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

Optimal. Leaf size=251 \[ -\frac {d^2 \left (a+b \text {csch}^{-1}(c x)\right )}{3 e^3 \left (d+e x^2\right )^{3/2}}+\frac {2 d \left (a+b \text {csch}^{-1}(c x)\right )}{e^3 \sqrt {d+e x^2}}+\frac {\sqrt {d+e x^2} \left (a+b \text {csch}^{-1}(c x)\right )}{e^3}+\frac {b x \tan ^{-1}\left (\frac {\sqrt {e} \sqrt {-c^2 x^2-1}}{c \sqrt {d+e x^2}}\right )}{e^{5/2} \sqrt {-c^2 x^2}}+\frac {8 b c \sqrt {d} x \tan ^{-1}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d} \sqrt {-c^2 x^2-1}}\right )}{3 e^3 \sqrt {-c^2 x^2}}+\frac {b c d x \sqrt {-c^2 x^2-1}}{3 e^2 \sqrt {-c^2 x^2} \left (c^2 d-e\right ) \sqrt {d+e x^2}} \]

[Out]

-1/3*d^2*(a+b*arccsch(c*x))/e^3/(e*x^2+d)^(3/2)+b*x*arctan(e^(1/2)*(-c^2*x^2-1)^(1/2)/c/(e*x^2+d)^(1/2))/e^(5/
2)/(-c^2*x^2)^(1/2)+8/3*b*c*x*arctan((e*x^2+d)^(1/2)/d^(1/2)/(-c^2*x^2-1)^(1/2))*d^(1/2)/e^3/(-c^2*x^2)^(1/2)+
2*d*(a+b*arccsch(c*x))/e^3/(e*x^2+d)^(1/2)+1/3*b*c*d*x*(-c^2*x^2-1)^(1/2)/(c^2*d-e)/e^2/(-c^2*x^2)^(1/2)/(e*x^
2+d)^(1/2)+(a+b*arccsch(c*x))*(e*x^2+d)^(1/2)/e^3

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Rubi [A]  time = 1.23, antiderivative size = 251, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 11, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.478, Rules used = {266, 43, 6302, 12, 1614, 157, 63, 217, 203, 93, 204} \[ -\frac {d^2 \left (a+b \text {csch}^{-1}(c x)\right )}{3 e^3 \left (d+e x^2\right )^{3/2}}+\frac {2 d \left (a+b \text {csch}^{-1}(c x)\right )}{e^3 \sqrt {d+e x^2}}+\frac {\sqrt {d+e x^2} \left (a+b \text {csch}^{-1}(c x)\right )}{e^3}+\frac {b c d x \sqrt {-c^2 x^2-1}}{3 e^2 \sqrt {-c^2 x^2} \left (c^2 d-e\right ) \sqrt {d+e x^2}}+\frac {8 b c \sqrt {d} x \tan ^{-1}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d} \sqrt {-c^2 x^2-1}}\right )}{3 e^3 \sqrt {-c^2 x^2}}+\frac {b x \tan ^{-1}\left (\frac {\sqrt {e} \sqrt {-c^2 x^2-1}}{c \sqrt {d+e x^2}}\right )}{e^{5/2} \sqrt {-c^2 x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(b*c*d*x*Sqrt[-1 - c^2*x^2])/(3*(c^2*d - e)*e^2*Sqrt[-(c^2*x^2)]*Sqrt[d + e*x^2]) - (d^2*(a + b*ArcCsch[c*x]))
/(3*e^3*(d + e*x^2)^(3/2)) + (2*d*(a + b*ArcCsch[c*x]))/(e^3*Sqrt[d + e*x^2]) + (Sqrt[d + e*x^2]*(a + b*ArcCsc
h[c*x]))/e^3 + (b*x*ArcTan[(Sqrt[e]*Sqrt[-1 - c^2*x^2])/(c*Sqrt[d + e*x^2])])/(e^(5/2)*Sqrt[-(c^2*x^2)]) + (8*
b*c*Sqrt[d]*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 43

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 63

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 93

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 157

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 203

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

Rule 204

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

Rule 217

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 266

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 1614

Int[(Px_)*((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> With[{
Qx = PolynomialQuotient[Px, a + b*x, x], R = PolynomialRemainder[Px, a + b*x, x]}, Simp[(b*R*(a + b*x)^(m + 1)
*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/((m + 1)*(b*c - a*d)*(b*e - a*f)), x] + Dist[1/((m + 1)*(b*c - a*d)*(b*e
- a*f)), Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*ExpandToSum[(m + 1)*(b*c - a*d)*(b*e - a*f)*Qx + a*d*f
*R*(m + 1) - b*R*(d*e*(m + n + 2) + c*f*(m + p + 2)) - b*d*f*R*(m + n + p + 3)*x, x], x], x]] /; FreeQ[{a, b,
c, d, e, f, n, p}, x] && PolyQ[Px, x] && LtQ[m, -1] && IntegersQ[2*m, 2*n, 2*p]

Rule 6302

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*} \int \frac {x^5 \left (a+b \text {csch}^{-1}(c x)\right )}{\left (d+e x^2\right )^{5/2}} \, dx &=-\frac {d^2 \left (a+b \text {csch}^{-1}(c x)\right )}{3 e^3 \left (d+e x^2\right )^{3/2}}+\frac {2 d \left (a+b \text {csch}^{-1}(c x)\right )}{e^3 \sqrt {d+e x^2}}+\frac {\sqrt {d+e x^2} \left (a+b \text {csch}^{-1}(c x)\right )}{e^3}-\frac {(b c x) \int \frac {8 d^2+12 d e x^2+3 e^2 x^4}{3 e^3 x \sqrt {-1-c^2 x^2} \left (d+e x^2\right )^{3/2}} \, dx}{\sqrt {-c^2 x^2}}\\ &=-\frac {d^2 \left (a+b \text {csch}^{-1}(c x)\right )}{3 e^3 \left (d+e x^2\right )^{3/2}}+\frac {2 d \left (a+b \text {csch}^{-1}(c x)\right )}{e^3 \sqrt {d+e x^2}}+\frac {\sqrt {d+e x^2} \left (a+b \text {csch}^{-1}(c x)\right )}{e^3}-\frac {(b c x) \int \frac {8 d^2+12 d e x^2+3 e^2 x^4}{x \sqrt {-1-c^2 x^2} \left (d+e x^2\right )^{3/2}} \, dx}{3 e^3 \sqrt {-c^2 x^2}}\\ &=-\frac {d^2 \left (a+b \text {csch}^{-1}(c x)\right )}{3 e^3 \left (d+e x^2\right )^{3/2}}+\frac {2 d \left (a+b \text {csch}^{-1}(c x)\right )}{e^3 \sqrt {d+e x^2}}+\frac {\sqrt {d+e x^2} \left (a+b \text {csch}^{-1}(c x)\right )}{e^3}-\frac {(b c x) \operatorname {Subst}\left (\int \frac {8 d^2+12 d e x+3 e^2 x^2}{x \sqrt {-1-c^2 x} (d+e x)^{3/2}} \, dx,x,x^2\right )}{6 e^3 \sqrt {-c^2 x^2}}\\ &=\frac {b c d x \sqrt {-1-c^2 x^2}}{3 \left (c^2 d-e\right ) e^2 \sqrt {-c^2 x^2} \sqrt {d+e x^2}}-\frac {d^2 \left (a+b \text {csch}^{-1}(c x)\right )}{3 e^3 \left (d+e x^2\right )^{3/2}}+\frac {2 d \left (a+b \text {csch}^{-1}(c x)\right )}{e^3 \sqrt {d+e x^2}}+\frac {\sqrt {d+e x^2} \left (a+b \text {csch}^{-1}(c x)\right )}{e^3}-\frac {(b c x) \operatorname {Subst}\left (\int \frac {4 d^2 \left (c^2 d-e\right )+\frac {3}{2} d \left (c^2 d-e\right ) e x}{x \sqrt {-1-c^2 x} \sqrt {d+e x}} \, dx,x,x^2\right )}{3 d \left (c^2 d-e\right ) e^3 \sqrt {-c^2 x^2}}\\ &=\frac {b c d x \sqrt {-1-c^2 x^2}}{3 \left (c^2 d-e\right ) e^2 \sqrt {-c^2 x^2} \sqrt {d+e x^2}}-\frac {d^2 \left (a+b \text {csch}^{-1}(c x)\right )}{3 e^3 \left (d+e x^2\right )^{3/2}}+\frac {2 d \left (a+b \text {csch}^{-1}(c x)\right )}{e^3 \sqrt {d+e x^2}}+\frac {\sqrt {d+e x^2} \left (a+b \text {csch}^{-1}(c x)\right )}{e^3}-\frac {(4 b c d x) \operatorname {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 {(b c x) \operatorname {Subst}\left (\int \frac {1}{\sqrt {-1-c^2 x} \sqrt {d+e x}} \, dx,x,x^2\right )}{2 e^2 \sqrt {-c^2 x^2}}\\ &=\frac {b c d x \sqrt {-1-c^2 x^2}}{3 \left (c^2 d-e\right ) e^2 \sqrt {-c^2 x^2} \sqrt {d+e x^2}}-\frac {d^2 \left (a+b \text {csch}^{-1}(c x)\right )}{3 e^3 \left (d+e x^2\right )^{3/2}}+\frac {2 d \left (a+b \text {csch}^{-1}(c x)\right )}{e^3 \sqrt {d+e x^2}}+\frac {\sqrt {d+e x^2} \left (a+b \text {csch}^{-1}(c x)\right )}{e^3}-\frac {(8 b c d x) \operatorname {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 {(b x) \operatorname {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 )}{c e^2 \sqrt {-c^2 x^2}}\\ &=\frac {b c d x \sqrt {-1-c^2 x^2}}{3 \left (c^2 d-e\right ) e^2 \sqrt {-c^2 x^2} \sqrt {d+e x^2}}-\frac {d^2 \left (a+b \text {csch}^{-1}(c x)\right )}{3 e^3 \left (d+e x^2\right )^{3/2}}+\frac {2 d \left (a+b \text {csch}^{-1}(c x)\right )}{e^3 \sqrt {d+e x^2}}+\frac {\sqrt {d+e x^2} \left (a+b \text {csch}^{-1}(c x)\right )}{e^3}+\frac {8 b c \sqrt {d} x \tan ^{-1}\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 x) \operatorname {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 )}{c e^2 \sqrt {-c^2 x^2}}\\ &=\frac {b c d x \sqrt {-1-c^2 x^2}}{3 \left (c^2 d-e\right ) e^2 \sqrt {-c^2 x^2} \sqrt {d+e x^2}}-\frac {d^2 \left (a+b \text {csch}^{-1}(c x)\right )}{3 e^3 \left (d+e x^2\right )^{3/2}}+\frac {2 d \left (a+b \text {csch}^{-1}(c x)\right )}{e^3 \sqrt {d+e x^2}}+\frac {\sqrt {d+e x^2} \left (a+b \text {csch}^{-1}(c x)\right )}{e^3}+\frac {b x \tan ^{-1}\left (\frac {\sqrt {e} \sqrt {-1-c^2 x^2}}{c \sqrt {d+e x^2}}\right )}{e^{5/2} \sqrt {-c^2 x^2}}+\frac {8 b c \sqrt {d} x \tan ^{-1}\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*}

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Mathematica [A]  time = 0.56, size = 327, normalized size = 1.30 \[ \frac {a \left (c^2 d-e\right ) \left (8 d^2+12 d e x^2+3 e^2 x^4\right )+b \left (c^2 d-e\right ) \text {csch}^{-1}(c x) \left (8 d^2+12 d e x^2+3 e^2 x^4\right )+b c d e x \sqrt {\frac {1}{c^2 x^2}+1} \left (d+e x^2\right )}{3 e^3 \left (c^2 d-e\right ) \left (d+e x^2\right )^{3/2}}-\frac {b x \sqrt {\frac {1}{c^2 x^2}+1} \left (8 c^3 \sqrt {d} \sqrt {-d-e x^2} \tan ^{-1}\left (\frac {\sqrt {d} \sqrt {c^2 x^2+1}}{\sqrt {-d-e x^2}}\right )-3 \sqrt {c^2} \sqrt {e} \sqrt {c^2 d-e} \sqrt {\frac {c^2 \left (d+e x^2\right )}{c^2 d-e}} \sinh ^{-1}\left (\frac {c \sqrt {e} \sqrt {c^2 x^2+1}}{\sqrt {c^2} \sqrt {c^2 d-e}}\right )\right )}{3 c^2 e^3 \sqrt {c^2 x^2+1} \sqrt {d+e x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(b*c*d*e*Sqrt[1 + 1/(c^2*x^2)]*x*(d + e*x^2) + a*(c^2*d - e)*(8*d^2 + 12*d*e*x^2 + 3*e^2*x^4) + b*(c^2*d - e)*
(8*d^2 + 12*d*e*x^2 + 3*e^2*x^4)*ArcCsch[c*x])/(3*(c^2*d - e)*e^3*(d + e*x^2)^(3/2)) - (b*Sqrt[1 + 1/(c^2*x^2)
]*x*(-3*Sqrt[c^2]*Sqrt[c^2*d - e]*Sqrt[e]*Sqrt[(c^2*(d + e*x^2))/(c^2*d - e)]*ArcSinh[(c*Sqrt[e]*Sqrt[1 + c^2*
x^2])/(Sqrt[c^2]*Sqrt[c^2*d - e])] + 8*c^3*Sqrt[d]*Sqrt[-d - e*x^2]*ArcTan[(Sqrt[d]*Sqrt[1 + c^2*x^2])/Sqrt[-d
- e*x^2]]))/(3*c^2*e^3*Sqrt[1 + c^2*x^2]*Sqrt[d + e*x^2])

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fricas [B]  time = 1.11, size = 2421, normalized size = 9.65 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/12*(3*(b*c^2*d^3 + (b*c^2*d*e^2 - b*e^3)*x^4 - b*d^2*e + 2*(b*c^2*d^2*e - b*d*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) + 4*(8*b*c^3*d^3 - 8*b*c*d^2*e + 3*(b*c^3*d*e^2 - b*c*e^3)*x^4 +
12*(b*c^3*d^2*e - b*c*d*e^2)*x^2)*sqrt(e*x^2 + d)*log((c*x*sqrt((c^2*x^2 + 1)/(c^2*x^2)) + 1)/(c*x)) + 8*(b*c
^3*d^3 - b*c*d^2*e + (b*c^3*d*e^2 - b*c*e^3)*x^4 + 2*(b*c^3*d^2*e - b*c*d*e^2)*x^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*(8*a*c^3*d^3 - 8*a*c*d^2*e + 3*(a*c^3*d*e^2 - a*c*e^3)*x^4 + 12*(a*c^3*
d^2*e - a*c*d*e^2)*x^2 + (b*c^2*d*e^2*x^3 + b*c^2*d^2*e*x)*sqrt((c^2*x^2 + 1)/(c^2*x^2)))*sqrt(e*x^2 + d))/(c^
3*d^3*e^3 - c*d^2*e^4 + (c^3*d*e^5 - c*e^6)*x^4 + 2*(c^3*d^2*e^4 - c*d*e^5)*x^2), -1/6*(3*(b*c^2*d^3 + (b*c^2*
d*e^2 - b*e^3)*x^4 - b*d^2*e + 2*(b*c^2*d^2*e - b*d*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)) - 2*(8*b*c
^3*d^3 - 8*b*c*d^2*e + 3*(b*c^3*d*e^2 - b*c*e^3)*x^4 + 12*(b*c^3*d^2*e - b*c*d*e^2)*x^2)*sqrt(e*x^2 + d)*log((
c*x*sqrt((c^2*x^2 + 1)/(c^2*x^2)) + 1)/(c*x)) - 4*(b*c^3*d^3 - b*c*d^2*e + (b*c^3*d*e^2 - b*c*e^3)*x^4 + 2*(b*
c^3*d^2*e - b*c*d*e^2)*x^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*(8*a*c^3*d^3 - 8*
a*c*d^2*e + 3*(a*c^3*d*e^2 - a*c*e^3)*x^4 + 12*(a*c^3*d^2*e - a*c*d*e^2)*x^2 + (b*c^2*d*e^2*x^3 + b*c^2*d^2*e*
x)*sqrt((c^2*x^2 + 1)/(c^2*x^2)))*sqrt(e*x^2 + d))/(c^3*d^3*e^3 - c*d^2*e^4 + (c^3*d*e^5 - c*e^6)*x^4 + 2*(c^3
*d^2*e^4 - c*d*e^5)*x^2), 1/12*(16*(b*c^3*d^3 - b*c*d^2*e + (b*c^3*d*e^2 - b*c*e^3)*x^4 + 2*(b*c^3*d^2*e - b*c
*d*e^2)*x^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)) + 3*(b*c^2*d^3 + (b*c^2*d*e^2 - b*e^3)*x^4 - b*d^2*e + 2*(b
*c^2*d^2*e - b*d*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) + 4*(8*b*c^3*d^3 -
8*b*c*d^2*e + 3*(b*c^3*d*e^2 - b*c*e^3)*x^4 + 12*(b*c^3*d^2*e - b*c*d*e^2)*x^2)*sqrt(e*x^2 + d)*log((c*x*sqrt
((c^2*x^2 + 1)/(c^2*x^2)) + 1)/(c*x)) + 4*(8*a*c^3*d^3 - 8*a*c*d^2*e + 3*(a*c^3*d*e^2 - a*c*e^3)*x^4 + 12*(a*c
^3*d^2*e - a*c*d*e^2)*x^2 + (b*c^2*d*e^2*x^3 + b*c^2*d^2*e*x)*sqrt((c^2*x^2 + 1)/(c^2*x^2)))*sqrt(e*x^2 + d))/
(c^3*d^3*e^3 - c*d^2*e^4 + (c^3*d*e^5 - c*e^6)*x^4 + 2*(c^3*d^2*e^4 - c*d*e^5)*x^2), 1/6*(8*(b*c^3*d^3 - b*c*d
^2*e + (b*c^3*d*e^2 - b*c*e^3)*x^4 + 2*(b*c^3*d^2*e - b*c*d*e^2)*x^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)) -
3*(b*c^2*d^3 + (b*c^2*d*e^2 - b*e^3)*x^4 - b*d^2*e + 2*(b*c^2*d^2*e - b*d*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)) + 2*(8*b*c^3*d^3 - 8*b*c*d^2*e + 3*(b*c^3*d*e^2 - b*c*e^3)*x^4 + 12*(b*c^3*d^2*e - b*c*d*e^2)*x^2)
*sqrt(e*x^2 + d)*log((c*x*sqrt((c^2*x^2 + 1)/(c^2*x^2)) + 1)/(c*x)) + 2*(8*a*c^3*d^3 - 8*a*c*d^2*e + 3*(a*c^3*
d*e^2 - a*c*e^3)*x^4 + 12*(a*c^3*d^2*e - a*c*d*e^2)*x^2 + (b*c^2*d*e^2*x^3 + b*c^2*d^2*e*x)*sqrt((c^2*x^2 + 1)
/(c^2*x^2)))*sqrt(e*x^2 + d))/(c^3*d^3*e^3 - c*d^2*e^4 + (c^3*d*e^5 - c*e^6)*x^4 + 2*(c^3*d^2*e^4 - c*d*e^5)*x
^2)]

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b \operatorname {arcsch}\left (c x\right ) + a\right )} x^{5}}{{\left (e x^{2} + d\right )}^{\frac {5}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [F]  time = 0.44, size = 0, normalized size = 0.00 \[ \int \frac {x^{5} \left (a +b \,\mathrm {arccsch}\left (c x \right )\right )}{\left (e \,x^{2}+d \right )^{\frac {5}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {1}{3} \, {\left (\frac {3 \, x^{4}}{{\left (e x^{2} + d\right )}^{\frac {3}{2}} e} + \frac {12 \, d x^{2}}{{\left (e x^{2} + d\right )}^{\frac {3}{2}} e^{2}} + \frac {8 \, d^{2}}{{\left (e x^{2} + d\right )}^{\frac {3}{2}} e^{3}}\right )} a + \frac {1}{3} \, b {\left (\frac {{\left (3 \, e^{2} x^{4} + 12 \, d e x^{2} + 8 \, d^{2}\right )} \log \left (\sqrt {c^{2} x^{2} + 1} + 1\right )}{{\left (e^{4} x^{2} + d e^{3}\right )} \sqrt {e x^{2} + d}} + 3 \, \int \frac {3 \, c^{2} e^{2} x^{5} + 12 \, c^{2} d e x^{3} + 8 \, c^{2} d^{2} x}{3 \, {\left ({\left (c^{2} e^{4} x^{4} + d e^{3} + {\left (c^{2} d e^{3} + e^{4}\right )} x^{2}\right )} \sqrt {c^{2} x^{2} + 1} \sqrt {e x^{2} + d} + {\left (c^{2} e^{4} x^{4} + d e^{3} + {\left (c^{2} d e^{3} + e^{4}\right )} x^{2}\right )} \sqrt {e x^{2} + d}\right )}}\,{d x} - 3 \, \int \frac {3 \, {\left (e^{3} \log \relax (c) + e^{3}\right )} c^{2} x^{7} + 20 \, c^{2} d^{2} e x^{3} + 8 \, c^{2} d^{3} x + 3 \, {\left (5 \, c^{2} d e^{2} + e^{3} \log \relax (c)\right )} x^{5} + 3 \, {\left (c^{2} e^{3} x^{7} + e^{3} x^{5}\right )} \log \relax (x)}{3 \, {\left (c^{2} e^{5} x^{6} + d^{2} e^{3} + {\left (2 \, c^{2} d e^{4} + e^{5}\right )} x^{4} + {\left (c^{2} d^{2} e^{3} + 2 \, d e^{4}\right )} x^{2}\right )} \sqrt {e x^{2} + d}}\,{d x}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*(3*x^4/((e*x^2 + d)^(3/2)*e) + 12*d*x^2/((e*x^2 + d)^(3/2)*e^2) + 8*d^2/((e*x^2 + d)^(3/2)*e^3))*a + 1/3*b
*((3*e^2*x^4 + 12*d*e*x^2 + 8*d^2)*log(sqrt(c^2*x^2 + 1) + 1)/((e^4*x^2 + d*e^3)*sqrt(e*x^2 + d)) + 3*integrat
e(1/3*(3*c^2*e^2*x^5 + 12*c^2*d*e*x^3 + 8*c^2*d^2*x)/((c^2*e^4*x^4 + d*e^3 + (c^2*d*e^3 + e^4)*x^2)*sqrt(c^2*x
^2 + 1)*sqrt(e*x^2 + d) + (c^2*e^4*x^4 + d*e^3 + (c^2*d*e^3 + e^4)*x^2)*sqrt(e*x^2 + d)), x) - 3*integrate(1/3
*(3*(e^3*log(c) + e^3)*c^2*x^7 + 20*c^2*d^2*e*x^3 + 8*c^2*d^3*x + 3*(5*c^2*d*e^2 + e^3*log(c))*x^5 + 3*(c^2*e^
3*x^7 + e^3*x^5)*log(x))/((c^2*e^5*x^6 + d^2*e^3 + (2*c^2*d*e^4 + e^5)*x^4 + (c^2*d^2*e^3 + 2*d*e^4)*x^2)*sqrt
(e*x^2 + d)), x))

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {x^5\,\left (a+b\,\mathrm {asinh}\left (\frac {1}{c\,x}\right )\right )}{{\left (e\,x^2+d\right )}^{5/2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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