\(\int \frac {a+b \sec ^{-1}(c x)}{(d+e x)^{5/2}} \, dx\) [67]

Optimal result

Integrand size = 18, antiderivative size = 298 \[ \int \frac {a+b \sec ^{-1}(c x)}{(d+e x)^{5/2}} \, dx=-\frac {4 b e \left (1-c^2 x^2\right )}{3 c d \left (c^2 d^2-e^2\right ) \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {d+e x}}-\frac {2 \left (a+b \sec ^{-1}(c x)\right )}{3 e (d+e x)^{3/2}}+\frac {4 b \sqrt {d+e x} \sqrt {1-c^2 x^2} E\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{3 d \left (c^2 d^2-e^2\right ) \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {\frac {c (d+e x)}{c d+e}}}-\frac {4 b \sqrt {\frac {c (d+e x)}{c d+e}} \sqrt {1-c^2 x^2} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{3 c d e \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {d+e x}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.30 (sec) , antiderivative size = 298, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.611, Rules used = {5334, 1588, 972, 759, 21, 733, 435, 947, 174, 552, 551} \[ \int \frac {a+b \sec ^{-1}(c x)}{(d+e x)^{5/2}} \, dx=-\frac {2 \left (a+b \sec ^{-1}(c x)\right )}{3 e (d+e x)^{3/2}}+\frac {4 b \sqrt {1-c^2 x^2} \sqrt {d+e x} E\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{3 d x \sqrt {1-\frac {1}{c^2 x^2}} \left (c^2 d^2-e^2\right ) \sqrt {\frac {c (d+e x)}{c d+e}}}-\frac {4 b \sqrt {1-c^2 x^2} \sqrt {\frac {c (d+e x)}{c d+e}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{3 c d e x \sqrt {1-\frac {1}{c^2 x^2}} \sqrt {d+e x}}-\frac {4 b e \left (1-c^2 x^2\right )}{3 c d x \sqrt {1-\frac {1}{c^2 x^2}} \left (c^2 d^2-e^2\right ) \sqrt {d+e x}} \]

[In]

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

[Out]

(-4*b*e*(1 - c^2*x^2))/(3*c*d*(c^2*d^2 - e^2)*Sqrt[1 - 1/(c^2*x^2)]*x*Sqrt[d + e*x]) - (2*(a + b*ArcSec[c*x]))
/(3*e*(d + e*x)^(3/2)) + (4*b*Sqrt[d + e*x]*Sqrt[1 - c^2*x^2]*EllipticE[ArcSin[Sqrt[1 - c*x]/Sqrt[2]], (2*e)/(
c*d + e)])/(3*d*(c^2*d^2 - e^2)*Sqrt[1 - 1/(c^2*x^2)]*x*Sqrt[(c*(d + e*x))/(c*d + e)]) - (4*b*Sqrt[(c*(d + e*x
))/(c*d + e)]*Sqrt[1 - c^2*x^2]*EllipticPi[2, ArcSin[Sqrt[1 - c*x]/Sqrt[2]], (2*e)/(c*d + e)])/(3*c*d*e*Sqrt[1
- 1/(c^2*x^2)]*x*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 174

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 435

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

Rule 551

Int[1/(((a_) + (b_.)*(x_)^2)*Sqrt[(c_) + (d_.)*(x_)^2]*Sqrt[(e_) + (f_.)*(x_)^2]), x_Symbol] :> Simp[(1/(a*Sqr
t[c]*Sqrt[e]*Rt[-d/c, 2]))*EllipticPi[b*(c/(a*d)), ArcSin[Rt[-d/c, 2]*x], c*(f/(d*e))], 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 552

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

Rule 733

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 759

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 947

Int[1/(((d_.) + (e_.)*(x_))*Sqrt[(f_.) + (g_.)*(x_)]*Sqrt[(a_) + (c_.)*(x_)^2]), x_Symbol] :> With[{q = Rt[-c/
a, 2]}, Dist[Sqrt[1 + c*(x^2/a)]/Sqrt[a + c*x^2], 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 972

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 1588

Int[(x_)^(m_.)*((a_.) + (c_.)*(x_)^(mn2_.))^(p_)*((d_) + (e_.)*(x_)^(n_.))^(q_.), x_Symbol] :> Dist[x^(2*n*Fra
cPart[p])*((a + c/x^(2*n))^FracPart[p]/(c + a*x^(2*n))^FracPart[p]), Int[x^(m - 2*n*p)*(d + e*x^n)^q*(c + a*x^
(2*n))^p, x], x] /; FreeQ[{a, c, d, e, m, n, p, q}, x] && EqQ[mn2, -2*n] &&  !IntegerQ[p] &&  !IntegerQ[q] &&
PosQ[n]

Rule 5334

Int[((a_.) + ArcSec[(c_.)*(x_)]*(b_.))*((d_.) + (e_.)*(x_))^(m_.), x_Symbol] :> Simp[(d + e*x)^(m + 1)*((a + b
*ArcSec[c*x])/(e*(m + 1))), x] - Dist[b/(c*e*(m + 1)), Int[(d + e*x)^(m + 1)/(x^2*Sqrt[1 - 1/(c^2*x^2)]), x],
x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[m, -1]

Rubi steps \begin{align*} \text {integral}& = -\frac {2 \left (a+b \sec ^{-1}(c x)\right )}{3 e (d+e x)^{3/2}}+\frac {(2 b) \int \frac {1}{\sqrt {1-\frac {1}{c^2 x^2}} x^2 (d+e x)^{3/2}} \, dx}{3 c e} \\ & = -\frac {2 \left (a+b \sec ^{-1}(c x)\right )}{3 e (d+e x)^{3/2}}+\frac {\left (2 b \sqrt {-\frac {1}{c^2}+x^2}\right ) \int \frac {1}{x (d+e x)^{3/2} \sqrt {-\frac {1}{c^2}+x^2}} \, dx}{3 c e \sqrt {1-\frac {1}{c^2 x^2}} x} \\ & = -\frac {2 \left (a+b \sec ^{-1}(c x)\right )}{3 e (d+e x)^{3/2}}+\frac {\left (2 b \sqrt {-\frac {1}{c^2}+x^2}\right ) \int \left (-\frac {e}{d (d+e x)^{3/2} \sqrt {-\frac {1}{c^2}+x^2}}+\frac {1}{d x \sqrt {d+e x} \sqrt {-\frac {1}{c^2}+x^2}}\right ) \, dx}{3 c e \sqrt {1-\frac {1}{c^2 x^2}} x} \\ & = -\frac {2 \left (a+b \sec ^{-1}(c x)\right )}{3 e (d+e x)^{3/2}}-\frac {\left (2 b \sqrt {-\frac {1}{c^2}+x^2}\right ) \int \frac {1}{(d+e x)^{3/2} \sqrt {-\frac {1}{c^2}+x^2}} \, dx}{3 c d \sqrt {1-\frac {1}{c^2 x^2}} x}+\frac {\left (2 b \sqrt {-\frac {1}{c^2}+x^2}\right ) \int \frac {1}{x \sqrt {d+e x} \sqrt {-\frac {1}{c^2}+x^2}} \, dx}{3 c d e \sqrt {1-\frac {1}{c^2 x^2}} x} \\ & = -\frac {4 b e \left (1-c^2 x^2\right )}{3 c d \left (c^2 d^2-e^2\right ) \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {d+e x}}-\frac {2 \left (a+b \sec ^{-1}(c x)\right )}{3 e (d+e x)^{3/2}}+\frac {\left (4 b \sqrt {-\frac {1}{c^2}+x^2}\right ) \int \frac {-\frac {d}{2}-\frac {e x}{2}}{\sqrt {d+e x} \sqrt {-\frac {1}{c^2}+x^2}} \, dx}{3 c d \left (d^2-\frac {e^2}{c^2}\right ) \sqrt {1-\frac {1}{c^2 x^2}} x}+\frac {\left (2 b \sqrt {1-c^2 x^2}\right ) \int \frac {1}{x \sqrt {1-c x} \sqrt {1+c x} \sqrt {d+e x}} \, dx}{3 c d e \sqrt {1-\frac {1}{c^2 x^2}} x} \\ & = -\frac {4 b e \left (1-c^2 x^2\right )}{3 c d \left (c^2 d^2-e^2\right ) \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {d+e x}}-\frac {2 \left (a+b \sec ^{-1}(c x)\right )}{3 e (d+e x)^{3/2}}-\frac {\left (2 b \sqrt {-\frac {1}{c^2}+x^2}\right ) \int \frac {\sqrt {d+e x}}{\sqrt {-\frac {1}{c^2}+x^2}} \, dx}{3 c d \left (d^2-\frac {e^2}{c^2}\right ) \sqrt {1-\frac {1}{c^2 x^2}} x}-\frac {\left (4 b \sqrt {1-c^2 x^2}\right ) \text {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 c d e \sqrt {1-\frac {1}{c^2 x^2}} x} \\ & = -\frac {4 b e \left (1-c^2 x^2\right )}{3 c d \left (c^2 d^2-e^2\right ) \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {d+e x}}-\frac {2 \left (a+b \sec ^{-1}(c x)\right )}{3 e (d+e x)^{3/2}}-\frac {\left (4 b \sqrt {\frac {c (d+e x)}{c d+e}} \sqrt {1-c^2 x^2}\right ) \text {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 c d e \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {d+e x}}+\frac {\left (4 b \sqrt {d+e x} \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int \frac {\sqrt {1-\frac {2 e x^2}{c \left (d+\frac {e}{c}\right )}}}{\sqrt {1-x^2}} \, dx,x,\frac {\sqrt {1-c x}}{\sqrt {2}}\right )}{3 c^2 d \left (d^2-\frac {e^2}{c^2}\right ) \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {\frac {d+e x}{d+\frac {e}{c}}}} \\ & = -\frac {4 b e \left (1-c^2 x^2\right )}{3 c d \left (c^2 d^2-e^2\right ) \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {d+e x}}-\frac {2 \left (a+b \sec ^{-1}(c x)\right )}{3 e (d+e x)^{3/2}}+\frac {4 b \sqrt {d+e x} \sqrt {1-c^2 x^2} E\left (\arcsin \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 {1-\frac {1}{c^2 x^2}} x \sqrt {\frac {c (d+e x)}{c d+e}}}-\frac {4 b \sqrt {\frac {c (d+e x)}{c d+e}} \sqrt {1-c^2 x^2} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{3 c d e \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {d+e x}} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 30.38 (sec) , antiderivative size = 326, normalized size of antiderivative = 1.09 \[ \int \frac {a+b \sec ^{-1}(c x)}{(d+e x)^{5/2}} \, dx=\frac {2 \left (-\frac {a}{(d+e x)^{3/2}}+\frac {2 b c e^2 \sqrt {1-\frac {1}{c^2 x^2}} x}{\left (c^2 d^3-d e^2\right ) \sqrt {d+e x}}-\frac {b \sec ^{-1}(c x)}{(d+e x)^{3/2}}-\frac {2 i b \sqrt {\frac {e (1+c x)}{-c d+e}} \sqrt {\frac {e-c e x}{c d+e}} \left (-c d E\left (i \text {arcsinh}\left (\sqrt {-\frac {c}{c d+e}} \sqrt {d+e x}\right )|\frac {c d+e}{c d-e}\right )+c d \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {-\frac {c}{c d+e}} \sqrt {d+e x}\right ),\frac {c d+e}{c d-e}\right )+(c d+e) \operatorname {EllipticPi}\left (1+\frac {e}{c d},i \text {arcsinh}\left (\sqrt {-\frac {c}{c d+e}} \sqrt {d+e x}\right ),\frac {c d+e}{c d-e}\right )\right )}{d^2 \left (-\frac {c}{c d+e}\right )^{3/2} (c d+e)^2 \sqrt {1-\frac {1}{c^2 x^2}} x}\right )}{3 e} \]

[In]

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

[Out]

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(874\) vs. \(2(270)=540\).

Time = 7.33 (sec) , antiderivative size = 875, normalized size of antiderivative = 2.94

method	result	size
derivativedivides	\(\frac {-\frac {2 a}{3 \left (e x +d \right )^{\frac {3}{2}}}+2 b \left (-\frac {\operatorname {arcsec}\left (c x \right )}{3 \left (e x +d \right )^{\frac {3}{2}}}-\frac {2 \left (\sqrt {\frac {-c \left (e x +d \right )+c d -e}{c d -e}}\, \sqrt {\frac {-c \left (e x +d \right )+c d +e}{c d +e}}\, \operatorname {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} \sqrt {e x +d}-\sqrt {\frac {-c \left (e x +d \right )+c d -e}{c d -e}}\, \sqrt {\frac {-c \left (e x +d \right )+c d +e}{c d +e}}\, \operatorname {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} \sqrt {e x +d}+\sqrt {\frac {-c \left (e x +d \right )+c d -e}{c d -e}}\, \sqrt {\frac {-c \left (e x +d \right )+c d +e}{c d +e}}\, \operatorname {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} \sqrt {e x +d}-\sqrt {\frac {c}{c d -e}}\, c^{2} d \left (e x +d \right )^{2}+\sqrt {\frac {-c \left (e x +d \right )+c d -e}{c d -e}}\, \sqrt {\frac {-c \left (e x +d \right )+c d +e}{c d +e}}\, \operatorname {EllipticF}\left (\sqrt {e x +d}\, \sqrt {\frac {c}{c d -e}}, \sqrt {\frac {c d -e}{c d +e}}\right ) c d e \sqrt {e x +d}-\sqrt {\frac {-c \left (e x +d \right )+c d -e}{c d -e}}\, \sqrt {\frac {-c \left (e x +d \right )+c d +e}{c d +e}}\, \operatorname {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 {e x +d}+2 \sqrt {\frac {c}{c d -e}}\, c^{2} d^{2} \left (e x +d \right )-\sqrt {\frac {-c \left (e x +d \right )+c d -e}{c d -e}}\, \sqrt {\frac {-c \left (e x +d \right )+c d +e}{c d +e}}\, \operatorname {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 ) e^{2} \sqrt {e x +d}-\sqrt {\frac {c}{c d -e}}\, c^{2} d^{3}+\sqrt {\frac {c}{c d -e}}\, d \,e^{2}\right )}{3 c \left (c d -e \right ) \sqrt {\frac {c}{c d -e}}\, \sqrt {e x +d}\, \left (c d +e \right ) d^{2} x \sqrt {\frac {c^{2} \left (e x +d \right )^{2}-2 c^{2} d \left (e x +d \right )+c^{2} d^{2}-e^{2}}{c^{2} e^{2} x^{2}}}}\right )}{e}\)	\(875\)
default	\(\frac {-\frac {2 a}{3 \left (e x +d \right )^{\frac {3}{2}}}+2 b \left (-\frac {\operatorname {arcsec}\left (c x \right )}{3 \left (e x +d \right )^{\frac {3}{2}}}-\frac {2 \left (\sqrt {\frac {-c \left (e x +d \right )+c d -e}{c d -e}}\, \sqrt {\frac {-c \left (e x +d \right )+c d +e}{c d +e}}\, \operatorname {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} \sqrt {e x +d}-\sqrt {\frac {-c \left (e x +d \right )+c d -e}{c d -e}}\, \sqrt {\frac {-c \left (e x +d \right )+c d +e}{c d +e}}\, \operatorname {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} \sqrt {e x +d}+\sqrt {\frac {-c \left (e x +d \right )+c d -e}{c d -e}}\, \sqrt {\frac {-c \left (e x +d \right )+c d +e}{c d +e}}\, \operatorname {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} \sqrt {e x +d}-\sqrt {\frac {c}{c d -e}}\, c^{2} d \left (e x +d \right )^{2}+\sqrt {\frac {-c \left (e x +d \right )+c d -e}{c d -e}}\, \sqrt {\frac {-c \left (e x +d \right )+c d +e}{c d +e}}\, \operatorname {EllipticF}\left (\sqrt {e x +d}\, \sqrt {\frac {c}{c d -e}}, \sqrt {\frac {c d -e}{c d +e}}\right ) c d e \sqrt {e x +d}-\sqrt {\frac {-c \left (e x +d \right )+c d -e}{c d -e}}\, \sqrt {\frac {-c \left (e x +d \right )+c d +e}{c d +e}}\, \operatorname {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 {e x +d}+2 \sqrt {\frac {c}{c d -e}}\, c^{2} d^{2} \left (e x +d \right )-\sqrt {\frac {-c \left (e x +d \right )+c d -e}{c d -e}}\, \sqrt {\frac {-c \left (e x +d \right )+c d +e}{c d +e}}\, \operatorname {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 ) e^{2} \sqrt {e x +d}-\sqrt {\frac {c}{c d -e}}\, c^{2} d^{3}+\sqrt {\frac {c}{c d -e}}\, d \,e^{2}\right )}{3 c \left (c d -e \right ) \sqrt {\frac {c}{c d -e}}\, \sqrt {e x +d}\, \left (c d +e \right ) d^{2} x \sqrt {\frac {c^{2} \left (e x +d \right )^{2}-2 c^{2} d \left (e x +d \right )+c^{2} d^{2}-e^{2}}{c^{2} e^{2} x^{2}}}}\right )}{e}\)	\(875\)
parts	\(-\frac {2 a}{3 \left (e x +d \right )^{\frac {3}{2}} e}+\frac {2 b \left (-\frac {\operatorname {arcsec}\left (c x \right )}{3 \left (e x +d \right )^{\frac {3}{2}}}+\frac {\frac {2 \sqrt {\frac {c}{c d -e}}\, c^{2} d \left (e x +d \right )^{2}}{3}-\frac {2 \sqrt {-\frac {c \left (e x +d \right )-c d +e}{c d -e}}\, \sqrt {-\frac {c \left (e x +d \right )-c d -e}{c d +e}}\, \operatorname {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}}{3}+\frac {2 \sqrt {-\frac {c \left (e x +d \right )-c d +e}{c d -e}}\, \sqrt {-\frac {c \left (e x +d \right )-c d -e}{c d +e}}\, \operatorname {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^{2} d^{2}}{3}-\frac {2 \sqrt {-\frac {c \left (e x +d \right )-c d +e}{c d -e}}\, \sqrt {-\frac {c \left (e x +d \right )-c d -e}{c d +e}}\, \operatorname {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}}{3}-\frac {4 \sqrt {\frac {c}{c d -e}}\, c^{2} d^{2} \left (e x +d \right )}{3}-\frac {2 \sqrt {-\frac {c \left (e x +d \right )-c d +e}{c d -e}}\, \sqrt {-\frac {c \left (e x +d \right )-c d -e}{c d +e}}\, \operatorname {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}{3}+\frac {2 \sqrt {-\frac {c \left (e x +d \right )-c d +e}{c d -e}}\, \sqrt {-\frac {c \left (e x +d \right )-c d -e}{c d +e}}\, \operatorname {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}{3}+\frac {2 \sqrt {\frac {c}{c d -e}}\, c^{2} d^{3}}{3}+\frac {2 \sqrt {-\frac {c \left (e x +d \right )-c d +e}{c d -e}}\, \sqrt {-\frac {c \left (e x +d \right )-c d -e}{c d +e}}\, \operatorname {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}}{3}-\frac {2 \sqrt {\frac {c}{c d -e}}\, d \,e^{2}}{3}}{c \left (c d -e \right ) \sqrt {\frac {c}{c d -e}}\, \sqrt {e x +d}\, \left (c d +e \right ) d^{2} x \sqrt {\frac {c^{2} \left (e x +d \right )^{2}-2 c^{2} d \left (e x +d \right )+c^{2} d^{2}-e^{2}}{c^{2} e^{2} x^{2}}}}\right )}{e}\)	\(888\)

[In]

int((a+b*arcsec(c*x))/(e*x+d)^(5/2),x,method=_RETURNVERBOSE)

[Out]

2/e*(-1/3*a/(e*x+d)^(3/2)+b*(-1/3/(e*x+d)^(3/2)*arcsec(c*x)-2/3/c*(((-c*(e*x+d)+c*d-e)/(c*d-e))^(1/2)*((-c*(e*
x+d)+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*(e*x+d)^
(1/2)-((-c*(e*x+d)+c*d-e)/(c*d-e))^(1/2)*((-c*(e*x+d)+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*(e*x+d)^(1/2)+((-c*(e*x+d)+c*d-e)/(c*d-e))^(1/2)*((-c*(e*x+d)+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*(e*x+d)^(1/2)-(c/(c*d-e))^(1/2)*c^2*d*(e*x+d)^2+((-c*(e*x+d)+c*d-e)/(c*d-e))^(1/2)*((-c*(e*x+d)+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*(e*x+d)^(1/2)-((-c*(e*
x+d)+c*d-e)/(c*d-e))^(1/2)*((-c*(e*x+d)+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*(e*x+d)^(1/2)+2*(c/(c*d-e))^(1/2)*c^2*d^2*(e*x+d)-((-c*(e*x+d)+c*d-e)/(c*d-e))^(1/2)*
((-c*(e*x+d)+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^2*(e*x+d)^(1/2)-(c/(c*d-e))^(1/2)*c^2*d^3+(c/(c*d-e))^(1/2)*d*e^2)/(c*d-e)/(c/(c*d-e))^(1
/2)/(e*x+d)^(1/2)/(c*d+e)/d^2/x/((c^2*(e*x+d)^2-2*c^2*d*(e*x+d)+c^2*d^2-e^2)/c^2/e^2/x^2)^(1/2)))

Fricas [F]

\[ \int \frac {a+b \sec ^{-1}(c x)}{(d+e x)^{5/2}} \, dx=\int { \frac {b \operatorname {arcsec}\left (c x\right ) + a}{{\left (e x + d\right )}^{\frac {5}{2}}} \,d x } \]

[In]

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

[Out]

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

Sympy [F]

\[ \int \frac {a+b \sec ^{-1}(c x)}{(d+e x)^{5/2}} \, dx=\int \frac {a + b \operatorname {asec}{\left (c x \right )}}{\left (d + e x\right )^{\frac {5}{2}}}\, dx \]

[In]

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

[Out]

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

Maxima [F(-2)]

Exception generated. \[ \int \frac {a+b \sec ^{-1}(c x)}{(d+e x)^{5/2}} \, dx=\text {Exception raised: ValueError} \]

[In]

integrate((a+b*arcsec(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(e+c*d>0)', see `assume?` for m
ore details)

Giac [F]

\[ \int \frac {a+b \sec ^{-1}(c x)}{(d+e x)^{5/2}} \, dx=\int { \frac {b \operatorname {arcsec}\left (c x\right ) + a}{{\left (e x + d\right )}^{\frac {5}{2}}} \,d x } \]

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {a+b \sec ^{-1}(c x)}{(d+e x)^{5/2}} \, dx=\int \frac {a+b\,\mathrm {acos}\left (\frac {1}{c\,x}\right )}{{\left (d+e\,x\right )}^{5/2}} \,d x \]

[In]

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

[Out]

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