3.81 \(\int (d+e x)^{3/2} (a+b \text {sech}^{-1}(c x)) \, dx\)

Optimal. Leaf size=343 \[ \frac {2 (d+e x)^{5/2} \left (a+b \text {sech}^{-1}(c x)\right )}{5 e}-\frac {4 b e \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \sqrt {1-c^2 x^2} \sqrt {d+e x}}{15 c^2}-\frac {4 b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \left (2 c^2 d^2+e^2\right ) \sqrt {\frac {c (d+e x)}{c d+e}} F\left (\sin ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{15 c^3 \sqrt {d+e x}}-\frac {4 b d^3 \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \sqrt {\frac {c (d+e x)}{c d+e}} \Pi \left (2;\sin ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{5 e \sqrt {d+e x}}-\frac {28 b d \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \sqrt {d+e x} E\left (\sin ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{15 c \sqrt {\frac {c (d+e x)}{c d+e}}} \]

[Out]

2/5*(e*x+d)^(5/2)*(a+b*arcsech(c*x))/e-28/15*b*d*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)/c/(c*(e*x+d)/(c*d+e))^(1/2)-4/15*b*(2*c^2*d^2+e^2)*EllipticF(
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)*(c*(e*x+d)/(c*d+e))^(1/2
)/c^3/(e*x+d)^(1/2)-4/5*b*d^3*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)/e/(e*x+d)^(1/2)-4/15*b*e*(1/(c*x+1))^(1/2)*(c*x+1)^(1/2)*(e*x+d)^
(1/2)*(-c^2*x^2+1)^(1/2)/c^2

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Rubi [A]  time = 0.62, antiderivative size = 343, normalized size of antiderivative = 1.00, number of steps used = 21, number of rules used = 12, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {6288, 958, 719, 419, 932, 168, 538, 537, 844, 424, 931, 1584} \[ \frac {2 (d+e x)^{5/2} \left (a+b \text {sech}^{-1}(c x)\right )}{5 e}-\frac {4 b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \left (2 c^2 d^2+e^2\right ) \sqrt {\frac {c (d+e x)}{c d+e}} F\left (\sin ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{15 c^3 \sqrt {d+e x}}-\frac {4 b e \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \sqrt {1-c^2 x^2} \sqrt {d+e x}}{15 c^2}-\frac {4 b d^3 \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \sqrt {\frac {c (d+e x)}{c d+e}} \Pi \left (2;\sin ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{5 e \sqrt {d+e x}}-\frac {28 b d \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \sqrt {d+e x} E\left (\sin ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{15 c \sqrt {\frac {c (d+e x)}{c d+e}}} \]

Antiderivative was successfully verified.

[In]

Int[(d + e*x)^(3/2)*(a + b*ArcSech[c*x]),x]

[Out]

(-4*b*e*Sqrt[(1 + c*x)^(-1)]*Sqrt[1 + c*x]*Sqrt[d + e*x]*Sqrt[1 - c^2*x^2])/(15*c^2) + (2*(d + e*x)^(5/2)*(a +
b*ArcSech[c*x]))/(5*e) - (28*b*d*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)])/(15*c*Sqrt[(c*(d + e*x))/(c*d + e)]) - (4*b*(2*c^2*d^2 + e^2)*Sqrt[(1 + c*x)^(
-1)]*Sqrt[1 + c*x]*Sqrt[(c*(d + e*x))/(c*d + e)]*EllipticF[ArcSin[Sqrt[1 - c*x]/Sqrt[2]], (2*e)/(c*d + e)])/(1
5*c^3*Sqrt[d + e*x]) - (4*b*d^3*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)])/(5*e*Sqrt[d + e*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 419

Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> Simp[(1*EllipticF[ArcSin[Rt[-(d/c),
2]*x], (b*c)/(a*d)])/(Sqrt[a]*Sqrt[c]*Rt[-(d/c), 2]), x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] &
& GtQ[a, 0] &&  !(NegQ[b/a] && SimplerSqrtQ[-(b/a), -(d/c)])

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 844

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Dist[g/e, Int[(d
+ e*x)^(m + 1)*(a + c*x^2)^p, x], x] + Dist[(e*f - d*g)/e, Int[(d + e*x)^m*(a + c*x^2)^p, x], x] /; FreeQ[{a,
c, d, e, f, g, m, p}, x] && NeQ[c*d^2 + a*e^2, 0] &&  !IGtQ[m, 0]

Rule 931

Int[((d_.) + (e_.)*(x_))^(m_)/(Sqrt[(f_.) + (g_.)*(x_)]*Sqrt[(a_) + (c_.)*(x_)^2]), x_Symbol] :> Simp[(2*e^2*(
d + e*x)^(m - 2)*Sqrt[f + g*x]*Sqrt[a + c*x^2])/(c*g*(2*m - 1)), x] - Dist[1/(c*g*(2*m - 1)), Int[((d + e*x)^(
m - 3)*Simp[a*e^2*(d*g + 2*e*f*(m - 2)) - c*d^3*g*(2*m - 1) + e*(e*(a*e*g*(2*m - 3)) + c*d*(2*e*f - 3*d*g*(2*m
- 1)))*x + 2*e^2*(c*e*f - 3*c*d*g)*(m - 1)*x^2, x])/(Sqrt[f + g*x]*Sqrt[a + c*x^2]), 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[2*m] && GeQ[m, 2]

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 1584

Int[(u_.)*(x_)^(m_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.))^(n_.), x_Symbol] :> Int[u*x^(m + n*p)*(a + b*x^(q -
p))^n, x] /; FreeQ[{a, b, m, p, q}, x] && IntegerQ[n] && PosQ[q - p]

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 (d+e x)^{3/2} \left (a+b \text {sech}^{-1}(c x)\right ) \, dx &=\frac {2 (d+e x)^{5/2} \left (a+b \text {sech}^{-1}(c x)\right )}{5 e}+\frac {\left (2 b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \int \frac {(d+e x)^{5/2}}{x \sqrt {1-c^2 x^2}} \, dx}{5 e}\\ &=\frac {2 (d+e x)^{5/2} \left (a+b \text {sech}^{-1}(c x)\right )}{5 e}+\frac {\left (2 b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \int \left (\frac {3 d^2 e}{\sqrt {d+e x} \sqrt {1-c^2 x^2}}+\frac {d^3}{x \sqrt {d+e x} \sqrt {1-c^2 x^2}}+\frac {3 d e^2 x}{\sqrt {d+e x} \sqrt {1-c^2 x^2}}+\frac {e^3 x^2}{\sqrt {d+e x} \sqrt {1-c^2 x^2}}\right ) \, dx}{5 e}\\ &=\frac {2 (d+e x)^{5/2} \left (a+b \text {sech}^{-1}(c x)\right )}{5 e}+\frac {1}{5} \left (6 b d^2 \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {1-c^2 x^2}} \, dx+\frac {\left (2 b d^3 \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}{5 e}+\frac {1}{5} \left (6 b d e \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \int \frac {x}{\sqrt {d+e x} \sqrt {1-c^2 x^2}} \, dx+\frac {1}{5} \left (2 b e^2 \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \int \frac {x^2}{\sqrt {d+e x} \sqrt {1-c^2 x^2}} \, dx\\ &=-\frac {4 b e \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {d+e x} \sqrt {1-c^2 x^2}}{15 c^2}+\frac {2 (d+e x)^{5/2} \left (a+b \text {sech}^{-1}(c x)\right )}{5 e}+\frac {1}{5} \left (6 b d \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \int \frac {\sqrt {d+e x}}{\sqrt {1-c^2 x^2}} \, dx-\frac {1}{5} \left (6 b d^2 \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {1-c^2 x^2}} \, dx+\frac {\left (2 b d^3 \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}{5 e}+\frac {\left (2 b e \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \int \frac {e x-2 c^2 d x^2}{x \sqrt {d+e x} \sqrt {1-c^2 x^2}} \, dx}{15 c^2}-\frac {\left (12 b d^2 \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {-\frac {c^2 (d+e x)}{-c^2 d-c e}}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-x^2} \sqrt {1+\frac {2 c e x^2}{-c^2 d-c e}}} \, dx,x,\frac {\sqrt {1-c x}}{\sqrt {2}}\right )}{5 c \sqrt {d+e x}}\\ &=-\frac {4 b e \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {d+e x} \sqrt {1-c^2 x^2}}{15 c^2}+\frac {2 (d+e x)^{5/2} \left (a+b \text {sech}^{-1}(c x)\right )}{5 e}-\frac {12 b d^2 \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {\frac {c (d+e x)}{c d+e}} F\left (\sin ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{5 c \sqrt {d+e x}}-\frac {\left (4 b d^3 \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 )}{5 e}+\frac {\left (2 b e \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \int \frac {e-2 c^2 d x}{\sqrt {d+e x} \sqrt {1-c^2 x^2}} \, dx}{15 c^2}-\frac {\left (12 b d \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 )}{5 c \sqrt {-\frac {c^2 (d+e x)}{-c^2 d-c e}}}+\frac {\left (12 b d^2 \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {-\frac {c^2 (d+e x)}{-c^2 d-c e}}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-x^2} \sqrt {1+\frac {2 c e x^2}{-c^2 d-c e}}} \, dx,x,\frac {\sqrt {1-c x}}{\sqrt {2}}\right )}{5 c \sqrt {d+e x}}\\ &=-\frac {4 b e \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {d+e x} \sqrt {1-c^2 x^2}}{15 c^2}+\frac {2 (d+e x)^{5/2} \left (a+b \text {sech}^{-1}(c x)\right )}{5 e}-\frac {12 b d \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 )}{5 c \sqrt {\frac {c (d+e x)}{c d+e}}}-\frac {1}{15} \left (4 b d \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \int \frac {\sqrt {d+e x}}{\sqrt {1-c^2 x^2}} \, dx+\frac {\left (2 b \left (2 c^2 d^2+e^2\right ) \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {1-c^2 x^2}} \, dx}{15 c^2}-\frac {\left (4 b d^3 \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 )}{5 e \sqrt {d+e x}}\\ &=-\frac {4 b e \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {d+e x} \sqrt {1-c^2 x^2}}{15 c^2}+\frac {2 (d+e x)^{5/2} \left (a+b \text {sech}^{-1}(c x)\right )}{5 e}-\frac {12 b d \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 )}{5 c \sqrt {\frac {c (d+e x)}{c d+e}}}-\frac {4 b d^3 \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 )}{5 e \sqrt {d+e x}}+\frac {\left (8 b d \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 )}{15 c \sqrt {-\frac {c^2 (d+e x)}{-c^2 d-c e}}}-\frac {\left (4 b \left (2 c^2 d^2+e^2\right ) \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {-\frac {c^2 (d+e x)}{-c^2 d-c e}}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-x^2} \sqrt {1+\frac {2 c e x^2}{-c^2 d-c e}}} \, dx,x,\frac {\sqrt {1-c x}}{\sqrt {2}}\right )}{15 c^3 \sqrt {d+e x}}\\ &=-\frac {4 b e \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {d+e x} \sqrt {1-c^2 x^2}}{15 c^2}+\frac {2 (d+e x)^{5/2} \left (a+b \text {sech}^{-1}(c x)\right )}{5 e}-\frac {28 b d \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 )}{15 c \sqrt {\frac {c (d+e x)}{c d+e}}}-\frac {4 b \left (2 c^2 d^2+e^2\right ) \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {\frac {c (d+e x)}{c d+e}} F\left (\sin ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{15 c^3 \sqrt {d+e x}}-\frac {4 b d^3 \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 )}{5 e \sqrt {d+e x}}\\ \end {align*}

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Mathematica [C]  time = 10.78, size = 2653, normalized size = 7.73 \[ \text {Result too large to show} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[(d + e*x)^(3/2)*(a + b*ArcSech[c*x]),x]

[Out]

Sqrt[(1 - c*x)/(1 + c*x)]*Sqrt[d + e*x]*((-4*b*e)/(15*c^2) - (4*b*e*x)/(15*c)) + Sqrt[d + e*x]*((2*a*d^2)/(5*e
) + (4*a*d*x)/5 + (2*a*e*x^2)/5) + (2*b*(d + e*x)^(5/2)*ArcSech[c*x])/(5*e) - (4*b*(7*c*d*e*Sqrt[(1 - c*x)/(1
+ c*x)]*(e - (e*(1 - c*x))/(1 + c*x) + c*d*(1 + (1 - c*x)/(1 + c*x))) + ((7*I)*c^2*d^2*e*(c*d + e)*Sqrt[1 + (1
- c*x)/(1 + c*x)]*Sqrt[(e - (e*(1 - c*x))/(1 + c*x) + c*d*(1 + (1 - c*x)/(1 + c*x)))/(c*d + e)]*(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) - ((7*I)*c*d*e^2*(c*d + e)*Sqrt[1 + (1 - c*x)/(1 + c*x)]*Sqrt[(e - (e*(1 - c*x))
/(1 + c*x) + c*d*(1 + (1 - c*x)/(1 + c*x)))/(c*d + e)]*(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) + (3*I)*c^3*
d^3*Sqrt[1 + (1 - c*x)/(1 + c*x)]*Sqrt[(e - (e*(1 - c*x))/(1 + c*x) + c*d*(1 + (1 - c*x)/(1 + c*x)))/(c*d + e)
]*EllipticF[I*ArcSinh[Sqrt[(1 - c*x)/(1 + c*x)]], (c*d - e)/(c*d + e)] - (2*I)*c^2*d^2*e*Sqrt[1 + (1 - c*x)/(1
+ c*x)]*Sqrt[(e - (e*(1 - c*x))/(1 + c*x) + c*d*(1 + (1 - c*x)/(1 + c*x)))/(c*d + e)]*EllipticF[I*ArcSinh[Sqr
t[(1 - c*x)/(1 + c*x)]], (c*d - e)/(c*d + e)] - I*e^3*Sqrt[1 + (1 - c*x)/(1 + c*x)]*Sqrt[(e - (e*(1 - c*x))/(1
+ c*x) + c*d*(1 + (1 - c*x)/(1 + c*x)))/(c*d + e)]*EllipticF[I*ArcSinh[Sqrt[(1 - c*x)/(1 + c*x)]], (c*d - e)/
(c*d + e)] + ((3 + 3*I)*c^3*d^3*(-I + Sqrt[(1 - c*x)/(1 + c*x)])*(I + Sqrt[(1 - c*x)/(1 + c*x)])*Sqrt[((-I)*(S
qrt[-(c*d) - e]*Sqrt[c*d - e] + c*d*Sqrt[(1 - c*x)/(1 + c*x)] - e*Sqrt[(1 - c*x)/(1 + c*x)]))/(((-I)*c*d + Sqr
t[-(c*d) - e]*Sqrt[c*d - e] + I*e)*(-I + Sqrt[(1 - c*x)/(1 + c*x)]))]*Sqrt[((-I)*(Sqrt[-(c*d) - e]*Sqrt[c*d -
e] - c*d*Sqrt[(1 - c*x)/(1 + c*x)] + e*Sqrt[(1 - c*x)/(1 + c*x)]))/((I*c*d + Sqrt[-(c*d) - e]*Sqrt[c*d - e] -
I*e)*(-I + Sqrt[(1 - c*x)/(1 + c*x)]))]*(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] - (1 - I)*EllipticPi[(I*Sqrt[-(c*d) - e] -
Sqrt[c*d - e])/(Sqrt[-(c*d) - e] - I*Sqrt[c*d - e]), ArcSin[Sqrt[((Sqrt[-(c*d) - e] - I*Sqrt[c*d - e])*(I + S
qrt[(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]))/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)]))
] + ((3 + 3*I)*c^3*d^3*(1 + I*Sqrt[(1 - c*x)/(1 + c*x)])*(I + Sqrt[(1 - c*x)/(1 + c*x)])*Sqrt[((-I)*(Sqrt[-(c*
d) - e]*Sqrt[c*d - e] + c*d*Sqrt[(1 - c*x)/(1 + c*x)] - e*Sqrt[(1 - c*x)/(1 + c*x)]))/(((-I)*c*d + Sqrt[-(c*d)
- e]*Sqrt[c*d - e] + I*e)*(-I + Sqrt[(1 - c*x)/(1 + c*x)]))]*Sqrt[((-I)*(Sqrt[-(c*d) - e]*Sqrt[c*d - e] - c*d
*Sqrt[(1 - c*x)/(1 + c*x)] + e*Sqrt[(1 - c*x)/(1 + c*x)]))/((I*c*d + Sqrt[-(c*d) - e]*Sqrt[c*d - e] - I*e)*(-I
+ Sqrt[(1 - c*x)/(1 + c*x)]))]*(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] - (1 + I)*EllipticPi[((-I)*Sqrt[-(c*d) - e] + Sqrt
[c*d - e])/(Sqrt[-(c*d) - e] - I*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]))/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)]))]))/(
15*c^3*e*(1 + (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))])

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fricas [F]  time = 173.84, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (a e x + a d + {\left (b e x + b d\right )} \operatorname {arsech}\left (c x\right )\right )} \sqrt {e x + d}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(3/2)*(a+b*arcsech(c*x)),x, algorithm="fricas")

[Out]

integral((a*e*x + a*d + (b*e*x + b*d)*arcsech(c*x))*sqrt(e*x + d), x)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (e x + d\right )}^{\frac {3}{2}} {\left (b \operatorname {arsech}\left (c x\right ) + a\right )}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(3/2)*(a+b*arcsech(c*x)),x, algorithm="giac")

[Out]

integrate((e*x + d)^(3/2)*(b*arcsech(c*x) + a), x)

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maple [B]  time = 0.23, size = 830, normalized size = 2.42 \[ \frac {\frac {2 \left (e x +d \right )^{\frac {5}{2}} a}{5}+2 b \left (\frac {\left (e x +d \right )^{\frac {5}{2}} \mathrm {arcsech}\left (c x \right )}{5}-\frac {2 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 )^{\frac {5}{2}} c^{2}+9 \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 ) c^{2} d^{2}-7 \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 ) c^{2} d^{2}-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 ) c^{2} d^{2}-2 \sqrt {\frac {c}{c d +e}}\, \left (e x +d \right )^{\frac {3}{2}} c^{2} d -7 \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 ) c d e +7 \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 ) c d e +\sqrt {\frac {c}{c d +e}}\, \sqrt {e x +d}\, 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 ) e^{2}-\sqrt {\frac {c}{c d +e}}\, \sqrt {e x +d}\, e^{2}\right )}{15 c \sqrt {\frac {c}{c d +e}}\, \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((e*x+d)^(3/2)*(a+b*arcsech(c*x)),x)

[Out]

2/e*(1/5*(e*x+d)^(5/2)*a+b*(1/5*(e*x+d)^(5/2)*arcsech(c*x)-2/15/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)^(5/2)*c^2+9*(-((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))*c^2*d^2-7*(-((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))*c^2*d^2-3*(-((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))*c^2*d^2-2*(c/(c*d+e))^(1/2
)*(e*x+d)^(3/2)*c^2*d-7*(-((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))*c*d*e+7*(-((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))*c*d*e+(c/(c*d+e))^(1/2)*
(e*x+d)^(1/2)*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^2-(c/(c*d+e))^(1/2)*(e*x+d)^(1/2)*e^2)/(c/(c*d+e))^(1/2)/((
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((e*x+d)^(3/2)*(a+b*arcsech(c*x)),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 \[ \int \left (a+b\,\mathrm {acosh}\left (\frac {1}{c\,x}\right )\right )\,{\left (d+e\,x\right )}^{3/2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*acosh(1/(c*x)))*(d + e*x)^(3/2),x)

[Out]

int((a + b*acosh(1/(c*x)))*(d + e*x)^(3/2), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (a + b \operatorname {asech}{\left (c x \right )}\right ) \left (d + e x\right )^{\frac {3}{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)**(3/2)*(a+b*asech(c*x)),x)

[Out]

Integral((a + b*asech(c*x))*(d + e*x)**(3/2), x)

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