3.151 \(\int \frac{x^3 (a+b \text{sech}^{-1}(c x))}{\sqrt{d+e x^2}} \, dx\)

Optimal. Leaf size=251 \[ -\frac{d \sqrt{d+e x^2} \left (a+b \text{sech}^{-1}(c x)\right )}{e^2}+\frac{\left (d+e x^2\right )^{3/2} \left (a+b \text{sech}^{-1}(c x)\right )}{3 e^2}+\frac{2 b d^{3/2} \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \tanh ^{-1}\left (\frac{\sqrt{d+e x^2}}{\sqrt{d} \sqrt{1-c^2 x^2}}\right )}{3 e^2}+\frac{b \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \left (3 c^2 d-e\right ) \tan ^{-1}\left (\frac{\sqrt{e} \sqrt{1-c^2 x^2}}{c \sqrt{d+e x^2}}\right )}{6 c^3 e^{3/2}}-\frac{b \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \sqrt{1-c^2 x^2} \sqrt{d+e x^2}}{6 c^2 e} \]

[Out]

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

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Rubi [A]  time = 0.330097, antiderivative size = 251, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 12, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.522, Rules used = {266, 43, 6301, 12, 573, 154, 157, 63, 217, 203, 93, 207} \[ -\frac{d \sqrt{d+e x^2} \left (a+b \text{sech}^{-1}(c x)\right )}{e^2}+\frac{\left (d+e x^2\right )^{3/2} \left (a+b \text{sech}^{-1}(c x)\right )}{3 e^2}+\frac{2 b d^{3/2} \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \tanh ^{-1}\left (\frac{\sqrt{d+e x^2}}{\sqrt{d} \sqrt{1-c^2 x^2}}\right )}{3 e^2}+\frac{b \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \left (3 c^2 d-e\right ) \tan ^{-1}\left (\frac{\sqrt{e} \sqrt{1-c^2 x^2}}{c \sqrt{d+e x^2}}\right )}{6 c^3 e^{3/2}}-\frac{b \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \sqrt{1-c^2 x^2} \sqrt{d+e x^2}}{6 c^2 e} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 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 6301

Int[((a_.) + ArcSech[(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*ArcSech[c*x], u, x] + Dist[b*Sqrt[1 + c*x]*Sqrt[1/(1 + c*x)],
Int[SimplifyIntegrand[u/(x*Sqrt[1 - c*x]*Sqrt[1 + c*x]), 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 12

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

Rule 573

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

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 207

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

Rubi steps

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

Mathematica [A]  time = 1.26632, size = 406, normalized size = 1.62 \[ -\frac{\sqrt{d+e x^2} \left (2 a c^2 \left (2 d-e x^2\right )+2 b c^2 \text{sech}^{-1}(c x) \left (2 d-e x^2\right )+b e \sqrt{\frac{1-c x}{c x+1}} (c x+1)\right )}{6 c^2 e^2}-\frac{b \sqrt{\frac{1-c x}{c x+1}} \sqrt{1-c^2 x^2} \left (4 c^5 d^{3/2} \sqrt{-d-e x^2} \tan ^{-1}\left (\frac{\sqrt{d} \sqrt{1-c^2 x^2}}{\sqrt{-d-e x^2}}\right )+\sqrt{-c^2} e^{3/2} \sqrt{c^2 (-d)-e} \sqrt{\frac{c^2 \left (d+e x^2\right )}{c^2 d+e}} \sin ^{-1}\left (\frac{\sqrt{-c^2} \sqrt{e} \sqrt{1-c^2 x^2}}{c \sqrt{c^2 (-d)-e}}\right )-3 \left (-c^2\right )^{3/2} d \sqrt{e} \sqrt{c^2 (-d)-e} \sqrt{\frac{c^2 \left (d+e x^2\right )}{c^2 d+e}} \sin ^{-1}\left (\frac{c \sqrt{e} \sqrt{1-c^2 x^2}}{\sqrt{-c^2} \sqrt{c^2 (-d)-e}}\right )\right )}{6 c^5 e^2 (c x-1) \sqrt{d+e x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(Sqrt[d + e*x^2]*(b*e*Sqrt[(1 - c*x)/(1 + c*x)]*(1 + c*x) + 2*a*c^2*(2*d - e*x^2) + 2*b*c^2*(2*d - e*x^2)*Arc
Sech[c*x]))/(6*c^2*e^2) - (b*Sqrt[(1 - c*x)/(1 + c*x)]*Sqrt[1 - c^2*x^2]*(-3*(-c^2)^(3/2)*d*Sqrt[-(c^2*d) - e]
*Sqrt[e]*Sqrt[(c^2*(d + e*x^2))/(c^2*d + e)]*ArcSin[(c*Sqrt[e]*Sqrt[1 - c^2*x^2])/(Sqrt[-c^2]*Sqrt[-(c^2*d) -
e])] + Sqrt[-c^2]*Sqrt[-(c^2*d) - e]*e^(3/2)*Sqrt[(c^2*(d + e*x^2))/(c^2*d + e)]*ArcSin[(Sqrt[-c^2]*Sqrt[e]*Sq
rt[1 - c^2*x^2])/(c*Sqrt[-(c^2*d) - e])] + 4*c^5*d^(3/2)*Sqrt[-d - e*x^2]*ArcTan[(Sqrt[d]*Sqrt[1 - c^2*x^2])/S
qrt[-d - e*x^2]]))/(6*c^5*e^2*(-1 + c*x)*Sqrt[d + e*x^2])

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Maple [F]  time = 1.794, size = 0, normalized size = 0. \begin{align*} \int{{x}^{3} \left ( a+b{\rm arcsech} \left (cx\right ) \right ){\frac{1}{\sqrt{e{x}^{2}+d}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(a+b*arcsech(c*x))/(e*x^2+d)^(1/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 5.22589, size = 3032, normalized size = 12.08 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(a+b*arcsech(c*x))/(e*x^2+d)^(1/2),x, algorithm="fricas")

[Out]

[1/24*(4*b*c^3*d^(3/2)*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) + (3*b*c^2*d - b*e)*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*x^2 - 2*b*c^3*d)*sqrt(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*x^2 - b*c^2*e*x*sqrt(-(c^2*x^2 - 1)/(c^2*x^2)) - 4*a
*c^3*d)*sqrt(e*x^2 + d))/(c^3*e^2), 1/12*(2*b*c^3*d^(3/2)*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
) + (3*b*c^2*d - b*e)*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*x^2 - 2*b*c^3*d)*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*x^2 - b*c^2*e*x*sqrt(-(c^2*x^2 - 1)/(c^2*x^2)) -
4*a*c^3*d)*sqrt(e*x^2 + d))/(c^3*e^2), 1/24*(8*b*c^3*sqrt(-d)*d*arctan(-1/2*((c^3*d - c*e)*x^3 - 2*c*d*x)*sqr
t(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 -
b*e)*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*x^2 - 2*b*c^3*d)*sqrt(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*x^2 - b*c^2*e*x*sqrt(-(c^2*x^2 -
1)/(c^2*x^2)) - 4*a*c^3*d)*sqrt(e*x^2 + d))/(c^3*e^2), 1/12*(4*b*c^3*sqrt(-d)*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 - b*e)*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*x^2 - 2*b*c^3*d)*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*x^2 - b*c^2*e*x*sqrt(-(c^2*x^2 - 1)/(c^2*x^2)) -
4*a*c^3*d)*sqrt(e*x^2 + d))/(c^3*e^2)]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**3*(a+b*asech(c*x))/(e*x**2+d)**(1/2),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(a+b*arcsech(c*x))/(e*x^2+d)^(1/2),x, algorithm="giac")

[Out]

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