3.119 \(\int \frac{\sqrt{d+e x^2} (a+b \sec ^{-1}(c x))}{x^4} \, dx\)

Optimal. Leaf size=328 \[ \frac{b x \sqrt{1-c^2 x^2} \left (c^2 d+e\right ) \left (2 c^2 d+3 e\right ) \sqrt{\frac{e x^2}{d}+1} \text{EllipticF}\left (\sin ^{-1}(c x),-\frac{e}{c^2 d}\right )}{9 d \sqrt{c^2 x^2} \sqrt{c^2 x^2-1} \sqrt{d+e x^2}}-\frac{\left (d+e x^2\right )^{3/2} \left (a+b \sec ^{-1}(c x)\right )}{3 d x^3}+\frac{2 b c \sqrt{c^2 x^2-1} \left (c^2 d+2 e\right ) \sqrt{d+e x^2}}{9 d \sqrt{c^2 x^2}}+\frac{b c \sqrt{c^2 x^2-1} \sqrt{d+e x^2}}{9 x^2 \sqrt{c^2 x^2}}-\frac{2 b c^2 x \sqrt{1-c^2 x^2} \left (c^2 d+2 e\right ) \sqrt{d+e x^2} E\left (\sin ^{-1}(c x)|-\frac{e}{c^2 d}\right )}{9 d \sqrt{c^2 x^2} \sqrt{c^2 x^2-1} \sqrt{\frac{e x^2}{d}+1}} \]

[Out]

(2*b*c*(c^2*d + 2*e)*Sqrt[-1 + c^2*x^2]*Sqrt[d + e*x^2])/(9*d*Sqrt[c^2*x^2]) + (b*c*Sqrt[-1 + c^2*x^2]*Sqrt[d
+ e*x^2])/(9*x^2*Sqrt[c^2*x^2]) - ((d + e*x^2)^(3/2)*(a + b*ArcSec[c*x]))/(3*d*x^3) - (2*b*c^2*(c^2*d + 2*e)*x
*Sqrt[1 - c^2*x^2]*Sqrt[d + e*x^2]*EllipticE[ArcSin[c*x], -(e/(c^2*d))])/(9*d*Sqrt[c^2*x^2]*Sqrt[-1 + c^2*x^2]
*Sqrt[1 + (e*x^2)/d]) + (b*(c^2*d + e)*(2*c^2*d + 3*e)*x*Sqrt[1 - c^2*x^2]*Sqrt[1 + (e*x^2)/d]*EllipticF[ArcSi
n[c*x], -(e/(c^2*d))])/(9*d*Sqrt[c^2*x^2]*Sqrt[-1 + c^2*x^2]*Sqrt[d + e*x^2])

________________________________________________________________________________________

Rubi [A]  time = 0.424043, antiderivative size = 328, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 11, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.478, Rules used = {264, 5238, 12, 474, 583, 524, 427, 426, 424, 421, 419} \[ -\frac{\left (d+e x^2\right )^{3/2} \left (a+b \sec ^{-1}(c x)\right )}{3 d x^3}+\frac{2 b c \sqrt{c^2 x^2-1} \left (c^2 d+2 e\right ) \sqrt{d+e x^2}}{9 d \sqrt{c^2 x^2}}+\frac{b c \sqrt{c^2 x^2-1} \sqrt{d+e x^2}}{9 x^2 \sqrt{c^2 x^2}}+\frac{b x \sqrt{1-c^2 x^2} \left (c^2 d+e\right ) \left (2 c^2 d+3 e\right ) \sqrt{\frac{e x^2}{d}+1} F\left (\sin ^{-1}(c x)|-\frac{e}{c^2 d}\right )}{9 d \sqrt{c^2 x^2} \sqrt{c^2 x^2-1} \sqrt{d+e x^2}}-\frac{2 b c^2 x \sqrt{1-c^2 x^2} \left (c^2 d+2 e\right ) \sqrt{d+e x^2} E\left (\sin ^{-1}(c x)|-\frac{e}{c^2 d}\right )}{9 d \sqrt{c^2 x^2} \sqrt{c^2 x^2-1} \sqrt{\frac{e x^2}{d}+1}} \]

Antiderivative was successfully verified.

[In]

Int[(Sqrt[d + e*x^2]*(a + b*ArcSec[c*x]))/x^4,x]

[Out]

(2*b*c*(c^2*d + 2*e)*Sqrt[-1 + c^2*x^2]*Sqrt[d + e*x^2])/(9*d*Sqrt[c^2*x^2]) + (b*c*Sqrt[-1 + c^2*x^2]*Sqrt[d
+ e*x^2])/(9*x^2*Sqrt[c^2*x^2]) - ((d + e*x^2)^(3/2)*(a + b*ArcSec[c*x]))/(3*d*x^3) - (2*b*c^2*(c^2*d + 2*e)*x
*Sqrt[1 - c^2*x^2]*Sqrt[d + e*x^2]*EllipticE[ArcSin[c*x], -(e/(c^2*d))])/(9*d*Sqrt[c^2*x^2]*Sqrt[-1 + c^2*x^2]
*Sqrt[1 + (e*x^2)/d]) + (b*(c^2*d + e)*(2*c^2*d + 3*e)*x*Sqrt[1 - c^2*x^2]*Sqrt[1 + (e*x^2)/d]*EllipticF[ArcSi
n[c*x], -(e/(c^2*d))])/(9*d*Sqrt[c^2*x^2]*Sqrt[-1 + c^2*x^2]*Sqrt[d + e*x^2])

Rule 264

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

Rule 5238

Int[((a_.) + ArcSec[(c_.)*(x_)]*(b_.))*((f_.)*(x_))^(m_.)*((d_.) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> With[{u =
IntHide[(f*x)^m*(d + e*x^2)^p, x]}, Dist[a + b*ArcSec[c*x], u, x] - Dist[(b*c*x)/Sqrt[c^2*x^2], Int[SimplifyI
ntegrand[u/(x*Sqrt[c^2*x^2 - 1]), x], x], x]] /; FreeQ[{a, b, c, d, e, f, m, p}, x] && ((IGtQ[p, 0] &&  !(ILtQ
[(m - 1)/2, 0] && GtQ[m + 2*p + 3, 0])) || (IGtQ[(m + 1)/2, 0] &&  !(ILtQ[p, 0] && GtQ[m + 2*p + 3, 0])) || (I
LtQ[(m + 2*p + 1)/2, 0] &&  !ILtQ[(m - 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 474

Int[((e_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[(c*(e*x)^
(m + 1)*(a + b*x^n)^(p + 1)*(c + d*x^n)^(q - 1))/(a*e*(m + 1)), x] - Dist[1/(a*e^n*(m + 1)), Int[(e*x)^(m + n)
*(a + b*x^n)^p*(c + d*x^n)^(q - 2)*Simp[c*(c*b - a*d)*(m + 1) + c*n*(b*c*(p + 1) + a*d*(q - 1)) + d*((c*b - a*
d)*(m + 1) + c*b*n*(p + q))*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0]
&& GtQ[q, 1] && LtQ[m, -1] && IntBinomialQ[a, b, c, d, e, m, n, p, q, x]

Rule 583

Int[((g_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.)*((e_) + (f_.)*(x_)^(n_)),
x_Symbol] :> Simp[(e*(g*x)^(m + 1)*(a + b*x^n)^(p + 1)*(c + d*x^n)^(q + 1))/(a*c*g*(m + 1)), x] + Dist[1/(a*c*
g^n*(m + 1)), Int[(g*x)^(m + n)*(a + b*x^n)^p*(c + d*x^n)^q*Simp[a*f*c*(m + 1) - e*(b*c + a*d)*(m + n + 1) - e
*n*(b*c*p + a*d*q) - b*e*d*(m + n*(p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p, q}, x] &&
IGtQ[n, 0] && LtQ[m, -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 427

Int[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[Sqrt[a + b*x^2]/Sqrt[1 + (d*x^2)/c], x], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] &&  !GtQ[c, 0]

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{\sqrt{d+e x^2} \left (a+b \sec ^{-1}(c x)\right )}{x^4} \, dx &=-\frac{\left (d+e x^2\right )^{3/2} \left (a+b \sec ^{-1}(c x)\right )}{3 d x^3}-\frac{(b c x) \int -\frac{\left (d+e x^2\right )^{3/2}}{3 d x^4 \sqrt{-1+c^2 x^2}} \, dx}{\sqrt{c^2 x^2}}\\ &=-\frac{\left (d+e x^2\right )^{3/2} \left (a+b \sec ^{-1}(c x)\right )}{3 d x^3}+\frac{(b c x) \int \frac{\left (d+e x^2\right )^{3/2}}{x^4 \sqrt{-1+c^2 x^2}} \, dx}{3 d \sqrt{c^2 x^2}}\\ &=\frac{b c \sqrt{-1+c^2 x^2} \sqrt{d+e x^2}}{9 x^2 \sqrt{c^2 x^2}}-\frac{\left (d+e x^2\right )^{3/2} \left (a+b \sec ^{-1}(c x)\right )}{3 d x^3}-\frac{(b c x) \int \frac{-2 d \left (c^2 d+2 e\right )-e \left (c^2 d+3 e\right ) x^2}{x^2 \sqrt{-1+c^2 x^2} \sqrt{d+e x^2}} \, dx}{9 d \sqrt{c^2 x^2}}\\ &=\frac{2 b c \left (c^2 d+2 e\right ) \sqrt{-1+c^2 x^2} \sqrt{d+e x^2}}{9 d \sqrt{c^2 x^2}}+\frac{b c \sqrt{-1+c^2 x^2} \sqrt{d+e x^2}}{9 x^2 \sqrt{c^2 x^2}}-\frac{\left (d+e x^2\right )^{3/2} \left (a+b \sec ^{-1}(c x)\right )}{3 d x^3}-\frac{(b c x) \int \frac{-d e \left (c^2 d+3 e\right )+2 c^2 d e \left (c^2 d+2 e\right ) x^2}{\sqrt{-1+c^2 x^2} \sqrt{d+e x^2}} \, dx}{9 d^2 \sqrt{c^2 x^2}}\\ &=\frac{2 b c \left (c^2 d+2 e\right ) \sqrt{-1+c^2 x^2} \sqrt{d+e x^2}}{9 d \sqrt{c^2 x^2}}+\frac{b c \sqrt{-1+c^2 x^2} \sqrt{d+e x^2}}{9 x^2 \sqrt{c^2 x^2}}-\frac{\left (d+e x^2\right )^{3/2} \left (a+b \sec ^{-1}(c x)\right )}{3 d x^3}-\frac{\left (2 b c^3 \left (c^2 d+2 e\right ) x\right ) \int \frac{\sqrt{d+e x^2}}{\sqrt{-1+c^2 x^2}} \, dx}{9 d \sqrt{c^2 x^2}}+\frac{\left (b c \left (c^2 d+e\right ) \left (2 c^2 d+3 e\right ) x\right ) \int \frac{1}{\sqrt{-1+c^2 x^2} \sqrt{d+e x^2}} \, dx}{9 d \sqrt{c^2 x^2}}\\ &=\frac{2 b c \left (c^2 d+2 e\right ) \sqrt{-1+c^2 x^2} \sqrt{d+e x^2}}{9 d \sqrt{c^2 x^2}}+\frac{b c \sqrt{-1+c^2 x^2} \sqrt{d+e x^2}}{9 x^2 \sqrt{c^2 x^2}}-\frac{\left (d+e x^2\right )^{3/2} \left (a+b \sec ^{-1}(c x)\right )}{3 d x^3}-\frac{\left (2 b c^3 \left (c^2 d+2 e\right ) x \sqrt{1-c^2 x^2}\right ) \int \frac{\sqrt{d+e x^2}}{\sqrt{1-c^2 x^2}} \, dx}{9 d \sqrt{c^2 x^2} \sqrt{-1+c^2 x^2}}+\frac{\left (b c \left (c^2 d+e\right ) \left (2 c^2 d+3 e\right ) 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}{9 d \sqrt{c^2 x^2} \sqrt{d+e x^2}}\\ &=\frac{2 b c \left (c^2 d+2 e\right ) \sqrt{-1+c^2 x^2} \sqrt{d+e x^2}}{9 d \sqrt{c^2 x^2}}+\frac{b c \sqrt{-1+c^2 x^2} \sqrt{d+e x^2}}{9 x^2 \sqrt{c^2 x^2}}-\frac{\left (d+e x^2\right )^{3/2} \left (a+b \sec ^{-1}(c x)\right )}{3 d x^3}-\frac{\left (2 b c^3 \left (c^2 d+2 e\right ) x \sqrt{1-c^2 x^2} \sqrt{d+e x^2}\right ) \int \frac{\sqrt{1+\frac{e x^2}{d}}}{\sqrt{1-c^2 x^2}} \, dx}{9 d \sqrt{c^2 x^2} \sqrt{-1+c^2 x^2} \sqrt{1+\frac{e x^2}{d}}}+\frac{\left (b c \left (c^2 d+e\right ) \left (2 c^2 d+3 e\right ) x \sqrt{1-c^2 x^2} \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}{9 d \sqrt{c^2 x^2} \sqrt{-1+c^2 x^2} \sqrt{d+e x^2}}\\ &=\frac{2 b c \left (c^2 d+2 e\right ) \sqrt{-1+c^2 x^2} \sqrt{d+e x^2}}{9 d \sqrt{c^2 x^2}}+\frac{b c \sqrt{-1+c^2 x^2} \sqrt{d+e x^2}}{9 x^2 \sqrt{c^2 x^2}}-\frac{\left (d+e x^2\right )^{3/2} \left (a+b \sec ^{-1}(c x)\right )}{3 d x^3}-\frac{2 b c^2 \left (c^2 d+2 e\right ) x \sqrt{1-c^2 x^2} \sqrt{d+e x^2} E\left (\sin ^{-1}(c x)|-\frac{e}{c^2 d}\right )}{9 d \sqrt{c^2 x^2} \sqrt{-1+c^2 x^2} \sqrt{1+\frac{e x^2}{d}}}+\frac{b \left (c^2 d+e\right ) \left (2 c^2 d+3 e\right ) x \sqrt{1-c^2 x^2} \sqrt{1+\frac{e x^2}{d}} F\left (\sin ^{-1}(c x)|-\frac{e}{c^2 d}\right )}{9 d \sqrt{c^2 x^2} \sqrt{-1+c^2 x^2} \sqrt{d+e x^2}}\\ \end{align*}

Mathematica [C]  time = 0.612809, size = 247, normalized size = 0.75 \[ \frac{\sqrt{d+e x^2} \left (-3 a \left (d+e x^2\right )+b c x \sqrt{1-\frac{1}{c^2 x^2}} \left (2 c^2 d x^2+d+4 e x^2\right )-3 b \sec ^{-1}(c x) \left (d+e x^2\right )\right )}{9 d x^3}-\frac{i b c x \sqrt{1-\frac{1}{c^2 x^2}} \sqrt{\frac{e x^2}{d}+1} \left (2 c^2 d \left (c^2 d+2 e\right ) E\left (i \sinh ^{-1}\left (\sqrt{-c^2} x\right )|-\frac{e}{c^2 d}\right )-\left (2 c^4 d^2+5 c^2 d e+3 e^2\right ) \text{EllipticF}\left (i \sinh ^{-1}\left (\sqrt{-c^2} x\right ),-\frac{e}{c^2 d}\right )\right )}{9 \sqrt{-c^2} d \sqrt{1-c^2 x^2} \sqrt{d+e x^2}} \]

Antiderivative was successfully verified.

[In]

Integrate[(Sqrt[d + e*x^2]*(a + b*ArcSec[c*x]))/x^4,x]

[Out]

(Sqrt[d + e*x^2]*(-3*a*(d + e*x^2) + b*c*Sqrt[1 - 1/(c^2*x^2)]*x*(d + 2*c^2*d*x^2 + 4*e*x^2) - 3*b*(d + e*x^2)
*ArcSec[c*x]))/(9*d*x^3) - ((I/9)*b*c*Sqrt[1 - 1/(c^2*x^2)]*x*Sqrt[1 + (e*x^2)/d]*(2*c^2*d*(c^2*d + 2*e)*Ellip
ticE[I*ArcSinh[Sqrt[-c^2]*x], -(e/(c^2*d))] - (2*c^4*d^2 + 5*c^2*d*e + 3*e^2)*EllipticF[I*ArcSinh[Sqrt[-c^2]*x
], -(e/(c^2*d))]))/(Sqrt[-c^2]*d*Sqrt[1 - c^2*x^2]*Sqrt[d + e*x^2])

________________________________________________________________________________________

Maple [F]  time = 1.7, size = 0, normalized size = 0. \begin{align*} \int{\frac{a+b{\rm arcsec} \left (cx\right )}{{x}^{4}}\sqrt{e{x}^{2}+d}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*arcsec(c*x))*(e*x^2+d)^(1/2)/x^4,x)

[Out]

int((a+b*arcsec(c*x))*(e*x^2+d)^(1/2)/x^4,x)

________________________________________________________________________________________

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*arcsec(c*x))*(e*x^2+d)^(1/2)/x^4,x, algorithm="maxima")

[Out]

Exception raised: ValueError

________________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{e x^{2} + d}{\left (b \operatorname{arcsec}\left (c x\right ) + a\right )}}{x^{4}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arcsec(c*x))*(e*x^2+d)^(1/2)/x^4,x, algorithm="fricas")

[Out]

integral(sqrt(e*x^2 + d)*(b*arcsec(c*x) + a)/x^4, 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*asec(c*x))*(e*x**2+d)**(1/2)/x**4,x)

[Out]

Timed out

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{e x^{2} + d}{\left (b \operatorname{arcsec}\left (c x\right ) + a\right )}}{x^{4}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arcsec(c*x))*(e*x^2+d)^(1/2)/x^4,x, algorithm="giac")

[Out]

integrate(sqrt(e*x^2 + d)*(b*arcsec(c*x) + a)/x^4, x)