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

Optimal result

Integrand size = 23, antiderivative size = 256 \[ \int \frac {x^5 \left (a+b \text {csch}^{-1}(c x)\right )}{\left (d+e x^2\right )^{3/2}} \, dx=\frac {b x \sqrt {-1-c^2 x^2} \sqrt {d+e x^2}}{6 c e^2 \sqrt {-c^2 x^2}}-\frac {d^2 \left (a+b \text {csch}^{-1}(c x)\right )}{e^3 \sqrt {d+e x^2}}-\frac {2 d \sqrt {d+e x^2} \left (a+b \text {csch}^{-1}(c x)\right )}{e^3}+\frac {\left (d+e x^2\right )^{3/2} \left (a+b \text {csch}^{-1}(c x)\right )}{3 e^3}-\frac {b \left (9 c^2 d+e\right ) x \arctan \left (\frac {\sqrt {e} \sqrt {-1-c^2 x^2}}{c \sqrt {d+e x^2}}\right )}{6 c^2 e^{5/2} \sqrt {-c^2 x^2}}-\frac {8 b c d^{3/2} x \arctan \left (\frac {\sqrt {d+e x^2}}{\sqrt {d} \sqrt {-1-c^2 x^2}}\right )}{3 e^3 \sqrt {-c^2 x^2}} \]

[Out]

1/3*(e*x^2+d)^(3/2)*(a+b*arccsch(c*x))/e^3-1/6*b*(9*c^2*d+e)*x*arctan(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)-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)-d^2*(a+b*arccsch(c*x))/e^3/(e*x^2+d)^(1/2)-2*d*(a+b*arccsch(c*x))*(e*x^2+d)^(1/2)/e^3+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)

Rubi [A] (verified)

Time = 0.78 (sec) , antiderivative size = 256, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.478, Rules used = {272, 45, 6437, 12, 1629, 163, 65, 223, 209, 95, 210} \[ \int \frac {x^5 \left (a+b \text {csch}^{-1}(c x)\right )}{\left (d+e x^2\right )^{3/2}} \, dx=-\frac {d^2 \left (a+b \text {csch}^{-1}(c x)\right )}{e^3 \sqrt {d+e x^2}}-\frac {2 d \sqrt {d+e x^2} \left (a+b \text {csch}^{-1}(c x)\right )}{e^3}+\frac {\left (d+e x^2\right )^{3/2} \left (a+b \text {csch}^{-1}(c x)\right )}{3 e^3}-\frac {8 b c d^{3/2} x \arctan \left (\frac {\sqrt {d+e x^2}}{\sqrt {d} \sqrt {-c^2 x^2-1}}\right )}{3 e^3 \sqrt {-c^2 x^2}}-\frac {b x \left (9 c^2 d+e\right ) \arctan \left (\frac {\sqrt {e} \sqrt {-c^2 x^2-1}}{c \sqrt {d+e x^2}}\right )}{6 c^2 e^{5/2} \sqrt {-c^2 x^2}}+\frac {b x \sqrt {-c^2 x^2-1} \sqrt {d+e x^2}}{6 c e^2 \sqrt {-c^2 x^2}} \]

[In]

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

[Out]

(b*x*Sqrt[-1 - c^2*x^2]*Sqrt[d + e*x^2])/(6*c*e^2*Sqrt[-(c^2*x^2)]) - (d^2*(a + b*ArcCsch[c*x]))/(e^3*Sqrt[d +
e*x^2]) - (2*d*Sqrt[d + e*x^2]*(a + b*ArcCsch[c*x]))/e^3 + ((d + e*x^2)^(3/2)*(a + b*ArcCsch[c*x]))/(3*e^3) -
(b*(9*c^2*d + e)*x*ArcTan[(Sqrt[e]*Sqrt[-1 - c^2*x^2])/(c*Sqrt[d + e*x^2])])/(6*c^2*e^(5/2)*Sqrt[-(c^2*x^2)])
- (8*b*c*d^(3/2)*x*ArcTan[Sqrt[d + e*x^2]/(Sqrt[d]*Sqrt[-1 - c^2*x^2])])/(3*e^3*Sqrt[-(c^2*x^2)])

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 45

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

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 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 209

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

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 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 272

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

Rule 1629

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, Expon[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] + Dist[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]] /; F
reeQ[{a, b, c, d, e, f, m, n, p}, x] && PolyQ[Px, x]

Rule 6437

Int[((a_.) + ArcCsch[(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]}, Dist[a + b*ArcCsch[c*x], u, x] - Dist[b*c*(x/Sqrt[(-c^2)*x^2]), Int[Simp
lifyIntegrand[u/(x*Sqrt[-1 - c^2*x^2]), 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]))

Rubi steps \begin{align*} \text {integral}& = -\frac {d^2 \left (a+b \text {csch}^{-1}(c x)\right )}{e^3 \sqrt {d+e x^2}}-\frac {2 d \sqrt {d+e x^2} \left (a+b \text {csch}^{-1}(c x)\right )}{e^3}+\frac {\left (d+e x^2\right )^{3/2} \left (a+b \text {csch}^{-1}(c x)\right )}{3 e^3}-\frac {(b c x) \int \frac {-8 d^2-4 d e x^2+e^2 x^4}{3 e^3 x \sqrt {-1-c^2 x^2} \sqrt {d+e x^2}} \, dx}{\sqrt {-c^2 x^2}} \\ & = -\frac {d^2 \left (a+b \text {csch}^{-1}(c x)\right )}{e^3 \sqrt {d+e x^2}}-\frac {2 d \sqrt {d+e x^2} \left (a+b \text {csch}^{-1}(c x)\right )}{e^3}+\frac {\left (d+e x^2\right )^{3/2} \left (a+b \text {csch}^{-1}(c x)\right )}{3 e^3}-\frac {(b c x) \int \frac {-8 d^2-4 d e x^2+e^2 x^4}{x \sqrt {-1-c^2 x^2} \sqrt {d+e x^2}} \, dx}{3 e^3 \sqrt {-c^2 x^2}} \\ & = -\frac {d^2 \left (a+b \text {csch}^{-1}(c x)\right )}{e^3 \sqrt {d+e x^2}}-\frac {2 d \sqrt {d+e x^2} \left (a+b \text {csch}^{-1}(c x)\right )}{e^3}+\frac {\left (d+e x^2\right )^{3/2} \left (a+b \text {csch}^{-1}(c x)\right )}{3 e^3}-\frac {(b c x) \text {Subst}\left (\int \frac {-8 d^2-4 d e x+e^2 x^2}{x \sqrt {-1-c^2 x} \sqrt {d+e x}} \, dx,x,x^2\right )}{6 e^3 \sqrt {-c^2 x^2}} \\ & = \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 \text {csch}^{-1}(c x)\right )}{e^3 \sqrt {d+e x^2}}-\frac {2 d \sqrt {d+e x^2} \left (a+b \text {csch}^{-1}(c x)\right )}{e^3}+\frac {\left (d+e x^2\right )^{3/2} \left (a+b \text {csch}^{-1}(c x)\right )}{3 e^3}+\frac {(b x) \text {Subst}\left (\int \frac {8 c^2 d^2 e+\frac {1}{2} e^2 \left (9 c^2 d+e\right ) x}{x \sqrt {-1-c^2 x} \sqrt {d+e x}} \, dx,x,x^2\right )}{6 c e^4 \sqrt {-c^2 x^2}} \\ & = \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 \text {csch}^{-1}(c x)\right )}{e^3 \sqrt {d+e x^2}}-\frac {2 d \sqrt {d+e x^2} \left (a+b \text {csch}^{-1}(c x)\right )}{e^3}+\frac {\left (d+e x^2\right )^{3/2} \left (a+b \text {csch}^{-1}(c x)\right )}{3 e^3}+\frac {\left (4 b c d^2 x\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {-1-c^2 x} \sqrt {d+e x}} \, dx,x,x^2\right )}{3 e^3 \sqrt {-c^2 x^2}}+\frac {\left (b \left (9 c^2 d+e\right ) x\right ) \text {Subst}\left (\int \frac {1}{\sqrt {-1-c^2 x} \sqrt {d+e x}} \, dx,x,x^2\right )}{12 c e^2 \sqrt {-c^2 x^2}} \\ & = \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 \text {csch}^{-1}(c x)\right )}{e^3 \sqrt {d+e x^2}}-\frac {2 d \sqrt {d+e x^2} \left (a+b \text {csch}^{-1}(c x)\right )}{e^3}+\frac {\left (d+e x^2\right )^{3/2} \left (a+b \text {csch}^{-1}(c x)\right )}{3 e^3}+\frac {\left (8 b c d^2 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 )}{3 e^3 \sqrt {-c^2 x^2}}-\frac {\left (b \left (9 c^2 d+e\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 )}{6 c^3 e^2 \sqrt {-c^2 x^2}} \\ & = \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 \text {csch}^{-1}(c x)\right )}{e^3 \sqrt {d+e x^2}}-\frac {2 d \sqrt {d+e x^2} \left (a+b \text {csch}^{-1}(c x)\right )}{e^3}+\frac {\left (d+e x^2\right )^{3/2} \left (a+b \text {csch}^{-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 {\left (b \left (9 c^2 d+e\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 )}{6 c^3 e^2 \sqrt {-c^2 x^2}} \\ & = \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 \text {csch}^{-1}(c x)\right )}{e^3 \sqrt {d+e x^2}}-\frac {2 d \sqrt {d+e x^2} \left (a+b \text {csch}^{-1}(c x)\right )}{e^3}+\frac {\left (d+e x^2\right )^{3/2} \left (a+b \text {csch}^{-1}(c x)\right )}{3 e^3}-\frac {b \left (9 c^2 d+e\right ) x \arctan \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}}-\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}} \\ \end{align*}

Mathematica [C] (verified)

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

Time = 1.84 (sec) , antiderivative size = 260, normalized size of antiderivative = 1.02 \[ \int \frac {x^5 \left (a+b \text {csch}^{-1}(c x)\right )}{\left (d+e x^2\right )^{3/2}} \, dx=\frac {16 b d^2 \sqrt {1+\frac {d}{e x^2}} \sqrt {1+c^2 x^2} \operatorname {AppellF1}\left (1,\frac {1}{2},\frac {1}{2},2,-\frac {1}{c^2 x^2},-\frac {d}{e x^2}\right )-b e \left (9 c^2 d+e\right ) \sqrt {1+\frac {1}{c^2 x^2}} x^4 \sqrt {1+\frac {e x^2}{d}} \operatorname {AppellF1}\left (1,\frac {1}{2},\frac {1}{2},2,-c^2 x^2,-\frac {e x^2}{d}\right )+2 x \sqrt {1+c^2 x^2} \left (b e \sqrt {1+\frac {1}{c^2 x^2}} x \left (d+e x^2\right )-2 a c \left (8 d^2+4 d e x^2-e^2 x^4\right )-2 b c \left (8 d^2+4 d e x^2-e^2 x^4\right ) \text {csch}^{-1}(c x)\right )}{12 c e^3 x \sqrt {1+c^2 x^2} \sqrt {d+e x^2}} \]

[In]

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

[Out]

(16*b*d^2*Sqrt[1 + d/(e*x^2)]*Sqrt[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 + (e*x^2)/d]*AppellF1[1, 1/2, 1/2, 2, -(c^2*x^2), -((e*x^2)/d)]
+ 2*x*Sqrt[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)*ArcCsch[c*x]))/(12*c*e^3*x*Sqrt[1 + c^2*x^2]*Sqrt[d + e*x^2])

Maple [F]

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

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.44 (sec) , antiderivative size = 1719, normalized size of antiderivative = 6.71 \[ \int \frac {x^5 \left (a+b \text {csch}^{-1}(c x)\right )}{\left (d+e x^2\right )^{3/2}} \, dx=\text {Too large to display} \]

[In]

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

[Out]

[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^4*e*x^3 + (c^4*d + c^2*e)*x)*sqrt(e*x^2 + d)*sqrt(e)*sqrt((c^2*x^2 + 1)/(c^2*x
^2)) + e^2) + 8*(b*c^3*e^2*x^4 - 4*b*c^3*d*e*x^2 - 8*b*c^3*d^2)*sqrt(e*x^2 + d)*log((c*x*sqrt((c^2*x^2 + 1)/(c
^2*x^2)) + 1)/(c*x)) + 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*((c^3*d + c*e)*x^3 + 2*c*d*x)*sqrt(e*x^2 + d)*sqrt(d)*sqrt((c^2*x^2 + 1)/(c^2*x^2)) + 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 + (b*c^2*e^2*x^3 + b*c^2*d*e*x)*sqrt((c^2*x^2 + 1)/
(c^2*x^2)))*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)*arctan(1/2*(2*c^2*e*x^3 + (c^2*d + e)*x)*sqrt(e*x^2 + d)*sqrt(-e)*sqrt((c^2*x^2 + 1)/(c^2*x^2))/(c^
2*e^2*x^4 + (c^2*d*e + e^2)*x^2 + d*e)) + 4*(b*c^3*e^2*x^4 - 4*b*c^3*d*e*x^2 - 8*b*c^3*d^2)*sqrt(e*x^2 + d)*lo
g((c*x*sqrt((c^2*x^2 + 1)/(c^2*x^2)) + 1)/(c*x)) + 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 + d*e)*x^2 + 4*((c^3*d + c*e)*x^3 + 2*c*d*x)*sqrt(e*x^2 + d)*sqrt(d)*sqrt((c^2*x^
2 + 1)/(c^2*x^2)) + 8*d^2)/x^4) + 2*(2*a*c^3*e^2*x^4 - 8*a*c^3*d*e*x^2 - 16*a*c^3*d^2 + (b*c^2*e^2*x^3 + b*c^2
*d*e*x)*sqrt((c^2*x^2 + 1)/(c^2*x^2)))*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*((c^3*d + c*e)*x^3 + 2*c*d*x)*sqrt(e*x^2 + d)*sqrt(-d)*sqrt((c^2*x^2 + 1)/(c^2*
x^2))/(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)*l
og(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^4*e*x^3 + (c^4*d + c^2*e)*x)*sqrt(
e*x^2 + d)*sqrt(e)*sqrt((c^2*x^2 + 1)/(c^2*x^2)) + e^2) - 8*(b*c^3*e^2*x^4 - 4*b*c^3*d*e*x^2 - 8*b*c^3*d^2)*sq
rt(e*x^2 + d)*log((c*x*sqrt((c^2*x^2 + 1)/(c^2*x^2)) + 1)/(c*x)) - 4*(2*a*c^3*e^2*x^4 - 8*a*c^3*d*e*x^2 - 16*a
*c^3*d^2 + (b*c^2*e^2*x^3 + b*c^2*d*e*x)*sqrt((c^2*x^2 + 1)/(c^2*x^2)))*sqrt(e*x^2 + d))/(c^3*e^4*x^2 + c^3*d*
e^3), -1/12*(16*(b*c^3*d*e*x^2 + b*c^3*d^2)*sqrt(-d)*arctan(1/2*((c^3*d + c*e)*x^3 + 2*c*d*x)*sqrt(e*x^2 + d)*
sqrt(-d)*sqrt((c^2*x^2 + 1)/(c^2*x^2))/(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)*arctan(1/2*(2*c^2*e*x^3 + (c^2*d + e)*x)*sqrt(e*x^2 + d)*sqrt(-e)*sqrt((c^2*x
^2 + 1)/(c^2*x^2))/(c^2*e^2*x^4 + (c^2*d*e + e^2)*x^2 + d*e)) - 4*(b*c^3*e^2*x^4 - 4*b*c^3*d*e*x^2 - 8*b*c^3*d
^2)*sqrt(e*x^2 + d)*log((c*x*sqrt((c^2*x^2 + 1)/(c^2*x^2)) + 1)/(c*x)) - 2*(2*a*c^3*e^2*x^4 - 8*a*c^3*d*e*x^2
- 16*a*c^3*d^2 + (b*c^2*e^2*x^3 + b*c^2*d*e*x)*sqrt((c^2*x^2 + 1)/(c^2*x^2)))*sqrt(e*x^2 + d))/(c^3*e^4*x^2 +
c^3*d*e^3)]

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F(-2)]

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

[In]

integrate(x^5*(a+b*arccsch(c*x))/(e*x^2+d)^(3/2),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>0)', see `assume?` for more
details)Is e

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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