3.81 \(\int (d+e x)^{3/2} (a+b \text{sech}^{-1}(c x)) \, dx\)
Optimal. Leaf size=343 \[ -\frac{4 b \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \left (2 c^2 d^2+e^2\right ) \sqrt{\frac{c (d+e x)}{c d+e}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{2}}\right ),\frac{2 e}{c d+e}\right )}{15 c^3 \sqrt{d+e x}}+\frac{2 (d+e x)^{5/2} \left (a+b \text{sech}^{-1}(c x)\right )}{5 e}-\frac{4 b e \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \sqrt{1-c^2 x^2} \sqrt{d+e x}}{15 c^2}-\frac{4 b d^3 \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \sqrt{\frac{c (d+e x)}{c d+e}} \Pi \left (2;\sin ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{2}}\right )|\frac{2 e}{c d+e}\right )}{5 e \sqrt{d+e x}}-\frac{28 b d \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \sqrt{d+e x} E\left (\sin ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{2}}\right )|\frac{2 e}{c d+e}\right )}{15 c \sqrt{\frac{c (d+e x)}{c d+e}}} \]
[Out]
(-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])
________________________________________________________________________________________
Rubi [A] time = 0.621292, antiderivative size = 343, normalized size of antiderivative =
1., 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 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 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 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 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)
])
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 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 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 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]
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*}
Mathematica [C] time = 9.87643, 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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Maple [B] time = 0.395, size = 830, normalized size = 2.4 \begin{align*} \text{result too large to display} \end{align*}
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)))
________________________________________________________________________________________
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((e*x+d)^(3/2)*(a+b*arcsech(c*x)),x, algorithm="maxima")
[Out]
Exception raised: ValueError
________________________________________________________________________________________
Fricas [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((e*x+d)^(3/2)*(a+b*arcsech(c*x)),x, algorithm="fricas")
[Out]
Timed out
________________________________________________________________________________________
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((e*x+d)**(3/2)*(a+b*asech(c*x)),x)
[Out]
Timed out
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (e x + d\right )}^{\frac{3}{2}}{\left (b \operatorname{arsech}\left (c x\right ) + a\right )}\,{d x} \end{align*}
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)