3.176 \(\int \frac{a+b \text{sech}^{-1}(c x)}{(d+e x^2)^{5/2}} \, dx\)

Optimal. Leaf size=266 \[ \frac{2 b \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \sqrt{\frac{e x^2}{d}+1} \text{EllipticF}\left (\sin ^{-1}(c x),-\frac{e}{c^2 d}\right )}{3 c d^2 \sqrt{d+e x^2}}+\frac{2 x \left (a+b \text{sech}^{-1}(c x)\right )}{3 d^2 \sqrt{d+e x^2}}+\frac{x \left (a+b \text{sech}^{-1}(c x)\right )}{3 d \left (d+e x^2\right )^{3/2}}+\frac{b e x \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \sqrt{1-c^2 x^2}}{3 d^2 \left (c^2 d+e\right ) \sqrt{d+e x^2}}+\frac{b c \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \sqrt{d+e x^2} E\left (\sin ^{-1}(c x)|-\frac{e}{c^2 d}\right )}{3 d^2 \left (c^2 d+e\right ) \sqrt{\frac{e x^2}{d}+1}} \]

[Out]

(b*e*x*Sqrt[(1 + c*x)^(-1)]*Sqrt[1 + c*x]*Sqrt[1 - c^2*x^2])/(3*d^2*(c^2*d + e)*Sqrt[d + e*x^2]) + (x*(a + b*A
rcSech[c*x]))/(3*d*(d + e*x^2)^(3/2)) + (2*x*(a + b*ArcSech[c*x]))/(3*d^2*Sqrt[d + e*x^2]) + (b*c*Sqrt[(1 + c*
x)^(-1)]*Sqrt[1 + c*x]*Sqrt[d + e*x^2]*EllipticE[ArcSin[c*x], -(e/(c^2*d))])/(3*d^2*(c^2*d + e)*Sqrt[1 + (e*x^
2)/d]) + (2*b*Sqrt[(1 + c*x)^(-1)]*Sqrt[1 + c*x]*Sqrt[1 + (e*x^2)/d]*EllipticF[ArcSin[c*x], -(e/(c^2*d))])/(3*
c*d^2*Sqrt[d + e*x^2])

________________________________________________________________________________________

Rubi [A]  time = 0.20608, antiderivative size = 266, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 10, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {192, 191, 6291, 12, 527, 524, 426, 424, 421, 419} \[ \frac{2 x \left (a+b \text{sech}^{-1}(c x)\right )}{3 d^2 \sqrt{d+e x^2}}+\frac{x \left (a+b \text{sech}^{-1}(c x)\right )}{3 d \left (d+e x^2\right )^{3/2}}+\frac{b e x \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \sqrt{1-c^2 x^2}}{3 d^2 \left (c^2 d+e\right ) \sqrt{d+e x^2}}+\frac{2 b \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \sqrt{\frac{e x^2}{d}+1} F\left (\sin ^{-1}(c x)|-\frac{e}{c^2 d}\right )}{3 c d^2 \sqrt{d+e x^2}}+\frac{b c \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \sqrt{d+e x^2} E\left (\sin ^{-1}(c x)|-\frac{e}{c^2 d}\right )}{3 d^2 \left (c^2 d+e\right ) \sqrt{\frac{e x^2}{d}+1}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(b*e*x*Sqrt[(1 + c*x)^(-1)]*Sqrt[1 + c*x]*Sqrt[1 - c^2*x^2])/(3*d^2*(c^2*d + e)*Sqrt[d + e*x^2]) + (x*(a + b*A
rcSech[c*x]))/(3*d*(d + e*x^2)^(3/2)) + (2*x*(a + b*ArcSech[c*x]))/(3*d^2*Sqrt[d + e*x^2]) + (b*c*Sqrt[(1 + c*
x)^(-1)]*Sqrt[1 + c*x]*Sqrt[d + e*x^2]*EllipticE[ArcSin[c*x], -(e/(c^2*d))])/(3*d^2*(c^2*d + e)*Sqrt[1 + (e*x^
2)/d]) + (2*b*Sqrt[(1 + c*x)^(-1)]*Sqrt[1 + c*x]*Sqrt[1 + (e*x^2)/d]*EllipticF[ArcSin[c*x], -(e/(c^2*d))])/(3*
c*d^2*Sqrt[d + e*x^2])

Rule 192

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> -Simp[(x*(a + b*x^n)^(p + 1))/(a*n*(p + 1)), x] + Dist[(n*(p +
 1) + 1)/(a*n*(p + 1)), Int[(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b, n, p}, x] && ILtQ[Simplify[1/n + p + 1
], 0] && NeQ[p, -1]

Rule 191

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(x*(a + b*x^n)^(p + 1))/a, x] /; FreeQ[{a, b, n, p}, x] &
& EqQ[1/n + p + 1, 0]

Rule 6291

Int[((a_.) + ArcSech[(c_.)*(x_)]*(b_.))*((d_.) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> With[{u = IntHide[(d + e*x^
2)^p, x]}, Dist[a + b*ArcSech[c*x], u, x] + Dist[b*Sqrt[1 + c*x]*Sqrt[1/(1 + c*x)], Int[SimplifyIntegrand[u/(x
*Sqrt[1 - c*x]*Sqrt[1 + c*x]), x], x], x]] /; FreeQ[{a, b, c, d, e}, x] && (IGtQ[p, 0] || ILtQ[p + 1/2, 0])

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 527

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_.)*((e_) + (f_.)*(x_)^(n_)), x_Symbol] :> -Simp[
((b*e - a*f)*x*(a + b*x^n)^(p + 1)*(c + d*x^n)^(q + 1))/(a*n*(b*c - a*d)*(p + 1)), x] + Dist[1/(a*n*(b*c - a*d
)*(p + 1)), Int[(a + b*x^n)^(p + 1)*(c + d*x^n)^q*Simp[c*(b*e - a*f) + e*n*(b*c - a*d)*(p + 1) + d*(b*e - a*f)
*(n*(p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, q}, x] && LtQ[p, -1]

Rule 524

Int[((e_) + (f_.)*(x_)^(n_))/(Sqrt[(a_) + (b_.)*(x_)^(n_)]*Sqrt[(c_) + (d_.)*(x_)^(n_)]), x_Symbol] :> Dist[f/
b, Int[Sqrt[a + b*x^n]/Sqrt[c + d*x^n], x], x] + Dist[(b*e - a*f)/b, Int[1/(Sqrt[a + b*x^n]*Sqrt[c + d*x^n]),
x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] &&  !(EqQ[n, 2] && ((PosQ[b/a] && PosQ[d/c]) || (NegQ[b/a] && (PosQ[
d/c] || (GtQ[a, 0] && ( !GtQ[c, 0] || SimplerSqrtQ[-(b/a), -(d/c)]))))))

Rule 426

Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Dist[Sqrt[a + b*x^2]/Sqrt[1 + (b*x^2)/a]
, Int[Sqrt[1 + (b*x^2)/a]/Sqrt[c + d*x^2], x], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] &&  !GtQ
[a, 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 421

Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> Dist[Sqrt[1 + (d*x^2)/c]/Sqrt[c + d*
x^2], Int[1/(Sqrt[a + b*x^2]*Sqrt[1 + (d*x^2)/c]), x], x] /; FreeQ[{a, b, c, d}, x] &&  !GtQ[c, 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)])

Rubi steps

\begin{align*} \int \frac{a+b \text{sech}^{-1}(c x)}{\left (d+e x^2\right )^{5/2}} \, dx &=\frac{x \left (a+b \text{sech}^{-1}(c x)\right )}{3 d \left (d+e x^2\right )^{3/2}}+\frac{2 x \left (a+b \text{sech}^{-1}(c x)\right )}{3 d^2 \sqrt{d+e x^2}}+\left (b \sqrt{\frac{1}{1+c x}} \sqrt{1+c x}\right ) \int \frac{3 d+2 e x^2}{3 d^2 \sqrt{1-c^2 x^2} \left (d+e x^2\right )^{3/2}} \, dx\\ &=\frac{x \left (a+b \text{sech}^{-1}(c x)\right )}{3 d \left (d+e x^2\right )^{3/2}}+\frac{2 x \left (a+b \text{sech}^{-1}(c x)\right )}{3 d^2 \sqrt{d+e x^2}}+\frac{\left (b \sqrt{\frac{1}{1+c x}} \sqrt{1+c x}\right ) \int \frac{3 d+2 e x^2}{\sqrt{1-c^2 x^2} \left (d+e x^2\right )^{3/2}} \, dx}{3 d^2}\\ &=\frac{b e x \sqrt{\frac{1}{1+c x}} \sqrt{1+c x} \sqrt{1-c^2 x^2}}{3 d^2 \left (c^2 d+e\right ) \sqrt{d+e x^2}}+\frac{x \left (a+b \text{sech}^{-1}(c x)\right )}{3 d \left (d+e x^2\right )^{3/2}}+\frac{2 x \left (a+b \text{sech}^{-1}(c x)\right )}{3 d^2 \sqrt{d+e x^2}}-\frac{\left (b \sqrt{\frac{1}{1+c x}} \sqrt{1+c x}\right ) \int \frac{-d \left (3 c^2 d+2 e\right )-c^2 d e x^2}{\sqrt{1-c^2 x^2} \sqrt{d+e x^2}} \, dx}{3 d^3 \left (c^2 d+e\right )}\\ &=\frac{b e x \sqrt{\frac{1}{1+c x}} \sqrt{1+c x} \sqrt{1-c^2 x^2}}{3 d^2 \left (c^2 d+e\right ) \sqrt{d+e x^2}}+\frac{x \left (a+b \text{sech}^{-1}(c x)\right )}{3 d \left (d+e x^2\right )^{3/2}}+\frac{2 x \left (a+b \text{sech}^{-1}(c x)\right )}{3 d^2 \sqrt{d+e x^2}}+\frac{\left (2 b \sqrt{\frac{1}{1+c x}} \sqrt{1+c x}\right ) \int \frac{1}{\sqrt{1-c^2 x^2} \sqrt{d+e x^2}} \, dx}{3 d^2}+\frac{\left (b c^2 \sqrt{\frac{1}{1+c x}} \sqrt{1+c x}\right ) \int \frac{\sqrt{d+e x^2}}{\sqrt{1-c^2 x^2}} \, dx}{3 d^2 \left (c^2 d+e\right )}\\ &=\frac{b e x \sqrt{\frac{1}{1+c x}} \sqrt{1+c x} \sqrt{1-c^2 x^2}}{3 d^2 \left (c^2 d+e\right ) \sqrt{d+e x^2}}+\frac{x \left (a+b \text{sech}^{-1}(c x)\right )}{3 d \left (d+e x^2\right )^{3/2}}+\frac{2 x \left (a+b \text{sech}^{-1}(c x)\right )}{3 d^2 \sqrt{d+e x^2}}+\frac{\left (b c^2 \sqrt{\frac{1}{1+c x}} \sqrt{1+c x} \sqrt{d+e x^2}\right ) \int \frac{\sqrt{1+\frac{e x^2}{d}}}{\sqrt{1-c^2 x^2}} \, dx}{3 d^2 \left (c^2 d+e\right ) \sqrt{1+\frac{e x^2}{d}}}+\frac{\left (2 b \sqrt{\frac{1}{1+c x}} \sqrt{1+c x} \sqrt{1+\frac{e x^2}{d}}\right ) \int \frac{1}{\sqrt{1-c^2 x^2} \sqrt{1+\frac{e x^2}{d}}} \, dx}{3 d^2 \sqrt{d+e x^2}}\\ &=\frac{b e x \sqrt{\frac{1}{1+c x}} \sqrt{1+c x} \sqrt{1-c^2 x^2}}{3 d^2 \left (c^2 d+e\right ) \sqrt{d+e x^2}}+\frac{x \left (a+b \text{sech}^{-1}(c x)\right )}{3 d \left (d+e x^2\right )^{3/2}}+\frac{2 x \left (a+b \text{sech}^{-1}(c x)\right )}{3 d^2 \sqrt{d+e x^2}}+\frac{b c \sqrt{\frac{1}{1+c x}} \sqrt{1+c x} \sqrt{d+e x^2} E\left (\sin ^{-1}(c x)|-\frac{e}{c^2 d}\right )}{3 d^2 \left (c^2 d+e\right ) \sqrt{1+\frac{e x^2}{d}}}+\frac{2 b \sqrt{\frac{1}{1+c x}} \sqrt{1+c x} \sqrt{1+\frac{e x^2}{d}} F\left (\sin ^{-1}(c x)|-\frac{e}{c^2 d}\right )}{3 c d^2 \sqrt{d+e x^2}}\\ \end{align*}

Mathematica [C]  time = 5.08504, size = 517, normalized size = 1.94 \[ \frac{-\frac{i b \sqrt{\frac{1-c x}{c x+1}} (c x+1) \left (d+e x^2\right ) \sqrt{\frac{c \left (\sqrt{d}-i \sqrt{e} x\right )}{(c x+1) \left (c \sqrt{d}-i \sqrt{e}\right )}} \sqrt{\frac{c \left (\sqrt{d}+i \sqrt{e} x\right )}{(c x+1) \left (c \sqrt{d}+i \sqrt{e}\right )}} \left (\left (c \sqrt{d}-i \sqrt{e}\right ) E\left (i \sinh ^{-1}\left (\sqrt{\frac{\left (d c^2+e\right ) (1-c x)}{\left (\sqrt{d} c+i \sqrt{e}\right )^2 (c x+1)}}\right )|\frac{\left (\sqrt{d} c+i \sqrt{e}\right )^2}{\left (c \sqrt{d}-i \sqrt{e}\right )^2}\right )-2 \left (3 c \sqrt{d}+2 i \sqrt{e}\right ) \text{EllipticF}\left (i \sinh ^{-1}\left (\sqrt{\frac{(1-c x) \left (c^2 d+e\right )}{(c x+1) \left (c \sqrt{d}+i \sqrt{e}\right )^2}}\right ),\frac{\left (c \sqrt{d}+i \sqrt{e}\right )^2}{\left (c \sqrt{d}-i \sqrt{e}\right )^2}\right )\right )}{c \left (c \sqrt{d}+i \sqrt{e}\right ) \sqrt{-\frac{(c x-1) \left (c \sqrt{d}-i \sqrt{e}\right )}{(c x+1) \left (c \sqrt{d}+i \sqrt{e}\right )}}}+a x \left (3 d+2 e x^2\right )+\frac{b \sqrt{\frac{1-c x}{c x+1}} \left (d+e x^2\right ) (e x-c d)}{c^2 d+e}+b x \text{sech}^{-1}(c x) \left (3 d+2 e x^2\right )}{3 d^2 \left (d+e x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((b*Sqrt[(1 - c*x)/(1 + c*x)]*(-(c*d) + e*x)*(d + e*x^2))/(c^2*d + e) + a*x*(3*d + 2*e*x^2) + b*x*(3*d + 2*e*x
^2)*ArcSech[c*x] - (I*b*Sqrt[(1 - c*x)/(1 + c*x)]*(1 + c*x)*Sqrt[(c*(Sqrt[d] - I*Sqrt[e]*x))/((c*Sqrt[d] - I*S
qrt[e])*(1 + c*x))]*Sqrt[(c*(Sqrt[d] + I*Sqrt[e]*x))/((c*Sqrt[d] + I*Sqrt[e])*(1 + c*x))]*(d + e*x^2)*((c*Sqrt
[d] - I*Sqrt[e])*EllipticE[I*ArcSinh[Sqrt[((c^2*d + e)*(1 - c*x))/((c*Sqrt[d] + I*Sqrt[e])^2*(1 + c*x))]], (c*
Sqrt[d] + I*Sqrt[e])^2/(c*Sqrt[d] - I*Sqrt[e])^2] - 2*(3*c*Sqrt[d] + (2*I)*Sqrt[e])*EllipticF[I*ArcSinh[Sqrt[(
(c^2*d + e)*(1 - c*x))/((c*Sqrt[d] + I*Sqrt[e])^2*(1 + c*x))]], (c*Sqrt[d] + I*Sqrt[e])^2/(c*Sqrt[d] - I*Sqrt[
e])^2]))/(c*(c*Sqrt[d] + I*Sqrt[e])*Sqrt[-(((c*Sqrt[d] - I*Sqrt[e])*(-1 + c*x))/((c*Sqrt[d] + I*Sqrt[e])*(1 +
c*x)))]))/(3*d^2*(d + e*x^2)^(3/2))

________________________________________________________________________________________

Maple [F]  time = 1.012, size = 0, normalized size = 0. \begin{align*} \int{(a+b{\rm arcsech} \left (cx\right )) \left ( e{x}^{2}+d \right ) ^{-{\frac{5}{2}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{1}{3} \, a{\left (\frac{2 \, x}{\sqrt{e x^{2} + d} d^{2}} + \frac{x}{{\left (e x^{2} + d\right )}^{\frac{3}{2}} d}\right )} + b \int \frac{\log \left (\sqrt{\frac{1}{c x} + 1} \sqrt{\frac{1}{c x} - 1} + \frac{1}{c x}\right )}{{\left (e x^{2} + 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^2+d)^(5/2),x, algorithm="maxima")

[Out]

1/3*a*(2*x/(sqrt(e*x^2 + d)*d^2) + x/((e*x^2 + d)^(3/2)*d)) + b*integrate(log(sqrt(1/(c*x) + 1)*sqrt(1/(c*x) -
 1) + 1/(c*x))/(e*x^2 + d)^(5/2), x)

________________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{e x^{2} + d}{\left (b \operatorname{arsech}\left (c x\right ) + a\right )}}{e^{3} x^{6} + 3 \, d e^{2} x^{4} + 3 \, d^{2} e x^{2} + 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^2+d)^(5/2),x, algorithm="fricas")

[Out]

integral(sqrt(e*x^2 + d)*(b*arcsech(c*x) + a)/(e^3*x^6 + 3*d*e^2*x^4 + 3*d^2*e*x^2 + d^3), x)

________________________________________________________________________________________

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**2+d)**(5/2),x)

[Out]

Timed out

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{b \operatorname{arsech}\left (c x\right ) + a}{{\left (e x^{2} + 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^2+d)^(5/2),x, algorithm="giac")

[Out]

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