3.101 \(\int \frac{\sqrt{d^2-e^2 x^2}}{x^5 (d+e x)} \, dx\)

Optimal. Leaf size=143 \[ \frac{2 e^3 \sqrt{d^2-e^2 x^2}}{3 d^4 x}-\frac{3 e^2 \sqrt{d^2-e^2 x^2}}{8 d^3 x^2}+\frac{e \sqrt{d^2-e^2 x^2}}{3 d^2 x^3}-\frac{\sqrt{d^2-e^2 x^2}}{4 d x^4}-\frac{3 e^4 \tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{d}\right )}{8 d^4} \]

[Out]

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

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Rubi [A]  time = 0.135082, antiderivative size = 143, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {850, 835, 807, 266, 63, 208} \[ \frac{2 e^3 \sqrt{d^2-e^2 x^2}}{3 d^4 x}-\frac{3 e^2 \sqrt{d^2-e^2 x^2}}{8 d^3 x^2}+\frac{e \sqrt{d^2-e^2 x^2}}{3 d^2 x^3}-\frac{\sqrt{d^2-e^2 x^2}}{4 d x^4}-\frac{3 e^4 \tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{d}\right )}{8 d^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 850

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

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{\sqrt{d^2-e^2 x^2}}{x^5 (d+e x)} \, dx &=\int \frac{d-e x}{x^5 \sqrt{d^2-e^2 x^2}} \, dx\\ &=-\frac{\sqrt{d^2-e^2 x^2}}{4 d x^4}-\frac{\int \frac{4 d^2 e-3 d e^2 x}{x^4 \sqrt{d^2-e^2 x^2}} \, dx}{4 d^2}\\ &=-\frac{\sqrt{d^2-e^2 x^2}}{4 d x^4}+\frac{e \sqrt{d^2-e^2 x^2}}{3 d^2 x^3}+\frac{\int \frac{9 d^3 e^2-8 d^2 e^3 x}{x^3 \sqrt{d^2-e^2 x^2}} \, dx}{12 d^4}\\ &=-\frac{\sqrt{d^2-e^2 x^2}}{4 d x^4}+\frac{e \sqrt{d^2-e^2 x^2}}{3 d^2 x^3}-\frac{3 e^2 \sqrt{d^2-e^2 x^2}}{8 d^3 x^2}-\frac{\int \frac{16 d^4 e^3-9 d^3 e^4 x}{x^2 \sqrt{d^2-e^2 x^2}} \, dx}{24 d^6}\\ &=-\frac{\sqrt{d^2-e^2 x^2}}{4 d x^4}+\frac{e \sqrt{d^2-e^2 x^2}}{3 d^2 x^3}-\frac{3 e^2 \sqrt{d^2-e^2 x^2}}{8 d^3 x^2}+\frac{2 e^3 \sqrt{d^2-e^2 x^2}}{3 d^4 x}+\frac{\left (3 e^4\right ) \int \frac{1}{x \sqrt{d^2-e^2 x^2}} \, dx}{8 d^3}\\ &=-\frac{\sqrt{d^2-e^2 x^2}}{4 d x^4}+\frac{e \sqrt{d^2-e^2 x^2}}{3 d^2 x^3}-\frac{3 e^2 \sqrt{d^2-e^2 x^2}}{8 d^3 x^2}+\frac{2 e^3 \sqrt{d^2-e^2 x^2}}{3 d^4 x}+\frac{\left (3 e^4\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{d^2-e^2 x}} \, dx,x,x^2\right )}{16 d^3}\\ &=-\frac{\sqrt{d^2-e^2 x^2}}{4 d x^4}+\frac{e \sqrt{d^2-e^2 x^2}}{3 d^2 x^3}-\frac{3 e^2 \sqrt{d^2-e^2 x^2}}{8 d^3 x^2}+\frac{2 e^3 \sqrt{d^2-e^2 x^2}}{3 d^4 x}-\frac{\left (3 e^2\right ) \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 )}{8 d^3}\\ &=-\frac{\sqrt{d^2-e^2 x^2}}{4 d x^4}+\frac{e \sqrt{d^2-e^2 x^2}}{3 d^2 x^3}-\frac{3 e^2 \sqrt{d^2-e^2 x^2}}{8 d^3 x^2}+\frac{2 e^3 \sqrt{d^2-e^2 x^2}}{3 d^4 x}-\frac{3 e^4 \tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{d}\right )}{8 d^4}\\ \end{align*}

Mathematica [A]  time = 0.136423, size = 95, normalized size = 0.66 \[ \frac{\sqrt{d^2-e^2 x^2} \left (8 d^2 e x-6 d^3-9 d e^2 x^2+16 e^3 x^3\right )-9 e^4 x^4 \log \left (\sqrt{d^2-e^2 x^2}+d\right )+9 e^4 x^4 \log (x)}{24 d^4 x^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[d^2 - e^2*x^2]*(-6*d^3 + 8*d^2*e*x - 9*d*e^2*x^2 + 16*e^3*x^3) + 9*e^4*x^4*Log[x] - 9*e^4*x^4*Log[d + Sq
rt[d^2 - e^2*x^2]])/(24*d^4*x^4)

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Maple [B]  time = 0.077, size = 304, normalized size = 2.1 \begin{align*}{\frac{3\,{e}^{4}}{8\,{d}^{5}}\sqrt{-{x}^{2}{e}^{2}+{d}^{2}}}-{\frac{3\,{e}^{4}}{8\,{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}{3\,{d}^{4}{x}^{3}} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{{\frac{3}{2}}}}-{\frac{{e}^{4}}{{d}^{5}}\sqrt{- \left ({\frac{d}{e}}+x \right ) ^{2}{e}^{2}+2\,de \left ({\frac{d}{e}}+x \right ) }}-{\frac{{e}^{5}}{{d}^{4}}\arctan \left ({x\sqrt{{e}^{2}}{\frac{1}{\sqrt{- \left ({\frac{d}{e}}+x \right ) ^{2}{e}^{2}+2\,de \left ({\frac{d}{e}}+x \right ) }}}} \right ){\frac{1}{\sqrt{{e}^{2}}}}}-{\frac{1}{4\,{d}^{3}{x}^{4}} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{{\frac{3}{2}}}}-{\frac{5\,{e}^{2}}{8\,{d}^{5}{x}^{2}} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{{\frac{3}{2}}}}+{\frac{{e}^{3}}{{d}^{6}x} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{{\frac{3}{2}}}}+{\frac{{e}^{5}x}{{d}^{6}}\sqrt{-{x}^{2}{e}^{2}+{d}^{2}}}+{\frac{{e}^{5}}{{d}^{4}}\arctan \left ({x\sqrt{{e}^{2}}{\frac{1}{\sqrt{-{x}^{2}{e}^{2}+{d}^{2}}}}} \right ){\frac{1}{\sqrt{{e}^{2}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/8*e^4/d^5*(-e^2*x^2+d^2)^(1/2)-3/8*e^4/d^3/(d^2)^(1/2)*ln((2*d^2+2*(d^2)^(1/2)*(-e^2*x^2+d^2)^(1/2))/x)+1/3*
e/d^4/x^3*(-e^2*x^2+d^2)^(3/2)-e^4/d^5*(-(d/e+x)^2*e^2+2*d*e*(d/e+x))^(1/2)-e^5/d^4/(e^2)^(1/2)*arctan((e^2)^(
1/2)*x/(-(d/e+x)^2*e^2+2*d*e*(d/e+x))^(1/2))-1/4/d^3/x^4*(-e^2*x^2+d^2)^(3/2)-5/8/d^5*e^2/x^2*(-e^2*x^2+d^2)^(
3/2)+e^3/d^6/x*(-e^2*x^2+d^2)^(3/2)+e^5/d^6*x*(-e^2*x^2+d^2)^(1/2)+e^5/d^4/(e^2)^(1/2)*arctan((e^2)^(1/2)*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{\sqrt{-e^{2} x^{2} + d^{2}}}{{\left (e x + d\right )} x^{5}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.75541, size = 180, normalized size = 1.26 \begin{align*} \frac{9 \, e^{4} x^{4} \log \left (-\frac{d - \sqrt{-e^{2} x^{2} + d^{2}}}{x}\right ) +{\left (16 \, e^{3} x^{3} - 9 \, d e^{2} x^{2} + 8 \, d^{2} e x - 6 \, d^{3}\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{24 \, d^{4} x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/24*(9*e^4*x^4*log(-(d - sqrt(-e^2*x^2 + d^2))/x) + (16*e^3*x^3 - 9*d*e^2*x^2 + 8*d^2*e*x - 6*d^3)*sqrt(-e^2*
x^2 + d^2))/(d^4*x^4)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

Exception raised: NotImplementedError