3.1.57 \(\int \frac {x^3 (a+b \text {csch}^{-1}(c x))}{\sqrt {d+e x}} \, dx\) [57]

Optimal. Leaf size=939 \[ \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 c d^2 \sqrt {d+e x} \sqrt {1+c^2 x^2} E\left (\text {ArcSin}\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 \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 c \left (2 c^2 d^2+9 e^2\right ) \sqrt {d+e x} \sqrt {1+c^2 x^2} E\left (\text {ArcSin}\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}}}-\frac {64 b c d^3 \sqrt {\frac {c^2 (d+e x)}{c^2 d-\sqrt {-c^2} e}} \sqrt {1+c^2 x^2} F\left (\text {ArcSin}\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 \left (-c^2\right )^{3/2} e^3 \sqrt {1+\frac {1}{c^2 x^2}} x \sqrt {d+e x}}-\frac {32 b c 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 (\text {ArcSin}\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 {d+e x}}+\frac {64 b d^4 \sqrt {\frac {\sqrt {-c^2} (d+e x)}{\sqrt {-c^2} d+e}} \sqrt {1+c^2 x^2} \Pi \left (2;\text {ArcSin}\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}} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 1.97, antiderivative size = 939, normalized size of antiderivative = 1.00, number of steps used = 27, number of rules used = 17, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.810, Rules used = {45, 6445, 12, 6853, 6874, 733, 435, 958, 430, 946, 174, 552, 551, 847, 858, 956, 1668} \begin {gather*} \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;\text {ArcSin}\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 (\text {ArcSin}\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 (\text {ArcSin}\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 (\text {ArcSin}\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 (\text {ArcSin}\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}}} \end {gather*}

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 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 174

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 430

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

Rule 435

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

Rule 551

Int[1/(((a_) + (b_.)*(x_)^2)*Sqrt[(c_) + (d_.)*(x_)^2]*Sqrt[(e_) + (f_.)*(x_)^2]), x_Symbol] :> Simp[(1/(a*Sqr
t[c]*Sqrt[e]*Rt[-d/c, 2]))*EllipticPi[b*(c/(a*d)), ArcSin[Rt[-d/c, 2]*x], c*(f/(d*e))], 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 552

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

Rule 733

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 847

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 858

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 946

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 956

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)
/(Sqrt[f + g*x]*Sqrt[a + c*x^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], 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 958

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 1668

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]))

Rule 6445

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 6853

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

Rule 6874

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

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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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*}

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Mathematica [C] Result contains higher order function than in optimal. Order 9 vs. order 4 in optimal.
time = 29.46, size = 1098, normalized size = 1.17 \begin {gather*} \frac {a d^4 \sqrt {1+\frac {e x}{d}} B_{-\frac {e x}{d}}\left (4,\frac {1}{2}\right )}{e^4 \sqrt {d+e x}}+\frac {b \left (-\frac {c \left (e+\frac {d}{x}\right ) x \left (\frac {4 \left (-16 c^2 d^2+9 e^2\right ) \sqrt {1+\frac {1}{c^2 x^2}}}{105 e^3}+\frac {32 c^3 d^3 \text {csch}^{-1}(c x)}{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}\right )}{\sqrt {d+e x}}+\frac {2 \sqrt {e+\frac {d}{x}} \sqrt {c x} \left (-\frac {\sqrt {2} \left (40 c^3 d^3 e-8 c d e^3\right ) \sqrt {1+i c x} (i+c x) \sqrt {\frac {c d+c e x}{c d-i e}} F\left (\text {ArcSin}\left (\sqrt {-\frac {e (i+c x)}{c d-i e}}\right )|\frac {i c d+e}{2 e}\right )}{\sqrt {1+\frac {1}{c^2 x^2}} \sqrt {e+\frac {d}{x}} (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 d^2 e^2+9 e^4\right ) \sqrt {1+i c x} \sqrt {\frac {e (i+c x) (c d+c e x)}{(i c d+e)^2}} \Pi \left (1+\frac {i c d}{e};\text {ArcSin}\left (\sqrt {-\frac {e (i+c x)}{c d-i e}}\right )|\frac {i c d+e}{2 e}\right )}{e \sqrt {1+\frac {1}{c^2 x^2}} \sqrt {e+\frac {d}{x}} (c x)^{3/2}}-\frac {2 \left (-16 c^3 d^3 e+9 c d e^3\right ) \cosh \left (2 \text {csch}^{-1}(c x)\right ) \left (-\left ((c d+c e x) \left (1+c^2 x^2\right )\right )+\frac {c x \left (c d \sqrt {2+2 i c x} (i+c x) \sqrt {\frac {c d+c e x}{c d-i e}} F\left (\text {ArcSin}\left (\sqrt {-\frac {e (i+c x)}{c d-i e}}\right )|\frac {i c d+e}{2 e}\right )+2 \sqrt {-\frac {e (-i+c x)}{c d+i e}} (i+c x) \sqrt {\frac {c d+c e x}{c d-i e}} \left ((c d+i e) E\left (\text {ArcSin}\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 F\left (\text {ArcSin}\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+2 i c x} \sqrt {-\frac {e (i+c x)}{c d-i e}} \sqrt {\frac {e (i+c x) (c d+c e x)}{(i c d+e)^2}} \Pi \left (1+\frac {i c d}{e};\text {ArcSin}\left (\sqrt {-\frac {e (i+c x)}{c d-i e}}\right )|\frac {i c d+e}{2 e}\right )\right )}{2 \sqrt {-\frac {e (i+c x)}{c d-i e}}}\right )}{c d \sqrt {1+\frac {1}{c^2 x^2}} \sqrt {e+\frac {d}{x}} \sqrt {c x} \left (2+c^2 x^2\right )}\right )}{105 e^4 \sqrt {d+e x}}\right )}{c^4} \end {gather*}

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] Result contains complex when optimal does not.
time = 0.93, size = 2545, normalized size = 2.71

method	result	size
derivativedivides	\(\text {Expression too large to display}\)	\(2545\)
default	\(\text {Expression too large to display}\)	\(2545\)

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,method=_RETURNVERBOSE)

[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)+2/1
05/c^4*(8*I*((I*e+c*d)*c/(c^2*d^2+e^2))^(1/2)*c^3*d^3*e*(e*x+d)^(1/2)+3*((I*e+c*d)*c/(c^2*d^2+e^2))^(1/2)*c^4*
d*(e*x+d)^(7/2)-8*I*(-(I*c*(e*x+d)*e+c^2*d*(e*x+d)-c^2*d^2-e^2)/(c^2*d^2+e^2))^(1/2)*((I*c*(e*x+d)*e-c^2*d*(e*
x+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*d*e^3-14*((I*e+c*d)*c/(c^2*d^2+e^2))^(1/2)*c^4*d^2*(e*x+d)^(5/2)-19*I*((I
*e+c*d)*c/(c^2*d^2+e^2))^(1/2)*c^3*d^2*e*(e*x+d)^(3/2)-24*(-(I*c*(e*x+d)*e+c^2*d*(e*x+d)-c^2*d^2-e^2)/(c^2*d^2
+e^2))^(1/2)*((I*c*(e*x+d)*e-c^2*d*(e*x+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*c*(e*x+d)*e+c^2*d*(e*
x+d)-c^2*d^2-e^2)/(c^2*d^2+e^2))^(1/2)*((I*c*(e*x+d)*e-c^2*d*(e*x+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))*c^4*d^4+40*
I*(-(I*c*(e*x+d)*e+c^2*d*(e*x+d)-c^2*d^2-e^2)/(c^2*d^2+e^2))^(1/2)*((I*c*(e*x+d)*e-c^2*d*(e*x+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+48*(-(I*c*(e*x+d)*e+c^2*d*(e*x+d)-c^2*d^2-e^2)/(c^2*d^2+e^2))^(1/2)*((I*c*(e*x+d)*e
-c^2*d*(e*x+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+14*I
*((I*e+c*d)*c/(c^2*d^2+e^2))^(1/2)*c^3*d*e*(e*x+d)^(5/2)+19*((I*e+c*d)*c/(c^2*d^2+e^2))^(1/2)*c^4*d^3*(e*x+d)^
(3/2)+8*I*((I*e+c*d)*c/(c^2*d^2+e^2))^(1/2)*c*d*e^3*(e*x+d)^(1/2)-8*((I*e+c*d)*c/(c^2*d^2+e^2))^(1/2)*c^4*d^4*
(e*x+d)^(1/2)-3*I*((I*e+c*d)*c/(c^2*d^2+e^2))^(1/2)*c*e^3*(e*x+d)^(3/2)+15*(-(I*c*(e*x+d)*e+c^2*d*(e*x+d)-c^2*
d^2-e^2)/(c^2*d^2+e^2))^(1/2)*((I*c*(e*x+d)*e-c^2*d*(e*x+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*c
*(e*x+d)*e+c^2*d*(e*x+d)-c^2*d^2-e^2)/(c^2*d^2+e^2))^(1/2)*((I*c*(e*x+d)*e-c^2*d*(e*x+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^2-48*I*(-(I*c*(e*x+d)*e+c^2*d*(e*x+d)-c^2*d^2-e^2)/(c^2*d^2+e^2))^(1/2)*((I*c*(e*x+d)*e-c^2
*d*(e*x+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+3*((I*
e+c*d)*c/(c^2*d^2+e^2))^(1/2)*c^2*d*e^2*(e*x+d)^(3/2)-3*I*((I*e+c*d)*c/(c^2*d^2+e^2))^(1/2)*c^3*e*(e*x+d)^(7/2
)-8*((I*e+c*d)*c/(c^2*d^2+e^2))^(1/2)*c^2*d^2*e^2*(e*x+d)^(1/2)-9*(-(I*c*(e*x+d)*e+c^2*d*(e*x+d)-c^2*d^2-e^2)/
(c^2*d^2+e^2))^(1/2)*((I*c*(e*x+d)*e-c^2*d*(e*x+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*c*(e*x+d)*e+c^2*d*
(e*x+d)-c^2*d^2-e^2)/(c^2*d^2+e^2))^(1/2)*((I*c*(e*x+d)*e-c^2*d*(e*x+d)+c^2*d^2+e^2)/(c^2*d^2+e^2))^(1/2)*Elli
pticE((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)/((c
^2*(e*x+d)^2-2*c^2*d*(e*x+d)+c^2*d^2+e^2)/c^2/e^2/x^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]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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]

2/35*(5*(x*e + d)^(7/2)*e^(-4) - 21*(x*e + d)^(5/2)*d*e^(-4) + 35*(x*e + d)^(3/2)*d^2*e^(-4) - 35*sqrt(x*e + d
)*d^3*e^(-4))*a + 1/35*(2*(5*x^4*e^4 - d*x^3*e^3 + 2*d^2*x^2*e^2 - 8*d^3*x*e - 16*d^4)*e^(-4)*log(sqrt(c^2*x^2
+ 1) + 1)/sqrt(x*e + d) + 35*integrate(2/35*(5*c^2*x^5*e^4 - c^2*d*x^4*e^3 + 2*c^2*d^2*x^3*e^2 - 8*c^2*d^3*x^
2*e - 16*c^2*d^4*x)/((c^2*x^2*e^4 + e^4)*sqrt(c^2*x^2 + 1)*sqrt(x*e + d) + (c^2*x^2*e^4 + e^4)*sqrt(x*e + d)),
x) - 35*integrate(1/35*(5*c^2*x^5*(7*log(c) + 2)*e^4 - 2*c^2*d*x^4*e^3 - 16*c^2*d^3*x^2*e - 32*c^2*d^4*x + (4
*c^2*d^2*e^2 + 35*e^4*log(c))*x^3 + 35*(c^2*x^5*e^4 + x^3*e^4)*log(x))/((c^2*x^2*e^4 + e^4)*sqrt(x*e + d)), x)
)*b

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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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(x*e + d), x)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{3} \left (a + b \operatorname {acsch}{\left (c x \right )}\right )}{\sqrt {d + e x}}\, dx \end {gather*}

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]

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

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {x^3\,\left (a+b\,\mathrm {asinh}\left (\frac {1}{c\,x}\right )\right )}{\sqrt {d+e\,x}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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