3.1.57 \(\int \frac {x^3 (a+b \csc ^{-1}(c x))}{\sqrt {d+e x}} \, dx\) [57]

Optimal. Leaf size=714 \[ -\frac {4 b \sqrt {d+e x} \left (1-c^2 x^2\right )}{35 c^3 e \sqrt {1-\frac {1}{c^2 x^2}}}+\frac {4 b d \sqrt {d+e x} \left (1-c^2 x^2\right )}{21 c^3 e^2 \sqrt {1-\frac {1}{c^2 x^2}} x}-\frac {2 d^3 \sqrt {d+e x} \left (a+b \csc ^{-1}(c x)\right )}{e^4}+\frac {2 d^2 (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{e^4}-\frac {6 d (d+e x)^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{5 e^4}+\frac {2 (d+e x)^{7/2} \left (a+b \csc ^{-1}(c x)\right )}{7 e^4}-\frac {24 b d^2 \sqrt {d+e x} \sqrt {1-c^2 x^2} E\left (\text {ArcSin}\left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{35 c^2 e^3 \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {\frac {c (d+e x)}{c d+e}}}+\frac {4 b \left (2 c^2 d^2-9 e^2\right ) \sqrt {d+e x} \sqrt {1-c^2 x^2} E\left (\text {ArcSin}\left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{105 c^4 e^3 \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {\frac {c (d+e x)}{c d+e}}}+\frac {64 b d^3 \sqrt {\frac {c (d+e x)}{c d+e}} \sqrt {1-c^2 x^2} F\left (\text {ArcSin}\left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{35 c^2 e^3 \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {d+e x}}-\frac {32 b d (c d-e) (c d+e) \sqrt {\frac {c (d+e x)}{c d+e}} \sqrt {1-c^2 x^2} F\left (\text {ArcSin}\left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{105 c^4 e^3 \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {d+e x}}+\frac {64 b d^4 \sqrt {\frac {c (d+e x)}{c d+e}} \sqrt {1-c^2 x^2} \Pi \left (2;\text {ArcSin}\left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{35 c e^4 \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {d+e x}} \]

[Out]

2*d^2*(e*x+d)^(3/2)*(a+b*arccsc(c*x))/e^4-6/5*d*(e*x+d)^(5/2)*(a+b*arccsc(c*x))/e^4+2/7*(e*x+d)^(7/2)*(a+b*arc
csc(c*x))/e^4-2*d^3*(a+b*arccsc(c*x))*(e*x+d)^(1/2)/e^4-4/35*b*(-c^2*x^2+1)*(e*x+d)^(1/2)/c^3/e/(1-1/c^2/x^2)^
(1/2)+4/21*b*d*(-c^2*x^2+1)*(e*x+d)^(1/2)/c^3/e^2/x/(1-1/c^2/x^2)^(1/2)-24/35*b*d^2*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^3/x/(1-1/c^2/x^2)^(1/2)/(c*(e*x+d
)/(c*d+e))^(1/2)+4/105*b*(2*c^2*d^2-9*e^2)*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^4/e^3/x/(1-1/c^2/x^2)^(1/2)/(c*(e*x+d)/(c*d+e))^(1/2)+64/35*b*d^3*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^3/x/(1-
1/c^2/x^2)^(1/2)/(e*x+d)^(1/2)-32/105*b*d*(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^3/x/(1-1/c^2/x^2)^(1/2)/(e*x+d)^(1/2)+64/35*b*d
^4*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^4/x/(1-1/c^2/x^2)^(1/2)/(e*x+d)^(1/2)

________________________________________________________________________________________

Rubi [A]
time = 1.91, antiderivative size = 714, normalized size of antiderivative = 1.00, number of steps used = 27, number of rules used = 17, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.810, Rules used = {45, 5355, 12, 6853, 6874, 733, 435, 958, 430, 946, 174, 552, 551, 847, 858, 956, 1668} \begin {gather*} -\frac {2 d^3 \sqrt {d+e x} \left (a+b \csc ^{-1}(c x)\right )}{e^4}+\frac {2 d^2 (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{e^4}-\frac {6 d (d+e x)^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{5 e^4}+\frac {2 (d+e x)^{7/2} \left (a+b \csc ^{-1}(c x)\right )}{7 e^4}+\frac {64 b d^4 \sqrt {1-c^2 x^2} \sqrt {\frac {c (d+e x)}{c d+e}} \Pi \left (2;\text {ArcSin}\left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{35 c e^4 x \sqrt {1-\frac {1}{c^2 x^2}} \sqrt {d+e x}}+\frac {64 b d^3 \sqrt {1-c^2 x^2} \sqrt {\frac {c (d+e x)}{c d+e}} F\left (\text {ArcSin}\left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{35 c^2 e^3 x \sqrt {1-\frac {1}{c^2 x^2}} \sqrt {d+e x}}-\frac {24 b d^2 \sqrt {1-c^2 x^2} \sqrt {d+e x} E\left (\text {ArcSin}\left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{35 c^2 e^3 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} \left (2 c^2 d^2-9 e^2\right ) \sqrt {d+e x} E\left (\text {ArcSin}\left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{105 c^4 e^3 x \sqrt {1-\frac {1}{c^2 x^2}} \sqrt {\frac {c (d+e x)}{c d+e}}}-\frac {32 b d \sqrt {1-c^2 x^2} (c d-e) (c d+e) \sqrt {\frac {c (d+e x)}{c d+e}} F\left (\text {ArcSin}\left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{105 c^4 e^3 x \sqrt {1-\frac {1}{c^2 x^2}} \sqrt {d+e x}}+\frac {4 b d \left (1-c^2 x^2\right ) \sqrt {d+e x}}{21 c^3 e^2 x \sqrt {1-\frac {1}{c^2 x^2}}}-\frac {4 b \left (1-c^2 x^2\right ) \sqrt {d+e x}}{35 c^3 e \sqrt {1-\frac {1}{c^2 x^2}}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(x^3*(a + b*ArcCsc[c*x]))/Sqrt[d + e*x],x]

[Out]

(-4*b*Sqrt[d + e*x]*(1 - c^2*x^2))/(35*c^3*e*Sqrt[1 - 1/(c^2*x^2)]) + (4*b*d*Sqrt[d + e*x]*(1 - c^2*x^2))/(21*
c^3*e^2*Sqrt[1 - 1/(c^2*x^2)]*x) - (2*d^3*Sqrt[d + e*x]*(a + b*ArcCsc[c*x]))/e^4 + (2*d^2*(d + e*x)^(3/2)*(a +
b*ArcCsc[c*x]))/e^4 - (6*d*(d + e*x)^(5/2)*(a + b*ArcCsc[c*x]))/(5*e^4) + (2*(d + e*x)^(7/2)*(a + b*ArcCsc[c*
x]))/(7*e^4) - (24*b*d^2*Sqrt[d + e*x]*Sqrt[1 - c^2*x^2]*EllipticE[ArcSin[Sqrt[1 - c*x]/Sqrt[2]], (2*e)/(c*d +
e)])/(35*c^2*e^3*Sqrt[1 - 1/(c^2*x^2)]*x*Sqrt[(c*(d + e*x))/(c*d + e)]) + (4*b*(2*c^2*d^2 - 9*e^2)*Sqrt[d + e
*x]*Sqrt[1 - c^2*x^2]*EllipticE[ArcSin[Sqrt[1 - c*x]/Sqrt[2]], (2*e)/(c*d + e)])/(105*c^4*e^3*Sqrt[1 - 1/(c^2*
x^2)]*x*Sqrt[(c*(d + e*x))/(c*d + e)]) + (64*b*d^3*Sqrt[(c*(d + e*x))/(c*d + e)]*Sqrt[1 - c^2*x^2]*EllipticF[A
rcSin[Sqrt[1 - c*x]/Sqrt[2]], (2*e)/(c*d + e)])/(35*c^2*e^3*Sqrt[1 - 1/(c^2*x^2)]*x*Sqrt[d + e*x]) - (32*b*d*(
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)])/(105*c^4*e^3*Sqrt[1 - 1/(c^2*x^2)]*x*Sqrt[d + e*x]) + (64*b*d^4*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)])/(35*c*e^4*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 956

Int[(((d_.) + (e_.)*(x_))^(m_)*Sqrt[(f_.) + (g_.)*(x_)])/Sqrt[(a_) + (c_.)*(x_)^2], x_Symbol] :> Simp[2*e*(d +
e*x)^(m - 1)*Sqrt[f + g*x]*(Sqrt[a + c*x^2]/(c*(2*m + 1))), x] - Dist[1/(c*(2*m + 1)), Int[((d + e*x)^(m - 2)
/(Sqrt[f + g*x]*Sqrt[a + c*x^2]))*Simp[a*e*(d*g + 2*e*f*(m - 1)) - c*d^2*f*(2*m + 1) + (a*e^2*g*(2*m - 1) - c*
d*(4*e*f*m + d*g*(2*m + 1)))*x - c*e*(e*f + d*g*(4*m - 1))*x^2, 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[2*m] && GtQ[m, 1]

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 1668

Int[(Pq_)*((d_) + (e_.)*(x_))^(m_.)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{q = Expon[Pq, x], f = Coeff
[Pq, x, Expon[Pq, x]]}, Simp[f*(d + e*x)^(m + q - 1)*((a + c*x^2)^(p + 1)/(c*e^(q - 1)*(m + q + 2*p + 1))), x]
+ Dist[1/(c*e^q*(m + q + 2*p + 1)), Int[(d + e*x)^m*(a + c*x^2)^p*ExpandToSum[c*e^q*(m + q + 2*p + 1)*Pq - c*
f*(m + q + 2*p + 1)*(d + e*x)^q - f*(d + e*x)^(q - 2)*(a*e^2*(m + q - 1) - c*d^2*(m + q + 2*p + 1) - 2*c*d*e*(
m + q + p)*x), x], x], x] /; GtQ[q, 1] && NeQ[m + q + 2*p + 1, 0]] /; FreeQ[{a, c, d, e, m, p}, x] && PolyQ[Pq
, x] && NeQ[c*d^2 + a*e^2, 0] &&  !(EqQ[d, 0] && True) &&  !(IGtQ[m, 0] && RationalQ[a, c, d, e] && (IntegerQ[
p] || ILtQ[p + 1/2, 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*} \int \frac {x^3 \left (a+b \csc ^{-1}(c x)\right )}{\sqrt {d+e x}} \, dx &=-\frac {2 d^3 \sqrt {d+e x} \left (a+b \csc ^{-1}(c x)\right )}{e^4}+\frac {2 d^2 (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{e^4}-\frac {6 d (d+e x)^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{5 e^4}+\frac {2 (d+e x)^{7/2} \left (a+b \csc ^{-1}(c x)\right )}{7 e^4}+\frac {b \int \frac {2 \sqrt {d+e x} \left (-16 d^3+8 d^2 e x-6 d e^2 x^2+5 e^3 x^3\right )}{35 e^4 \sqrt {1-\frac {1}{c^2 x^2}} x^2} \, dx}{c}\\ &=-\frac {2 d^3 \sqrt {d+e x} \left (a+b \csc ^{-1}(c x)\right )}{e^4}+\frac {2 d^2 (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{e^4}-\frac {6 d (d+e x)^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{5 e^4}+\frac {2 (d+e x)^{7/2} \left (a+b \csc ^{-1}(c x)\right )}{7 e^4}+\frac {(2 b) \int \frac {\sqrt {d+e x} \left (-16 d^3+8 d^2 e x-6 d e^2 x^2+5 e^3 x^3\right )}{\sqrt {1-\frac {1}{c^2 x^2}} x^2} \, dx}{35 c e^4}\\ &=-\frac {2 d^3 \sqrt {d+e x} \left (a+b \csc ^{-1}(c x)\right )}{e^4}+\frac {2 d^2 (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{e^4}-\frac {6 d (d+e x)^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{5 e^4}+\frac {2 (d+e x)^{7/2} \left (a+b \csc ^{-1}(c x)\right )}{7 e^4}+\frac {\left (2 b \sqrt {1-c^2 x^2}\right ) \int \frac {\sqrt {d+e x} \left (-16 d^3+8 d^2 e x-6 d e^2 x^2+5 e^3 x^3\right )}{x \sqrt {1-c^2 x^2}} \, dx}{35 c e^4 \sqrt {1-\frac {1}{c^2 x^2}} x}\\ &=-\frac {2 d^3 \sqrt {d+e x} \left (a+b \csc ^{-1}(c x)\right )}{e^4}+\frac {2 d^2 (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{e^4}-\frac {6 d (d+e x)^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{5 e^4}+\frac {2 (d+e x)^{7/2} \left (a+b \csc ^{-1}(c x)\right )}{7 e^4}+\frac {\left (2 b \sqrt {1-c^2 x^2}\right ) \int \left (\frac {8 d^2 e \sqrt {d+e x}}{\sqrt {1-c^2 x^2}}-\frac {16 d^3 \sqrt {d+e x}}{x \sqrt {1-c^2 x^2}}-\frac {6 d e^2 x \sqrt {d+e x}}{\sqrt {1-c^2 x^2}}+\frac {5 e^3 x^2 \sqrt {d+e x}}{\sqrt {1-c^2 x^2}}\right ) \, dx}{35 c e^4 \sqrt {1-\frac {1}{c^2 x^2}} x}\\ &=-\frac {2 d^3 \sqrt {d+e x} \left (a+b \csc ^{-1}(c x)\right )}{e^4}+\frac {2 d^2 (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{e^4}-\frac {6 d (d+e x)^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{5 e^4}+\frac {2 (d+e x)^{7/2} \left (a+b \csc ^{-1}(c x)\right )}{7 e^4}-\frac {\left (32 b d^3 \sqrt {1-c^2 x^2}\right ) \int \frac {\sqrt {d+e x}}{x \sqrt {1-c^2 x^2}} \, dx}{35 c e^4 \sqrt {1-\frac {1}{c^2 x^2}} x}+\frac {\left (16 b d^2 \sqrt {1-c^2 x^2}\right ) \int \frac {\sqrt {d+e x}}{\sqrt {1-c^2 x^2}} \, dx}{35 c e^3 \sqrt {1-\frac {1}{c^2 x^2}} x}-\frac {\left (12 b d \sqrt {1-c^2 x^2}\right ) \int \frac {x \sqrt {d+e x}}{\sqrt {1-c^2 x^2}} \, dx}{35 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^2 \sqrt {d+e x}}{\sqrt {1-c^2 x^2}} \, dx}{7 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 )}{35 c^3 e \sqrt {1-\frac {1}{c^2 x^2}}}+\frac {8 b d \sqrt {d+e x} \left (1-c^2 x^2\right )}{35 c^3 e^2 \sqrt {1-\frac {1}{c^2 x^2}} x}-\frac {2 d^3 \sqrt {d+e x} \left (a+b \csc ^{-1}(c x)\right )}{e^4}+\frac {2 d^2 (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{e^4}-\frac {6 d (d+e x)^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{5 e^4}+\frac {2 (d+e x)^{7/2} \left (a+b \csc ^{-1}(c x)\right )}{7 e^4}-\frac {\left (32 b d^4 \sqrt {1-c^2 x^2}\right ) \int \frac {1}{x \sqrt {d+e x} \sqrt {1-c^2 x^2}} \, dx}{35 c e^4 \sqrt {1-\frac {1}{c^2 x^2}} x}-\frac {\left (32 b d^3 \sqrt {1-c^2 x^2}\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {1-c^2 x^2}} \, dx}{35 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 {-\frac {e}{2}-\frac {1}{2} c^2 d x}{\sqrt {d+e x} \sqrt {1-c^2 x^2}} \, dx}{35 c^3 e^2 \sqrt {1-\frac {1}{c^2 x^2}} x}+\frac {\left (2 b \sqrt {1-c^2 x^2}\right ) \int \frac {2 d+3 e x+c^2 d x^2}{\sqrt {d+e x} \sqrt {1-c^2 x^2}} \, dx}{35 c^3 e \sqrt {1-\frac {1}{c^2 x^2}} x}-\frac {\left (32 b d^2 \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 )}{35 c^2 e^3 \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 )}{35 c^3 e \sqrt {1-\frac {1}{c^2 x^2}}}+\frac {4 b d \sqrt {d+e x} \left (1-c^2 x^2\right )}{21 c^3 e^2 \sqrt {1-\frac {1}{c^2 x^2}} x}-\frac {2 d^3 \sqrt {d+e x} \left (a+b \csc ^{-1}(c x)\right )}{e^4}+\frac {2 d^2 (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{e^4}-\frac {6 d (d+e x)^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{5 e^4}+\frac {2 (d+e x)^{7/2} \left (a+b \csc ^{-1}(c x)\right )}{7 e^4}-\frac {32 b d^2 \sqrt {d+e x} \sqrt {1-c^2 x^2} E\left (\sin ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{35 c^2 e^3 \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {\frac {c (d+e x)}{c d+e}}}-\frac {\left (32 b d^4 \sqrt {1-c^2 x^2}\right ) \int \frac {1}{x \sqrt {1-c x} \sqrt {1+c x} \sqrt {d+e x}} \, dx}{35 c e^4 \sqrt {1-\frac {1}{c^2 x^2}} x}-\frac {\left (4 b \sqrt {1-c^2 x^2}\right ) \int \frac {-\frac {7}{2} c^2 d e^2+\frac {1}{2} c^2 e \left (2 c^2 d^2-9 e^2\right ) x}{\sqrt {d+e x} \sqrt {1-c^2 x^2}} \, dx}{105 c^5 e^3 \sqrt {1-\frac {1}{c^2 x^2}} x}-\frac {\left (4 b d^2 \sqrt {1-c^2 x^2}\right ) \int \frac {\sqrt {d+e x}}{\sqrt {1-c^2 x^2}} \, dx}{35 c e^3 \sqrt {1-\frac {1}{c^2 x^2}} x}+\frac {\left (4 b d (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}{35 c^3 e^3 \sqrt {1-\frac {1}{c^2 x^2}} x}+\frac {\left (64 b d^3 \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 )}{35 c^2 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 )}{35 c^3 e \sqrt {1-\frac {1}{c^2 x^2}}}+\frac {4 b d \sqrt {d+e x} \left (1-c^2 x^2\right )}{21 c^3 e^2 \sqrt {1-\frac {1}{c^2 x^2}} x}-\frac {2 d^3 \sqrt {d+e x} \left (a+b \csc ^{-1}(c x)\right )}{e^4}+\frac {2 d^2 (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{e^4}-\frac {6 d (d+e x)^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{5 e^4}+\frac {2 (d+e x)^{7/2} \left (a+b \csc ^{-1}(c x)\right )}{7 e^4}-\frac {32 b d^2 \sqrt {d+e x} \sqrt {1-c^2 x^2} E\left (\sin ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{35 c^2 e^3 \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {\frac {c (d+e x)}{c d+e}}}+\frac {64 b d^3 \sqrt {\frac {c (d+e x)}{c d+e}} \sqrt {1-c^2 x^2} F\left (\sin ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{35 c^2 e^3 \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {d+e x}}+\frac {\left (64 b d^4 \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 )}{35 c e^4 \sqrt {1-\frac {1}{c^2 x^2}} x}+\frac {\left (4 b d (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}{105 c^3 e^3 \sqrt {1-\frac {1}{c^2 x^2}} x}-\frac {\left (2 b \left (2 c^2 d^2-9 e^2\right ) \sqrt {1-c^2 x^2}\right ) \int \frac {\sqrt {d+e x}}{\sqrt {1-c^2 x^2}} \, dx}{105 c^3 e^3 \sqrt {1-\frac {1}{c^2 x^2}} x}+\frac {\left (8 b d^2 \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 )}{35 c^2 e^3 \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {-\frac {c^2 (d+e x)}{-c^2 d-c e}}}-\frac {\left (8 b d (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 )}{35 c^4 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 )}{35 c^3 e \sqrt {1-\frac {1}{c^2 x^2}}}+\frac {4 b d \sqrt {d+e x} \left (1-c^2 x^2\right )}{21 c^3 e^2 \sqrt {1-\frac {1}{c^2 x^2}} x}-\frac {2 d^3 \sqrt {d+e x} \left (a+b \csc ^{-1}(c x)\right )}{e^4}+\frac {2 d^2 (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{e^4}-\frac {6 d (d+e x)^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{5 e^4}+\frac {2 (d+e x)^{7/2} \left (a+b \csc ^{-1}(c x)\right )}{7 e^4}-\frac {24 b d^2 \sqrt {d+e x} \sqrt {1-c^2 x^2} E\left (\sin ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{35 c^2 e^3 \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {\frac {c (d+e x)}{c d+e}}}+\frac {64 b d^3 \sqrt {\frac {c (d+e x)}{c d+e}} \sqrt {1-c^2 x^2} F\left (\sin ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{35 c^2 e^3 \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {d+e x}}-\frac {8 b d (c d-e) (c d+e) \sqrt {\frac {c (d+e x)}{c d+e}} \sqrt {1-c^2 x^2} F\left (\sin ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{35 c^4 e^3 \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {d+e x}}+\frac {\left (64 b d^4 \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 )}{35 c e^4 \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {d+e x}}+\frac {\left (4 b \left (2 c^2 d^2-9 e^2\right ) \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 )}{105 c^4 e^3 \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {-\frac {c^2 (d+e x)}{-c^2 d-c e}}}-\frac {\left (8 b d (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 )}{105 c^4 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 )}{35 c^3 e \sqrt {1-\frac {1}{c^2 x^2}}}+\frac {4 b d \sqrt {d+e x} \left (1-c^2 x^2\right )}{21 c^3 e^2 \sqrt {1-\frac {1}{c^2 x^2}} x}-\frac {2 d^3 \sqrt {d+e x} \left (a+b \csc ^{-1}(c x)\right )}{e^4}+\frac {2 d^2 (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{e^4}-\frac {6 d (d+e x)^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{5 e^4}+\frac {2 (d+e x)^{7/2} \left (a+b \csc ^{-1}(c x)\right )}{7 e^4}-\frac {24 b d^2 \sqrt {d+e x} \sqrt {1-c^2 x^2} E\left (\sin ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{35 c^2 e^3 \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {\frac {c (d+e x)}{c d+e}}}+\frac {4 b \left (2 c^2 d^2-9 e^2\right ) \sqrt {d+e x} \sqrt {1-c^2 x^2} E\left (\sin ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{105 c^4 e^3 \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {\frac {c (d+e x)}{c d+e}}}+\frac {64 b d^3 \sqrt {\frac {c (d+e x)}{c d+e}} \sqrt {1-c^2 x^2} F\left (\sin ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{35 c^2 e^3 \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {d+e x}}-\frac {32 b d (c d-e) (c d+e) \sqrt {\frac {c (d+e x)}{c d+e}} \sqrt {1-c^2 x^2} F\left (\sin ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{105 c^4 e^3 \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {d+e x}}+\frac {64 b d^4 \sqrt {\frac {c (d+e x)}{c d+e}} \sqrt {1-c^2 x^2} \Pi \left (2;\sin ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{35 c e^4 \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {d+e x}}\\ \end {align*}

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Mathematica [C] Result contains higher order function than in optimal. Order 9 vs. order 4 in optimal.
time = 30.53, size = 873, normalized size = 1.22 \begin {gather*} \frac {a d^4 \sqrt {1+\frac {e x}{d}} B_{-\frac {e x}{d}}\left (4,\frac {1}{2}\right )}{e^4 \sqrt {d+e x}}+\frac {b \left (-\frac {c \left (e+\frac {d}{x}\right ) x \left (-\frac {4 \left (16 c^2 d^2+9 e^2\right ) \sqrt {1-\frac {1}{c^2 x^2}}}{105 e^3}+\frac {32 c^3 d^3 \csc ^{-1}(c x)}{35 e^4}-\frac {2 c^3 x^3 \csc ^{-1}(c x)}{7 e}-\frac {4 c^2 x^2 \left (e \sqrt {1-\frac {1}{c^2 x^2}}-3 c d \csc ^{-1}(c x)\right )}{35 e^2}+\frac {4 c x \left (5 c d e \sqrt {1-\frac {1}{c^2 x^2}}-12 c^2 d^2 \csc ^{-1}(c x)\right )}{105 e^3}\right )}{\sqrt {d+e x}}+\frac {2 \sqrt {e+\frac {d}{x}} \sqrt {c x} \left (\frac {2 \left (40 c^3 d^3 e+8 c d e^3\right ) \sqrt {\frac {c d+c e x}{c d+e}} \sqrt {1-c^2 x^2} F\left (\text {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 (48 c^4 d^4+16 c^2 d^2 e^2+9 e^4\right ) \sqrt {\frac {c d+c e x}{c d+e}} \sqrt {1-c^2 x^2} \Pi \left (2;\text {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 (-16 c^3 d^3 e-9 c d e^3\right ) \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} F\left (\text {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 (\text {ArcSin}\left (\sqrt {\frac {c d+c e x}{c d-e}}\right )|\frac {c d-e}{c d+e}\right )-e F\left (\text {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} \Pi \left (2;\text {ArcSin}\left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )\right )}{c d \sqrt {1-\frac {1}{c^2 x^2}} \sqrt {e+\frac {d}{x}} \sqrt {c x} \left (-2+c^2 x^2\right )}\right )}{105 e^4 \sqrt {d+e x}}\right )}{c^4} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

Integrate[(x^3*(a + b*ArcCsc[c*x]))/Sqrt[d + e*x],x]

[Out]

(a*d^4*Sqrt[1 + (e*x)/d]*Beta[-((e*x)/d), 4, 1/2])/(e^4*Sqrt[d + e*x]) + (b*(-((c*(e + d/x)*x*((-4*(16*c^2*d^2
+ 9*e^2)*Sqrt[1 - 1/(c^2*x^2)])/(105*e^3) + (32*c^3*d^3*ArcCsc[c*x])/(35*e^4) - (2*c^3*x^3*ArcCsc[c*x])/(7*e)
- (4*c^2*x^2*(e*Sqrt[1 - 1/(c^2*x^2)] - 3*c*d*ArcCsc[c*x]))/(35*e^2) + (4*c*x*(5*c*d*e*Sqrt[1 - 1/(c^2*x^2)]
- 12*c^2*d^2*ArcCsc[c*x]))/(105*e^3)))/Sqrt[d + e*x]) + (2*Sqrt[e + d/x]*Sqrt[c*x]*((2*(40*c^3*d^3*e + 8*c*d*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*(48*c^4*d^4 + 16*c^2*d^2*e^2 + 9*e^4)*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]*(c*x)^(3/2)) + (2*(-16*c^3*d^3*e - 9*c*d*e^3)*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)*Sqrt[(e - c*e*x)/(c*d + e)]*Sqrt[(c*d + c*e*x)/(c*d - e)]*((c*d + e)*EllipticE[A
rcSin[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)]))/(c*d*Sqrt[1 - 1/(c^2*x^2)]*Sqrt[e + d/x]*S
qrt[c*x]*(-2 + c^2*x^2))))/(105*e^4*Sqrt[d + e*x])))/c^4

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Maple [A]
time = 0.62, size = 1233, normalized size = 1.73 Too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*(a+b*arccsc(c*x))/(e*x+d)^(1/2),x,method=_RETURNVERBOSE)

[Out]

2/e^4*(-a*(-1/7*(e*x+d)^(7/2)+3/5*d*(e*x+d)^(5/2)-d^2*(e*x+d)^(3/2)+d^3*(e*x+d)^(1/2))-b*(-1/7*arccsc(c*x)*(e*
x+d)^(7/2)+3/5*arccsc(c*x)*d*(e*x+d)^(5/2)-arccsc(c*x)*d^2*(e*x+d)^(3/2)+arccsc(c*x)*d^3*(e*x+d)^(1/2)+2/105/c
^4*(-3*(c/(c*d-e))^(1/2)*c^3*(e*x+d)^(7/2)+14*(c/(c*d-e))^(1/2)*c^3*d*(e*x+d)^(5/2)-19*(c/(c*d-e))^(1/2)*c^3*d
^2*(e*x+d)^(3/2)+16*EllipticE((e*x+d)^(1/2)*(c/(c*d-e))^(1/2),((c*d-e)/(c*d+e))^(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)*c^3*d^3-48*d^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)*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^3+24*((-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^3*d^3+8*(c/(c*d-e))^(1/2)*c^3*d^3*(e*x+d)^(1/2)+16*EllipticE((e*x+d)
^(1/2)*(c/(c*d-e))^(1/2),((c*d-e)/(c*d+e))^(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)*c^2*d^2*e-16*((-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+3*(c/(c*d-e))^(1/2)*c*e^2*(e*x+d)^(3/2)+9*Elliptic
E((e*x+d)^(1/2)*(c/(c*d-e))^(1/2),((c*d-e)/(c*d+e))^(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)*c*d*e^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^2-8*(c/(c*d-e))^(1/2)*c*d*e^2*(e*x+d)^(1/2)+9*Elli
pticE((e*x+d)^(1/2)*(c/(c*d-e))^(1/2),((c*d-e)/(c*d+e))^(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)*e^3-9*((-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^3)/(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)))

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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(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
more details

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Fricas [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(a+b*arccsc(c*x))/(e*x+d)^(1/2),x, algorithm="fricas")

[Out]

Timed out

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**3*(a+b*acsc(c*x))/(e*x+d)**(1/2),x)

[Out]

Timed out

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(a+b*arccsc(c*x))/(e*x+d)^(1/2),x, algorithm="giac")

[Out]

integrate((b*arccsc(c*x) + a)*x^3/sqrt(e*x + d), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {x^3\,\left (a+b\,\mathrm {asin}\left (\frac {1}{c\,x}\right )\right )}{\sqrt {d+e\,x}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^3*(a + b*asin(1/(c*x))))/(d + e*x)^(1/2),x)

[Out]

int((x^3*(a + b*asin(1/(c*x))))/(d + e*x)^(1/2), x)

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