3.85 \(\int \frac{a+b \text{sech}^{-1}(c x)}{(d+e x)^{5/2}} \, dx\)
Optimal. Leaf size=278 \[ -\frac{2 \left (a+b \text{sech}^{-1}(c x)\right )}{3 e (d+e x)^{3/2}}+\frac{4 b e \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \sqrt{1-c^2 x^2}}{3 d \left (c^2 d^2-e^2\right ) \sqrt{d+e x}}-\frac{4 b c \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \sqrt{d+e x} E\left (\sin ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{2}}\right )|\frac{2 e}{c d+e}\right )}{3 d \left (c^2 d^2-e^2\right ) \sqrt{\frac{c (d+e x)}{c d+e}}}+\frac{4 b \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \sqrt{\frac{c (d+e x)}{c d+e}} \Pi \left (2;\sin ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{2}}\right )|\frac{2 e}{c d+e}\right )}{3 d e \sqrt{d+e x}} \]
[Out]
(4*b*e*Sqrt[(1 + c*x)^(-1)]*Sqrt[1 + c*x]*Sqrt[1 - c^2*x^2])/(3*d*(c^2*d^2 - e^2)*Sqrt[d + e*x]) - (2*(a + b*A
rcSech[c*x]))/(3*e*(d + e*x)^(3/2)) - (4*b*c*Sqrt[(1 + c*x)^(-1)]*Sqrt[1 + c*x]*Sqrt[d + e*x]*EllipticE[ArcSin
[Sqrt[1 - c*x]/Sqrt[2]], (2*e)/(c*d + e)])/(3*d*(c^2*d^2 - e^2)*Sqrt[(c*(d + e*x))/(c*d + e)]) + (4*b*Sqrt[(1
+ c*x)^(-1)]*Sqrt[1 + c*x]*Sqrt[(c*(d + e*x))/(c*d + e)]*EllipticPi[2, ArcSin[Sqrt[1 - c*x]/Sqrt[2]], (2*e)/(c
*d + e)])/(3*d*e*Sqrt[d + e*x])
________________________________________________________________________________________
Rubi [A] time = 0.31253, antiderivative size = 278, normalized size of antiderivative = 1.,
number of steps used = 11, number of rules used = 10, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.556, Rules used
= {6288, 958, 745, 21, 719, 424, 932, 168, 538, 537} \[ -\frac{2 \left (a+b \text{sech}^{-1}(c x)\right )}{3 e (d+e x)^{3/2}}+\frac{4 b e \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \sqrt{1-c^2 x^2}}{3 d \left (c^2 d^2-e^2\right ) \sqrt{d+e x}}-\frac{4 b c \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \sqrt{d+e x} E\left (\sin ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{2}}\right )|\frac{2 e}{c d+e}\right )}{3 d \left (c^2 d^2-e^2\right ) \sqrt{\frac{c (d+e x)}{c d+e}}}+\frac{4 b \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \sqrt{\frac{c (d+e x)}{c d+e}} \Pi \left (2;\sin ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{2}}\right )|\frac{2 e}{c d+e}\right )}{3 d e \sqrt{d+e x}} \]
Antiderivative was successfully verified.
[In]
Int[(a + b*ArcSech[c*x])/(d + e*x)^(5/2),x]
[Out]
(4*b*e*Sqrt[(1 + c*x)^(-1)]*Sqrt[1 + c*x]*Sqrt[1 - c^2*x^2])/(3*d*(c^2*d^2 - e^2)*Sqrt[d + e*x]) - (2*(a + b*A
rcSech[c*x]))/(3*e*(d + e*x)^(3/2)) - (4*b*c*Sqrt[(1 + c*x)^(-1)]*Sqrt[1 + c*x]*Sqrt[d + e*x]*EllipticE[ArcSin
[Sqrt[1 - c*x]/Sqrt[2]], (2*e)/(c*d + e)])/(3*d*(c^2*d^2 - e^2)*Sqrt[(c*(d + e*x))/(c*d + e)]) + (4*b*Sqrt[(1
+ c*x)^(-1)]*Sqrt[1 + c*x]*Sqrt[(c*(d + e*x))/(c*d + e)]*EllipticPi[2, ArcSin[Sqrt[1 - c*x]/Sqrt[2]], (2*e)/(c
*d + e)])/(3*d*e*Sqrt[d + e*x])
Rule 6288
Int[((a_.) + ArcSech[(c_.)*(x_)]*(b_.))*((d_.) + (e_.)*(x_))^(m_.), x_Symbol] :> Simp[((d + e*x)^(m + 1)*(a +
b*ArcSech[c*x]))/(e*(m + 1)), x] + Dist[(b*Sqrt[1 + c*x]*Sqrt[1/(1 + c*x)])/(e*(m + 1)), Int[(d + e*x)^(m + 1)
/(x*Sqrt[1 - c^2*x^2]), x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[m, -1]
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 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 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 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 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 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 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 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 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)
])
Rubi steps
\begin{align*} \int \frac{a+b \text{sech}^{-1}(c x)}{(d+e x)^{5/2}} \, dx &=-\frac{2 \left (a+b \text{sech}^{-1}(c x)\right )}{3 e (d+e x)^{3/2}}-\frac{\left (2 b \sqrt{\frac{1}{1+c x}} \sqrt{1+c x}\right ) \int \frac{1}{x (d+e x)^{3/2} \sqrt{1-c^2 x^2}} \, dx}{3 e}\\ &=-\frac{2 \left (a+b \text{sech}^{-1}(c x)\right )}{3 e (d+e x)^{3/2}}-\frac{\left (2 b \sqrt{\frac{1}{1+c x}} \sqrt{1+c x}\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 e}\\ &=-\frac{2 \left (a+b \text{sech}^{-1}(c x)\right )}{3 e (d+e x)^{3/2}}+\frac{\left (2 b \sqrt{\frac{1}{1+c x}} \sqrt{1+c x}\right ) \int \frac{1}{(d+e x)^{3/2} \sqrt{1-c^2 x^2}} \, dx}{3 d}-\frac{\left (2 b \sqrt{\frac{1}{1+c x}} \sqrt{1+c x}\right ) \int \frac{1}{x \sqrt{d+e x} \sqrt{1-c^2 x^2}} \, dx}{3 d e}\\ &=\frac{4 b e \sqrt{\frac{1}{1+c x}} \sqrt{1+c x} \sqrt{1-c^2 x^2}}{3 d \left (c^2 d^2-e^2\right ) \sqrt{d+e x}}-\frac{2 \left (a+b \text{sech}^{-1}(c x)\right )}{3 e (d+e x)^{3/2}}-\frac{\left (2 b \sqrt{\frac{1}{1+c x}} \sqrt{1+c x}\right ) \int \frac{1}{x \sqrt{1-c x} \sqrt{1+c x} \sqrt{d+e x}} \, dx}{3 d e}-\frac{\left (4 b c^2 \sqrt{\frac{1}{1+c x}} \sqrt{1+c x}\right ) \int \frac{-\frac{d}{2}-\frac{e x}{2}}{\sqrt{d+e x} \sqrt{1-c^2 x^2}} \, dx}{3 d \left (c^2 d^2-e^2\right )}\\ &=\frac{4 b e \sqrt{\frac{1}{1+c x}} \sqrt{1+c x} \sqrt{1-c^2 x^2}}{3 d \left (c^2 d^2-e^2\right ) \sqrt{d+e x}}-\frac{2 \left (a+b \text{sech}^{-1}(c x)\right )}{3 e (d+e x)^{3/2}}+\frac{\left (4 b \sqrt{\frac{1}{1+c x}} \sqrt{1+c x}\right ) \operatorname{Subst}\left (\int \frac{1}{\left (1-x^2\right ) \sqrt{2-x^2} \sqrt{d+\frac{e}{c}-\frac{e x^2}{c}}} \, dx,x,\sqrt{1-c x}\right )}{3 d e}+\frac{\left (2 b c^2 \sqrt{\frac{1}{1+c x}} \sqrt{1+c x}\right ) \int \frac{\sqrt{d+e x}}{\sqrt{1-c^2 x^2}} \, dx}{3 d \left (c^2 d^2-e^2\right )}\\ &=\frac{4 b e \sqrt{\frac{1}{1+c x}} \sqrt{1+c x} \sqrt{1-c^2 x^2}}{3 d \left (c^2 d^2-e^2\right ) \sqrt{d+e x}}-\frac{2 \left (a+b \text{sech}^{-1}(c x)\right )}{3 e (d+e x)^{3/2}}+\frac{\left (4 b \sqrt{\frac{1}{1+c x}} \sqrt{1+c x} \sqrt{\frac{c (d+e x)}{c d+e}}\right ) \operatorname{Subst}\left (\int \frac{1}{\left (1-x^2\right ) \sqrt{2-x^2} \sqrt{1-\frac{e x^2}{c \left (d+\frac{e}{c}\right )}}} \, dx,x,\sqrt{1-c x}\right )}{3 d e \sqrt{d+e x}}-\frac{\left (4 b c \sqrt{\frac{1}{1+c x}} \sqrt{1+c x} \sqrt{d+e x}\right ) \operatorname{Subst}\left (\int \frac{\sqrt{1+\frac{2 c e x^2}{-c^2 d-c e}}}{\sqrt{1-x^2}} \, dx,x,\frac{\sqrt{1-c x}}{\sqrt{2}}\right )}{3 d \left (c^2 d^2-e^2\right ) \sqrt{-\frac{c^2 (d+e x)}{-c^2 d-c e}}}\\ &=\frac{4 b e \sqrt{\frac{1}{1+c x}} \sqrt{1+c x} \sqrt{1-c^2 x^2}}{3 d \left (c^2 d^2-e^2\right ) \sqrt{d+e x}}-\frac{2 \left (a+b \text{sech}^{-1}(c x)\right )}{3 e (d+e x)^{3/2}}-\frac{4 b c \sqrt{\frac{1}{1+c x}} \sqrt{1+c x} \sqrt{d+e x} E\left (\sin ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{2}}\right )|\frac{2 e}{c d+e}\right )}{3 d \left (c^2 d^2-e^2\right ) \sqrt{\frac{c (d+e x)}{c d+e}}}+\frac{4 b \sqrt{\frac{1}{1+c x}} \sqrt{1+c x} \sqrt{\frac{c (d+e x)}{c d+e}} \Pi \left (2;\sin ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{2}}\right )|\frac{2 e}{c d+e}\right )}{3 d e \sqrt{d+e x}}\\ \end{align*}
Mathematica [C] time = 12.6615, size = 4527, normalized size = 16.28 \[ \text{Result too large to show} \]
Warning: Unable to verify antiderivative.
[In]
Integrate[(a + b*ArcSech[c*x])/(d + e*x)^(5/2),x]
[Out]
(-2*a)/(3*e*(d + e*x)^(3/2)) + Sqrt[(1 - c*x)/(1 + c*x)]*Sqrt[d + e*x]*((4*b*c)/(3*d*(c^2*d^2 - e^2)) - (4*b)/
(3*d*(c*d + e)*(d + e*x))) - (2*b*ArcSech[c*x])/(3*e*(d + e*x)^(3/2)) - (4*b*((e*Sqrt[(1 - c*x)/(1 + c*x)]*Sqr
t[c*(1 + (1 - c*x)/(1 + c*x))]*(c*d + e + (c*d*(1 - c*x))/(1 + c*x) - (e*(1 - c*x))/(1 + c*x)))/((1 + (1 - c*x
)/(1 + c*x))*Sqrt[c + (c*(1 - c*x))/(1 + c*x)]*Sqrt[(c*d + e + (c*d*(1 - c*x))/(1 + c*x) - (e*(1 - c*x))/(1 +
c*x))/(c + (c*(1 - c*x))/(1 + c*x))]) - ((c*d - e)*Sqrt[c*(1 + (1 - c*x)/(1 + c*x))]*Sqrt[c*(1 + (1 - c*x)/(1
+ c*x))*(c*d + e + (c*d*(1 - c*x))/(1 + c*x) - (e*(1 - c*x))/(1 + c*x))]*((I*(-(c*d) - e)*e*Sqrt[1 + (1 - c*x)
/(1 + c*x)]*Sqrt[1 - ((c*d - e)*(1 - c*x))/((-(c*d) - e)*(1 + c*x))]*(EllipticE[I*ArcSinh[Sqrt[(1 - c*x)/(1 +
c*x)]], -((c*d - e)/(-(c*d) - e))] - EllipticF[I*ArcSinh[Sqrt[(1 - c*x)/(1 + c*x)]], -((c*d - e)/(-(c*d) - e))
]))/((c*d - e)*Sqrt[c*(1 + (1 - c*x)/(1 + c*x))*(c*d + e + ((c*d - e)*(1 - c*x))/(1 + c*x))]) + (I*c*d*Sqrt[1
+ (1 - c*x)/(1 + c*x)]*Sqrt[1 - ((c*d - e)*(1 - c*x))/((-(c*d) - e)*(1 + c*x))]*EllipticF[I*ArcSinh[Sqrt[(1 -
c*x)/(1 + c*x)]], -((c*d - e)/(-(c*d) - e))])/Sqrt[c*(1 + (1 - c*x)/(1 + c*x))*(c*d + e + ((c*d - e)*(1 - c*x)
)/(1 + c*x))] + (I*e*Sqrt[1 + (1 - c*x)/(1 + c*x)]*Sqrt[1 - ((c*d - e)*(1 - c*x))/((-(c*d) - e)*(1 + c*x))]*El
lipticF[I*ArcSinh[Sqrt[(1 - c*x)/(1 + c*x)]], -((c*d - e)/(-(c*d) - e))])/Sqrt[c*(1 + (1 - c*x)/(1 + c*x))*(c*
d + e + ((c*d - e)*(1 - c*x))/(1 + c*x))] - (I*c*d*(I + Sqrt[-(c*d) - e]/Sqrt[c*d - e])*(-I + Sqrt[(1 - c*x)/(
1 + c*x)])^2*Sqrt[((Sqrt[-(c*d) - e] - I*Sqrt[c*d - e])*(I + Sqrt[(1 - c*x)/(1 + c*x)]))/((Sqrt[-(c*d) - e] +
I*Sqrt[c*d - e])*(-I + Sqrt[(1 - c*x)/(1 + c*x)]))]*Sqrt[(I*(-(Sqrt[-(c*d) - e]/Sqrt[c*d - e]) + Sqrt[(1 - c*x
)/(1 + c*x)]))/((I + Sqrt[-(c*d) - e]/Sqrt[c*d - e])*(-I + Sqrt[(1 - c*x)/(1 + c*x)]))]*Sqrt[(I*(Sqrt[-(c*d) -
e]/Sqrt[c*d - e] + Sqrt[(1 - c*x)/(1 + c*x)]))/((I - Sqrt[-(c*d) - e]/Sqrt[c*d - e])*(-I + Sqrt[(1 - c*x)/(1
+ c*x)]))]*((1 + I)*EllipticF[ArcSin[Sqrt[((Sqrt[-(c*d) - e] - I*Sqrt[c*d - e])*(I + Sqrt[(1 - c*x)/(1 + c*x)]
))/((Sqrt[-(c*d) - e] + I*Sqrt[c*d - e])*(-I + Sqrt[(1 - c*x)/(1 + c*x)]))]], (Sqrt[-(c*d) - e] + I*Sqrt[c*d -
e])^2/(Sqrt[-(c*d) - e] - I*Sqrt[c*d - e])^2] - (2*I)*EllipticPi[((-I)*(I + Sqrt[-(c*d) - e]/Sqrt[c*d - e]))/
(-I + Sqrt[-(c*d) - e]/Sqrt[c*d - e]), ArcSin[Sqrt[((Sqrt[-(c*d) - e] - I*Sqrt[c*d - e])*(I + Sqrt[(1 - c*x)/(
1 + c*x)]))/((Sqrt[-(c*d) - e] + I*Sqrt[c*d - e])*(-I + Sqrt[(1 - c*x)/(1 + c*x)]))]], (Sqrt[-(c*d) - e] + I*S
qrt[c*d - e])^2/(Sqrt[-(c*d) - e] - I*Sqrt[c*d - e])^2]))/((I - Sqrt[-(c*d) - e]/Sqrt[c*d - e])*Sqrt[c*(1 + (1
- c*x)/(1 + c*x))*(c*d + e + ((c*d - e)*(1 - c*x))/(1 + c*x))]) - (I*(I + Sqrt[-(c*d) - e]/Sqrt[c*d - e])*e*(
-I + Sqrt[(1 - c*x)/(1 + c*x)])^2*Sqrt[((Sqrt[-(c*d) - e] - I*Sqrt[c*d - e])*(I + Sqrt[(1 - c*x)/(1 + c*x)]))/
((Sqrt[-(c*d) - e] + I*Sqrt[c*d - e])*(-I + Sqrt[(1 - c*x)/(1 + c*x)]))]*Sqrt[(I*(-(Sqrt[-(c*d) - e]/Sqrt[c*d
- e]) + Sqrt[(1 - c*x)/(1 + c*x)]))/((I + Sqrt[-(c*d) - e]/Sqrt[c*d - e])*(-I + Sqrt[(1 - c*x)/(1 + c*x)]))]*S
qrt[(I*(Sqrt[-(c*d) - e]/Sqrt[c*d - e] + Sqrt[(1 - c*x)/(1 + c*x)]))/((I - Sqrt[-(c*d) - e]/Sqrt[c*d - e])*(-I
+ Sqrt[(1 - c*x)/(1 + c*x)]))]*((1 + I)*EllipticF[ArcSin[Sqrt[((Sqrt[-(c*d) - e] - I*Sqrt[c*d - e])*(I + Sqrt
[(1 - c*x)/(1 + c*x)]))/((Sqrt[-(c*d) - e] + I*Sqrt[c*d - e])*(-I + Sqrt[(1 - c*x)/(1 + c*x)]))]], (Sqrt[-(c*d
) - e] + I*Sqrt[c*d - e])^2/(Sqrt[-(c*d) - e] - I*Sqrt[c*d - e])^2] - (2*I)*EllipticPi[((-I)*(I + Sqrt[-(c*d)
- e]/Sqrt[c*d - e]))/(-I + Sqrt[-(c*d) - e]/Sqrt[c*d - e]), ArcSin[Sqrt[((Sqrt[-(c*d) - e] - I*Sqrt[c*d - e])*
(I + Sqrt[(1 - c*x)/(1 + c*x)]))/((Sqrt[-(c*d) - e] + I*Sqrt[c*d - e])*(-I + Sqrt[(1 - c*x)/(1 + c*x)]))]], (S
qrt[-(c*d) - e] + I*Sqrt[c*d - e])^2/(Sqrt[-(c*d) - e] - I*Sqrt[c*d - e])^2]))/((I - Sqrt[-(c*d) - e]/Sqrt[c*d
- e])*Sqrt[c*(1 + (1 - c*x)/(1 + c*x))*(c*d + e + ((c*d - e)*(1 - c*x))/(1 + c*x))]) + (I*c*d*(I + Sqrt[-(c*d
) - e]/Sqrt[c*d - e])*(-I + Sqrt[(1 - c*x)/(1 + c*x)])^2*Sqrt[((Sqrt[-(c*d) - e] - I*Sqrt[c*d - e])*(I + Sqrt[
(1 - c*x)/(1 + c*x)]))/((Sqrt[-(c*d) - e] + I*Sqrt[c*d - e])*(-I + Sqrt[(1 - c*x)/(1 + c*x)]))]*Sqrt[(I*(-(Sqr
t[-(c*d) - e]/Sqrt[c*d - e]) + Sqrt[(1 - c*x)/(1 + c*x)]))/((I + Sqrt[-(c*d) - e]/Sqrt[c*d - e])*(-I + Sqrt[(1
- c*x)/(1 + c*x)]))]*Sqrt[(I*(Sqrt[-(c*d) - e]/Sqrt[c*d - e] + Sqrt[(1 - c*x)/(1 + c*x)]))/((I - Sqrt[-(c*d)
- e]/Sqrt[c*d - e])*(-I + Sqrt[(1 - c*x)/(1 + c*x)]))]*((-1 + I)*EllipticF[ArcSin[Sqrt[((Sqrt[-(c*d) - e] - I*
Sqrt[c*d - e])*(I + Sqrt[(1 - c*x)/(1 + c*x)]))/((Sqrt[-(c*d) - e] + I*Sqrt[c*d - e])*(-I + Sqrt[(1 - c*x)/(1
+ c*x)]))]], (Sqrt[-(c*d) - e] + I*Sqrt[c*d - e])^2/(Sqrt[-(c*d) - e] - I*Sqrt[c*d - e])^2] - (2*I)*EllipticPi
[(I*(I + Sqrt[-(c*d) - e]/Sqrt[c*d - e]))/(-I + Sqrt[-(c*d) - e]/Sqrt[c*d - e]), ArcSin[Sqrt[((Sqrt[-(c*d) - e
] - I*Sqrt[c*d - e])*(I + Sqrt[(1 - c*x)/(1 + c*x)]))/((Sqrt[-(c*d) - e] + I*Sqrt[c*d - e])*(-I + Sqrt[(1 - c*
x)/(1 + c*x)]))]], (Sqrt[-(c*d) - e] + I*Sqrt[c*d - e])^2/(Sqrt[-(c*d) - e] - I*Sqrt[c*d - e])^2]))/((I - Sqrt
[-(c*d) - e]/Sqrt[c*d - e])*Sqrt[c*(1 + (1 - c*x)/(1 + c*x))*(c*d + e + ((c*d - e)*(1 - c*x))/(1 + c*x))]) + (
I*(I + Sqrt[-(c*d) - e]/Sqrt[c*d - e])*e*(-I + Sqrt[(1 - c*x)/(1 + c*x)])^2*Sqrt[((Sqrt[-(c*d) - e] - I*Sqrt[c
*d - e])*(I + Sqrt[(1 - c*x)/(1 + c*x)]))/((Sqrt[-(c*d) - e] + I*Sqrt[c*d - e])*(-I + Sqrt[(1 - c*x)/(1 + c*x)
]))]*Sqrt[(I*(-(Sqrt[-(c*d) - e]/Sqrt[c*d - e]) + Sqrt[(1 - c*x)/(1 + c*x)]))/((I + Sqrt[-(c*d) - e]/Sqrt[c*d
- e])*(-I + Sqrt[(1 - c*x)/(1 + c*x)]))]*Sqrt[(I*(Sqrt[-(c*d) - e]/Sqrt[c*d - e] + Sqrt[(1 - c*x)/(1 + c*x)]))
/((I - Sqrt[-(c*d) - e]/Sqrt[c*d - e])*(-I + Sqrt[(1 - c*x)/(1 + c*x)]))]*((-1 + I)*EllipticF[ArcSin[Sqrt[((Sq
rt[-(c*d) - e] - I*Sqrt[c*d - e])*(I + Sqrt[(1 - c*x)/(1 + c*x)]))/((Sqrt[-(c*d) - e] + I*Sqrt[c*d - e])*(-I +
Sqrt[(1 - c*x)/(1 + c*x)]))]], (Sqrt[-(c*d) - e] + I*Sqrt[c*d - e])^2/(Sqrt[-(c*d) - e] - I*Sqrt[c*d - e])^2]
- (2*I)*EllipticPi[(I*(I + Sqrt[-(c*d) - e]/Sqrt[c*d - e]))/(-I + Sqrt[-(c*d) - e]/Sqrt[c*d - e]), ArcSin[Sqr
t[((Sqrt[-(c*d) - e] - I*Sqrt[c*d - e])*(I + Sqrt[(1 - c*x)/(1 + c*x)]))/((Sqrt[-(c*d) - e] + I*Sqrt[c*d - e])
*(-I + Sqrt[(1 - c*x)/(1 + c*x)]))]], (Sqrt[-(c*d) - e] + I*Sqrt[c*d - e])^2/(Sqrt[-(c*d) - e] - I*Sqrt[c*d -
e])^2]))/((I - Sqrt[-(c*d) - e]/Sqrt[c*d - e])*Sqrt[c*(1 + (1 - c*x)/(1 + c*x))*(c*d + e + ((c*d - e)*(1 - c*x
))/(1 + c*x))])))/((1 + (1 - c*x)/(1 + c*x))*Sqrt[c + (c*(1 - c*x))/(1 + c*x)]*Sqrt[(c*d + e + (c*d*(1 - c*x))
/(1 + c*x) - (e*(1 - c*x))/(1 + c*x))/(c + (c*(1 - c*x))/(1 + c*x))])))/(3*d*e*(c^2*d^2 - e^2))
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Maple [B] time = 0.335, size = 902, normalized size = 3.2 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((a+b*arcsech(c*x))/(e*x+d)^(5/2),x)
[Out]
2/e*(-1/3*a/(e*x+d)^(3/2)+b*(-1/3/(e*x+d)^(3/2)*arcsech(c*x)+2/3*c*e^2*(-((e*x+d)*c-c*d-e)/c/x/e)^(1/2)*x*(((e
*x+d)*c-c*d+e)/c/x/e)^(1/2)*((c/(c*d+e))^(1/2)*(e*x+d)^2*c^2*d-(-((e*x+d)*c-c*d-e)/(c*d+e))^(1/2)*(-((e*x+d)*c
-c*d+e)/(c*d-e))^(1/2)*EllipticF((e*x+d)^(1/2)*(c/(c*d+e))^(1/2),((c*d+e)/(c*d-e))^(1/2))*(e*x+d)^(1/2)*c^2*d^
2+c^2*(-((e*x+d)*c-c*d-e)/(c*d+e))^(1/2)*(-((e*x+d)*c-c*d+e)/(c*d-e))^(1/2)*EllipticE((e*x+d)^(1/2)*(c/(c*d+e)
)^(1/2),((c*d+e)/(c*d-e))^(1/2))*d^2*(e*x+d)^(1/2)-(-((e*x+d)*c-c*d-e)/(c*d+e))^(1/2)*(-((e*x+d)*c-c*d+e)/(c*d
-e))^(1/2)*EllipticPi((e*x+d)^(1/2)*(c/(c*d+e))^(1/2),1/c*(c*d+e)/d,(c/(c*d-e))^(1/2)/(c/(c*d+e))^(1/2))*(e*x+
d)^(1/2)*c^2*d^2-2*(c/(c*d+e))^(1/2)*(e*x+d)*c^2*d^2+(-((e*x+d)*c-c*d-e)/(c*d+e))^(1/2)*(-((e*x+d)*c-c*d+e)/(c
*d-e))^(1/2)*EllipticF((e*x+d)^(1/2)*(c/(c*d+e))^(1/2),((c*d+e)/(c*d-e))^(1/2))*(e*x+d)^(1/2)*c*d*e-(-((e*x+d)
*c-c*d-e)/(c*d+e))^(1/2)*(-((e*x+d)*c-c*d+e)/(c*d-e))^(1/2)*EllipticE((e*x+d)^(1/2)*(c/(c*d+e))^(1/2),((c*d+e)
/(c*d-e))^(1/2))*(e*x+d)^(1/2)*c*d*e+(c/(c*d+e))^(1/2)*c^2*d^3+(-((e*x+d)*c-c*d-e)/(c*d+e))^(1/2)*(-((e*x+d)*c
-c*d+e)/(c*d-e))^(1/2)*EllipticPi((e*x+d)^(1/2)*(c/(c*d+e))^(1/2),1/c*(c*d+e)/d,(c/(c*d-e))^(1/2)/(c/(c*d+e))^
(1/2))*(e*x+d)^(1/2)*e^2-(c/(c*d+e))^(1/2)*d*e^2)/(c/(c*d+e))^(1/2)/(e*x+d)^(1/2)/d^2/(c*d+e)/(c*d-e)/((e*x+d)
^2*c^2-2*(e*x+d)*c^2*d+c^2*d^2-e^2)))
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((a+b*arcsech(c*x))/(e*x+d)^(5/2),x, algorithm="maxima")
[Out]
Exception raised: ValueError
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{e x + d}{\left (b \operatorname{arsech}\left (c x\right ) + a\right )}}{e^{3} x^{3} + 3 \, d e^{2} x^{2} + 3 \, d^{2} e x + d^{3}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((a+b*arcsech(c*x))/(e*x+d)^(5/2),x, algorithm="fricas")
[Out]
integral(sqrt(e*x + d)*(b*arcsech(c*x) + a)/(e^3*x^3 + 3*d*e^2*x^2 + 3*d^2*e*x + d^3), x)
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((a+b*asech(c*x))/(e*x+d)**(5/2),x)
[Out]
Timed out
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{b \operatorname{arsech}\left (c x\right ) + a}{{\left (e x + d\right )}^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((a+b*arcsech(c*x))/(e*x+d)^(5/2),x, algorithm="giac")
[Out]
integrate((b*arcsech(c*x) + a)/(e*x + d)^(5/2), x)