\(\int \frac {x^2 (a+b \csc ^{-1}(c x))}{\sqrt {d+e x}} \, dx\) [58]

Optimal result

Integrand size = 21, antiderivative size = 530 \[ \int \frac {x^2 \left (a+b \csc ^{-1}(c x)\right )}{\sqrt {d+e x}} \, dx=-\frac {4 b \sqrt {d+e x} \left (1-c^2 x^2\right )}{15 c^3 e \sqrt {1-\frac {1}{c^2 x^2}} x}+\frac {2 d^2 \sqrt {d+e x} \left (a+b \csc ^{-1}(c x)\right )}{e^3}-\frac {4 d (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e^3}+\frac {2 (d+e x)^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{5 e^3}+\frac {4 b d \sqrt {d+e x} \sqrt {1-c^2 x^2} E\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{5 c^2 e^2 \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {\frac {c (d+e x)}{c d+e}}}-\frac {32 b d^2 \sqrt {\frac {c (d+e x)}{c d+e}} \sqrt {1-c^2 x^2} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{15 c^2 e^2 \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {d+e x}}+\frac {4 b (c d-e) (c d+e) \sqrt {\frac {c (d+e x)}{c d+e}} \sqrt {1-c^2 x^2} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{15 c^4 e^2 \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {d+e x}}-\frac {32 b d^3 \sqrt {\frac {c (d+e x)}{c d+e}} \sqrt {1-c^2 x^2} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{15 c e^3 \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {d+e x}} \]

[Out]

-4/3*d*(e*x+d)^(3/2)*(a+b*arccsc(c*x))/e^3+2/5*(e*x+d)^(5/2)*(a+b*arccsc(c*x))/e^3+2*d^2*(a+b*arccsc(c*x))*(e*
x+d)^(1/2)/e^3-4/15*b*(-c^2*x^2+1)*(e*x+d)^(1/2)/c^3/e/x/(1-1/c^2/x^2)^(1/2)+4/5*b*d*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)/c^2/e^2/x/(1-1/c^2/x^2)^(1/2)/(c*(e*x+
d)/(c*d+e))^(1/2)-32/15*b*d^2*EllipticF(1/2*(-c*x+1)^(1/2)*2^(1/2),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^2/e^2/x/(1-1/c^2/x^2)^(1/2)/(e*x+d)^(1/2)+4/15*b*(c*d-e)*(c*d+e)*EllipticF(1/2*
(-c*x+1)^(1/2)*2^(1/2),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^4/e^2/x/(1-1/
c^2/x^2)^(1/2)/(e*x+d)^(1/2)-32/15*b*d^3*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/e^3/x/(1-1/c^2/x^2)^(1/2)/(e*x+d)^(1/2)

Rubi [A] (verified)

Time = 1.48 (sec) , antiderivative size = 530, normalized size of antiderivative = 1.00, number of steps used = 20, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.714, Rules used = {45, 5355, 12, 6853, 6874, 733, 435, 958, 430, 946, 174, 552, 551, 847, 858} \[ \int \frac {x^2 \left (a+b \csc ^{-1}(c x)\right )}{\sqrt {d+e x}} \, dx=\frac {2 d^2 \sqrt {d+e x} \left (a+b \csc ^{-1}(c x)\right )}{e^3}-\frac {4 d (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e^3}+\frac {2 (d+e x)^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{5 e^3}-\frac {32 b d^3 \sqrt {1-c^2 x^2} \sqrt {\frac {c (d+e x)}{c d+e}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{15 c e^3 x \sqrt {1-\frac {1}{c^2 x^2}} \sqrt {d+e x}}-\frac {32 b d^2 \sqrt {1-c^2 x^2} \sqrt {\frac {c (d+e x)}{c d+e}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{15 c^2 e^2 x \sqrt {1-\frac {1}{c^2 x^2}} \sqrt {d+e x}}+\frac {4 b d \sqrt {1-c^2 x^2} \sqrt {d+e x} E\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{5 c^2 e^2 x \sqrt {1-\frac {1}{c^2 x^2}} \sqrt {\frac {c (d+e x)}{c d+e}}}+\frac {4 b \sqrt {1-c^2 x^2} (c d-e) (c d+e) \sqrt {\frac {c (d+e x)}{c d+e}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{15 c^4 e^2 x \sqrt {1-\frac {1}{c^2 x^2}} \sqrt {d+e x}}-\frac {4 b \left (1-c^2 x^2\right ) \sqrt {d+e x}}{15 c^3 e x \sqrt {1-\frac {1}{c^2 x^2}}} \]

[In]

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

[Out]

(-4*b*Sqrt[d + e*x]*(1 - c^2*x^2))/(15*c^3*e*Sqrt[1 - 1/(c^2*x^2)]*x) + (2*d^2*Sqrt[d + e*x]*(a + b*ArcCsc[c*x
]))/e^3 - (4*d*(d + e*x)^(3/2)*(a + b*ArcCsc[c*x]))/(3*e^3) + (2*(d + e*x)^(5/2)*(a + b*ArcCsc[c*x]))/(5*e^3)
+ (4*b*d*Sqrt[d + e*x]*Sqrt[1 - c^2*x^2]*EllipticE[ArcSin[Sqrt[1 - c*x]/Sqrt[2]], (2*e)/(c*d + e)])/(5*c^2*e^2
*Sqrt[1 - 1/(c^2*x^2)]*x*Sqrt[(c*(d + e*x))/(c*d + e)]) - (32*b*d^2*Sqrt[(c*(d + e*x))/(c*d + e)]*Sqrt[1 - c^2
*x^2]*EllipticF[ArcSin[Sqrt[1 - c*x]/Sqrt[2]], (2*e)/(c*d + e)])/(15*c^2*e^2*Sqrt[1 - 1/(c^2*x^2)]*x*Sqrt[d +
e*x]) + (4*b*(c*d - e)*(c*d + e)*Sqrt[(c*(d + e*x))/(c*d + e)]*Sqrt[1 - c^2*x^2]*EllipticF[ArcSin[Sqrt[1 - c*x
]/Sqrt[2]], (2*e)/(c*d + e)])/(15*c^4*e^2*Sqrt[1 - 1/(c^2*x^2)]*x*Sqrt[d + e*x]) - (32*b*d^3*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)])/(15*c*e^3*Sqrt[1
- 1/(c^2*x^2)]*x*Sqrt[d + e*x])

Rule 12

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

Rule 45

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

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 430

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

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 847

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

Rule 858

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 946

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

Rule 5355

Int[((a_.) + ArcCsc[(c_.)*(x_)]*(b_.))*(u_), x_Symbol] :> With[{v = IntHide[u, x]}, Dist[a + b*ArcCsc[c*x], v,
x] + Dist[b/c, Int[SimplifyIntegrand[v/(x^2*Sqrt[1 - 1/(c^2*x^2)]), x], x], x] /; InverseFunctionFreeQ[v, x]]
/; FreeQ[{a, b, c}, x]

Rule 6853

Int[(u_.)*((a_.) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[b^IntPart[p]*((a + b*x^n)^FracPart[p]/(x^(n*FracPa
rt[p])*(1 + a*(1/(x^n*b)))^FracPart[p])), Int[u*x^(n*p)*(1 + a*(1/(x^n*b)))^p, x], x] /; FreeQ[{a, b, p}, x] &
&  !IntegerQ[p] && ILtQ[n, 0] &&  !RationalFunctionQ[u, x] && IntegerQ[p + 1/2]

Rule 6874

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rubi steps \begin{align*} \text {integral}& = \frac {2 d^2 \sqrt {d+e x} \left (a+b \csc ^{-1}(c x)\right )}{e^3}-\frac {4 d (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e^3}+\frac {2 (d+e x)^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{5 e^3}+\frac {b \int \frac {2 \sqrt {d+e x} \left (8 d^2-4 d e x+3 e^2 x^2\right )}{15 e^3 \sqrt {1-\frac {1}{c^2 x^2}} x^2} \, dx}{c} \\ & = \frac {2 d^2 \sqrt {d+e x} \left (a+b \csc ^{-1}(c x)\right )}{e^3}-\frac {4 d (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e^3}+\frac {2 (d+e x)^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{5 e^3}+\frac {(2 b) \int \frac {\sqrt {d+e x} \left (8 d^2-4 d e x+3 e^2 x^2\right )}{\sqrt {1-\frac {1}{c^2 x^2}} x^2} \, dx}{15 c e^3} \\ & = \frac {2 d^2 \sqrt {d+e x} \left (a+b \csc ^{-1}(c x)\right )}{e^3}-\frac {4 d (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e^3}+\frac {2 (d+e x)^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{5 e^3}+\frac {\left (2 b \sqrt {1-c^2 x^2}\right ) \int \frac {\sqrt {d+e x} \left (8 d^2-4 d e x+3 e^2 x^2\right )}{x \sqrt {1-c^2 x^2}} \, dx}{15 c e^3 \sqrt {1-\frac {1}{c^2 x^2}} x} \\ & = \frac {2 d^2 \sqrt {d+e x} \left (a+b \csc ^{-1}(c x)\right )}{e^3}-\frac {4 d (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e^3}+\frac {2 (d+e x)^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{5 e^3}+\frac {\left (2 b \sqrt {1-c^2 x^2}\right ) \int \left (-\frac {4 d e \sqrt {d+e x}}{\sqrt {1-c^2 x^2}}+\frac {8 d^2 \sqrt {d+e x}}{x \sqrt {1-c^2 x^2}}+\frac {3 e^2 x \sqrt {d+e x}}{\sqrt {1-c^2 x^2}}\right ) \, dx}{15 c e^3 \sqrt {1-\frac {1}{c^2 x^2}} x} \\ & = \frac {2 d^2 \sqrt {d+e x} \left (a+b \csc ^{-1}(c x)\right )}{e^3}-\frac {4 d (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e^3}+\frac {2 (d+e x)^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{5 e^3}+\frac {\left (16 b d^2 \sqrt {1-c^2 x^2}\right ) \int \frac {\sqrt {d+e x}}{x \sqrt {1-c^2 x^2}} \, dx}{15 c e^3 \sqrt {1-\frac {1}{c^2 x^2}} x}-\frac {\left (8 b d \sqrt {1-c^2 x^2}\right ) \int \frac {\sqrt {d+e x}}{\sqrt {1-c^2 x^2}} \, dx}{15 c e^2 \sqrt {1-\frac {1}{c^2 x^2}} x}+\frac {\left (2 b \sqrt {1-c^2 x^2}\right ) \int \frac {x \sqrt {d+e x}}{\sqrt {1-c^2 x^2}} \, dx}{5 c e \sqrt {1-\frac {1}{c^2 x^2}} x} \\ & = -\frac {4 b \sqrt {d+e x} \left (1-c^2 x^2\right )}{15 c^3 e \sqrt {1-\frac {1}{c^2 x^2}} x}+\frac {2 d^2 \sqrt {d+e x} \left (a+b \csc ^{-1}(c x)\right )}{e^3}-\frac {4 d (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e^3}+\frac {2 (d+e x)^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{5 e^3}+\frac {\left (16 b d^3 \sqrt {1-c^2 x^2}\right ) \int \frac {1}{x \sqrt {d+e x} \sqrt {1-c^2 x^2}} \, dx}{15 c e^3 \sqrt {1-\frac {1}{c^2 x^2}} x}+\frac {\left (16 b d^2 \sqrt {1-c^2 x^2}\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {1-c^2 x^2}} \, dx}{15 c e^2 \sqrt {1-\frac {1}{c^2 x^2}} x}-\frac {\left (4 b \sqrt {1-c^2 x^2}\right ) \int \frac {-\frac {e}{2}-\frac {1}{2} c^2 d x}{\sqrt {d+e x} \sqrt {1-c^2 x^2}} \, dx}{15 c^3 e \sqrt {1-\frac {1}{c^2 x^2}} x}+\frac {\left (16 b d \sqrt {d+e x} \sqrt {1-c^2 x^2}\right ) \text {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^2 e^2 \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {-\frac {c^2 (d+e x)}{-c^2 d-c e}}} \\ & = -\frac {4 b \sqrt {d+e x} \left (1-c^2 x^2\right )}{15 c^3 e \sqrt {1-\frac {1}{c^2 x^2}} x}+\frac {2 d^2 \sqrt {d+e x} \left (a+b \csc ^{-1}(c x)\right )}{e^3}-\frac {4 d (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e^3}+\frac {2 (d+e x)^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{5 e^3}+\frac {16 b d \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 )}{15 c^2 e^2 \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {\frac {c (d+e x)}{c d+e}}}+\frac {\left (16 b d^3 \sqrt {1-c^2 x^2}\right ) \int \frac {1}{x \sqrt {1-c x} \sqrt {1+c x} \sqrt {d+e x}} \, dx}{15 c e^3 \sqrt {1-\frac {1}{c^2 x^2}} x}+\frac {\left (2 b d \sqrt {1-c^2 x^2}\right ) \int \frac {\sqrt {d+e x}}{\sqrt {1-c^2 x^2}} \, dx}{15 c e^2 \sqrt {1-\frac {1}{c^2 x^2}} x}-\frac {\left (2 b (c d-e) (c d+e) \sqrt {1-c^2 x^2}\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {1-c^2 x^2}} \, dx}{15 c^3 e^2 \sqrt {1-\frac {1}{c^2 x^2}} x}-\frac {\left (32 b d^2 \sqrt {-\frac {c^2 (d+e x)}{-c^2 d-c e}} \sqrt {1-c^2 x^2}\right ) \text {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^2 e^2 \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {d+e x}} \\ & = -\frac {4 b \sqrt {d+e x} \left (1-c^2 x^2\right )}{15 c^3 e \sqrt {1-\frac {1}{c^2 x^2}} x}+\frac {2 d^2 \sqrt {d+e x} \left (a+b \csc ^{-1}(c x)\right )}{e^3}-\frac {4 d (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e^3}+\frac {2 (d+e x)^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{5 e^3}+\frac {16 b d \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 )}{15 c^2 e^2 \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {\frac {c (d+e x)}{c d+e}}}-\frac {32 b d^2 \sqrt {\frac {c (d+e x)}{c d+e}} \sqrt {1-c^2 x^2} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{15 c^2 e^2 \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {d+e x}}-\frac {\left (32 b d^3 \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 )}{15 c e^3 \sqrt {1-\frac {1}{c^2 x^2}} x}-\frac {\left (4 b d \sqrt {d+e x} \sqrt {1-c^2 x^2}\right ) \text {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^2 e^2 \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {-\frac {c^2 (d+e x)}{-c^2 d-c e}}}+\frac {\left (4 b (c d-e) (c d+e) \sqrt {-\frac {c^2 (d+e x)}{-c^2 d-c e}} \sqrt {1-c^2 x^2}\right ) \text {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^4 e^2 \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {d+e x}} \\ & = -\frac {4 b \sqrt {d+e x} \left (1-c^2 x^2\right )}{15 c^3 e \sqrt {1-\frac {1}{c^2 x^2}} x}+\frac {2 d^2 \sqrt {d+e x} \left (a+b \csc ^{-1}(c x)\right )}{e^3}-\frac {4 d (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e^3}+\frac {2 (d+e x)^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{5 e^3}+\frac {4 b d \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 )}{5 c^2 e^2 \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {\frac {c (d+e x)}{c d+e}}}-\frac {32 b d^2 \sqrt {\frac {c (d+e x)}{c d+e}} \sqrt {1-c^2 x^2} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{15 c^2 e^2 \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {d+e x}}+\frac {4 b (c d-e) (c d+e) \sqrt {\frac {c (d+e x)}{c d+e}} \sqrt {1-c^2 x^2} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{15 c^4 e^2 \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {d+e x}}-\frac {\left (32 b d^3 \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 )}{15 c e^3 \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {d+e x}} \\ & = -\frac {4 b \sqrt {d+e x} \left (1-c^2 x^2\right )}{15 c^3 e \sqrt {1-\frac {1}{c^2 x^2}} x}+\frac {2 d^2 \sqrt {d+e x} \left (a+b \csc ^{-1}(c x)\right )}{e^3}-\frac {4 d (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e^3}+\frac {2 (d+e x)^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{5 e^3}+\frac {4 b d \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 )}{5 c^2 e^2 \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {\frac {c (d+e x)}{c d+e}}}-\frac {32 b d^2 \sqrt {\frac {c (d+e x)}{c d+e}} \sqrt {1-c^2 x^2} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{15 c^2 e^2 \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {d+e x}}+\frac {4 b (c d-e) (c d+e) \sqrt {\frac {c (d+e x)}{c d+e}} \sqrt {1-c^2 x^2} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{15 c^4 e^2 \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {d+e x}}-\frac {32 b d^3 \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 )}{15 c e^3 \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {d+e x}} \\ \end{align*}

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 4 in optimal.

Time = 34.01 (sec) , antiderivative size = 784, normalized size of antiderivative = 1.48 \[ \int \frac {x^2 \left (a+b \csc ^{-1}(c x)\right )}{\sqrt {d+e x}} \, dx=-\frac {a d^3 \sqrt {1+\frac {e x}{d}} B_{-\frac {e x}{d}}\left (3,\frac {1}{2}\right )}{e^3 \sqrt {d+e x}}+\frac {b \left (-\frac {c \left (e+\frac {d}{x}\right ) x \left (\frac {4 c d \sqrt {1-\frac {1}{c^2 x^2}}}{5 e^2}-\frac {16 c^2 d^2 \csc ^{-1}(c x)}{15 e^3}-\frac {2 c^2 x^2 \csc ^{-1}(c x)}{5 e}-\frac {4 c x \left (e \sqrt {1-\frac {1}{c^2 x^2}}-2 c d \csc ^{-1}(c x)\right )}{15 e^2}\right )}{\sqrt {d+e x}}-\frac {2 \sqrt {e+\frac {d}{x}} \sqrt {c x} \left (\frac {2 \left (7 c^2 d^2 e+e^3\right ) \sqrt {\frac {c d+c e x}{c d+e}} \sqrt {1-c^2 x^2} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{\sqrt {1-\frac {1}{c^2 x^2}} \sqrt {e+\frac {d}{x}} (c x)^{3/2}}+\frac {2 \left (8 c^3 d^3+3 c d e^2\right ) \sqrt {\frac {c d+c 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 )}{\sqrt {1-\frac {1}{c^2 x^2}} \sqrt {e+\frac {d}{x}} (c x)^{3/2}}-\frac {6 c d e \cos \left (2 \csc ^{-1}(c x)\right ) \left ((c d+c e x) \left (-1+c^2 x^2\right )+c^2 d x \sqrt {\frac {c d+c e x}{c d+e}} \sqrt {1-c^2 x^2} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )-\frac {c x (1+c x) \sqrt {\frac {e-c e x}{c d+e}} \sqrt {\frac {c d+c e x}{c d-e}} \left ((c d+e) E\left (\arcsin \left (\sqrt {\frac {c d+c e x}{c d-e}}\right )|\frac {c d-e}{c d+e}\right )-e \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {c d+c e x}{c d-e}}\right ),\frac {c d-e}{c d+e}\right )\right )}{\sqrt {\frac {e (1+c x)}{-c d+e}}}+c e x \sqrt {\frac {c d+c 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 )\right )}{\sqrt {1-\frac {1}{c^2 x^2}} \sqrt {e+\frac {d}{x}} \sqrt {c x} \left (-2+c^2 x^2\right )}\right )}{15 e^3 \sqrt {d+e x}}\right )}{c^3} \]

[In]

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

[Out]

-((a*d^3*Sqrt[1 + (e*x)/d]*Beta[-((e*x)/d), 3, 1/2])/(e^3*Sqrt[d + e*x])) + (b*(-((c*(e + d/x)*x*((4*c*d*Sqrt[
1 - 1/(c^2*x^2)])/(5*e^2) - (16*c^2*d^2*ArcCsc[c*x])/(15*e^3) - (2*c^2*x^2*ArcCsc[c*x])/(5*e) - (4*c*x*(e*Sqrt
[1 - 1/(c^2*x^2)] - 2*c*d*ArcCsc[c*x]))/(15*e^2)))/Sqrt[d + e*x]) - (2*Sqrt[e + d/x]*Sqrt[c*x]*((2*(7*c^2*d^2*
e + e^3)*Sqrt[(c*d + c*e*x)/(c*d + e)]*Sqrt[1 - c^2*x^2]*EllipticF[ArcSin[Sqrt[1 - c*x]/Sqrt[2]], (2*e)/(c*d +
e)])/(Sqrt[1 - 1/(c^2*x^2)]*Sqrt[e + d/x]*(c*x)^(3/2)) + (2*(8*c^3*d^3 + 3*c*d*e^2)*Sqrt[(c*d + c*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)])/(Sqrt[1 - 1/(c^2*x^2)]*S
qrt[e + d/x]*(c*x)^(3/2)) - (6*c*d*e*Cos[2*ArcCsc[c*x]]*((c*d + c*e*x)*(-1 + c^2*x^2) + c^2*d*x*Sqrt[(c*d + c*
e*x)/(c*d + e)]*Sqrt[1 - c^2*x^2]*EllipticF[ArcSin[Sqrt[1 - c*x]/Sqrt[2]], (2*e)/(c*d + e)] - (c*x*(1 + c*x)*S
qrt[(e - c*e*x)/(c*d + e)]*Sqrt[(c*d + c*e*x)/(c*d - e)]*((c*d + e)*EllipticE[ArcSin[Sqrt[(c*d + c*e*x)/(c*d -
e)]], (c*d - e)/(c*d + e)] - e*EllipticF[ArcSin[Sqrt[(c*d + c*e*x)/(c*d - e)]], (c*d - e)/(c*d + e)]))/Sqrt[(
e*(1 + c*x))/(-(c*d) + e)] + c*e*x*Sqrt[(c*d + c*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)]))/(Sqrt[1 - 1/(c^2*x^2)]*Sqrt[e + d/x]*Sqrt[c*x]*(-2 + c^2*x^2))))/(15*e^3*
Sqrt[d + e*x])))/c^3

Maple [A] (verified)

Time = 9.16 (sec) , antiderivative size = 850, normalized size of antiderivative = 1.60

method	result	size
derivativedivides	\(\frac {2 a \left (\frac {\left (e x +d \right )^{\frac {5}{2}}}{5}-\frac {2 \left (e x +d \right )^{\frac {3}{2}} d}{3}+d^{2} \sqrt {e x +d}\right )+2 b \left (\frac {\operatorname {arccsc}\left (c x \right ) \left (e x +d \right )^{\frac {5}{2}}}{5}-\frac {2 \,\operatorname {arccsc}\left (c x \right ) \left (e x +d \right )^{\frac {3}{2}} d}{3}+\operatorname {arccsc}\left (c x \right ) d^{2} \sqrt {e x +d}+\frac {\frac {2 \sqrt {\frac {c}{c d -e}}\, c^{2} \left (e x +d \right )^{\frac {5}{2}}}{15}+\frac {8 d^{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^{2}}{15}+\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 ) c^{2} d^{2}}{5}-\frac {16 d^{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 ) c^{2}}{15}-\frac {4 \sqrt {\frac {c}{c d -e}}\, c^{2} d \left (e x +d \right )^{\frac {3}{2}}}{15}-\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 ) c d e}{5}+\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 ) c d e}{5}+\frac {2 \sqrt {\frac {c}{c d -e}}\, c^{2} d^{2} \sqrt {e x +d}}{15}+\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 ) e^{2}}{15}-\frac {2 \sqrt {\frac {c}{c d -e}}\, e^{2} \sqrt {e x +d}}{15}}{c^{3} \sqrt {\frac {c}{c d -e}}\, 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^{3}}\)	\(850\)
default	\(\frac {2 a \left (\frac {\left (e x +d \right )^{\frac {5}{2}}}{5}-\frac {2 \left (e x +d \right )^{\frac {3}{2}} d}{3}+d^{2} \sqrt {e x +d}\right )+2 b \left (\frac {\operatorname {arccsc}\left (c x \right ) \left (e x +d \right )^{\frac {5}{2}}}{5}-\frac {2 \,\operatorname {arccsc}\left (c x \right ) \left (e x +d \right )^{\frac {3}{2}} d}{3}+\operatorname {arccsc}\left (c x \right ) d^{2} \sqrt {e x +d}+\frac {\frac {2 \sqrt {\frac {c}{c d -e}}\, c^{2} \left (e x +d \right )^{\frac {5}{2}}}{15}+\frac {8 d^{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^{2}}{15}+\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 ) c^{2} d^{2}}{5}-\frac {16 d^{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 ) c^{2}}{15}-\frac {4 \sqrt {\frac {c}{c d -e}}\, c^{2} d \left (e x +d \right )^{\frac {3}{2}}}{15}-\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 ) c d e}{5}+\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 ) c d e}{5}+\frac {2 \sqrt {\frac {c}{c d -e}}\, c^{2} d^{2} \sqrt {e x +d}}{15}+\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 ) e^{2}}{15}-\frac {2 \sqrt {\frac {c}{c d -e}}\, e^{2} \sqrt {e x +d}}{15}}{c^{3} \sqrt {\frac {c}{c d -e}}\, 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^{3}}\)	\(850\)
parts	\(\frac {2 a \left (\frac {\left (e x +d \right )^{\frac {5}{2}}}{5}-\frac {2 \left (e x +d \right )^{\frac {3}{2}} d}{3}+d^{2} \sqrt {e x +d}\right )}{e^{3}}+\frac {2 b \left (\frac {\operatorname {arccsc}\left (c x \right ) \left (e x +d \right )^{\frac {5}{2}}}{5}-\frac {2 \,\operatorname {arccsc}\left (c x \right ) \left (e x +d \right )^{\frac {3}{2}} d}{3}+\operatorname {arccsc}\left (c x \right ) d^{2} \sqrt {e x +d}+\frac {\frac {2 \sqrt {\frac {c}{c d -e}}\, c^{2} \left (e x +d \right )^{\frac {5}{2}}}{15}-\frac {4 \sqrt {\frac {c}{c d -e}}\, c^{2} d \left (e x +d \right )^{\frac {3}{2}}}{15}+\frac {8 d^{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^{2}}{15}+\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 ) c^{2} d^{2}}{5}-\frac {16 d^{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 ) c^{2}}{15}+\frac {2 \sqrt {\frac {c}{c d -e}}\, c^{2} d^{2} \sqrt {e x +d}}{15}-\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 ) c d e}{5}+\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 ) c d e}{5}+\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 ) e^{2}}{15}-\frac {2 \sqrt {\frac {c}{c d -e}}\, e^{2} \sqrt {e x +d}}{15}}{c^{3} \sqrt {\frac {c}{c d -e}}\, 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^{3}}\)	\(865\)

[In]

int(x^2*(a+b*arccsc(c*x))/(e*x+d)^(1/2),x,method=_RETURNVERBOSE)

[Out]

2/e^3*(a*(1/5*(e*x+d)^(5/2)-2/3*(e*x+d)^(3/2)*d+d^2*(e*x+d)^(1/2))+b*(1/5*arccsc(c*x)*(e*x+d)^(5/2)-2/3*arccsc
(c*x)*(e*x+d)^(3/2)*d+arccsc(c*x)*d^2*(e*x+d)^(1/2)+2/15/c^3*((c/(c*d-e))^(1/2)*c^2*(e*x+d)^(5/2)+4*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^2+3*((-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^2-8*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)*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^2-2*(c/(c*d-e))^(1/2)*c^2*d*(e*x+d)^(3/2)-3*((-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+3*((-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+(c/(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)*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^2*(
e*x+d)^(1/2))/(c/(c*d-e))^(1/2)/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(-1)]

Timed out. \[ \int \frac {x^2 \left (a+b \csc ^{-1}(c x)\right )}{\sqrt {d+e x}} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Sympy [F]

\[ \int \frac {x^2 \left (a+b \csc ^{-1}(c x)\right )}{\sqrt {d+e x}} \, dx=\int \frac {x^{2} \left (a + b \operatorname {acsc}{\left (c x \right )}\right )}{\sqrt {d + e x}}\, dx \]

[In]

integrate(x**2*(a+b*acsc(c*x))/(e*x+d)**(1/2),x)

[Out]

Integral(x**2*(a + b*acsc(c*x))/sqrt(d + e*x), x)

Maxima [F(-2)]

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

[In]

integrate(x^2*(a+b*arccsc(c*x))/(e*x+d)^(1/2),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]

\[ \int \frac {x^2 \left (a+b \csc ^{-1}(c x)\right )}{\sqrt {d+e x}} \, dx=\int { \frac {{\left (b \operatorname {arccsc}\left (c x\right ) + a\right )} x^{2}}{\sqrt {e x + d}} \,d x } \]

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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