\(\int \frac {\sqrt {c^2-d^2 x^2}}{x^5 (c+d x)} \, dx\) [126]

Optimal result

Integrand size = 27, antiderivative size = 143 \[ \int \frac {\sqrt {c^2-d^2 x^2}}{x^5 (c+d x)} \, dx=-\frac {\sqrt {c^2-d^2 x^2}}{4 c x^4}+\frac {d \sqrt {c^2-d^2 x^2}}{3 c^2 x^3}-\frac {3 d^2 \sqrt {c^2-d^2 x^2}}{8 c^3 x^2}+\frac {2 d^3 \sqrt {c^2-d^2 x^2}}{3 c^4 x}-\frac {3 d^4 \text {arctanh}\left (\frac {\sqrt {c^2-d^2 x^2}}{c}\right )}{8 c^4} \] Output:

-1/4*(-d^2*x^2+c^2)^(1/2)/c/x^4+1/3*d*(-d^2*x^2+c^2)^(1/2)/c^2/x^3-3/8*d^2 
*(-d^2*x^2+c^2)^(1/2)/c^3/x^2+2/3*d^3*(-d^2*x^2+c^2)^(1/2)/c^4/x-3/8*d^4*a 
rctanh((-d^2*x^2+c^2)^(1/2)/c)/c^4
 

Mathematica [A] (verified)

Time = 0.19 (sec) , antiderivative size = 120, normalized size of antiderivative = 0.84 \[ \int \frac {\sqrt {c^2-d^2 x^2}}{x^5 (c+d x)} \, dx=\frac {\sqrt {c^2-d^2 x^2} \left (-6 c^3+8 c^2 d x-9 c d^2 x^2+16 d^3 x^3\right )}{24 c^4 x^4}-\frac {3 \sqrt {c^2} d^4 \log (x)}{8 c^5}+\frac {3 \sqrt {c^2} d^4 \log \left (\sqrt {c^2}-\sqrt {c^2-d^2 x^2}\right )}{8 c^5} \] Input:

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

Output:

(Sqrt[c^2 - d^2*x^2]*(-6*c^3 + 8*c^2*d*x - 9*c*d^2*x^2 + 16*d^3*x^3))/(24* 
c^4*x^4) - (3*Sqrt[c^2]*d^4*Log[x])/(8*c^5) + (3*Sqrt[c^2]*d^4*Log[Sqrt[c^ 
2] - Sqrt[c^2 - d^2*x^2]])/(8*c^5)
 

Rubi [A] (verified)

Time = 0.53 (sec) , antiderivative size = 157, normalized size of antiderivative = 1.10, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.407, Rules used = {566, 539, 27, 539, 27, 539, 27, 534, 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.

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

\(\Big \downarrow \) 566

\(\displaystyle \int \frac {c-d x}{x^5 \sqrt {c^2-d^2 x^2}}dx\)

\(\Big \downarrow \) 539

\(\displaystyle -\frac {\int \frac {c d (4 c-3 d x)}{x^4 \sqrt {c^2-d^2 x^2}}dx}{4 c^2}-\frac {\sqrt {c^2-d^2 x^2}}{4 c x^4}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {d \int \frac {4 c-3 d x}{x^4 \sqrt {c^2-d^2 x^2}}dx}{4 c}-\frac {\sqrt {c^2-d^2 x^2}}{4 c x^4}\)

\(\Big \downarrow \) 539

\(\displaystyle -\frac {d \left (-\frac {\int \frac {c d (9 c-8 d x)}{x^3 \sqrt {c^2-d^2 x^2}}dx}{3 c^2}-\frac {4 \sqrt {c^2-d^2 x^2}}{3 c x^3}\right )}{4 c}-\frac {\sqrt {c^2-d^2 x^2}}{4 c x^4}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {d \left (-\frac {d \int \frac {9 c-8 d x}{x^3 \sqrt {c^2-d^2 x^2}}dx}{3 c}-\frac {4 \sqrt {c^2-d^2 x^2}}{3 c x^3}\right )}{4 c}-\frac {\sqrt {c^2-d^2 x^2}}{4 c x^4}\)

\(\Big \downarrow \) 539

\(\displaystyle -\frac {d \left (-\frac {d \left (-\frac {\int \frac {c d (16 c-9 d x)}{x^2 \sqrt {c^2-d^2 x^2}}dx}{2 c^2}-\frac {9 \sqrt {c^2-d^2 x^2}}{2 c x^2}\right )}{3 c}-\frac {4 \sqrt {c^2-d^2 x^2}}{3 c x^3}\right )}{4 c}-\frac {\sqrt {c^2-d^2 x^2}}{4 c x^4}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {d \left (-\frac {d \left (-\frac {d \int \frac {16 c-9 d x}{x^2 \sqrt {c^2-d^2 x^2}}dx}{2 c}-\frac {9 \sqrt {c^2-d^2 x^2}}{2 c x^2}\right )}{3 c}-\frac {4 \sqrt {c^2-d^2 x^2}}{3 c x^3}\right )}{4 c}-\frac {\sqrt {c^2-d^2 x^2}}{4 c x^4}\)

\(\Big \downarrow \) 534

\(\displaystyle -\frac {d \left (-\frac {d \left (-\frac {d \left (-9 d \int \frac {1}{x \sqrt {c^2-d^2 x^2}}dx-\frac {16 \sqrt {c^2-d^2 x^2}}{c x}\right )}{2 c}-\frac {9 \sqrt {c^2-d^2 x^2}}{2 c x^2}\right )}{3 c}-\frac {4 \sqrt {c^2-d^2 x^2}}{3 c x^3}\right )}{4 c}-\frac {\sqrt {c^2-d^2 x^2}}{4 c x^4}\)

\(\Big \downarrow \) 243

\(\displaystyle -\frac {d \left (-\frac {d \left (-\frac {d \left (-\frac {9}{2} d \int \frac {1}{x^2 \sqrt {c^2-d^2 x^2}}dx^2-\frac {16 \sqrt {c^2-d^2 x^2}}{c x}\right )}{2 c}-\frac {9 \sqrt {c^2-d^2 x^2}}{2 c x^2}\right )}{3 c}-\frac {4 \sqrt {c^2-d^2 x^2}}{3 c x^3}\right )}{4 c}-\frac {\sqrt {c^2-d^2 x^2}}{4 c x^4}\)

\(\Big \downarrow \) 73

\(\displaystyle -\frac {d \left (-\frac {d \left (-\frac {d \left (\frac {9 \int \frac {1}{\frac {c^2}{d^2}-\frac {x^4}{d^2}}d\sqrt {c^2-d^2 x^2}}{d}-\frac {16 \sqrt {c^2-d^2 x^2}}{c x}\right )}{2 c}-\frac {9 \sqrt {c^2-d^2 x^2}}{2 c x^2}\right )}{3 c}-\frac {4 \sqrt {c^2-d^2 x^2}}{3 c x^3}\right )}{4 c}-\frac {\sqrt {c^2-d^2 x^2}}{4 c x^4}\)

\(\Big \downarrow \) 221

\(\displaystyle -\frac {d \left (-\frac {d \left (-\frac {d \left (\frac {9 d \text {arctanh}\left (\frac {\sqrt {c^2-d^2 x^2}}{c}\right )}{c}-\frac {16 \sqrt {c^2-d^2 x^2}}{c x}\right )}{2 c}-\frac {9 \sqrt {c^2-d^2 x^2}}{2 c x^2}\right )}{3 c}-\frac {4 \sqrt {c^2-d^2 x^2}}{3 c x^3}\right )}{4 c}-\frac {\sqrt {c^2-d^2 x^2}}{4 c x^4}\)

Input:

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

Output:

-1/4*Sqrt[c^2 - d^2*x^2]/(c*x^4) - (d*((-4*Sqrt[c^2 - d^2*x^2])/(3*c*x^3) 
- (d*((-9*Sqrt[c^2 - d^2*x^2])/(2*c*x^2) - (d*((-16*Sqrt[c^2 - d^2*x^2])/( 
c*x) + (9*d*ArcTanh[Sqrt[c^2 - d^2*x^2]/c])/c))/(2*c)))/(3*c)))/(4*c)
 

Defintions of rubi rules used

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 534

Int[(x_)^(m_)*((c_) + (d_.)*(x_))*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> 
Simp[(-c)*x^(m + 1)*((a + b*x^2)^(p + 1)/(2*a*(p + 1))), x] + Simp[d   Int[ 
x^(m + 1)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, m, p}, x] && ILtQ[m, 
0] && GtQ[p, -1] && EqQ[m + 2*p + 3, 0]
 

rule 539

Int[(x_)^(m_)*((c_) + (d_.)*(x_))*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> 
Simp[c*x^(m + 1)*((a + b*x^2)^(p + 1)/(a*(m + 1))), x] + Simp[1/(a*(m + 1)) 
   Int[x^(m + 1)*(a + b*x^2)^p*(a*d*(m + 1) - b*c*(m + 2*p + 3)*x), x], x] 
/; FreeQ[{a, b, c, d, p}, x] && ILtQ[m, -1] && GtQ[p, -1] && IntegerQ[2*p]
 

rule 566

Int[((x_)^(m_)*((a_) + (b_.)*(x_)^2)^(p_))/((c_) + (d_.)*(x_)), x_Symbol] : 
> Int[x^m*(a/c + b*(x/d))*(a + b*x^2)^(p - 1), x] /; FreeQ[{a, b, c, d}, x] 
 && EqQ[b*c^2 + a*d^2, 0] && GtQ[p, 0]
 

Maple [A] (verified)

Time = 0.23 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.69

method	result	size
risch	\(-\frac {\sqrt {-d^{2} x^{2}+c^{2}}\, \left (-16 d^{3} x^{3}+9 c \,d^{2} x^{2}-8 c^{2} d x +6 c^{3}\right )}{24 c^{4} x^{4}}-\frac {3 d^{4} \ln \left (\frac {2 c^{2}+2 \sqrt {c^{2}}\, \sqrt {-d^{2} x^{2}+c^{2}}}{x}\right )}{8 c^{3} \sqrt {c^{2}}}\)	\(99\)
default	\(\frac {-\frac {\left (-d^{2} x^{2}+c^{2}\right )^{\frac {3}{2}}}{4 c^{2} x^{4}}+\frac {d^{2} \left (-\frac {\left (-d^{2} x^{2}+c^{2}\right )^{\frac {3}{2}}}{2 c^{2} x^{2}}-\frac {d^{2} \left (\sqrt {-d^{2} x^{2}+c^{2}}-\frac {c^{2} \ln \left (\frac {2 c^{2}+2 \sqrt {c^{2}}\, \sqrt {-d^{2} x^{2}+c^{2}}}{x}\right )}{\sqrt {c^{2}}}\right )}{2 c^{2}}\right )}{4 c^{2}}}{c}+\frac {d^{4} \left (\sqrt {-d^{2} x^{2}+c^{2}}-\frac {c^{2} \ln \left (\frac {2 c^{2}+2 \sqrt {c^{2}}\, \sqrt {-d^{2} x^{2}+c^{2}}}{x}\right )}{\sqrt {c^{2}}}\right )}{c^{5}}+\frac {d^{2} \left (-\frac {\left (-d^{2} x^{2}+c^{2}\right )^{\frac {3}{2}}}{2 c^{2} x^{2}}-\frac {d^{2} \left (\sqrt {-d^{2} x^{2}+c^{2}}-\frac {c^{2} \ln \left (\frac {2 c^{2}+2 \sqrt {c^{2}}\, \sqrt {-d^{2} x^{2}+c^{2}}}{x}\right )}{\sqrt {c^{2}}}\right )}{2 c^{2}}\right )}{c^{3}}+\frac {d \left (-d^{2} x^{2}+c^{2}\right )^{\frac {3}{2}}}{3 c^{4} x^{3}}-\frac {d^{3} \left (-\frac {\left (-d^{2} x^{2}+c^{2}\right )^{\frac {3}{2}}}{c^{2} x}-\frac {2 d^{2} \left (\frac {x \sqrt {-d^{2} x^{2}+c^{2}}}{2}+\frac {c^{2} \arctan \left (\frac {\sqrt {d^{2}}\, x}{\sqrt {-d^{2} x^{2}+c^{2}}}\right )}{2 \sqrt {d^{2}}}\right )}{c^{2}}\right )}{c^{4}}-\frac {d^{4} \left (\sqrt {-d^{2} \left (x +\frac {c}{d}\right )^{2}+2 c d \left (x +\frac {c}{d}\right )}+\frac {c d \arctan \left (\frac {\sqrt {d^{2}}\, x}{\sqrt {-d^{2} \left (x +\frac {c}{d}\right )^{2}+2 c d \left (x +\frac {c}{d}\right )}}\right )}{\sqrt {d^{2}}}\right )}{c^{5}}\)	\(477\)

Input:

int((-d^2*x^2+c^2)^(1/2)/x^5/(d*x+c),x,method=_RETURNVERBOSE)
 

Output:

-1/24*(-d^2*x^2+c^2)^(1/2)*(-16*d^3*x^3+9*c*d^2*x^2-8*c^2*d*x+6*c^3)/c^4/x 
^4-3/8/c^3*d^4/(c^2)^(1/2)*ln((2*c^2+2*(c^2)^(1/2)*(-d^2*x^2+c^2)^(1/2))/x 
)
 

Fricas [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.60 \[ \int \frac {\sqrt {c^2-d^2 x^2}}{x^5 (c+d x)} \, dx=\frac {9 \, d^{4} x^{4} \log \left (-\frac {c - \sqrt {-d^{2} x^{2} + c^{2}}}{x}\right ) + {\left (16 \, d^{3} x^{3} - 9 \, c d^{2} x^{2} + 8 \, c^{2} d x - 6 \, c^{3}\right )} \sqrt {-d^{2} x^{2} + c^{2}}}{24 \, c^{4} x^{4}} \] Input:

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

Output:

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

Sympy [F]

\[ \int \frac {\sqrt {c^2-d^2 x^2}}{x^5 (c+d x)} \, dx=\int \frac {\sqrt {- \left (- c + d x\right ) \left (c + d x\right )}}{x^{5} \left (c + d x\right )}\, dx \] Input:

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

Output:

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

Maxima [F]

\[ \int \frac {\sqrt {c^2-d^2 x^2}}{x^5 (c+d x)} \, dx=\int { \frac {\sqrt {-d^{2} x^{2} + c^{2}}}{{\left (d x + c\right )} x^{5}} \,d x } \] Input:

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

Output:

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 329 vs. \(2 (123) = 246\).

Time = 0.13 (sec) , antiderivative size = 329, normalized size of antiderivative = 2.30 \[ \int \frac {\sqrt {c^2-d^2 x^2}}{x^5 (c+d x)} \, dx=\frac {{\left (3 \, d^{5} - \frac {8 \, {\left (c d + \sqrt {-d^{2} x^{2} + c^{2}} {\left | d \right |}\right )} d^{3}}{x} + \frac {24 \, {\left (c d + \sqrt {-d^{2} x^{2} + c^{2}} {\left | d \right |}\right )}^{2} d}{x^{2}} - \frac {72 \, {\left (c d + \sqrt {-d^{2} x^{2} + c^{2}} {\left | d \right |}\right )}^{3}}{d x^{3}}\right )} d^{8} x^{4}}{192 \, {\left (c d + \sqrt {-d^{2} x^{2} + c^{2}} {\left | d \right |}\right )}^{4} c^{4} {\left | d \right |}} - \frac {3 \, d^{5} \log \left (\frac {{\left | -2 \, c d - 2 \, \sqrt {-d^{2} x^{2} + c^{2}} {\left | d \right |} \right |}}{2 \, d^{2} {\left | x \right |}}\right )}{8 \, c^{4} {\left | d \right |}} + \frac {\frac {72 \, {\left (c d + \sqrt {-d^{2} x^{2} + c^{2}} {\left | d \right |}\right )} c^{12} d^{5} {\left | d \right |}}{x} - \frac {24 \, {\left (c d + \sqrt {-d^{2} x^{2} + c^{2}} {\left | d \right |}\right )}^{2} c^{12} d^{3} {\left | d \right |}}{x^{2}} + \frac {8 \, {\left (c d + \sqrt {-d^{2} x^{2} + c^{2}} {\left | d \right |}\right )}^{3} c^{12} d {\left | d \right |}}{x^{3}} - \frac {3 \, {\left (c d + \sqrt {-d^{2} x^{2} + c^{2}} {\left | d \right |}\right )}^{4} c^{12} {\left | d \right |}}{d x^{4}}}{192 \, c^{16} d^{4}} \] Input:

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

Output:

1/192*(3*d^5 - 8*(c*d + sqrt(-d^2*x^2 + c^2)*abs(d))*d^3/x + 24*(c*d + sqr 
t(-d^2*x^2 + c^2)*abs(d))^2*d/x^2 - 72*(c*d + sqrt(-d^2*x^2 + c^2)*abs(d)) 
^3/(d*x^3))*d^8*x^4/((c*d + sqrt(-d^2*x^2 + c^2)*abs(d))^4*c^4*abs(d)) - 3 
/8*d^5*log(1/2*abs(-2*c*d - 2*sqrt(-d^2*x^2 + c^2)*abs(d))/(d^2*abs(x)))/( 
c^4*abs(d)) + 1/192*(72*(c*d + sqrt(-d^2*x^2 + c^2)*abs(d))*c^12*d^5*abs(d 
)/x - 24*(c*d + sqrt(-d^2*x^2 + c^2)*abs(d))^2*c^12*d^3*abs(d)/x^2 + 8*(c* 
d + sqrt(-d^2*x^2 + c^2)*abs(d))^3*c^12*d*abs(d)/x^3 - 3*(c*d + sqrt(-d^2* 
x^2 + c^2)*abs(d))^4*c^12*abs(d)/(d*x^4))/(c^16*d^4)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt {c^2-d^2 x^2}}{x^5 (c+d x)} \, dx=\int \frac {\sqrt {c^2-d^2\,x^2}}{x^5\,\left (c+d\,x\right )} \,d x \] Input:

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.20 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.76 \[ \int \frac {\sqrt {c^2-d^2 x^2}}{x^5 (c+d x)} \, dx=\frac {-6 \sqrt {-d^{2} x^{2}+c^{2}}\, c^{3}+8 \sqrt {-d^{2} x^{2}+c^{2}}\, c^{2} d x -9 \sqrt {-d^{2} x^{2}+c^{2}}\, c \,d^{2} x^{2}+16 \sqrt {-d^{2} x^{2}+c^{2}}\, d^{3} x^{3}+9 \,\mathrm {log}\left (\tan \left (\frac {\mathit {asin} \left (\frac {d x}{c}\right )}{2}\right )\right ) d^{4} x^{4}}{24 c^{4} x^{4}} \] Input:

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

Output:

( - 6*sqrt(c**2 - d**2*x**2)*c**3 + 8*sqrt(c**2 - d**2*x**2)*c**2*d*x - 9* 
sqrt(c**2 - d**2*x**2)*c*d**2*x**2 + 16*sqrt(c**2 - d**2*x**2)*d**3*x**3 + 
 9*log(tan(asin((d*x)/c)/2))*d**4*x**4)/(24*c**4*x**4)