\(\int (d+e x)^{3/2} (a+b \csc ^{-1}(c x)) \, dx\) [56]

Optimal result

Integrand size = 18, antiderivative size = 372 \[ \int (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right ) \, dx=-\frac {4 b e \sqrt {d+e x} \left (1-c^2 x^2\right )}{15 c^3 \sqrt {1-\frac {1}{c^2 x^2}} x}+\frac {2 (d+e x)^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{5 e}-\frac {28 b d \sqrt {d+e x} \sqrt {1-c^2 x^2} E\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{15 c^2 \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {\frac {c (d+e x)}{c d+e}}}-\frac {4 b \left (2 c^2 d^2+e^2\right ) \sqrt {\frac {c (d+e x)}{c d+e}} \sqrt {1-c^2 x^2} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{15 c^4 \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {d+e x}}-\frac {4 b d^3 \sqrt {\frac {c (d+e x)}{c d+e}} \sqrt {1-c^2 x^2} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{5 c e \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {d+e x}} \]

[Out]

2/5*(e*x+d)^(5/2)*(a+b*arccsc(c*x))/e-4/15*b*e*(-c^2*x^2+1)*(e*x+d)^(1/2)/c^3/x/(1-1/c^2/x^2)^(1/2)-28/15*b*d*
EllipticE(1/2*(-c*x+1)^(1/2)*2^(1/2),2^(1/2)*(e/(c*d+e))^(1/2))*(e*x+d)^(1/2)*(-c^2*x^2+1)^(1/2)/c^2/x/(1-1/c^
2/x^2)^(1/2)/(c*(e*x+d)/(c*d+e))^(1/2)-4/15*b*(2*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/x/(1-1/c^2/x^2)^(1/2)/(e*x+d)^(1/2)-4/5*b*d^3
*EllipticPi(1/2*(-c*x+1)^(1/2)*2^(1/2),2,2^(1/2)*(e/(c*d+e))^(1/2))*(c*(e*x+d)/(c*d+e))^(1/2)*(-c^2*x^2+1)^(1/
2)/c/e/x/(1-1/c^2/x^2)^(1/2)/(e*x+d)^(1/2)

Rubi [A] (verified)

Time = 0.54 (sec) , antiderivative size = 372, normalized size of antiderivative = 1.00, number of steps used = 22, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.722, Rules used = {5335, 1588, 972, 733, 430, 947, 174, 552, 551, 858, 435, 945, 1598} \[ \int (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right ) \, dx=\frac {2 (d+e x)^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{5 e}-\frac {4 b d^3 \sqrt {1-c^2 x^2} \sqrt {\frac {c (d+e x)}{c d+e}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{5 c e x \sqrt {1-\frac {1}{c^2 x^2}} \sqrt {d+e x}}-\frac {28 b d \sqrt {1-c^2 x^2} \sqrt {d+e x} E\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{15 c^2 x \sqrt {1-\frac {1}{c^2 x^2}} \sqrt {\frac {c (d+e x)}{c d+e}}}-\frac {4 b \sqrt {1-c^2 x^2} \left (2 c^2 d^2+e^2\right ) \sqrt {\frac {c (d+e x)}{c d+e}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{15 c^4 x \sqrt {1-\frac {1}{c^2 x^2}} \sqrt {d+e x}}-\frac {4 b e \left (1-c^2 x^2\right ) \sqrt {d+e x}}{15 c^3 x \sqrt {1-\frac {1}{c^2 x^2}}} \]

[In]

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

[Out]

(-4*b*e*Sqrt[d + e*x]*(1 - c^2*x^2))/(15*c^3*Sqrt[1 - 1/(c^2*x^2)]*x) + (2*(d + e*x)^(5/2)*(a + b*ArcCsc[c*x])
)/(5*e) - (28*b*d*Sqrt[d + e*x]*Sqrt[1 - c^2*x^2]*EllipticE[ArcSin[Sqrt[1 - c*x]/Sqrt[2]], (2*e)/(c*d + e)])/(
15*c^2*Sqrt[1 - 1/(c^2*x^2)]*x*Sqrt[(c*(d + e*x))/(c*d + e)]) - (4*b*(2*c^2*d^2 + e^2)*Sqrt[(c*(d + e*x))/(c*d
+ e)]*Sqrt[1 - c^2*x^2]*EllipticF[ArcSin[Sqrt[1 - c*x]/Sqrt[2]], (2*e)/(c*d + e)])/(15*c^4*Sqrt[1 - 1/(c^2*x^
2)]*x*Sqrt[d + e*x]) - (4*b*d^3*Sqrt[(c*(d + e*x))/(c*d + e)]*Sqrt[1 - c^2*x^2]*EllipticPi[2, ArcSin[Sqrt[1 -
c*x]/Sqrt[2]], (2*e)/(c*d + e)])/(5*c*e*Sqrt[1 - 1/(c^2*x^2)]*x*Sqrt[d + e*x])

Rule 174

Int[1/(((a_.) + (b_.)*(x_))*Sqrt[(c_.) + (d_.)*(x_)]*Sqrt[(e_.) + (f_.)*(x_)]*Sqrt[(g_.) + (h_.)*(x_)]), x_Sym
bol] :> Dist[-2, Subst[Int[1/(Simp[b*c - a*d - b*x^2, x]*Sqrt[Simp[(d*e - c*f)/d + f*(x^2/d), x]]*Sqrt[Simp[(d
*g - c*h)/d + h*(x^2/d), x]]), x], x, Sqrt[c + d*x]], x] /; FreeQ[{a, b, c, d, e, f, g, h}, x] && GtQ[(d*e - c
*f)/d, 0]

Rule 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 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 945

Int[((d_.) + (e_.)*(x_))^(m_)/(Sqrt[(f_.) + (g_.)*(x_)]*Sqrt[(a_) + (c_.)*(x_)^2]), x_Symbol] :> Simp[2*e^2*(d
+ e*x)^(m - 2)*Sqrt[f + g*x]*(Sqrt[a + c*x^2]/(c*g*(2*m - 1))), x] - Dist[1/(c*g*(2*m - 1)), Int[((d + e*x)^(
m - 3)/(Sqrt[f + g*x]*Sqrt[a + c*x^2]))*Simp[a*e^2*(d*g + 2*e*f*(m - 2)) - c*d^3*g*(2*m - 1) + e*(e*(a*e*g*(2*
m - 3)) + c*d*(2*e*f - 3*d*g*(2*m - 1)))*x + 2*e^2*(c*e*f - 3*c*d*g)*(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] && GeQ[m, 2]

Rule 947

Int[1/(((d_.) + (e_.)*(x_))*Sqrt[(f_.) + (g_.)*(x_)]*Sqrt[(a_) + (c_.)*(x_)^2]), x_Symbol] :> With[{q = Rt[-c/
a, 2]}, Dist[Sqrt[1 + c*(x^2/a)]/Sqrt[a + c*x^2], Int[1/((d + e*x)*Sqrt[f + g*x]*Sqrt[1 - q*x]*Sqrt[1 + q*x]),
x], x]] /; FreeQ[{a, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] && NeQ[c*d^2 + a*e^2, 0] &&  !GtQ[a, 0]

Rule 972

Int[((f_.) + (g_.)*(x_))^(n_)/(((d_.) + (e_.)*(x_))*Sqrt[(a_) + (c_.)*(x_)^2]), x_Symbol] :> Int[ExpandIntegra
nd[1/(Sqrt[f + g*x]*Sqrt[a + c*x^2]), (f + g*x)^(n + 1/2)/(d + e*x), x], x] /; FreeQ[{a, c, d, e, f, g}, x] &&
NeQ[e*f - d*g, 0] && NeQ[c*d^2 + a*e^2, 0] && IntegerQ[n + 1/2]

Rule 1588

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

Rule 1598

Int[(u_.)*(x_)^(m_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.))^(n_.), x_Symbol] :> Int[u*x^(m + n*p)*(a + b*x^(q -
p))^n, x] /; FreeQ[{a, b, m, p, q}, x] && IntegerQ[n] && PosQ[q - p]

Rule 5335

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

Rubi steps \begin{align*} \text {integral}& = \frac {2 (d+e x)^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{5 e}+\frac {(2 b) \int \frac {(d+e x)^{5/2}}{\sqrt {1-\frac {1}{c^2 x^2}} x^2} \, dx}{5 c e} \\ & = \frac {2 (d+e x)^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{5 e}+\frac {\left (2 b \sqrt {-\frac {1}{c^2}+x^2}\right ) \int \frac {(d+e x)^{5/2}}{x \sqrt {-\frac {1}{c^2}+x^2}} \, dx}{5 c e \sqrt {1-\frac {1}{c^2 x^2}} x} \\ & = \frac {2 (d+e x)^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{5 e}+\frac {\left (2 b \sqrt {-\frac {1}{c^2}+x^2}\right ) \int \left (\frac {3 d^2 e}{\sqrt {d+e x} \sqrt {-\frac {1}{c^2}+x^2}}+\frac {d^3}{x \sqrt {d+e x} \sqrt {-\frac {1}{c^2}+x^2}}+\frac {3 d e^2 x}{\sqrt {d+e x} \sqrt {-\frac {1}{c^2}+x^2}}+\frac {e^3 x^2}{\sqrt {d+e x} \sqrt {-\frac {1}{c^2}+x^2}}\right ) \, dx}{5 c e \sqrt {1-\frac {1}{c^2 x^2}} x} \\ & = \frac {2 (d+e x)^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{5 e}+\frac {\left (6 b d^2 \sqrt {-\frac {1}{c^2}+x^2}\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {-\frac {1}{c^2}+x^2}} \, dx}{5 c \sqrt {1-\frac {1}{c^2 x^2}} x}+\frac {\left (2 b d^3 \sqrt {-\frac {1}{c^2}+x^2}\right ) \int \frac {1}{x \sqrt {d+e x} \sqrt {-\frac {1}{c^2}+x^2}} \, dx}{5 c e \sqrt {1-\frac {1}{c^2 x^2}} x}+\frac {\left (6 b d e \sqrt {-\frac {1}{c^2}+x^2}\right ) \int \frac {x}{\sqrt {d+e x} \sqrt {-\frac {1}{c^2}+x^2}} \, dx}{5 c \sqrt {1-\frac {1}{c^2 x^2}} x}+\frac {\left (2 b e^2 \sqrt {-\frac {1}{c^2}+x^2}\right ) \int \frac {x^2}{\sqrt {d+e x} \sqrt {-\frac {1}{c^2}+x^2}} \, dx}{5 c \sqrt {1-\frac {1}{c^2 x^2}} x} \\ & = -\frac {4 b e \sqrt {d+e x} \left (1-c^2 x^2\right )}{15 c^3 \sqrt {1-\frac {1}{c^2 x^2}} x}+\frac {2 (d+e x)^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{5 e}+\frac {\left (6 b d \sqrt {-\frac {1}{c^2}+x^2}\right ) \int \frac {\sqrt {d+e x}}{\sqrt {-\frac {1}{c^2}+x^2}} \, dx}{5 c \sqrt {1-\frac {1}{c^2 x^2}} x}-\frac {\left (6 b d^2 \sqrt {-\frac {1}{c^2}+x^2}\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {-\frac {1}{c^2}+x^2}} \, dx}{5 c \sqrt {1-\frac {1}{c^2 x^2}} x}-\frac {\left (2 b e \sqrt {-\frac {1}{c^2}+x^2}\right ) \int \frac {-\frac {e x}{c^2}+2 d x^2}{x \sqrt {d+e x} \sqrt {-\frac {1}{c^2}+x^2}} \, dx}{15 c \sqrt {1-\frac {1}{c^2 x^2}} x}+\frac {\left (2 b d^3 \sqrt {1-c^2 x^2}\right ) \int \frac {1}{x \sqrt {1-c x} \sqrt {1+c x} \sqrt {d+e x}} \, dx}{5 c e \sqrt {1-\frac {1}{c^2 x^2}} x}-\frac {\left (12 b d^2 \sqrt {\frac {d+e x}{d+\frac {e}{c}}} \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1-x^2} \sqrt {1-\frac {2 e x^2}{c \left (d+\frac {e}{c}\right )}}} \, dx,x,\frac {\sqrt {1-c x}}{\sqrt {2}}\right )}{5 c^2 \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {d+e x}} \\ & = -\frac {4 b e \sqrt {d+e x} \left (1-c^2 x^2\right )}{15 c^3 \sqrt {1-\frac {1}{c^2 x^2}} x}+\frac {2 (d+e x)^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{5 e}-\frac {12 b d^2 \sqrt {\frac {c (d+e x)}{c d+e}} \sqrt {1-c^2 x^2} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{5 c^2 \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {d+e x}}-\frac {\left (2 b e \sqrt {-\frac {1}{c^2}+x^2}\right ) \int \frac {-\frac {e}{c^2}+2 d x}{\sqrt {d+e x} \sqrt {-\frac {1}{c^2}+x^2}} \, dx}{15 c \sqrt {1-\frac {1}{c^2 x^2}} x}-\frac {\left (4 b d^3 \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int \frac {1}{\left (1-x^2\right ) \sqrt {2-x^2} \sqrt {d+\frac {e}{c}-\frac {e x^2}{c}}} \, dx,x,\sqrt {1-c x}\right )}{5 c e \sqrt {1-\frac {1}{c^2 x^2}} x}-\frac {\left (12 b d \sqrt {d+e x} \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int \frac {\sqrt {1-\frac {2 e x^2}{c \left (d+\frac {e}{c}\right )}}}{\sqrt {1-x^2}} \, dx,x,\frac {\sqrt {1-c x}}{\sqrt {2}}\right )}{5 c^2 \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {\frac {d+e x}{d+\frac {e}{c}}}}+\frac {\left (12 b d^2 \sqrt {\frac {d+e x}{d+\frac {e}{c}}} \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1-x^2} \sqrt {1-\frac {2 e x^2}{c \left (d+\frac {e}{c}\right )}}} \, dx,x,\frac {\sqrt {1-c x}}{\sqrt {2}}\right )}{5 c^2 \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {d+e x}} \\ & = -\frac {4 b e \sqrt {d+e x} \left (1-c^2 x^2\right )}{15 c^3 \sqrt {1-\frac {1}{c^2 x^2}} x}+\frac {2 (d+e x)^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{5 e}-\frac {12 b d \sqrt {d+e x} \sqrt {1-c^2 x^2} E\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{5 c^2 \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {\frac {c (d+e x)}{c d+e}}}-\frac {\left (4 b d \sqrt {-\frac {1}{c^2}+x^2}\right ) \int \frac {\sqrt {d+e x}}{\sqrt {-\frac {1}{c^2}+x^2}} \, dx}{15 c \sqrt {1-\frac {1}{c^2 x^2}} x}-\frac {\left (2 b \left (-2 d^2-\frac {e^2}{c^2}\right ) \sqrt {-\frac {1}{c^2}+x^2}\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {-\frac {1}{c^2}+x^2}} \, dx}{15 c \sqrt {1-\frac {1}{c^2 x^2}} x}-\frac {\left (4 b d^3 \sqrt {\frac {c (d+e x)}{c d+e}} \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int \frac {1}{\left (1-x^2\right ) \sqrt {2-x^2} \sqrt {1-\frac {e x^2}{c \left (d+\frac {e}{c}\right )}}} \, dx,x,\sqrt {1-c x}\right )}{5 c e \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {d+e x}} \\ & = -\frac {4 b e \sqrt {d+e x} \left (1-c^2 x^2\right )}{15 c^3 \sqrt {1-\frac {1}{c^2 x^2}} x}+\frac {2 (d+e x)^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{5 e}-\frac {12 b d \sqrt {d+e x} \sqrt {1-c^2 x^2} E\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{5 c^2 \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {\frac {c (d+e x)}{c d+e}}}-\frac {4 b d^3 \sqrt {\frac {c (d+e x)}{c d+e}} \sqrt {1-c^2 x^2} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{5 c e \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {d+e x}}+\frac {\left (8 b d \sqrt {d+e x} \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int \frac {\sqrt {1-\frac {2 e x^2}{c \left (d+\frac {e}{c}\right )}}}{\sqrt {1-x^2}} \, dx,x,\frac {\sqrt {1-c x}}{\sqrt {2}}\right )}{15 c^2 \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {\frac {d+e x}{d+\frac {e}{c}}}}+\frac {\left (4 b \left (-2 d^2-\frac {e^2}{c^2}\right ) \sqrt {\frac {d+e x}{d+\frac {e}{c}}} \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1-x^2} \sqrt {1-\frac {2 e x^2}{c \left (d+\frac {e}{c}\right )}}} \, dx,x,\frac {\sqrt {1-c x}}{\sqrt {2}}\right )}{15 c^2 \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {d+e x}} \\ & = -\frac {4 b e \sqrt {d+e x} \left (1-c^2 x^2\right )}{15 c^3 \sqrt {1-\frac {1}{c^2 x^2}} x}+\frac {2 (d+e x)^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{5 e}-\frac {28 b d \sqrt {d+e x} \sqrt {1-c^2 x^2} E\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{15 c^2 \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 {\frac {c (d+e x)}{c d+e}} \sqrt {1-c^2 x^2} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{15 c^4 \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {d+e x}}-\frac {4 b d^3 \sqrt {\frac {c (d+e x)}{c d+e}} \sqrt {1-c^2 x^2} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{5 c e \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {d+e x}} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 12.33 (sec) , antiderivative size = 333, normalized size of antiderivative = 0.90 \[ \int (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right ) \, dx=\frac {1}{15} \left (\frac {4 b e \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {d+e x}}{c}+\frac {6 a (d+e x)^{5/2}}{e}+\frac {6 b (d+e x)^{5/2} \csc ^{-1}(c x)}{e}-\frac {4 i b \sqrt {\frac {e (1+c x)}{-c d+e}} \sqrt {\frac {e-c e x}{c d+e}} \left (-7 c d (c d-e) E\left (i \text {arcsinh}\left (\sqrt {-\frac {c}{c d+e}} \sqrt {d+e x}\right )|\frac {c d+e}{c d-e}\right )+\left (9 c^2 d^2-7 c d e+e^2\right ) \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {-\frac {c}{c d+e}} \sqrt {d+e x}\right ),\frac {c d+e}{c d-e}\right )-3 c^2 d^2 \operatorname {EllipticPi}\left (1+\frac {e}{c d},i \text {arcsinh}\left (\sqrt {-\frac {c}{c d+e}} \sqrt {d+e x}\right ),\frac {c d+e}{c d-e}\right )\right )}{c^3 e \sqrt {-\frac {c}{c d+e}} \sqrt {1-\frac {1}{c^2 x^2}} x}\right ) \]

[In]

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

[Out]

((4*b*e*Sqrt[1 - 1/(c^2*x^2)]*x*Sqrt[d + e*x])/c + (6*a*(d + e*x)^(5/2))/e + (6*b*(d + e*x)^(5/2)*ArcCsc[c*x])
/e - ((4*I)*b*Sqrt[(e*(1 + c*x))/(-(c*d) + e)]*Sqrt[(e - c*e*x)/(c*d + e)]*(-7*c*d*(c*d - e)*EllipticE[I*ArcSi
nh[Sqrt[-(c/(c*d + e))]*Sqrt[d + e*x]], (c*d + e)/(c*d - e)] + (9*c^2*d^2 - 7*c*d*e + e^2)*EllipticF[I*ArcSinh
[Sqrt[-(c/(c*d + e))]*Sqrt[d + e*x]], (c*d + e)/(c*d - e)] - 3*c^2*d^2*EllipticPi[1 + e/(c*d), I*ArcSinh[Sqrt[
-(c/(c*d + e))]*Sqrt[d + e*x]], (c*d + e)/(c*d - e)]))/(c^3*e*Sqrt[-(c/(c*d + e))]*Sqrt[1 - 1/(c^2*x^2)]*x))/1
5

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(797\) vs. \(2(335)=670\).

Time = 9.27 (sec) , antiderivative size = 798, normalized size of antiderivative = 2.15

method	result	size
derivativedivides	\(\frac {\frac {2 a \left (e x +d \right )^{\frac {5}{2}}}{5}+2 b \left (\frac {\operatorname {arccsc}\left (c x \right ) \left (e x +d \right )^{\frac {5}{2}}}{5}+\frac {\frac {2 \sqrt {\frac {c}{c d -e}}\, c^{2} \left (e x +d \right )^{\frac {5}{2}}}{15}+\frac {6 d^{2} \sqrt {\frac {-c \left (e x +d \right )+c d -e}{c d -e}}\, \sqrt {\frac {-c \left (e x +d \right )+c d +e}{c d +e}}\, \operatorname {EllipticF}\left (\sqrt {e x +d}\, \sqrt {\frac {c}{c d -e}}, \sqrt {\frac {c d -e}{c d +e}}\right ) c^{2}}{5}-\frac {14 \sqrt {\frac {-c \left (e x +d \right )+c d -e}{c d -e}}\, \sqrt {\frac {-c \left (e x +d \right )+c d +e}{c d +e}}\, \operatorname {EllipticE}\left (\sqrt {e x +d}\, \sqrt {\frac {c}{c d -e}}, \sqrt {\frac {c d -e}{c d +e}}\right ) c^{2} d^{2}}{15}-\frac {2 d^{2} \sqrt {\frac {-c \left (e x +d \right )+c d -e}{c d -e}}\, \sqrt {\frac {-c \left (e x +d \right )+c d +e}{c d +e}}\, \operatorname {EllipticPi}\left (\sqrt {e x +d}\, \sqrt {\frac {c}{c d -e}}, \frac {c d -e}{c d}, \frac {\sqrt {\frac {c}{c d +e}}}{\sqrt {\frac {c}{c d -e}}}\right ) c^{2}}{5}-\frac {4 \sqrt {\frac {c}{c d -e}}\, c^{2} d \left (e x +d \right )^{\frac {3}{2}}}{15}+\frac {14 \sqrt {\frac {-c \left (e x +d \right )+c d -e}{c d -e}}\, \sqrt {\frac {-c \left (e x +d \right )+c d +e}{c d +e}}\, \operatorname {EllipticF}\left (\sqrt {e x +d}\, \sqrt {\frac {c}{c d -e}}, \sqrt {\frac {c d -e}{c d +e}}\right ) c d e}{15}-\frac {14 \sqrt {\frac {-c \left (e x +d \right )+c d -e}{c d -e}}\, \sqrt {\frac {-c \left (e x +d \right )+c d +e}{c d +e}}\, \operatorname {EllipticE}\left (\sqrt {e x +d}\, \sqrt {\frac {c}{c d -e}}, \sqrt {\frac {c d -e}{c d +e}}\right ) c d e}{15}+\frac {2 \sqrt {\frac {c}{c d -e}}\, c^{2} d^{2} \sqrt {e x +d}}{15}+\frac {2 \sqrt {\frac {-c \left (e x +d \right )+c d -e}{c d -e}}\, \sqrt {\frac {-c \left (e x +d \right )+c d +e}{c d +e}}\, \operatorname {EllipticF}\left (\sqrt {e x +d}\, \sqrt {\frac {c}{c d -e}}, \sqrt {\frac {c d -e}{c d +e}}\right ) e^{2}}{15}-\frac {2 \sqrt {\frac {c}{c d -e}}\, e^{2} \sqrt {e x +d}}{15}}{c^{3} \sqrt {\frac {c}{c d -e}}\, x \sqrt {\frac {c^{2} \left (e x +d \right )^{2}-2 c^{2} d \left (e x +d \right )+c^{2} d^{2}-e^{2}}{c^{2} e^{2} x^{2}}}}\right )}{e}\)	\(798\)
default	\(\frac {\frac {2 a \left (e x +d \right )^{\frac {5}{2}}}{5}+2 b \left (\frac {\operatorname {arccsc}\left (c x \right ) \left (e x +d \right )^{\frac {5}{2}}}{5}+\frac {\frac {2 \sqrt {\frac {c}{c d -e}}\, c^{2} \left (e x +d \right )^{\frac {5}{2}}}{15}+\frac {6 d^{2} \sqrt {\frac {-c \left (e x +d \right )+c d -e}{c d -e}}\, \sqrt {\frac {-c \left (e x +d \right )+c d +e}{c d +e}}\, \operatorname {EllipticF}\left (\sqrt {e x +d}\, \sqrt {\frac {c}{c d -e}}, \sqrt {\frac {c d -e}{c d +e}}\right ) c^{2}}{5}-\frac {14 \sqrt {\frac {-c \left (e x +d \right )+c d -e}{c d -e}}\, \sqrt {\frac {-c \left (e x +d \right )+c d +e}{c d +e}}\, \operatorname {EllipticE}\left (\sqrt {e x +d}\, \sqrt {\frac {c}{c d -e}}, \sqrt {\frac {c d -e}{c d +e}}\right ) c^{2} d^{2}}{15}-\frac {2 d^{2} \sqrt {\frac {-c \left (e x +d \right )+c d -e}{c d -e}}\, \sqrt {\frac {-c \left (e x +d \right )+c d +e}{c d +e}}\, \operatorname {EllipticPi}\left (\sqrt {e x +d}\, \sqrt {\frac {c}{c d -e}}, \frac {c d -e}{c d}, \frac {\sqrt {\frac {c}{c d +e}}}{\sqrt {\frac {c}{c d -e}}}\right ) c^{2}}{5}-\frac {4 \sqrt {\frac {c}{c d -e}}\, c^{2} d \left (e x +d \right )^{\frac {3}{2}}}{15}+\frac {14 \sqrt {\frac {-c \left (e x +d \right )+c d -e}{c d -e}}\, \sqrt {\frac {-c \left (e x +d \right )+c d +e}{c d +e}}\, \operatorname {EllipticF}\left (\sqrt {e x +d}\, \sqrt {\frac {c}{c d -e}}, \sqrt {\frac {c d -e}{c d +e}}\right ) c d e}{15}-\frac {14 \sqrt {\frac {-c \left (e x +d \right )+c d -e}{c d -e}}\, \sqrt {\frac {-c \left (e x +d \right )+c d +e}{c d +e}}\, \operatorname {EllipticE}\left (\sqrt {e x +d}\, \sqrt {\frac {c}{c d -e}}, \sqrt {\frac {c d -e}{c d +e}}\right ) c d e}{15}+\frac {2 \sqrt {\frac {c}{c d -e}}\, c^{2} d^{2} \sqrt {e x +d}}{15}+\frac {2 \sqrt {\frac {-c \left (e x +d \right )+c d -e}{c d -e}}\, \sqrt {\frac {-c \left (e x +d \right )+c d +e}{c d +e}}\, \operatorname {EllipticF}\left (\sqrt {e x +d}\, \sqrt {\frac {c}{c d -e}}, \sqrt {\frac {c d -e}{c d +e}}\right ) e^{2}}{15}-\frac {2 \sqrt {\frac {c}{c d -e}}\, e^{2} \sqrt {e x +d}}{15}}{c^{3} \sqrt {\frac {c}{c d -e}}\, x \sqrt {\frac {c^{2} \left (e x +d \right )^{2}-2 c^{2} d \left (e x +d \right )+c^{2} d^{2}-e^{2}}{c^{2} e^{2} x^{2}}}}\right )}{e}\)	\(798\)
parts	\(\frac {2 a \left (e x +d \right )^{\frac {5}{2}}}{5 e}+\frac {2 b \left (\frac {\operatorname {arccsc}\left (c x \right ) \left (e x +d \right )^{\frac {5}{2}}}{5}+\frac {\frac {2 \sqrt {\frac {c}{c d -e}}\, c^{2} \left (e x +d \right )^{\frac {5}{2}}}{15}-\frac {4 \sqrt {\frac {c}{c d -e}}\, c^{2} d \left (e x +d \right )^{\frac {3}{2}}}{15}+\frac {6 d^{2} \sqrt {-\frac {c \left (e x +d \right )-c d +e}{c d -e}}\, \sqrt {-\frac {c \left (e x +d \right )-c d -e}{c d +e}}\, \operatorname {EllipticF}\left (\sqrt {e x +d}\, \sqrt {\frac {c}{c d -e}}, \sqrt {\frac {c d -e}{c d +e}}\right ) c^{2}}{5}-\frac {14 \sqrt {-\frac {c \left (e x +d \right )-c d +e}{c d -e}}\, \sqrt {-\frac {c \left (e x +d \right )-c d -e}{c d +e}}\, \operatorname {EllipticE}\left (\sqrt {e x +d}\, \sqrt {\frac {c}{c d -e}}, \sqrt {\frac {c d -e}{c d +e}}\right ) c^{2} d^{2}}{15}-\frac {2 d^{2} \sqrt {-\frac {c \left (e x +d \right )-c d +e}{c d -e}}\, \sqrt {-\frac {c \left (e x +d \right )-c d -e}{c d +e}}\, \operatorname {EllipticPi}\left (\sqrt {e x +d}\, \sqrt {\frac {c}{c d -e}}, \frac {c d -e}{c d}, \frac {\sqrt {\frac {c}{c d +e}}}{\sqrt {\frac {c}{c d -e}}}\right ) c^{2}}{5}+\frac {2 \sqrt {\frac {c}{c d -e}}\, c^{2} d^{2} \sqrt {e x +d}}{15}+\frac {14 \sqrt {-\frac {c \left (e x +d \right )-c d +e}{c d -e}}\, \sqrt {-\frac {c \left (e x +d \right )-c d -e}{c d +e}}\, \operatorname {EllipticF}\left (\sqrt {e x +d}\, \sqrt {\frac {c}{c d -e}}, \sqrt {\frac {c d -e}{c d +e}}\right ) c d e}{15}-\frac {14 \sqrt {-\frac {c \left (e x +d \right )-c d +e}{c d -e}}\, \sqrt {-\frac {c \left (e x +d \right )-c d -e}{c d +e}}\, \operatorname {EllipticE}\left (\sqrt {e x +d}\, \sqrt {\frac {c}{c d -e}}, \sqrt {\frac {c d -e}{c d +e}}\right ) c d e}{15}+\frac {2 \sqrt {-\frac {c \left (e x +d \right )-c d +e}{c d -e}}\, \sqrt {-\frac {c \left (e x +d \right )-c d -e}{c d +e}}\, \operatorname {EllipticF}\left (\sqrt {e x +d}\, \sqrt {\frac {c}{c d -e}}, \sqrt {\frac {c d -e}{c d +e}}\right ) e^{2}}{15}-\frac {2 \sqrt {\frac {c}{c d -e}}\, e^{2} \sqrt {e x +d}}{15}}{c^{3} \sqrt {\frac {c}{c d -e}}\, x \sqrt {\frac {c^{2} \left (e x +d \right )^{2}-2 c^{2} d \left (e x +d \right )+c^{2} d^{2}-e^{2}}{c^{2} e^{2} x^{2}}}}\right )}{e}\)	\(812\)

[In]

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

[Out]

2/e*(1/5*a*(e*x+d)^(5/2)+b*(1/5*arccsc(c*x)*(e*x+d)^(5/2)+2/15/c^3*((c/(c*d-e))^(1/2)*c^2*(e*x+d)^(5/2)+9*d^2*
((-c*(e*x+d)+c*d-e)/(c*d-e))^(1/2)*((-c*(e*x+d)+c*d+e)/(c*d+e))^(1/2)*EllipticF((e*x+d)^(1/2)*(c/(c*d-e))^(1/2
),((c*d-e)/(c*d+e))^(1/2))*c^2-7*((-c*(e*x+d)+c*d-e)/(c*d-e))^(1/2)*((-c*(e*x+d)+c*d+e)/(c*d+e))^(1/2)*Ellipti
cE((e*x+d)^(1/2)*(c/(c*d-e))^(1/2),((c*d-e)/(c*d+e))^(1/2))*c^2*d^2-3*d^2*((-c*(e*x+d)+c*d-e)/(c*d-e))^(1/2)*(
(-c*(e*x+d)+c*d+e)/(c*d+e))^(1/2)*EllipticPi((e*x+d)^(1/2)*(c/(c*d-e))^(1/2),1/c*(c*d-e)/d,(c/(c*d+e))^(1/2)/(
c/(c*d-e))^(1/2))*c^2-2*(c/(c*d-e))^(1/2)*c^2*d*(e*x+d)^(3/2)+7*((-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-7*((-c*(e*x+d
)+c*d-e)/(c*d-e))^(1/2)*((-c*(e*x+d)+c*d+e)/(c*d+e))^(1/2)*EllipticE((e*x+d)^(1/2)*(c/(c*d-e))^(1/2),((c*d-e)/
(c*d+e))^(1/2))*c*d*e+(c/(c*d-e))^(1/2)*c^2*d^2*(e*x+d)^(1/2)+((-c*(e*x+d)+c*d-e)/(c*d-e))^(1/2)*((-c*(e*x+d)+
c*d+e)/(c*d+e))^(1/2)*EllipticF((e*x+d)^(1/2)*(c/(c*d-e))^(1/2),((c*d-e)/(c*d+e))^(1/2))*e^2-(c/(c*d-e))^(1/2)
*e^2*(e*x+d)^(1/2))/(c/(c*d-e))^(1/2)/x/((c^2*(e*x+d)^2-2*c^2*d*(e*x+d)+c^2*d^2-e^2)/c^2/e^2/x^2)^(1/2)))

Fricas [F(-1)]

Timed out. \[ \int (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right ) \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Sympy [F]

\[ \int (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right ) \, dx=\int \left (a + b \operatorname {acsc}{\left (c x \right )}\right ) \left (d + e x\right )^{\frac {3}{2}}\, dx \]

[In]

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

[Out]

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

Maxima [F(-2)]

Exception generated. \[ \int (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right ) \, dx=\text {Exception raised: ValueError} \]

[In]

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

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(e+c*d>0)', see `assume?` for m
ore details)

Giac [F]

\[ \int (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right ) \, dx=\int { {\left (e x + d\right )}^{\frac {3}{2}} {\left (b \operatorname {arccsc}\left (c x\right ) + a\right )} \,d x } \]

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right ) \, dx=\int \left (a+b\,\mathrm {asin}\left (\frac {1}{c\,x}\right )\right )\,{\left (d+e\,x\right )}^{3/2} \,d x \]

[In]

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

[Out]

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