3.1.72 \(\int \frac {a+b \text {csch}^{-1}(c x)}{(d+e x)^{5/2}} \, dx\) [72]

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

[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 = 0.38, 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 = {6425, 1588, 972, 759, 21, 733, 435, 947, 174, 552, 551} \begin {gather*} -\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 {c^2 x^2+1} \sqrt {d+e x} E\left (\text {ArcSin}\left (\frac {\sqrt {1-\sqrt {-c^2} x}}{\sqrt {2}}\right )|-\frac {2 \sqrt {-c^2} e}{c^2 d-\sqrt {-c^2} e}\right )}{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;\text {ArcSin}\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}}-\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}} \end {gather*}

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 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 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 947

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 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 1588

Int[(x_)^(m_.)*((a_.) + (c_.)*(x_)^(mn2_.))^(p_)*((d_) + (e_.)*(x_)^(n_.))^(q_.), x_Symbol] :> Dist[x^(2*n*Fra
cPart[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 6425

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 ) \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 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 ) \text {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 ) \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 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] Result contains complex when optimal does not.
time = 18.87, size = 784, normalized size = 2.12 \begin {gather*} \frac {-\frac {2 a (d+e x)}{e}+\frac {4 b c \sqrt {1+\frac {1}{c^2 x^2}} (d+e x)^3}{d \left (c^2 d^2+e^2\right )}-\frac {2 b e x^2 (d+e x) \text {csch}^{-1}(c x)}{d^2}-\frac {2 b (d+e x)^3 \text {csch}^{-1}(c x)}{d^2 e}+\frac {4 b x (d+e x)^2 \left (-c d e \sqrt {1+\frac {1}{c^2 x^2}}+\left (c^2 d^2+e^2\right ) \text {csch}^{-1}(c x)\right )}{d^2 \left (c^2 d^2+e^2\right )}+\frac {2 i b c d \sqrt {2+2 i c x} (d+e x)^2 \sqrt {\frac {c e (i+c x) (d+e x)}{(i c d+e)^2}} \Pi \left (1+\frac {i c d}{e};\text {ArcSin}\left (\sqrt {-\frac {e (i+c x)}{c d-i e}}\right )|\frac {i c d+e}{2 e}\right )}{(c d+i e) e^2 \sqrt {1+\frac {1}{c^2 x^2}} x}+\frac {4 b (d+e x)^2 \cosh \left (2 \text {csch}^{-1}(c x)\right ) \left (-c (d+e x) \left (1+c^2 x^2\right )+\frac {c x \left (c d \sqrt {2+2 i c x} (i+c x) \sqrt {\frac {c (d+e x)}{c d-i e}} F\left (\text {ArcSin}\left (\sqrt {-\frac {e (i+c x)}{c d-i e}}\right )|\frac {i c d+e}{2 e}\right )+2 \sqrt {\frac {e (1+i c x)}{-i c d+e}} (i+c x) \sqrt {\frac {c (d+e x)}{c d-i e}} \left ((c d+i e) E\left (\text {ArcSin}\left (\sqrt {\frac {c (d+e x)}{c d-i e}}\right )|\frac {c d-i e}{c d+i e}\right )-i e F\left (\text {ArcSin}\left (\sqrt {\frac {c (d+e x)}{c d-i e}}\right )|\frac {c d-i e}{c d+i e}\right )\right )+(i c d+e) \sqrt {\frac {e (1-i c x)}{i c d+e}} \sqrt {2+2 i c x} \sqrt {\frac {c e (i+c x) (d+e x)}{(i c d+e)^2}} \Pi \left (1+\frac {i c d}{e};\text {ArcSin}\left (\sqrt {-\frac {e (i+c x)}{c d-i e}}\right )|\frac {i c d+e}{2 e}\right )\right )}{2 \sqrt {\frac {e (1-i c x)}{i c d+e}}}\right )}{d \left (c^2 d^2+e^2\right ) \sqrt {1+\frac {1}{c^2 x^2}} \left (2+c^2 x^2\right )}}{3 (d+e x)^{5/2}} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

((-2*a*(d + e*x))/e + (4*b*c*Sqrt[1 + 1/(c^2*x^2)]*(d + e*x)^3)/(d*(c^2*d^2 + e^2)) - (2*b*e*x^2*(d + e*x)*Arc
Csch[c*x])/d^2 - (2*b*(d + e*x)^3*ArcCsch[c*x])/(d^2*e) + (4*b*x*(d + e*x)^2*(-(c*d*e*Sqrt[1 + 1/(c^2*x^2)]) +
(c^2*d^2 + e^2)*ArcCsch[c*x]))/(d^2*(c^2*d^2 + e^2)) + ((2*I)*b*c*d*Sqrt[2 + (2*I)*c*x]*(d + e*x)^2*Sqrt[(c*e
*(I + c*x)*(d + 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)])/((c*d + I*e)*e^2*Sqrt[1 + 1/(c^2*x^2)]*x) + (4*b*(d + e*x)^2*Cosh[2*ArcCsch[c*x]]*(-(c*(d + e
*x)*(1 + c^2*x^2)) + (c*x*(c*d*Sqrt[2 + (2*I)*c*x]*(I + c*x)*Sqrt[(c*(d + 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*(1 + I*c*x))/((-I)*c*d + e)]*(I + c*x)*Sqr
t[(c*(d + e*x))/(c*d - I*e)]*((c*d + I*e)*EllipticE[ArcSin[Sqrt[(c*(d + e*x))/(c*d - I*e)]], (c*d - I*e)/(c*d
+ I*e)] - I*e*EllipticF[ArcSin[Sqrt[(c*(d + e*x))/(c*d - I*e)]], (c*d - I*e)/(c*d + I*e)]) + (I*c*d + e)*Sqrt[
(e*(1 - I*c*x))/(I*c*d + e)]*Sqrt[2 + (2*I)*c*x]*Sqrt[(c*e*(I + c*x)*(d + 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*(1 - I*c*x))/(I*c*d + e
)])))/(d*(c^2*d^2 + e^2)*Sqrt[1 + 1/(c^2*x^2)]*(2 + c^2*x^2)))/(3*(d + e*x)^(5/2))

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Maple [C] Result contains complex when optimal does not.
time = 0.80, size = 2079, normalized size = 5.63

method	result	size
derivativedivides	\(\text {Expression too large to display}\)	\(2079\)
default	\(\text {Expression too large to display}\)	\(2079\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/e*(-1/3*a/(e*x+d)^(3/2)+b*(-1/3/(e*x+d)^(3/2)*arccsch(c*x)-2/3/c*(-2*I*((I*e+c*d)*c/(c^2*d^2+e^2))^(1/2)*c^2
*d^2*e*(e*x+d)-I*(-(I*c*(e*x+d)*e+c^2*d*(e*x+d)-c^2*d^2-e^2)/(c^2*d^2+e^2))^(1/2)*((I*c*(e*x+d)*e-c^2*d*(e*x+d
)+c^2*d^2+e^2)/(c^2*d^2+e^2))^(1/2)*EllipticPi((e*x+d)^(1/2)*((I*e+c*d)*c/(c^2*d^2+e^2))^(1/2),1/(I*e+c*d)/c*(
c^2*d^2+e^2)/d,(-(I*e-c*d)*c/(c^2*d^2+e^2))^(1/2)/((I*e+c*d)*c/(c^2*d^2+e^2))^(1/2))*c^2*d^2*e*(e*x+d)^(1/2)-(
(I*e+c*d)*c/(c^2*d^2+e^2))^(1/2)*c^3*d^2*(e*x+d)^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))*(-(I*c*(e*
x+d)*e+c^2*d*(e*x+d)-c^2*d^2-e^2)/(c^2*d^2+e^2))^(1/2)*((I*c*(e*x+d)*e-c^2*d*(e*x+d)+c^2*d^2+e^2)/(c^2*d^2+e^2
))^(1/2)*c^3*d^3*(e*x+d)^(1/2)+(-(I*c*(e*x+d)*e+c^2*d*(e*x+d)-c^2*d^2-e^2)/(c^2*d^2+e^2))^(1/2)*((I*c*(e*x+d)*
e-c^2*d*(e*x+d)+c^2*d^2+e^2)/(c^2*d^2+e^2))^(1/2)*EllipticF((e*x+d)^(1/2)*((I*e+c*d)*c/(c^2*d^2+e^2))^(1/2),(-
(2*I*c*d*e-c^2*d^2+e^2)/(c^2*d^2+e^2))^(1/2))*c^3*d^3*(e*x+d)^(1/2)-(-(I*c*(e*x+d)*e+c^2*d*(e*x+d)-c^2*d^2-e^2
)/(c^2*d^2+e^2))^(1/2)*((I*c*(e*x+d)*e-c^2*d*(e*x+d)+c^2*d^2+e^2)/(c^2*d^2+e^2))^(1/2)*EllipticE((e*x+d)^(1/2)
*((I*e+c*d)*c/(c^2*d^2+e^2))^(1/2),(-(2*I*c*d*e-c^2*d^2+e^2)/(c^2*d^2+e^2))^(1/2))*c^3*d^3*(e*x+d)^(1/2)+I*((I
*e+c*d)*c/(c^2*d^2+e^2))^(1/2)*c^2*d*e*(e*x+d)^2+2*((I*e+c*d)*c/(c^2*d^2+e^2))^(1/2)*c^3*d^3*(e*x+d)-I*(-(I*c*
(e*x+d)*e+c^2*d*(e*x+d)-c^2*d^2-e^2)/(c^2*d^2+e^2))^(1/2)*((I*c*(e*x+d)*e-c^2*d*(e*x+d)+c^2*d^2+e^2)/(c^2*d^2+
e^2))^(1/2)*EllipticPi((e*x+d)^(1/2)*((I*e+c*d)*c/(c^2*d^2+e^2))^(1/2),1/(I*e+c*d)/c*(c^2*d^2+e^2)/d,(-(I*e-c*
d)*c/(c^2*d^2+e^2))^(1/2)/((I*e+c*d)*c/(c^2*d^2+e^2))^(1/2))*e^3*(e*x+d)^(1/2)+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+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))*(-(I
*c*(e*x+d)*e+c^2*d*(e*x+d)-c^2*d^2-e^2)/(c^2*d^2+e^2))^(1/2)*((I*c*(e*x+d)*e-c^2*d*(e*x+d)+c^2*d^2+e^2)/(c^2*d
^2+e^2))^(1/2)*c*d*e^2*(e*x+d)^(1/2)+(-(I*c*(e*x+d)*e+c^2*d*(e*x+d)-c^2*d^2-e^2)/(c^2*d^2+e^2))^(1/2)*((I*c*(e
*x+d)*e-c^2*d*(e*x+d)+c^2*d^2+e^2)/(c^2*d^2+e^2))^(1/2)*EllipticF((e*x+d)^(1/2)*((I*e+c*d)*c/(c^2*d^2+e^2))^(1
/2),(-(2*I*c*d*e-c^2*d^2+e^2)/(c^2*d^2+e^2))^(1/2))*c*d*e^2*(e*x+d)^(1/2)-(-(I*c*(e*x+d)*e+c^2*d*(e*x+d)-c^2*d
^2-e^2)/(c^2*d^2+e^2))^(1/2)*((I*c*(e*x+d)*e-c^2*d*(e*x+d)+c^2*d^2+e^2)/(c^2*d^2+e^2))^(1/2)*EllipticE((e*x+d)
^(1/2)*((I*e+c*d)*c/(c^2*d^2+e^2))^(1/2),(-(2*I*c*d*e-c^2*d^2+e^2)/(c^2*d^2+e^2))^(1/2))*c*d*e^2*(e*x+d)^(1/2)
+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)/((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/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 \begin {gather*} \text {Failed to integrate} \end {gather*}

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

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Fricas [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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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