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

Optimal. Leaf size=230 \[ \frac{\left (d+e x^2\right )^3 \left (a+b \text{sech}^{-1}(c x)\right )}{6 e}-\frac{b \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \sqrt{1-c^2 x^2} \left (3 c^4 d^2+3 c^2 d e+e^2\right )}{6 c^6}-\frac{b d^3 \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \tanh ^{-1}\left (\sqrt{1-c^2 x^2}\right )}{6 e}+\frac{b e \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \left (1-c^2 x^2\right )^{3/2} \left (3 c^2 d+2 e\right )}{18 c^6}-\frac{b e^2 \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \left (1-c^2 x^2\right )^{5/2}}{30 c^6} \]

[Out]

-(b*(3*c^4*d^2 + 3*c^2*d*e + e^2)*Sqrt[(1 + c*x)^(-1)]*Sqrt[1 + c*x]*Sqrt[1 - c^2*x^2])/(6*c^6) + (b*e*(3*c^2*
d + 2*e)*Sqrt[(1 + c*x)^(-1)]*Sqrt[1 + c*x]*(1 - c^2*x^2)^(3/2))/(18*c^6) - (b*e^2*Sqrt[(1 + c*x)^(-1)]*Sqrt[1
+ c*x]*(1 - c^2*x^2)^(5/2))/(30*c^6) + ((d + e*x^2)^3*(a + b*ArcSech[c*x]))/(6*e) - (b*d^3*Sqrt[(1 + c*x)^(-1
)]*Sqrt[1 + c*x]*ArcTanh[Sqrt[1 - c^2*x^2]])/(6*e)

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Rubi [A]  time = 0.252243, antiderivative size = 230, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.316, Rules used = {6299, 517, 446, 88, 63, 208} \[ \frac{\left (d+e x^2\right )^3 \left (a+b \text{sech}^{-1}(c x)\right )}{6 e}-\frac{b \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \sqrt{1-c^2 x^2} \left (3 c^4 d^2+3 c^2 d e+e^2\right )}{6 c^6}-\frac{b d^3 \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \tanh ^{-1}\left (\sqrt{1-c^2 x^2}\right )}{6 e}+\frac{b e \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \left (1-c^2 x^2\right )^{3/2} \left (3 c^2 d+2 e\right )}{18 c^6}-\frac{b e^2 \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \left (1-c^2 x^2\right )^{5/2}}{30 c^6} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(b*(3*c^4*d^2 + 3*c^2*d*e + e^2)*Sqrt[(1 + c*x)^(-1)]*Sqrt[1 + c*x]*Sqrt[1 - c^2*x^2])/(6*c^6) + (b*e*(3*c^2*
d + 2*e)*Sqrt[(1 + c*x)^(-1)]*Sqrt[1 + c*x]*(1 - c^2*x^2)^(3/2))/(18*c^6) - (b*e^2*Sqrt[(1 + c*x)^(-1)]*Sqrt[1
+ c*x]*(1 - c^2*x^2)^(5/2))/(30*c^6) + ((d + e*x^2)^3*(a + b*ArcSech[c*x]))/(6*e) - (b*d^3*Sqrt[(1 + c*x)^(-1
)]*Sqrt[1 + c*x]*ArcTanh[Sqrt[1 - c^2*x^2]])/(6*e)

Rule 6299

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

Rule 517

Int[(u_.)*((c_) + (d_.)*(x_)^(n_.))^(q_.)*((a1_) + (b1_.)*(x_)^(non2_.))^(p_.)*((a2_) + (b2_.)*(x_)^(non2_.))^
(p_.), x_Symbol] :> Int[u*(a1*a2 + b1*b2*x^n)^p*(c + d*x^n)^q, x] /; FreeQ[{a1, b1, a2, b2, c, d, n, p, q}, x]
&& EqQ[non2, n/2] && EqQ[a2*b1 + a1*b2, 0] && (IntegerQ[p] || (GtQ[a1, 0] && GtQ[a2, 0]))

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 88

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

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 208

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

Rubi steps

\begin{align*} \int x \left (d+e x^2\right )^2 \left (a+b \text{sech}^{-1}(c x)\right ) \, dx &=\frac{\left (d+e x^2\right )^3 \left (a+b \text{sech}^{-1}(c x)\right )}{6 e}+\frac{\left (b \sqrt{\frac{1}{1+c x}} \sqrt{1+c x}\right ) \int \frac{\left (d+e x^2\right )^3}{x \sqrt{1-c x} \sqrt{1+c x}} \, dx}{6 e}\\ &=\frac{\left (d+e x^2\right )^3 \left (a+b \text{sech}^{-1}(c x)\right )}{6 e}+\frac{\left (b \sqrt{\frac{1}{1+c x}} \sqrt{1+c x}\right ) \int \frac{\left (d+e x^2\right )^3}{x \sqrt{1-c^2 x^2}} \, dx}{6 e}\\ &=\frac{\left (d+e x^2\right )^3 \left (a+b \text{sech}^{-1}(c x)\right )}{6 e}+\frac{\left (b \sqrt{\frac{1}{1+c x}} \sqrt{1+c x}\right ) \operatorname{Subst}\left (\int \frac{(d+e x)^3}{x \sqrt{1-c^2 x}} \, dx,x,x^2\right )}{12 e}\\ &=\frac{\left (d+e x^2\right )^3 \left (a+b \text{sech}^{-1}(c x)\right )}{6 e}+\frac{\left (b \sqrt{\frac{1}{1+c x}} \sqrt{1+c x}\right ) \operatorname{Subst}\left (\int \left (\frac{e \left (3 c^4 d^2+3 c^2 d e+e^2\right )}{c^4 \sqrt{1-c^2 x}}+\frac{d^3}{x \sqrt{1-c^2 x}}-\frac{e^2 \left (3 c^2 d+2 e\right ) \sqrt{1-c^2 x}}{c^4}+\frac{e^3 \left (1-c^2 x\right )^{3/2}}{c^4}\right ) \, dx,x,x^2\right )}{12 e}\\ &=-\frac{b \left (3 c^4 d^2+3 c^2 d e+e^2\right ) \sqrt{\frac{1}{1+c x}} \sqrt{1+c x} \sqrt{1-c^2 x^2}}{6 c^6}+\frac{b e \left (3 c^2 d+2 e\right ) \sqrt{\frac{1}{1+c x}} \sqrt{1+c x} \left (1-c^2 x^2\right )^{3/2}}{18 c^6}-\frac{b e^2 \sqrt{\frac{1}{1+c x}} \sqrt{1+c x} \left (1-c^2 x^2\right )^{5/2}}{30 c^6}+\frac{\left (d+e x^2\right )^3 \left (a+b \text{sech}^{-1}(c x)\right )}{6 e}+\frac{\left (b d^3 \sqrt{\frac{1}{1+c x}} \sqrt{1+c x}\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{1-c^2 x}} \, dx,x,x^2\right )}{12 e}\\ &=-\frac{b \left (3 c^4 d^2+3 c^2 d e+e^2\right ) \sqrt{\frac{1}{1+c x}} \sqrt{1+c x} \sqrt{1-c^2 x^2}}{6 c^6}+\frac{b e \left (3 c^2 d+2 e\right ) \sqrt{\frac{1}{1+c x}} \sqrt{1+c x} \left (1-c^2 x^2\right )^{3/2}}{18 c^6}-\frac{b e^2 \sqrt{\frac{1}{1+c x}} \sqrt{1+c x} \left (1-c^2 x^2\right )^{5/2}}{30 c^6}+\frac{\left (d+e x^2\right )^3 \left (a+b \text{sech}^{-1}(c x)\right )}{6 e}-\frac{\left (b d^3 \sqrt{\frac{1}{1+c x}} \sqrt{1+c x}\right ) \operatorname{Subst}\left (\int \frac{1}{\frac{1}{c^2}-\frac{x^2}{c^2}} \, dx,x,\sqrt{1-c^2 x^2}\right )}{6 c^2 e}\\ &=-\frac{b \left (3 c^4 d^2+3 c^2 d e+e^2\right ) \sqrt{\frac{1}{1+c x}} \sqrt{1+c x} \sqrt{1-c^2 x^2}}{6 c^6}+\frac{b e \left (3 c^2 d+2 e\right ) \sqrt{\frac{1}{1+c x}} \sqrt{1+c x} \left (1-c^2 x^2\right )^{3/2}}{18 c^6}-\frac{b e^2 \sqrt{\frac{1}{1+c x}} \sqrt{1+c x} \left (1-c^2 x^2\right )^{5/2}}{30 c^6}+\frac{\left (d+e x^2\right )^3 \left (a+b \text{sech}^{-1}(c x)\right )}{6 e}-\frac{b d^3 \sqrt{\frac{1}{1+c x}} \sqrt{1+c x} \tanh ^{-1}\left (\sqrt{1-c^2 x^2}\right )}{6 e}\\ \end{align*}

Mathematica [A]  time = 0.257237, size = 139, normalized size = 0.6 \[ \frac{1}{6} a x^2 \left (3 d^2+3 d e x^2+e^2 x^4\right )-\frac{b \sqrt{\frac{1-c x}{c x+1}} (c x+1) \left (3 c^4 \left (15 d^2+5 d e x^2+e^2 x^4\right )+2 c^2 e \left (15 d+2 e x^2\right )+8 e^2\right )}{90 c^6}+\frac{1}{6} b x^2 \text{sech}^{-1}(c x) \left (3 d^2+3 d e x^2+e^2 x^4\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a*x^2*(3*d^2 + 3*d*e*x^2 + e^2*x^4))/6 - (b*Sqrt[(1 - c*x)/(1 + c*x)]*(1 + c*x)*(8*e^2 + 2*c^2*e*(15*d + 2*e*
x^2) + 3*c^4*(15*d^2 + 5*d*e*x^2 + e^2*x^4)))/(90*c^6) + (b*x^2*(3*d^2 + 3*d*e*x^2 + e^2*x^4)*ArcSech[c*x])/6

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Maple [A]  time = 0.175, size = 180, normalized size = 0.8 \begin{align*}{\frac{1}{{c}^{2}} \left ({\frac{a}{{c}^{4}} \left ({\frac{{e}^{2}{c}^{6}{x}^{6}}{6}}+{\frac{{c}^{6}{x}^{4}de}{2}}+{\frac{{x}^{2}{c}^{6}{d}^{2}}{2}} \right ) }+{\frac{b}{{c}^{4}} \left ({\frac{{\rm arcsech} \left (cx\right ){e}^{2}{c}^{6}{x}^{6}}{6}}+{\frac{{\rm arcsech} \left (cx\right ){c}^{6}{x}^{4}de}{2}}+{\frac{{\rm arcsech} \left (cx\right ){c}^{6}{x}^{2}{d}^{2}}{2}}-{\frac{cx \left ( 3\,{c}^{4}{e}^{2}{x}^{4}+15\,{c}^{4}de{x}^{2}+45\,{d}^{2}{c}^{4}+4\,{c}^{2}{e}^{2}{x}^{2}+30\,{c}^{2}de+8\,{e}^{2} \right ) }{90}\sqrt{-{\frac{cx-1}{cx}}}\sqrt{{\frac{cx+1}{cx}}}} \right ) } \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/c^2*(a/c^4*(1/6*e^2*c^6*x^6+1/2*c^6*x^4*d*e+1/2*x^2*c^6*d^2)+b/c^4*(1/6*arcsech(c*x)*e^2*c^6*x^6+1/2*arcsech
(c*x)*c^6*x^4*d*e+1/2*arcsech(c*x)*c^6*x^2*d^2-1/90*(-(c*x-1)/c/x)^(1/2)*c*x*((c*x+1)/c/x)^(1/2)*(3*c^4*e^2*x^
4+15*c^4*d*e*x^2+45*c^4*d^2+4*c^2*e^2*x^2+30*c^2*d*e+8*e^2)))

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Maxima [A]  time = 0.991712, size = 250, normalized size = 1.09 \begin{align*} \frac{1}{6} \, a e^{2} x^{6} + \frac{1}{2} \, a d e x^{4} + \frac{1}{2} \, a d^{2} x^{2} + \frac{1}{2} \,{\left (x^{2} \operatorname{arsech}\left (c x\right ) - \frac{x \sqrt{\frac{1}{c^{2} x^{2}} - 1}}{c}\right )} b d^{2} + \frac{1}{6} \,{\left (3 \, x^{4} \operatorname{arsech}\left (c x\right ) + \frac{c^{2} x^{3}{\left (\frac{1}{c^{2} x^{2}} - 1\right )}^{\frac{3}{2}} - 3 \, x \sqrt{\frac{1}{c^{2} x^{2}} - 1}}{c^{3}}\right )} b d e + \frac{1}{90} \,{\left (15 \, x^{6} \operatorname{arsech}\left (c x\right ) - \frac{3 \, c^{4} x^{5}{\left (\frac{1}{c^{2} x^{2}} - 1\right )}^{\frac{5}{2}} - 10 \, c^{2} x^{3}{\left (\frac{1}{c^{2} x^{2}} - 1\right )}^{\frac{3}{2}} + 15 \, x \sqrt{\frac{1}{c^{2} x^{2}} - 1}}{c^{5}}\right )} b e^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*a*e^2*x^6 + 1/2*a*d*e*x^4 + 1/2*a*d^2*x^2 + 1/2*(x^2*arcsech(c*x) - x*sqrt(1/(c^2*x^2) - 1)/c)*b*d^2 + 1/6
*(3*x^4*arcsech(c*x) + (c^2*x^3*(1/(c^2*x^2) - 1)^(3/2) - 3*x*sqrt(1/(c^2*x^2) - 1))/c^3)*b*d*e + 1/90*(15*x^6
*arcsech(c*x) - (3*c^4*x^5*(1/(c^2*x^2) - 1)^(5/2) - 10*c^2*x^3*(1/(c^2*x^2) - 1)^(3/2) + 15*x*sqrt(1/(c^2*x^2
) - 1))/c^5)*b*e^2

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Fricas [A]  time = 2.01406, size = 413, normalized size = 1.8 \begin{align*} \frac{15 \, a c^{5} e^{2} x^{6} + 45 \, a c^{5} d e x^{4} + 45 \, a c^{5} d^{2} x^{2} + 15 \,{\left (b c^{5} e^{2} x^{6} + 3 \, b c^{5} d e x^{4} + 3 \, b c^{5} d^{2} x^{2}\right )} \log \left (\frac{c x \sqrt{-\frac{c^{2} x^{2} - 1}{c^{2} x^{2}}} + 1}{c x}\right ) -{\left (3 \, b c^{4} e^{2} x^{5} +{\left (15 \, b c^{4} d e + 4 \, b c^{2} e^{2}\right )} x^{3} +{\left (45 \, b c^{4} d^{2} + 30 \, b c^{2} d e + 8 \, b e^{2}\right )} x\right )} \sqrt{-\frac{c^{2} x^{2} - 1}{c^{2} x^{2}}}}{90 \, c^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/90*(15*a*c^5*e^2*x^6 + 45*a*c^5*d*e*x^4 + 45*a*c^5*d^2*x^2 + 15*(b*c^5*e^2*x^6 + 3*b*c^5*d*e*x^4 + 3*b*c^5*d
^2*x^2)*log((c*x*sqrt(-(c^2*x^2 - 1)/(c^2*x^2)) + 1)/(c*x)) - (3*b*c^4*e^2*x^5 + (15*b*c^4*d*e + 4*b*c^2*e^2)*
x^3 + (45*b*c^4*d^2 + 30*b*c^2*d*e + 8*b*e^2)*x)*sqrt(-(c^2*x^2 - 1)/(c^2*x^2)))/c^5

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Sympy [A]  time = 22.6777, size = 252, normalized size = 1.1 \begin{align*} \begin{cases} \frac{a d^{2} x^{2}}{2} + \frac{a d e x^{4}}{2} + \frac{a e^{2} x^{6}}{6} + \frac{b d^{2} x^{2} \operatorname{asech}{\left (c x \right )}}{2} + \frac{b d e x^{4} \operatorname{asech}{\left (c x \right )}}{2} + \frac{b e^{2} x^{6} \operatorname{asech}{\left (c x \right )}}{6} - \frac{b d^{2} \sqrt{- c^{2} x^{2} + 1}}{2 c^{2}} - \frac{b d e x^{2} \sqrt{- c^{2} x^{2} + 1}}{6 c^{2}} - \frac{b e^{2} x^{4} \sqrt{- c^{2} x^{2} + 1}}{30 c^{2}} - \frac{b d e \sqrt{- c^{2} x^{2} + 1}}{3 c^{4}} - \frac{2 b e^{2} x^{2} \sqrt{- c^{2} x^{2} + 1}}{45 c^{4}} - \frac{4 b e^{2} \sqrt{- c^{2} x^{2} + 1}}{45 c^{6}} & \text{for}\: c \neq 0 \\\left (a + \infty b\right ) \left (\frac{d^{2} x^{2}}{2} + \frac{d e x^{4}}{2} + \frac{e^{2} x^{6}}{6}\right ) & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((a*d**2*x**2/2 + a*d*e*x**4/2 + a*e**2*x**6/6 + b*d**2*x**2*asech(c*x)/2 + b*d*e*x**4*asech(c*x)/2 +
b*e**2*x**6*asech(c*x)/6 - b*d**2*sqrt(-c**2*x**2 + 1)/(2*c**2) - b*d*e*x**2*sqrt(-c**2*x**2 + 1)/(6*c**2) -
b*e**2*x**4*sqrt(-c**2*x**2 + 1)/(30*c**2) - b*d*e*sqrt(-c**2*x**2 + 1)/(3*c**4) - 2*b*e**2*x**2*sqrt(-c**2*x*
*2 + 1)/(45*c**4) - 4*b*e**2*sqrt(-c**2*x**2 + 1)/(45*c**6), Ne(c, 0)), ((a + oo*b)*(d**2*x**2/2 + d*e*x**4/2
+ e**2*x**6/6), True))

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

[Out]

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