Integrand size = 19, antiderivative size = 238 \[ \int \frac {x \left (a+b \csc ^{-1}(c x)\right )}{(d+e x)^{3/2}} \, dx=\frac {2 d \left (a+b \csc ^{-1}(c x)\right )}{e^2 \sqrt {d+e x}}+\frac {2 \sqrt {d+e x} \left (a+b \csc ^{-1}(c x)\right )}{e^2}-\frac {4 b \sqrt {\frac {c (d+e x)}{c d+e}} \sqrt {1-c^2 x^2} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{c^2 e \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {d+e x}}-\frac {8 b d \sqrt {\frac {c (d+e x)}{c d+e}} \sqrt {1-c^2 x^2} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{c e^2 \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {d+e x}} \]
2*d*(a+b*arccsc(c*x))/e^2/(e*x+d)^(1/2)+2*(a+b*arccsc(c*x))*(e*x+d)^(1/2)/ e^2-4*b*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/x/(1-1/c^2/x^2)^(1/2)/(e* x+d)^(1/2)-8*b*d*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^2/x/(1-1/c^2/x^ 2)^(1/2)/(e*x+d)^(1/2)
Result contains complex when optimal does not.
Time = 12.57 (sec) , antiderivative size = 226, normalized size of antiderivative = 0.95 \[ \int \frac {x \left (a+b \csc ^{-1}(c x)\right )}{(d+e x)^{3/2}} \, dx=\frac {2 \left (\frac {a (2 d+e x)}{\sqrt {d+e x}}+\frac {b (2 d+e x) \csc ^{-1}(c x)}{\sqrt {d+e x}}-\frac {2 i b \sqrt {\frac {e (1+c x)}{-c d+e}} \sqrt {\frac {e-c e x}{c d+e}} \left (\operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {-\frac {c}{c d+e}} \sqrt {d+e x}\right ),\frac {c d+e}{c d-e}\right )-2 \operatorname {EllipticPi}\left (1+\frac {e}{c d},i \text {arcsinh}\left (\sqrt {-\frac {c}{c d+e}} \sqrt {d+e x}\right ),\frac {c d+e}{c d-e}\right )\right )}{c \sqrt {-\frac {c}{c d+e}} \sqrt {1-\frac {1}{c^2 x^2}} x}\right )}{e^2} \]
(2*((a*(2*d + e*x))/Sqrt[d + e*x] + (b*(2*d + e*x)*ArcCsc[c*x])/Sqrt[d + e *x] - ((2*I)*b*Sqrt[(e*(1 + c*x))/(-(c*d) + e)]*Sqrt[(e - c*e*x)/(c*d + e) ]*(EllipticF[I*ArcSinh[Sqrt[-(c/(c*d + e))]*Sqrt[d + e*x]], (c*d + e)/(c*d - e)] - 2*EllipticPi[1 + e/(c*d), I*ArcSinh[Sqrt[-(c/(c*d + e))]*Sqrt[d + e*x]], (c*d + e)/(c*d - e)]))/(c*Sqrt[-(c/(c*d + e))]*Sqrt[1 - 1/(c^2*x^2 )]*x)))/e^2
Time = 1.53 (sec) , antiderivative size = 225, normalized size of antiderivative = 0.95, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.579, Rules used = {5770, 27, 7272, 2351, 27, 511, 321, 632, 186, 413, 412}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {x \left (a+b \csc ^{-1}(c x)\right )}{(d+e x)^{3/2}} \, dx\) |
| \(\Big \downarrow \) 5770 |
| \(\displaystyle \frac {b \int \frac {2 (2 d+e x)}{e^2 \sqrt {1-\frac {1}{c^2 x^2}} x^2 \sqrt {d+e x}}dx}{c}+\frac {2 \sqrt {d+e x} \left (a+b \csc ^{-1}(c x)\right )}{e^2}+\frac {2 d \left (a+b \csc ^{-1}(c x)\right )}{e^2 \sqrt {d+e x}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 b \int \frac {2 d+e x}{\sqrt {1-\frac {1}{c^2 x^2}} x^2 \sqrt {d+e x}}dx}{c e^2}+\frac {2 \sqrt {d+e x} \left (a+b \csc ^{-1}(c x)\right )}{e^2}+\frac {2 d \left (a+b \csc ^{-1}(c x)\right )}{e^2 \sqrt {d+e x}}\) |
| \(\Big \downarrow \) 7272 |
| \(\displaystyle \frac {2 b \sqrt {1-c^2 x^2} \int \frac {2 d+e x}{x \sqrt {d+e x} \sqrt {1-c^2 x^2}}dx}{c e^2 x \sqrt {1-\frac {1}{c^2 x^2}}}+\frac {2 \sqrt {d+e x} \left (a+b \csc ^{-1}(c x)\right )}{e^2}+\frac {2 d \left (a+b \csc ^{-1}(c x)\right )}{e^2 \sqrt {d+e x}}\) |
| \(\Big \downarrow \) 2351 |
| \(\displaystyle \frac {2 b \sqrt {1-c^2 x^2} \left (\int \frac {e}{\sqrt {d+e x} \sqrt {1-c^2 x^2}}dx+2 d \int \frac {1}{x \sqrt {d+e x} \sqrt {1-c^2 x^2}}dx\right )}{c e^2 x \sqrt {1-\frac {1}{c^2 x^2}}}+\frac {2 \sqrt {d+e x} \left (a+b \csc ^{-1}(c x)\right )}{e^2}+\frac {2 d \left (a+b \csc ^{-1}(c x)\right )}{e^2 \sqrt {d+e x}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 b \sqrt {1-c^2 x^2} \left (e \int \frac {1}{\sqrt {d+e x} \sqrt {1-c^2 x^2}}dx+2 d \int \frac {1}{x \sqrt {d+e x} \sqrt {1-c^2 x^2}}dx\right )}{c e^2 x \sqrt {1-\frac {1}{c^2 x^2}}}+\frac {2 \sqrt {d+e x} \left (a+b \csc ^{-1}(c x)\right )}{e^2}+\frac {2 d \left (a+b \csc ^{-1}(c x)\right )}{e^2 \sqrt {d+e x}}\) |
| \(\Big \downarrow \) 511 |
| \(\displaystyle \frac {2 b \sqrt {1-c^2 x^2} \left (2 d \int \frac {1}{x \sqrt {d+e x} \sqrt {1-c^2 x^2}}dx-\frac {2 e \sqrt {\frac {c (d+e x)}{c d+e}} \int \frac {1}{\sqrt {1-\frac {e (1-c x)}{c d+e}} \sqrt {\frac {1}{2} (c x-1)+1}}d\frac {\sqrt {1-c x}}{\sqrt {2}}}{c \sqrt {d+e x}}\right )}{c e^2 x \sqrt {1-\frac {1}{c^2 x^2}}}+\frac {2 \sqrt {d+e x} \left (a+b \csc ^{-1}(c x)\right )}{e^2}+\frac {2 d \left (a+b \csc ^{-1}(c x)\right )}{e^2 \sqrt {d+e x}}\) |
| \(\Big \downarrow \) 321 |
| \(\displaystyle \frac {2 b \sqrt {1-c^2 x^2} \left (2 d \int \frac {1}{x \sqrt {d+e x} \sqrt {1-c^2 x^2}}dx-\frac {2 e \sqrt {\frac {c (d+e x)}{c d+e}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{c \sqrt {d+e x}}\right )}{c e^2 x \sqrt {1-\frac {1}{c^2 x^2}}}+\frac {2 \sqrt {d+e x} \left (a+b \csc ^{-1}(c x)\right )}{e^2}+\frac {2 d \left (a+b \csc ^{-1}(c x)\right )}{e^2 \sqrt {d+e x}}\) |
| \(\Big \downarrow \) 632 |
| \(\displaystyle \frac {2 b \sqrt {1-c^2 x^2} \left (2 d \int \frac {1}{x \sqrt {1-c x} \sqrt {c x+1} \sqrt {d+e x}}dx-\frac {2 e \sqrt {\frac {c (d+e x)}{c d+e}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{c \sqrt {d+e x}}\right )}{c e^2 x \sqrt {1-\frac {1}{c^2 x^2}}}+\frac {2 \sqrt {d+e x} \left (a+b \csc ^{-1}(c x)\right )}{e^2}+\frac {2 d \left (a+b \csc ^{-1}(c x)\right )}{e^2 \sqrt {d+e x}}\) |
| \(\Big \downarrow \) 186 |
| \(\displaystyle \frac {2 b \sqrt {1-c^2 x^2} \left (-4 d \int \frac {1}{c x \sqrt {c x+1} \sqrt {d+\frac {e}{c}-\frac {e (1-c x)}{c}}}d\sqrt {1-c x}-\frac {2 e \sqrt {\frac {c (d+e x)}{c d+e}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{c \sqrt {d+e x}}\right )}{c e^2 x \sqrt {1-\frac {1}{c^2 x^2}}}+\frac {2 \sqrt {d+e x} \left (a+b \csc ^{-1}(c x)\right )}{e^2}+\frac {2 d \left (a+b \csc ^{-1}(c x)\right )}{e^2 \sqrt {d+e x}}\) |
| \(\Big \downarrow \) 413 |
| \(\displaystyle \frac {2 b \sqrt {1-c^2 x^2} \left (-\frac {4 d \sqrt {1-\frac {e (1-c x)}{c d+e}} \int \frac {1}{c x \sqrt {c x+1} \sqrt {1-\frac {e (1-c x)}{c d+e}}}d\sqrt {1-c x}}{\sqrt {-\frac {e (1-c x)}{c}+\frac {e}{c}+d}}-\frac {2 e \sqrt {\frac {c (d+e x)}{c d+e}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{c \sqrt {d+e x}}\right )}{c e^2 x \sqrt {1-\frac {1}{c^2 x^2}}}+\frac {2 \sqrt {d+e x} \left (a+b \csc ^{-1}(c x)\right )}{e^2}+\frac {2 d \left (a+b \csc ^{-1}(c x)\right )}{e^2 \sqrt {d+e x}}\) |
| \(\Big \downarrow \) 412 |
| \(\displaystyle \frac {2 \sqrt {d+e x} \left (a+b \csc ^{-1}(c x)\right )}{e^2}+\frac {2 d \left (a+b \csc ^{-1}(c x)\right )}{e^2 \sqrt {d+e x}}+\frac {2 b \sqrt {1-c^2 x^2} \left (-\frac {2 e \sqrt {\frac {c (d+e x)}{c d+e}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{c \sqrt {d+e x}}-\frac {4 d \sqrt {1-\frac {e (1-c x)}{c d+e}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{\sqrt {-\frac {e (1-c x)}{c}+\frac {e}{c}+d}}\right )}{c e^2 x \sqrt {1-\frac {1}{c^2 x^2}}}\) |
(2*d*(a + b*ArcCsc[c*x]))/(e^2*Sqrt[d + e*x]) + (2*Sqrt[d + e*x]*(a + b*Ar cCsc[c*x]))/e^2 + (2*b*Sqrt[1 - c^2*x^2]*((-2*e*Sqrt[(c*(d + e*x))/(c*d + e)]*EllipticF[ArcSin[Sqrt[1 - c*x]/Sqrt[2]], (2*e)/(c*d + e)])/(c*Sqrt[d + e*x]) - (4*d*Sqrt[1 - (e*(1 - c*x))/(c*d + e)]*EllipticPi[2, ArcSin[Sqrt[ 1 - c*x]/Sqrt[2]], (2*e)/(c*d + e)])/Sqrt[d + e/c - (e*(1 - c*x))/c]))/(c* e^2*Sqrt[1 - 1/(c^2*x^2)]*x)
3.1.65.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[1/(((a_.) + (b_.)*(x_))*Sqrt[(c_.) + (d_.)*(x_)]*Sqrt[(e_.) + (f_.)*(x_ )]*Sqrt[(g_.) + (h_.)*(x_)]), x_] :> Simp[-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]
Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> S imp[(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] && GtQ[a, 0] && !(NegQ[b/a] && SimplerSqrtQ[-b/a, -d/c])
Int[1/(((a_) + (b_.)*(x_)^2)*Sqrt[(c_) + (d_.)*(x_)^2]*Sqrt[(e_) + (f_.)*(x _)^2]), x_Symbol] :> Simp[(1/(a*Sqrt[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] && S implerSqrtQ[-f/e, -d/c])
Int[1/(((a_) + (b_.)*(x_)^2)*Sqrt[(c_) + (d_.)*(x_)^2]*Sqrt[(e_) + (f_.)*(x _)^2]), x_Symbol] :> Simp[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]
Int[1/(Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(a_) + (b_.)*(x_)^2]), x_Symbol] :> Wit h[{q = Rt[-b/a, 2]}, Simp[-2*(Sqrt[q*((c + d*x)/(d + c*q))]/(Sqrt[a]*q*Sqrt [c + d*x])) Subst[Int[1/(Sqrt[1 - 2*d*(x^2/(d + c*q))]*Sqrt[1 - x^2]), x] , x, Sqrt[(1 - q*x)/2]], x]] /; FreeQ[{a, b, c, d}, x] && NegQ[b/a] && GtQ[ a, 0]
Int[1/((x_)*Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(a_) + (b_.)*(x_)^2]), x_Symbol] : > With[{q = Rt[-b/a, 2]}, Simp[1/Sqrt[a] Int[1/(x*Sqrt[c + d*x]*Sqrt[1 - q*x]*Sqrt[1 + q*x]), x], x]] /; FreeQ[{a, b, c, d}, x] && NegQ[b/a] && GtQ[ a, 0]
Int[((Px_)*((c_) + (d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_)^2)^(p_.))/(x_), x_S ymbol] :> Int[PolynomialQuotient[Px, x, x]*(c + d*x)^n*(a + b*x^2)^p, x] + Simp[PolynomialRemainder[Px, x, x] Int[(c + d*x)^n*((a + b*x^2)^p/x), x], x] /; FreeQ[{a, b, c, d, n, p}, x] && PolynomialQ[Px, x]
Int[((a_.) + ArcCsc[(c_.)*(x_)]*(b_.))*(u_), x_Symbol] :> With[{v = IntHide [u, x]}, Simp[(a + b*ArcCsc[c*x]) v, x] + Simp[b/c Int[SimplifyIntegran d[v/(x^2*Sqrt[1 - 1/(c^2*x^2)]), x], x], x] /; InverseFunctionFreeQ[v, x]] /; FreeQ[{a, b, c}, x]
Int[(u_.)*((a_.) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[b^IntPart[p]*(( a + b*x^n)^FracPart[p]/(x^(n*FracPart[p])*(1 + a*(1/(x^n*b)))^FracPart[p])) Int[u*x^(n*p)*(1 + a*(1/(x^n*b)))^p, x], x] /; FreeQ[{a, b, p}, x] && ! IntegerQ[p] && ILtQ[n, 0] && !RationalFunctionQ[u, x] && IntegerQ[p + 1/2]
Time = 6.61 (sec) , antiderivative size = 280, normalized size of antiderivative = 1.18
| method | result | size |
| parts | \(\frac {2 a \left (\sqrt {e x +d}+\frac {d}{\sqrt {e x +d}}\right )}{e^{2}}+\frac {2 b \left (\sqrt {e x +d}\, \operatorname {arccsc}\left (c x \right )+\frac {\operatorname {arccsc}\left (c x \right ) d}{\sqrt {e x +d}}+\frac {2 \sqrt {-\frac {c \left (e x +d \right )-c d +e}{c d -e}}\, \sqrt {-\frac {c \left (e x +d \right )-c d -e}{c d +e}}\, \left (\operatorname {EllipticF}\left (\sqrt {e x +d}\, \sqrt {\frac {c}{c d -e}}, \sqrt {\frac {c d -e}{c d +e}}\right )-2 \operatorname {EllipticPi}\left (\sqrt {e x +d}\, \sqrt {\frac {c}{c d -e}}, \frac {c d -e}{c d}, \frac {\sqrt {\frac {c}{c d +e}}}{\sqrt {\frac {c}{c d -e}}}\right )\right )}{c \sqrt {\frac {c^{2} \left (e x +d \right )^{2}-2 c^{2} d \left (e x +d \right )+c^{2} d^{2}-e^{2}}{c^{2} e^{2} x^{2}}}\, x \sqrt {\frac {c}{c d -e}}}\right )}{e^{2}}\) | \(280\) |
| derivativedivides | \(\frac {-2 a \left (-\sqrt {e x +d}-\frac {d}{\sqrt {e x +d}}\right )-2 b \left (-\sqrt {e x +d}\, \operatorname {arccsc}\left (c x \right )-\frac {\operatorname {arccsc}\left (c x \right ) d}{\sqrt {e x +d}}-\frac {2 \sqrt {\frac {-c \left (e x +d \right )+c d -e}{c d -e}}\, \sqrt {\frac {-c \left (e x +d \right )+c d +e}{c d +e}}\, \left (\operatorname {EllipticF}\left (\sqrt {e x +d}\, \sqrt {\frac {c}{c d -e}}, \sqrt {\frac {c d -e}{c d +e}}\right )-2 \operatorname {EllipticPi}\left (\sqrt {e x +d}\, \sqrt {\frac {c}{c d -e}}, \frac {c d -e}{c d}, \frac {\sqrt {\frac {c}{c d +e}}}{\sqrt {\frac {c}{c d -e}}}\right )\right )}{c \sqrt {\frac {c^{2} \left (e x +d \right )^{2}-2 c^{2} d \left (e x +d \right )+c^{2} d^{2}-e^{2}}{c^{2} e^{2} x^{2}}}\, x \sqrt {\frac {c}{c d -e}}}\right )}{e^{2}}\) | \(282\) |
| default | \(\frac {-2 a \left (-\sqrt {e x +d}-\frac {d}{\sqrt {e x +d}}\right )-2 b \left (-\sqrt {e x +d}\, \operatorname {arccsc}\left (c x \right )-\frac {\operatorname {arccsc}\left (c x \right ) d}{\sqrt {e x +d}}-\frac {2 \sqrt {\frac {-c \left (e x +d \right )+c d -e}{c d -e}}\, \sqrt {\frac {-c \left (e x +d \right )+c d +e}{c d +e}}\, \left (\operatorname {EllipticF}\left (\sqrt {e x +d}\, \sqrt {\frac {c}{c d -e}}, \sqrt {\frac {c d -e}{c d +e}}\right )-2 \operatorname {EllipticPi}\left (\sqrt {e x +d}\, \sqrt {\frac {c}{c d -e}}, \frac {c d -e}{c d}, \frac {\sqrt {\frac {c}{c d +e}}}{\sqrt {\frac {c}{c d -e}}}\right )\right )}{c \sqrt {\frac {c^{2} \left (e x +d \right )^{2}-2 c^{2} d \left (e x +d \right )+c^{2} d^{2}-e^{2}}{c^{2} e^{2} x^{2}}}\, x \sqrt {\frac {c}{c d -e}}}\right )}{e^{2}}\) | \(282\) |
2*a/e^2*((e*x+d)^(1/2)+d/(e*x+d)^(1/2))+2*b/e^2*((e*x+d)^(1/2)*arccsc(c*x) +arccsc(c*x)*d/(e*x+d)^(1/2)+2/c*(-(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))-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*(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)/x/(c/(c*d-e))^(1/2))
\[ \int \frac {x \left (a+b \csc ^{-1}(c x)\right )}{(d+e x)^{3/2}} \, dx=\int { \frac {{\left (b \operatorname {arccsc}\left (c x\right ) + a\right )} x}{{\left (e x + d\right )}^{\frac {3}{2}}} \,d x } \]
\[ \int \frac {x \left (a+b \csc ^{-1}(c x)\right )}{(d+e x)^{3/2}} \, dx=\int \frac {x \left (a + b \operatorname {acsc}{\left (c x \right )}\right )}{\left (d + e x\right )^{\frac {3}{2}}}\, dx \]
Exception generated. \[ \int \frac {x \left (a+b \csc ^{-1}(c x)\right )}{(d+e x)^{3/2}} \, dx=\text {Exception raised: ValueError} \]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(e+c*d>0)', see `assume?` for mor e details)
\[ \int \frac {x \left (a+b \csc ^{-1}(c x)\right )}{(d+e x)^{3/2}} \, dx=\int { \frac {{\left (b \operatorname {arccsc}\left (c x\right ) + a\right )} x}{{\left (e x + d\right )}^{\frac {3}{2}}} \,d x } \]
Timed out. \[ \int \frac {x \left (a+b \csc ^{-1}(c x)\right )}{(d+e x)^{3/2}} \, dx=\int \frac {x\,\left (a+b\,\mathrm {asin}\left (\frac {1}{c\,x}\right )\right )}{{\left (d+e\,x\right )}^{3/2}} \,d x \]