\(\int \frac {1}{x (d+e x)^2 (d^2-e^2 x^2)^{3/2}} \, dx\) [176]

Optimal result

Integrand size = 27, antiderivative size = 118 \[ \int \frac {1}{x (d+e x)^2 \left (d^2-e^2 x^2\right )^{3/2}} \, dx=\frac {2 (d-e x)}{5 d \left (d^2-e^2 x^2\right )^{5/2}}+\frac {5 d-8 e x}{15 d^3 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {15 d-16 e x}{15 d^5 \sqrt {d^2-e^2 x^2}}-\frac {\text {arctanh}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{d^5} \]

[Out]

2/5*(-e*x+d)/d/(-e^2*x^2+d^2)^(5/2)+1/15*(-8*e*x+5*d)/d^3/(-e^2*x^2+d^2)^(3/2)-arctanh((-e^2*x^2+d^2)^(1/2)/d)
/d^5+1/15*(-16*e*x+15*d)/d^5/(-e^2*x^2+d^2)^(1/2)

Rubi [A] (verified)

Time = 0.11 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.259, Rules used = {866, 1819, 837, 12, 272, 65, 214} \[ \int \frac {1}{x (d+e x)^2 \left (d^2-e^2 x^2\right )^{3/2}} \, dx=-\frac {\text {arctanh}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{d^5}+\frac {2 (d-e x)}{5 d \left (d^2-e^2 x^2\right )^{5/2}}+\frac {15 d-16 e x}{15 d^5 \sqrt {d^2-e^2 x^2}}+\frac {5 d-8 e x}{15 d^3 \left (d^2-e^2 x^2\right )^{3/2}} \]

[In]

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

[Out]

(2*(d - e*x))/(5*d*(d^2 - e^2*x^2)^(5/2)) + (5*d - 8*e*x)/(15*d^3*(d^2 - e^2*x^2)^(3/2)) + (15*d - 16*e*x)/(15
*d^5*Sqrt[d^2 - e^2*x^2]) - ArcTanh[Sqrt[d^2 - e^2*x^2]/d]/d^5

Rule 12

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

Rule 65

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 214

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x/Rt[-a/b, 2]], x] /; FreeQ[{a, b},
x] && NegQ[a/b]

Rule 272

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 837

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] +
Dist[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 866

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

Rule 1819

Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = PolynomialQuotient[(c*x)^m*Pq,
a + b*x^2, x], f = Coeff[PolynomialRemainder[(c*x)^m*Pq, a + b*x^2, x], x, 0], g = Coeff[PolynomialRemainder[
(c*x)^m*Pq, a + b*x^2, x], x, 1]}, Simp[(a*g - b*f*x)*((a + b*x^2)^(p + 1)/(2*a*b*(p + 1))), x] + Dist[1/(2*a*
(p + 1)), Int[(c*x)^m*(a + b*x^2)^(p + 1)*ExpandToSum[(2*a*(p + 1)*Q)/(c*x)^m + (f*(2*p + 3))/(c*x)^m, x], x],
x]] /; FreeQ[{a, b, c}, x] && PolyQ[Pq, x] && LtQ[p, -1] && ILtQ[m, 0]

Rubi steps \begin{align*} \text {integral}& = \int \frac {(d-e x)^2}{x \left (d^2-e^2 x^2\right )^{7/2}} \, dx \\ & = \frac {2 (d-e x)}{5 d \left (d^2-e^2 x^2\right )^{5/2}}-\frac {\int \frac {-5 d^2+8 d e x}{x \left (d^2-e^2 x^2\right )^{5/2}} \, dx}{5 d^2} \\ & = \frac {2 (d-e x)}{5 d \left (d^2-e^2 x^2\right )^{5/2}}+\frac {5 d-8 e x}{15 d^3 \left (d^2-e^2 x^2\right )^{3/2}}-\frac {\int \frac {-15 d^4 e^2+16 d^3 e^3 x}{x \left (d^2-e^2 x^2\right )^{3/2}} \, dx}{15 d^6 e^2} \\ & = \frac {2 (d-e x)}{5 d \left (d^2-e^2 x^2\right )^{5/2}}+\frac {5 d-8 e x}{15 d^3 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {15 d-16 e x}{15 d^5 \sqrt {d^2-e^2 x^2}}-\frac {\int -\frac {15 d^6 e^4}{x \sqrt {d^2-e^2 x^2}} \, dx}{15 d^{10} e^4} \\ & = \frac {2 (d-e x)}{5 d \left (d^2-e^2 x^2\right )^{5/2}}+\frac {5 d-8 e x}{15 d^3 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {15 d-16 e x}{15 d^5 \sqrt {d^2-e^2 x^2}}+\frac {\int \frac {1}{x \sqrt {d^2-e^2 x^2}} \, dx}{d^4} \\ & = \frac {2 (d-e x)}{5 d \left (d^2-e^2 x^2\right )^{5/2}}+\frac {5 d-8 e x}{15 d^3 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {15 d-16 e x}{15 d^5 \sqrt {d^2-e^2 x^2}}+\frac {\text {Subst}\left (\int \frac {1}{x \sqrt {d^2-e^2 x}} \, dx,x,x^2\right )}{2 d^4} \\ & = \frac {2 (d-e x)}{5 d \left (d^2-e^2 x^2\right )^{5/2}}+\frac {5 d-8 e x}{15 d^3 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {15 d-16 e x}{15 d^5 \sqrt {d^2-e^2 x^2}}-\frac {\text {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^2} \\ & = \frac {2 (d-e x)}{5 d \left (d^2-e^2 x^2\right )^{5/2}}+\frac {5 d-8 e x}{15 d^3 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {15 d-16 e x}{15 d^5 \sqrt {d^2-e^2 x^2}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{d^5} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.47 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.91 \[ \int \frac {1}{x (d+e x)^2 \left (d^2-e^2 x^2\right )^{3/2}} \, dx=\frac {\frac {\sqrt {d^2-e^2 x^2} \left (26 d^3+22 d^2 e x-17 d e^2 x^2-16 e^3 x^3\right )}{(d-e x) (d+e x)^3}+30 \text {arctanh}\left (\frac {\sqrt {-e^2} x-\sqrt {d^2-e^2 x^2}}{d}\right )}{15 d^5} \]

[In]

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

[Out]

((Sqrt[d^2 - e^2*x^2]*(26*d^3 + 22*d^2*e*x - 17*d*e^2*x^2 - 16*e^3*x^3))/((d - e*x)*(d + e*x)^3) + 30*ArcTanh[
(Sqrt[-e^2]*x - Sqrt[d^2 - e^2*x^2])/d])/(15*d^5)

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(329\) vs. \(2(104)=208\).

Time = 0.41 (sec) , antiderivative size = 330, normalized size of antiderivative = 2.80

method	result	size
default	\(\frac {\frac {1}{d^{2} \sqrt {-e^{2} x^{2}+d^{2}}}-\frac {\ln \left (\frac {2 d^{2}+2 \sqrt {d^{2}}\, \sqrt {-e^{2} x^{2}+d^{2}}}{x}\right )}{d^{2} \sqrt {d^{2}}}}{d^{2}}-\frac {-\frac {1}{5 d e \left (x +\frac {d}{e}\right )^{2} \sqrt {-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )}}+\frac {3 e \left (-\frac {1}{3 d e \left (x +\frac {d}{e}\right ) \sqrt {-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )}}-\frac {-2 \left (x +\frac {d}{e}\right ) e^{2}+2 d e}{3 e \,d^{3} \sqrt {-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )}}\right )}{5 d}}{e d}-\frac {-\frac {1}{3 d e \left (x +\frac {d}{e}\right ) \sqrt {-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )}}-\frac {-2 \left (x +\frac {d}{e}\right ) e^{2}+2 d e}{3 e \,d^{3} \sqrt {-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )}}}{d^{2}}\)	\(330\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 168, normalized size of antiderivative = 1.42 \[ \int \frac {1}{x (d+e x)^2 \left (d^2-e^2 x^2\right )^{3/2}} \, dx=\frac {26 \, e^{4} x^{4} + 52 \, d e^{3} x^{3} - 52 \, d^{3} e x - 26 \, d^{4} + 15 \, {\left (e^{4} x^{4} + 2 \, d e^{3} x^{3} - 2 \, d^{3} e x - d^{4}\right )} \log \left (-\frac {d - \sqrt {-e^{2} x^{2} + d^{2}}}{x}\right ) + {\left (16 \, e^{3} x^{3} + 17 \, d e^{2} x^{2} - 22 \, d^{2} e x - 26 \, d^{3}\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{15 \, {\left (d^{5} e^{4} x^{4} + 2 \, d^{6} e^{3} x^{3} - 2 \, d^{8} e x - d^{9}\right )}} \]

[In]

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

[Out]

1/15*(26*e^4*x^4 + 52*d*e^3*x^3 - 52*d^3*e*x - 26*d^4 + 15*(e^4*x^4 + 2*d*e^3*x^3 - 2*d^3*e*x - d^4)*log(-(d -
sqrt(-e^2*x^2 + d^2))/x) + (16*e^3*x^3 + 17*d*e^2*x^2 - 22*d^2*e*x - 26*d^3)*sqrt(-e^2*x^2 + d^2))/(d^5*e^4*x
^4 + 2*d^6*e^3*x^3 - 2*d^8*e*x - d^9)

Sympy [F]

\[ \int \frac {1}{x (d+e x)^2 \left (d^2-e^2 x^2\right )^{3/2}} \, dx=\int \frac {1}{x \left (- \left (- d + e x\right ) \left (d + e x\right )\right )^{\frac {3}{2}} \left (d + e x\right )^{2}}\, dx \]

[In]

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

[Out]

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

Maxima [F]

\[ \int \frac {1}{x (d+e x)^2 \left (d^2-e^2 x^2\right )^{3/2}} \, dx=\int { \frac {1}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} {\left (e x + d\right )}^{2} x} \,d x } \]

[In]

integrate(1/x/(e*x+d)^2/(-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)^2*x), x)

Giac [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.32 (sec) , antiderivative size = 284, normalized size of antiderivative = 2.41 \[ \int \frac {1}{x (d+e x)^2 \left (d^2-e^2 x^2\right )^{3/2}} \, dx=-\frac {{\left (e^{4} {\left (\frac {120 \, \log \left (\sqrt {\frac {2 \, d}{e x + d} - 1} + 1\right )}{d^{5} e^{4} \mathrm {sgn}\left (\frac {1}{e x + d}\right ) \mathrm {sgn}\left (e\right )} - \frac {120 \, \log \left ({\left | \sqrt {\frac {2 \, d}{e x + d} - 1} - 1 \right |}\right )}{d^{5} e^{4} \mathrm {sgn}\left (\frac {1}{e x + d}\right ) \mathrm {sgn}\left (e\right )} - \frac {15}{d^{5} e^{4} \sqrt {\frac {2 \, d}{e x + d} - 1} \mathrm {sgn}\left (\frac {1}{e x + d}\right ) \mathrm {sgn}\left (e\right )} - \frac {3 \, d^{20} e^{16} {\left (\frac {2 \, d}{e x + d} - 1\right )}^{\frac {5}{2}} \mathrm {sgn}\left (\frac {1}{e x + d}\right )^{4} \mathrm {sgn}\left (e\right )^{4} + 25 \, d^{20} e^{16} {\left (\frac {2 \, d}{e x + d} - 1\right )}^{\frac {3}{2}} \mathrm {sgn}\left (\frac {1}{e x + d}\right )^{4} \mathrm {sgn}\left (e\right )^{4} + 165 \, d^{20} e^{16} \sqrt {\frac {2 \, d}{e x + d} - 1} \mathrm {sgn}\left (\frac {1}{e x + d}\right )^{4} \mathrm {sgn}\left (e\right )^{4}}{d^{25} e^{20} \mathrm {sgn}\left (\frac {1}{e x + d}\right )^{5} \mathrm {sgn}\left (e\right )^{5}}\right )} + \frac {4 \, {\left (15 \, \log \left (2\right ) - 30 \, \log \left (i + 1\right ) + 32 i\right )} \mathrm {sgn}\left (\frac {1}{e x + d}\right ) \mathrm {sgn}\left (e\right )}{d^{5}}\right )} e}{120 \, {\left | e \right |}} \]

[In]

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

[Out]

-1/120*(e^4*(120*log(sqrt(2*d/(e*x + d) - 1) + 1)/(d^5*e^4*sgn(1/(e*x + d))*sgn(e)) - 120*log(abs(sqrt(2*d/(e*
x + d) - 1) - 1))/(d^5*e^4*sgn(1/(e*x + d))*sgn(e)) - 15/(d^5*e^4*sqrt(2*d/(e*x + d) - 1)*sgn(1/(e*x + d))*sgn
(e)) - (3*d^20*e^16*(2*d/(e*x + d) - 1)^(5/2)*sgn(1/(e*x + d))^4*sgn(e)^4 + 25*d^20*e^16*(2*d/(e*x + d) - 1)^(
3/2)*sgn(1/(e*x + d))^4*sgn(e)^4 + 165*d^20*e^16*sqrt(2*d/(e*x + d) - 1)*sgn(1/(e*x + d))^4*sgn(e)^4)/(d^25*e^
20*sgn(1/(e*x + d))^5*sgn(e)^5)) + 4*(15*log(2) - 30*log(I + 1) + 32*I)*sgn(1/(e*x + d))*sgn(e)/d^5)*e/abs(e)

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{x (d+e x)^2 \left (d^2-e^2 x^2\right )^{3/2}} \, dx=\int \frac {1}{x\,{\left (d^2-e^2\,x^2\right )}^{3/2}\,{\left (d+e\,x\right )}^2} \,d x \]

[In]

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

[Out]

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