3.26 \(\int \frac{\tanh ^{-1}(\frac{\sqrt{e} x}{\sqrt{d+e x^2}})}{x^{5/2}} \, dx\)

Optimal. Leaf size=272 \[ \frac{2 e^{3/4} \left (\sqrt{d}+\sqrt{e} x\right ) \sqrt{\frac{d+e x^2}{\left (\sqrt{d}+\sqrt{e} x\right )^2}} \text{EllipticF}\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{e} \sqrt{x}}{\sqrt [4]{d}}\right ),\frac{1}{2}\right )}{3 d^{3/4} \sqrt{d+e x^2}}-\frac{4 e^{3/4} \left (\sqrt{d}+\sqrt{e} x\right ) \sqrt{\frac{d+e x^2}{\left (\sqrt{d}+\sqrt{e} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{e} \sqrt{x}}{\sqrt [4]{d}}\right )|\frac{1}{2}\right )}{3 d^{3/4} \sqrt{d+e x^2}}+\frac{4 e \sqrt{x} \sqrt{d+e x^2}}{3 d \left (\sqrt{d}+\sqrt{e} x\right )}-\frac{4 \sqrt{e} \sqrt{d+e x^2}}{3 d \sqrt{x}}-\frac{2 \tanh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d+e x^2}}\right )}{3 x^{3/2}} \]

[Out]

(-4*Sqrt[e]*Sqrt[d + e*x^2])/(3*d*Sqrt[x]) + (4*e*Sqrt[x]*Sqrt[d + e*x^2])/(3*d*(Sqrt[d] + Sqrt[e]*x)) - (2*Ar
cTanh[(Sqrt[e]*x)/Sqrt[d + e*x^2]])/(3*x^(3/2)) - (4*e^(3/4)*(Sqrt[d] + Sqrt[e]*x)*Sqrt[(d + e*x^2)/(Sqrt[d] +
Sqrt[e]*x)^2]*EllipticE[2*ArcTan[(e^(1/4)*Sqrt[x])/d^(1/4)], 1/2])/(3*d^(3/4)*Sqrt[d + e*x^2]) + (2*e^(3/4)*(
Sqrt[d] + Sqrt[e]*x)*Sqrt[(d + e*x^2)/(Sqrt[d] + Sqrt[e]*x)^2]*EllipticF[2*ArcTan[(e^(1/4)*Sqrt[x])/d^(1/4)],
1/2])/(3*d^(3/4)*Sqrt[d + e*x^2])

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Rubi [A]  time = 0.161741, antiderivative size = 272, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.24, Rules used = {6221, 325, 329, 305, 220, 1196} \[ \frac{2 e^{3/4} \left (\sqrt{d}+\sqrt{e} x\right ) \sqrt{\frac{d+e x^2}{\left (\sqrt{d}+\sqrt{e} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{e} \sqrt{x}}{\sqrt [4]{d}}\right )|\frac{1}{2}\right )}{3 d^{3/4} \sqrt{d+e x^2}}-\frac{4 e^{3/4} \left (\sqrt{d}+\sqrt{e} x\right ) \sqrt{\frac{d+e x^2}{\left (\sqrt{d}+\sqrt{e} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{e} \sqrt{x}}{\sqrt [4]{d}}\right )|\frac{1}{2}\right )}{3 d^{3/4} \sqrt{d+e x^2}}+\frac{4 e \sqrt{x} \sqrt{d+e x^2}}{3 d \left (\sqrt{d}+\sqrt{e} x\right )}-\frac{4 \sqrt{e} \sqrt{d+e x^2}}{3 d \sqrt{x}}-\frac{2 \tanh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d+e x^2}}\right )}{3 x^{3/2}} \]

Antiderivative was successfully verified.

[In]

Int[ArcTanh[(Sqrt[e]*x)/Sqrt[d + e*x^2]]/x^(5/2),x]

[Out]

(-4*Sqrt[e]*Sqrt[d + e*x^2])/(3*d*Sqrt[x]) + (4*e*Sqrt[x]*Sqrt[d + e*x^2])/(3*d*(Sqrt[d] + Sqrt[e]*x)) - (2*Ar
cTanh[(Sqrt[e]*x)/Sqrt[d + e*x^2]])/(3*x^(3/2)) - (4*e^(3/4)*(Sqrt[d] + Sqrt[e]*x)*Sqrt[(d + e*x^2)/(Sqrt[d] +
Sqrt[e]*x)^2]*EllipticE[2*ArcTan[(e^(1/4)*Sqrt[x])/d^(1/4)], 1/2])/(3*d^(3/4)*Sqrt[d + e*x^2]) + (2*e^(3/4)*(
Sqrt[d] + Sqrt[e]*x)*Sqrt[(d + e*x^2)/(Sqrt[d] + Sqrt[e]*x)^2]*EllipticF[2*ArcTan[(e^(1/4)*Sqrt[x])/d^(1/4)],
1/2])/(3*d^(3/4)*Sqrt[d + e*x^2])

Rule 6221

Int[ArcTanh[((c_.)*(x_))/Sqrt[(a_.) + (b_.)*(x_)^2]]*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[((d*x)^(m + 1)*ArcT
anh[(c*x)/Sqrt[a + b*x^2]])/(d*(m + 1)), x] - Dist[c/(d*(m + 1)), Int[(d*x)^(m + 1)/Sqrt[a + b*x^2], x], x] /;
FreeQ[{a, b, c, d, m}, x] && EqQ[b, c^2] && NeQ[m, -1]

Rule 325

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] - Dist[(b*(m + n*(p + 1) + 1))/(a*c^n*(m + 1)), Int[(c*x)^(m + n)*(a + b*x^n)^p, x], x] /; Free
Q[{a, b, c, p}, x] && IGtQ[n, 0] && LtQ[m, -1] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 329

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = Denominator[m]}, Dist[k/c, Subst[I
nt[x^(k*(m + 1) - 1)*(a + (b*x^(k*n))/c^n)^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0]
&& FractionQ[m] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 305

Int[(x_)^2/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> With[{q = Rt[b/a, 2]}, Dist[1/q, Int[1/Sqrt[a + b*x^4], x],
x] - Dist[1/q, Int[(1 - q*x^2)/Sqrt[a + b*x^4], x], x]] /; FreeQ[{a, b}, x] && PosQ[b/a]

Rule 220

Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> With[{q = Rt[b/a, 4]}, Simp[((1 + q^2*x^2)*Sqrt[(a + b*x^4)/(a*(
1 + q^2*x^2)^2)]*EllipticF[2*ArcTan[q*x], 1/2])/(2*q*Sqrt[a + b*x^4]), x]] /; FreeQ[{a, b}, x] && PosQ[b/a]

Rule 1196

Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[c/a, 4]}, -Simp[(d*x*Sqrt[a + c
*x^4])/(a*(1 + q^2*x^2)), x] + Simp[(d*(1 + q^2*x^2)*Sqrt[(a + c*x^4)/(a*(1 + q^2*x^2)^2)]*EllipticE[2*ArcTan[
q*x], 1/2])/(q*Sqrt[a + c*x^4]), x] /; EqQ[e + d*q^2, 0]] /; FreeQ[{a, c, d, e}, x] && PosQ[c/a]

Rubi steps

\begin{align*} \int \frac{\tanh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d+e x^2}}\right )}{x^{5/2}} \, dx &=-\frac{2 \tanh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d+e x^2}}\right )}{3 x^{3/2}}+\frac{1}{3} \left (2 \sqrt{e}\right ) \int \frac{1}{x^{3/2} \sqrt{d+e x^2}} \, dx\\ &=-\frac{4 \sqrt{e} \sqrt{d+e x^2}}{3 d \sqrt{x}}-\frac{2 \tanh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d+e x^2}}\right )}{3 x^{3/2}}+\frac{\left (2 e^{3/2}\right ) \int \frac{\sqrt{x}}{\sqrt{d+e x^2}} \, dx}{3 d}\\ &=-\frac{4 \sqrt{e} \sqrt{d+e x^2}}{3 d \sqrt{x}}-\frac{2 \tanh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d+e x^2}}\right )}{3 x^{3/2}}+\frac{\left (4 e^{3/2}\right ) \operatorname{Subst}\left (\int \frac{x^2}{\sqrt{d+e x^4}} \, dx,x,\sqrt{x}\right )}{3 d}\\ &=-\frac{4 \sqrt{e} \sqrt{d+e x^2}}{3 d \sqrt{x}}-\frac{2 \tanh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d+e x^2}}\right )}{3 x^{3/2}}+\frac{(4 e) \operatorname{Subst}\left (\int \frac{1}{\sqrt{d+e x^4}} \, dx,x,\sqrt{x}\right )}{3 \sqrt{d}}-\frac{(4 e) \operatorname{Subst}\left (\int \frac{1-\frac{\sqrt{e} x^2}{\sqrt{d}}}{\sqrt{d+e x^4}} \, dx,x,\sqrt{x}\right )}{3 \sqrt{d}}\\ &=-\frac{4 \sqrt{e} \sqrt{d+e x^2}}{3 d \sqrt{x}}+\frac{4 e \sqrt{x} \sqrt{d+e x^2}}{3 d \left (\sqrt{d}+\sqrt{e} x\right )}-\frac{2 \tanh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d+e x^2}}\right )}{3 x^{3/2}}-\frac{4 e^{3/4} \left (\sqrt{d}+\sqrt{e} x\right ) \sqrt{\frac{d+e x^2}{\left (\sqrt{d}+\sqrt{e} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{e} \sqrt{x}}{\sqrt [4]{d}}\right )|\frac{1}{2}\right )}{3 d^{3/4} \sqrt{d+e x^2}}+\frac{2 e^{3/4} \left (\sqrt{d}+\sqrt{e} x\right ) \sqrt{\frac{d+e x^2}{\left (\sqrt{d}+\sqrt{e} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{e} \sqrt{x}}{\sqrt [4]{d}}\right )|\frac{1}{2}\right )}{3 d^{3/4} \sqrt{d+e x^2}}\\ \end{align*}

Mathematica [C]  time = 0.104923, size = 118, normalized size = 0.43 \[ \frac{4 e^{3/2} x^{3/2} \sqrt{\frac{e x^2}{d}+1} \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{3}{4},\frac{7}{4},-\frac{e x^2}{d}\right )}{9 d \sqrt{d+e x^2}}-\frac{4 \sqrt{e} \sqrt{d+e x^2}}{3 d \sqrt{x}}-\frac{2 \tanh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d+e x^2}}\right )}{3 x^{3/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[ArcTanh[(Sqrt[e]*x)/Sqrt[d + e*x^2]]/x^(5/2),x]

[Out]

(-4*Sqrt[e]*Sqrt[d + e*x^2])/(3*d*Sqrt[x]) - (2*ArcTanh[(Sqrt[e]*x)/Sqrt[d + e*x^2]])/(3*x^(3/2)) + (4*e^(3/2)
*x^(3/2)*Sqrt[1 + (e*x^2)/d]*Hypergeometric2F1[1/2, 3/4, 7/4, -((e*x^2)/d)])/(9*d*Sqrt[d + e*x^2])

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Maple [F]  time = 0.819, size = 0, normalized size = 0. \begin{align*} \int{{\it Artanh} \left ({x\sqrt{e}{\frac{1}{\sqrt{e{x}^{2}+d}}}} \right ){x}^{-{\frac{5}{2}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(arctanh(x*e^(1/2)/(e*x^2+d)^(1/2))/x^(5/2),x)

[Out]

int(arctanh(x*e^(1/2)/(e*x^2+d)^(1/2))/x^(5/2),x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arctanh(x*e^(1/2)/(e*x^2+d)^(1/2))/x^(5/2),x, algorithm="maxima")

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\operatorname{artanh}\left (\frac{\sqrt{e} x}{\sqrt{e x^{2} + d}}\right )}{x^{\frac{5}{2}}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arctanh(x*e^(1/2)/(e*x^2+d)^(1/2))/x^(5/2),x, algorithm="fricas")

[Out]

integral(arctanh(sqrt(e)*x/sqrt(e*x^2 + d))/x^(5/2), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{atanh}{\left (\frac{\sqrt{e} x}{\sqrt{d + e x^{2}}} \right )}}{x^{\frac{5}{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(atanh(x*e**(1/2)/(e*x**2+d)**(1/2))/x**(5/2),x)

[Out]

Integral(atanh(sqrt(e)*x/sqrt(d + e*x**2))/x**(5/2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arctanh(x*e^(1/2)/(e*x^2+d)^(1/2))/x^(5/2),x, algorithm="giac")

[Out]

integrate(arctanh(sqrt(e)*x/sqrt(e*x^2 + d))/x^(5/2), x)