\(\int x (d+e x^2)^{3/2} (a+b \csc ^{-1}(c x)) \, dx\) [129]

Optimal result

Integrand size = 21, antiderivative size = 262 \[ \int x \left (d+e x^2\right )^{3/2} \left (a+b \csc ^{-1}(c x)\right ) \, dx=\frac {b \left (7 c^2 d+3 e\right ) x \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}{40 c^3 \sqrt {c^2 x^2}}+\frac {b x \sqrt {-1+c^2 x^2} \left (d+e x^2\right )^{3/2}}{20 c \sqrt {c^2 x^2}}+\frac {\left (d+e x^2\right )^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{5 e}-\frac {b c d^{5/2} x \arctan \left (\frac {\sqrt {d+e x^2}}{\sqrt {d} \sqrt {-1+c^2 x^2}}\right )}{5 e \sqrt {c^2 x^2}}+\frac {b \left (15 c^4 d^2+10 c^2 d e+3 e^2\right ) x \text {arctanh}\left (\frac {\sqrt {e} \sqrt {-1+c^2 x^2}}{c \sqrt {d+e x^2}}\right )}{40 c^4 \sqrt {e} \sqrt {c^2 x^2}} \]

[Out]

1/5*(e*x^2+d)^(5/2)*(a+b*arccsc(c*x))/e-1/5*b*c*d^(5/2)*x*arctan((e*x^2+d)^(1/2)/d^(1/2)/(c^2*x^2-1)^(1/2))/e/
(c^2*x^2)^(1/2)+1/40*b*(15*c^4*d^2+10*c^2*d*e+3*e^2)*x*arctanh(e^(1/2)*(c^2*x^2-1)^(1/2)/c/(e*x^2+d)^(1/2))/c^
4/e^(1/2)/(c^2*x^2)^(1/2)+1/20*b*x*(e*x^2+d)^(3/2)*(c^2*x^2-1)^(1/2)/c/(c^2*x^2)^(1/2)+1/40*b*(7*c^2*d+3*e)*x*
(c^2*x^2-1)^(1/2)*(e*x^2+d)^(1/2)/c^3/(c^2*x^2)^(1/2)

Rubi [A] (verified)

Time = 0.22 (sec) , antiderivative size = 262, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.476, Rules used = {5345, 457, 104, 159, 163, 65, 223, 212, 95, 210} \[ \int x \left (d+e x^2\right )^{3/2} \left (a+b \csc ^{-1}(c x)\right ) \, dx=\frac {\left (d+e x^2\right )^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{5 e}-\frac {b c d^{5/2} x \arctan \left (\frac {\sqrt {d+e x^2}}{\sqrt {d} \sqrt {c^2 x^2-1}}\right )}{5 e \sqrt {c^2 x^2}}+\frac {b x \left (15 c^4 d^2+10 c^2 d e+3 e^2\right ) \text {arctanh}\left (\frac {\sqrt {e} \sqrt {c^2 x^2-1}}{c \sqrt {d+e x^2}}\right )}{40 c^4 \sqrt {e} \sqrt {c^2 x^2}}+\frac {b x \sqrt {c^2 x^2-1} \left (d+e x^2\right )^{3/2}}{20 c \sqrt {c^2 x^2}}+\frac {b x \sqrt {c^2 x^2-1} \left (7 c^2 d+3 e\right ) \sqrt {d+e x^2}}{40 c^3 \sqrt {c^2 x^2}} \]

[In]

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

[Out]

(b*(7*c^2*d + 3*e)*x*Sqrt[-1 + c^2*x^2]*Sqrt[d + e*x^2])/(40*c^3*Sqrt[c^2*x^2]) + (b*x*Sqrt[-1 + c^2*x^2]*(d +
e*x^2)^(3/2))/(20*c*Sqrt[c^2*x^2]) + ((d + e*x^2)^(5/2)*(a + b*ArcCsc[c*x]))/(5*e) - (b*c*d^(5/2)*x*ArcTan[Sq
rt[d + e*x^2]/(Sqrt[d]*Sqrt[-1 + c^2*x^2])])/(5*e*Sqrt[c^2*x^2]) + (b*(15*c^4*d^2 + 10*c^2*d*e + 3*e^2)*x*ArcT
anh[(Sqrt[e]*Sqrt[-1 + c^2*x^2])/(c*Sqrt[d + e*x^2])])/(40*c^4*Sqrt[e]*Sqrt[c^2*x^2])

Rule 65

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 95

Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x_)), x_Symbol] :> With[{q = Denomin
ator[m]}, Dist[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] && LtQ[-1, m, 0] && SimplerQ[
a + b*x, c + d*x]

Rule 104

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[b*(a +
b*x)^(m - 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d*f*(m + n + p + 1))), x] + Dist[1/(d*f*(m + n + p + 1)), I
nt[(a + b*x)^(m - 2)*(c + d*x)^n*(e + f*x)^p*Simp[a^2*d*f*(m + n + p + 1) - b*(b*c*e*(m - 1) + a*(d*e*(n + 1)
+ c*f*(p + 1))) + b*(a*d*f*(2*m + n + p) - b*(d*e*(m + n) + c*f*(m + p)))*x, x], x], x] /; FreeQ[{a, b, c, d,
e, f, n, p}, x] && GtQ[m, 1] && NeQ[m + n + p + 1, 0] && IntegersQ[2*m, 2*n, 2*p]

Rule 159

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)), x_Symb
ol] :> Simp[h*(a + b*x)^m*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d*f*(m + n + p + 2))), x] + Dist[1/(d*f*(m + n
+ p + 2)), Int[(a + b*x)^(m - 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*g*(m + n + p + 2) - h*(b*c*e*m + a*(d*e*(
n + 1) + c*f*(p + 1))) + (b*d*f*g*(m + n + p + 2) + h*(a*d*f*m - b*(d*e*(m + n + 1) + c*f*(m + p + 1))))*x, x]
, x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && GtQ[m, 0] && NeQ[m + n + p + 2, 0] && IntegersQ[2*m, 2
*n, 2*p]

Rule 163

Int[(((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)))/((a_.) + (b_.)*(x_)), x_Symbol]
:> Dist[h/b, Int[(c + d*x)^n*(e + f*x)^p, x], x] + Dist[(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 210

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 212

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

Rule 223

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

Rule 457

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Dist[1/n, Subst[Int
[x^(Simplify[(m + 1)/n] - 1)*(a + b*x)^p*(c + d*x)^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p, q}, x] &&
NeQ[b*c - a*d, 0] && IntegerQ[Simplify[(m + 1)/n]]

Rule 5345

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

Rubi steps \begin{align*} \text {integral}& = \frac {\left (d+e x^2\right )^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{5 e}+\frac {(b c x) \int \frac {\left (d+e x^2\right )^{5/2}}{x \sqrt {-1+c^2 x^2}} \, dx}{5 e \sqrt {c^2 x^2}} \\ & = \frac {\left (d+e x^2\right )^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{5 e}+\frac {(b c x) \text {Subst}\left (\int \frac {(d+e x)^{5/2}}{x \sqrt {-1+c^2 x}} \, dx,x,x^2\right )}{10 e \sqrt {c^2 x^2}} \\ & = \frac {b x \sqrt {-1+c^2 x^2} \left (d+e x^2\right )^{3/2}}{20 c \sqrt {c^2 x^2}}+\frac {\left (d+e x^2\right )^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{5 e}+\frac {(b x) \text {Subst}\left (\int \frac {\sqrt {d+e x} \left (2 c^2 d^2+\frac {1}{2} e \left (7 c^2 d+3 e\right ) x\right )}{x \sqrt {-1+c^2 x}} \, dx,x,x^2\right )}{20 c e \sqrt {c^2 x^2}} \\ & = \frac {b \left (7 c^2 d+3 e\right ) x \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}{40 c^3 \sqrt {c^2 x^2}}+\frac {b x \sqrt {-1+c^2 x^2} \left (d+e x^2\right )^{3/2}}{20 c \sqrt {c^2 x^2}}+\frac {\left (d+e x^2\right )^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{5 e}+\frac {(b x) \text {Subst}\left (\int \frac {2 c^4 d^3+\frac {1}{4} e \left (15 c^4 d^2+10 c^2 d e+3 e^2\right ) x}{x \sqrt {-1+c^2 x} \sqrt {d+e x}} \, dx,x,x^2\right )}{20 c^3 e \sqrt {c^2 x^2}} \\ & = \frac {b \left (7 c^2 d+3 e\right ) x \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}{40 c^3 \sqrt {c^2 x^2}}+\frac {b x \sqrt {-1+c^2 x^2} \left (d+e x^2\right )^{3/2}}{20 c \sqrt {c^2 x^2}}+\frac {\left (d+e x^2\right )^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{5 e}+\frac {\left (b c d^3 x\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {-1+c^2 x} \sqrt {d+e x}} \, dx,x,x^2\right )}{10 e \sqrt {c^2 x^2}}+\frac {\left (b \left (15 c^4 d^2+10 c^2 d e+3 e^2\right ) x\right ) \text {Subst}\left (\int \frac {1}{\sqrt {-1+c^2 x} \sqrt {d+e x}} \, dx,x,x^2\right )}{80 c^3 \sqrt {c^2 x^2}} \\ & = \frac {b \left (7 c^2 d+3 e\right ) x \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}{40 c^3 \sqrt {c^2 x^2}}+\frac {b x \sqrt {-1+c^2 x^2} \left (d+e x^2\right )^{3/2}}{20 c \sqrt {c^2 x^2}}+\frac {\left (d+e x^2\right )^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{5 e}+\frac {\left (b c d^3 x\right ) \text {Subst}\left (\int \frac {1}{-d-x^2} \, dx,x,\frac {\sqrt {d+e x^2}}{\sqrt {-1+c^2 x^2}}\right )}{5 e \sqrt {c^2 x^2}}+\frac {\left (b \left (15 c^4 d^2+10 c^2 d e+3 e^2\right ) x\right ) \text {Subst}\left (\int \frac {1}{\sqrt {d+\frac {e}{c^2}+\frac {e x^2}{c^2}}} \, dx,x,\sqrt {-1+c^2 x^2}\right )}{40 c^5 \sqrt {c^2 x^2}} \\ & = \frac {b \left (7 c^2 d+3 e\right ) x \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}{40 c^3 \sqrt {c^2 x^2}}+\frac {b x \sqrt {-1+c^2 x^2} \left (d+e x^2\right )^{3/2}}{20 c \sqrt {c^2 x^2}}+\frac {\left (d+e x^2\right )^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{5 e}-\frac {b c d^{5/2} x \arctan \left (\frac {\sqrt {d+e x^2}}{\sqrt {d} \sqrt {-1+c^2 x^2}}\right )}{5 e \sqrt {c^2 x^2}}+\frac {\left (b \left (15 c^4 d^2+10 c^2 d e+3 e^2\right ) x\right ) \text {Subst}\left (\int \frac {1}{1-\frac {e x^2}{c^2}} \, dx,x,\frac {\sqrt {-1+c^2 x^2}}{\sqrt {d+e x^2}}\right )}{40 c^5 \sqrt {c^2 x^2}} \\ & = \frac {b \left (7 c^2 d+3 e\right ) x \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}{40 c^3 \sqrt {c^2 x^2}}+\frac {b x \sqrt {-1+c^2 x^2} \left (d+e x^2\right )^{3/2}}{20 c \sqrt {c^2 x^2}}+\frac {\left (d+e x^2\right )^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{5 e}-\frac {b c d^{5/2} x \arctan \left (\frac {\sqrt {d+e x^2}}{\sqrt {d} \sqrt {-1+c^2 x^2}}\right )}{5 e \sqrt {c^2 x^2}}+\frac {b \left (15 c^4 d^2+10 c^2 d e+3 e^2\right ) x \text {arctanh}\left (\frac {\sqrt {e} \sqrt {-1+c^2 x^2}}{c \sqrt {d+e x^2}}\right )}{40 c^4 \sqrt {e} \sqrt {c^2 x^2}} \\ \end{align*}

Mathematica [C] (verified)

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

Time = 1.32 (sec) , antiderivative size = 248, normalized size of antiderivative = 0.95 \[ \int x \left (d+e x^2\right )^{3/2} \left (a+b \csc ^{-1}(c x)\right ) \, dx=\frac {\frac {16 a \left (d+e x^2\right )^3}{e}+\frac {2 b \sqrt {1-\frac {1}{c^2 x^2}} x \left (d+e x^2\right ) \left (3 e+c^2 \left (9 d+2 e x^2\right )\right )}{c^3}+\frac {b \left (-\frac {8 c^2 d^3 \sqrt {1+\frac {d}{e x^2}} \operatorname {AppellF1}\left (1,\frac {1}{2},\frac {1}{2},2,\frac {1}{c^2 x^2},-\frac {d}{e x^2}\right )}{e}-\frac {\left (15 c^4 d^2+10 c^2 d e+3 e^2\right ) \sqrt {1-\frac {1}{c^2 x^2}} x^4 \sqrt {1+\frac {e x^2}{d}} \operatorname {AppellF1}\left (1,\frac {1}{2},\frac {1}{2},2,c^2 x^2,-\frac {e x^2}{d}\right )}{\sqrt {1-c^2 x^2}}\right )}{c^3 x}+\frac {16 b \left (d+e x^2\right )^3 \csc ^{-1}(c x)}{e}}{80 \sqrt {d+e x^2}} \]

[In]

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

[Out]

((16*a*(d + e*x^2)^3)/e + (2*b*Sqrt[1 - 1/(c^2*x^2)]*x*(d + e*x^2)*(3*e + c^2*(9*d + 2*e*x^2)))/c^3 + (b*((-8*
c^2*d^3*Sqrt[1 + d/(e*x^2)]*AppellF1[1, 1/2, 1/2, 2, 1/(c^2*x^2), -(d/(e*x^2))])/e - ((15*c^4*d^2 + 10*c^2*d*e
+ 3*e^2)*Sqrt[1 - 1/(c^2*x^2)]*x^4*Sqrt[1 + (e*x^2)/d]*AppellF1[1, 1/2, 1/2, 2, c^2*x^2, -((e*x^2)/d)])/Sqrt[
1 - c^2*x^2]))/(c^3*x) + (16*b*(d + e*x^2)^3*ArcCsc[c*x])/e)/(80*Sqrt[d + e*x^2])

Maple [F]

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

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.98 (sec) , antiderivative size = 1375, normalized size of antiderivative = 5.25 \[ \int x \left (d+e x^2\right )^{3/2} \left (a+b \csc ^{-1}(c x)\right ) \, dx=\text {Too large to display} \]

[In]

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

[Out]

[1/160*(8*b*c^5*sqrt(-d)*d^2*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) + (15*b*c^4*d^2 + 10*b*c^2*d*e + 3*b*e^2)*sqr
t(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*(8*a*c^5*e^2*x^4 + 16*a*c^5*d*e*x^2 + 8*a*c^5*d^2 + 8*(b*c^5*e
^2*x^4 + 2*b*c^5*d*e*x^2 + b*c^5*d^2)*arccsc(c*x) + (2*b*c^3*e^2*x^2 + 9*b*c^3*d*e + 3*b*c*e^2)*sqrt(c^2*x^2 -
1))*sqrt(e*x^2 + d))/(c^5*e), -1/160*(16*b*c^5*d^(5/2)*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)) - (15*b*c^4*d^2 + 10*b*c^2*d*e + 3*b*e^2)*s
qrt(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)*sqr
t(c^2*x^2 - 1)*sqrt(e*x^2 + d)*sqrt(e) + e^2) - 4*(8*a*c^5*e^2*x^4 + 16*a*c^5*d*e*x^2 + 8*a*c^5*d^2 + 8*(b*c^5
*e^2*x^4 + 2*b*c^5*d*e*x^2 + b*c^5*d^2)*arccsc(c*x) + (2*b*c^3*e^2*x^2 + 9*b*c^3*d*e + 3*b*c*e^2)*sqrt(c^2*x^2
- 1))*sqrt(e*x^2 + d))/(c^5*e), 1/80*(4*b*c^5*sqrt(-d)*d^2*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) - (15*b*c^4*d^
2 + 10*b*c^2*d*e + 3*b*e^2)*sqrt(-e)*arctan(1/2*(2*c^2*e*x^2 + c^2*d - e)*sqrt(c^2*x^2 - 1)*sqrt(e*x^2 + d)*sq
rt(-e)/(c^3*e^2*x^4 - c*d*e + (c^3*d*e - c*e^2)*x^2)) + 2*(8*a*c^5*e^2*x^4 + 16*a*c^5*d*e*x^2 + 8*a*c^5*d^2 +
8*(b*c^5*e^2*x^4 + 2*b*c^5*d*e*x^2 + b*c^5*d^2)*arccsc(c*x) + (2*b*c^3*e^2*x^2 + 9*b*c^3*d*e + 3*b*c*e^2)*sqrt
(c^2*x^2 - 1))*sqrt(e*x^2 + d))/(c^5*e), -1/80*(8*b*c^5*d^(5/2)*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)) + (15*b*c^4*d^2 + 10*b*c^2*d*e + 3*
b*e^2)*sqrt(-e)*arctan(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)) - 2*(8*a*c^5*e^2*x^4 + 16*a*c^5*d*e*x^2 + 8*a*c^5*d^2 + 8*(b*c^5*e^2*x^4 + 2*
b*c^5*d*e*x^2 + b*c^5*d^2)*arccsc(c*x) + (2*b*c^3*e^2*x^2 + 9*b*c^3*d*e + 3*b*c*e^2)*sqrt(c^2*x^2 - 1))*sqrt(e
*x^2 + d))/(c^5*e)]

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

1/5*(e*x^2 + d)^(5/2)*a/e + 1/5*(5*e*integrate(1/5*(c^2*e^2*x^5 + 2*c^2*d*e*x^3 + c^2*d^2*x)*e^(1/2*log(e*x^2
+ d) + 1/2*log(c*x + 1) + 1/2*log(c*x - 1))/(c^2*e*x^2 + (c^2*e*x^2 - e)*e^(log(c*x + 1) + log(c*x - 1)) - e),
x) + (e^2*x^4*arctan2(1, sqrt(c*x + 1)*sqrt(c*x - 1)) + 2*d*e*x^2*arctan2(1, sqrt(c*x + 1)*sqrt(c*x - 1)) + d
^2*arctan2(1, sqrt(c*x + 1)*sqrt(c*x - 1)))*sqrt(e*x^2 + d))*b/e

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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