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

Optimal. Leaf size=113 \[ \frac{2 e \sqrt{d^2-e^2 x^2}}{d^4 x}+\frac{\sqrt{d^2-e^2 x^2}}{d^2 x^2 (d+e x)}-\frac{3 \sqrt{d^2-e^2 x^2}}{2 d^3 x^2}-\frac{3 e^2 \tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{d}\right )}{2 d^4} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0907612, antiderivative size = 113, 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, 835, 807, 266, 63, 208} \[ \frac{2 e \sqrt{d^2-e^2 x^2}}{d^4 x}+\frac{\sqrt{d^2-e^2 x^2}}{d^2 x^2 (d+e x)}-\frac{3 \sqrt{d^2-e^2 x^2}}{2 d^3 x^2}-\frac{3 e^2 \tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{d}\right )}{2 d^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 835

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))/((m + 1)*(c*d^2 + a*e^2)), x] + Dist[1/((m + 1)*(c*d^2 + a*e^2)), Int[
(d + e*x)^(m + 1)*(a + c*x^2)^p*Simp[(c*d*f + a*e*g)*(m + 1) - c*(e*f - d*g)*(m + 2*p + 3)*x, x], x], x] /; Fr
eeQ[{a, c, d, e, f, g, p}, x] && NeQ[c*d^2 + a*e^2, 0] && LtQ[m, -1] && (IntegerQ[m] || IntegerQ[p] || Integer
sQ[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^3 (d+e x) \sqrt{d^2-e^2 x^2}} \, dx &=\frac{\sqrt{d^2-e^2 x^2}}{d^2 x^2 (d+e x)}-\frac{\int \frac{-3 d e^2+2 e^3 x}{x^3 \sqrt{d^2-e^2 x^2}} \, dx}{d^2 e^2}\\ &=-\frac{3 \sqrt{d^2-e^2 x^2}}{2 d^3 x^2}+\frac{\sqrt{d^2-e^2 x^2}}{d^2 x^2 (d+e x)}+\frac{\int \frac{-4 d^2 e^3+3 d e^4 x}{x^2 \sqrt{d^2-e^2 x^2}} \, dx}{2 d^4 e^2}\\ &=-\frac{3 \sqrt{d^2-e^2 x^2}}{2 d^3 x^2}+\frac{2 e \sqrt{d^2-e^2 x^2}}{d^4 x}+\frac{\sqrt{d^2-e^2 x^2}}{d^2 x^2 (d+e x)}+\frac{\left (3 e^2\right ) \int \frac{1}{x \sqrt{d^2-e^2 x^2}} \, dx}{2 d^3}\\ &=-\frac{3 \sqrt{d^2-e^2 x^2}}{2 d^3 x^2}+\frac{2 e \sqrt{d^2-e^2 x^2}}{d^4 x}+\frac{\sqrt{d^2-e^2 x^2}}{d^2 x^2 (d+e x)}+\frac{\left (3 e^2\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{d^2-e^2 x}} \, dx,x,x^2\right )}{4 d^3}\\ &=-\frac{3 \sqrt{d^2-e^2 x^2}}{2 d^3 x^2}+\frac{2 e \sqrt{d^2-e^2 x^2}}{d^4 x}+\frac{\sqrt{d^2-e^2 x^2}}{d^2 x^2 (d+e x)}-\frac{3 \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 )}{2 d^3}\\ &=-\frac{3 \sqrt{d^2-e^2 x^2}}{2 d^3 x^2}+\frac{2 e \sqrt{d^2-e^2 x^2}}{d^4 x}+\frac{\sqrt{d^2-e^2 x^2}}{d^2 x^2 (d+e x)}-\frac{3 e^2 \tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{d}\right )}{2 d^4}\\ \end{align*}

Mathematica [A]  time = 0.338108, size = 127, normalized size = 1.12 \[ -\frac{e^2 \tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{d}\right )}{d^4}-\frac{d e^2 x^2 \sqrt{1-\frac{e^2 x^2}{d^2}} \tanh ^{-1}\left (\sqrt{1-\frac{e^2 x^2}{d^2}}\right )-2 d^2 e x+d^3-3 d e^2 x^2+4 e^3 x^3}{2 d^4 x^2 \sqrt{d^2-e^2 x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [A]  time = 0.068, size = 133, normalized size = 1.2 \begin{align*} -{\frac{3\,{e}^{2}}{2\,{d}^{3}}\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{e}{{d}^{4}}\sqrt{- \left ({\frac{d}{e}}+x \right ) ^{2}{e}^{2}+2\,de \left ({\frac{d}{e}}+x \right ) } \left ({\frac{d}{e}}+x \right ) ^{-1}}-{\frac{1}{2\,{d}^{3}{x}^{2}}\sqrt{-{x}^{2}{e}^{2}+{d}^{2}}}+{\frac{e}{{d}^{4}x}\sqrt{-{x}^{2}{e}^{2}+{d}^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [A]  time = 1.59989, size = 220, normalized size = 1.95 \begin{align*} \frac{2 \, e^{3} x^{3} + 2 \, d e^{2} x^{2} + 3 \,{\left (e^{3} x^{3} + d e^{2} x^{2}\right )} \log \left (-\frac{d - \sqrt{-e^{2} x^{2} + d^{2}}}{x}\right ) +{\left (4 \, e^{2} x^{2} + d e x - d^{2}\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{2 \,{\left (d^{4} e x^{3} + d^{5} x^{2}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: NotImplementedError