\(\int \frac {x^5 (a+b \csc ^{-1}(c x))}{(d+e x^2)^{3/2}} \, dx\) [147]

Optimal result

Integrand size = 23, antiderivative size = 252 \[ \int \frac {x^5 \left (a+b \csc ^{-1}(c x)\right )}{\left (d+e x^2\right )^{3/2}} \, dx=\frac {b x \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}{6 c e^2 \sqrt {c^2 x^2}}-\frac {d^2 \left (a+b \csc ^{-1}(c x)\right )}{e^3 \sqrt {d+e x^2}}-\frac {2 d \sqrt {d+e x^2} \left (a+b \csc ^{-1}(c x)\right )}{e^3}+\frac {\left (d+e x^2\right )^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e^3}+\frac {8 b c d^{3/2} x \arctan \left (\frac {\sqrt {d+e x^2}}{\sqrt {d} \sqrt {-1+c^2 x^2}}\right )}{3 e^3 \sqrt {c^2 x^2}}-\frac {b \left (9 c^2 d-e\right ) x \text {arctanh}\left (\frac {\sqrt {e} \sqrt {-1+c^2 x^2}}{c \sqrt {d+e x^2}}\right )}{6 c^2 e^{5/2} \sqrt {c^2 x^2}} \] Output:

1/6*b*x*(c^2*x^2-1)^(1/2)*(e*x^2+d)^(1/2)/c/e^2/(c^2*x^2)^(1/2)-d^2*(a+b*a 
rccsc(c*x))/e^3/(e*x^2+d)^(1/2)-2*d*(e*x^2+d)^(1/2)*(a+b*arccsc(c*x))/e^3+ 
1/3*(e*x^2+d)^(3/2)*(a+b*arccsc(c*x))/e^3+8/3*b*c*d^(3/2)*x*arctan((e*x^2+ 
d)^(1/2)/d^(1/2)/(c^2*x^2-1)^(1/2))/e^3/(c^2*x^2)^(1/2)-1/6*b*(9*c^2*d-e)* 
x*arctanh(e^(1/2)*(c^2*x^2-1)^(1/2)/c/(e*x^2+d)^(1/2))/c^2/e^(5/2)/(c^2*x^ 
2)^(1/2)
 

Mathematica [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 6 vs. order 3 in optimal.

Time = 1.43 (sec) , antiderivative size = 263, normalized size of antiderivative = 1.04 \[ \int \frac {x^5 \left (a+b \csc ^{-1}(c x)\right )}{\left (d+e x^2\right )^{3/2}} \, dx=\frac {16 b d^2 \sqrt {1+\frac {d}{e x^2}} \left (-1+c^2 x^2\right ) \operatorname {AppellF1}\left (1,\frac {1}{2},\frac {1}{2},2,\frac {1}{c^2 x^2},-\frac {d}{e x^2}\right )+b e \left (-9 c^2 d+e\right ) \sqrt {1-\frac {1}{c^2 x^2}} x^4 \sqrt {1-c^2 x^2} \sqrt {1+\frac {e x^2}{d}} \operatorname {AppellF1}\left (1,\frac {1}{2},\frac {1}{2},2,c^2 x^2,-\frac {e x^2}{d}\right )+2 x \left (-1+c^2 x^2\right ) \left (b e \sqrt {1-\frac {1}{c^2 x^2}} x \left (d+e x^2\right )-2 a c \left (8 d^2+4 d e x^2-e^2 x^4\right )-2 b c \left (8 d^2+4 d e x^2-e^2 x^4\right ) \csc ^{-1}(c x)\right )}{12 c e^3 x \left (-1+c^2 x^2\right ) \sqrt {d+e x^2}} \] Input:

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

Output:

(16*b*d^2*Sqrt[1 + d/(e*x^2)]*(-1 + c^2*x^2)*AppellF1[1, 1/2, 1/2, 2, 1/(c 
^2*x^2), -(d/(e*x^2))] + b*e*(-9*c^2*d + e)*Sqrt[1 - 1/(c^2*x^2)]*x^4*Sqrt 
[1 - c^2*x^2]*Sqrt[1 + (e*x^2)/d]*AppellF1[1, 1/2, 1/2, 2, c^2*x^2, -((e*x 
^2)/d)] + 2*x*(-1 + c^2*x^2)*(b*e*Sqrt[1 - 1/(c^2*x^2)]*x*(d + e*x^2) - 2* 
a*c*(8*d^2 + 4*d*e*x^2 - e^2*x^4) - 2*b*c*(8*d^2 + 4*d*e*x^2 - e^2*x^4)*Ar 
cCsc[c*x]))/(12*c*e^3*x*(-1 + c^2*x^2)*Sqrt[d + e*x^2])
 

Rubi [A] (verified)

Time = 1.37 (sec) , antiderivative size = 234, normalized size of antiderivative = 0.93, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.435, Rules used = {5762, 27, 7282, 2118, 27, 175, 66, 104, 217, 221}

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.

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

\(\Big \downarrow \) 5762

\(\displaystyle \frac {b c x \int -\frac {-e^2 x^4+4 d e x^2+8 d^2}{3 e^3 x \sqrt {c^2 x^2-1} \sqrt {e x^2+d}}dx}{\sqrt {c^2 x^2}}-\frac {d^2 \left (a+b \csc ^{-1}(c x)\right )}{e^3 \sqrt {d+e x^2}}-\frac {2 d \sqrt {d+e x^2} \left (a+b \csc ^{-1}(c x)\right )}{e^3}+\frac {\left (d+e x^2\right )^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e^3}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {b c x \int \frac {-e^2 x^4+4 d e x^2+8 d^2}{x \sqrt {c^2 x^2-1} \sqrt {e x^2+d}}dx}{3 e^3 \sqrt {c^2 x^2}}-\frac {d^2 \left (a+b \csc ^{-1}(c x)\right )}{e^3 \sqrt {d+e x^2}}-\frac {2 d \sqrt {d+e x^2} \left (a+b \csc ^{-1}(c x)\right )}{e^3}+\frac {\left (d+e x^2\right )^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e^3}\)

\(\Big \downarrow \) 7282

\(\displaystyle -\frac {b c x \int \frac {-e^2 x^4+4 d e x^2+8 d^2}{x^2 \sqrt {c^2 x^2-1} \sqrt {e x^2+d}}dx^2}{6 e^3 \sqrt {c^2 x^2}}-\frac {d^2 \left (a+b \csc ^{-1}(c x)\right )}{e^3 \sqrt {d+e x^2}}-\frac {2 d \sqrt {d+e x^2} \left (a+b \csc ^{-1}(c x)\right )}{e^3}+\frac {\left (d+e x^2\right )^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e^3}\)

\(\Big \downarrow \) 2118

\(\displaystyle -\frac {b c x \left (\frac {\int \frac {e \left (16 c^2 d^2+\left (9 c^2 d-e\right ) e x^2\right )}{2 x^2 \sqrt {c^2 x^2-1} \sqrt {e x^2+d}}dx^2}{c^2 e}-\frac {e \sqrt {c^2 x^2-1} \sqrt {d+e x^2}}{c^2}\right )}{6 e^3 \sqrt {c^2 x^2}}-\frac {d^2 \left (a+b \csc ^{-1}(c x)\right )}{e^3 \sqrt {d+e x^2}}-\frac {2 d \sqrt {d+e x^2} \left (a+b \csc ^{-1}(c x)\right )}{e^3}+\frac {\left (d+e x^2\right )^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e^3}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {b c x \left (\frac {\int \frac {16 c^2 d^2+\left (9 c^2 d-e\right ) e x^2}{x^2 \sqrt {c^2 x^2-1} \sqrt {e x^2+d}}dx^2}{2 c^2}-\frac {e \sqrt {c^2 x^2-1} \sqrt {d+e x^2}}{c^2}\right )}{6 e^3 \sqrt {c^2 x^2}}-\frac {d^2 \left (a+b \csc ^{-1}(c x)\right )}{e^3 \sqrt {d+e x^2}}-\frac {2 d \sqrt {d+e x^2} \left (a+b \csc ^{-1}(c x)\right )}{e^3}+\frac {\left (d+e x^2\right )^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e^3}\)

\(\Big \downarrow \) 175

\(\displaystyle -\frac {b c x \left (\frac {16 c^2 d^2 \int \frac {1}{x^2 \sqrt {c^2 x^2-1} \sqrt {e x^2+d}}dx^2+e \left (9 c^2 d-e\right ) \int \frac {1}{\sqrt {c^2 x^2-1} \sqrt {e x^2+d}}dx^2}{2 c^2}-\frac {e \sqrt {c^2 x^2-1} \sqrt {d+e x^2}}{c^2}\right )}{6 e^3 \sqrt {c^2 x^2}}-\frac {d^2 \left (a+b \csc ^{-1}(c x)\right )}{e^3 \sqrt {d+e x^2}}-\frac {2 d \sqrt {d+e x^2} \left (a+b \csc ^{-1}(c x)\right )}{e^3}+\frac {\left (d+e x^2\right )^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e^3}\)

\(\Big \downarrow \) 66

\(\displaystyle -\frac {b c x \left (\frac {16 c^2 d^2 \int \frac {1}{x^2 \sqrt {c^2 x^2-1} \sqrt {e x^2+d}}dx^2+2 e \left (9 c^2 d-e\right ) \int \frac {1}{c^2-e x^4}d\frac {\sqrt {c^2 x^2-1}}{\sqrt {e x^2+d}}}{2 c^2}-\frac {e \sqrt {c^2 x^2-1} \sqrt {d+e x^2}}{c^2}\right )}{6 e^3 \sqrt {c^2 x^2}}-\frac {d^2 \left (a+b \csc ^{-1}(c x)\right )}{e^3 \sqrt {d+e x^2}}-\frac {2 d \sqrt {d+e x^2} \left (a+b \csc ^{-1}(c x)\right )}{e^3}+\frac {\left (d+e x^2\right )^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e^3}\)

\(\Big \downarrow \) 104

\(\displaystyle -\frac {b c x \left (\frac {32 c^2 d^2 \int \frac {1}{-x^4-d}d\frac {\sqrt {e x^2+d}}{\sqrt {c^2 x^2-1}}+2 e \left (9 c^2 d-e\right ) \int \frac {1}{c^2-e x^4}d\frac {\sqrt {c^2 x^2-1}}{\sqrt {e x^2+d}}}{2 c^2}-\frac {e \sqrt {c^2 x^2-1} \sqrt {d+e x^2}}{c^2}\right )}{6 e^3 \sqrt {c^2 x^2}}-\frac {d^2 \left (a+b \csc ^{-1}(c x)\right )}{e^3 \sqrt {d+e x^2}}-\frac {2 d \sqrt {d+e x^2} \left (a+b \csc ^{-1}(c x)\right )}{e^3}+\frac {\left (d+e x^2\right )^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e^3}\)

\(\Big \downarrow \) 217

\(\displaystyle -\frac {b c x \left (\frac {2 e \left (9 c^2 d-e\right ) \int \frac {1}{c^2-e x^4}d\frac {\sqrt {c^2 x^2-1}}{\sqrt {e x^2+d}}-32 c^2 d^{3/2} \arctan \left (\frac {\sqrt {d+e x^2}}{\sqrt {d} \sqrt {c^2 x^2-1}}\right )}{2 c^2}-\frac {e \sqrt {c^2 x^2-1} \sqrt {d+e x^2}}{c^2}\right )}{6 e^3 \sqrt {c^2 x^2}}-\frac {d^2 \left (a+b \csc ^{-1}(c x)\right )}{e^3 \sqrt {d+e x^2}}-\frac {2 d \sqrt {d+e x^2} \left (a+b \csc ^{-1}(c x)\right )}{e^3}+\frac {\left (d+e x^2\right )^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e^3}\)

\(\Big \downarrow \) 221

\(\displaystyle -\frac {d^2 \left (a+b \csc ^{-1}(c x)\right )}{e^3 \sqrt {d+e x^2}}-\frac {2 d \sqrt {d+e x^2} \left (a+b \csc ^{-1}(c x)\right )}{e^3}+\frac {\left (d+e x^2\right )^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e^3}-\frac {b c x \left (\frac {\frac {2 \sqrt {e} \left (9 c^2 d-e\right ) \text {arctanh}\left (\frac {\sqrt {e} \sqrt {c^2 x^2-1}}{c \sqrt {d+e x^2}}\right )}{c}-32 c^2 d^{3/2} \arctan \left (\frac {\sqrt {d+e x^2}}{\sqrt {d} \sqrt {c^2 x^2-1}}\right )}{2 c^2}-\frac {e \sqrt {c^2 x^2-1} \sqrt {d+e x^2}}{c^2}\right )}{6 e^3 \sqrt {c^2 x^2}}\)

Input:

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

Output:

-((d^2*(a + b*ArcCsc[c*x]))/(e^3*Sqrt[d + e*x^2])) - (2*d*Sqrt[d + e*x^2]* 
(a + b*ArcCsc[c*x]))/e^3 + ((d + e*x^2)^(3/2)*(a + b*ArcCsc[c*x]))/(3*e^3) 
 - (b*c*x*(-((e*Sqrt[-1 + c^2*x^2]*Sqrt[d + e*x^2])/c^2) + (-32*c^2*d^(3/2 
)*ArcTan[Sqrt[d + e*x^2]/(Sqrt[d]*Sqrt[-1 + c^2*x^2])] + (2*(9*c^2*d - e)* 
Sqrt[e]*ArcTanh[(Sqrt[e]*Sqrt[-1 + c^2*x^2])/(c*Sqrt[d + e*x^2])])/c)/(2*c 
^2)))/(6*e^3*Sqrt[c^2*x^2])
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 66

Int[1/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Simp[ 
2   Subst[Int[1/(b - d*x^2), x], x, Sqrt[a + b*x]/Sqrt[c + d*x]], x] /; Fre 
eQ[{a, b, c, d}, x] &&  !GtQ[c - a*(d/b), 0]
 

rule 104

Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x 
_)), x_] :> With[{q = Denominator[m]}, Simp[q   Subst[Int[x^(q*(m + 1) - 1) 
/(b*e - a*f - (d*e - c*f)*x^q), x], x, (a + b*x)^(1/q)/(c + d*x)^(1/q)], x] 
] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[m + n + 1, 0] && RationalQ[n] && L 
tQ[-1, m, 0] && SimplerQ[a + b*x, c + d*x]
 

rule 175

Int[(((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_ 
)))/((a_.) + (b_.)*(x_)), x_] :> Simp[h/b   Int[(c + d*x)^n*(e + f*x)^p, x] 
, x] + Simp[(b*g - a*h)/b   Int[(c + d*x)^n*((e + f*x)^p/(a + b*x)), x], x] 
 /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x]
 

rule 217

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^( 
-1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & 
& (LtQ[a, 0] || LtQ[b, 0])
 

rule 221

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x 
/Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
 

rule 2118

Int[(Px_)*((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f 
_.)*(x_))^(p_.), x_Symbol] :> With[{q = Expon[Px, x], k = Coeff[Px, x, Expo 
n[Px, x]]}, Simp[k*(a + b*x)^(m + q - 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 
1)/(d*f*b^(q - 1)*(m + n + p + q + 1))), x] + Simp[1/(d*f*b^q*(m + n + p + 
q + 1))   Int[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p*ExpandToSum[d*f*b^q*(m + 
n + p + q + 1)*Px - d*f*k*(m + n + p + q + 1)*(a + b*x)^q + k*(a + b*x)^(q 
- 2)*(a^2*d*f*(m + n + p + q + 1) - b*(b*c*e*(m + q - 1) + a*(d*e*(n + 1) + 
 c*f*(p + 1))) + b*(a*d*f*(2*(m + q) + n + p) - b*(d*e*(m + q + n) + c*f*(m 
 + q + p)))*x), x], x], x] /; NeQ[m + n + p + q + 1, 0]] /; FreeQ[{a, b, c, 
 d, e, f, m, n, p}, x] && PolyQ[Px, x]
 

rule 5762

Int[((a_.) + ArcCsc[(c_.)*(x_)]*(b_.))*((f_.)*(x_))^(m_.)*((d_.) + (e_.)*(x 
_)^2)^(p_.), x_Symbol] :> With[{u = IntHide[(f*x)^m*(d + e*x^2)^p, x]}, Sim 
p[(a + b*ArcCsc[c*x])   u, x] + Simp[b*c*(x/Sqrt[c^2*x^2])   Int[SimplifyIn 
tegrand[u/(x*Sqrt[c^2*x^2 - 1]), x], x], x]] /; FreeQ[{a, b, c, d, e, f, m, 
 p}, x] && ((IGtQ[p, 0] &&  !(ILtQ[(m - 1)/2, 0] && GtQ[m + 2*p + 3, 0])) | 
| (IGtQ[(m + 1)/2, 0] &&  !(ILtQ[p, 0] && GtQ[m + 2*p + 3, 0])) || (ILtQ[(m 
 + 2*p + 1)/2, 0] &&  !ILtQ[(m - 1)/2, 0]))
 

rule 7282

Int[(u_)/(x_), x_Symbol] :> With[{lst = PowerVariableExpn[u, 0, x]}, Simp[1 
/lst[[2]]   Subst[Int[NormalizeIntegrand[Simplify[lst[[1]]/x], x], x], x, ( 
lst[[3]]*x)^lst[[2]]], x] /;  !FalseQ[lst] && NeQ[lst[[2]], 0]] /; NonsumQ[ 
u] &&  !RationalFunctionQ[u, x]
 

Maple [F]

\[\int \frac {x^{5} \left (a +b \,\operatorname {arccsc}\left (c x \right )\right )}{\left (e \,x^{2}+d \right )^{\frac {3}{2}}}d x\]

Input:

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

Output:

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

Fricas [A] (verification not implemented)

Time = 0.37 (sec) , antiderivative size = 1480, normalized size of antiderivative = 5.87 \[ \int \frac {x^5 \left (a+b \csc ^{-1}(c x)\right )}{\left (d+e x^2\right )^{3/2}} \, dx=\text {Too large to display} \] Input:

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

Output:

[-1/24*((9*b*c^2*d^2 - b*d*e + (9*b*c^2*d*e - b*e^2)*x^2)*sqrt(e)*log(8*c^ 
4*e^2*x^4 + c^4*d^2 - 6*c^2*d*e + 8*(c^4*d*e - c^2*e^2)*x^2 + 4*(2*c^3*e*x 
^2 + c^3*d - c*e)*sqrt(c^2*x^2 - 1)*sqrt(e*x^2 + d)*sqrt(e) + e^2) - 16*(b 
*c^3*d*e*x^2 + b*c^3*d^2)*sqrt(-d)*log(((c^4*d^2 - 6*c^2*d*e + e^2)*x^4 - 
8*(c^2*d^2 - d*e)*x^2 - 4*sqrt(c^2*x^2 - 1)*((c^2*d - e)*x^2 - 2*d)*sqrt(e 
*x^2 + d)*sqrt(-d) + 8*d^2)/x^4) - 4*(2*a*c^3*e^2*x^4 - 8*a*c^3*d*e*x^2 - 
16*a*c^3*d^2 + 2*(b*c^3*e^2*x^4 - 4*b*c^3*d*e*x^2 - 8*b*c^3*d^2)*arccsc(c* 
x) + (b*c*e^2*x^2 + b*c*d*e)*sqrt(c^2*x^2 - 1))*sqrt(e*x^2 + d))/(c^3*e^4* 
x^2 + c^3*d*e^3), 1/24*(32*(b*c^3*d*e*x^2 + b*c^3*d^2)*sqrt(d)*arctan(-1/2 
*sqrt(c^2*x^2 - 1)*((c^2*d - e)*x^2 - 2*d)*sqrt(e*x^2 + d)*sqrt(d)/(c^2*d* 
e*x^4 + (c^2*d^2 - d*e)*x^2 - d^2)) - (9*b*c^2*d^2 - b*d*e + (9*b*c^2*d*e 
- b*e^2)*x^2)*sqrt(e)*log(8*c^4*e^2*x^4 + c^4*d^2 - 6*c^2*d*e + 8*(c^4*d*e 
 - c^2*e^2)*x^2 + 4*(2*c^3*e*x^2 + c^3*d - c*e)*sqrt(c^2*x^2 - 1)*sqrt(e*x 
^2 + d)*sqrt(e) + e^2) + 4*(2*a*c^3*e^2*x^4 - 8*a*c^3*d*e*x^2 - 16*a*c^3*d 
^2 + 2*(b*c^3*e^2*x^4 - 4*b*c^3*d*e*x^2 - 8*b*c^3*d^2)*arccsc(c*x) + (b*c* 
e^2*x^2 + b*c*d*e)*sqrt(c^2*x^2 - 1))*sqrt(e*x^2 + d))/(c^3*e^4*x^2 + c^3* 
d*e^3), 1/12*((9*b*c^2*d^2 - b*d*e + (9*b*c^2*d*e - b*e^2)*x^2)*sqrt(-e)*a 
rctan(1/2*(2*c^2*e*x^2 + c^2*d - e)*sqrt(c^2*x^2 - 1)*sqrt(e*x^2 + d)*sqrt 
(-e)/(c^3*e^2*x^4 - c*d*e + (c^3*d*e - c*e^2)*x^2)) + 8*(b*c^3*d*e*x^2 + b 
*c^3*d^2)*sqrt(-d)*log(((c^4*d^2 - 6*c^2*d*e + e^2)*x^4 - 8*(c^2*d^2 - ...
 

Sympy [F]

\[ \int \frac {x^5 \left (a+b \csc ^{-1}(c x)\right )}{\left (d+e x^2\right )^{3/2}} \, dx=\int \frac {x^{5} \left (a + b \operatorname {acsc}{\left (c x \right )}\right )}{\left (d + e x^{2}\right )^{\frac {3}{2}}}\, dx \] Input:

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

Output:

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

Maxima [F(-2)]

Exception generated. \[ \int \frac {x^5 \left (a+b \csc ^{-1}(c x)\right )}{\left (d+e x^2\right )^{3/2}} \, dx=\text {Exception raised: ValueError} \] Input:

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

Output:

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>0)', see `assume?` for more de 
tails)Is e
 

Giac [F]

\[ \int \frac {x^5 \left (a+b \csc ^{-1}(c x)\right )}{\left (d+e x^2\right )^{3/2}} \, dx=\int { \frac {{\left (b \operatorname {arccsc}\left (c x\right ) + a\right )} x^{5}}{{\left (e x^{2} + d\right )}^{\frac {3}{2}}} \,d x } \] Input:

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

Output:

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

Mupad [F(-1)]

Timed out. \[ \int \frac {x^5 \left (a+b \csc ^{-1}(c x)\right )}{\left (d+e x^2\right )^{3/2}} \, dx=\int \frac {x^5\,\left (a+b\,\mathrm {asin}\left (\frac {1}{c\,x}\right )\right )}{{\left (e\,x^2+d\right )}^{3/2}} \,d x \] Input:

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

Output:

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

Reduce [F]

\[ \int \frac {x^5 \left (a+b \csc ^{-1}(c x)\right )}{\left (d+e x^2\right )^{3/2}} \, dx=\frac {-8 \sqrt {e \,x^{2}+d}\, a \,d^{2}-4 \sqrt {e \,x^{2}+d}\, a d e \,x^{2}+\sqrt {e \,x^{2}+d}\, a \,e^{2} x^{4}+3 \left (\int \frac {\mathit {acsc} \left (c x \right ) x^{5}}{\sqrt {e \,x^{2}+d}\, d +\sqrt {e \,x^{2}+d}\, e \,x^{2}}d x \right ) b d \,e^{3}+3 \left (\int \frac {\mathit {acsc} \left (c x \right ) x^{5}}{\sqrt {e \,x^{2}+d}\, d +\sqrt {e \,x^{2}+d}\, e \,x^{2}}d x \right ) b \,e^{4} x^{2}}{3 e^{3} \left (e \,x^{2}+d \right )} \] Input:

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

Output:

( - 8*sqrt(d + e*x**2)*a*d**2 - 4*sqrt(d + e*x**2)*a*d*e*x**2 + sqrt(d + e 
*x**2)*a*e**2*x**4 + 3*int((acsc(c*x)*x**5)/(sqrt(d + e*x**2)*d + sqrt(d + 
 e*x**2)*e*x**2),x)*b*d*e**3 + 3*int((acsc(c*x)*x**5)/(sqrt(d + e*x**2)*d 
+ sqrt(d + e*x**2)*e*x**2),x)*b*e**4*x**2)/(3*e**3*(d + e*x**2))