\(\int \sqrt {d+e x} (a+b \csc ^{-1}(c x)) \, dx\) [53]

Optimal result

Integrand size = 18, antiderivative size = 315 \[ \int \sqrt {d+e x} \left (a+b \csc ^{-1}(c x)\right ) \, dx=\frac {2 (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e}-\frac {4 b \sqrt {d+e x} \sqrt {1-c^2 x^2} E\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{3 c^2 \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {\frac {c (d+e x)}{c d+e}}}-\frac {4 b d \sqrt {\frac {c (d+e x)}{c d+e}} \sqrt {1-c^2 x^2} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{3 c^2 \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {d+e x}}-\frac {4 b d^2 \sqrt {\frac {c (d+e x)}{c d+e}} \sqrt {1-c^2 x^2} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{3 c e \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {d+e x}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.34 (sec) , antiderivative size = 315, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.611, Rules used = {5335, 1588, 972, 733, 430, 947, 174, 552, 551, 858, 435} \[ \int \sqrt {d+e x} \left (a+b \csc ^{-1}(c x)\right ) \, dx=\frac {2 (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e}-\frac {4 b d^2 \sqrt {1-c^2 x^2} \sqrt {\frac {c (d+e x)}{c d+e}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{3 c e x \sqrt {1-\frac {1}{c^2 x^2}} \sqrt {d+e x}}-\frac {4 b d \sqrt {1-c^2 x^2} \sqrt {\frac {c (d+e x)}{c d+e}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{3 c^2 x \sqrt {1-\frac {1}{c^2 x^2}} \sqrt {d+e x}}-\frac {4 b \sqrt {1-c^2 x^2} \sqrt {d+e x} E\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{3 c^2 x \sqrt {1-\frac {1}{c^2 x^2}} \sqrt {\frac {c (d+e x)}{c d+e}}} \]

[In]

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

[Out]

(2*(d + e*x)^(3/2)*(a + b*ArcCsc[c*x]))/(3*e) - (4*b*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*Sqrt[1 - 1/(c^2*x^2)]*x*Sqrt[(c*(d + e*x))/(c*d + e)]) - (4*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*S
qrt[1 - 1/(c^2*x^2)]*x*Sqrt[d + e*x]) - (4*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*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 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 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)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e}+\frac {(2 b) \int \frac {(d+e x)^{3/2}}{\sqrt {1-\frac {1}{c^2 x^2}} x^2} \, dx}{3 c e} \\ & = \frac {2 (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e}+\frac {\left (2 b \sqrt {-\frac {1}{c^2}+x^2}\right ) \int \frac {(d+e x)^{3/2}}{x \sqrt {-\frac {1}{c^2}+x^2}} \, dx}{3 c e \sqrt {1-\frac {1}{c^2 x^2}} x} \\ & = \frac {2 (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e}+\frac {\left (2 b \sqrt {-\frac {1}{c^2}+x^2}\right ) \int \left (\frac {2 d e}{\sqrt {d+e x} \sqrt {-\frac {1}{c^2}+x^2}}+\frac {d^2}{x \sqrt {d+e x} \sqrt {-\frac {1}{c^2}+x^2}}+\frac {e^2 x}{\sqrt {d+e x} \sqrt {-\frac {1}{c^2}+x^2}}\right ) \, dx}{3 c e \sqrt {1-\frac {1}{c^2 x^2}} x} \\ & = \frac {2 (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e}+\frac {\left (4 b d \sqrt {-\frac {1}{c^2}+x^2}\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {-\frac {1}{c^2}+x^2}} \, dx}{3 c \sqrt {1-\frac {1}{c^2 x^2}} x}+\frac {\left (2 b d^2 \sqrt {-\frac {1}{c^2}+x^2}\right ) \int \frac {1}{x \sqrt {d+e x} \sqrt {-\frac {1}{c^2}+x^2}} \, dx}{3 c e \sqrt {1-\frac {1}{c^2 x^2}} x}+\frac {\left (2 b e \sqrt {-\frac {1}{c^2}+x^2}\right ) \int \frac {x}{\sqrt {d+e x} \sqrt {-\frac {1}{c^2}+x^2}} \, dx}{3 c \sqrt {1-\frac {1}{c^2 x^2}} x} \\ & = \frac {2 (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e}+\frac {\left (2 b \sqrt {-\frac {1}{c^2}+x^2}\right ) \int \frac {\sqrt {d+e x}}{\sqrt {-\frac {1}{c^2}+x^2}} \, dx}{3 c \sqrt {1-\frac {1}{c^2 x^2}} x}-\frac {\left (2 b d \sqrt {-\frac {1}{c^2}+x^2}\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {-\frac {1}{c^2}+x^2}} \, dx}{3 c \sqrt {1-\frac {1}{c^2 x^2}} x}+\frac {\left (2 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 \sqrt {1-\frac {1}{c^2 x^2}} x}-\frac {\left (8 b d \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 )}{3 c^2 \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {d+e x}} \\ & = \frac {2 (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e}-\frac {8 b d \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 )}{3 c^2 \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {d+e x}}-\frac {\left (4 b d^2 \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 )}{3 c e \sqrt {1-\frac {1}{c^2 x^2}} x}-\frac {\left (4 b \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 )}{3 c^2 \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {\frac {d+e x}{d+\frac {e}{c}}}}+\frac {\left (4 b d \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 )}{3 c^2 \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {d+e x}} \\ & = \frac {2 (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e}-\frac {4 b \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 )}{3 c^2 \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {\frac {c (d+e x)}{c d+e}}}-\frac {4 b d \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 )}{3 c^2 \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {d+e x}}-\frac {\left (4 b d^2 \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 )}{3 c e \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {d+e x}} \\ & = \frac {2 (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e}-\frac {4 b \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 )}{3 c^2 \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {\frac {c (d+e x)}{c d+e}}}-\frac {4 b d \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 )}{3 c^2 \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {d+e x}}-\frac {4 b d^2 \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 )}{3 c e \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {d+e x}} \\ \end{align*}

Mathematica [B] (warning: unable to verify)

Leaf count is larger than twice the leaf count of optimal. \(657\) vs. \(2(315)=630\).

Time = 32.09 (sec) , antiderivative size = 657, normalized size of antiderivative = 2.09 \[ \int \sqrt {d+e x} \left (a+b \csc ^{-1}(c x)\right ) \, dx=\frac {2 a (d+e x)^{3/2}}{3 e}-\frac {b (c d+c e x) \left (-\frac {2 \left (2 e \sqrt {1-\frac {1}{c^2 x^2}}+c d \csc ^{-1}(c x)+c e x \csc ^{-1}(c x)\right )}{e}+\frac {4 d \sqrt {-c^2 \left (1-\frac {1}{c^2 x^2}\right ) x^2} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{(c d+e) \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {\frac {c d+c e x}{c d+e}}}-\frac {4 (-c d+e) \sqrt {-c^2 \left (1-\frac {1}{c^2 x^2}\right ) x^2} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{c e \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {\frac {c d+c e x}{c d+e}}}+\frac {\left (c^2 \left (1-\frac {1}{c^2 x^2}\right ) x^2 (c d+c e x)+c^2 d x \sqrt {-c^2 \left (1-\frac {1}{c^2 x^2}\right ) x^2} \sqrt {\frac {c d+c e x}{c d+e}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )-\frac {c x (1+c x) \sqrt {\frac {e-c e x}{c d+e}} \sqrt {\frac {c d+c e x}{c d-e}} \left ((c d+e) E\left (\arcsin \left (\sqrt {\frac {c d+c e x}{c d-e}}\right )|\frac {c d-e}{c d+e}\right )-e \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {c d+c e x}{c d-e}}\right ),\frac {c d-e}{c d+e}\right )\right )}{\sqrt {\frac {e (1+c x)}{-c d+e}}}+c e x \sqrt {-c^2 \left (1-\frac {1}{c^2 x^2}\right ) x^2} \sqrt {\frac {c d+c e x}{c d+e}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )\right ) \sec \left (2 \csc ^{-1}(c x)\right ) \sin \left (4 \csc ^{-1}(c x)\right )}{c^2 \left (1-\frac {1}{c^2 x^2}\right ) \left (e+\frac {d}{x}\right ) x^2}\right )}{3 c^2 \sqrt {d+e x}} \]

[In]

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

[Out]

(2*a*(d + e*x)^(3/2))/(3*e) - (b*(c*d + c*e*x)*((-2*(2*e*Sqrt[1 - 1/(c^2*x^2)] + c*d*ArcCsc[c*x] + c*e*x*ArcCs
c[c*x]))/e + (4*d*Sqrt[-(c^2*(1 - 1/(c^2*x^2))*x^2)]*EllipticF[ArcSin[Sqrt[1 - c*x]/Sqrt[2]], (2*e)/(c*d + e)]
)/((c*d + e)*Sqrt[1 - 1/(c^2*x^2)]*x*Sqrt[(c*d + c*e*x)/(c*d + e)]) - (4*(-(c*d) + e)*Sqrt[-(c^2*(1 - 1/(c^2*x
^2))*x^2)]*EllipticPi[2, ArcSin[Sqrt[1 - c*x]/Sqrt[2]], (2*e)/(c*d + e)])/(c*e*Sqrt[1 - 1/(c^2*x^2)]*x*Sqrt[(c
*d + c*e*x)/(c*d + e)]) + ((c^2*(1 - 1/(c^2*x^2))*x^2*(c*d + c*e*x) + c^2*d*x*Sqrt[-(c^2*(1 - 1/(c^2*x^2))*x^2
)]*Sqrt[(c*d + c*e*x)/(c*d + e)]*EllipticF[ArcSin[Sqrt[1 - c*x]/Sqrt[2]], (2*e)/(c*d + e)] - (c*x*(1 + c*x)*Sq
rt[(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^2*(1 - 1/(c^2*x^2))*x^2)]*Sqrt[(c*d + c*e*x)/(c*d + e)]*EllipticPi[
2, ArcSin[Sqrt[1 - c*x]/Sqrt[2]], (2*e)/(c*d + e)])*Sec[2*ArcCsc[c*x]]*Sin[4*ArcCsc[c*x]])/(c^2*(1 - 1/(c^2*x^
2))*(e + d/x)*x^2)))/(3*c^2*Sqrt[d + e*x])

Maple [A] (verified)

Time = 8.52 (sec) , antiderivative size = 386, normalized size of antiderivative = 1.23

method	result	size
derivativedivides	\(\frac {\frac {2 \left (e x +d \right )^{\frac {3}{2}} a}{3}+2 b \left (\frac {\left (e x +d \right )^{\frac {3}{2}} \operatorname {arccsc}\left (c x \right )}{3}+\frac {2 \left (2 d \operatorname {EllipticF}\left (\sqrt {e x +d}\, \sqrt {\frac {c}{c d -e}}, \sqrt {\frac {c d -e}{c d +e}}\right ) c -\operatorname {EllipticE}\left (\sqrt {e x +d}\, \sqrt {\frac {c}{c d -e}}, \sqrt {\frac {c d -e}{c d +e}}\right ) c d -d \operatorname {EllipticPi}\left (\sqrt {e x +d}\, \sqrt {\frac {c}{c d -e}}, \frac {c d -e}{c d}, \frac {\sqrt {\frac {c}{c d +e}}}{\sqrt {\frac {c}{c d -e}}}\right ) c +\operatorname {EllipticF}\left (\sqrt {e x +d}\, \sqrt {\frac {c}{c d -e}}, \sqrt {\frac {c d -e}{c d +e}}\right ) e -\operatorname {EllipticE}\left (\sqrt {e x +d}\, \sqrt {\frac {c}{c d -e}}, \sqrt {\frac {c d -e}{c d +e}}\right ) e \right ) \sqrt {\frac {-c \left (e x +d \right )+c d +e}{c d +e}}\, \sqrt {\frac {-c \left (e x +d \right )+c d -e}{c d -e}}}{3 c^{2} \sqrt {\frac {c}{c d -e}}\, x \sqrt {\frac {c^{2} \left (e x +d \right )^{2}-2 c^{2} d \left (e x +d \right )+c^{2} d^{2}-e^{2}}{c^{2} e^{2} x^{2}}}}\right )}{e}\)	\(386\)
default	\(\frac {\frac {2 \left (e x +d \right )^{\frac {3}{2}} a}{3}+2 b \left (\frac {\left (e x +d \right )^{\frac {3}{2}} \operatorname {arccsc}\left (c x \right )}{3}+\frac {2 \left (2 d \operatorname {EllipticF}\left (\sqrt {e x +d}\, \sqrt {\frac {c}{c d -e}}, \sqrt {\frac {c d -e}{c d +e}}\right ) c -\operatorname {EllipticE}\left (\sqrt {e x +d}\, \sqrt {\frac {c}{c d -e}}, \sqrt {\frac {c d -e}{c d +e}}\right ) c d -d \operatorname {EllipticPi}\left (\sqrt {e x +d}\, \sqrt {\frac {c}{c d -e}}, \frac {c d -e}{c d}, \frac {\sqrt {\frac {c}{c d +e}}}{\sqrt {\frac {c}{c d -e}}}\right ) c +\operatorname {EllipticF}\left (\sqrt {e x +d}\, \sqrt {\frac {c}{c d -e}}, \sqrt {\frac {c d -e}{c d +e}}\right ) e -\operatorname {EllipticE}\left (\sqrt {e x +d}\, \sqrt {\frac {c}{c d -e}}, \sqrt {\frac {c d -e}{c d +e}}\right ) e \right ) \sqrt {\frac {-c \left (e x +d \right )+c d +e}{c d +e}}\, \sqrt {\frac {-c \left (e x +d \right )+c d -e}{c d -e}}}{3 c^{2} \sqrt {\frac {c}{c d -e}}\, x \sqrt {\frac {c^{2} \left (e x +d \right )^{2}-2 c^{2} d \left (e x +d \right )+c^{2} d^{2}-e^{2}}{c^{2} e^{2} x^{2}}}}\right )}{e}\)	\(386\)
parts	\(\frac {2 a \left (e x +d \right )^{\frac {3}{2}}}{3 e}+\frac {2 b \left (\frac {\left (e x +d \right )^{\frac {3}{2}} \operatorname {arccsc}\left (c x \right )}{3}+\frac {2 \left (2 d \operatorname {EllipticF}\left (\sqrt {e x +d}\, \sqrt {\frac {c}{c d -e}}, \sqrt {\frac {c d -e}{c d +e}}\right ) c -\operatorname {EllipticE}\left (\sqrt {e x +d}\, \sqrt {\frac {c}{c d -e}}, \sqrt {\frac {c d -e}{c d +e}}\right ) c d -d \operatorname {EllipticPi}\left (\sqrt {e x +d}\, \sqrt {\frac {c}{c d -e}}, \frac {c d -e}{c d}, \frac {\sqrt {\frac {c}{c d +e}}}{\sqrt {\frac {c}{c d -e}}}\right ) c +\operatorname {EllipticF}\left (\sqrt {e x +d}\, \sqrt {\frac {c}{c d -e}}, \sqrt {\frac {c d -e}{c d +e}}\right ) e -\operatorname {EllipticE}\left (\sqrt {e x +d}\, \sqrt {\frac {c}{c d -e}}, \sqrt {\frac {c d -e}{c d +e}}\right ) e \right ) \sqrt {-\frac {c \left (e x +d \right )-c d -e}{c d +e}}\, \sqrt {-\frac {c \left (e x +d \right )-c d +e}{c d -e}}}{3 c^{2} \sqrt {\frac {c}{c d -e}}\, x \sqrt {\frac {c^{2} \left (e x +d \right )^{2}-2 c^{2} d \left (e x +d \right )+c^{2} d^{2}-e^{2}}{c^{2} e^{2} x^{2}}}}\right )}{e}\)	\(390\)

[In]

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

[Out]

2/e*(1/3*(e*x+d)^(3/2)*a+b*(1/3*(e*x+d)^(3/2)*arccsc(c*x)+2/3/c^2*(2*d*EllipticF((e*x+d)^(1/2)*(c/(c*d-e))^(1/
2),((c*d-e)/(c*d+e))^(1/2))*c-EllipticE((e*x+d)^(1/2)*(c/(c*d-e))^(1/2),((c*d-e)/(c*d+e))^(1/2))*c*d-d*Ellipti
cPi((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+EllipticF((e*x+d)^(1/
2)*(c/(c*d-e))^(1/2),((c*d-e)/(c*d+e))^(1/2))*e-EllipticE((e*x+d)^(1/2)*(c/(c*d-e))^(1/2),((c*d-e)/(c*d+e))^(1
/2))*e)*((-c*(e*x+d)+c*d+e)/(c*d+e))^(1/2)*((-c*(e*x+d)+c*d-e)/(c*d-e))^(1/2)/(c/(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]

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

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F(-2)]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

integrate((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)

Mupad [F(-1)]

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

[In]

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

[Out]

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