3.2.94 \(\int \frac {\sqrt {d^2-e^2 x^2}}{x (d+e x)^4} \, dx\) [194]

3.2.94.1 Optimal result

Integrand size = 27, antiderivative size = 110 \[ \int \frac {\sqrt {d^2-e^2 x^2}}{x (d+e x)^4} \, dx=\frac {8 d (d-e x)}{5 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {4 e x}{5 d \left (d^2-e^2 x^2\right )^{3/2}}+\frac {5 d-8 e x}{5 d^3 \sqrt {d^2-e^2 x^2}}-\frac {\text {arctanh}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{d^3} \]

output

8/5*d*(-e*x+d)/(-e^2*x^2+d^2)^(5/2)-4/5*e*x/d/(-e^2*x^2+d^2)^(3/2)-arctanh 
((-e^2*x^2+d^2)^(1/2)/d)/d^3+1/5*(-8*e*x+5*d)/d^3/(-e^2*x^2+d^2)^(1/2)
 

3.2.94.2 Mathematica [A] (verified)

Time = 0.48 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.80 \[ \int \frac {\sqrt {d^2-e^2 x^2}}{x (d+e x)^4} \, dx=\frac {\frac {\sqrt {d^2-e^2 x^2} \left (13 d^2+19 d e x+8 e^2 x^2\right )}{(d+e x)^3}+10 \text {arctanh}\left (\frac {\sqrt {-e^2} x-\sqrt {d^2-e^2 x^2}}{d}\right )}{5 d^3} \]

input

Integrate[Sqrt[d^2 - e^2*x^2]/(x*(d + e*x)^4),x]
 

output

((Sqrt[d^2 - e^2*x^2]*(13*d^2 + 19*d*e*x + 8*e^2*x^2))/(d + e*x)^3 + 10*Ar 
cTanh[(Sqrt[-e^2]*x - Sqrt[d^2 - e^2*x^2])/d])/(5*d^3)
 

3.2.94.3 Rubi [A] (verified)

Time = 0.38 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.04, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.370, Rules used = {570, 532, 25, 2336, 27, 532, 27, 243, 73, 221}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {\sqrt {d^2-e^2 x^2}}{x (d+e x)^4} \, dx\)

\(\Big \downarrow \) 570

\(\displaystyle \int \frac {(d-e x)^4}{x \left (d^2-e^2 x^2\right )^{7/2}}dx\)

\(\Big \downarrow \) 532

\(\displaystyle \frac {8 d (d-e x)}{5 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {\int -\frac {5 d^4-12 e x d^3-5 e^2 x^2 d^2}{x \left (d^2-e^2 x^2\right )^{5/2}}dx}{5 d^2}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\int \frac {5 d^4-12 e x d^3-5 e^2 x^2 d^2}{x \left (d^2-e^2 x^2\right )^{5/2}}dx}{5 d^2}+\frac {8 d (d-e x)}{5 \left (d^2-e^2 x^2\right )^{5/2}}\)

\(\Big \downarrow \) 2336

\(\displaystyle \frac {-\frac {\int -\frac {3 d^3 (5 d-8 e x)}{x \left (d^2-e^2 x^2\right )^{3/2}}dx}{3 d^2}-\frac {4 d e x}{\left (d^2-e^2 x^2\right )^{3/2}}}{5 d^2}+\frac {8 d (d-e x)}{5 \left (d^2-e^2 x^2\right )^{5/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {d \int \frac {5 d-8 e x}{x \left (d^2-e^2 x^2\right )^{3/2}}dx-\frac {4 d e x}{\left (d^2-e^2 x^2\right )^{3/2}}}{5 d^2}+\frac {8 d (d-e x)}{5 \left (d^2-e^2 x^2\right )^{5/2}}\)

\(\Big \downarrow \) 532

\(\displaystyle \frac {d \left (\frac {5 d-8 e x}{d^2 \sqrt {d^2-e^2 x^2}}-\frac {\int -\frac {5 d}{x \sqrt {d^2-e^2 x^2}}dx}{d^2}\right )-\frac {4 d e x}{\left (d^2-e^2 x^2\right )^{3/2}}}{5 d^2}+\frac {8 d (d-e x)}{5 \left (d^2-e^2 x^2\right )^{5/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {d \left (\frac {5 \int \frac {1}{x \sqrt {d^2-e^2 x^2}}dx}{d}+\frac {5 d-8 e x}{d^2 \sqrt {d^2-e^2 x^2}}\right )-\frac {4 d e x}{\left (d^2-e^2 x^2\right )^{3/2}}}{5 d^2}+\frac {8 d (d-e x)}{5 \left (d^2-e^2 x^2\right )^{5/2}}\)

\(\Big \downarrow \) 243

\(\displaystyle \frac {d \left (\frac {5 \int \frac {1}{x^2 \sqrt {d^2-e^2 x^2}}dx^2}{2 d}+\frac {5 d-8 e x}{d^2 \sqrt {d^2-e^2 x^2}}\right )-\frac {4 d e x}{\left (d^2-e^2 x^2\right )^{3/2}}}{5 d^2}+\frac {8 d (d-e x)}{5 \left (d^2-e^2 x^2\right )^{5/2}}\)

\(\Big \downarrow \) 73

\(\displaystyle \frac {d \left (\frac {5 d-8 e x}{d^2 \sqrt {d^2-e^2 x^2}}-\frac {5 \int \frac {1}{\frac {d^2}{e^2}-\frac {x^4}{e^2}}d\sqrt {d^2-e^2 x^2}}{d e^2}\right )-\frac {4 d e x}{\left (d^2-e^2 x^2\right )^{3/2}}}{5 d^2}+\frac {8 d (d-e x)}{5 \left (d^2-e^2 x^2\right )^{5/2}}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {d \left (\frac {5 d-8 e x}{d^2 \sqrt {d^2-e^2 x^2}}-\frac {5 \text {arctanh}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{d^2}\right )-\frac {4 d e x}{\left (d^2-e^2 x^2\right )^{3/2}}}{5 d^2}+\frac {8 d (d-e x)}{5 \left (d^2-e^2 x^2\right )^{5/2}}\)

input

Int[Sqrt[d^2 - e^2*x^2]/(x*(d + e*x)^4),x]
 

output

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

3.2.94.3.1 Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 73

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ 
{p = Denominator[m]}, Simp[p/b   Subst[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] && Lt 
Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL 
inearQ[a, b, c, d, m, n, x]
 

rule 221

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 243

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[1/2   Subst[In 
t[x^((m - 1)/2)*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, m, p}, x] && I 
ntegerQ[(m - 1)/2]
 

rule 532

Int[(x_)^(m_)*((c_) + (d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbo 
l] :> With[{Qx = PolynomialQuotient[x^m*(c + d*x)^n, a + b*x^2, x], e = Coe 
ff[PolynomialRemainder[x^m*(c + d*x)^n, a + b*x^2, x], x, 0], f = Coeff[Pol 
ynomialRemainder[x^m*(c + d*x)^n, a + b*x^2, x], x, 1]}, Simp[(a*f - b*e*x) 
*((a + b*x^2)^(p + 1)/(2*a*b*(p + 1))), x] + Simp[1/(2*a*(p + 1))   Int[x^m 
*(a + b*x^2)^(p + 1)*ExpandToSum[2*a*(p + 1)*(Qx/x^m) + e*((2*p + 3)/x^m), 
x], x], x]] /; FreeQ[{a, b, c, d}, x] && IGtQ[n, 0] && ILtQ[m, 0] && LtQ[p, 
 -1] && IntegerQ[2*p]
 

rule 570

Int[((e_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), 
x_Symbol] :> Simp[c^(2*n)/a^n   Int[(e*x)^m*((a + b*x^2)^(n + p)/(c - d*x)^ 
n), x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && EqQ[b*c^2 + a*d^2, 0] && I 
LtQ[n, -1] &&  !(IGtQ[m, 0] && ILtQ[m + n, 0] &&  !GtQ[p, 1])
 

rule 2336

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[PolynomialRema 
inder[(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] + Simp[1/(2*a*(p + 1))   Int[(c*x)^m*(a + b*x^2)^(p + 1)*Ex 
pandToSum[(2*a*(p + 1)*Q)/(c*x)^m + (f*(2*p + 3))/(c*x)^m, x], x], x]] /; F 
reeQ[{a, b, c}, x] && PolyQ[Pq, x] && LtQ[p, -1] && ILtQ[m, 0]
 

3.2.94.4 Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(414\) vs. \(2(96)=192\).

Time = 0.44 (sec) , antiderivative size = 415, normalized size of antiderivative = 3.77

method	result	size
default	\(\frac {\sqrt {-e^{2} x^{2}+d^{2}}-\frac {d^{2} \ln \left (\frac {2 d^{2}+2 \sqrt {d^{2}}\, \sqrt {-e^{2} x^{2}+d^{2}}}{x}\right )}{\sqrt {d^{2}}}}{d^{4}}-\frac {-\frac {\left (-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {3}{2}}}{5 d e \left (x +\frac {d}{e}\right )^{4}}-\frac {\left (-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {3}{2}}}{15 d^{2} \left (x +\frac {d}{e}\right )^{3}}}{e^{3} d}-\frac {-\frac {\left (-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {3}{2}}}{d e \left (x +\frac {d}{e}\right )^{2}}-\frac {e \left (\sqrt {-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )}+\frac {d e \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )}}\right )}{\sqrt {e^{2}}}\right )}{d}}{e \,d^{3}}-\frac {\sqrt {-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )}+\frac {d e \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )}}\right )}{\sqrt {e^{2}}}}{d^{4}}+\frac {\left (-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {3}{2}}}{3 e^{3} d^{3} \left (x +\frac {d}{e}\right )^{3}}\)	\(415\)

input

int((-e^2*x^2+d^2)^(1/2)/x/(e*x+d)^4,x,method=_RETURNVERBOSE)
 

output

1/d^4*((-e^2*x^2+d^2)^(1/2)-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^3/d*(-1/5/d/e/(x+d/e)^4*(-(x+d/e)^2*e^2+2*d*e*(x+d 
/e))^(3/2)-1/15/d^2/(x+d/e)^3*(-(x+d/e)^2*e^2+2*d*e*(x+d/e))^(3/2))-1/e/d^ 
3*(-1/d/e/(x+d/e)^2*(-(x+d/e)^2*e^2+2*d*e*(x+d/e))^(3/2)-e/d*((-(x+d/e)^2* 
e^2+2*d*e*(x+d/e))^(1/2)+d*e/(e^2)^(1/2)*arctan((e^2)^(1/2)*x/(-(x+d/e)^2* 
e^2+2*d*e*(x+d/e))^(1/2))))-1/d^4*((-(x+d/e)^2*e^2+2*d*e*(x+d/e))^(1/2)+d* 
e/(e^2)^(1/2)*arctan((e^2)^(1/2)*x/(-(x+d/e)^2*e^2+2*d*e*(x+d/e))^(1/2)))+ 
1/3/e^3/d^3/(x+d/e)^3*(-(x+d/e)^2*e^2+2*d*e*(x+d/e))^(3/2)
 

3.2.94.5 Fricas [A] (verification not implemented)

Time = 0.27 (sec) , antiderivative size = 153, normalized size of antiderivative = 1.39 \[ \int \frac {\sqrt {d^2-e^2 x^2}}{x (d+e x)^4} \, dx=\frac {13 \, e^{3} x^{3} + 39 \, d e^{2} x^{2} + 39 \, d^{2} e x + 13 \, d^{3} + 5 \, {\left (e^{3} x^{3} + 3 \, d e^{2} x^{2} + 3 \, d^{2} e x + d^{3}\right )} \log \left (-\frac {d - \sqrt {-e^{2} x^{2} + d^{2}}}{x}\right ) + {\left (8 \, e^{2} x^{2} + 19 \, d e x + 13 \, d^{2}\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{5 \, {\left (d^{3} e^{3} x^{3} + 3 \, d^{4} e^{2} x^{2} + 3 \, d^{5} e x + d^{6}\right )}} \]

input

integrate((-e^2*x^2+d^2)^(1/2)/x/(e*x+d)^4,x, algorithm="fricas")
 

output

1/5*(13*e^3*x^3 + 39*d*e^2*x^2 + 39*d^2*e*x + 13*d^3 + 5*(e^3*x^3 + 3*d*e^ 
2*x^2 + 3*d^2*e*x + d^3)*log(-(d - sqrt(-e^2*x^2 + d^2))/x) + (8*e^2*x^2 + 
 19*d*e*x + 13*d^2)*sqrt(-e^2*x^2 + d^2))/(d^3*e^3*x^3 + 3*d^4*e^2*x^2 + 3 
*d^5*e*x + d^6)
 

3.2.94.6 Sympy [F]

\[ \int \frac {\sqrt {d^2-e^2 x^2}}{x (d+e x)^4} \, dx=\int \frac {\sqrt {- \left (- d + e x\right ) \left (d + e x\right )}}{x \left (d + e x\right )^{4}}\, dx \]

input

integrate((-e**2*x**2+d**2)**(1/2)/x/(e*x+d)**4,x)
 

output

Integral(sqrt(-(-d + e*x)*(d + e*x))/(x*(d + e*x)**4), x)
 

3.2.94.7 Maxima [F]

\[ \int \frac {\sqrt {d^2-e^2 x^2}}{x (d+e x)^4} \, dx=\int { \frac {\sqrt {-e^{2} x^{2} + d^{2}}}{{\left (e x + d\right )}^{4} x} \,d x } \]

input

integrate((-e^2*x^2+d^2)^(1/2)/x/(e*x+d)^4,x, algorithm="maxima")
 

output

integrate(sqrt(-e^2*x^2 + d^2)/((e*x + d)^4*x), x)
 

3.2.94.8 Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 212 vs. \(2 (97) = 194\).

Time = 0.30 (sec) , antiderivative size = 212, normalized size of antiderivative = 1.93 \[ \int \frac {\sqrt {d^2-e^2 x^2}}{x (d+e x)^4} \, dx=-\frac {e \log \left (\frac {{\left | -2 \, d e - 2 \, \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |} \right |}}{2 \, e^{2} {\left | x \right |}}\right )}{d^{3} {\left | e \right |}} - \frac {2 \, {\left (13 \, e + \frac {45 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}}{e x} + \frac {75 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{2}}{e^{3} x^{2}} + \frac {55 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{3}}{e^{5} x^{3}} + \frac {20 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{4}}{e^{7} x^{4}}\right )}}{5 \, d^{3} {\left (\frac {d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}}{e^{2} x} + 1\right )}^{5} {\left | e \right |}} \]

input

integrate((-e^2*x^2+d^2)^(1/2)/x/(e*x+d)^4,x, algorithm="giac")
 

output

-e*log(1/2*abs(-2*d*e - 2*sqrt(-e^2*x^2 + d^2)*abs(e))/(e^2*abs(x)))/(d^3* 
abs(e)) - 2/5*(13*e + 45*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))/(e*x) + 75*(d 
*e + sqrt(-e^2*x^2 + d^2)*abs(e))^2/(e^3*x^2) + 55*(d*e + sqrt(-e^2*x^2 + 
d^2)*abs(e))^3/(e^5*x^3) + 20*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^4/(e^7*x 
^4))/(d^3*((d*e + sqrt(-e^2*x^2 + d^2)*abs(e))/(e^2*x) + 1)^5*abs(e))
 

3.2.94.9 Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt {d^2-e^2 x^2}}{x (d+e x)^4} \, dx=\int \frac {\sqrt {d^2-e^2\,x^2}}{x\,{\left (d+e\,x\right )}^4} \,d x \]

input

int((d^2 - e^2*x^2)^(1/2)/(x*(d + e*x)^4),x)
 

output

int((d^2 - e^2*x^2)^(1/2)/(x*(d + e*x)^4), x)