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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]

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

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Rubi [A]  time = 0.31, antiderivative size = 278, normalized size of antiderivative = 1.00, 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 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 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 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]

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*}

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Mathematica [C]  time = 13.74, 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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fricas [F]  time = 5.01, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {e x + d} {\left (b \operatorname {arsech}\left (c x\right ) + a\right )}}{e^{3} x^{3} + 3 \, d e^{2} x^{2} + 3 \, d^{2} e x + d^{3}}, x\right ) \]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [B]  time = 0.12, size = 902, normalized size = 3.24 \[ \frac {-\frac {2 a}{3 \left (e x +d \right )^{\frac {3}{2}}}+2 b \left (-\frac {\mathrm {arcsech}\left (c x \right )}{3 \left (e x +d \right )^{\frac {3}{2}}}+\frac {2 c \,e^{2} \sqrt {-\frac {\left (e x +d \right ) c -c d -e}{c x e}}\, x \sqrt {\frac {\left (e x +d \right ) c -c d +e}{c x e}}\, \left (\sqrt {\frac {c}{c d +e}}\, \left (e x +d \right )^{2} c^{2} d -\sqrt {-\frac {\left (e x +d \right ) c -c d -e}{c d +e}}\, \sqrt {-\frac {\left (e x +d \right ) c -c d +e}{c d -e}}\, \EllipticF \left (\sqrt {e x +d}\, \sqrt {\frac {c}{c d +e}}, \sqrt {\frac {c d +e}{c d -e}}\right ) \sqrt {e x +d}\, c^{2} d^{2}+c^{2} \sqrt {-\frac {\left (e x +d \right ) c -c d -e}{c d +e}}\, \sqrt {-\frac {\left (e x +d \right ) c -c d +e}{c d -e}}\, \EllipticE \left (\sqrt {e x +d}\, \sqrt {\frac {c}{c d +e}}, \sqrt {\frac {c d +e}{c d -e}}\right ) d^{2} \sqrt {e x +d}-\sqrt {-\frac {\left (e x +d \right ) c -c d -e}{c d +e}}\, \sqrt {-\frac {\left (e x +d \right ) c -c d +e}{c d -e}}\, \EllipticPi \left (\sqrt {e x +d}\, \sqrt {\frac {c}{c d +e}}, \frac {c d +e}{c d}, \frac {\sqrt {\frac {c}{c d -e}}}{\sqrt {\frac {c}{c d +e}}}\right ) \sqrt {e x +d}\, c^{2} d^{2}-2 \sqrt {\frac {c}{c d +e}}\, \left (e x +d \right ) c^{2} d^{2}+\sqrt {-\frac {\left (e x +d \right ) c -c d -e}{c d +e}}\, \sqrt {-\frac {\left (e x +d \right ) c -c d +e}{c d -e}}\, \EllipticF \left (\sqrt {e x +d}\, \sqrt {\frac {c}{c d +e}}, \sqrt {\frac {c d +e}{c d -e}}\right ) \sqrt {e x +d}\, c d e -\sqrt {-\frac {\left (e x +d \right ) c -c d -e}{c d +e}}\, \sqrt {-\frac {\left (e x +d \right ) c -c d +e}{c d -e}}\, \EllipticE \left (\sqrt {e x +d}\, \sqrt {\frac {c}{c d +e}}, \sqrt {\frac {c d +e}{c d -e}}\right ) \sqrt {e x +d}\, c d e +\sqrt {\frac {c}{c d +e}}\, c^{2} d^{3}+\sqrt {-\frac {\left (e x +d \right ) c -c d -e}{c d +e}}\, \sqrt {-\frac {\left (e x +d \right ) c -c d +e}{c d -e}}\, \EllipticPi \left (\sqrt {e x +d}\, \sqrt {\frac {c}{c d +e}}, \frac {c d +e}{c d}, \frac {\sqrt {\frac {c}{c d -e}}}{\sqrt {\frac {c}{c d +e}}}\right ) \sqrt {e x +d}\, e^{2}-\sqrt {\frac {c}{c d +e}}\, d \,e^{2}\right )}{3 \sqrt {e x +d}\, \left (c d -e \right ) \left (c d +e \right ) \sqrt {\frac {c}{c d +e}}\, d^{2} \left (\left (e x +d \right )^{2} c^{2}-2 \left (e x +d \right ) c^{2} d +c^{2} d^{2}-e^{2}\right )}\right )}{e} \]

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/(e*x+d)^(3/2)*a+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)/(e*x+d)^(1/2)/(c*d-e)/(c*d+e)/(c/(c*d+e))^(1/2)/d^2/((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 = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]

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 >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(c*d-e>0)', see `assume?` for m
ore details)Is c*d-e positive or negative?

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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