3.129 \(\int x (d+e x^2)^{3/2} (a+b \text{csch}^{-1}(c x)) \, dx\)

Optimal. Leaf size=270 \[ \frac{\left (d+e x^2\right )^{5/2} \left (a+b \text{csch}^{-1}(c x)\right )}{5 e}+\frac{b x \left (15 c^4 d^2-10 c^2 d e+3 e^2\right ) \tan ^{-1}\left (\frac{\sqrt{e} \sqrt{-c^2 x^2-1}}{c \sqrt{d+e x^2}}\right )}{40 c^4 \sqrt{e} \sqrt{-c^2 x^2}}+\frac{b c d^{5/2} x \tan ^{-1}\left (\frac{\sqrt{d+e x^2}}{\sqrt{d} \sqrt{-c^2 x^2-1}}\right )}{5 e \sqrt{-c^2 x^2}}+\frac{b x \sqrt{-c^2 x^2-1} \left (d+e x^2\right )^{3/2}}{20 c \sqrt{-c^2 x^2}}+\frac{b x \sqrt{-c^2 x^2-1} \left (7 c^2 d-3 e\right ) \sqrt{d+e x^2}}{40 c^3 \sqrt{-c^2 x^2}} \]

[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*ArcCsch[c*x]))/(5*e) + (b*(15*c^4*d^2 -
10*c^2*d*e + 3*e^2)*x*ArcTan[(Sqrt[e]*Sqrt[-1 - c^2*x^2])/(c*Sqrt[d + e*x^2])])/(40*c^4*Sqrt[e]*Sqrt[-(c^2*x^2
)]) + (b*c*d^(5/2)*x*ArcTan[Sqrt[d + e*x^2]/(Sqrt[d]*Sqrt[-1 - c^2*x^2])])/(5*e*Sqrt[-(c^2*x^2)])

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Rubi [A]  time = 0.281469, antiderivative size = 270, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 10, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.476, Rules used = {6300, 446, 102, 154, 157, 63, 217, 203, 93, 204} \[ \frac{\left (d+e x^2\right )^{5/2} \left (a+b \text{csch}^{-1}(c x)\right )}{5 e}+\frac{b x \left (15 c^4 d^2-10 c^2 d e+3 e^2\right ) \tan ^{-1}\left (\frac{\sqrt{e} \sqrt{-c^2 x^2-1}}{c \sqrt{d+e x^2}}\right )}{40 c^4 \sqrt{e} \sqrt{-c^2 x^2}}+\frac{b c d^{5/2} x \tan ^{-1}\left (\frac{\sqrt{d+e x^2}}{\sqrt{d} \sqrt{-c^2 x^2-1}}\right )}{5 e \sqrt{-c^2 x^2}}+\frac{b x \sqrt{-c^2 x^2-1} \left (d+e x^2\right )^{3/2}}{20 c \sqrt{-c^2 x^2}}+\frac{b x \sqrt{-c^2 x^2-1} \left (7 c^2 d-3 e\right ) \sqrt{d+e x^2}}{40 c^3 \sqrt{-c^2 x^2}} \]

Antiderivative was successfully verified.

[In]

Int[x*(d + e*x^2)^(3/2)*(a + b*ArcCsch[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*ArcCsch[c*x]))/(5*e) + (b*(15*c^4*d^2 -
10*c^2*d*e + 3*e^2)*x*ArcTan[(Sqrt[e]*Sqrt[-1 - c^2*x^2])/(c*Sqrt[d + e*x^2])])/(40*c^4*Sqrt[e]*Sqrt[-(c^2*x^2
)]) + (b*c*d^(5/2)*x*ArcTan[Sqrt[d + e*x^2]/(Sqrt[d]*Sqrt[-1 - c^2*x^2])])/(5*e*Sqrt[-(c^2*x^2)])

Rule 6300

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

Rule 446

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 102

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 154

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 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 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 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 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 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 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])

Rubi steps

\begin{align*} \int x \left (d+e x^2\right )^{3/2} \left (a+b \text{csch}^{-1}(c x)\right ) \, dx &=\frac{\left (d+e x^2\right )^{5/2} \left (a+b \text{csch}^{-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 \text{csch}^{-1}(c x)\right )}{5 e}-\frac{(b c x) \operatorname{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 \text{csch}^{-1}(c x)\right )}{5 e}+\frac{(b x) \operatorname{Subst}\left (\int \frac{\sqrt{d+e x} \left (-2 c^2 d^2-\frac{1}{2} \left (7 c^2 d-3 e\right ) e 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 \text{csch}^{-1}(c x)\right )}{5 e}-\frac{(b x) \operatorname{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 \text{csch}^{-1}(c x)\right )}{5 e}-\frac{\left (b c d^3 x\right ) \operatorname{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 ) \operatorname{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 \text{csch}^{-1}(c x)\right )}{5 e}-\frac{\left (b c d^3 x\right ) \operatorname{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 ) \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 )}{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 \text{csch}^{-1}(c x)\right )}{5 e}+\frac{b c d^{5/2} x \tan ^{-1}\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 ) \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 )}{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 \text{csch}^{-1}(c x)\right )}{5 e}+\frac{b \left (15 c^4 d^2-10 c^2 d e+3 e^2\right ) x \tan ^{-1}\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}}+\frac{b c d^{5/2} x \tan ^{-1}\left (\frac{\sqrt{d+e x^2}}{\sqrt{d} \sqrt{-1-c^2 x^2}}\right )}{5 e \sqrt{-c^2 x^2}}\\ \end{align*}

Mathematica [A]  time = 0.727555, size = 314, normalized size = 1.16 \[ \frac{\sqrt{d+e x^2} \left (8 a c^3 \left (d+e x^2\right )^2+b e x \sqrt{\frac{1}{c^2 x^2}+1} \left (c^2 \left (9 d+2 e x^2\right )-3 e\right )+8 b c^3 \text{csch}^{-1}(c x) \left (d+e x^2\right )^2\right )}{40 c^3 e}-\frac{b x \sqrt{\frac{1}{c^2 x^2}+1} \left (\sqrt{c^2} \sqrt{e} \sqrt{c^2 d-e} \left (-15 c^4 d^2+10 c^2 d e-3 e^2\right ) \sqrt{\frac{c^2 \left (d+e x^2\right )}{c^2 d-e}} \sinh ^{-1}\left (\frac{c \sqrt{e} \sqrt{c^2 x^2+1}}{\sqrt{c^2} \sqrt{c^2 d-e}}\right )+8 c^7 d^{5/2} \sqrt{-d-e x^2} \tan ^{-1}\left (\frac{\sqrt{d} \sqrt{c^2 x^2+1}}{\sqrt{-d-e x^2}}\right )\right )}{40 c^6 e \sqrt{c^2 x^2+1} \sqrt{d+e x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[d + e*x^2]*(8*a*c^3*(d + e*x^2)^2 + b*e*Sqrt[1 + 1/(c^2*x^2)]*x*(-3*e + c^2*(9*d + 2*e*x^2)) + 8*b*c^3*(
d + e*x^2)^2*ArcCsch[c*x]))/(40*c^3*e) - (b*Sqrt[1 + 1/(c^2*x^2)]*x*(Sqrt[c^2]*Sqrt[c^2*d - e]*Sqrt[e]*(-15*c^
4*d^2 + 10*c^2*d*e - 3*e^2)*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^7*d^(5/2)*Sqrt[-d - e*x^2]*ArcTan[(Sqrt[d]*Sqrt[1 + c^2*x^2])/Sqrt[-d - e*x^2]]))/(
40*c^6*e*Sqrt[1 + c^2*x^2]*Sqrt[d + e*x^2])

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Maple [F]  time = 0.447, size = 0, normalized size = 0. \begin{align*} \int x \left ( e{x}^{2}+d \right ) ^{{\frac{3}{2}}} \left ( a+b{\rm arccsch} \left (cx\right ) \right ) \, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{{\left (e x^{2} + d\right )}^{\frac{5}{2}} a}{5 \, e} + \frac{1}{5} \,{\left (\frac{{\left (e^{2} x^{4} + 2 \, d e x^{2} + d^{2}\right )} \sqrt{e x^{2} + d} \log \left (\sqrt{c^{2} x^{2} + 1} + 1\right )}{e} + 5 \, \int \frac{{\left (c^{2} e^{2} x^{5} + 2 \, c^{2} d e x^{3} + c^{2} d^{2} x\right )} \sqrt{e x^{2} + d}}{5 \,{\left (c^{2} e x^{2} +{\left (c^{2} e x^{2} + e\right )} \sqrt{c^{2} x^{2} + 1} + e\right )}}\,{d x} - 5 \, \int \frac{{\left ({\left (5 \, e^{2} \log \left (c\right ) + e^{2}\right )} c^{2} x^{5} +{\left ({\left (5 \, d e \log \left (c\right ) + 2 \, d e\right )} c^{2} + 5 \, e^{2} \log \left (c\right )\right )} x^{3} +{\left (c^{2} d^{2} + 5 \, d e \log \left (c\right )\right )} x + 5 \,{\left (c^{2} e^{2} x^{5} +{\left (c^{2} d e + e^{2}\right )} x^{3} + d e x\right )} \log \left (x\right )\right )} \sqrt{e x^{2} + d}}{5 \,{\left (c^{2} e x^{2} + e\right )}}\,{d x}\right )} b \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 10.5, size = 3579, normalized size = 13.26 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/160*(8*b*c^5*d^(5/2)*log(((c^4*d^2 + 6*c^2*d*e + e^2)*x^4 + 8*(c^2*d^2 + d*e)*x^2 - 4*((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) + (15*b*c^4*d^2 - 10*b*c^2*d*e +
3*b*e^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) + 32*(b*c^5*e^2*x^4 + 2*b*c^5*d*e*x^
2 + b*c^5*d^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^5*e^2*x^4 + 16*a*
c^5*d*e*x^2 + 8*a*c^5*d^2 + (2*b*c^4*e^2*x^3 + 3*(3*b*c^4*d*e - b*c^2*e^2)*x)*sqrt((c^2*x^2 + 1)/(c^2*x^2)))*s
qrt(e*x^2 + d))/(c^5*e), 1/80*(4*b*c^5*d^(5/2)*log(((c^4*d^2 + 6*c^2*d*e + e^2)*x^4 + 8*(c^2*d^2 + d*e)*x^2 -
4*((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) - (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^3 + (c^2*d + e)*x)*sqrt(e*x^2 + d)*sqrt(-e)*sqr
t((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)) + 16*(b*c^5*e^2*x^4 + 2*b*c^5*d*e*x^2 +
b*c^5*d^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^5*e^2*x^4 + 16*a*c^5*
d*e*x^2 + 8*a*c^5*d^2 + (2*b*c^4*e^2*x^3 + 3*(3*b*c^4*d*e - b*c^2*e^2)*x)*sqrt((c^2*x^2 + 1)/(c^2*x^2)))*sqrt(
e*x^2 + d))/(c^5*e), 1/160*(16*b*c^5*sqrt(-d)*d^2*arctan(1/2*((c^3*d + c*e)*x^3 + 2*c*d*x)*sqrt(e*x^2 + d)*sqr
t(-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)) + (15*b*c^4*d^2 - 10*b*c^2*d*e
+ 3*b*e^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) + 32*(b*c^5*e^2*x^4 + 2*b*c^5*d*e*
x^2 + b*c^5*d^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^5*e^2*x^4 + 16*
a*c^5*d*e*x^2 + 8*a*c^5*d^2 + (2*b*c^4*e^2*x^3 + 3*(3*b*c^4*d*e - b*c^2*e^2)*x)*sqrt((c^2*x^2 + 1)/(c^2*x^2)))
*sqrt(e*x^2 + d))/(c^5*e), 1/80*(8*b*c^5*sqrt(-d)*d^2*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)) - (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^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)) + 16*(b*c^5*e^2*x^4 + 2*b*c^5*d*e*x^2 + b*c^5*d^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^5*e^2*x^4 + 16*a*c^5*d*e*x^2 + 8*a*c^5
*d^2 + (2*b*c^4*e^2*x^3 + 3*(3*b*c^4*d*e - b*c^2*e^2)*x)*sqrt((c^2*x^2 + 1)/(c^2*x^2)))*sqrt(e*x^2 + d))/(c^5*
e)]

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (e x^{2} + d\right )}^{\frac{3}{2}}{\left (b \operatorname{arcsch}\left (c x\right ) + a\right )} x\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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