\(\int \frac {\sqrt {a+\frac {c}{x^2}}}{d+e x} \, dx\) [284]

Optimal result

Integrand size = 19, antiderivative size = 121 \[ \int \frac {\sqrt {a+\frac {c}{x^2}}}{d+e x} \, dx=\frac {\sqrt {a} \text {arctanh}\left (\frac {\sqrt {a+\frac {c}{x^2}}}{\sqrt {a}}\right )}{e}-\frac {\sqrt {a d^2+c e^2} \text {arctanh}\left (\frac {a d-\frac {c e}{x}}{\sqrt {a d^2+c e^2} \sqrt {a+\frac {c}{x^2}}}\right )}{d e}-\frac {\sqrt {c} \text {arctanh}\left (\frac {\sqrt {c}}{\sqrt {a+\frac {c}{x^2}} x}\right )}{d} \] Output:

a^(1/2)*arctanh((a+c/x^2)^(1/2)/a^(1/2))/e-(a*d^2+c*e^2)^(1/2)*arctanh((a* 
d-c*e/x)/(a*d^2+c*e^2)^(1/2)/(a+c/x^2)^(1/2))/d/e-c^(1/2)*arctanh(c^(1/2)/ 
(a+c/x^2)^(1/2)/x)/d
                                                                                    
                                                                                    
 

Mathematica [A] (verified)

Time = 0.42 (sec) , antiderivative size = 160, normalized size of antiderivative = 1.32 \[ \int \frac {\sqrt {a+\frac {c}{x^2}}}{d+e x} \, dx=-\frac {\sqrt {a+\frac {c}{x^2}} x \left (2 \sqrt {-a d^2-c e^2} \arctan \left (\frac {\sqrt {a} (d+e x)-e \sqrt {c+a x^2}}{\sqrt {-a d^2-c e^2}}\right )-2 \sqrt {c} e \text {arctanh}\left (\frac {\sqrt {a} x-\sqrt {c+a x^2}}{\sqrt {c}}\right )+\sqrt {a} d \log \left (-\sqrt {a} x+\sqrt {c+a x^2}\right )\right )}{d e \sqrt {c+a x^2}} \] Input:

Integrate[Sqrt[a + c/x^2]/(d + e*x),x]
 

Output:

-((Sqrt[a + c/x^2]*x*(2*Sqrt[-(a*d^2) - c*e^2]*ArcTan[(Sqrt[a]*(d + e*x) - 
 e*Sqrt[c + a*x^2])/Sqrt[-(a*d^2) - c*e^2]] - 2*Sqrt[c]*e*ArcTanh[(Sqrt[a] 
*x - Sqrt[c + a*x^2])/Sqrt[c]] + Sqrt[a]*d*Log[-(Sqrt[a]*x) + Sqrt[c + a*x 
^2]]))/(d*e*Sqrt[c + a*x^2]))
 

Rubi [A] (verified)

Time = 0.60 (sec) , antiderivative size = 124, normalized size of antiderivative = 1.02, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.579, Rules used = {1774, 1803, 606, 243, 73, 221, 719, 224, 219, 488, 219}

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.

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

\(\Big \downarrow \) 1774

\(\displaystyle \int \frac {\sqrt {a+\frac {c}{x^2}}}{x \left (\frac {d}{x}+e\right )}dx\)

\(\Big \downarrow \) 1803

\(\displaystyle -\int \frac {\sqrt {a+\frac {c}{x^2}} x}{\frac {d}{x}+e}d\frac {1}{x}\)

\(\Big \downarrow \) 606

\(\displaystyle \frac {\int \frac {a d-\frac {c e}{x}}{\sqrt {a+\frac {c}{x^2}} \left (\frac {d}{x}+e\right )}d\frac {1}{x}}{e}-\frac {a \int \frac {x}{\sqrt {a+\frac {c}{x^2}}}d\frac {1}{x}}{e}\)

\(\Big \downarrow \) 243

\(\displaystyle \frac {\int \frac {a d-\frac {c e}{x}}{\sqrt {a+\frac {c}{x^2}} \left (\frac {d}{x}+e\right )}d\frac {1}{x}}{e}-\frac {a \int \frac {x}{\sqrt {a+\frac {c}{x^2}}}d\frac {1}{x^2}}{2 e}\)

\(\Big \downarrow \) 73

\(\displaystyle \frac {\int \frac {a d-\frac {c e}{x}}{\sqrt {a+\frac {c}{x^2}} \left (\frac {d}{x}+e\right )}d\frac {1}{x}}{e}-\frac {a \int \frac {1}{\frac {\sqrt {a+\frac {c}{x^2}}}{c}-\frac {a}{c}}d\sqrt {a+\frac {c}{x^2}}}{c e}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {\int \frac {a d-\frac {c e}{x}}{\sqrt {a+\frac {c}{x^2}} \left (\frac {d}{x}+e\right )}d\frac {1}{x}}{e}+\frac {\sqrt {a} \text {arctanh}\left (\frac {\sqrt {a+\frac {c}{x^2}}}{\sqrt {a}}\right )}{e}\)

\(\Big \downarrow \) 719

\(\displaystyle \frac {\frac {\left (a d^2+c e^2\right ) \int \frac {1}{\sqrt {a+\frac {c}{x^2}} \left (\frac {d}{x}+e\right )}d\frac {1}{x}}{d}-\frac {c e \int \frac {1}{\sqrt {a+\frac {c}{x^2}}}d\frac {1}{x}}{d}}{e}+\frac {\sqrt {a} \text {arctanh}\left (\frac {\sqrt {a+\frac {c}{x^2}}}{\sqrt {a}}\right )}{e}\)

\(\Big \downarrow \) 224

\(\displaystyle \frac {\frac {\left (a d^2+c e^2\right ) \int \frac {1}{\sqrt {a+\frac {c}{x^2}} \left (\frac {d}{x}+e\right )}d\frac {1}{x}}{d}-\frac {c e \int \frac {1}{1-\frac {c}{x^2}}d\frac {1}{\sqrt {a+\frac {c}{x^2}} x}}{d}}{e}+\frac {\sqrt {a} \text {arctanh}\left (\frac {\sqrt {a+\frac {c}{x^2}}}{\sqrt {a}}\right )}{e}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {\frac {\left (a d^2+c e^2\right ) \int \frac {1}{\sqrt {a+\frac {c}{x^2}} \left (\frac {d}{x}+e\right )}d\frac {1}{x}}{d}-\frac {\sqrt {c} e \text {arctanh}\left (\frac {\sqrt {c}}{x \sqrt {a+\frac {c}{x^2}}}\right )}{d}}{e}+\frac {\sqrt {a} \text {arctanh}\left (\frac {\sqrt {a+\frac {c}{x^2}}}{\sqrt {a}}\right )}{e}\)

\(\Big \downarrow \) 488

\(\displaystyle \frac {-\frac {\left (a d^2+c e^2\right ) \int \frac {1}{a d^2+c e^2-\frac {1}{x^2}}d\frac {a d-\frac {c e}{x}}{\sqrt {a+\frac {c}{x^2}}}}{d}-\frac {\sqrt {c} e \text {arctanh}\left (\frac {\sqrt {c}}{x \sqrt {a+\frac {c}{x^2}}}\right )}{d}}{e}+\frac {\sqrt {a} \text {arctanh}\left (\frac {\sqrt {a+\frac {c}{x^2}}}{\sqrt {a}}\right )}{e}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {-\frac {\sqrt {a d^2+c e^2} \text {arctanh}\left (\frac {a d-\frac {c e}{x}}{\sqrt {a+\frac {c}{x^2}} \sqrt {a d^2+c e^2}}\right )}{d}-\frac {\sqrt {c} e \text {arctanh}\left (\frac {\sqrt {c}}{x \sqrt {a+\frac {c}{x^2}}}\right )}{d}}{e}+\frac {\sqrt {a} \text {arctanh}\left (\frac {\sqrt {a+\frac {c}{x^2}}}{\sqrt {a}}\right )}{e}\)

Input:

Int[Sqrt[a + c/x^2]/(d + e*x),x]
 

Output:

(Sqrt[a]*ArcTanh[Sqrt[a + c/x^2]/Sqrt[a]])/e + (-((Sqrt[a*d^2 + c*e^2]*Arc 
Tanh[(a*d - (c*e)/x)/(Sqrt[a*d^2 + c*e^2]*Sqrt[a + c/x^2])])/d) - (Sqrt[c] 
*e*ArcTanh[Sqrt[c]/(Sqrt[a + c/x^2]*x)])/d)/e
 

Defintions of rubi rules used

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 219

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* 
ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt 
Q[a, 0] || LtQ[b, 0])
 

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 224

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], 
x, x/Sqrt[a + b*x^2]] /; FreeQ[{a, b}, x] &&  !GtQ[a, 0]
 

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 488

Int[1/(((c_) + (d_.)*(x_))*Sqrt[(a_) + (b_.)*(x_)^2]), x_Symbol] :> -Subst[ 
Int[1/(b*c^2 + a*d^2 - x^2), x], x, (a*d - b*c*x)/Sqrt[a + b*x^2]] /; FreeQ 
[{a, b, c, d}, x]
 

rule 606

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

rule 719

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p 
_.), x_Symbol] :> Simp[g/e   Int[(d + e*x)^(m + 1)*(a + c*x^2)^p, x], x] + 
Simp[(e*f - d*g)/e   Int[(d + e*x)^m*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, 
d, e, f, g, m, p}, x] &&  !IGtQ[m, 0]
 

rule 1774

Int[((d_) + (e_.)*(x_)^(mn_.))^(q_.)*((a_) + (c_.)*(x_)^(n2_.))^(p_.), x_Sy 
mbol] :> Int[x^(mn*q)*(e + d/x^mn)^q*(a + c*x^n2)^p, x] /; FreeQ[{a, c, d, 
e, mn, p}, x] && EqQ[n2, -2*mn] && IntegerQ[q] && (PosQ[n2] ||  !IntegerQ[p 
])
 

rule 1803

Int[(x_)^(m_.)*((a_) + (c_.)*(x_)^(n2_.))^(p_.)*((d_) + (e_.)*(x_)^(n_))^(q 
_.), x_Symbol] :> Simp[1/n   Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(d + e*x 
)^q*(a + c*x^2)^p, x], x, x^n], x] /; FreeQ[{a, c, d, e, m, n, p, q}, x] && 
 EqQ[n2, 2*n] && IntegerQ[Simplify[(m + 1)/n]]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(246\) vs. \(2(103)=206\).

Time = 0.10 (sec) , antiderivative size = 247, normalized size of antiderivative = 2.04

method	result	size
default	\(-\frac {\sqrt {\frac {a \,x^{2}+c}{x^{2}}}\, x \left (\ln \left (\frac {2 c +2 \sqrt {c}\, \sqrt {a \,x^{2}+c}}{x}\right ) \sqrt {\frac {a \,d^{2}+c \,e^{2}}{e^{2}}}\, \sqrt {c}\, e^{2}-\sqrt {a}\, d \ln \left (\frac {\sqrt {a \,x^{2}+c}\, \sqrt {a}+a x}{\sqrt {a}}\right ) e \sqrt {\frac {a \,d^{2}+c \,e^{2}}{e^{2}}}-\ln \left (\frac {2 \sqrt {a \,x^{2}+c}\, \sqrt {\frac {a \,d^{2}+c \,e^{2}}{e^{2}}}\, e -2 a d x +2 e c}{e x +d}\right ) a \,d^{2}-\ln \left (\frac {2 \sqrt {a \,x^{2}+c}\, \sqrt {\frac {a \,d^{2}+c \,e^{2}}{e^{2}}}\, e -2 a d x +2 e c}{e x +d}\right ) c \,e^{2}\right )}{\sqrt {a \,x^{2}+c}\, d \,e^{2} \sqrt {\frac {a \,d^{2}+c \,e^{2}}{e^{2}}}}\)	\(247\)

Input:

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

Output:

-((a*x^2+c)/x^2)^(1/2)*x*(ln(2*(c^(1/2)*(a*x^2+c)^(1/2)+c)/x)*((a*d^2+c*e^ 
2)/e^2)^(1/2)*c^(1/2)*e^2-a^(1/2)*d*ln(((a*x^2+c)^(1/2)*a^(1/2)+a*x)/a^(1/ 
2))*e*((a*d^2+c*e^2)/e^2)^(1/2)-ln(2*((a*x^2+c)^(1/2)*((a*d^2+c*e^2)/e^2)^ 
(1/2)*e-a*d*x+e*c)/(e*x+d))*a*d^2-ln(2*((a*x^2+c)^(1/2)*((a*d^2+c*e^2)/e^2 
)^(1/2)*e-a*d*x+e*c)/(e*x+d))*c*e^2)/(a*x^2+c)^(1/2)/d/e^2/((a*d^2+c*e^2)/ 
e^2)^(1/2)
 

Fricas [A] (verification not implemented)

Time = 0.54 (sec) , antiderivative size = 1508, normalized size of antiderivative = 12.46 \[ \int \frac {\sqrt {a+\frac {c}{x^2}}}{d+e x} \, dx=\text {Too large to display} \] Input:

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

Output:

[1/2*(sqrt(a)*d*log(-2*a*x^2 - 2*sqrt(a)*x^2*sqrt((a*x^2 + c)/x^2) - c) + 
sqrt(c)*e*log(-(a*x^2 - 2*sqrt(c)*x*sqrt((a*x^2 + c)/x^2) + 2*c)/x^2) + sq 
rt(a*d^2 + c*e^2)*log((2*a*c*d*e*x - a*c*d^2 - 2*c^2*e^2 - (2*a^2*d^2 + a* 
c*e^2)*x^2 + 2*(a*d*x^2 - c*e*x)*sqrt(a*d^2 + c*e^2)*sqrt((a*x^2 + c)/x^2) 
)/(e^2*x^2 + 2*d*e*x + d^2)))/(d*e), -1/2*(2*sqrt(-a)*d*arctan(sqrt(-a)*x^ 
2*sqrt((a*x^2 + c)/x^2)/(a*x^2 + c)) - sqrt(c)*e*log(-(a*x^2 - 2*sqrt(c)*x 
*sqrt((a*x^2 + c)/x^2) + 2*c)/x^2) - sqrt(a*d^2 + c*e^2)*log((2*a*c*d*e*x 
- a*c*d^2 - 2*c^2*e^2 - (2*a^2*d^2 + a*c*e^2)*x^2 + 2*(a*d*x^2 - c*e*x)*sq 
rt(a*d^2 + c*e^2)*sqrt((a*x^2 + c)/x^2))/(e^2*x^2 + 2*d*e*x + d^2)))/(d*e) 
, 1/2*(sqrt(a)*d*log(-2*a*x^2 - 2*sqrt(a)*x^2*sqrt((a*x^2 + c)/x^2) - c) + 
 sqrt(c)*e*log(-(a*x^2 - 2*sqrt(c)*x*sqrt((a*x^2 + c)/x^2) + 2*c)/x^2) + 2 
*sqrt(-a*d^2 - c*e^2)*arctan((a*d*x^2 - c*e*x)*sqrt(-a*d^2 - c*e^2)*sqrt(( 
a*x^2 + c)/x^2)/(a*c*d^2 + c^2*e^2 + (a^2*d^2 + a*c*e^2)*x^2)))/(d*e), -1/ 
2*(2*sqrt(-a)*d*arctan(sqrt(-a)*x^2*sqrt((a*x^2 + c)/x^2)/(a*x^2 + c)) - s 
qrt(c)*e*log(-(a*x^2 - 2*sqrt(c)*x*sqrt((a*x^2 + c)/x^2) + 2*c)/x^2) - 2*s 
qrt(-a*d^2 - c*e^2)*arctan((a*d*x^2 - c*e*x)*sqrt(-a*d^2 - c*e^2)*sqrt((a* 
x^2 + c)/x^2)/(a*c*d^2 + c^2*e^2 + (a^2*d^2 + a*c*e^2)*x^2)))/(d*e), 1/2*( 
2*sqrt(-c)*e*arctan(sqrt(-c)*x*sqrt((a*x^2 + c)/x^2)/c) + sqrt(a)*d*log(-2 
*a*x^2 - 2*sqrt(a)*x^2*sqrt((a*x^2 + c)/x^2) - c) + sqrt(a*d^2 + c*e^2)*lo 
g((2*a*c*d*e*x - a*c*d^2 - 2*c^2*e^2 - (2*a^2*d^2 + a*c*e^2)*x^2 + 2*(a...
 

Sympy [F]

\[ \int \frac {\sqrt {a+\frac {c}{x^2}}}{d+e x} \, dx=\int \frac {\sqrt {a + \frac {c}{x^{2}}}}{d + e x}\, dx \] Input:

integrate((a+c/x**2)**(1/2)/(e*x+d),x)
 

Output:

Integral(sqrt(a + c/x**2)/(d + e*x), x)
 

Maxima [F]

\[ \int \frac {\sqrt {a+\frac {c}{x^2}}}{d+e x} \, dx=\int { \frac {\sqrt {a + \frac {c}{x^{2}}}}{e x + d} \,d x } \] Input:

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

Output:

integrate(sqrt(a + c/x^2)/(e*x + d), x)
 

Giac [F(-2)]

Exception generated. \[ \int \frac {\sqrt {a+\frac {c}{x^2}}}{d+e x} \, dx=\text {Exception raised: TypeError} \] Input:

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

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:index.cc index_m i_lex_is_greater E 
rror: Bad Argument Value
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt {a+\frac {c}{x^2}}}{d+e x} \, dx=\int \frac {\sqrt {a+\frac {c}{x^2}}}{d+e\,x} \,d x \] Input:

int((a + c/x^2)^(1/2)/(d + e*x),x)
 

Output:

int((a + c/x^2)^(1/2)/(d + e*x), x)
 

Reduce [B] (verification not implemented)

Time = 0.20 (sec) , antiderivative size = 148, normalized size of antiderivative = 1.22 \[ \int \frac {\sqrt {a+\frac {c}{x^2}}}{d+e x} \, dx=\frac {2 \sqrt {a \,d^{2}+c \,e^{2}}\, \mathrm {log}\left (-\sqrt {a \,x^{2}+c}\, \sqrt {a \,d^{2}+c \,e^{2}}+a d x -c e \right )-2 \sqrt {a \,d^{2}+c \,e^{2}}\, \mathrm {log}\left (e x +d \right )-\sqrt {a}\, \mathrm {log}\left (\sqrt {a \,x^{2}+c}-\sqrt {a}\, x \right ) d +\sqrt {a}\, \mathrm {log}\left (\sqrt {a \,x^{2}+c}+\sqrt {a}\, x \right ) d +\sqrt {c}\, \mathrm {log}\left (\sqrt {a \,x^{2}+c}-\sqrt {c}\right ) e -\sqrt {c}\, \mathrm {log}\left (\sqrt {a \,x^{2}+c}+\sqrt {c}\right ) e}{2 d e} \] Input:

int((a+c/x^2)^(1/2)/(e*x+d),x)
 

Output:

(2*sqrt(a*d**2 + c*e**2)*log( - sqrt(a*x**2 + c)*sqrt(a*d**2 + c*e**2) + a 
*d*x - c*e) - 2*sqrt(a*d**2 + c*e**2)*log(d + e*x) - sqrt(a)*log(sqrt(a*x* 
*2 + c) - sqrt(a)*x)*d + sqrt(a)*log(sqrt(a*x**2 + c) + sqrt(a)*x)*d + sqr 
t(c)*log(sqrt(a*x**2 + c) - sqrt(c))*e - sqrt(c)*log(sqrt(a*x**2 + c) + sq 
rt(c))*e)/(2*d*e)