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

Optimal. Leaf size=499 \[ -\frac {2 d^2 \left (a+b \text {csch}^{-1}(c x)\right )}{e^3 \sqrt {d+e x}}-\frac {4 d \sqrt {d+e x} \left (a+b \text {csch}^{-1}(c x)\right )}{e^3}+\frac {2 (d+e x)^{3/2} \left (a+b \text {csch}^{-1}(c x)\right )}{3 e^3}+\frac {32 b d^2 \sqrt {c^2 x^2+1} \sqrt {\frac {\sqrt {-c^2} (d+e x)}{\sqrt {-c^2} d+e}} \Pi \left (2;\sin ^{-1}\left (\frac {\sqrt {1-\sqrt {-c^2} x}}{\sqrt {2}}\right )|\frac {2 e}{\sqrt {-c^2} d+e}\right )}{3 c e^3 x \sqrt {\frac {1}{c^2 x^2}+1} \sqrt {d+e x}}-\frac {20 b c d \sqrt {c^2 x^2+1} \sqrt {\frac {c^2 (d+e x)}{c^2 d-\sqrt {-c^2} e}} F\left (\sin ^{-1}\left (\frac {\sqrt {1-\sqrt {-c^2} x}}{\sqrt {2}}\right )|-\frac {2 \sqrt {-c^2} e}{c^2 d-\sqrt {-c^2} e}\right )}{3 \left (-c^2\right )^{3/2} e^2 x \sqrt {\frac {1}{c^2 x^2}+1} \sqrt {d+e x}}+\frac {4 b c \sqrt {c^2 x^2+1} \sqrt {d+e x} E\left (\sin ^{-1}\left (\frac {\sqrt {1-\sqrt {-c^2} x}}{\sqrt {2}}\right )|-\frac {2 \sqrt {-c^2} e}{c^2 d-\sqrt {-c^2} e}\right )}{3 \left (-c^2\right )^{3/2} e^2 x \sqrt {\frac {1}{c^2 x^2}+1} \sqrt {\frac {c^2 (d+e x)}{c^2 d-\sqrt {-c^2} e}}} \]

[Out]

2/3*(e*x+d)^(3/2)*(a+b*arccsch(c*x))/e^3-2*d^2*(a+b*arccsch(c*x))/e^3/(e*x+d)^(1/2)-4*d*(a+b*arccsch(c*x))*(e*
x+d)^(1/2)/e^3+32/3*b*d^2*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^3/x/(1+1/c^2/x^2)^(1/2)/(e*x+d)^(1/2
)+4/3*b*c*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^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)
-20/3*b*c*d*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^2/x/(1+1/c^2/x^2)^(1/2)/(e*x+d)^(1/
2)

________________________________________________________________________________________

Rubi [A]  time = 2.01, antiderivative size = 499, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 13, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.619, Rules used = {43, 6310, 12, 6721, 6742, 719, 419, 932, 168, 538, 537, 844, 424} \[ -\frac {2 d^2 \left (a+b \text {csch}^{-1}(c x)\right )}{e^3 \sqrt {d+e x}}-\frac {4 d \sqrt {d+e x} \left (a+b \text {csch}^{-1}(c x)\right )}{e^3}+\frac {2 (d+e x)^{3/2} \left (a+b \text {csch}^{-1}(c x)\right )}{3 e^3}+\frac {32 b d^2 \sqrt {c^2 x^2+1} \sqrt {\frac {\sqrt {-c^2} (d+e x)}{\sqrt {-c^2} d+e}} \Pi \left (2;\sin ^{-1}\left (\frac {\sqrt {1-\sqrt {-c^2} x}}{\sqrt {2}}\right )|\frac {2 e}{\sqrt {-c^2} d+e}\right )}{3 c e^3 x \sqrt {\frac {1}{c^2 x^2}+1} \sqrt {d+e x}}-\frac {20 b c d \sqrt {c^2 x^2+1} \sqrt {\frac {c^2 (d+e x)}{c^2 d-\sqrt {-c^2} e}} F\left (\sin ^{-1}\left (\frac {\sqrt {1-\sqrt {-c^2} x}}{\sqrt {2}}\right )|-\frac {2 \sqrt {-c^2} e}{c^2 d-\sqrt {-c^2} e}\right )}{3 \left (-c^2\right )^{3/2} e^2 x \sqrt {\frac {1}{c^2 x^2}+1} \sqrt {d+e x}}+\frac {4 b c \sqrt {c^2 x^2+1} \sqrt {d+e x} E\left (\sin ^{-1}\left (\frac {\sqrt {1-\sqrt {-c^2} x}}{\sqrt {2}}\right )|-\frac {2 \sqrt {-c^2} e}{c^2 d-\sqrt {-c^2} e}\right )}{3 \left (-c^2\right )^{3/2} e^2 x \sqrt {\frac {1}{c^2 x^2}+1} \sqrt {\frac {c^2 (d+e x)}{c^2 d-\sqrt {-c^2} e}}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*d^2*(a + b*ArcCsch[c*x]))/(e^3*Sqrt[d + e*x]) - (4*d*Sqrt[d + e*x]*(a + b*ArcCsch[c*x]))/e^3 + (2*(d + e*x
)^(3/2)*(a + b*ArcCsch[c*x]))/(3*e^3) + (4*b*c*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^2)^(3/2)*e^2*Sqrt[1 + 1/(c^2*x^2)]*x*Sqrt
[(c^2*(d + e*x))/(c^2*d - Sqrt[-c^2]*e)]) - (20*b*c*d*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)])/(3*(-c^2)^
(3/2)*e^2*Sqrt[1 + 1/(c^2*x^2)]*x*Sqrt[d + e*x]) + (32*b*d^2*Sqrt[(Sqrt[-c^2]*(d + e*x))/(Sqrt[-c^2]*d + e)]*S
qrt[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*Sqr
t[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 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 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 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 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 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 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 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 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 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 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 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]]

Rubi steps

\begin {align*} \int \frac {x^2 \left (a+b \text {csch}^{-1}(c x)\right )}{(d+e x)^{3/2}} \, dx &=-\frac {2 d^2 \left (a+b \text {csch}^{-1}(c x)\right )}{e^3 \sqrt {d+e x}}-\frac {4 d \sqrt {d+e x} \left (a+b \text {csch}^{-1}(c x)\right )}{e^3}+\frac {2 (d+e x)^{3/2} \left (a+b \text {csch}^{-1}(c x)\right )}{3 e^3}+\frac {b \int \frac {2 \left (-8 d^2-4 d e x+e^2 x^2\right )}{3 e^3 \sqrt {1+\frac {1}{c^2 x^2}} x^2 \sqrt {d+e x}} \, dx}{c}\\ &=-\frac {2 d^2 \left (a+b \text {csch}^{-1}(c x)\right )}{e^3 \sqrt {d+e x}}-\frac {4 d \sqrt {d+e x} \left (a+b \text {csch}^{-1}(c x)\right )}{e^3}+\frac {2 (d+e x)^{3/2} \left (a+b \text {csch}^{-1}(c x)\right )}{3 e^3}+\frac {(2 b) \int \frac {-8 d^2-4 d e x+e^2 x^2}{\sqrt {1+\frac {1}{c^2 x^2}} x^2 \sqrt {d+e x}} \, dx}{3 c e^3}\\ &=-\frac {2 d^2 \left (a+b \text {csch}^{-1}(c x)\right )}{e^3 \sqrt {d+e x}}-\frac {4 d \sqrt {d+e x} \left (a+b \text {csch}^{-1}(c x)\right )}{e^3}+\frac {2 (d+e x)^{3/2} \left (a+b \text {csch}^{-1}(c x)\right )}{3 e^3}+\frac {\left (2 b \sqrt {1+c^2 x^2}\right ) \int \frac {-8 d^2-4 d e x+e^2 x^2}{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 {2 d^2 \left (a+b \text {csch}^{-1}(c x)\right )}{e^3 \sqrt {d+e x}}-\frac {4 d \sqrt {d+e x} \left (a+b \text {csch}^{-1}(c x)\right )}{e^3}+\frac {2 (d+e x)^{3/2} \left (a+b \text {csch}^{-1}(c x)\right )}{3 e^3}+\frac {\left (2 b \sqrt {1+c^2 x^2}\right ) \int \left (-\frac {4 d e}{\sqrt {d+e x} \sqrt {1+c^2 x^2}}-\frac {8 d^2}{x \sqrt {d+e x} \sqrt {1+c^2 x^2}}+\frac {e^2 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 {2 d^2 \left (a+b \text {csch}^{-1}(c x)\right )}{e^3 \sqrt {d+e x}}-\frac {4 d \sqrt {d+e x} \left (a+b \text {csch}^{-1}(c x)\right )}{e^3}+\frac {2 (d+e x)^{3/2} \left (a+b \text {csch}^{-1}(c x)\right )}{3 e^3}-\frac {\left (16 b d^2 \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 (8 b d \sqrt {1+c^2 x^2}\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {1+c^2 x^2}} \, dx}{3 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}{\sqrt {d+e x} \sqrt {1+c^2 x^2}} \, dx}{3 c e \sqrt {1+\frac {1}{c^2 x^2}} x}\\ &=-\frac {2 d^2 \left (a+b \text {csch}^{-1}(c x)\right )}{e^3 \sqrt {d+e x}}-\frac {4 d \sqrt {d+e x} \left (a+b \text {csch}^{-1}(c x)\right )}{e^3}+\frac {2 (d+e x)^{3/2} \left (a+b \text {csch}^{-1}(c x)\right )}{3 e^3}-\frac {\left (16 b d^2 \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 (2 b \sqrt {1+c^2 x^2}\right ) \int \frac {\sqrt {d+e x}}{\sqrt {1+c^2 x^2}} \, dx}{3 c e^2 \sqrt {1+\frac {1}{c^2 x^2}} x}-\frac {\left (2 b d \sqrt {1+c^2 x^2}\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {1+c^2 x^2}} \, dx}{3 c e^2 \sqrt {1+\frac {1}{c^2 x^2}} x}-\frac {\left (16 b \sqrt {-c^2} d \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 )}{3 c^3 e^2 \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 )}{e^3 \sqrt {d+e x}}-\frac {4 d \sqrt {d+e x} \left (a+b \text {csch}^{-1}(c x)\right )}{e^3}+\frac {2 (d+e x)^{3/2} \left (a+b \text {csch}^{-1}(c x)\right )}{3 e^3}-\frac {16 b \sqrt {-c^2} d \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 )}{3 c^3 e^2 \sqrt {1+\frac {1}{c^2 x^2}} x \sqrt {d+e x}}+\frac {\left (32 b d^2 \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 )}{3 c e^3 \sqrt {1+\frac {1}{c^2 x^2}} x}+\frac {\left (4 b \sqrt {-c^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 )}{3 c^3 e^2 \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} d \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 )}{3 c^3 e^2 \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 )}{e^3 \sqrt {d+e x}}-\frac {4 d \sqrt {d+e x} \left (a+b \text {csch}^{-1}(c x)\right )}{e^3}+\frac {2 (d+e x)^{3/2} \left (a+b \text {csch}^{-1}(c x)\right )}{3 e^3}+\frac {4 b \sqrt {-c^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 )}{3 c^3 e^2 \sqrt {1+\frac {1}{c^2 x^2}} x \sqrt {\frac {c^2 (d+e x)}{c^2 d-\sqrt {-c^2} e}}}-\frac {20 b \sqrt {-c^2} d \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 )}{3 c^3 e^2 \sqrt {1+\frac {1}{c^2 x^2}} x \sqrt {d+e x}}+\frac {\left (32 b d^2 \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 )}{3 c e^3 \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 )}{e^3 \sqrt {d+e x}}-\frac {4 d \sqrt {d+e x} \left (a+b \text {csch}^{-1}(c x)\right )}{e^3}+\frac {2 (d+e x)^{3/2} \left (a+b \text {csch}^{-1}(c x)\right )}{3 e^3}+\frac {4 b \sqrt {-c^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 )}{3 c^3 e^2 \sqrt {1+\frac {1}{c^2 x^2}} x \sqrt {\frac {c^2 (d+e x)}{c^2 d-\sqrt {-c^2} e}}}-\frac {20 b \sqrt {-c^2} d \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 )}{3 c^3 e^2 \sqrt {1+\frac {1}{c^2 x^2}} x \sqrt {d+e x}}+\frac {32 b d^2 \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 )}{3 c e^3 \sqrt {1+\frac {1}{c^2 x^2}} x \sqrt {d+e x}}\\ \end {align*}

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Mathematica [C]  time = 14.49, size = 979, normalized size = 1.96 \[ \frac {b \left (\frac {2 \left (\frac {d}{x}+e\right )^{3/2} (c x)^{3/2} \left (-\frac {5 \sqrt {2} c d e \sqrt {i c x+1} (c x+i) \sqrt {\frac {c d+c e x}{c d-i e}} F\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 (8 c^2 d^2-e^2\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 e \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}} F\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 F\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 )}{\sqrt {1+\frac {1}{c^2 x^2}} \sqrt {\frac {d}{x}+e} \sqrt {c x} \left (c^2 x^2+2\right )}\right )}{3 e^3 (d+e x)^{3/2}}-\frac {c^2 \left (\frac {d}{x}+e\right )^2 x^2 \left (-\frac {2 c x \text {csch}^{-1}(c x)}{3 e^2}-\frac {2 c d \text {csch}^{-1}(c x)}{e^2 \left (\frac {d}{x}+e\right )}+\frac {16 c d \text {csch}^{-1}(c x)}{3 e^3}-\frac {4 \sqrt {1+\frac {1}{c^2 x^2}}}{3 e^2}\right )}{(d+e x)^{3/2}}\right )}{c^3}-\frac {a d^3 \left (\frac {e x}{d}+1\right )^{3/2} B_{-\frac {e x}{d}}\left (3,-\frac {1}{2}\right )}{e^3 (d+e x)^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((a*d^3*(1 + (e*x)/d)^(3/2)*Beta[-((e*x)/d), 3, -1/2])/(e^3*(d + e*x)^(3/2))) + (b*(-((c^2*(e + d/x)^2*x^2*((
-4*Sqrt[1 + 1/(c^2*x^2)])/(3*e^2) + (16*c*d*ArcCsch[c*x])/(3*e^3) - (2*c*d*ArcCsch[c*x])/(e^2*(e + d/x)) - (2*
c*x*ArcCsch[c*x])/(3*e^2)))/(d + e*x)^(3/2)) + (2*(e + d/x)^(3/2)*(c*x)^(3/2)*((-5*Sqrt[2]*c*d*e*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^2*d^2 - e^2)*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*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)*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))])))/(Sqrt[1 + 1/(c^2*x^2)]*Sqrt[e + d/x]*Sqrt[c*x]*(
2 + c^2*x^2))))/(3*e^3*(d + e*x)^(3/2))))/c^3

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fricas [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b \operatorname {arcsch}\left (c x\right ) + a\right )} x^{2}}{{\left (e x + d\right )}^{\frac {3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [C]  time = 0.07, size = 896, normalized size = 1.80 \[ \frac {2 a \left (\frac {\left (e x +d \right )^{\frac {3}{2}}}{3}-2 d \sqrt {e x +d}-\frac {d^{2}}{\sqrt {e x +d}}\right )+2 b \left (\frac {\left (e x +d \right )^{\frac {3}{2}} \mathrm {arccsch}\left (c x \right )}{3}-2 \,\mathrm {arccsch}\left (c x \right ) d \sqrt {e x +d}-\frac {\mathrm {arccsch}\left (c x \right ) d^{2}}{\sqrt {e x +d}}-\frac {2 \sqrt {-\frac {i \left (e x +d \right ) c e +\left (e x +d \right ) c^{2} d -c^{2} d^{2}-e^{2}}{c^{2} d^{2}+e^{2}}}\, \sqrt {\frac {i \left (e x +d \right ) c e -\left (e x +d \right ) c^{2} d +c^{2} d^{2}+e^{2}}{c^{2} d^{2}+e^{2}}}\, \left (5 i \EllipticF \left (\sqrt {e x +d}\, \sqrt {\frac {\left (c d +i e \right ) c}{c^{2} d^{2}+e^{2}}}, \sqrt {-\frac {-c^{2} d^{2}+2 i c d e +e^{2}}{c^{2} d^{2}+e^{2}}}\right ) c d e -4 \EllipticF \left (\sqrt {e x +d}\, \sqrt {\frac {\left (c d +i e \right ) c}{c^{2} d^{2}+e^{2}}}, \sqrt {-\frac {-c^{2} d^{2}+2 i c d e +e^{2}}{c^{2} d^{2}+e^{2}}}\right ) c^{2} d^{2}-\EllipticE \left (\sqrt {e x +d}\, \sqrt {\frac {\left (c d +i e \right ) c}{c^{2} d^{2}+e^{2}}}, \sqrt {-\frac {-c^{2} d^{2}+2 i c d e +e^{2}}{c^{2} d^{2}+e^{2}}}\right ) c^{2} d^{2}-8 i \EllipticPi \left (\sqrt {e x +d}\, \sqrt {\frac {\left (c d +i e \right ) c}{c^{2} d^{2}+e^{2}}}, \frac {c^{2} d^{2}+e^{2}}{\left (c d +i e \right ) c d}, \frac {\sqrt {-\frac {\left (-c d +i e \right ) c}{c^{2} d^{2}+e^{2}}}}{\sqrt {\frac {\left (c d +i e \right ) c}{c^{2} d^{2}+e^{2}}}}\right ) c d e +8 \EllipticPi \left (\sqrt {e x +d}\, \sqrt {\frac {\left (c d +i e \right ) c}{c^{2} d^{2}+e^{2}}}, \frac {c^{2} d^{2}+e^{2}}{\left (c d +i e \right ) c d}, \frac {\sqrt {-\frac {\left (-c d +i e \right ) c}{c^{2} d^{2}+e^{2}}}}{\sqrt {\frac {\left (c d +i e \right ) c}{c^{2} d^{2}+e^{2}}}}\right ) c^{2} d^{2}+\EllipticF \left (\sqrt {e x +d}\, \sqrt {\frac {\left (c d +i e \right ) c}{c^{2} d^{2}+e^{2}}}, \sqrt {-\frac {-c^{2} d^{2}+2 i c d e +e^{2}}{c^{2} d^{2}+e^{2}}}\right ) e^{2}-\EllipticE \left (\sqrt {e x +d}\, \sqrt {\frac {\left (c d +i e \right ) c}{c^{2} d^{2}+e^{2}}}, \sqrt {-\frac {-c^{2} d^{2}+2 i c d e +e^{2}}{c^{2} d^{2}+e^{2}}}\right ) e^{2}\right )}{3 c^{2} \sqrt {\frac {\left (e x +d \right )^{2} c^{2}-2 \left (e x +d \right ) c^{2} d +c^{2} d^{2}+e^{2}}{c^{2} x^{2} e^{2}}}\, x \sqrt {\frac {\left (c d +i e \right ) c}{c^{2} d^{2}+e^{2}}}\, \left (-c d +i e \right )}\right )}{e^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/e^3*(a*(1/3*(e*x+d)^(3/2)-2*d*(e*x+d)^(1/2)-d^2/(e*x+d)^(1/2))+b*(1/3*(e*x+d)^(3/2)*arccsch(c*x)-2*arccsch(c
*x)*d*(e*x+d)^(1/2)-arccsch(c*x)*d^2/(e*x+d)^(1/2)-2/3/c^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)*((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)*(5*I*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-4*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-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^2-8*I*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*d*e+8*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^2*d^2+Ellipti
cF((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^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^2)/(((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]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {2}{3} \, a {\left (\frac {{\left (e x + d\right )}^{\frac {3}{2}}}{e^{3}} - \frac {6 \, \sqrt {e x + d} d}{e^{3}} - \frac {3 \, d^{2}}{\sqrt {e x + d} e^{3}}\right )} + \frac {1}{3} \, b {\left (\frac {2 \, {\left (e^{2} x^{2} - 4 \, d e x - 8 \, d^{2}\right )} \log \left (\sqrt {c^{2} x^{2} + 1} + 1\right )}{\sqrt {e x + d} e^{3}} + 3 \, \int \frac {2 \, {\left (c^{2} e^{2} x^{3} - 4 \, c^{2} d e x^{2} - 8 \, c^{2} d^{2} x\right )}}{3 \, {\left ({\left (c^{2} e^{3} x^{2} + e^{3}\right )} \sqrt {c^{2} x^{2} + 1} \sqrt {e x + d} + {\left (c^{2} e^{3} x^{2} + e^{3}\right )} \sqrt {e x + d}\right )}}\,{d x} - 3 \, \int -\frac {6 \, c^{2} d e^{2} x^{3} - {\left (3 \, e^{3} \log \relax (c) + 2 \, e^{3}\right )} c^{2} x^{4} + 16 \, c^{2} d^{3} x + 3 \, {\left (8 \, c^{2} d^{2} e - e^{3} \log \relax (c)\right )} x^{2} - 3 \, {\left (c^{2} e^{3} x^{4} + e^{3} x^{2}\right )} \log \relax (x)}{3 \, {\left (c^{2} e^{4} x^{3} + c^{2} d e^{3} x^{2} + e^{4} x + d e^{3}\right )} \sqrt {e x + d}}\,{d x}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3*a*((e*x + d)^(3/2)/e^3 - 6*sqrt(e*x + d)*d/e^3 - 3*d^2/(sqrt(e*x + d)*e^3)) + 1/3*b*(2*(e^2*x^2 - 4*d*e*x
- 8*d^2)*log(sqrt(c^2*x^2 + 1) + 1)/(sqrt(e*x + d)*e^3) + 3*integrate(2/3*(c^2*e^2*x^3 - 4*c^2*d*e*x^2 - 8*c^2
*d^2*x)/((c^2*e^3*x^2 + e^3)*sqrt(c^2*x^2 + 1)*sqrt(e*x + d) + (c^2*e^3*x^2 + e^3)*sqrt(e*x + d)), x) - 3*inte
grate(-1/3*(6*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 + 3*(8*c^2*d^2*e - e^3*log(c))*x^2
- 3*(c^2*e^3*x^4 + e^3*x^2)*log(x))/((c^2*e^4*x^3 + c^2*d*e^3*x^2 + e^4*x + d*e^3)*sqrt(e*x + d)), x))

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {x^2\,\left (a+b\,\mathrm {asinh}\left (\frac {1}{c\,x}\right )\right )}{{\left (d+e\,x\right )}^{3/2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{2} \left (a + b \operatorname {acsch}{\left (c x \right )}\right )}{\left (d + e x\right )^{\frac {3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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