3.134 \(\int \frac{1}{x^2 (d+e x) (d^2-e^2 x^2)^{3/2}} \, dx\)

Optimal. Leaf size=120 \[ \frac{4 d-3 e x}{3 d^4 x \sqrt{d^2-e^2 x^2}}-\frac{8 \sqrt{d^2-e^2 x^2}}{3 d^5 x}+\frac{1}{3 d^2 x (d+e x) \sqrt{d^2-e^2 x^2}}+\frac{e \tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{d}\right )}{d^5} \]

[Out]

(4*d - 3*e*x)/(3*d^4*x*Sqrt[d^2 - e^2*x^2]) + 1/(3*d^2*x*(d + e*x)*Sqrt[d^2 - e^2*x^2]) - (8*Sqrt[d^2 - e^2*x^
2])/(3*d^5*x) + (e*ArcTanh[Sqrt[d^2 - e^2*x^2]/d])/d^5

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Rubi [A]  time = 0.102529, antiderivative size = 120, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {857, 823, 807, 266, 63, 208} \[ \frac{4 d-3 e x}{3 d^4 x \sqrt{d^2-e^2 x^2}}-\frac{8 \sqrt{d^2-e^2 x^2}}{3 d^5 x}+\frac{1}{3 d^2 x (d+e x) \sqrt{d^2-e^2 x^2}}+\frac{e \tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{d}\right )}{d^5} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^2*(d + e*x)*(d^2 - e^2*x^2)^(3/2)),x]

[Out]

(4*d - 3*e*x)/(3*d^4*x*Sqrt[d^2 - e^2*x^2]) + 1/(3*d^2*x*(d + e*x)*Sqrt[d^2 - e^2*x^2]) - (8*Sqrt[d^2 - e^2*x^
2])/(3*d^5*x) + (e*ArcTanh[Sqrt[d^2 - e^2*x^2]/d])/d^5

Rule 857

Int[(((f_.) + (g_.)*(x_))^(n_)*((a_) + (c_.)*(x_)^2)^(p_))/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[(d*(f + g*x)
^(n + 1)*(a + c*x^2)^(p + 1))/(2*a*p*(e*f - d*g)*(d + e*x)), x] + Dist[1/(p*(2*c*d)*(e*f - d*g)), Int[(f + g*x
)^n*(a + c*x^2)^p*(c*e*f*(2*p + 1) - c*d*g*(n + 2*p + 1) + c*e*g*(n + 2*p + 2)*x), x], x] /; FreeQ[{a, c, d, e
, f, g}, x] && NeQ[e*f - d*g, 0] && EqQ[c*d^2 + a*e^2, 0] &&  !IntegerQ[p] && ILtQ[n, 0] && ILtQ[n + 2*p, 0] &
&  !IGtQ[n, 0]

Rule 823

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> -Simp[((d + e*x)^(
m + 1)*(f*a*c*e - a*g*c*d + c*(c*d*f + a*e*g)*x)*(a + c*x^2)^(p + 1))/(2*a*c*(p + 1)*(c*d^2 + a*e^2)), x] + Di
st[1/(2*a*c*(p + 1)*(c*d^2 + a*e^2)), Int[(d + e*x)^m*(a + c*x^2)^(p + 1)*Simp[f*(c^2*d^2*(2*p + 3) + a*c*e^2*
(m + 2*p + 3)) - a*c*d*e*g*m + c*e*(c*d*f + a*e*g)*(m + 2*p + 4)*x, x], x], x] /; FreeQ[{a, c, d, e, f, g}, x]
&& NeQ[c*d^2 + a*e^2, 0] && LtQ[p, -1] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])

Rule 807

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> -Simp[((e*f - d*g
)*(d + e*x)^(m + 1)*(a + c*x^2)^(p + 1))/(2*(p + 1)*(c*d^2 + a*e^2)), x] + Dist[(c*d*f + a*e*g)/(c*d^2 + a*e^2
), Int[(d + e*x)^(m + 1)*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g, m, p}, x] && NeQ[c*d^2 + a*e^2, 0]
&& EqQ[Simplify[m + 2*p + 3], 0]

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 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 \frac{1}{x^2 (d+e x) \left (d^2-e^2 x^2\right )^{3/2}} \, dx &=\frac{1}{3 d^2 x (d+e x) \sqrt{d^2-e^2 x^2}}-\frac{\int \frac{-4 d e^2+3 e^3 x}{x^2 \left (d^2-e^2 x^2\right )^{3/2}} \, dx}{3 d^2 e^2}\\ &=\frac{4 d-3 e x}{3 d^4 x \sqrt{d^2-e^2 x^2}}+\frac{1}{3 d^2 x (d+e x) \sqrt{d^2-e^2 x^2}}-\frac{\int \frac{-8 d^3 e^4+3 d^2 e^5 x}{x^2 \sqrt{d^2-e^2 x^2}} \, dx}{3 d^6 e^4}\\ &=\frac{4 d-3 e x}{3 d^4 x \sqrt{d^2-e^2 x^2}}+\frac{1}{3 d^2 x (d+e x) \sqrt{d^2-e^2 x^2}}-\frac{8 \sqrt{d^2-e^2 x^2}}{3 d^5 x}-\frac{e \int \frac{1}{x \sqrt{d^2-e^2 x^2}} \, dx}{d^4}\\ &=\frac{4 d-3 e x}{3 d^4 x \sqrt{d^2-e^2 x^2}}+\frac{1}{3 d^2 x (d+e x) \sqrt{d^2-e^2 x^2}}-\frac{8 \sqrt{d^2-e^2 x^2}}{3 d^5 x}-\frac{e \operatorname{Subst}\left (\int \frac{1}{x \sqrt{d^2-e^2 x}} \, dx,x,x^2\right )}{2 d^4}\\ &=\frac{4 d-3 e x}{3 d^4 x \sqrt{d^2-e^2 x^2}}+\frac{1}{3 d^2 x (d+e x) \sqrt{d^2-e^2 x^2}}-\frac{8 \sqrt{d^2-e^2 x^2}}{3 d^5 x}+\frac{\operatorname{Subst}\left (\int \frac{1}{\frac{d^2}{e^2}-\frac{x^2}{e^2}} \, dx,x,\sqrt{d^2-e^2 x^2}\right )}{d^4 e}\\ &=\frac{4 d-3 e x}{3 d^4 x \sqrt{d^2-e^2 x^2}}+\frac{1}{3 d^2 x (d+e x) \sqrt{d^2-e^2 x^2}}-\frac{8 \sqrt{d^2-e^2 x^2}}{3 d^5 x}+\frac{e \tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{d}\right )}{d^5}\\ \end{align*}

Mathematica [A]  time = 0.125512, size = 101, normalized size = 0.84 \[ \frac{\frac{\sqrt{d^2-e^2 x^2} \left (7 d^2 e x+3 d^3-5 d e^2 x^2-8 e^3 x^3\right )}{x (e x-d) (d+e x)^2}+3 e \log \left (\sqrt{d^2-e^2 x^2}+d\right )-3 e \log (x)}{3 d^5} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^2*(d + e*x)*(d^2 - e^2*x^2)^(3/2)),x]

[Out]

((Sqrt[d^2 - e^2*x^2]*(3*d^3 + 7*d^2*e*x - 5*d*e^2*x^2 - 8*e^3*x^3))/(x*(-d + e*x)*(d + e*x)^2) - 3*e*Log[x] +
3*e*Log[d + Sqrt[d^2 - e^2*x^2]])/(3*d^5)

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Maple [A]  time = 0.062, size = 188, normalized size = 1.6 \begin{align*} -{\frac{e}{{d}^{4}}{\frac{1}{\sqrt{-{x}^{2}{e}^{2}+{d}^{2}}}}}+{\frac{e}{{d}^{4}}\ln \left ({\frac{1}{x} \left ( 2\,{d}^{2}+2\,\sqrt{{d}^{2}}\sqrt{-{x}^{2}{e}^{2}+{d}^{2}} \right ) } \right ){\frac{1}{\sqrt{{d}^{2}}}}}-{\frac{1}{3\,{d}^{3}} \left ({\frac{d}{e}}+x \right ) ^{-1}{\frac{1}{\sqrt{- \left ({\frac{d}{e}}+x \right ) ^{2}{e}^{2}+2\,de \left ({\frac{d}{e}}+x \right ) }}}}+{\frac{2\,{e}^{2}x}{3\,{d}^{5}}{\frac{1}{\sqrt{- \left ({\frac{d}{e}}+x \right ) ^{2}{e}^{2}+2\,de \left ({\frac{d}{e}}+x \right ) }}}}-{\frac{1}{{d}^{3}x}{\frac{1}{\sqrt{-{x}^{2}{e}^{2}+{d}^{2}}}}}+2\,{\frac{{e}^{2}x}{{d}^{5}\sqrt{-{x}^{2}{e}^{2}+{d}^{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^2/(e*x+d)/(-e^2*x^2+d^2)^(3/2),x)

[Out]

-e/d^4/(-e^2*x^2+d^2)^(1/2)+e/d^4/(d^2)^(1/2)*ln((2*d^2+2*(d^2)^(1/2)*(-e^2*x^2+d^2)^(1/2))/x)-1/3/d^3/(d/e+x)
/(-(d/e+x)^2*e^2+2*d*e*(d/e+x))^(1/2)+2/3*e^2/d^5/(-(d/e+x)^2*e^2+2*d*e*(d/e+x))^(1/2)*x-1/d^3/x/(-e^2*x^2+d^2
)^(1/2)+2/d^5*e^2*x/(-e^2*x^2+d^2)^(1/2)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^2/(e*x+d)/(-e^2*x^2+d^2)^(3/2),x, algorithm="maxima")

[Out]

integrate(1/((-e^2*x^2 + d^2)^(3/2)*(e*x + d)*x^2), x)

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Fricas [A]  time = 1.60876, size = 354, normalized size = 2.95 \begin{align*} -\frac{4 \, e^{4} x^{4} + 4 \, d e^{3} x^{3} - 4 \, d^{2} e^{2} x^{2} - 4 \, d^{3} e x + 3 \,{\left (e^{4} x^{4} + d e^{3} x^{3} - d^{2} e^{2} x^{2} - d^{3} e x\right )} \log \left (-\frac{d - \sqrt{-e^{2} x^{2} + d^{2}}}{x}\right ) +{\left (8 \, e^{3} x^{3} + 5 \, d e^{2} x^{2} - 7 \, d^{2} e x - 3 \, d^{3}\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{3 \,{\left (d^{5} e^{3} x^{4} + d^{6} e^{2} x^{3} - d^{7} e x^{2} - d^{8} x\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^2/(e*x+d)/(-e^2*x^2+d^2)^(3/2),x, algorithm="fricas")

[Out]

-1/3*(4*e^4*x^4 + 4*d*e^3*x^3 - 4*d^2*e^2*x^2 - 4*d^3*e*x + 3*(e^4*x^4 + d*e^3*x^3 - d^2*e^2*x^2 - d^3*e*x)*lo
g(-(d - sqrt(-e^2*x^2 + d^2))/x) + (8*e^3*x^3 + 5*d*e^2*x^2 - 7*d^2*e*x - 3*d^3)*sqrt(-e^2*x^2 + d^2))/(d^5*e^
3*x^4 + d^6*e^2*x^3 - d^7*e*x^2 - d^8*x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{x^{2} \left (- \left (- d + e x\right ) \left (d + e x\right )\right )^{\frac{3}{2}} \left (d + e x\right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**2/(e*x+d)/(-e**2*x**2+d**2)**(3/2),x)

[Out]

Integral(1/(x**2*(-(-d + e*x)*(d + e*x))**(3/2)*(d + e*x)), x)

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Giac [A]  time = 1.22125, size = 1, normalized size = 0.01 \begin{align*} +\infty \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^2/(e*x+d)/(-e^2*x^2+d^2)^(3/2),x, algorithm="giac")

[Out]

+Infinity