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

Optimal. Leaf size=393 \[ -\frac {2 \left (a+b \text {csch}^{-1}(c x)\right )}{e^2 \sqrt {d+e x}}+\frac {2 d \left (a+b \text {csch}^{-1}(c x)\right )}{3 e^2 (d+e x)^{3/2}}+\frac {4 b \left (c^2 x^2+1\right )}{3 c x \sqrt {\frac {1}{c^2 x^2}+1} \left (c^2 d^2+e^2\right ) \sqrt {d+e x}}-\frac {4 b \sqrt {-c^2} \sqrt {c^2 x^2+1} \sqrt {d+e x} E\left (\sin ^{-1}\left (\frac {\sqrt {1-\sqrt {-c^2} x}}{\sqrt {2}}\right )|-\frac {2 \sqrt {-c^2} e}{c^2 d-\sqrt {-c^2} e}\right )}{3 c e x \sqrt {\frac {1}{c^2 x^2}+1} \left (c^2 d^2+e^2\right ) \sqrt {\frac {c^2 (d+e x)}{c^2 d-\sqrt {-c^2} e}}}+\frac {8 b \sqrt {c^2 x^2+1} \sqrt {\frac {\sqrt {-c^2} (d+e x)}{\sqrt {-c^2} d+e}} \Pi \left (2;\sin ^{-1}\left (\frac {\sqrt {1-\sqrt {-c^2} x}}{\sqrt {2}}\right )|\frac {2 e}{\sqrt {-c^2} d+e}\right )}{3 c e^2 x \sqrt {\frac {1}{c^2 x^2}+1} \sqrt {d+e x}} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 2.19, antiderivative size = 393, normalized size of antiderivative = 1.00, number of steps used = 19, number of rules used = 14, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.737, Rules used = {43, 6310, 12, 6721, 6742, 745, 21, 719, 424, 958, 932, 168, 538, 537} \[ -\frac {2 \left (a+b \text {csch}^{-1}(c x)\right )}{e^2 \sqrt {d+e x}}+\frac {2 d \left (a+b \text {csch}^{-1}(c x)\right )}{3 e^2 (d+e x)^{3/2}}+\frac {4 b \left (c^2 x^2+1\right )}{3 c x \sqrt {\frac {1}{c^2 x^2}+1} \left (c^2 d^2+e^2\right ) \sqrt {d+e x}}-\frac {4 b \sqrt {-c^2} \sqrt {c^2 x^2+1} \sqrt {d+e x} E\left (\sin ^{-1}\left (\frac {\sqrt {1-\sqrt {-c^2} x}}{\sqrt {2}}\right )|-\frac {2 \sqrt {-c^2} e}{c^2 d-\sqrt {-c^2} e}\right )}{3 c e x \sqrt {\frac {1}{c^2 x^2}+1} \left (c^2 d^2+e^2\right ) \sqrt {\frac {c^2 (d+e x)}{c^2 d-\sqrt {-c^2} e}}}+\frac {8 b \sqrt {c^2 x^2+1} \sqrt {\frac {\sqrt {-c^2} (d+e x)}{\sqrt {-c^2} d+e}} \Pi \left (2;\sin ^{-1}\left (\frac {\sqrt {1-\sqrt {-c^2} x}}{\sqrt {2}}\right )|\frac {2 e}{\sqrt {-c^2} d+e}\right )}{3 c e^2 x \sqrt {\frac {1}{c^2 x^2}+1} \sqrt {d+e x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(4*b*(1 + c^2*x^2))/(3*c*(c^2*d^2 + e^2)*Sqrt[1 + 1/(c^2*x^2)]*x*Sqrt[d + e*x]) + (2*d*(a + b*ArcCsch[c*x]))/(
3*e^2*(d + e*x)^(3/2)) - (2*(a + b*ArcCsch[c*x]))/(e^2*Sqrt[d + e*x]) - (4*b*Sqrt[-c^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)])/(3*c*e*
(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)]) + (8*b*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^2*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 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 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 745

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 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 958

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 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 \left (a+b \text {csch}^{-1}(c x)\right )}{(d+e x)^{5/2}} \, dx &=\frac {2 d \left (a+b \text {csch}^{-1}(c x)\right )}{3 e^2 (d+e x)^{3/2}}-\frac {2 \left (a+b \text {csch}^{-1}(c x)\right )}{e^2 \sqrt {d+e x}}+\frac {b \int \frac {2 (-2 d-3 e x)}{3 e^2 \sqrt {1+\frac {1}{c^2 x^2}} x^2 (d+e x)^{3/2}} \, dx}{c}\\ &=\frac {2 d \left (a+b \text {csch}^{-1}(c x)\right )}{3 e^2 (d+e x)^{3/2}}-\frac {2 \left (a+b \text {csch}^{-1}(c x)\right )}{e^2 \sqrt {d+e x}}+\frac {(2 b) \int \frac {-2 d-3 e x}{\sqrt {1+\frac {1}{c^2 x^2}} x^2 (d+e x)^{3/2}} \, dx}{3 c e^2}\\ &=\frac {2 d \left (a+b \text {csch}^{-1}(c x)\right )}{3 e^2 (d+e x)^{3/2}}-\frac {2 \left (a+b \text {csch}^{-1}(c x)\right )}{e^2 \sqrt {d+e x}}+\frac {\left (2 b \sqrt {1+c^2 x^2}\right ) \int \frac {-2 d-3 e x}{x (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 {2 d \left (a+b \text {csch}^{-1}(c x)\right )}{3 e^2 (d+e x)^{3/2}}-\frac {2 \left (a+b \text {csch}^{-1}(c x)\right )}{e^2 \sqrt {d+e x}}+\frac {\left (2 b \sqrt {1+c^2 x^2}\right ) \int \left (-\frac {3 e}{(d+e x)^{3/2} \sqrt {1+c^2 x^2}}-\frac {2 d}{x (d+e x)^{3/2} \sqrt {1+c^2 x^2}}\right ) \, dx}{3 c e^2 \sqrt {1+\frac {1}{c^2 x^2}} x}\\ &=\frac {2 d \left (a+b \text {csch}^{-1}(c x)\right )}{3 e^2 (d+e x)^{3/2}}-\frac {2 \left (a+b \text {csch}^{-1}(c x)\right )}{e^2 \sqrt {d+e x}}-\frac {\left (4 b d \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^2 \sqrt {1+\frac {1}{c^2 x^2}} x}-\frac {\left (2 b \sqrt {1+c^2 x^2}\right ) \int \frac {1}{(d+e x)^{3/2} \sqrt {1+c^2 x^2}} \, dx}{c e \sqrt {1+\frac {1}{c^2 x^2}} x}\\ &=\frac {4 b \left (1+c^2 x^2\right )}{c \left (c^2 d^2+e^2\right ) \sqrt {1+\frac {1}{c^2 x^2}} x \sqrt {d+e x}}+\frac {2 d \left (a+b \text {csch}^{-1}(c x)\right )}{3 e^2 (d+e x)^{3/2}}-\frac {2 \left (a+b \text {csch}^{-1}(c x)\right )}{e^2 \sqrt {d+e x}}-\frac {\left (4 b d \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^2 \sqrt {1+\frac {1}{c^2 x^2}} x}+\frac {\left (4 b c \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 \left (c^2 d^2+e^2\right ) \sqrt {1+\frac {1}{c^2 x^2}} x}\\ &=\frac {4 b \left (1+c^2 x^2\right )}{c \left (c^2 d^2+e^2\right ) \sqrt {1+\frac {1}{c^2 x^2}} x \sqrt {d+e x}}+\frac {2 d \left (a+b \text {csch}^{-1}(c x)\right )}{3 e^2 (d+e x)^{3/2}}-\frac {2 \left (a+b \text {csch}^{-1}(c x)\right )}{e^2 \sqrt {d+e x}}-\frac {\left (4 b \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^2 \sqrt {1+\frac {1}{c^2 x^2}} x}+\frac {\left (4 b \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 \sqrt {1+\frac {1}{c^2 x^2}} x}-\frac {\left (2 b c \sqrt {1+c^2 x^2}\right ) \int \frac {\sqrt {d+e x}}{\sqrt {1+c^2 x^2}} \, dx}{e \left (c^2 d^2+e^2\right ) \sqrt {1+\frac {1}{c^2 x^2}} x}\\ &=\frac {4 b \left (1+c^2 x^2\right )}{3 c \left (c^2 d^2+e^2\right ) \sqrt {1+\frac {1}{c^2 x^2}} x \sqrt {d+e x}}+\frac {2 d \left (a+b \text {csch}^{-1}(c x)\right )}{3 e^2 (d+e x)^{3/2}}-\frac {2 \left (a+b \text {csch}^{-1}(c x)\right )}{e^2 \sqrt {d+e x}}-\frac {\left (4 b \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^2 \sqrt {1+\frac {1}{c^2 x^2}} x}-\frac {\left (8 b c \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 \left (c^2 d^2+e^2\right ) \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 )}{c e \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 \left (1+c^2 x^2\right )}{3 c \left (c^2 d^2+e^2\right ) \sqrt {1+\frac {1}{c^2 x^2}} x \sqrt {d+e x}}+\frac {2 d \left (a+b \text {csch}^{-1}(c x)\right )}{3 e^2 (d+e x)^{3/2}}-\frac {2 \left (a+b \text {csch}^{-1}(c x)\right )}{e^2 \sqrt {d+e x}}-\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 )}{c e \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 (8 b \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^2 \sqrt {1+\frac {1}{c^2 x^2}} x}+\frac {\left (4 b c \sqrt {1+c^2 x^2}\right ) \int \frac {\sqrt {d+e x}}{\sqrt {1+c^2 x^2}} \, dx}{3 e \left (c^2 d^2+e^2\right ) \sqrt {1+\frac {1}{c^2 x^2}} x}\\ &=\frac {4 b \left (1+c^2 x^2\right )}{3 c \left (c^2 d^2+e^2\right ) \sqrt {1+\frac {1}{c^2 x^2}} x \sqrt {d+e x}}+\frac {2 d \left (a+b \text {csch}^{-1}(c x)\right )}{3 e^2 (d+e x)^{3/2}}-\frac {2 \left (a+b \text {csch}^{-1}(c x)\right )}{e^2 \sqrt {d+e x}}-\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 )}{c e \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 (8 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 e \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 (8 b \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^2 \sqrt {1+\frac {1}{c^2 x^2}} x \sqrt {d+e x}}\\ &=\frac {4 b \left (1+c^2 x^2\right )}{3 c \left (c^2 d^2+e^2\right ) \sqrt {1+\frac {1}{c^2 x^2}} x \sqrt {d+e x}}+\frac {2 d \left (a+b \text {csch}^{-1}(c x)\right )}{3 e^2 (d+e x)^{3/2}}-\frac {2 \left (a+b \text {csch}^{-1}(c x)\right )}{e^2 \sqrt {d+e x}}-\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 e \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 {8 b \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^2 \sqrt {1+\frac {1}{c^2 x^2}} x \sqrt {d+e x}}\\ \end {align*}

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Mathematica [C]  time = 2.51, size = 390, normalized size = 0.99 \[ \frac {2}{3} \left (-\frac {a (2 d+3 e x)}{e^2 (d+e x)^{3/2}}+\frac {2 b c x \sqrt {\frac {1}{c^2 x^2}+1}}{\left (c^2 d^2+e^2\right ) \sqrt {d+e x}}+\frac {2 i b \sqrt {-\frac {c}{c d-i e}} \sqrt {-\frac {e (c x-i)}{c d+i e}} \sqrt {-\frac {e (c x+i)}{c d-i e}} \left (-c d F\left (i \sinh ^{-1}\left (\sqrt {-\frac {c}{c d-i e}} \sqrt {d+e x}\right )|\frac {c d-i e}{c d+i e}\right )+c d E\left (i \sinh ^{-1}\left (\sqrt {-\frac {c}{c d-i e}} \sqrt {d+e x}\right )|\frac {c d-i e}{c d+i e}\right )+2 (c d-i e) \Pi \left (1-\frac {i e}{c d};i \sinh ^{-1}\left (\sqrt {-\frac {c}{c d-i e}} \sqrt {d+e x}\right )|\frac {c d-i e}{c d+i e}\right )\right )}{c^2 d e^2 x \sqrt {\frac {1}{c^2 x^2}+1}}-\frac {b \text {csch}^{-1}(c x) (2 d+3 e x)}{e^2 (d+e x)^{3/2}}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*((2*b*c*Sqrt[1 + 1/(c^2*x^2)]*x)/((c^2*d^2 + e^2)*Sqrt[d + e*x]) - (a*(2*d + 3*e*x))/(e^2*(d + e*x)^(3/2))
- (b*(2*d + 3*e*x)*ArcCsch[c*x])/(e^2*(d + e*x)^(3/2)) + ((2*I)*b*Sqrt[-(c/(c*d - I*e))]*Sqrt[-((e*(-I + c*x))
/(c*d + I*e))]*Sqrt[-((e*(I + c*x))/(c*d - I*e))]*(c*d*EllipticE[I*ArcSinh[Sqrt[-(c/(c*d - I*e))]*Sqrt[d + e*x
]], (c*d - I*e)/(c*d + I*e)] - c*d*EllipticF[I*ArcSinh[Sqrt[-(c/(c*d - I*e))]*Sqrt[d + e*x]], (c*d - I*e)/(c*d
+ I*e)] + 2*(c*d - I*e)*EllipticPi[1 - (I*e)/(c*d), I*ArcSinh[Sqrt[-(c/(c*d - I*e))]*Sqrt[d + e*x]], (c*d - I
*e)/(c*d + I*e)]))/(c^2*d*e^2*Sqrt[1 + 1/(c^2*x^2)]*x)))/3

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fricas [F]  time = 8.18, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (b x \operatorname {arcsch}\left (c x\right ) + a x\right )} \sqrt {e x + d}}{e^{3} x^{3} + 3 \, d e^{2} x^{2} + 3 \, d^{2} e x + d^{3}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b \operatorname {arcsch}\left (c x\right ) + a\right )} x}{{\left (e x + d\right )}^{\frac {5}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [C]  time = 0.08, size = 2107, normalized size = 5.36 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ -\frac {1}{3} \, b {\left (\frac {2 \, {\left (3 \, e x + 2 \, d\right )} \log \left (\sqrt {c^{2} x^{2} + 1} + 1\right )}{{\left (e^{3} x + d e^{2}\right )} \sqrt {e x + d}} + 3 \, \int \frac {2 \, {\left (3 \, c^{2} e x^{2} + 2 \, c^{2} d x\right )}}{3 \, {\left ({\left (c^{2} e^{3} x^{3} + c^{2} d e^{2} x^{2} + e^{3} x + d e^{2}\right )} \sqrt {c^{2} x^{2} + 1} \sqrt {e x + d} + {\left (c^{2} e^{3} x^{3} + c^{2} d e^{2} x^{2} + e^{3} x + d e^{2}\right )} \sqrt {e x + d}\right )}}\,{d x} + 3 \, \int -\frac {10 \, c^{2} d e x^{2} - 3 \, {\left (e^{2} \log \relax (c) - 2 \, e^{2}\right )} c^{2} x^{3} + {\left (4 \, c^{2} d^{2} - 3 \, e^{2} \log \relax (c)\right )} x - 3 \, {\left (c^{2} e^{2} x^{3} + e^{2} x\right )} \log \relax (x)}{3 \, {\left (c^{2} e^{4} x^{4} + 2 \, c^{2} d e^{3} x^{3} + 2 \, d e^{3} x + d^{2} e^{2} + {\left (c^{2} d^{2} e^{2} + e^{4}\right )} x^{2}\right )} \sqrt {e x + d}}\,{d x}\right )} - \frac {2}{3} \, a {\left (\frac {3}{\sqrt {e x + d} e^{2}} - \frac {d}{{\left (e x + d\right )}^{\frac {3}{2}} e^{2}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*b*(2*(3*e*x + 2*d)*log(sqrt(c^2*x^2 + 1) + 1)/((e^3*x + d*e^2)*sqrt(e*x + d)) + 3*integrate(2/3*(3*c^2*e*
x^2 + 2*c^2*d*x)/((c^2*e^3*x^3 + c^2*d*e^2*x^2 + e^3*x + d*e^2)*sqrt(c^2*x^2 + 1)*sqrt(e*x + d) + (c^2*e^3*x^3
+ c^2*d*e^2*x^2 + e^3*x + d*e^2)*sqrt(e*x + d)), x) + 3*integrate(-1/3*(10*c^2*d*e*x^2 - 3*(e^2*log(c) - 2*e^
2)*c^2*x^3 + (4*c^2*d^2 - 3*e^2*log(c))*x - 3*(c^2*e^2*x^3 + e^2*x)*log(x))/((c^2*e^4*x^4 + 2*c^2*d*e^3*x^3 +
2*d*e^3*x + d^2*e^2 + (c^2*d^2*e^2 + e^4)*x^2)*sqrt(e*x + d)), x)) - 2/3*a*(3/(sqrt(e*x + d)*e^2) - d/((e*x +
d)^(3/2)*e^2))

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {x\,\left (a+b\,\mathrm {asinh}\left (\frac {1}{c\,x}\right )\right )}{{\left (d+e\,x\right )}^{5/2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x \left (a + b \operatorname {acsch}{\left (c x \right )}\right )}{\left (d + e x\right )^{\frac {5}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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