x} \sqrt{c^2 x^2+1} E\left (\sin ^{-1}\left (\frac{\sqrt{1-\sqrt{-c^2} x}}{\sqrt{2}}\right )|-\frac{2 \sqrt{-c^2} e}{c^2 d-\sqrt{-c^2} e}\right )}{105 \left (-c^2\right )^{5/2} e^3 \sqrt{1+\frac{1}{c^2 x^2}} x \sqrt{\frac{c^2 (d+e x)}{c^2 d-\sqrt{-c^2} e}}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(4*b*Sqrt[d + e*x]*(1 + c^2*x^2))/(35*c^3*e*Sqrt[1 + 1/(c^2*x^2)]) - (4*b*d*Sqrt[d + e*x]*(1 + c^2*x^2))/(21*c
^3*e^2*Sqrt[1 + 1/(c^2*x^2)]*x) - (2*d^3*Sqrt[d + e*x]*(a + b*ArcCsch[c*x]))/e^4 + (2*d^2*(d + e*x)^(3/2)*(a +
b*ArcCsch[c*x]))/e^4 - (6*d*(d + e*x)^(5/2)*(a + b*ArcCsch[c*x]))/(5*e^4) + (2*(d + e*x)^(7/2)*(a + b*ArcCsch
[c*x]))/(7*e^4) + (24*b*c*d^2*Sqrt[d + e*x]*Sqrt[1 + c^2*x^2]*EllipticE[ArcSin[Sqrt[1 - Sqrt[-c^2]*x]/Sqrt[2]]
, (-2*Sqrt[-c^2]*e)/(c^2*d - Sqrt[-c^2]*e)])/(35*(-c^2)^(3/2)*e^3*Sqrt[1 + 1/(c^2*x^2)]*x*Sqrt[(c^2*(d + e*x))
/(c^2*d - Sqrt[-c^2]*e)]) + (4*b*c*(2*c^2*d^2 + 9*e^2)*Sqrt[d + e*x]*Sqrt[1 + c^2*x^2]*EllipticE[ArcSin[Sqrt[1
- Sqrt[-c^2]*x]/Sqrt[2]], (-2*Sqrt[-c^2]*e)/(c^2*d - Sqrt[-c^2]*e)])/(105*(-c^2)^(5/2)*e^3*Sqrt[1 + 1/(c^2*x^
2)]*x*Sqrt[(c^2*(d + e*x))/(c^2*d - Sqrt[-c^2]*e)]) - (64*b*c*d^3*Sqrt[(c^2*(d + e*x))/(c^2*d - Sqrt[-c^2]*e)]
*Sqrt[1 + c^2*x^2]*EllipticF[ArcSin[Sqrt[1 - Sqrt[-c^2]*x]/Sqrt[2]], (-2*Sqrt[-c^2]*e)/(c^2*d - Sqrt[-c^2]*e)]
)/(35*(-c^2)^(3/2)*e^3*Sqrt[1 + 1/(c^2*x^2)]*x*Sqrt[d + e*x]) - (32*b*c*d*(c^2*d^2 + e^2)*Sqrt[(c^2*(d + e*x))
/(c^2*d - Sqrt[-c^2]*e)]*Sqrt[1 + c^2*x^2]*EllipticF[ArcSin[Sqrt[1 - Sqrt[-c^2]*x]/Sqrt[2]], (-2*Sqrt[-c^2]*e)
/(c^2*d - Sqrt[-c^2]*e)])/(105*(-c^2)^(5/2)*e^3*Sqrt[1 + 1/(c^2*x^2)]*x*Sqrt[d + e*x]) + (64*b*d^4*Sqrt[(Sqrt[
-c^2]*(d + e*x))/(Sqrt[-c^2]*d + e)]*Sqrt[1 + c^2*x^2]*EllipticPi[2, ArcSin[Sqrt[1 - Sqrt[-c^2]*x]/Sqrt[2]], (
2*e)/(Sqrt[-c^2]*d + e)])/(35*c*e^4*Sqrt[1 + 1/(c^2*x^2)]*x*Sqrt[d + e*x])

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 6310

Int[((a_.) + ArcCsch[(c_.)*(x_)]*(b_.))*(u_), x_Symbol] :> With[{v = IntHide[u, x]}, Dist[a + b*ArcCsch[c*x],
v, x] + Dist[b/c, Int[SimplifyIntegrand[v/(x^2*Sqrt[1 + 1/(c^2*x^2)]), x], x], x] /; InverseFunctionFreeQ[v, x
]] /; FreeQ[{a, b, c}, x]

Rule 12

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

Rule 6721

Int[(u_.)*((a_.) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[(b^IntPart[p]*(a + b*x^n)^FracPart[p])/(x^(n*FracP
art[p])*(1 + a/(x^n*b))^FracPart[p]), Int[u*x^(n*p)*(1 + a/(x^n*b))^p, x], x] /; FreeQ[{a, b, p}, x] &&  !Inte
gerQ[p] && ILtQ[n, 0] &&  !RationalFunctionQ[u, x] && IntegerQ[p + 1/2]

Rule 6742

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rule 719

Int[((d_) + (e_.)*(x_))^(m_)/Sqrt[(a_) + (c_.)*(x_)^2], x_Symbol] :> Dist[(2*a*Rt[-(c/a), 2]*(d + e*x)^m*Sqrt[
1 + (c*x^2)/a])/(c*Sqrt[a + c*x^2]*((c*(d + e*x))/(c*d - a*e*Rt[-(c/a), 2]))^m), Subst[Int[(1 + (2*a*e*Rt[-(c/
a), 2]*x^2)/(c*d - a*e*Rt[-(c/a), 2]))^m/Sqrt[1 - x^2], x], x, Sqrt[(1 - Rt[-(c/a), 2]*x)/2]], x] /; FreeQ[{a,
c, d, e}, x] && NeQ[c*d^2 + a*e^2, 0] && EqQ[m^2, 1/4]

Rule 424

Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[(Sqrt[a]*EllipticE[ArcSin[Rt[-(d/c)
, 2]*x], (b*c)/(a*d)])/(Sqrt[c]*Rt[-(d/c), 2]), x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[
a, 0]

Rule 944

Int[Sqrt[(f_.) + (g_.)*(x_)]/(((d_.) + (e_.)*(x_))*Sqrt[(a_) + (c_.)*(x_)^2]), x_Symbol] :> Dist[g/e, Int[1/(S
qrt[f + g*x]*Sqrt[a + c*x^2]), x], x] + Dist[(e*f - d*g)/e, Int[1/((d + e*x)*Sqrt[f + g*x]*Sqrt[a + c*x^2]), x
], x] /; FreeQ[{a, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] && NeQ[c*d^2 + a*e^2, 0]

Rule 419

Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> Simp[(1*EllipticF[ArcSin[Rt[-(d/c),
2]*x], (b*c)/(a*d)])/(Sqrt[a]*Sqrt[c]*Rt[-(d/c), 2]), x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] &
& GtQ[a, 0] &&  !(NegQ[b/a] && SimplerSqrtQ[-(b/a), -(d/c)])

Rule 932

Int[1/(((d_.) + (e_.)*(x_))*Sqrt[(f_.) + (g_.)*(x_)]*Sqrt[(a_) + (c_.)*(x_)^2]), x_Symbol] :> With[{q = Rt[-(c
/a), 2]}, Dist[1/Sqrt[a], Int[1/((d + e*x)*Sqrt[f + g*x]*Sqrt[1 - q*x]*Sqrt[1 + q*x]), x], x]] /; FreeQ[{a, c,
d, e, f, g}, x] && NeQ[e*f - d*g, 0] && NeQ[c*d^2 + a*e^2, 0] && GtQ[a, 0]

Rule 168

Int[1/(((a_.) + (b_.)*(x_))*Sqrt[(c_.) + (d_.)*(x_)]*Sqrt[(e_.) + (f_.)*(x_)]*Sqrt[(g_.) + (h_.)*(x_)]), x_Sym
bol] :> Dist[-2, Subst[Int[1/(Simp[b*c - a*d - b*x^2, x]*Sqrt[Simp[(d*e - c*f)/d + (f*x^2)/d, x]]*Sqrt[Simp[(d
*g - c*h)/d + (h*x^2)/d, x]]), x], x, Sqrt[c + d*x]], x] /; FreeQ[{a, b, c, d, e, f, g, h}, x] && GtQ[(d*e - c
*f)/d, 0]

Rule 538

Int[1/(((a_) + (b_.)*(x_)^2)*Sqrt[(c_) + (d_.)*(x_)^2]*Sqrt[(e_) + (f_.)*(x_)^2]), x_Symbol] :> Dist[Sqrt[1 +
(d*x^2)/c]/Sqrt[c + d*x^2], Int[1/((a + b*x^2)*Sqrt[1 + (d*x^2)/c]*Sqrt[e + f*x^2]), x], x] /; FreeQ[{a, b, c,
d, e, f}, x] &&  !GtQ[c, 0]

Rule 537

Int[1/(((a_) + (b_.)*(x_)^2)*Sqrt[(c_) + (d_.)*(x_)^2]*Sqrt[(e_) + (f_.)*(x_)^2]), x_Symbol] :> Simp[(1*Ellipt
icPi[(b*c)/(a*d), ArcSin[Rt[-(d/c), 2]*x], (c*f)/(d*e)])/(a*Sqrt[c]*Sqrt[e]*Rt[-(d/c), 2]), x] /; FreeQ[{a, b,
c, d, e, f}, x] &&  !GtQ[d/c, 0] && GtQ[c, 0] && GtQ[e, 0] &&  !( !GtQ[f/e, 0] && SimplerSqrtQ[-(f/e), -(d/c)
])

Rule 833

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(g*(d + e*x)
^m*(a + c*x^2)^(p + 1))/(c*(m + 2*p + 2)), x] + Dist[1/(c*(m + 2*p + 2)), Int[(d + e*x)^(m - 1)*(a + c*x^2)^p*
Simp[c*d*f*(m + 2*p + 2) - a*e*g*m + c*(e*f*(m + 2*p + 2) + d*g*m)*x, x], x], x] /; FreeQ[{a, c, d, e, f, g, p
}, x] && NeQ[c*d^2 + a*e^2, 0] && GtQ[m, 0] && NeQ[m + 2*p + 2, 0] && (IntegerQ[m] || IntegerQ[p] || IntegersQ
[2*m, 2*p]) &&  !(IGtQ[m, 0] && EqQ[f, 0])

Rule 844

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Dist[g/e, Int[(d
+ e*x)^(m + 1)*(a + c*x^2)^p, x], x] + Dist[(e*f - d*g)/e, Int[(d + e*x)^m*(a + c*x^2)^p, x], x] /; FreeQ[{a,
c, d, e, f, g, m, p}, x] && NeQ[c*d^2 + a*e^2, 0] &&  !IGtQ[m, 0]

Rule 942

Int[(((d_.) + (e_.)*(x_))^(m_)*Sqrt[(f_.) + (g_.)*(x_)])/Sqrt[(a_) + (c_.)*(x_)^2], x_Symbol] :> Simp[(2*e*(d
+ e*x)^(m - 1)*Sqrt[f + g*x]*Sqrt[a + c*x^2])/(c*(2*m + 1)), x] - Dist[1/(c*(2*m + 1)), Int[((d + e*x)^(m - 2)
*Simp[a*e*(d*g + 2*e*f*(m - 1)) - c*d^2*f*(2*m + 1) + (a*e^2*g*(2*m - 1) - c*d*(4*e*f*m + d*g*(2*m + 1)))*x -
c*e*(e*f + d*g*(4*m - 1))*x^2, x])/(Sqrt[f + g*x]*Sqrt[a + c*x^2]), x], x] /; FreeQ[{a, c, d, e, f, g}, x] &&
NeQ[e*f - d*g, 0] && NeQ[c*d^2 + a*e^2, 0] && IntegerQ[2*m] && GtQ[m, 1]

Rule 1654

Int[(Pq_)*((d_) + (e_.)*(x_))^(m_.)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{q = Expon[Pq, x], f = Coeff
[Pq, x, Expon[Pq, x]]}, Simp[(f*(d + e*x)^(m + q - 1)*(a + c*x^2)^(p + 1))/(c*e^(q - 1)*(m + q + 2*p + 1)), x]
+ Dist[1/(c*e^q*(m + q + 2*p + 1)), Int[(d + e*x)^m*(a + c*x^2)^p*ExpandToSum[c*e^q*(m + q + 2*p + 1)*Pq - c*
f*(m + q + 2*p + 1)*(d + e*x)^q - f*(d + e*x)^(q - 2)*(a*e^2*(m + q - 1) - c*d^2*(m + q + 2*p + 1) - 2*c*d*e*(
m + q + p)*x), x], x], x] /; GtQ[q, 1] && NeQ[m + q + 2*p + 1, 0]] /; FreeQ[{a, c, d, e, m, p}, x] && PolyQ[Pq
, x] && NeQ[c*d^2 + a*e^2, 0] &&  !(EqQ[d, 0] && True) &&  !(IGtQ[m, 0] && RationalQ[a, c, d, e] && (IntegerQ[
p] || ILtQ[p + 1/2, 0]))

Rubi steps

\begin{align*} \int \frac{x^3 \left (a+b \text{csch}^{-1}(c x)\right )}{\sqrt{d+e x}} \, dx &=-\frac{2 d^3 \sqrt{d+e x} \left (a+b \text{csch}^{-1}(c x)\right )}{e^4}+\frac{2 d^2 (d+e x)^{3/2} \left (a+b \text{csch}^{-1}(c x)\right )}{e^4}-\frac{6 d (d+e x)^{5/2} \left (a+b \text{csch}^{-1}(c x)\right )}{5 e^4}+\frac{2 (d+e x)^{7/2} \left (a+b \text{csch}^{-1}(c x)\right )}{7 e^4}+\frac{b \int \frac{2 \sqrt{d+e x} \left (-16 d^3+8 d^2 e x-6 d e^2 x^2+5 e^3 x^3\right )}{35 e^4 \sqrt{1+\frac{1}{c^2 x^2}} x^2} \, dx}{c}\\ &=-\frac{2 d^3 \sqrt{d+e x} \left (a+b \text{csch}^{-1}(c x)\right )}{e^4}+\frac{2 d^2 (d+e x)^{3/2} \left (a+b \text{csch}^{-1}(c x)\right )}{e^4}-\frac{6 d (d+e x)^{5/2} \left (a+b \text{csch}^{-1}(c x)\right )}{5 e^4}+\frac{2 (d+e x)^{7/2} \left (a+b \text{csch}^{-1}(c x)\right )}{7 e^4}+\frac{(2 b) \int \frac{\sqrt{d+e x} \left (-16 d^3+8 d^2 e x-6 d e^2 x^2+5 e^3 x^3\right )}{\sqrt{1+\frac{1}{c^2 x^2}} x^2} \, dx}{35 c e^4}\\ &=-\frac{2 d^3 \sqrt{d+e x} \left (a+b \text{csch}^{-1}(c x)\right )}{e^4}+\frac{2 d^2 (d+e x)^{3/2} \left (a+b \text{csch}^{-1}(c x)\right )}{e^4}-\frac{6 d (d+e x)^{5/2} \left (a+b \text{csch}^{-1}(c x)\right )}{5 e^4}+\frac{2 (d+e x)^{7/2} \left (a+b \text{csch}^{-1}(c x)\right )}{7 e^4}+\frac{\left (2 b \sqrt{1+c^2 x^2}\right ) \int \frac{\sqrt{d+e x} \left (-16 d^3+8 d^2 e x-6 d e^2 x^2+5 e^3 x^3\right )}{x \sqrt{1+c^2 x^2}} \, dx}{35 c e^4 \sqrt{1+\frac{1}{c^2 x^2}} x}\\ &=-\frac{2 d^3 \sqrt{d+e x} \left (a+b \text{csch}^{-1}(c x)\right )}{e^4}+\frac{2 d^2 (d+e x)^{3/2} \left (a+b \text{csch}^{-1}(c x)\right )}{e^4}-\frac{6 d (d+e x)^{5/2} \left (a+b \text{csch}^{-1}(c x)\right )}{5 e^4}+\frac{2 (d+e x)^{7/2} \left (a+b \text{csch}^{-1}(c x)\right )}{7 e^4}+\frac{\left (2 b \sqrt{1+c^2 x^2}\right ) \int \left (\frac{8 d^2 e \sqrt{d+e x}}{\sqrt{1+c^2 x^2}}-\frac{16 d^3 \sqrt{d+e x}}{x \sqrt{1+c^2 x^2}}-\frac{6 d e^2 x \sqrt{d+e x}}{\sqrt{1+c^2 x^2}}+\frac{5 e^3 x^2 \sqrt{d+e x}}{\sqrt{1+c^2 x^2}}\right ) \, dx}{35 c e^4 \sqrt{1+\frac{1}{c^2 x^2}} x}\\ &=-\frac{2 d^3 \sqrt{d+e x} \left (a+b \text{csch}^{-1}(c x)\right )}{e^4}+\frac{2 d^2 (d+e x)^{3/2} \left (a+b \text{csch}^{-1}(c x)\right )}{e^4}-\frac{6 d (d+e x)^{5/2} \left (a+b \text{csch}^{-1}(c x)\right )}{5 e^4}+\frac{2 (d+e x)^{7/2} \left (a+b \text{csch}^{-1}(c x)\right )}{7 e^4}-\frac{\left (32 b d^3 \sqrt{1+c^2 x^2}\right ) \int \frac{\sqrt{d+e x}}{x \sqrt{1+c^2 x^2}} \, dx}{35 c e^4 \sqrt{1+\frac{1}{c^2 x^2}} x}+\frac{\left (16 b d^2 \sqrt{1+c^2 x^2}\right ) \int \frac{\sqrt{d+e x}}{\sqrt{1+c^2 x^2}} \, dx}{35 c e^3 \sqrt{1+\frac{1}{c^2 x^2}} x}-\frac{\left (12 b d \sqrt{1+c^2 x^2}\right ) \int \frac{x \sqrt{d+e x}}{\sqrt{1+c^2 x^2}} \, dx}{35 c e^2 \sqrt{1+\frac{1}{c^2 x^2}} x}+\frac{\left (2 b \sqrt{1+c^2 x^2}\right ) \int \frac{x^2 \sqrt{d+e x}}{\sqrt{1+c^2 x^2}} \, dx}{7 c e \sqrt{1+\frac{1}{c^2 x^2}} x}\\ &=\frac{4 b \sqrt{d+e x} \left (1+c^2 x^2\right )}{35 c^3 e \sqrt{1+\frac{1}{c^2 x^2}}}-\frac{8 b d \sqrt{d+e x} \left (1+c^2 x^2\right )}{35 c^3 e^2 \sqrt{1+\frac{1}{c^2 x^2}} x}-\frac{2 d^3 \sqrt{d+e x} \left (a+b \text{csch}^{-1}(c x)\right )}{e^4}+\frac{2 d^2 (d+e x)^{3/2} \left (a+b \text{csch}^{-1}(c x)\right )}{e^4}-\frac{6 d (d+e x)^{5/2} \left (a+b \text{csch}^{-1}(c x)\right )}{5 e^4}+\frac{2 (d+e x)^{7/2} \left (a+b \text{csch}^{-1}(c x)\right )}{7 e^4}-\frac{\left (32 b d^4 \sqrt{1+c^2 x^2}\right ) \int \frac{1}{x \sqrt{d+e x} \sqrt{1+c^2 x^2}} \, dx}{35 c e^4 \sqrt{1+\frac{1}{c^2 x^2}} x}-\frac{\left (32 b d^3 \sqrt{1+c^2 x^2}\right ) \int \frac{1}{\sqrt{d+e x} \sqrt{1+c^2 x^2}} \, dx}{35 c e^3 \sqrt{1+\frac{1}{c^2 x^2}} x}-\frac{\left (8 b d \sqrt{1+c^2 x^2}\right ) \int \frac{-\frac{e}{2}+\frac{1}{2} c^2 d x}{\sqrt{d+e x} \sqrt{1+c^2 x^2}} \, dx}{35 c^3 e^2 \sqrt{1+\frac{1}{c^2 x^2}} x}-\frac{\left (2 b \sqrt{1+c^2 x^2}\right ) \int \frac{2 d+3 e x-c^2 d x^2}{\sqrt{d+e x} \sqrt{1+c^2 x^2}} \, dx}{35 c^3 e \sqrt{1+\frac{1}{c^2 x^2}} x}+\frac{\left (32 b \sqrt{-c^2} d^2 \sqrt{d+e x} \sqrt{1+c^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{\sqrt{1+\frac{2 \sqrt{-c^2} e x^2}{c^2 d-\sqrt{-c^2} e}}}{\sqrt{1-x^2}} \, dx,x,\frac{\sqrt{1-\sqrt{-c^2} x}}{\sqrt{2}}\right )}{35 c^3 e^3 \sqrt{1+\frac{1}{c^2 x^2}} x \sqrt{\frac{c^2 (d+e x)}{c^2 d-\sqrt{-c^2} e}}}\\ &=\frac{4 b \sqrt{d+e x} \left (1+c^2 x^2\right )}{35 c^3 e \sqrt{1+\frac{1}{c^2 x^2}}}-\frac{4 b d \sqrt{d+e x} \left (1+c^2 x^2\right )}{21 c^3 e^2 \sqrt{1+\frac{1}{c^2 x^2}} x}-\frac{2 d^3 \sqrt{d+e x} \left (a+b \text{csch}^{-1}(c x)\right )}{e^4}+\frac{2 d^2 (d+e x)^{3/2} \left (a+b \text{csch}^{-1}(c x)\right )}{e^4}-\frac{6 d (d+e x)^{5/2} \left (a+b \text{csch}^{-1}(c x)\right )}{5 e^4}+\frac{2 (d+e x)^{7/2} \left (a+b \text{csch}^{-1}(c x)\right )}{7 e^4}+\frac{32 b \sqrt{-c^2} d^2 \sqrt{d+e x} \sqrt{1+c^2 x^2} E\left (\sin ^{-1}\left (\frac{\sqrt{1-\sqrt{-c^2} x}}{\sqrt{2}}\right )|-\frac{2 \sqrt{-c^2} e}{c^2 d-\sqrt{-c^2} e}\right )}{35 c^3 e^3 \sqrt{1+\frac{1}{c^2 x^2}} x \sqrt{\frac{c^2 (d+e x)}{c^2 d-\sqrt{-c^2} e}}}-\frac{\left (32 b d^4 \sqrt{1+c^2 x^2}\right ) \int \frac{1}{x \sqrt{1-\sqrt{-c^2} x} \sqrt{1+\sqrt{-c^2} x} \sqrt{d+e x}} \, dx}{35 c e^4 \sqrt{1+\frac{1}{c^2 x^2}} x}-\frac{\left (4 b \sqrt{1+c^2 x^2}\right ) \int \frac{\frac{7}{2} c^2 d e^2+\frac{1}{2} c^2 e \left (2 c^2 d^2+9 e^2\right ) x}{\sqrt{d+e x} \sqrt{1+c^2 x^2}} \, dx}{105 c^5 e^3 \sqrt{1+\frac{1}{c^2 x^2}} x}-\frac{\left (4 b d^2 \sqrt{1+c^2 x^2}\right ) \int \frac{\sqrt{d+e x}}{\sqrt{1+c^2 x^2}} \, dx}{35 c e^3 \sqrt{1+\frac{1}{c^2 x^2}} x}+\frac{\left (4 b d \left (c^2 d^2+e^2\right ) \sqrt{1+c^2 x^2}\right ) \int \frac{1}{\sqrt{d+e x} \sqrt{1+c^2 x^2}} \, dx}{35 c^3 e^3 \sqrt{1+\frac{1}{c^2 x^2}} x}-\frac{\left (64 b \sqrt{-c^2} d^3 \sqrt{\frac{c^2 (d+e x)}{c^2 d-\sqrt{-c^2} e}} \sqrt{1+c^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-x^2} \sqrt{1+\frac{2 \sqrt{-c^2} e x^2}{c^2 d-\sqrt{-c^2} e}}} \, dx,x,\frac{\sqrt{1-\sqrt{-c^2} x}}{\sqrt{2}}\right )}{35 c^3 e^3 \sqrt{1+\frac{1}{c^2 x^2}} x \sqrt{d+e x}}\\ &=\frac{4 b \sqrt{d+e x} \left (1+c^2 x^2\right )}{35 c^3 e \sqrt{1+\frac{1}{c^2 x^2}}}-\frac{4 b d \sqrt{d+e x} \left (1+c^2 x^2\right )}{21 c^3 e^2 \sqrt{1+\frac{1}{c^2 x^2}} x}-\frac{2 d^3 \sqrt{d+e x} \left (a+b \text{csch}^{-1}(c x)\right )}{e^4}+\frac{2 d^2 (d+e x)^{3/2} \left (a+b \text{csch}^{-1}(c x)\right )}{e^4}-\frac{6 d (d+e x)^{5/2} \left (a+b \text{csch}^{-1}(c x)\right )}{5 e^4}+\frac{2 (d+e x)^{7/2} \left (a+b \text{csch}^{-1}(c x)\right )}{7 e^4}+\frac{32 b \sqrt{-c^2} d^2 \sqrt{d+e x} \sqrt{1+c^2 x^2} E\left (\sin ^{-1}\left (\frac{\sqrt{1-\sqrt{-c^2} x}}{\sqrt{2}}\right )|-\frac{2 \sqrt{-c^2} e}{c^2 d-\sqrt{-c^2} e}\right )}{35 c^3 e^3 \sqrt{1+\frac{1}{c^2 x^2}} x \sqrt{\frac{c^2 (d+e x)}{c^2 d-\sqrt{-c^2} e}}}-\frac{64 b \sqrt{-c^2} d^3 \sqrt{\frac{c^2 (d+e x)}{c^2 d-\sqrt{-c^2} e}} \sqrt{1+c^2 x^2} F\left (\sin ^{-1}\left (\frac{\sqrt{1-\sqrt{-c^2} x}}{\sqrt{2}}\right )|-\frac{2 \sqrt{-c^2} e}{c^2 d-\sqrt{-c^2} e}\right )}{35 c^3 e^3 \sqrt{1+\frac{1}{c^2 x^2}} x \sqrt{d+e x}}+\frac{\left (64 b d^4 \sqrt{1+c^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{1}{\left (1-x^2\right ) \sqrt{2-x^2} \sqrt{d+\frac{e}{\sqrt{-c^2}}-\frac{e x^2}{\sqrt{-c^2}}}} \, dx,x,\sqrt{1-\sqrt{-c^2} x}\right )}{35 c e^4 \sqrt{1+\frac{1}{c^2 x^2}} x}+\frac{\left (4 b d \left (c^2 d^2+e^2\right ) \sqrt{1+c^2 x^2}\right ) \int \frac{1}{\sqrt{d+e x} \sqrt{1+c^2 x^2}} \, dx}{105 c^3 e^3 \sqrt{1+\frac{1}{c^2 x^2}} x}-\frac{\left (2 b \left (2 c^2 d^2+9 e^2\right ) \sqrt{1+c^2 x^2}\right ) \int \frac{\sqrt{d+e x}}{\sqrt{1+c^2 x^2}} \, dx}{105 c^3 e^3 \sqrt{1+\frac{1}{c^2 x^2}} x}-\frac{\left (8 b \sqrt{-c^2} d^2 \sqrt{d+e x} \sqrt{1+c^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{\sqrt{1+\frac{2 \sqrt{-c^2} e x^2}{c^2 d-\sqrt{-c^2} e}}}{\sqrt{1-x^2}} \, dx,x,\frac{\sqrt{1-\sqrt{-c^2} x}}{\sqrt{2}}\right )}{35 c^3 e^3 \sqrt{1+\frac{1}{c^2 x^2}} x \sqrt{\frac{c^2 (d+e x)}{c^2 d-\sqrt{-c^2} e}}}+\frac{\left (8 b \sqrt{-c^2} d \left (c^2 d^2+e^2\right ) \sqrt{\frac{c^2 (d+e x)}{c^2 d-\sqrt{-c^2} e}} \sqrt{1+c^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-x^2} \sqrt{1+\frac{2 \sqrt{-c^2} e x^2}{c^2 d-\sqrt{-c^2} e}}} \, dx,x,\frac{\sqrt{1-\sqrt{-c^2} x}}{\sqrt{2}}\right )}{35 c^5 e^3 \sqrt{1+\frac{1}{c^2 x^2}} x \sqrt{d+e x}}\\ &=\frac{4 b \sqrt{d+e x} \left (1+c^2 x^2\right )}{35 c^3 e \sqrt{1+\frac{1}{c^2 x^2}}}-\frac{4 b d \sqrt{d+e x} \left (1+c^2 x^2\right )}{21 c^3 e^2 \sqrt{1+\frac{1}{c^2 x^2}} x}-\frac{2 d^3 \sqrt{d+e x} \left (a+b \text{csch}^{-1}(c x)\right )}{e^4}+\frac{2 d^2 (d+e x)^{3/2} \left (a+b \text{csch}^{-1}(c x)\right )}{e^4}-\frac{6 d (d+e x)^{5/2} \left (a+b \text{csch}^{-1}(c x)\right )}{5 e^4}+\frac{2 (d+e x)^{7/2} \left (a+b \text{csch}^{-1}(c x)\right )}{7 e^4}+\frac{24 b \sqrt{-c^2} d^2 \sqrt{d+e x} \sqrt{1+c^2 x^2} E\left (\sin ^{-1}\left (\frac{\sqrt{1-\sqrt{-c^2} x}}{\sqrt{2}}\right )|-\frac{2 \sqrt{-c^2} e}{c^2 d-\sqrt{-c^2} e}\right )}{35 c^3 e^3 \sqrt{1+\frac{1}{c^2 x^2}} x \sqrt{\frac{c^2 (d+e x)}{c^2 d-\sqrt{-c^2} e}}}-\frac{64 b \sqrt{-c^2} d^3 \sqrt{\frac{c^2 (d+e x)}{c^2 d-\sqrt{-c^2} e}} \sqrt{1+c^2 x^2} F\left (\sin ^{-1}\left (\frac{\sqrt{1-\sqrt{-c^2} x}}{\sqrt{2}}\right )|-\frac{2 \sqrt{-c^2} e}{c^2 d-\sqrt{-c^2} e}\right )}{35 c^3 e^3 \sqrt{1+\frac{1}{c^2 x^2}} x \sqrt{d+e x}}+\frac{8 b \sqrt{-c^2} d \left (c^2 d^2+e^2\right ) \sqrt{\frac{c^2 (d+e x)}{c^2 d-\sqrt{-c^2} e}} \sqrt{1+c^2 x^2} F\left (\sin ^{-1}\left (\frac{\sqrt{1-\sqrt{-c^2} x}}{\sqrt{2}}\right )|-\frac{2 \sqrt{-c^2} e}{c^2 d-\sqrt{-c^2} e}\right )}{35 c^5 e^3 \sqrt{1+\frac{1}{c^2 x^2}} x \sqrt{d+e x}}-\frac{\left (4 b \sqrt{-c^2} \left (2 c^2 d^2+9 e^2\right ) \sqrt{d+e x} \sqrt{1+c^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{\sqrt{1+\frac{2 \sqrt{-c^2} e x^2}{c^2 d-\sqrt{-c^2} e}}}{\sqrt{1-x^2}} \, dx,x,\frac{\sqrt{1-\sqrt{-c^2} x}}{\sqrt{2}}\right )}{105 c^5 e^3 \sqrt{1+\frac{1}{c^2 x^2}} x \sqrt{\frac{c^2 (d+e x)}{c^2 d-\sqrt{-c^2} e}}}+\frac{\left (8 b \sqrt{-c^2} d \left (c^2 d^2+e^2\right ) \sqrt{\frac{c^2 (d+e x)}{c^2 d-\sqrt{-c^2} e}} \sqrt{1+c^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-x^2} \sqrt{1+\frac{2 \sqrt{-c^2} e x^2}{c^2 d-\sqrt{-c^2} e}}} \, dx,x,\frac{\sqrt{1-\sqrt{-c^2} x}}{\sqrt{2}}\right )}{105 c^5 e^3 \sqrt{1+\frac{1}{c^2 x^2}} x \sqrt{d+e x}}+\frac{\left (64 b d^4 \sqrt{1+c^2 x^2} \sqrt{1+\frac{e \left (-1+\sqrt{-c^2} x\right )}{\sqrt{-c^2} d+e}}\right ) \operatorname{Subst}\left (\int \frac{1}{\left (1-x^2\right ) \sqrt{2-x^2} \sqrt{1-\frac{e x^2}{\sqrt{-c^2} \left (d+\frac{e}{\sqrt{-c^2}}\right )}}} \, dx,x,\sqrt{1-\sqrt{-c^2} x}\right )}{35 c e^4 \sqrt{1+\frac{1}{c^2 x^2}} x \sqrt{d+e x}}\\ &=\frac{4 b \sqrt{d+e x} \left (1+c^2 x^2\right )}{35 c^3 e \sqrt{1+\frac{1}{c^2 x^2}}}-\frac{4 b d \sqrt{d+e x} \left (1+c^2 x^2\right )}{21 c^3 e^2 \sqrt{1+\frac{1}{c^2 x^2}} x}-\frac{2 d^3 \sqrt{d+e x} \left (a+b \text{csch}^{-1}(c x)\right )}{e^4}+\frac{2 d^2 (d+e x)^{3/2} \left (a+b \text{csch}^{-1}(c x)\right )}{e^4}-\frac{6 d (d+e x)^{5/2} \left (a+b \text{csch}^{-1}(c x)\right )}{5 e^4}+\frac{2 (d+e x)^{7/2} \left (a+b \text{csch}^{-1}(c x)\right )}{7 e^4}+\frac{24 b \sqrt{-c^2} d^2 \sqrt{d+e x} \sqrt{1+c^2 x^2} E\left (\sin ^{-1}\left (\frac{\sqrt{1-\sqrt{-c^2} x}}{\sqrt{2}}\right )|-\frac{2 \sqrt{-c^2} e}{c^2 d-\sqrt{-c^2} e}\right )}{35 c^3 e^3 \sqrt{1+\frac{1}{c^2 x^2}} x \sqrt{\frac{c^2 (d+e x)}{c^2 d-\sqrt{-c^2} e}}}-\frac{4 b \sqrt{-c^2} \left (2 c^2 d^2+9 e^2\right ) \sqrt{d+e x} \sqrt{1+c^2 x^2} E\left (\sin ^{-1}\left (\frac{\sqrt{1-\sqrt{-c^2} x}}{\sqrt{2}}\right )|-\frac{2 \sqrt{-c^2} e}{c^2 d-\sqrt{-c^2} e}\right )}{105 c^5 e^3 \sqrt{1+\frac{1}{c^2 x^2}} x \sqrt{\frac{c^2 (d+e x)}{c^2 d-\sqrt{-c^2} e}}}-\frac{64 b \sqrt{-c^2} d^3 \sqrt{\frac{c^2 (d+e x)}{c^2 d-\sqrt{-c^2} e}} \sqrt{1+c^2 x^2} F\left (\sin ^{-1}\left (\frac{\sqrt{1-\sqrt{-c^2} x}}{\sqrt{2}}\right )|-\frac{2 \sqrt{-c^2} e}{c^2 d-\sqrt{-c^2} e}\right )}{35 c^3 e^3 \sqrt{1+\frac{1}{c^2 x^2}} x \sqrt{d+e x}}+\frac{32 b \sqrt{-c^2} d \left (c^2 d^2+e^2\right ) \sqrt{\frac{c^2 (d+e x)}{c^2 d-\sqrt{-c^2} e}} \sqrt{1+c^2 x^2} F\left (\sin ^{-1}\left (\frac{\sqrt{1-\sqrt{-c^2} x}}{\sqrt{2}}\right )|-\frac{2 \sqrt{-c^2} e}{c^2 d-\sqrt{-c^2} e}\right )}{105 c^5 e^3 \sqrt{1+\frac{1}{c^2 x^2}} x \sqrt{d+e x}}+\frac{64 b d^4 \sqrt{1+c^2 x^2} \sqrt{1-\frac{e \left (1-\sqrt{-c^2} x\right )}{\sqrt{-c^2} d+e}} \Pi \left (2;\sin ^{-1}\left (\frac{\sqrt{1-\sqrt{-c^2} x}}{\sqrt{2}}\right )|\frac{2 e}{\sqrt{-c^2} d+e}\right )}{35 c e^4 \sqrt{1+\frac{1}{c^2 x^2}} x \sqrt{d+e x}}\\ \end{align*}

Mathematica [C]  time = 14.1763, size = 1098, normalized size = 1.17 \[ \frac{a \sqrt{\frac{e x}{d}+1} B_{-\frac{e x}{d}}\left (4,\frac{1}{2}\right ) d^4}{e^4 \sqrt{d+e x}}+\frac{b \left (\frac{2 \sqrt{\frac{d}{x}+e} \sqrt{c x} \left (-\frac{\sqrt{2} \left (40 c^3 d^3 e-8 c d e^3\right ) \sqrt{i c x+1} (c x+i) \sqrt{\frac{c d+c e x}{c d-i e}} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{-\frac{e (c x+i)}{c d-i e}}\right ),\frac{i c d+e}{2 e}\right )}{\sqrt{1+\frac{1}{c^2 x^2}} \sqrt{\frac{d}{x}+e} (c x)^{3/2} \sqrt{\frac{e (1-i c x)}{i c d+e}}}+\frac{i \sqrt{2} (c d-i e) \left (48 c^4 d^4-16 c^2 e^2 d^2+9 e^4\right ) \sqrt{i c x+1} \sqrt{\frac{e (c x+i) (c d+c e x)}{(i c d+e)^2}} \Pi \left (\frac{i c d}{e}+1;\sin ^{-1}\left (\sqrt{-\frac{e (c x+i)}{c d-i e}}\right )|\frac{i c d+e}{2 e}\right )}{e \sqrt{1+\frac{1}{c^2 x^2}} \sqrt{\frac{d}{x}+e} (c x)^{3/2}}-\frac{2 \left (9 c d e^3-16 c^3 d^3 e\right ) \cosh \left (2 \text{csch}^{-1}(c x)\right ) \left (\frac{c x \left (c d \sqrt{2 i c x+2} (c x+i) \sqrt{\frac{c d+c e x}{c d-i e}} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{-\frac{e (c x+i)}{c d-i e}}\right ),\frac{i c d+e}{2 e}\right )+2 \sqrt{-\frac{e (c x-i)}{c d+i e}} (c x+i) \sqrt{\frac{c d+c e x}{c d-i e}} \left ((c d+i e) E\left (\sin ^{-1}\left (\sqrt{\frac{c d+c e x}{c d-i e}}\right )|\frac{c d-i e}{c d+i e}\right )-i e \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{c d+c e x}{c d-i e}}\right ),\frac{c d-i e}{c d+i e}\right )\right )+(i c d+e) \sqrt{2 i c x+2} \sqrt{-\frac{e (c x+i)}{c d-i e}} \sqrt{\frac{e (c x+i) (c d+c e x)}{(i c d+e)^2}} \Pi \left (\frac{i c d}{e}+1;\sin ^{-1}\left (\sqrt{-\frac{e (c x+i)}{c d-i e}}\right )|\frac{i c d+e}{2 e}\right )\right )}{2 \sqrt{-\frac{e (c x+i)}{c d-i e}}}-(c d+c e x) \left (c^2 x^2+1\right )\right )}{c d \sqrt{1+\frac{1}{c^2 x^2}} \sqrt{\frac{d}{x}+e} \sqrt{c x} \left (c^2 x^2+2\right )}\right )}{105 e^4 \sqrt{d+e x}}-\frac{c \left (\frac{d}{x}+e\right ) x \left (\frac{32 c^3 \text{csch}^{-1}(c x) d^3}{35 e^4}-\frac{2 c^3 x^3 \text{csch}^{-1}(c x)}{7 e}-\frac{4 c^2 x^2 \left (e \sqrt{1+\frac{1}{c^2 x^2}}-3 c d \text{csch}^{-1}(c x)\right )}{35 e^2}+\frac{4 c x \left (5 c d e \sqrt{1+\frac{1}{c^2 x^2}}-12 c^2 d^2 \text{csch}^{-1}(c x)\right )}{105 e^3}+\frac{4 \left (9 e^2-16 c^2 d^2\right ) \sqrt{1+\frac{1}{c^2 x^2}}}{105 e^3}\right )}{\sqrt{d+e x}}\right )}{c^4} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(a*d^4*Sqrt[1 + (e*x)/d]*Beta[-((e*x)/d), 4, 1/2])/(e^4*Sqrt[d + e*x]) + (b*(-((c*(e + d/x)*x*((4*(-16*c^2*d^2
+ 9*e^2)*Sqrt[1 + 1/(c^2*x^2)])/(105*e^3) + (32*c^3*d^3*ArcCsch[c*x])/(35*e^4) - (2*c^3*x^3*ArcCsch[c*x])/(7*
e) - (4*c^2*x^2*(e*Sqrt[1 + 1/(c^2*x^2)] - 3*c*d*ArcCsch[c*x]))/(35*e^2) + (4*c*x*(5*c*d*e*Sqrt[1 + 1/(c^2*x^2
)] - 12*c^2*d^2*ArcCsch[c*x]))/(105*e^3)))/Sqrt[d + e*x]) + (2*Sqrt[e + d/x]*Sqrt[c*x]*(-((Sqrt[2]*(40*c^3*d^3
*e - 8*c*d*e^3)*Sqrt[1 + I*c*x]*(I + c*x)*Sqrt[(c*d + c*e*x)/(c*d - I*e)]*EllipticF[ArcSin[Sqrt[-((e*(I + c*x)
)/(c*d - I*e))]], (I*c*d + e)/(2*e)])/(Sqrt[1 + 1/(c^2*x^2)]*Sqrt[e + d/x]*(c*x)^(3/2)*Sqrt[(e*(1 - I*c*x))/(I
*c*d + e)])) + (I*Sqrt[2]*(c*d - I*e)*(48*c^4*d^4 - 16*c^2*d^2*e^2 + 9*e^4)*Sqrt[1 + I*c*x]*Sqrt[(e*(I + c*x)*
(c*d + c*e*x))/(I*c*d + e)^2]*EllipticPi[1 + (I*c*d)/e, ArcSin[Sqrt[-((e*(I + c*x))/(c*d - I*e))]], (I*c*d + e
)/(2*e)])/(e*Sqrt[1 + 1/(c^2*x^2)]*Sqrt[e + d/x]*(c*x)^(3/2)) - (2*(-16*c^3*d^3*e + 9*c*d*e^3)*Cosh[2*ArcCsch[
c*x]]*(-((c*d + c*e*x)*(1 + c^2*x^2)) + (c*x*(c*d*Sqrt[2 + (2*I)*c*x]*(I + c*x)*Sqrt[(c*d + c*e*x)/(c*d - I*e)
]*EllipticF[ArcSin[Sqrt[-((e*(I + c*x))/(c*d - I*e))]], (I*c*d + e)/(2*e)] + 2*Sqrt[-((e*(-I + c*x))/(c*d + I*
e))]*(I + c*x)*Sqrt[(c*d + c*e*x)/(c*d - I*e)]*((c*d + I*e)*EllipticE[ArcSin[Sqrt[(c*d + c*e*x)/(c*d - I*e)]],
(c*d - I*e)/(c*d + I*e)] - I*e*EllipticF[ArcSin[Sqrt[(c*d + c*e*x)/(c*d - I*e)]], (c*d - I*e)/(c*d + I*e)]) +
(I*c*d + e)*Sqrt[2 + (2*I)*c*x]*Sqrt[-((e*(I + c*x))/(c*d - I*e))]*Sqrt[(e*(I + c*x)*(c*d + c*e*x))/(I*c*d +
e)^2]*EllipticPi[1 + (I*c*d)/e, ArcSin[Sqrt[-((e*(I + c*x))/(c*d - I*e))]], (I*c*d + e)/(2*e)]))/(2*Sqrt[-((e*
(I + c*x))/(c*d - I*e))])))/(c*d*Sqrt[1 + 1/(c^2*x^2)]*Sqrt[e + d/x]*Sqrt[c*x]*(2 + c^2*x^2))))/(105*e^4*Sqrt[
d + e*x])))/c^4

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Maple [C]  time = 0.321, size = 2543, normalized size = 2.7 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*(a+b*arccsch(c*x))/(e*x+d)^(1/2),x)

[Out]

2/e^4*(a*(1/7*(e*x+d)^(7/2)-3/5*d*(e*x+d)^(5/2)+d^2*(e*x+d)^(3/2)-d^3*(e*x+d)^(1/2))+b*(1/7*arccsch(c*x)*(e*x+
d)^(7/2)-3/5*arccsch(c*x)*d*(e*x+d)^(5/2)+arccsch(c*x)*d^2*(e*x+d)^(3/2)-arccsch(c*x)*d^3*(e*x+d)^(1/2)+2/105/
c^4*(3*I*((I*e+c*d)*c/(c^2*d^2+e^2))^(1/2)*(e*x+d)^(7/2)*c^3*e-3*((I*e+c*d)*c/(c^2*d^2+e^2))^(1/2)*(e*x+d)^(7/
2)*c^4*d+19*I*((I*e+c*d)*c/(c^2*d^2+e^2))^(1/2)*(e*x+d)^(3/2)*c^3*d^2*e+14*((I*e+c*d)*c/(c^2*d^2+e^2))^(1/2)*(
e*x+d)^(5/2)*c^4*d^2-14*I*((I*e+c*d)*c/(c^2*d^2+e^2))^(1/2)*(e*x+d)^(5/2)*c^3*d*e-19*((I*e+c*d)*c/(c^2*d^2+e^2
))^(1/2)*(e*x+d)^(3/2)*c^4*d^3-8*I*((I*e+c*d)*c/(c^2*d^2+e^2))^(1/2)*(e*x+d)^(1/2)*c*d*e^3+24*(-(I*(e*x+d)*c*e
+(e*x+d)*c^2*d-c^2*d^2-e^2)/(c^2*d^2+e^2))^(1/2)*((I*(e*x+d)*c*e-(e*x+d)*c^2*d+c^2*d^2+e^2)/(c^2*d^2+e^2))^(1/
2)*EllipticF((e*x+d)^(1/2)*((I*e+c*d)*c/(c^2*d^2+e^2))^(1/2),(-(2*I*c*d*e-c^2*d^2+e^2)/(c^2*d^2+e^2))^(1/2))*c
^4*d^4+16*(-(I*(e*x+d)*c*e+(e*x+d)*c^2*d-c^2*d^2-e^2)/(c^2*d^2+e^2))^(1/2)*((I*(e*x+d)*c*e-(e*x+d)*c^2*d+c^2*d
^2+e^2)/(c^2*d^2+e^2))^(1/2)*EllipticE((e*x+d)^(1/2)*((I*e+c*d)*c/(c^2*d^2+e^2))^(1/2),(-(2*I*c*d*e-c^2*d^2+e^
2)/(c^2*d^2+e^2))^(1/2))*c^4*d^4+3*I*((I*e+c*d)*c/(c^2*d^2+e^2))^(1/2)*(e*x+d)^(3/2)*c*e^3-48*(-(I*(e*x+d)*c*e
+(e*x+d)*c^2*d-c^2*d^2-e^2)/(c^2*d^2+e^2))^(1/2)*((I*(e*x+d)*c*e-(e*x+d)*c^2*d+c^2*d^2+e^2)/(c^2*d^2+e^2))^(1/
2)*EllipticPi((e*x+d)^(1/2)*((I*e+c*d)*c/(c^2*d^2+e^2))^(1/2),1/(I*e+c*d)/c*(c^2*d^2+e^2)/d,(-(I*e-c*d)*c/(c^2
*d^2+e^2))^(1/2)/((I*e+c*d)*c/(c^2*d^2+e^2))^(1/2))*c^4*d^4+48*I*(-(I*(e*x+d)*c*e+(e*x+d)*c^2*d-c^2*d^2-e^2)/(
c^2*d^2+e^2))^(1/2)*((I*(e*x+d)*c*e-(e*x+d)*c^2*d+c^2*d^2+e^2)/(c^2*d^2+e^2))^(1/2)*EllipticPi((e*x+d)^(1/2)*(
(I*e+c*d)*c/(c^2*d^2+e^2))^(1/2),1/(I*e+c*d)/c*(c^2*d^2+e^2)/d,(-(I*e-c*d)*c/(c^2*d^2+e^2))^(1/2)/((I*e+c*d)*c
/(c^2*d^2+e^2))^(1/2))*c^3*d^3*e+8*((I*e+c*d)*c/(c^2*d^2+e^2))^(1/2)*(e*x+d)^(1/2)*c^4*d^4-40*I*(-(I*(e*x+d)*c
*e+(e*x+d)*c^2*d-c^2*d^2-e^2)/(c^2*d^2+e^2))^(1/2)*((I*(e*x+d)*c*e-(e*x+d)*c^2*d+c^2*d^2+e^2)/(c^2*d^2+e^2))^(
1/2)*EllipticF((e*x+d)^(1/2)*((I*e+c*d)*c/(c^2*d^2+e^2))^(1/2),(-(2*I*c*d*e-c^2*d^2+e^2)/(c^2*d^2+e^2))^(1/2))
*c^3*d^3*e-3*((I*e+c*d)*c/(c^2*d^2+e^2))^(1/2)*(e*x+d)^(3/2)*c^2*d*e^2-8*I*((I*e+c*d)*c/(c^2*d^2+e^2))^(1/2)*(
e*x+d)^(1/2)*c^3*d^3*e-15*(-(I*(e*x+d)*c*e+(e*x+d)*c^2*d-c^2*d^2-e^2)/(c^2*d^2+e^2))^(1/2)*((I*(e*x+d)*c*e-(e*
x+d)*c^2*d+c^2*d^2+e^2)/(c^2*d^2+e^2))^(1/2)*EllipticF((e*x+d)^(1/2)*((I*e+c*d)*c/(c^2*d^2+e^2))^(1/2),(-(2*I*
c*d*e-c^2*d^2+e^2)/(c^2*d^2+e^2))^(1/2))*c^2*d^2*e^2+7*(-(I*(e*x+d)*c*e+(e*x+d)*c^2*d-c^2*d^2-e^2)/(c^2*d^2+e^
2))^(1/2)*((I*(e*x+d)*c*e-(e*x+d)*c^2*d+c^2*d^2+e^2)/(c^2*d^2+e^2))^(1/2)*EllipticE((e*x+d)^(1/2)*((I*e+c*d)*c
/(c^2*d^2+e^2))^(1/2),(-(2*I*c*d*e-c^2*d^2+e^2)/(c^2*d^2+e^2))^(1/2))*c^2*d^2*e^2+8*I*(-(I*(e*x+d)*c*e+(e*x+d)
*c^2*d-c^2*d^2-e^2)/(c^2*d^2+e^2))^(1/2)*((I*(e*x+d)*c*e-(e*x+d)*c^2*d+c^2*d^2+e^2)/(c^2*d^2+e^2))^(1/2)*Ellip
ticF((e*x+d)^(1/2)*((I*e+c*d)*c/(c^2*d^2+e^2))^(1/2),(-(2*I*c*d*e-c^2*d^2+e^2)/(c^2*d^2+e^2))^(1/2))*c*d*e^3+8
*((I*e+c*d)*c/(c^2*d^2+e^2))^(1/2)*(e*x+d)^(1/2)*c^2*d^2*e^2+9*(-(I*(e*x+d)*c*e+(e*x+d)*c^2*d-c^2*d^2-e^2)/(c^
2*d^2+e^2))^(1/2)*((I*(e*x+d)*c*e-(e*x+d)*c^2*d+c^2*d^2+e^2)/(c^2*d^2+e^2))^(1/2)*EllipticF((e*x+d)^(1/2)*((I*
e+c*d)*c/(c^2*d^2+e^2))^(1/2),(-(2*I*c*d*e-c^2*d^2+e^2)/(c^2*d^2+e^2))^(1/2))*e^4-9*(-(I*(e*x+d)*c*e+(e*x+d)*c
^2*d-c^2*d^2-e^2)/(c^2*d^2+e^2))^(1/2)*((I*(e*x+d)*c*e-(e*x+d)*c^2*d+c^2*d^2+e^2)/(c^2*d^2+e^2))^(1/2)*Ellipti
cE((e*x+d)^(1/2)*((I*e+c*d)*c/(c^2*d^2+e^2))^(1/2),(-(2*I*c*d*e-c^2*d^2+e^2)/(c^2*d^2+e^2))^(1/2))*e^4)/(((e*x
+d)^2*c^2-2*(e*x+d)*c^2*d+c^2*d^2+e^2)/c^2/x^2/e^2)^(1/2)/x/((I*e+c*d)*c/(c^2*d^2+e^2))^(1/2)/(I*e-c*d)))

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

[Out]

Exception raised: ValueError

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(a+b*arccsch(c*x))/(e*x+d)^(1/2),x, algorithm="fricas")

[Out]

integral((b*x^3*arccsch(c*x) + a*x^3)/sqrt(e*x + d), x)

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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*(a+b*acsch(c*x))/(e*x+d)**(1/2),x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(a+b*arccsch(c*x))/(e*x+d)^(1/2),x, algorithm="giac")

[Out]

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