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

Optimal. Leaf size=496 \[ \frac {2 d^2 (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e^3}-\frac {4 d (d+e x)^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{5 e^3}+\frac {2 (d+e x)^{7/2} \left (a+b \csc ^{-1}(c x)\right )}{7 e^3}-\frac {32 b d^4 \sqrt {1-c^2 x^2} \sqrt {\frac {c (d+e x)}{c d+e}} \Pi \left (2;\sin ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{105 c e^3 x \sqrt {1-\frac {1}{c^2 x^2}} \sqrt {d+e x}}-\frac {4 b d \sqrt {1-c^2 x^2} \left (9 c^2 d^2-e^2\right ) \sqrt {\frac {c (d+e x)}{c d+e}} F\left (\sin ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{105 c^4 e^2 x \sqrt {1-\frac {1}{c^2 x^2}} \sqrt {d+e x}}+\frac {4 b \sqrt {1-c^2 x^2} \left (5 c^2 d^2-9 e^2\right ) \sqrt {d+e x} E\left (\sin ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{105 c^4 e^2 x \sqrt {1-\frac {1}{c^2 x^2}} \sqrt {\frac {c (d+e x)}{c d+e}}}+\frac {4 b d \left (1-c^2 x^2\right ) \sqrt {d+e x}}{105 c^3 e x \sqrt {1-\frac {1}{c^2 x^2}}}-\frac {4 b \left (1-c^2 x^2\right ) (d+e x)^{3/2}}{35 c^3 e x \sqrt {1-\frac {1}{c^2 x^2}}} \]

[Out]

2/3*d^2*(e*x+d)^(3/2)*(a+b*arccsc(c*x))/e^3-4/5*d*(e*x+d)^(5/2)*(a+b*arccsc(c*x))/e^3+2/7*(e*x+d)^(7/2)*(a+b*a
rccsc(c*x))/e^3-4/35*b*(e*x+d)^(3/2)*(-c^2*x^2+1)/c^3/e/x/(1-1/c^2/x^2)^(1/2)+4/105*b*d*(-c^2*x^2+1)*(e*x+d)^(
1/2)/c^3/e/x/(1-1/c^2/x^2)^(1/2)+4/105*b*(5*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^2/x/(1-1/c^2/x^2)^(1/2)/(c*(e*x+d)/(c*d+e))^(1/2)-4/105*b*
d*(9*c^2*d^2-e^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^4/e^2/x/(1-1/c^2/x^2)^(1/2)/(e*x+d)^(1/2)-32/105*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^3/x/(1-1/c^2/x^2)^(1/2)/(e*x+d
)^(1/2)

________________________________________________________________________________________

Rubi [A]  time = 2.83, antiderivative size = 693, normalized size of antiderivative = 1.40, number of steps used = 31, number of rules used = 16, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.762, Rules used = {43, 5247, 12, 6721, 6742, 743, 844, 719, 424, 419, 958, 932, 168, 538, 537, 833} \[ \frac {2 d^2 (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e^3}-\frac {4 d (d+e x)^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{5 e^3}+\frac {2 (d+e x)^{7/2} \left (a+b \csc ^{-1}(c x)\right )}{7 e^3}-\frac {32 b d^3 \sqrt {1-c^2 x^2} \sqrt {\frac {c (d+e x)}{c d+e}} F\left (\sin ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{105 c^2 e^2 x \sqrt {1-\frac {1}{c^2 x^2}} \sqrt {d+e x}}+\frac {32 b d^2 \sqrt {1-c^2 x^2} \sqrt {d+e x} E\left (\sin ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{105 c^2 e^2 x \sqrt {1-\frac {1}{c^2 x^2}} \sqrt {\frac {c (d+e x)}{c d+e}}}-\frac {4 b \sqrt {1-c^2 x^2} \left (c^2 d^2+3 e^2\right ) \sqrt {d+e x} E\left (\sin ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{35 c^4 e^2 x \sqrt {1-\frac {1}{c^2 x^2}} \sqrt {\frac {c (d+e x)}{c d+e}}}-\frac {32 b d^4 \sqrt {1-c^2 x^2} \sqrt {\frac {c (d+e x)}{c d+e}} \Pi \left (2;\sin ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{105 c e^3 x \sqrt {1-\frac {1}{c^2 x^2}} \sqrt {d+e x}}-\frac {4 b d \sqrt {1-c^2 x^2} (c d-e) (c d+e) \sqrt {\frac {c (d+e x)}{c d+e}} F\left (\sin ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{105 c^4 e^2 x \sqrt {1-\frac {1}{c^2 x^2}} \sqrt {d+e x}}+\frac {4 b d \left (1-c^2 x^2\right ) \sqrt {d+e x}}{105 c^3 e x \sqrt {1-\frac {1}{c^2 x^2}}}-\frac {4 b \left (1-c^2 x^2\right ) (d+e x)^{3/2}}{35 c^3 e x \sqrt {1-\frac {1}{c^2 x^2}}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(4*b*d*Sqrt[d + e*x]*(1 - c^2*x^2))/(105*c^3*e*Sqrt[1 - 1/(c^2*x^2)]*x) - (4*b*(d + e*x)^(3/2)*(1 - c^2*x^2))/
(35*c^3*e*Sqrt[1 - 1/(c^2*x^2)]*x) + (2*d^2*(d + e*x)^(3/2)*(a + b*ArcCsc[c*x]))/(3*e^3) - (4*d*(d + e*x)^(5/2
)*(a + b*ArcCsc[c*x]))/(5*e^3) + (2*(d + e*x)^(7/2)*(a + b*ArcCsc[c*x]))/(7*e^3) + (32*b*d^2*Sqrt[d + e*x]*Sqr
t[1 - c^2*x^2]*EllipticE[ArcSin[Sqrt[1 - c*x]/Sqrt[2]], (2*e)/(c*d + e)])/(105*c^2*e^2*Sqrt[1 - 1/(c^2*x^2)]*x
*Sqrt[(c*(d + e*x))/(c*d + e)]) - (4*b*(c^2*d^2 + 3*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)])/(35*c^4*e^2*Sqrt[1 - 1/(c^2*x^2)]*x*Sqrt[(c*(d + e*x))/(c*d + e)]) - (32
*b*d^3*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^2*e^2*Sqrt[1 - 1/(c^2*x^2)]*x*Sqrt[d + e*x]) - (4*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^2*Sqrt[1 - 1/(c^2
*x^2)]*x*Sqrt[d + e*x]) - (32*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)])/(105*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 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 743

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))/(c*(m + 2*p + 1)), x] + Dist[1/(c*(m + 2*p + 1)), Int[(d + e*x)^(m - 2)*Simp[c*d^2*(m + 2*p + 1) - a*e^
2*(m - 1) + 2*c*d*e*(m + p)*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] && If[RationalQ[m], GtQ[m, 1], SumSimplerQ[m, -2]] && NeQ[m + 2*p + 1, 0] && IntQuadraticQ[a, 0, c, d, e, m
, p, x]

Rule 833

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 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 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 x^2 \sqrt {d+e x} \left (a+b \csc ^{-1}(c x)\right ) \, dx &=\frac {2 d^2 (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e^3}-\frac {4 d (d+e x)^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{5 e^3}+\frac {2 (d+e x)^{7/2} \left (a+b \csc ^{-1}(c x)\right )}{7 e^3}+\frac {b \int \frac {2 (d+e x)^{3/2} \left (8 d^2-12 d e x+15 e^2 x^2\right )}{105 e^3 \sqrt {1-\frac {1}{c^2 x^2}} x^2} \, dx}{c}\\ &=\frac {2 d^2 (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e^3}-\frac {4 d (d+e x)^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{5 e^3}+\frac {2 (d+e x)^{7/2} \left (a+b \csc ^{-1}(c x)\right )}{7 e^3}+\frac {(2 b) \int \frac {(d+e x)^{3/2} \left (8 d^2-12 d e x+15 e^2 x^2\right )}{\sqrt {1-\frac {1}{c^2 x^2}} x^2} \, dx}{105 c e^3}\\ &=\frac {2 d^2 (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e^3}-\frac {4 d (d+e x)^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{5 e^3}+\frac {2 (d+e x)^{7/2} \left (a+b \csc ^{-1}(c x)\right )}{7 e^3}+\frac {\left (2 b \sqrt {1-c^2 x^2}\right ) \int \frac {(d+e x)^{3/2} \left (8 d^2-12 d e x+15 e^2 x^2\right )}{x \sqrt {1-c^2 x^2}} \, dx}{105 c e^3 \sqrt {1-\frac {1}{c^2 x^2}} x}\\ &=\frac {2 d^2 (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e^3}-\frac {4 d (d+e x)^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{5 e^3}+\frac {2 (d+e x)^{7/2} \left (a+b \csc ^{-1}(c x)\right )}{7 e^3}+\frac {\left (2 b \sqrt {1-c^2 x^2}\right ) \int \left (-\frac {12 d e (d+e x)^{3/2}}{\sqrt {1-c^2 x^2}}+\frac {8 d^2 (d+e x)^{3/2}}{x \sqrt {1-c^2 x^2}}+\frac {15 e^2 x (d+e x)^{3/2}}{\sqrt {1-c^2 x^2}}\right ) \, dx}{105 c e^3 \sqrt {1-\frac {1}{c^2 x^2}} x}\\ &=\frac {2 d^2 (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e^3}-\frac {4 d (d+e x)^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{5 e^3}+\frac {2 (d+e x)^{7/2} \left (a+b \csc ^{-1}(c x)\right )}{7 e^3}+\frac {\left (16 b d^2 \sqrt {1-c^2 x^2}\right ) \int \frac {(d+e x)^{3/2}}{x \sqrt {1-c^2 x^2}} \, dx}{105 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 {(d+e x)^{3/2}}{\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 (d+e x)^{3/2}}{\sqrt {1-c^2 x^2}} \, dx}{7 c e \sqrt {1-\frac {1}{c^2 x^2}} x}\\ &=\frac {16 b d \sqrt {d+e x} \left (1-c^2 x^2\right )}{105 c^3 e \sqrt {1-\frac {1}{c^2 x^2}} x}-\frac {4 b (d+e x)^{3/2} \left (1-c^2 x^2\right )}{35 c^3 e \sqrt {1-\frac {1}{c^2 x^2}} x}+\frac {2 d^2 (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e^3}-\frac {4 d (d+e x)^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{5 e^3}+\frac {2 (d+e x)^{7/2} \left (a+b \csc ^{-1}(c x)\right )}{7 e^3}+\frac {\left (16 b d^2 \sqrt {1-c^2 x^2}\right ) \int \left (\frac {2 d e}{\sqrt {d+e x} \sqrt {1-c^2 x^2}}+\frac {d^2}{x \sqrt {d+e x} \sqrt {1-c^2 x^2}}+\frac {e^2 x}{\sqrt {d+e x} \sqrt {1-c^2 x^2}}\right ) \, dx}{105 c e^3 \sqrt {1-\frac {1}{c^2 x^2}} x}+\frac {\left (16 b d \sqrt {1-c^2 x^2}\right ) \int \frac {\frac {1}{2} \left (-3 c^2 d^2-e^2\right )-2 c^2 d e x}{\sqrt {d+e x} \sqrt {1-c^2 x^2}} \, dx}{105 c^3 e^2 \sqrt {1-\frac {1}{c^2 x^2}} x}-\frac {\left (4 b \sqrt {1-c^2 x^2}\right ) \int \frac {\left (-\frac {3 e}{2}-\frac {3}{2} c^2 d x\right ) \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 {4 b d \sqrt {d+e x} \left (1-c^2 x^2\right )}{105 c^3 e \sqrt {1-\frac {1}{c^2 x^2}} x}-\frac {4 b (d+e x)^{3/2} \left (1-c^2 x^2\right )}{35 c^3 e \sqrt {1-\frac {1}{c^2 x^2}} x}+\frac {2 d^2 (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e^3}-\frac {4 d (d+e x)^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{5 e^3}+\frac {2 (d+e x)^{7/2} \left (a+b \csc ^{-1}(c x)\right )}{7 e^3}+\frac {\left (16 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}{105 c e^3 \sqrt {1-\frac {1}{c^2 x^2}} x}-\frac {\left (32 b d^2 \sqrt {1-c^2 x^2}\right ) \int \frac {\sqrt {d+e x}}{\sqrt {1-c^2 x^2}} \, dx}{105 c e^2 \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}{105 c e^2 \sqrt {1-\frac {1}{c^2 x^2}} x}+\frac {\left (8 b \sqrt {1-c^2 x^2}\right ) \int \frac {3 c^2 d e+\frac {3}{4} c^2 \left (c^2 d^2+3 e^2\right ) x}{\sqrt {d+e x} \sqrt {1-c^2 x^2}} \, dx}{105 c^5 e \sqrt {1-\frac {1}{c^2 x^2}} x}+\frac {\left (16 b d^2 \sqrt {1-c^2 x^2}\right ) \int \frac {x}{\sqrt {d+e x} \sqrt {1-c^2 x^2}} \, dx}{105 c e \sqrt {1-\frac {1}{c^2 x^2}} x}+\frac {\left (8 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^2 \sqrt {1-\frac {1}{c^2 x^2}} x}\\ &=\frac {4 b d \sqrt {d+e x} \left (1-c^2 x^2\right )}{105 c^3 e \sqrt {1-\frac {1}{c^2 x^2}} x}-\frac {4 b (d+e x)^{3/2} \left (1-c^2 x^2\right )}{35 c^3 e \sqrt {1-\frac {1}{c^2 x^2}} x}+\frac {2 d^2 (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e^3}-\frac {4 d (d+e x)^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{5 e^3}+\frac {2 (d+e x)^{7/2} \left (a+b \csc ^{-1}(c x)\right )}{7 e^3}+\frac {\left (16 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}{105 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 {\sqrt {d+e x}}{\sqrt {1-c^2 x^2}} \, dx}{105 c e^2 \sqrt {1-\frac {1}{c^2 x^2}} x}-\frac {\left (16 b d^3 \sqrt {1-c^2 x^2}\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {1-c^2 x^2}} \, dx}{105 c e^2 \sqrt {1-\frac {1}{c^2 x^2}} x}+\frac {\left (2 b \left (c^2 d^2+3 e^2\right ) \sqrt {1-c^2 x^2}\right ) \int \frac {\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 (8 b \left (3 c^2 d e^2-\frac {3}{4} c^2 d \left (c^2 d^2+3 e^2\right )\right ) \sqrt {1-c^2 x^2}\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {1-c^2 x^2}} \, dx}{105 c^5 e^2 \sqrt {1-\frac {1}{c^2 x^2}} 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 )}{105 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 (64 b d^3 \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 )}{105 c^2 e^2 \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {d+e x}}-\frac {\left (16 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 )}{105 c^4 e^2 \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {d+e x}}\\ &=\frac {4 b d \sqrt {d+e x} \left (1-c^2 x^2\right )}{105 c^3 e \sqrt {1-\frac {1}{c^2 x^2}} x}-\frac {4 b (d+e x)^{3/2} \left (1-c^2 x^2\right )}{35 c^3 e \sqrt {1-\frac {1}{c^2 x^2}} x}+\frac {2 d^2 (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e^3}-\frac {4 d (d+e x)^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{5 e^3}+\frac {2 (d+e x)^{7/2} \left (a+b \csc ^{-1}(c x)\right )}{7 e^3}+\frac {64 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 )}{105 c^2 e^2 \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 )}{105 c^2 e^2 \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {d+e x}}-\frac {16 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^2 \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {d+e x}}-\frac {\left (32 b d^4 \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 )}{105 c e^3 \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 ) \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 )}{105 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 \left (c^2 d^2+3 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 )}{35 c^4 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 (32 b d^3 \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 )}{105 c^2 e^2 \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {d+e x}}-\frac {\left (16 b \left (3 c^2 d e^2-\frac {3}{4} c^2 d \left (c^2 d^2+3 e^2\right )\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 )}{105 c^6 e^2 \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {d+e x}}\\ &=\frac {4 b d \sqrt {d+e x} \left (1-c^2 x^2\right )}{105 c^3 e \sqrt {1-\frac {1}{c^2 x^2}} x}-\frac {4 b (d+e x)^{3/2} \left (1-c^2 x^2\right )}{35 c^3 e \sqrt {1-\frac {1}{c^2 x^2}} x}+\frac {2 d^2 (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e^3}-\frac {4 d (d+e x)^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{5 e^3}+\frac {2 (d+e x)^{7/2} \left (a+b \csc ^{-1}(c x)\right )}{7 e^3}+\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 )}{105 c^2 e^2 \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {\frac {c (d+e x)}{c d+e}}}-\frac {4 b \left (c^2 d^2+3 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 )}{35 c^4 e^2 \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {\frac {c (d+e x)}{c d+e}}}-\frac {32 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 )}{105 c^2 e^2 \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {d+e x}}-\frac {16 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^2 \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {d+e x}}+\frac {4 b d \left (c^2 d^2-e^2\right ) \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^2 \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {d+e x}}-\frac {\left (32 b d^4 \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 )}{105 c e^3 \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {d+e x}}\\ &=\frac {4 b d \sqrt {d+e x} \left (1-c^2 x^2\right )}{105 c^3 e \sqrt {1-\frac {1}{c^2 x^2}} x}-\frac {4 b (d+e x)^{3/2} \left (1-c^2 x^2\right )}{35 c^3 e \sqrt {1-\frac {1}{c^2 x^2}} x}+\frac {2 d^2 (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e^3}-\frac {4 d (d+e x)^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{5 e^3}+\frac {2 (d+e x)^{7/2} \left (a+b \csc ^{-1}(c x)\right )}{7 e^3}+\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 )}{105 c^2 e^2 \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {\frac {c (d+e x)}{c d+e}}}-\frac {4 b \left (c^2 d^2+3 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 )}{35 c^4 e^2 \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {\frac {c (d+e x)}{c d+e}}}-\frac {32 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 )}{105 c^2 e^2 \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {d+e x}}-\frac {16 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^2 \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {d+e x}}+\frac {4 b d \left (c^2 d^2-e^2\right ) \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^2 \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {d+e x}}-\frac {32 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 )}{105 c e^3 \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {d+e x}}\\ \end {align*}

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Mathematica [C]  time = 13.93, size = 870, normalized size = 1.75 \[ \frac {b \left (-\frac {c \left (\frac {d}{x}+e\right ) x \left (-\frac {16 c^3 \csc ^{-1}(c x) d^3}{105 e^3}-\frac {2}{7} c^3 x^3 \csc ^{-1}(c x)-\frac {2 c^2 x^2 \left (2 \sqrt {1-\frac {1}{c^2 x^2}} e+c d \csc ^{-1}(c x)\right )}{35 e}-\frac {8 c x \left (c d e \sqrt {1-\frac {1}{c^2 x^2}}-c^2 d^2 \csc ^{-1}(c x)\right )}{105 e^2}-\frac {4 \left (9 e^2-5 c^2 d^2\right ) \sqrt {1-\frac {1}{c^2 x^2}}}{105 e^2}\right )}{\sqrt {d+e x}}-\frac {2 \sqrt {\frac {d}{x}+e} \sqrt {c x} \left (\frac {2 \left (9 c^3 d^3 e-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 (8 c^4 d^4+5 c^2 e^2 d^2-9 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}}+\frac {2 \left (9 c d e^3-5 c^3 d^3 e\right ) \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 )}{c d \sqrt {1-\frac {1}{c^2 x^2}} \sqrt {\frac {d}{x}+e} \sqrt {c x} \left (c^2 x^2-2\right )}\right )}{105 e^3 \sqrt {d+e x}}\right )}{c^4}-\frac {a d^3 \sqrt {d+e x} B_{-\frac {e x}{d}}\left (3,\frac {3}{2}\right )}{e^3 \sqrt {\frac {e x}{d}+1}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

-((a*d^3*Sqrt[d + e*x]*Beta[-((e*x)/d), 3, 3/2])/(e^3*Sqrt[1 + (e*x)/d])) + (b*(-((c*(e + d/x)*x*((-4*(-5*c^2*
d^2 + 9*e^2)*Sqrt[1 - 1/(c^2*x^2)])/(105*e^2) - (16*c^3*d^3*ArcCsc[c*x])/(105*e^3) - (2*c^3*x^3*ArcCsc[c*x])/7
- (2*c^2*x^2*(2*e*Sqrt[1 - 1/(c^2*x^2)] + c*d*ArcCsc[c*x]))/(35*e) - (8*c*x*(c*d*e*Sqrt[1 - 1/(c^2*x^2)] - c^
2*d^2*ArcCsc[c*x]))/(105*e^2)))/Sqrt[d + e*x]) - (2*Sqrt[e + d/x]*Sqrt[c*x]*((2*(9*c^3*d^3*e - 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*(8*c^4*d^4 + 5*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*(-5*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[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]*EllipticP
i[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]*Sqrt[c*x]*(-2
+ c^2*x^2))))/(105*e^3*Sqrt[d + e*x])))/c^4

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fricas [F]  time = 1.43, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (b x^{2} \operatorname {arccsc}\left (c x\right ) + a x^{2}\right )} \sqrt {e x + d}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [B]  time = 0.09, size = 1222, normalized size = 2.46 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/e^3*(a*(1/7*(e*x+d)^(7/2)-2/5*(e*x+d)^(5/2)*d+1/3*(e*x+d)^(3/2)*d^2)+b*(1/7*arccsc(c*x)*(e*x+d)^(7/2)-2/5*ar
ccsc(c*x)*(e*x+d)^(5/2)*d+1/3*arccsc(c*x)*(e*x+d)^(3/2)*d^2+2/105/c^4*(3*(c/(c*d-e))^(1/2)*(e*x+d)^(7/2)*c^3-7
*(c/(c*d-e))^(1/2)*(e*x+d)^(5/2)*c^3*d+5*(c/(c*d-e))^(1/2)*(e*x+d)^(3/2)*c^3*d^2+4*(-((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))*
c^3*d^3+5*(-((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))*c^3*d^3-8*d^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)*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-(
c/(c*d-e))^(1/2)*(e*x+d)^(1/2)*c^3*d^3-5*(-((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))*c^2*d^2*e+5*(-((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))*c^2
*d^2*e-3*(c/(c*d-e))^(1/2)*(e*x+d)^(3/2)*c*e^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))*c*d*e^2-9*(-((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
))*c*d*e^2+(c/(c*d-e))^(1/2)*(e*x+d)^(1/2)*c*d*e^2+9*(-((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^3-9*(-((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^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 \[ \text {Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[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(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 \[ \int x^2\,\left (a+b\,\mathrm {asin}\left (\frac {1}{c\,x}\right )\right )\,\sqrt {d+e\,x} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{2} \left (a + b \operatorname {acsc}{\left (c x \right )}\right ) \sqrt {d + e x}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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