3.69 \(\int \frac {x^3 (a+b \csc ^{-1}(c x))}{(d+e x)^{5/2}} \, dx\)

Optimal. Leaf size=602 \[ \frac {2 d^3 \left (a+b \csc ^{-1}(c x)\right )}{3 e^4 (d+e x)^{3/2}}-\frac {6 d^2 \left (a+b \csc ^{-1}(c x)\right )}{e^4 \sqrt {d+e x}}-\frac {6 d \sqrt {d+e x} \left (a+b \csc ^{-1}(c x)\right )}{e^4}+\frac {2 (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e^4}+\frac {64 b d^2 \sqrt {1-c^2 x^2} \sqrt {\frac {c (d+e x)}{c d+e}} \Pi \left (2;\sin ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{3 c e^4 x \sqrt {1-\frac {1}{c^2 x^2}} \sqrt {d+e x}}-\frac {4 b d^2 \left (1-c^2 x^2\right )}{3 c e^2 x \sqrt {1-\frac {1}{c^2 x^2}} \left (c^2 d^2-e^2\right ) \sqrt {d+e x}}+\frac {8 b d^2 \sqrt {1-c^2 x^2} \sqrt {d+e x} E\left (\sin ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{3 e^3 x \sqrt {1-\frac {1}{c^2 x^2}} \left (c^2 d^2-e^2\right ) \sqrt {\frac {c (d+e x)}{c d+e}}}-\frac {4 b \sqrt {1-c^2 x^2} \left (2 c^2 d^2-e^2\right ) \sqrt {d+e x} E\left (\sin ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{3 c^2 e^3 x \sqrt {1-\frac {1}{c^2 x^2}} \left (c^2 d^2-e^2\right ) \sqrt {\frac {c (d+e x)}{c d+e}}}+\frac {32 b d \sqrt {1-c^2 x^2} \sqrt {\frac {c (d+e x)}{c d+e}} F\left (\sin ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{3 c^2 e^3 x \sqrt {1-\frac {1}{c^2 x^2}} \sqrt {d+e x}} \]

[Out]

2/3*d^3*(a+b*arccsc(c*x))/e^4/(e*x+d)^(3/2)+2/3*(e*x+d)^(3/2)*(a+b*arccsc(c*x))/e^4-6*d^2*(a+b*arccsc(c*x))/e^
4/(e*x+d)^(1/2)-4/3*b*d^2*(-c^2*x^2+1)/c/e^2/(c^2*d^2-e^2)/x/(1-1/c^2/x^2)^(1/2)/(e*x+d)^(1/2)-6*d*(a+b*arccsc
(c*x))*(e*x+d)^(1/2)/e^4+8/3*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)/e^3/(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*(2*c^2*d^2-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^2/e^3/(c^2*
d^2-e^2)/x/(1-1/c^2/x^2)^(1/2)/(c*(e*x+d)/(c*d+e))^(1/2)+32/3*b*d*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)+6
4/3*b*d^2*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 = 2.90, antiderivative size = 602, normalized size of antiderivative = 1.00, number of steps used = 31, number of rules used = 18, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.857, Rules used = {43, 5247, 12, 6721, 6742, 745, 21, 719, 424, 958, 932, 168, 538, 537, 835, 844, 419, 1651} \[ \frac {2 d^3 \left (a+b \csc ^{-1}(c x)\right )}{3 e^4 (d+e x)^{3/2}}-\frac {6 d^2 \left (a+b \csc ^{-1}(c x)\right )}{e^4 \sqrt {d+e x}}-\frac {6 d \sqrt {d+e x} \left (a+b \csc ^{-1}(c x)\right )}{e^4}+\frac {2 (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e^4}-\frac {4 b d^2 \left (1-c^2 x^2\right )}{3 c e^2 x \sqrt {1-\frac {1}{c^2 x^2}} \left (c^2 d^2-e^2\right ) \sqrt {d+e x}}+\frac {8 b d^2 \sqrt {1-c^2 x^2} \sqrt {d+e x} E\left (\sin ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{3 e^3 x \sqrt {1-\frac {1}{c^2 x^2}} \left (c^2 d^2-e^2\right ) \sqrt {\frac {c (d+e x)}{c d+e}}}-\frac {4 b \sqrt {1-c^2 x^2} \left (2 c^2 d^2-e^2\right ) \sqrt {d+e x} E\left (\sin ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{3 c^2 e^3 x \sqrt {1-\frac {1}{c^2 x^2}} \left (c^2 d^2-e^2\right ) \sqrt {\frac {c (d+e x)}{c d+e}}}+\frac {64 b d^2 \sqrt {1-c^2 x^2} \sqrt {\frac {c (d+e x)}{c d+e}} \Pi \left (2;\sin ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{3 c e^4 x \sqrt {1-\frac {1}{c^2 x^2}} \sqrt {d+e x}}+\frac {32 b d \sqrt {1-c^2 x^2} \sqrt {\frac {c (d+e x)}{c d+e}} F\left (\sin ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{3 c^2 e^3 x \sqrt {1-\frac {1}{c^2 x^2}} \sqrt {d+e x}} \]

Antiderivative was successfully verified.

[In]

Int[(x^3*(a + b*ArcCsc[c*x]))/(d + e*x)^(5/2),x]

[Out]

(-4*b*d^2*(1 - c^2*x^2))/(3*c*e^2*(c^2*d^2 - e^2)*Sqrt[1 - 1/(c^2*x^2)]*x*Sqrt[d + e*x]) + (2*d^3*(a + b*ArcCs
c[c*x]))/(3*e^4*(d + e*x)^(3/2)) - (6*d^2*(a + b*ArcCsc[c*x]))/(e^4*Sqrt[d + e*x]) - (6*d*Sqrt[d + e*x]*(a + b
*ArcCsc[c*x]))/e^4 + (2*(d + e*x)^(3/2)*(a + b*ArcCsc[c*x]))/(3*e^4) + (8*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)])/(3*e^3*(c^2*d^2 - e^2)*Sqrt[1 - 1/(c^2*x^2)]*x*Sq
rt[(c*(d + e*x))/(c*d + e)]) - (4*b*(2*c^2*d^2 - 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)])/(3*c^2*e^3*(c^2*d^2 - e^2)*Sqrt[1 - 1/(c^2*x^2)]*x*Sqrt[(c*(d + e*x))/(c*d
+ e)]) + (32*b*d*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)])/(3*c^2*e^3*Sqrt[1 - 1/(c^2*x^2)]*x*Sqrt[d + e*x]) + (64*b*d^2*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*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 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 43

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 168

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 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)])

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 537

Int[1/(((a_) + (b_.)*(x_)^2)*Sqrt[(c_) + (d_.)*(x_)^2]*Sqrt[(e_) + (f_.)*(x_)^2]), x_Symbol] :> Simp[(1*Ellipt
icPi[(b*c)/(a*d), ArcSin[Rt[-(d/c), 2]*x], (c*f)/(d*e)])/(a*Sqrt[c]*Sqrt[e]*Rt[-(d/c), 2]), 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 538

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

Rule 719

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 745

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 835

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[((e*f - d*g)
*(d + e*x)^(m + 1)*(a + c*x^2)^(p + 1))/((m + 1)*(c*d^2 + a*e^2)), x] + Dist[1/((m + 1)*(c*d^2 + a*e^2)), Int[
(d + e*x)^(m + 1)*(a + c*x^2)^p*Simp[(c*d*f + a*e*g)*(m + 1) - c*(e*f - d*g)*(m + 2*p + 3)*x, x], x], x] /; Fr
eeQ[{a, c, d, e, f, g, p}, x] && NeQ[c*d^2 + a*e^2, 0] && LtQ[m, -1] && (IntegerQ[m] || IntegerQ[p] || Integer
sQ[2*m, 2*p])

Rule 844

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 932

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[((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 1651

Int[(Pq_)*((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = PolynomialQuotient[Pq, d
+ e*x, x], R = PolynomialRemainder[Pq, d + e*x, x]}, Simp[(e*R*(d + e*x)^(m + 1)*(a + c*x^2)^(p + 1))/((m + 1
)*(c*d^2 + a*e^2)), x] + Dist[1/((m + 1)*(c*d^2 + a*e^2)), Int[(d + e*x)^(m + 1)*(a + c*x^2)^p*ExpandToSum[(m
+ 1)*(c*d^2 + a*e^2)*Q + c*d*R*(m + 1) - c*e*R*(m + 2*p + 3)*x, x], x], x]] /; FreeQ[{a, c, d, e, p}, x] && Po
lyQ[Pq, x] && NeQ[c*d^2 + a*e^2, 0] && LtQ[m, -1]

Rule 5247

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 6721

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

Rule 6742

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 )}{(d+e x)^{5/2}} \, dx &=\frac {2 d^3 \left (a+b \csc ^{-1}(c x)\right )}{3 e^4 (d+e x)^{3/2}}-\frac {6 d^2 \left (a+b \csc ^{-1}(c x)\right )}{e^4 \sqrt {d+e x}}-\frac {6 d \sqrt {d+e x} \left (a+b \csc ^{-1}(c x)\right )}{e^4}+\frac {2 (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e^4}+\frac {b \int \frac {2 \left (-16 d^3-24 d^2 e x-6 d e^2 x^2+e^3 x^3\right )}{3 e^4 \sqrt {1-\frac {1}{c^2 x^2}} x^2 (d+e x)^{3/2}} \, dx}{c}\\ &=\frac {2 d^3 \left (a+b \csc ^{-1}(c x)\right )}{3 e^4 (d+e x)^{3/2}}-\frac {6 d^2 \left (a+b \csc ^{-1}(c x)\right )}{e^4 \sqrt {d+e x}}-\frac {6 d \sqrt {d+e x} \left (a+b \csc ^{-1}(c x)\right )}{e^4}+\frac {2 (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e^4}+\frac {(2 b) \int \frac {-16 d^3-24 d^2 e x-6 d e^2 x^2+e^3 x^3}{\sqrt {1-\frac {1}{c^2 x^2}} x^2 (d+e x)^{3/2}} \, dx}{3 c e^4}\\ &=\frac {2 d^3 \left (a+b \csc ^{-1}(c x)\right )}{3 e^4 (d+e x)^{3/2}}-\frac {6 d^2 \left (a+b \csc ^{-1}(c x)\right )}{e^4 \sqrt {d+e x}}-\frac {6 d \sqrt {d+e x} \left (a+b \csc ^{-1}(c x)\right )}{e^4}+\frac {2 (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e^4}+\frac {\left (2 b \sqrt {1-c^2 x^2}\right ) \int \frac {-16 d^3-24 d^2 e x-6 d e^2 x^2+e^3 x^3}{x (d+e x)^{3/2} \sqrt {1-c^2 x^2}} \, dx}{3 c e^4 \sqrt {1-\frac {1}{c^2 x^2}} x}\\ &=\frac {2 d^3 \left (a+b \csc ^{-1}(c x)\right )}{3 e^4 (d+e x)^{3/2}}-\frac {6 d^2 \left (a+b \csc ^{-1}(c x)\right )}{e^4 \sqrt {d+e x}}-\frac {6 d \sqrt {d+e x} \left (a+b \csc ^{-1}(c x)\right )}{e^4}+\frac {2 (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e^4}+\frac {\left (2 b \sqrt {1-c^2 x^2}\right ) \int \left (-\frac {24 d^2 e}{(d+e x)^{3/2} \sqrt {1-c^2 x^2}}-\frac {16 d^3}{x (d+e x)^{3/2} \sqrt {1-c^2 x^2}}-\frac {6 d e^2 x}{(d+e x)^{3/2} \sqrt {1-c^2 x^2}}+\frac {e^3 x^2}{(d+e x)^{3/2} \sqrt {1-c^2 x^2}}\right ) \, dx}{3 c e^4 \sqrt {1-\frac {1}{c^2 x^2}} x}\\ &=\frac {2 d^3 \left (a+b \csc ^{-1}(c x)\right )}{3 e^4 (d+e x)^{3/2}}-\frac {6 d^2 \left (a+b \csc ^{-1}(c x)\right )}{e^4 \sqrt {d+e x}}-\frac {6 d \sqrt {d+e x} \left (a+b \csc ^{-1}(c x)\right )}{e^4}+\frac {2 (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e^4}-\frac {\left (32 b d^3 \sqrt {1-c^2 x^2}\right ) \int \frac {1}{x (d+e x)^{3/2} \sqrt {1-c^2 x^2}} \, dx}{3 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 {1}{(d+e x)^{3/2} \sqrt {1-c^2 x^2}} \, dx}{c e^3 \sqrt {1-\frac {1}{c^2 x^2}} x}-\frac {\left (4 b d \sqrt {1-c^2 x^2}\right ) \int \frac {x}{(d+e x)^{3/2} \sqrt {1-c^2 x^2}} \, dx}{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}{(d+e x)^{3/2} \sqrt {1-c^2 x^2}} \, dx}{3 c e \sqrt {1-\frac {1}{c^2 x^2}} x}\\ &=-\frac {68 b d^2 \left (1-c^2 x^2\right )}{3 c e^2 \left (c^2 d^2-e^2\right ) \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {d+e x}}+\frac {2 d^3 \left (a+b \csc ^{-1}(c x)\right )}{3 e^4 (d+e x)^{3/2}}-\frac {6 d^2 \left (a+b \csc ^{-1}(c x)\right )}{e^4 \sqrt {d+e x}}-\frac {6 d \sqrt {d+e x} \left (a+b \csc ^{-1}(c x)\right )}{e^4}+\frac {2 (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e^4}-\frac {\left (32 b d^3 \sqrt {1-c^2 x^2}\right ) \int \left (-\frac {e}{d (d+e x)^{3/2} \sqrt {1-c^2 x^2}}+\frac {1}{d x \sqrt {d+e x} \sqrt {1-c^2 x^2}}\right ) \, dx}{3 c e^4 \sqrt {1-\frac {1}{c^2 x^2}} x}+\frac {\left (32 b c d^2 \sqrt {1-c^2 x^2}\right ) \int \frac {-\frac {d}{2}-\frac {e x}{2}}{\sqrt {d+e x} \sqrt {1-c^2 x^2}} \, dx}{e^3 \left (c^2 d^2-e^2\right ) \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}{c e^2 \left (c^2 d^2-e^2\right ) \sqrt {1-\frac {1}{c^2 x^2}} x}+\frac {\left (4 b \sqrt {1-c^2 x^2}\right ) \int \frac {\frac {d}{2}+\frac {1}{2} \left (\frac {2 c^2 d^2}{e}-e\right ) x}{\sqrt {d+e x} \sqrt {1-c^2 x^2}} \, dx}{3 c e \left (c^2 d^2-e^2\right ) \sqrt {1-\frac {1}{c^2 x^2}} x}\\ &=-\frac {68 b d^2 \left (1-c^2 x^2\right )}{3 c e^2 \left (c^2 d^2-e^2\right ) \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {d+e x}}+\frac {2 d^3 \left (a+b \csc ^{-1}(c x)\right )}{3 e^4 (d+e x)^{3/2}}-\frac {6 d^2 \left (a+b \csc ^{-1}(c x)\right )}{e^4 \sqrt {d+e x}}-\frac {6 d \sqrt {d+e x} \left (a+b \csc ^{-1}(c x)\right )}{e^4}+\frac {2 (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e^4}-\frac {\left (32 b d^2 \sqrt {1-c^2 x^2}\right ) \int \frac {1}{x \sqrt {d+e x} \sqrt {1-c^2 x^2}} \, dx}{3 c e^4 \sqrt {1-\frac {1}{c^2 x^2}} x}+\frac {\left (32 b d^2 \sqrt {1-c^2 x^2}\right ) \int \frac {1}{(d+e x)^{3/2} \sqrt {1-c^2 x^2}} \, dx}{3 c e^3 \sqrt {1-\frac {1}{c^2 x^2}} x}+\frac {\left (4 b c d^2 \sqrt {1-c^2 x^2}\right ) \int \frac {\sqrt {d+e x}}{\sqrt {1-c^2 x^2}} \, dx}{e^3 \left (c^2 d^2-e^2\right ) \sqrt {1-\frac {1}{c^2 x^2}} x}-\frac {\left (16 b c d^2 \sqrt {1-c^2 x^2}\right ) \int \frac {\sqrt {d+e x}}{\sqrt {1-c^2 x^2}} \, dx}{e^3 \left (c^2 d^2-e^2\right ) \sqrt {1-\frac {1}{c^2 x^2}} x}+\frac {\left (4 b d \left (1-\frac {c^2 d^2}{e^2}\right ) \sqrt {1-c^2 x^2}\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {1-c^2 x^2}} \, dx}{3 c e \left (c^2 d^2-e^2\right ) \sqrt {1-\frac {1}{c^2 x^2}} x}+\frac {\left (2 b \left (-1+\frac {2 c^2 d^2}{e^2}\right ) \sqrt {1-c^2 x^2}\right ) \int \frac {\sqrt {d+e x}}{\sqrt {1-c^2 x^2}} \, dx}{3 c e \left (c^2 d^2-e^2\right ) \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}{c e^3 \left (c^2 d^2-e^2\right ) \sqrt {1-\frac {1}{c^2 x^2}} x}\\ &=-\frac {4 b d^2 \left (1-c^2 x^2\right )}{3 c e^2 \left (c^2 d^2-e^2\right ) \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {d+e x}}+\frac {2 d^3 \left (a+b \csc ^{-1}(c x)\right )}{3 e^4 (d+e x)^{3/2}}-\frac {6 d^2 \left (a+b \csc ^{-1}(c x)\right )}{e^4 \sqrt {d+e x}}-\frac {6 d \sqrt {d+e x} \left (a+b \csc ^{-1}(c x)\right )}{e^4}+\frac {2 (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e^4}-\frac {\left (32 b d^2 \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 e^4 \sqrt {1-\frac {1}{c^2 x^2}} x}-\frac {\left (64 b c d^2 \sqrt {1-c^2 x^2}\right ) \int \frac {-\frac {d}{2}-\frac {e x}{2}}{\sqrt {d+e x} \sqrt {1-c^2 x^2}} \, dx}{3 e^3 \left (c^2 d^2-e^2\right ) \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 ) \operatorname {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 )}{e^3 \left (c^2 d^2-e^2\right ) \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {-\frac {c^2 (d+e x)}{-c^2 d-c e}}}+\frac {\left (32 b d^2 \sqrt {d+e x} \sqrt {1-c^2 x^2}\right ) \operatorname {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 )}{e^3 \left (c^2 d^2-e^2\right ) \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 \left (-1+\frac {2 c^2 d^2}{e^2}\right ) \sqrt {d+e x} \sqrt {1-c^2 x^2}\right ) \operatorname {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 )}{3 c^2 e \left (c^2 d^2-e^2\right ) \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 \left (1-\frac {c^2 d^2}{e^2}\right ) \sqrt {-\frac {c^2 (d+e x)}{-c^2 d-c e}} \sqrt {1-c^2 x^2}\right ) \operatorname {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 )}{3 c^2 e \left (c^2 d^2-e^2\right ) \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {d+e x}}+\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 ) \operatorname {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 )}{c^2 e^3 \left (c^2 d^2-e^2\right ) \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {d+e x}}\\ &=-\frac {4 b d^2 \left (1-c^2 x^2\right )}{3 c e^2 \left (c^2 d^2-e^2\right ) \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {d+e x}}+\frac {2 d^3 \left (a+b \csc ^{-1}(c x)\right )}{3 e^4 (d+e x)^{3/2}}-\frac {6 d^2 \left (a+b \csc ^{-1}(c x)\right )}{e^4 \sqrt {d+e x}}-\frac {6 d \sqrt {d+e x} \left (a+b \csc ^{-1}(c x)\right )}{e^4}+\frac {2 (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 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 )}{e^3 \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 \left (2 c^2 d^2-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 )}{3 c^2 e^3 \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 {32 b d \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 )}{3 c^2 e^3 \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {d+e x}}+\frac {\left (64 b d^2 \sqrt {1-c^2 x^2}\right ) \operatorname {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 e^4 \sqrt {1-\frac {1}{c^2 x^2}} x}+\frac {\left (32 b c d^2 \sqrt {1-c^2 x^2}\right ) \int \frac {\sqrt {d+e x}}{\sqrt {1-c^2 x^2}} \, dx}{3 e^3 \left (c^2 d^2-e^2\right ) \sqrt {1-\frac {1}{c^2 x^2}} x}\\ &=-\frac {4 b d^2 \left (1-c^2 x^2\right )}{3 c e^2 \left (c^2 d^2-e^2\right ) \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {d+e x}}+\frac {2 d^3 \left (a+b \csc ^{-1}(c x)\right )}{3 e^4 (d+e x)^{3/2}}-\frac {6 d^2 \left (a+b \csc ^{-1}(c x)\right )}{e^4 \sqrt {d+e x}}-\frac {6 d \sqrt {d+e x} \left (a+b \csc ^{-1}(c x)\right )}{e^4}+\frac {2 (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 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 )}{e^3 \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 \left (2 c^2 d^2-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 )}{3 c^2 e^3 \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 {32 b d \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 )}{3 c^2 e^3 \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {d+e x}}+\frac {\left (64 b d^2 \sqrt {\frac {c (d+e x)}{c d+e}} \sqrt {1-c^2 x^2}\right ) \operatorname {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 e^4 \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {d+e x}}-\frac {\left (64 b d^2 \sqrt {d+e x} \sqrt {1-c^2 x^2}\right ) \operatorname {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 )}{3 e^3 \left (c^2 d^2-e^2\right ) \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {-\frac {c^2 (d+e x)}{-c^2 d-c e}}}\\ &=-\frac {4 b d^2 \left (1-c^2 x^2\right )}{3 c e^2 \left (c^2 d^2-e^2\right ) \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {d+e x}}+\frac {2 d^3 \left (a+b \csc ^{-1}(c x)\right )}{3 e^4 (d+e x)^{3/2}}-\frac {6 d^2 \left (a+b \csc ^{-1}(c x)\right )}{e^4 \sqrt {d+e x}}-\frac {6 d \sqrt {d+e x} \left (a+b \csc ^{-1}(c x)\right )}{e^4}+\frac {2 (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e^4}+\frac {8 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 )}{3 e^3 \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 \left (2 c^2 d^2-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 )}{3 c^2 e^3 \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 {32 b d \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 )}{3 c^2 e^3 \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {d+e x}}+\frac {64 b d^2 \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 )}{3 c e^4 \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {d+e x}}\\ \end {align*}

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Mathematica [C]  time = 14.10, size = 887, normalized size = 1.47 \[ \frac {a \left (\frac {e x}{d}+1\right )^{5/2} B_{-\frac {e x}{d}}\left (4,-\frac {3}{2}\right ) d^4}{e^4 (d+e x)^{5/2}}+\frac {b \left (\frac {2 \left (\frac {d}{x}+e\right )^{5/2} (c x)^{5/2} \left (\frac {2 \cos \left (2 \csc ^{-1}(c x)\right ) \left (d x \sqrt {\frac {c d+c 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 ) c^2-\frac {x (c x+1) \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 (\sin ^{-1}\left (\sqrt {\frac {c d+c e x}{c d-e}}\right )|\frac {c d-e}{c d+e}\right )-e F\left (\sin ^{-1}\left (\sqrt {\frac {c d+c e x}{c d-e}}\right )|\frac {c d-e}{c d+e}\right )\right ) c}{\sqrt {\frac {e (c x+1)}{e-c d}}}+e x \sqrt {\frac {c d+c 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 ) c+(c d+c e x) \left (c^2 x^2-1\right )\right ) e^3}{\sqrt {1-\frac {1}{c^2 x^2}} \sqrt {\frac {d}{x}+e} \sqrt {c x} \left (c^2 x^2-2\right )}+\frac {2 \left (8 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 (\sin ^{-1}\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 {\frac {d}{x}+e} (c x)^{3/2}}+\frac {2 \left (16 c^4 d^4-16 c^2 e^2 d^2-e^4\right ) \sqrt {\frac {c d+c 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 )}{\sqrt {1-\frac {1}{c^2 x^2}} \sqrt {\frac {d}{x}+e} (c x)^{3/2}}\right )}{3 (c d-e) e^4 (c d+e) (d+e x)^{5/2}}-\frac {c^3 \left (\frac {d}{x}+e\right )^3 x^3 \left (-\frac {2 c x \csc ^{-1}(c x)}{3 e^3}-\frac {2 c d \csc ^{-1}(c x)}{3 e^2 \left (\frac {d}{x}+e\right )^2}+\frac {32 c d \csc ^{-1}(c x)}{3 e^4}-\frac {2 \left (-7 c^3 \csc ^{-1}(c x) d^3-2 c^2 e \sqrt {1-\frac {1}{c^2 x^2}} d^2+7 c e^2 \csc ^{-1}(c x) d\right )}{3 e^3 \left (e^2-c^2 d^2\right ) \left (\frac {d}{x}+e\right )}-\frac {4 \sqrt {1-\frac {1}{c^2 x^2}}}{3 e \left (e^2-c^2 d^2\right )}\right )}{(d+e x)^{5/2}}\right )}{c^4} \]

Antiderivative was successfully verified.

[In]

Integrate[(x^3*(a + b*ArcCsc[c*x]))/(d + e*x)^(5/2),x]

[Out]

(a*d^4*(1 + (e*x)/d)^(5/2)*Beta[-((e*x)/d), 4, -3/2])/(e^4*(d + e*x)^(5/2)) + (b*(-((c^3*(e + d/x)^3*x^3*((-4*
Sqrt[1 - 1/(c^2*x^2)])/(3*e*(-(c^2*d^2) + e^2)) + (32*c*d*ArcCsc[c*x])/(3*e^4) - (2*c*d*ArcCsc[c*x])/(3*e^2*(e
+ d/x)^2) - (2*c*x*ArcCsc[c*x])/(3*e^3) - (2*(-2*c^2*d^2*e*Sqrt[1 - 1/(c^2*x^2)] - 7*c^3*d^3*ArcCsc[c*x] + 7*
c*d*e^2*ArcCsc[c*x]))/(3*e^3*(-(c^2*d^2) + e^2)*(e + d/x))))/(d + e*x)^(5/2)) + (2*(e + d/x)^(5/2)*(c*x)^(5/2)
*((2*(8*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*(16*c^4*d^4 - 16*c^2*d^2*e^
2 - 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*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[
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))))/(3*(c*d - e)*e^4*(c*d + e)*(d + e*x)^(5/2))))/c^4

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fricas [F]  time = 1.49, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (b x^{3} \operatorname {arccsc}\left (c x\right ) + a x^{3}\right )} \sqrt {e x + d}}{e^{3} x^{3} + 3 \, d e^{2} x^{2} + 3 \, d^{2} e x + d^{3}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(a+b*arccsc(c*x))/(e*x+d)^(5/2),x, algorithm="fricas")

[Out]

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

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b \operatorname {arccsc}\left (c x\right ) + a\right )} x^{3}}{{\left (e x + d\right )}^{\frac {5}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(a+b*arccsc(c*x))/(e*x+d)^(5/2),x, algorithm="giac")

[Out]

integrate((b*arccsc(c*x) + a)*x^3/(e*x + d)^(5/2), x)

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maple [A]  time = 0.09, size = 1077, normalized size = 1.79 \[ \frac {2 a \left (\frac {\left (e x +d \right )^{\frac {3}{2}}}{3}-3 d \sqrt {e x +d}+\frac {d^{3}}{3 \left (e x +d \right )^{\frac {3}{2}}}-\frac {3 d^{2}}{\sqrt {e x +d}}\right )+2 b \left (\frac {\left (e x +d \right )^{\frac {3}{2}} \mathrm {arccsc}\left (c x \right )}{3}-3 \,\mathrm {arccsc}\left (c x \right ) d \sqrt {e x +d}+\frac {\mathrm {arccsc}\left (c x \right ) d^{3}}{3 \left (e x +d \right )^{\frac {3}{2}}}-\frac {3 \,\mathrm {arccsc}\left (c x \right ) d^{2}}{\sqrt {e x +d}}+\frac {\frac {2 \sqrt {\frac {c}{d c -e}}\, \left (e x +d \right )^{2} c^{3} d^{2}}{3}-\frac {16 \sqrt {-\frac {\left (e x +d \right ) c -d c +e}{d c -e}}\, \sqrt {-\frac {\left (e x +d \right ) c -d c -e}{d c +e}}\, \EllipticF \left (\sqrt {e x +d}\, \sqrt {\frac {c}{d c -e}}, \sqrt {\frac {d c -e}{d c +e}}\right ) \sqrt {e x +d}\, c^{3} d^{3}}{3}+\frac {32 \sqrt {-\frac {\left (e x +d \right ) c -d c +e}{d c -e}}\, \sqrt {-\frac {\left (e x +d \right ) c -d c -e}{d c +e}}\, \EllipticPi \left (\sqrt {e x +d}\, \sqrt {\frac {c}{d c -e}}, \frac {d c -e}{c d}, \frac {\sqrt {\frac {c}{d c +e}}}{\sqrt {\frac {c}{d c -e}}}\right ) \sqrt {e x +d}\, c^{3} d^{3}}{3}-\frac {4 \sqrt {\frac {c}{d c -e}}\, \left (e x +d \right ) c^{3} d^{3}}{3}+\frac {2 \sqrt {\frac {c}{d c -e}}\, c^{3} d^{4}}{3}+\frac {14 \sqrt {-\frac {\left (e x +d \right ) c -d c +e}{d c -e}}\, \sqrt {-\frac {\left (e x +d \right ) c -d c -e}{d c +e}}\, \EllipticF \left (\sqrt {e x +d}\, \sqrt {\frac {c}{d c -e}}, \sqrt {\frac {d c -e}{d c +e}}\right ) \sqrt {e x +d}\, c d \,e^{2}}{3}+\frac {2 \sqrt {-\frac {\left (e x +d \right ) c -d c +e}{d c -e}}\, \sqrt {-\frac {\left (e x +d \right ) c -d c -e}{d c +e}}\, \EllipticE \left (\sqrt {e x +d}\, \sqrt {\frac {c}{d c -e}}, \sqrt {\frac {d c -e}{d c +e}}\right ) \sqrt {e x +d}\, c d \,e^{2}}{3}-\frac {32 \sqrt {-\frac {\left (e x +d \right ) c -d c +e}{d c -e}}\, \sqrt {-\frac {\left (e x +d \right ) c -d c -e}{d c +e}}\, \EllipticPi \left (\sqrt {e x +d}\, \sqrt {\frac {c}{d c -e}}, \frac {d c -e}{c d}, \frac {\sqrt {\frac {c}{d c +e}}}{\sqrt {\frac {c}{d c -e}}}\right ) \sqrt {e x +d}\, c d \,e^{2}}{3}-\frac {2 \sqrt {-\frac {\left (e x +d \right ) c -d c +e}{d c -e}}\, \sqrt {-\frac {\left (e x +d \right ) c -d c -e}{d c +e}}\, \EllipticF \left (\sqrt {e x +d}\, \sqrt {\frac {c}{d c -e}}, \sqrt {\frac {d c -e}{d c +e}}\right ) \sqrt {e x +d}\, e^{3}}{3}+\frac {2 \sqrt {-\frac {\left (e x +d \right ) c -d c +e}{d c -e}}\, \sqrt {-\frac {\left (e x +d \right ) c -d c -e}{d c +e}}\, \EllipticE \left (\sqrt {e x +d}\, \sqrt {\frac {c}{d c -e}}, \sqrt {\frac {d c -e}{d c +e}}\right ) \sqrt {e x +d}\, e^{3}}{3}-\frac {2 \sqrt {\frac {c}{d c -e}}\, c \,d^{2} e^{2}}{3}}{c^{2} \left (d c -e \right ) \sqrt {\frac {c}{d c -e}}\, \sqrt {e x +d}\, \left (d c +e \right ) 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^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*(a+b*arccsc(c*x))/(e*x+d)^(5/2),x)

[Out]

2/e^4*(a*(1/3*(e*x+d)^(3/2)-3*d*(e*x+d)^(1/2)+1/3*d^3/(e*x+d)^(3/2)-3*d^2/(e*x+d)^(1/2))+b*(1/3*(e*x+d)^(3/2)*
arccsc(c*x)-3*arccsc(c*x)*d*(e*x+d)^(1/2)+1/3*arccsc(c*x)*d^3/(e*x+d)^(3/2)-3*arccsc(c*x)*d^2/(e*x+d)^(1/2)+2/
3/c^2*((c/(c*d-e))^(1/2)*(e*x+d)^2*c^3*d^2-8*(-((e*x+d)*c-d*c+e)/(c*d-e))^(1/2)*(-((e*x+d)*c-d*c-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*x+d)^(1/2)*c^3*d^3+16*(-((e*x+d)*c-
d*c+e)/(c*d-e))^(1/2)*(-((e*x+d)*c-d*c-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*x+d)^(1/2)*c^3*d^3-2*(c/(c*d-e))^(1/2)*(e*x+d)*c^3*d^3+(c/(c*d-e))
^(1/2)*c^3*d^4+7*(-((e*x+d)*c-d*c+e)/(c*d-e))^(1/2)*(-((e*x+d)*c-d*c-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*x+d)^(1/2)*c*d*e^2+(-((e*x+d)*c-d*c+e)/(c*d-e))^(1/2)*(-((e*x+d
)*c-d*c-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))*(e*x+d)^(1/2)*c*d
*e^2-16*(-((e*x+d)*c-d*c+e)/(c*d-e))^(1/2)*(-((e*x+d)*c-d*c-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*x+d)^(1/2)*c*d*e^2-(-((e*x+d)*c-d*c+e)/(c*d-e
))^(1/2)*(-((e*x+d)*c-d*c-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*x+d)^(1/2)*e^3+(-((e*x+d)*c-d*c+e)/(c*d-e))^(1/2)*(-((e*x+d)*c-d*c-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))*(e*x+d)^(1/2)*e^3-(c/(c*d-e))^(1/2)*c*d^2*e^2)/(c*d-e)/(c/(c*d-e
))^(1/2)/(e*x+d)^(1/2)/(c*d+e)/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 \[ \text {Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(a+b*arccsc(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(c*d-e>0)', see `assume?` for m
ore details)Is c*d-e positive or negative?

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {x^3\,\left (a+b\,\mathrm {asin}\left (\frac {1}{c\,x}\right )\right )}{{\left (d+e\,x\right )}^{5/2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^3*(a + b*asin(1/(c*x))))/(d + e*x)^(5/2),x)

[Out]

int((x^3*(a + b*asin(1/(c*x))))/(d + e*x)^(5/2), x)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**3*(a+b*acsc(c*x))/(e*x+d)**(5/2),x)

[Out]

Timed out

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