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

Optimal. Leaf size=369 \[ -\frac {2 \left (a+b \text {csch}^{-1}(c x)\right )}{3 e (d+e x)^{3/2}}-\frac {4 b e \left (c^2 x^2+1\right )}{3 c d x \sqrt {\frac {1}{c^2 x^2}+1} \left (c^2 d^2+e^2\right ) \sqrt {d+e x}}+\frac {4 b \sqrt {-c^2} \sqrt {c^2 x^2+1} \sqrt {d+e x} E\left (\sin ^{-1}\left (\frac {\sqrt {1-\sqrt {-c^2} x}}{\sqrt {2}}\right )|-\frac {2 \sqrt {-c^2} e}{c^2 d-\sqrt {-c^2} e}\right )}{3 c d x \sqrt {\frac {1}{c^2 x^2}+1} \left (c^2 d^2+e^2\right ) \sqrt {\frac {d+e x}{\frac {e}{\sqrt {-c^2}}+d}}}+\frac {4 b \sqrt {c^2 x^2+1} \sqrt {\frac {\sqrt {-c^2} (d+e x)}{\sqrt {-c^2} d+e}} \Pi \left (2;\sin ^{-1}\left (\frac {\sqrt {1-\sqrt {-c^2} x}}{\sqrt {2}}\right )|\frac {2 e}{\sqrt {-c^2} d+e}\right )}{3 c d e x \sqrt {\frac {1}{c^2 x^2}+1} \sqrt {d+e x}} \]

[Out]

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

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Rubi [A]  time = 0.57, antiderivative size = 369, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 11, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.611, Rules used = {6290, 1574, 958, 745, 21, 719, 424, 933, 168, 538, 537} \[ -\frac {2 \left (a+b \text {csch}^{-1}(c x)\right )}{3 e (d+e x)^{3/2}}-\frac {4 b e \left (c^2 x^2+1\right )}{3 c d x \sqrt {\frac {1}{c^2 x^2}+1} \left (c^2 d^2+e^2\right ) \sqrt {d+e x}}+\frac {4 b \sqrt {-c^2} \sqrt {c^2 x^2+1} \sqrt {d+e x} E\left (\sin ^{-1}\left (\frac {\sqrt {1-\sqrt {-c^2} x}}{\sqrt {2}}\right )|-\frac {2 \sqrt {-c^2} e}{c^2 d-\sqrt {-c^2} e}\right )}{3 c d x \sqrt {\frac {1}{c^2 x^2}+1} \left (c^2 d^2+e^2\right ) \sqrt {\frac {d+e x}{\frac {e}{\sqrt {-c^2}}+d}}}+\frac {4 b \sqrt {c^2 x^2+1} \sqrt {\frac {\sqrt {-c^2} (d+e x)}{\sqrt {-c^2} d+e}} \Pi \left (2;\sin ^{-1}\left (\frac {\sqrt {1-\sqrt {-c^2} x}}{\sqrt {2}}\right )|\frac {2 e}{\sqrt {-c^2} d+e}\right )}{3 c d e x \sqrt {\frac {1}{c^2 x^2}+1} \sqrt {d+e x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-4*b*e*(1 + c^2*x^2))/(3*c*d*(c^2*d^2 + e^2)*Sqrt[1 + 1/(c^2*x^2)]*x*Sqrt[d + e*x]) - (2*(a + b*ArcCsch[c*x])
)/(3*e*(d + e*x)^(3/2)) + (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*d*(c^2*d^2 + e^2)*Sqrt[1 + 1/(c^2*x^2)]*x*Sqrt[(
d + e*x)/(d + e/Sqrt[-c^2])]) + (4*b*Sqrt[(Sqrt[-c^2]*(d + e*x))/(Sqrt[-c^2]*d + e)]*Sqrt[1 + c^2*x^2]*Ellipti
cPi[2, ArcSin[Sqrt[1 - Sqrt[-c^2]*x]/Sqrt[2]], (2*e)/(Sqrt[-c^2]*d + e)])/(3*c*d*e*Sqrt[1 + 1/(c^2*x^2)]*x*Sqr
t[d + e*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 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 933

Int[1/(((d_.) + (e_.)*(x_))*Sqrt[(f_.) + (g_.)*(x_)]*Sqrt[(a_) + (c_.)*(x_)^2]), x_Symbol] :> With[{q = Rt[-(c
/a), 2]}, Dist[Sqrt[1 + (c*x^2)/a]/Sqrt[a + c*x^2], 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 1574

Int[(x_)^(m_.)*((a_.) + (c_.)*(x_)^(mn2_.))^(p_)*((d_) + (e_.)*(x_)^(n_.))^(q_.), x_Symbol] :> Dist[(x^(2*n*Fr
acPart[p])*(a + c/x^(2*n))^FracPart[p])/(c + a*x^(2*n))^FracPart[p], Int[x^(m - 2*n*p)*(d + e*x^n)^q*(c + a*x^
(2*n))^p, x], x] /; FreeQ[{a, c, d, e, m, n, p, q}, x] && EqQ[mn2, -2*n] &&  !IntegerQ[p] &&  !IntegerQ[q] &&
PosQ[n]

Rule 6290

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

Rubi steps

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

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Mathematica [C]  time = 14.10, size = 892, normalized size = 2.42 \[ \frac {b \left (\frac {2 \left (\frac {d}{x}+e\right )^{5/2} (c x)^{5/2} \left (\frac {i \sqrt {2} c d (c d-i e) \sqrt {i c x+1} \sqrt {\frac {e (c x+i) (c d+c e x)}{(i c d+e)^2}} \Pi \left (\frac {i c d}{e}+1;\sin ^{-1}\left (\sqrt {-\frac {e (c x+i)}{c d-i e}}\right )|\frac {i c d+e}{2 e}\right )}{e \sqrt {1+\frac {1}{c^2 x^2}} \sqrt {\frac {d}{x}+e} (c x)^{3/2}}+\frac {2 e \cosh \left (2 \text {csch}^{-1}(c x)\right ) \left (\frac {c x \left (c d \sqrt {2 i c x+2} (c x+i) \sqrt {\frac {c d+c e x}{c d-i e}} F\left (\sin ^{-1}\left (\sqrt {-\frac {e (c x+i)}{c d-i e}}\right )|\frac {i c d+e}{2 e}\right )+2 \sqrt {-\frac {e (c x-i)}{c d+i e}} (c x+i) \sqrt {\frac {c d+c e x}{c d-i e}} \left ((c d+i e) E\left (\sin ^{-1}\left (\sqrt {\frac {c d+c e x}{c d-i e}}\right )|\frac {c d-i e}{c d+i e}\right )-i e F\left (\sin ^{-1}\left (\sqrt {\frac {c d+c e x}{c d-i e}}\right )|\frac {c d-i e}{c d+i e}\right )\right )+(i c d+e) \sqrt {2 i c x+2} \sqrt {-\frac {e (c x+i)}{c d-i e}} \sqrt {\frac {e (c x+i) (c d+c e x)}{(i c d+e)^2}} \Pi \left (\frac {i c d}{e}+1;\sin ^{-1}\left (\sqrt {-\frac {e (c x+i)}{c d-i e}}\right )|\frac {i c d+e}{2 e}\right )\right )}{2 \sqrt {-\frac {e (c x+i)}{c d-i e}}}-(c d+c e x) \left (c^2 x^2+1\right )\right )}{c d \sqrt {1+\frac {1}{c^2 x^2}} \sqrt {\frac {d}{x}+e} \sqrt {c x} \left (c^2 x^2+2\right )}\right )}{3 e \left (c^2 d^2+e^2\right ) (d+e x)^{5/2}}-\frac {c^3 \left (\frac {d}{x}+e\right )^3 x^3 \left (\frac {2 \text {csch}^{-1}(c x)}{3 c^2 d^2 e}+\frac {2 e \text {csch}^{-1}(c x)}{3 c^2 d^2 \left (\frac {d}{x}+e\right )^2}-\frac {4 \left (c^2 \text {csch}^{-1}(c x) d^2-c e \sqrt {1+\frac {1}{c^2 x^2}} d+e^2 \text {csch}^{-1}(c x)\right )}{3 c^2 d^2 \left (c^2 d^2+e^2\right ) \left (\frac {d}{x}+e\right )}-\frac {4 \sqrt {1+\frac {1}{c^2 x^2}}}{3 c d \left (c^2 d^2+e^2\right )}\right )}{(d+e x)^{5/2}}\right )}{c}-\frac {2 a}{3 e (d+e x)^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*a)/(3*e*(d + e*x)^(3/2)) + (b*(-((c^3*(e + d/x)^3*x^3*((-4*Sqrt[1 + 1/(c^2*x^2)])/(3*c*d*(c^2*d^2 + e^2))
+ (2*ArcCsch[c*x])/(3*c^2*d^2*e) + (2*e*ArcCsch[c*x])/(3*c^2*d^2*(e + d/x)^2) - (4*(-(c*d*e*Sqrt[1 + 1/(c^2*x^
2)]) + c^2*d^2*ArcCsch[c*x] + e^2*ArcCsch[c*x]))/(3*c^2*d^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)*((I*Sqrt[2]*c*d*(c*d - I*e)*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[Arc
Sin[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))))/(3*e*(c^2*d^2 + e^2)*(d + e*x)^(5/2))))/c

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [C]  time = 0.07, size = 2079, normalized size = 5.63 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/e*(-1/3/(e*x+d)^(3/2)*a+b*(-1/3/(e*x+d)^(3/2)*arccsch(c*x)-2/3/c*(-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*e+c*d)*c/(c^2*d^2+e^2))^(1/2)*(e*x+d)^2*c^3*d^2+I*((I*e+c*d
)*c/(c^2*d^2+e^2))^(1/2)*d*e^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)*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)*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^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-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)*Ellipt
icPi((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*e+c*d)*c/(c^2*d^2+e^2))^(1/2)*c^3*d^4+I
*((I*e+c*d)*c/(c^2*d^2+e^2))^(1/2)*c^2*d^3*e+(-(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)*(e*x+d)^2*c^2*d*e-((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^2/((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 \[ -\frac {1}{3} \, {\left (6 \, c^{2} \int \frac {x}{3 \, {\left ({\left (c^{2} e^{2} x^{3} + c^{2} d e x^{2} + e^{2} x + d e\right )} \sqrt {c^{2} x^{2} + 1} \sqrt {e x + d} + {\left (c^{2} e^{2} x^{3} + c^{2} d e x^{2} + e^{2} x + d e\right )} \sqrt {e x + d}\right )}}\,{d x} + \frac {2 \, \log \left (\sqrt {c^{2} x^{2} + 1} + 1\right )}{{\left (e^{2} x + d e\right )} \sqrt {e x + d}} + 3 \, \int \frac {{\left (3 \, e \log \relax (c) - 2 \, e\right )} c^{2} x^{2} - 2 \, c^{2} d x + 3 \, e \log \relax (c) + 3 \, {\left (c^{2} e x^{2} + e\right )} \log \relax (x)}{3 \, {\left (c^{2} e^{3} x^{4} + 2 \, c^{2} d e^{2} x^{3} + 2 \, d e^{2} x + d^{2} e + {\left (c^{2} d^{2} e + e^{3}\right )} x^{2}\right )} \sqrt {e x + d}}\,{d x}\right )} b - \frac {2 \, a}{3 \, {\left (e x + d\right )}^{\frac {3}{2}} e} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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