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

Optimal. Leaf size=359 \[ \frac{b x \sqrt{d+e x^2} \text{EllipticF}\left (\tan ^{-1}(c x),1-\frac{e}{c^2 d}\right )}{3 d^2 \sqrt{-c^2 x^2} \sqrt{-c^2 x^2-1} \left (c^2 d-e\right ) \sqrt{\frac{d+e x^2}{d \left (c^2 x^2+1\right )}}}+\frac{x^3 \left (a+b \text{csch}^{-1}(c x)\right )}{3 d \left (d+e x^2\right )^{3/2}}+\frac{b c^3 x^2 \sqrt{d+e x^2}}{3 d e \sqrt{-c^2 x^2} \sqrt{-c^2 x^2-1} \left (c^2 d-e\right )}+\frac{b c x^2 \sqrt{-c^2 x^2-1}}{3 d \sqrt{-c^2 x^2} \left (c^2 d-e\right ) \sqrt{d+e x^2}}-\frac{b c^2 x \sqrt{d+e x^2} E\left (\tan ^{-1}(c x)|1-\frac{e}{c^2 d}\right )}{3 d e \sqrt{-c^2 x^2} \sqrt{-c^2 x^2-1} \left (c^2 d-e\right ) \sqrt{\frac{d+e x^2}{d \left (c^2 x^2+1\right )}}} \]

[Out]

(b*c*x^2*Sqrt[-1 - c^2*x^2])/(3*d*(c^2*d - e)*Sqrt[-(c^2*x^2)]*Sqrt[d + e*x^2]) + (b*c^3*x^2*Sqrt[d + e*x^2])/
(3*d*(c^2*d - e)*e*Sqrt[-(c^2*x^2)]*Sqrt[-1 - c^2*x^2]) + (x^3*(a + b*ArcCsch[c*x]))/(3*d*(d + e*x^2)^(3/2)) -
(b*c^2*x*Sqrt[d + e*x^2]*EllipticE[ArcTan[c*x], 1 - e/(c^2*d)])/(3*d*(c^2*d - e)*e*Sqrt[-(c^2*x^2)]*Sqrt[-1 -
c^2*x^2]*Sqrt[(d + e*x^2)/(d*(1 + c^2*x^2))]) + (b*x*Sqrt[d + e*x^2]*EllipticF[ArcTan[c*x], 1 - e/(c^2*d)])/(
3*d^2*(c^2*d - e)*Sqrt[-(c^2*x^2)]*Sqrt[-1 - c^2*x^2]*Sqrt[(d + e*x^2)/(d*(1 + c^2*x^2))])

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Rubi [A]  time = 0.307725, antiderivative size = 359, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 8, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.348, Rules used = {264, 6302, 12, 471, 422, 418, 492, 411} \[ \frac{x^3 \left (a+b \text{csch}^{-1}(c x)\right )}{3 d \left (d+e x^2\right )^{3/2}}+\frac{b x \sqrt{d+e x^2} F\left (\tan ^{-1}(c x)|1-\frac{e}{c^2 d}\right )}{3 d^2 \sqrt{-c^2 x^2} \sqrt{-c^2 x^2-1} \left (c^2 d-e\right ) \sqrt{\frac{d+e x^2}{d \left (c^2 x^2+1\right )}}}+\frac{b c^3 x^2 \sqrt{d+e x^2}}{3 d e \sqrt{-c^2 x^2} \sqrt{-c^2 x^2-1} \left (c^2 d-e\right )}+\frac{b c x^2 \sqrt{-c^2 x^2-1}}{3 d \sqrt{-c^2 x^2} \left (c^2 d-e\right ) \sqrt{d+e x^2}}-\frac{b c^2 x \sqrt{d+e x^2} E\left (\tan ^{-1}(c x)|1-\frac{e}{c^2 d}\right )}{3 d e \sqrt{-c^2 x^2} \sqrt{-c^2 x^2-1} \left (c^2 d-e\right ) \sqrt{\frac{d+e x^2}{d \left (c^2 x^2+1\right )}}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(b*c*x^2*Sqrt[-1 - c^2*x^2])/(3*d*(c^2*d - e)*Sqrt[-(c^2*x^2)]*Sqrt[d + e*x^2]) + (b*c^3*x^2*Sqrt[d + e*x^2])/
(3*d*(c^2*d - e)*e*Sqrt[-(c^2*x^2)]*Sqrt[-1 - c^2*x^2]) + (x^3*(a + b*ArcCsch[c*x]))/(3*d*(d + e*x^2)^(3/2)) -
(b*c^2*x*Sqrt[d + e*x^2]*EllipticE[ArcTan[c*x], 1 - e/(c^2*d)])/(3*d*(c^2*d - e)*e*Sqrt[-(c^2*x^2)]*Sqrt[-1 -
c^2*x^2]*Sqrt[(d + e*x^2)/(d*(1 + c^2*x^2))]) + (b*x*Sqrt[d + e*x^2]*EllipticF[ArcTan[c*x], 1 - e/(c^2*d)])/(
3*d^2*(c^2*d - e)*Sqrt[-(c^2*x^2)]*Sqrt[-1 - c^2*x^2]*Sqrt[(d + e*x^2)/(d*(1 + c^2*x^2))])

Rule 264

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[((c*x)^(m + 1)*(a + b*x^n)^(p + 1))/(a
*c*(m + 1)), x] /; FreeQ[{a, b, c, m, n, p}, x] && EqQ[(m + 1)/n + p + 1, 0] && NeQ[m, -1]

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]))

Rule 12

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

Rule 471

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[(e^(n -
1)*(e*x)^(m - n + 1)*(a + b*x^n)^(p + 1)*(c + d*x^n)^(q + 1))/(n*(b*c - a*d)*(p + 1)), x] - Dist[e^n/(n*(b*c -
a*d)*(p + 1)), Int[(e*x)^(m - n)*(a + b*x^n)^(p + 1)*(c + d*x^n)^q*Simp[c*(m - n + 1) + d*(m + n*(p + q + 1)
+ 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, q}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && LtQ[p, -1] && GeQ[n
, m - n + 1] && GtQ[m - n + 1, 0] && IntBinomialQ[a, b, c, d, e, m, n, p, q, x]

Rule 422

Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Dist[a, Int[1/(Sqrt[a + b*x^2]*Sqrt[c +
d*x^2]), x], x] + Dist[b, Int[x^2/(Sqrt[a + b*x^2]*Sqrt[c + d*x^2]), x], x] /; FreeQ[{a, b, c, d}, x] && PosQ[
d/c] && PosQ[b/a]

Rule 418

Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> Simp[(Sqrt[a + b*x^2]*EllipticF[ArcT
an[Rt[d/c, 2]*x], 1 - (b*c)/(a*d)])/(a*Rt[d/c, 2]*Sqrt[c + d*x^2]*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]), x] /
; FreeQ[{a, b, c, d}, x] && PosQ[d/c] && PosQ[b/a] &&  !SimplerSqrtQ[b/a, d/c]

Rule 492

Int[(x_)^2/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> Simp[(x*Sqrt[a + b*x^2])/(b*Sqr
t[c + d*x^2]), x] - Dist[c/b, Int[Sqrt[a + b*x^2]/(c + d*x^2)^(3/2), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b
*c - a*d, 0] && PosQ[b/a] && PosQ[d/c] &&  !SimplerSqrtQ[b/a, d/c]

Rule 411

Int[Sqrt[(a_) + (b_.)*(x_)^2]/((c_) + (d_.)*(x_)^2)^(3/2), x_Symbol] :> Simp[(Sqrt[a + b*x^2]*EllipticE[ArcTan
[Rt[d/c, 2]*x], 1 - (b*c)/(a*d)])/(c*Rt[d/c, 2]*Sqrt[c + d*x^2]*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]), x] /;
FreeQ[{a, b, c, d}, x] && PosQ[b/a] && PosQ[d/c]

Rubi steps

\begin{align*} \int \frac{x^2 \left (a+b \text{csch}^{-1}(c x)\right )}{\left (d+e x^2\right )^{5/2}} \, dx &=\frac{x^3 \left (a+b \text{csch}^{-1}(c x)\right )}{3 d \left (d+e x^2\right )^{3/2}}-\frac{(b c x) \int \frac{x^2}{3 d \sqrt{-1-c^2 x^2} \left (d+e x^2\right )^{3/2}} \, dx}{\sqrt{-c^2 x^2}}\\ &=\frac{x^3 \left (a+b \text{csch}^{-1}(c x)\right )}{3 d \left (d+e x^2\right )^{3/2}}-\frac{(b c x) \int \frac{x^2}{\sqrt{-1-c^2 x^2} \left (d+e x^2\right )^{3/2}} \, dx}{3 d \sqrt{-c^2 x^2}}\\ &=\frac{b c x^2 \sqrt{-1-c^2 x^2}}{3 d \left (c^2 d-e\right ) \sqrt{-c^2 x^2} \sqrt{d+e x^2}}+\frac{x^3 \left (a+b \text{csch}^{-1}(c x)\right )}{3 d \left (d+e x^2\right )^{3/2}}+\frac{(b c x) \int \frac{\sqrt{-1-c^2 x^2}}{\sqrt{d+e x^2}} \, dx}{3 d \left (-c^2 d+e\right ) \sqrt{-c^2 x^2}}\\ &=\frac{b c x^2 \sqrt{-1-c^2 x^2}}{3 d \left (c^2 d-e\right ) \sqrt{-c^2 x^2} \sqrt{d+e x^2}}+\frac{x^3 \left (a+b \text{csch}^{-1}(c x)\right )}{3 d \left (d+e x^2\right )^{3/2}}-\frac{(b c x) \int \frac{1}{\sqrt{-1-c^2 x^2} \sqrt{d+e x^2}} \, dx}{3 d \left (-c^2 d+e\right ) \sqrt{-c^2 x^2}}-\frac{\left (b c^3 x\right ) \int \frac{x^2}{\sqrt{-1-c^2 x^2} \sqrt{d+e x^2}} \, dx}{3 d \left (-c^2 d+e\right ) \sqrt{-c^2 x^2}}\\ &=\frac{b c x^2 \sqrt{-1-c^2 x^2}}{3 d \left (c^2 d-e\right ) \sqrt{-c^2 x^2} \sqrt{d+e x^2}}+\frac{b c^3 x^2 \sqrt{d+e x^2}}{3 d \left (c^2 d-e\right ) e \sqrt{-c^2 x^2} \sqrt{-1-c^2 x^2}}+\frac{x^3 \left (a+b \text{csch}^{-1}(c x)\right )}{3 d \left (d+e x^2\right )^{3/2}}+\frac{b x \sqrt{d+e x^2} F\left (\tan ^{-1}(c x)|1-\frac{e}{c^2 d}\right )}{3 d^2 \left (c^2 d-e\right ) \sqrt{-c^2 x^2} \sqrt{-1-c^2 x^2} \sqrt{\frac{d+e x^2}{d \left (1+c^2 x^2\right )}}}-\frac{\left (b c^3 x\right ) \int \frac{\sqrt{d+e x^2}}{\left (-1-c^2 x^2\right )^{3/2}} \, dx}{3 d e \left (-c^2 d+e\right ) \sqrt{-c^2 x^2}}\\ &=\frac{b c x^2 \sqrt{-1-c^2 x^2}}{3 d \left (c^2 d-e\right ) \sqrt{-c^2 x^2} \sqrt{d+e x^2}}+\frac{b c^3 x^2 \sqrt{d+e x^2}}{3 d \left (c^2 d-e\right ) e \sqrt{-c^2 x^2} \sqrt{-1-c^2 x^2}}+\frac{x^3 \left (a+b \text{csch}^{-1}(c x)\right )}{3 d \left (d+e x^2\right )^{3/2}}-\frac{b c^2 x \sqrt{d+e x^2} E\left (\tan ^{-1}(c x)|1-\frac{e}{c^2 d}\right )}{3 d \left (c^2 d-e\right ) e \sqrt{-c^2 x^2} \sqrt{-1-c^2 x^2} \sqrt{\frac{d+e x^2}{d \left (1+c^2 x^2\right )}}}+\frac{b x \sqrt{d+e x^2} F\left (\tan ^{-1}(c x)|1-\frac{e}{c^2 d}\right )}{3 d^2 \left (c^2 d-e\right ) \sqrt{-c^2 x^2} \sqrt{-1-c^2 x^2} \sqrt{\frac{d+e x^2}{d \left (1+c^2 x^2\right )}}}\\ \end{align*}

Mathematica [A]  time = 0.280522, size = 189, normalized size = 0.53 \[ \frac{x^2 \left (a x \left (c^2 d-e\right )+b c \sqrt{\frac{1}{c^2 x^2}+1} \left (d+e x^2\right )+b x \left (c^2 d-e\right ) \text{csch}^{-1}(c x)\right )}{3 d \left (c^2 d-e\right ) \left (d+e x^2\right )^{3/2}}-\frac{b c x \sqrt{\frac{1}{c^2 x^2}+1} \sqrt{\frac{e x^2}{d}+1} E\left (\sin ^{-1}\left (\sqrt{-\frac{e}{d}} x\right )|\frac{c^2 d}{e}\right )}{3 d \sqrt{c^2 x^2+1} \sqrt{-\frac{e}{d}} \left (c^2 d-e\right ) \sqrt{d+e x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(x^2*(a*(c^2*d - e)*x + b*c*Sqrt[1 + 1/(c^2*x^2)]*(d + e*x^2) + b*(c^2*d - e)*x*ArcCsch[c*x]))/(3*d*(c^2*d - e
)*(d + e*x^2)^(3/2)) - (b*c*Sqrt[1 + 1/(c^2*x^2)]*x*Sqrt[1 + (e*x^2)/d]*EllipticE[ArcSin[Sqrt[-(e/d)]*x], (c^2
*d)/e])/(3*d*(c^2*d - e)*Sqrt[-(e/d)]*Sqrt[1 + c^2*x^2]*Sqrt[d + e*x^2])

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Maple [F]  time = 0.484, size = 0, normalized size = 0. \begin{align*} \int{{x}^{2} \left ( a+b{\rm arccsch} \left (cx\right ) \right ) \left ( e{x}^{2}+d \right ) ^{-{\frac{5}{2}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (b x^{2} \operatorname{arcsch}\left (c x\right ) + a x^{2}\right )} \sqrt{e x^{2} + d}}{e^{3} x^{6} + 3 \, d e^{2} x^{4} + 3 \, d^{2} e x^{2} + d^{3}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((b*x^2*arccsch(c*x) + a*x^2)*sqrt(e*x^2 + d)/(e^3*x^6 + 3*d*e^2*x^4 + 3*d^2*e*x^2 + d^3), x)

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

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b \operatorname{arcsch}\left (c x\right ) + a\right )} x^{2}}{{\left (e x^{2} + d\right )}^{\frac{5}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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