\(\int \frac {x^2 (a+b \text {csch}^{-1}(c x))}{(d+e x)^{5/2}} \, dx\) [70]

Optimal result

Integrand size = 21, antiderivative size = 569 \[ \int \frac {x^2 \left (a+b \text {csch}^{-1}(c x)\right )}{(d+e x)^{5/2}} \, dx=-\frac {4 b d \left (1+c^2 x^2\right )}{3 c e \left (c^2 d^2+e^2\right ) \sqrt {1+\frac {1}{c^2 x^2}} x \sqrt {d+e x}}-\frac {2 d^2 \left (a+b \text {csch}^{-1}(c x)\right )}{3 e^3 (d+e x)^{3/2}}+\frac {4 d \left (a+b \text {csch}^{-1}(c x)\right )}{e^3 \sqrt {d+e x}}+\frac {2 \sqrt {d+e x} \left (a+b \text {csch}^{-1}(c x)\right )}{e^3}+\frac {4 b \sqrt {-c^2} d \sqrt {d+e x} \sqrt {1+c^2 x^2} E\left (\arcsin \left (\frac {\sqrt {1-\sqrt {-c^2} x}}{\sqrt {2}}\right )|-\frac {2 \sqrt {-c^2} e}{c^2 d-\sqrt {-c^2} e}\right )}{3 c e^2 \left (c^2 d^2+e^2\right ) \sqrt {1+\frac {1}{c^2 x^2}} x \sqrt {\frac {c^2 (d+e x)}{c^2 d-\sqrt {-c^2} e}}}+\frac {4 b c \sqrt {\frac {c^2 (d+e x)}{c^2 d-\sqrt {-c^2} e}} \sqrt {1+c^2 x^2} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\sqrt {-c^2} x}}{\sqrt {2}}\right ),-\frac {2 \sqrt {-c^2} e}{c^2 d-\sqrt {-c^2} e}\right )}{\left (-c^2\right )^{3/2} e^2 \sqrt {1+\frac {1}{c^2 x^2}} x \sqrt {d+e x}}-\frac {32 b d \sqrt {\frac {\sqrt {-c^2} (d+e x)}{\sqrt {-c^2} d+e}} \sqrt {1+c^2 x^2} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-\sqrt {-c^2} x}}{\sqrt {2}}\right ),\frac {2 e}{\sqrt {-c^2} d+e}\right )}{3 c e^3 \sqrt {1+\frac {1}{c^2 x^2}} x \sqrt {d+e x}} \]

[Out]

-2/3*d^2*(a+b*arccsch(c*x))/e^3/(e*x+d)^(3/2)+4*d*(a+b*arccsch(c*x))/e^3/(e*x+d)^(1/2)-4/3*b*d*(c^2*x^2+1)/c/e
/(c^2*d^2+e^2)/x/(1+1/c^2/x^2)^(1/2)/(e*x+d)^(1/2)+2*(a+b*arccsch(c*x))*(e*x+d)^(1/2)/e^3-32/3*b*d*EllipticPi(
1/2*(1-x*(-c^2)^(1/2))^(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^3/x/(1+1/c^2/x^2)^(1/2)/(e*x+d)^(1/2)+4/3*b*d*EllipticE(1/2*(1-x*(-c^2)^(
1/2))^(1/2)*2^(1/2),(-2*e*(-c^2)^(1/2)/(c^2*d-e*(-c^2)^(1/2)))^(1/2))*(-c^2)^(1/2)*(e*x+d)^(1/2)*(c^2*x^2+1)^(
1/2)/c/e^2/(c^2*d^2+e^2)/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*b*c*EllipticF(1/2*
(1-x*(-c^2)^(1/2))^(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^2/x/(1+1/c^2/x^2)^(1/2)/(e*x+d)^(1/2)

Rubi [A] (verified)

Time = 1.73 (sec) , antiderivative size = 569, normalized size of antiderivative = 1.00, number of steps used = 25, number of rules used = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.810, Rules used = {45, 6445, 12, 6853, 6874, 759, 21, 733, 435, 972, 946, 174, 552, 551, 849, 858, 430} \[ \int \frac {x^2 \left (a+b \text {csch}^{-1}(c x)\right )}{(d+e x)^{5/2}} \, dx=-\frac {2 d^2 \left (a+b \text {csch}^{-1}(c x)\right )}{3 e^3 (d+e x)^{3/2}}+\frac {4 d \left (a+b \text {csch}^{-1}(c x)\right )}{e^3 \sqrt {d+e x}}+\frac {2 \sqrt {d+e x} \left (a+b \text {csch}^{-1}(c x)\right )}{e^3}+\frac {4 b \sqrt {-c^2} d \sqrt {c^2 x^2+1} \sqrt {d+e x} E\left (\arcsin \left (\frac {\sqrt {1-\sqrt {-c^2} x}}{\sqrt {2}}\right )|-\frac {2 \sqrt {-c^2} e}{c^2 d-\sqrt {-c^2} e}\right )}{3 c e^2 x \sqrt {\frac {1}{c^2 x^2}+1} \left (c^2 d^2+e^2\right ) \sqrt {\frac {c^2 (d+e x)}{c^2 d-\sqrt {-c^2} e}}}-\frac {32 b d \sqrt {c^2 x^2+1} \sqrt {\frac {\sqrt {-c^2} (d+e x)}{\sqrt {-c^2} d+e}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-\sqrt {-c^2} x}}{\sqrt {2}}\right ),\frac {2 e}{\sqrt {-c^2} d+e}\right )}{3 c e^3 x \sqrt {\frac {1}{c^2 x^2}+1} \sqrt {d+e x}}+\frac {4 b c \sqrt {c^2 x^2+1} \sqrt {\frac {c^2 (d+e x)}{c^2 d-\sqrt {-c^2} e}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\sqrt {-c^2} x}}{\sqrt {2}}\right ),-\frac {2 \sqrt {-c^2} e}{c^2 d-\sqrt {-c^2} e}\right )}{\left (-c^2\right )^{3/2} e^2 x \sqrt {\frac {1}{c^2 x^2}+1} \sqrt {d+e x}}-\frac {4 b d \left (c^2 x^2+1\right )}{3 c e x \sqrt {\frac {1}{c^2 x^2}+1} \left (c^2 d^2+e^2\right ) \sqrt {d+e x}} \]

[In]

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

[Out]

(-4*b*d*(1 + c^2*x^2))/(3*c*e*(c^2*d^2 + e^2)*Sqrt[1 + 1/(c^2*x^2)]*x*Sqrt[d + e*x]) - (2*d^2*(a + b*ArcCsch[c
*x]))/(3*e^3*(d + e*x)^(3/2)) + (4*d*(a + b*ArcCsch[c*x]))/(e^3*Sqrt[d + e*x]) + (2*Sqrt[d + e*x]*(a + b*ArcCs
ch[c*x]))/e^3 + (4*b*Sqrt[-c^2]*d*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)])/(3*c*e^2*(c^2*d^2 + e^2)*Sqrt[1 + 1/(c^2*x^2)]*x*Sqrt[(c^2*(d
+ e*x))/(c^2*d - Sqrt[-c^2]*e)]) + (4*b*c*Sqrt[(c^2*(d + e*x))/(c^2*d - Sqrt[-c^2]*e)]*Sqrt[1 + c^2*x^2]*Elli
pticF[ArcSin[Sqrt[1 - Sqrt[-c^2]*x]/Sqrt[2]], (-2*Sqrt[-c^2]*e)/(c^2*d - Sqrt[-c^2]*e)])/((-c^2)^(3/2)*e^2*Sqr
t[1 + 1/(c^2*x^2)]*x*Sqrt[d + e*x]) - (32*b*d*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)])/(3*c*e^3*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 21

Int[(u_.)*((a_) + (b_.)*(v_))^(m_.)*((c_) + (d_.)*(v_))^(n_.), x_Symbol] :> Dist[(b/d)^m, Int[u*(c + d*v)^(m +
n), x], x] /; FreeQ[{a, b, c, d, n}, x] && EqQ[b*c - a*d, 0] && IntegerQ[m] && ( !IntegerQ[n] || SimplerQ[c +
d*x, a + 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 759

Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[e*(d + e*x)^(m + 1)*((a + c*x^2)^(p
+ 1)/((m + 1)*(c*d^2 + a*e^2))), x] + Dist[c/((m + 1)*(c*d^2 + a*e^2)), Int[(d + e*x)^(m + 1)*Simp[d*(m + 1)
- e*(m + 2*p + 3)*x, x]*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, m, p}, x] && NeQ[c*d^2 + a*e^2, 0] && NeQ[
m, -1] && ((LtQ[m, -1] && IntQuadraticQ[a, 0, c, d, e, m, p, x]) || (SumSimplerQ[m, 1] && IntegerQ[p]) || ILtQ
[Simplify[m + 2*p + 3], 0])

Rule 849

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(e*f - d*g)*
(d + e*x)^(m + 1)*((a + c*x^2)^(p + 1)/((m + 1)*(c*d^2 + a*e^2))), x] + Dist[1/((m + 1)*(c*d^2 + a*e^2)), Int[
(d + e*x)^(m + 1)*(a + c*x^2)^p*Simp[(c*d*f + a*e*g)*(m + 1) - c*(e*f - d*g)*(m + 2*p + 3)*x, x], x], x] /; Fr
eeQ[{a, c, d, e, f, g, p}, x] && NeQ[c*d^2 + a*e^2, 0] && LtQ[m, -1] && (IntegerQ[m] || IntegerQ[p] || Integer
sQ[2*m, 2*p])

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 972

Int[((f_.) + (g_.)*(x_))^(n_)/(((d_.) + (e_.)*(x_))*Sqrt[(a_) + (c_.)*(x_)^2]), x_Symbol] :> Int[ExpandIntegra
nd[1/(Sqrt[f + g*x]*Sqrt[a + c*x^2]), (f + g*x)^(n + 1/2)/(d + e*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[n + 1/2]

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*} \text {integral}& = -\frac {2 d^2 \left (a+b \text {csch}^{-1}(c x)\right )}{3 e^3 (d+e x)^{3/2}}+\frac {4 d \left (a+b \text {csch}^{-1}(c x)\right )}{e^3 \sqrt {d+e x}}+\frac {2 \sqrt {d+e x} \left (a+b \text {csch}^{-1}(c x)\right )}{e^3}+\frac {b \int \frac {2 \left (8 d^2+12 d e x+3 e^2 x^2\right )}{3 e^3 \sqrt {1+\frac {1}{c^2 x^2}} x^2 (d+e x)^{3/2}} \, dx}{c} \\ & = -\frac {2 d^2 \left (a+b \text {csch}^{-1}(c x)\right )}{3 e^3 (d+e x)^{3/2}}+\frac {4 d \left (a+b \text {csch}^{-1}(c x)\right )}{e^3 \sqrt {d+e x}}+\frac {2 \sqrt {d+e x} \left (a+b \text {csch}^{-1}(c x)\right )}{e^3}+\frac {(2 b) \int \frac {8 d^2+12 d e x+3 e^2 x^2}{\sqrt {1+\frac {1}{c^2 x^2}} x^2 (d+e x)^{3/2}} \, dx}{3 c e^3} \\ & = -\frac {2 d^2 \left (a+b \text {csch}^{-1}(c x)\right )}{3 e^3 (d+e x)^{3/2}}+\frac {4 d \left (a+b \text {csch}^{-1}(c x)\right )}{e^3 \sqrt {d+e x}}+\frac {2 \sqrt {d+e x} \left (a+b \text {csch}^{-1}(c x)\right )}{e^3}+\frac {\left (2 b \sqrt {1+c^2 x^2}\right ) \int \frac {8 d^2+12 d e x+3 e^2 x^2}{x (d+e x)^{3/2} \sqrt {1+c^2 x^2}} \, dx}{3 c e^3 \sqrt {1+\frac {1}{c^2 x^2}} x} \\ & = -\frac {2 d^2 \left (a+b \text {csch}^{-1}(c x)\right )}{3 e^3 (d+e x)^{3/2}}+\frac {4 d \left (a+b \text {csch}^{-1}(c x)\right )}{e^3 \sqrt {d+e x}}+\frac {2 \sqrt {d+e x} \left (a+b \text {csch}^{-1}(c x)\right )}{e^3}+\frac {\left (2 b \sqrt {1+c^2 x^2}\right ) \int \left (\frac {12 d e}{(d+e x)^{3/2} \sqrt {1+c^2 x^2}}+\frac {8 d^2}{x (d+e x)^{3/2} \sqrt {1+c^2 x^2}}+\frac {3 e^2 x}{(d+e x)^{3/2} \sqrt {1+c^2 x^2}}\right ) \, dx}{3 c e^3 \sqrt {1+\frac {1}{c^2 x^2}} x} \\ & = -\frac {2 d^2 \left (a+b \text {csch}^{-1}(c x)\right )}{3 e^3 (d+e x)^{3/2}}+\frac {4 d \left (a+b \text {csch}^{-1}(c x)\right )}{e^3 \sqrt {d+e x}}+\frac {2 \sqrt {d+e x} \left (a+b \text {csch}^{-1}(c x)\right )}{e^3}+\frac {\left (16 b d^2 \sqrt {1+c^2 x^2}\right ) \int \frac {1}{x (d+e x)^{3/2} \sqrt {1+c^2 x^2}} \, dx}{3 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 {1}{(d+e x)^{3/2} \sqrt {1+c^2 x^2}} \, dx}{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}{(d+e x)^{3/2} \sqrt {1+c^2 x^2}} \, dx}{c e \sqrt {1+\frac {1}{c^2 x^2}} x} \\ & = -\frac {12 b d \left (1+c^2 x^2\right )}{c e \left (c^2 d^2+e^2\right ) \sqrt {1+\frac {1}{c^2 x^2}} x \sqrt {d+e x}}-\frac {2 d^2 \left (a+b \text {csch}^{-1}(c x)\right )}{3 e^3 (d+e x)^{3/2}}+\frac {4 d \left (a+b \text {csch}^{-1}(c x)\right )}{e^3 \sqrt {d+e x}}+\frac {2 \sqrt {d+e x} \left (a+b \text {csch}^{-1}(c x)\right )}{e^3}+\frac {\left (16 b d^2 \sqrt {1+c^2 x^2}\right ) \int \left (-\frac {e}{d (d+e x)^{3/2} \sqrt {1+c^2 x^2}}+\frac {1}{d x \sqrt {d+e x} \sqrt {1+c^2 x^2}}\right ) \, dx}{3 c e^3 \sqrt {1+\frac {1}{c^2 x^2}} x}-\frac {\left (16 b c d \sqrt {1+c^2 x^2}\right ) \int \frac {-\frac {d}{2}-\frac {e x}{2}}{\sqrt {d+e x} \sqrt {1+c^2 x^2}} \, dx}{e^2 \left (c^2 d^2+e^2\right ) \sqrt {1+\frac {1}{c^2 x^2}} x}-\frac {\left (4 b \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}{c e \left (c^2 d^2+e^2\right ) \sqrt {1+\frac {1}{c^2 x^2}} x} \\ & = -\frac {12 b d \left (1+c^2 x^2\right )}{c e \left (c^2 d^2+e^2\right ) \sqrt {1+\frac {1}{c^2 x^2}} x \sqrt {d+e x}}-\frac {2 d^2 \left (a+b \text {csch}^{-1}(c x)\right )}{3 e^3 (d+e x)^{3/2}}+\frac {4 d \left (a+b \text {csch}^{-1}(c x)\right )}{e^3 \sqrt {d+e x}}+\frac {2 \sqrt {d+e x} \left (a+b \text {csch}^{-1}(c x)\right )}{e^3}+\frac {\left (16 b d \sqrt {1+c^2 x^2}\right ) \int \frac {1}{x \sqrt {d+e x} \sqrt {1+c^2 x^2}} \, dx}{3 c e^3 \sqrt {1+\frac {1}{c^2 x^2}} x}+\frac {\left (2 b \sqrt {1+c^2 x^2}\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {1+c^2 x^2}} \, dx}{c e^2 \sqrt {1+\frac {1}{c^2 x^2}} x}-\frac {\left (16 b d \sqrt {1+c^2 x^2}\right ) \int \frac {1}{(d+e x)^{3/2} \sqrt {1+c^2 x^2}} \, dx}{3 c e^2 \sqrt {1+\frac {1}{c^2 x^2}} x}-\frac {\left (2 b c d \sqrt {1+c^2 x^2}\right ) \int \frac {\sqrt {d+e x}}{\sqrt {1+c^2 x^2}} \, dx}{e^2 \left (c^2 d^2+e^2\right ) \sqrt {1+\frac {1}{c^2 x^2}} x}+\frac {\left (8 b c d \sqrt {1+c^2 x^2}\right ) \int \frac {\sqrt {d+e x}}{\sqrt {1+c^2 x^2}} \, dx}{e^2 \left (c^2 d^2+e^2\right ) \sqrt {1+\frac {1}{c^2 x^2}} x} \\ & = -\frac {4 b d \left (1+c^2 x^2\right )}{3 c e \left (c^2 d^2+e^2\right ) \sqrt {1+\frac {1}{c^2 x^2}} x \sqrt {d+e x}}-\frac {2 d^2 \left (a+b \text {csch}^{-1}(c x)\right )}{3 e^3 (d+e x)^{3/2}}+\frac {4 d \left (a+b \text {csch}^{-1}(c x)\right )}{e^3 \sqrt {d+e x}}+\frac {2 \sqrt {d+e x} \left (a+b \text {csch}^{-1}(c x)\right )}{e^3}+\frac {\left (16 b d \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}{3 c e^3 \sqrt {1+\frac {1}{c^2 x^2}} x}+\frac {\left (32 b c d \sqrt {1+c^2 x^2}\right ) \int \frac {-\frac {d}{2}-\frac {e x}{2}}{\sqrt {d+e x} \sqrt {1+c^2 x^2}} \, dx}{3 e^2 \left (c^2 d^2+e^2\right ) \sqrt {1+\frac {1}{c^2 x^2}} x}-\frac {\left (4 b \sqrt {-c^2} d \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 )}{c e^2 \left (c^2 d^2+e^2\right ) \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 (16 b \sqrt {-c^2} d \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 )}{c e^2 \left (c^2 d^2+e^2\right ) \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 (4 b \sqrt {-c^2} \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 )}{c^3 e^2 \sqrt {1+\frac {1}{c^2 x^2}} x \sqrt {d+e x}} \\ & = -\frac {4 b d \left (1+c^2 x^2\right )}{3 c e \left (c^2 d^2+e^2\right ) \sqrt {1+\frac {1}{c^2 x^2}} x \sqrt {d+e x}}-\frac {2 d^2 \left (a+b \text {csch}^{-1}(c x)\right )}{3 e^3 (d+e x)^{3/2}}+\frac {4 d \left (a+b \text {csch}^{-1}(c x)\right )}{e^3 \sqrt {d+e x}}+\frac {2 \sqrt {d+e x} \left (a+b \text {csch}^{-1}(c x)\right )}{e^3}+\frac {12 b \sqrt {-c^2} d \sqrt {d+e x} \sqrt {1+c^2 x^2} E\left (\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 )}{c e^2 \left (c^2 d^2+e^2\right ) \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} \sqrt {\frac {c^2 (d+e x)}{c^2 d-\sqrt {-c^2} e}} \sqrt {1+c^2 x^2} \operatorname {EllipticF}\left (\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 )}{c^3 e^2 \sqrt {1+\frac {1}{c^2 x^2}} x \sqrt {d+e x}}-\frac {\left (32 b d \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 )}{3 c e^3 \sqrt {1+\frac {1}{c^2 x^2}} x}-\frac {\left (16 b c d \sqrt {1+c^2 x^2}\right ) \int \frac {\sqrt {d+e x}}{\sqrt {1+c^2 x^2}} \, dx}{3 e^2 \left (c^2 d^2+e^2\right ) \sqrt {1+\frac {1}{c^2 x^2}} x} \\ & = -\frac {4 b d \left (1+c^2 x^2\right )}{3 c e \left (c^2 d^2+e^2\right ) \sqrt {1+\frac {1}{c^2 x^2}} x \sqrt {d+e x}}-\frac {2 d^2 \left (a+b \text {csch}^{-1}(c x)\right )}{3 e^3 (d+e x)^{3/2}}+\frac {4 d \left (a+b \text {csch}^{-1}(c x)\right )}{e^3 \sqrt {d+e x}}+\frac {2 \sqrt {d+e x} \left (a+b \text {csch}^{-1}(c x)\right )}{e^3}+\frac {12 b \sqrt {-c^2} d \sqrt {d+e x} \sqrt {1+c^2 x^2} E\left (\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 )}{c e^2 \left (c^2 d^2+e^2\right ) \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} \sqrt {\frac {c^2 (d+e x)}{c^2 d-\sqrt {-c^2} e}} \sqrt {1+c^2 x^2} \operatorname {EllipticF}\left (\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 )}{c^3 e^2 \sqrt {1+\frac {1}{c^2 x^2}} x \sqrt {d+e x}}-\frac {\left (32 b \sqrt {-c^2} d \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 )}{3 c e^2 \left (c^2 d^2+e^2\right ) \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 \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 )}{3 c e^3 \sqrt {1+\frac {1}{c^2 x^2}} x \sqrt {d+e x}} \\ & = -\frac {4 b d \left (1+c^2 x^2\right )}{3 c e \left (c^2 d^2+e^2\right ) \sqrt {1+\frac {1}{c^2 x^2}} x \sqrt {d+e x}}-\frac {2 d^2 \left (a+b \text {csch}^{-1}(c x)\right )}{3 e^3 (d+e x)^{3/2}}+\frac {4 d \left (a+b \text {csch}^{-1}(c x)\right )}{e^3 \sqrt {d+e x}}+\frac {2 \sqrt {d+e x} \left (a+b \text {csch}^{-1}(c x)\right )}{e^3}+\frac {4 b \sqrt {-c^2} d \sqrt {d+e x} \sqrt {1+c^2 x^2} E\left (\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 )}{3 c e^2 \left (c^2 d^2+e^2\right ) \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} \sqrt {\frac {c^2 (d+e x)}{c^2 d-\sqrt {-c^2} e}} \sqrt {1+c^2 x^2} \operatorname {EllipticF}\left (\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 )}{c^3 e^2 \sqrt {1+\frac {1}{c^2 x^2}} x \sqrt {d+e x}}-\frac {32 b d \sqrt {1+c^2 x^2} \sqrt {1-\frac {e \left (1-\sqrt {-c^2} x\right )}{\sqrt {-c^2} d+e}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-\sqrt {-c^2} x}}{\sqrt {2}}\right ),\frac {2 e}{\sqrt {-c^2} d+e}\right )}{3 c e^3 \sqrt {1+\frac {1}{c^2 x^2}} x \sqrt {d+e x}} \\ \end{align*}

Mathematica [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 4 in optimal.

Time = 35.05 (sec) , antiderivative size = 1076, normalized size of antiderivative = 1.89 \[ \int \frac {x^2 \left (a+b \text {csch}^{-1}(c x)\right )}{(d+e x)^{5/2}} \, dx=-\frac {a d^3 \left (1+\frac {e x}{d}\right )^{5/2} B_{-\frac {e x}{d}}\left (3,-\frac {3}{2}\right )}{e^3 (d+e x)^{5/2}}+\frac {b \left (-\frac {c^3 \left (e+\frac {d}{x}\right )^3 x^3 \left (-\frac {4 c d \sqrt {1+\frac {1}{c^2 x^2}}}{3 e^2 \left (c^2 d^2+e^2\right )}-\frac {16 \text {csch}^{-1}(c x)}{3 e^3}+\frac {2 \text {csch}^{-1}(c x)}{3 e \left (e+\frac {d}{x}\right )^2}+\frac {4 \left (c d e \sqrt {1+\frac {1}{c^2 x^2}}+2 c^2 d^2 \text {csch}^{-1}(c x)+2 e^2 \text {csch}^{-1}(c x)\right )}{3 e^2 \left (c^2 d^2+e^2\right ) \left (e+\frac {d}{x}\right )}\right )}{(d+e x)^{5/2}}-\frac {2 \left (e+\frac {d}{x}\right )^{5/2} (c x)^{5/2} \left (-\frac {\sqrt {2} \left (3 c^2 d^2 e+3 e^3\right ) \sqrt {1+i c x} (i+c x) \sqrt {\frac {c d+c e x}{c d-i e}} \operatorname {EllipticF}\left (\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 (8 c^3 d^3+9 c d e^2\right ) \sqrt {1+i c x} \sqrt {\frac {e (i+c x) (c d+c e x)}{(i c d+e)^2}} \operatorname {EllipticPi}\left (1+\frac {i c d}{e},\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 c d e \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}} \operatorname {EllipticF}\left (\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 (\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 \operatorname {EllipticF}\left (\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}} \operatorname {EllipticPi}\left (1+\frac {i c d}{e},\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 )}{\sqrt {1+\frac {1}{c^2 x^2}} \sqrt {e+\frac {d}{x}} \sqrt {c x} \left (2+c^2 x^2\right )}\right )}{3 e^3 \left (c^2 d^2+e^2\right ) (d+e x)^{5/2}}\right )}{c^3} \]

[In]

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

[Out]

-((a*d^3*(1 + (e*x)/d)^(5/2)*Beta[-((e*x)/d), 3, -3/2])/(e^3*(d + e*x)^(5/2))) + (b*(-((c^3*(e + d/x)^3*x^3*((
-4*c*d*Sqrt[1 + 1/(c^2*x^2)])/(3*e^2*(c^2*d^2 + e^2)) - (16*ArcCsch[c*x])/(3*e^3) + (2*ArcCsch[c*x])/(3*e*(e +
d/x)^2) + (4*(c*d*e*Sqrt[1 + 1/(c^2*x^2)] + 2*c^2*d^2*ArcCsch[c*x] + 2*e^2*ArcCsch[c*x]))/(3*e^2*(c^2*d^2 + e
^2)*(e + d/x))))/(d + e*x)^(5/2)) - (2*(e + d/x)^(5/2)*(c*x)^(5/2)*(-((Sqrt[2]*(3*c^2*d^2*e + 3*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)*(8*c^3*d^3 + 9*c*d*e^2)*Sqrt[1 + I*c*x]*Sqrt[(e*(I + c*x)*(c*d + c*e*x))/(I*c*d + e)^2]*EllipticP
i[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*c*d*e*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)*E
llipticE[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))])))/(Sqrt[1 + 1/(c^2*x^2)]*Sqrt[e + d
/x]*Sqrt[c*x]*(2 + c^2*x^2))))/(3*e^3*(c^2*d^2 + e^2)*(d + e*x)^(5/2))))/c^3

Maple [C] (verified)

Result contains complex when optimal does not.

Time = 8.88 (sec) , antiderivative size = 2492, normalized size of antiderivative = 4.38

method	result	size
derivativedivides	\(\text {Expression too large to display}\)	\(2492\)
default	\(\text {Expression too large to display}\)	\(2492\)
parts	\(\text {Expression too large to display}\)	\(2500\)

[In]

int(x^2*(a+b*arccsch(c*x))/(e*x+d)^(5/2),x,method=_RETURNVERBOSE)

[Out]

2/e^3*(a*((e*x+d)^(1/2)-1/3*d^2/(e*x+d)^(3/2)+2*d/(e*x+d)^(1/2))+b*((e*x+d)^(1/2)*arccsch(c*x)-1/3*arccsch(c*x
)*d^2/(e*x+d)^(3/2)+2*arccsch(c*x)*d/(e*x+d)^(1/2)+2/3/c*(-8*I*(-(I*c*e*(e*x+d)+c^2*d*(e*x+d)-c^2*d^2-e^2)/(c^
2*d^2+e^2))^(1/2)*((I*c*e*(e*x+d)-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)*((c
*d+I*e)*c/(c^2*d^2+e^2))^(1/2),1/(c*d+I*e)/c*(c^2*d^2+e^2)/d,(-(I*e-c*d)*c/(c^2*d^2+e^2))^(1/2)/((c*d+I*e)*c/(
c^2*d^2+e^2))^(1/2))*c^2*d^2*e*(e*x+d)^(1/2)-4*(-(I*c*e*(e*x+d)+c^2*d*(e*x+d)-c^2*d^2-e^2)/(c^2*d^2+e^2))^(1/2
)*((I*c*e*(e*x+d)-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)*((c*d+I*e)*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*x+d)^(1/2)+(-(I*c*e*(e*x+d)+c^2*d*(e*
x+d)-c^2*d^2-e^2)/(c^2*d^2+e^2))^(1/2)*((I*c*e*(e*x+d)-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)*((c*d+I*e)*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*
x+d)^(1/2)-8*I*(-(I*c*e*(e*x+d)+c^2*d*(e*x+d)-c^2*d^2-e^2)/(c^2*d^2+e^2))^(1/2)*((I*c*e*(e*x+d)-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)*((c*d+I*e)*c/(c^2*d^2+e^2))^(1/2),1/(c*d+I*e)/c*(c^
2*d^2+e^2)/d,(-(I*e-c*d)*c/(c^2*d^2+e^2))^(1/2)/((c*d+I*e)*c/(c^2*d^2+e^2))^(1/2))*e^3*(e*x+d)^(1/2)+8*(-(I*c*
e*(e*x+d)+c^2*d*(e*x+d)-c^2*d^2-e^2)/(c^2*d^2+e^2))^(1/2)*((I*c*e*(e*x+d)-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)*((c*d+I*e)*c/(c^2*d^2+e^2))^(1/2),1/(c*d+I*e)/c*(c^2*d^2+e^2)/d,(-(I*e-c*
d)*c/(c^2*d^2+e^2))^(1/2)/((c*d+I*e)*c/(c^2*d^2+e^2))^(1/2))*c^3*d^3*(e*x+d)^(1/2)-I*((c*d+I*e)*c/(c^2*d^2+e^2
))^(1/2)*d*e^3+((c*d+I*e)*c/(c^2*d^2+e^2))^(1/2)*c^3*d^2*(e*x+d)^2+3*I*(-(I*c*e*(e*x+d)+c^2*d*(e*x+d)-c^2*d^2-
e^2)/(c^2*d^2+e^2))^(1/2)*((I*c*e*(e*x+d)-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)*((c*d+I*e)*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^3*(e*x+d)^(1/2)-2*((c*
d+I*e)*c/(c^2*d^2+e^2))^(1/2)*c^3*d^3*(e*x+d)-I*((c*d+I*e)*c/(c^2*d^2+e^2))^(1/2)*c^2*d^3*e-4*(-(I*c*e*(e*x+d)
+c^2*d*(e*x+d)-c^2*d^2-e^2)/(c^2*d^2+e^2))^(1/2)*((I*c*e*(e*x+d)-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)*((c*d+I*e)*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^2*(e*x+d)^(1/2)+(-(I*c*e*(e*x+d)+c^2*d*(e*x+d)-c^2*d^2-e^2)/(c^2*d^2+e^2))^(1/2)*((I*c*e*(e*x+d)-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)*((c*d+I*e)*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^2*(e*x+d)^(1/2)-I*((c*d+I*e)*c/(c^2*d^2+e^2))^(1/2)*c^2*d*e*(e*x+d)^
2+8*(-(I*c*e*(e*x+d)+c^2*d*(e*x+d)-c^2*d^2-e^2)/(c^2*d^2+e^2))^(1/2)*((I*c*e*(e*x+d)-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)*((c*d+I*e)*c/(c^2*d^2+e^2))^(1/2),1/(c*d+I*e)/c*(c^2*d^2+e^2)/
d,(-(I*e-c*d)*c/(c^2*d^2+e^2))^(1/2)/((c*d+I*e)*c/(c^2*d^2+e^2))^(1/2))*c*d*e^2*(e*x+d)^(1/2)+3*I*(-(I*c*e*(e*
x+d)+c^2*d*(e*x+d)-c^2*d^2-e^2)/(c^2*d^2+e^2))^(1/2)*((I*c*e*(e*x+d)-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)*((c*d+I*e)*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*(e*x+d)^(1/2)+((c*d+I*e)*c/(c^2*d^2+e^2))^(1/2)*c^3*d^4+2*I*((c*d+I*e)*c/(c^2*d^2+e^2))^(1/2)*c^2
*d^2*e*(e*x+d)+((c*d+I*e)*c/(c^2*d^2+e^2))^(1/2)*c*d^2*e^2)/((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/(c^2*d^2+e^2)/(e*x+d)^(1/2)/((c*d+I*e)*c/(c^2*d^2+e^2))^(1/2)/(I*e-c*d)))

Fricas [F]

\[ \int \frac {x^2 \left (a+b \text {csch}^{-1}(c x)\right )}{(d+e x)^{5/2}} \, dx=\int { \frac {{\left (b \operatorname {arcsch}\left (c x\right ) + a\right )} x^{2}}{{\left (e x + d\right )}^{\frac {5}{2}}} \,d x } \]

[In]

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

[Out]

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

Sympy [F]

\[ \int \frac {x^2 \left (a+b \text {csch}^{-1}(c x)\right )}{(d+e x)^{5/2}} \, dx=\int \frac {x^{2} \left (a + b \operatorname {acsch}{\left (c x \right )}\right )}{\left (d + e x\right )^{\frac {5}{2}}}\, dx \]

[In]

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

[Out]

Integral(x**2*(a + b*acsch(c*x))/(d + e*x)**(5/2), x)

Maxima [F]

\[ \int \frac {x^2 \left (a+b \text {csch}^{-1}(c x)\right )}{(d+e x)^{5/2}} \, dx=\int { \frac {{\left (b \operatorname {arcsch}\left (c x\right ) + a\right )} x^{2}}{{\left (e x + d\right )}^{\frac {5}{2}}} \,d x } \]

[In]

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

[Out]

1/3*b*(2*(3*e^2*x^2 + 12*d*e*x + 8*d^2)*log(sqrt(c^2*x^2 + 1) + 1)/((e^4*x + d*e^3)*sqrt(e*x + d)) + 3*integra
te(2/3*(3*c^2*e^2*x^3 + 12*c^2*d*e*x^2 + 8*c^2*d^2*x)/((c^2*e^4*x^3 + c^2*d*e^3*x^2 + e^4*x + d*e^3)*sqrt(c^2*
x^2 + 1)*sqrt(e*x + d) + (c^2*e^4*x^3 + c^2*d*e^3*x^2 + e^4*x + d*e^3)*sqrt(e*x + d)), x) - 3*integrate(1/3*(3
0*c^2*d*e^2*x^3 + 3*(e^3*log(c) + 2*e^3)*c^2*x^4 + 16*c^2*d^3*x + (40*c^2*d^2*e + 3*e^3*log(c))*x^2 + 3*(c^2*e
^3*x^4 + e^3*x^2)*log(x))/((c^2*e^5*x^4 + 2*c^2*d*e^4*x^3 + 2*d*e^4*x + d^2*e^3 + (c^2*d^2*e^3 + e^5)*x^2)*sqr
t(e*x + d)), x)) + 2/3*a*(3*sqrt(e*x + d)/e^3 + 6*d/(sqrt(e*x + d)*e^3) - d^2/((e*x + d)^(3/2)*e^3))

Giac [F]

\[ \int \frac {x^2 \left (a+b \text {csch}^{-1}(c x)\right )}{(d+e x)^{5/2}} \, dx=\int { \frac {{\left (b \operatorname {arcsch}\left (c x\right ) + a\right )} x^{2}}{{\left (e x + d\right )}^{\frac {5}{2}}} \,d x } \]

[In]

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

[Out]

integrate((b*arccsch(c*x) + a)*x^2/(e*x + d)^(5/2), x)

Mupad [F(-1)]

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

[In]

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

[Out]

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