\(\int \frac {1}{x \sqrt {c+d x} (a-b x^2)} \, dx\) [589]

Optimal result

Integrand size = 23, antiderivative size = 155 \[ \int \frac {1}{x \sqrt {c+d x} \left (a-b x^2\right )} \, dx=-\frac {2 \text {arctanh}\left (\frac {\sqrt {c+d x}}{\sqrt {c}}\right )}{a \sqrt {c}}+\frac {\sqrt [4]{b} \text {arctanh}\left (\frac {\sqrt [4]{b} \sqrt {c+d x}}{\sqrt {\sqrt {b} c-\sqrt {a} d}}\right )}{a \sqrt {\sqrt {b} c-\sqrt {a} d}}+\frac {\sqrt [4]{b} \text {arctanh}\left (\frac {\sqrt [4]{b} \sqrt {c+d x}}{\sqrt {\sqrt {b} c+\sqrt {a} d}}\right )}{a \sqrt {\sqrt {b} c+\sqrt {a} d}} \] Output:

-2*arctanh((d*x+c)^(1/2)/c^(1/2))/a/c^(1/2)+b^(1/4)*arctanh(b^(1/4)*(d*x+c 
)^(1/2)/(b^(1/2)*c-a^(1/2)*d)^(1/2))/a/(b^(1/2)*c-a^(1/2)*d)^(1/2)+b^(1/4) 
*arctanh(b^(1/4)*(d*x+c)^(1/2)/(b^(1/2)*c+a^(1/2)*d)^(1/2))/a/(b^(1/2)*c+a 
^(1/2)*d)^(1/2)
 

Mathematica [A] (verified)

Time = 0.44 (sec) , antiderivative size = 183, normalized size of antiderivative = 1.18 \[ \int \frac {1}{x \sqrt {c+d x} \left (a-b x^2\right )} \, dx=\frac {\frac {\sqrt {b} \arctan \left (\frac {\sqrt {-b c-\sqrt {a} \sqrt {b} d} \sqrt {c+d x}}{\sqrt {b} c+\sqrt {a} d}\right )}{\sqrt {-b c-\sqrt {a} \sqrt {b} d}}+\frac {\sqrt {b} \arctan \left (\frac {\sqrt {-b c+\sqrt {a} \sqrt {b} d} \sqrt {c+d x}}{\sqrt {b} c-\sqrt {a} d}\right )}{\sqrt {-b c+\sqrt {a} \sqrt {b} d}}-\frac {2 \text {arctanh}\left (\frac {\sqrt {c+d x}}{\sqrt {c}}\right )}{\sqrt {c}}}{a} \] Input:

Integrate[1/(x*Sqrt[c + d*x]*(a - b*x^2)),x]
 

Output:

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

Rubi [A] (verified)

Time = 0.60 (sec) , antiderivative size = 162, normalized size of antiderivative = 1.05, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {561, 25, 27, 1484, 2009}

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.

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

\(\Big \downarrow \) 561

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1484

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

\(\Big \downarrow \) 2009

\(\displaystyle -2 \left (-\frac {\sqrt [4]{b} \text {arctanh}\left (\frac {\sqrt [4]{b} \sqrt {c+d x}}{\sqrt {\sqrt {b} c-\sqrt {a} d}}\right )}{2 a \sqrt {\sqrt {b} c-\sqrt {a} d}}-\frac {\sqrt [4]{b} \text {arctanh}\left (\frac {\sqrt [4]{b} \sqrt {c+d x}}{\sqrt {\sqrt {a} d+\sqrt {b} c}}\right )}{2 a \sqrt {\sqrt {a} d+\sqrt {b} c}}+\frac {\text {arctanh}\left (\frac {\sqrt {c+d x}}{\sqrt {c}}\right )}{a \sqrt {c}}\right )\)

Input:

Int[1/(x*Sqrt[c + d*x]*(a - b*x^2)),x]
 

Output:

-2*(ArcTanh[Sqrt[c + d*x]/Sqrt[c]]/(a*Sqrt[c]) - (b^(1/4)*ArcTanh[(b^(1/4) 
*Sqrt[c + d*x])/Sqrt[Sqrt[b]*c - Sqrt[a]*d]])/(2*a*Sqrt[Sqrt[b]*c - Sqrt[a 
]*d]) - (b^(1/4)*ArcTanh[(b^(1/4)*Sqrt[c + d*x])/Sqrt[Sqrt[b]*c + Sqrt[a]* 
d]])/(2*a*Sqrt[Sqrt[b]*c + Sqrt[a]*d]))
 

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 561

Int[(x_)^(m_.)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbo 
l] :> With[{k = Denominator[n]}, Simp[k/d   Subst[Int[x^(k*(n + 1) - 1)*(-c 
/d + x^k/d)^m*Simp[(b*c^2 + a*d^2)/d^2 - 2*b*c*(x^k/d^2) + b*(x^(2*k)/d^2), 
 x]^p, x], x, (c + d*x)^(1/k)], x]] /; FreeQ[{a, b, c, d, m, p}, x] && Frac 
tionQ[n] && IntegerQ[p] && IntegerQ[m]
 

rule 1484

Int[((d_) + (e_.)*(x_)^2)^(q_)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symb 
ol] :> Int[ExpandIntegrand[(d + e*x^2)^q/(a + b*x^2 + c*x^4), x], x] /; Fre 
eQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 
 0] && IntegerQ[q]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

Maple [A] (verified)

Time = 0.40 (sec) , antiderivative size = 114, normalized size of antiderivative = 0.74

method	result	size
pseudoelliptic	\(\frac {\frac {b \,\operatorname {arctanh}\left (\frac {b \sqrt {d x +c}}{\sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{\sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}}-\frac {b \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{\sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}}-\frac {2 \,\operatorname {arctanh}\left (\frac {\sqrt {d x +c}}{\sqrt {c}}\right )}{\sqrt {c}}}{a}\)	\(114\)
derivativedivides	\(-2 d^{2} \left (\frac {\operatorname {arctanh}\left (\frac {\sqrt {d x +c}}{\sqrt {c}}\right )}{a \,d^{2} \sqrt {c}}-\frac {b^{2} \left (\frac {\operatorname {arctanh}\left (\frac {b \sqrt {d x +c}}{\sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{2 b \sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}}-\frac {\arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{2 b \sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{a \,d^{2}}\right )\)	\(137\)
default	\(2 d^{2} \left (-\frac {\operatorname {arctanh}\left (\frac {\sqrt {d x +c}}{\sqrt {c}}\right )}{a \,d^{2} \sqrt {c}}+\frac {b^{2} \left (\frac {\operatorname {arctanh}\left (\frac {b \sqrt {d x +c}}{\sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{2 b \sqrt {\left (b c +\sqrt {a b \,d^{2}}\right ) b}}-\frac {\arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{2 b \sqrt {\left (-b c +\sqrt {a b \,d^{2}}\right ) b}}\right )}{a \,d^{2}}\right )\)	\(137\)

Input:

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

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1012 vs. \(2 (113) = 226\).

Time = 0.13 (sec) , antiderivative size = 2033, normalized size of antiderivative = 13.12 \[ \int \frac {1}{x \sqrt {c+d x} \left (a-b x^2\right )} \, dx=\text {Too large to display} \] Input:

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

Output:

[1/2*(a*c*sqrt((b*c + (a^2*b*c^2 - a^3*d^2)*sqrt(b*d^2/(a^3*b^2*c^4 - 2*a^ 
4*b*c^2*d^2 + a^5*d^4)))/(a^2*b*c^2 - a^3*d^2))*log(sqrt(d*x + c)*b + (a*b 
*c - (a^3*b*c^2 - a^4*d^2)*sqrt(b*d^2/(a^3*b^2*c^4 - 2*a^4*b*c^2*d^2 + a^5 
*d^4)))*sqrt((b*c + (a^2*b*c^2 - a^3*d^2)*sqrt(b*d^2/(a^3*b^2*c^4 - 2*a^4* 
b*c^2*d^2 + a^5*d^4)))/(a^2*b*c^2 - a^3*d^2))) - a*c*sqrt((b*c + (a^2*b*c^ 
2 - a^3*d^2)*sqrt(b*d^2/(a^3*b^2*c^4 - 2*a^4*b*c^2*d^2 + a^5*d^4)))/(a^2*b 
*c^2 - a^3*d^2))*log(sqrt(d*x + c)*b - (a*b*c - (a^3*b*c^2 - a^4*d^2)*sqrt 
(b*d^2/(a^3*b^2*c^4 - 2*a^4*b*c^2*d^2 + a^5*d^4)))*sqrt((b*c + (a^2*b*c^2 
- a^3*d^2)*sqrt(b*d^2/(a^3*b^2*c^4 - 2*a^4*b*c^2*d^2 + a^5*d^4)))/(a^2*b*c 
^2 - a^3*d^2))) + a*c*sqrt((b*c - (a^2*b*c^2 - a^3*d^2)*sqrt(b*d^2/(a^3*b^ 
2*c^4 - 2*a^4*b*c^2*d^2 + a^5*d^4)))/(a^2*b*c^2 - a^3*d^2))*log(sqrt(d*x + 
 c)*b + (a*b*c + (a^3*b*c^2 - a^4*d^2)*sqrt(b*d^2/(a^3*b^2*c^4 - 2*a^4*b*c 
^2*d^2 + a^5*d^4)))*sqrt((b*c - (a^2*b*c^2 - a^3*d^2)*sqrt(b*d^2/(a^3*b^2* 
c^4 - 2*a^4*b*c^2*d^2 + a^5*d^4)))/(a^2*b*c^2 - a^3*d^2))) - a*c*sqrt((b*c 
 - (a^2*b*c^2 - a^3*d^2)*sqrt(b*d^2/(a^3*b^2*c^4 - 2*a^4*b*c^2*d^2 + a^5*d 
^4)))/(a^2*b*c^2 - a^3*d^2))*log(sqrt(d*x + c)*b - (a*b*c + (a^3*b*c^2 - a 
^4*d^2)*sqrt(b*d^2/(a^3*b^2*c^4 - 2*a^4*b*c^2*d^2 + a^5*d^4)))*sqrt((b*c - 
 (a^2*b*c^2 - a^3*d^2)*sqrt(b*d^2/(a^3*b^2*c^4 - 2*a^4*b*c^2*d^2 + a^5*d^4 
)))/(a^2*b*c^2 - a^3*d^2))) + 2*sqrt(c)*log((d*x - 2*sqrt(d*x + c)*sqrt(c) 
 + 2*c)/x))/(a*c), 1/2*(a*c*sqrt((b*c + (a^2*b*c^2 - a^3*d^2)*sqrt(b*d^...
 

Sympy [F]

\[ \int \frac {1}{x \sqrt {c+d x} \left (a-b x^2\right )} \, dx=- \int \frac {1}{- a x \sqrt {c + d x} + b x^{3} \sqrt {c + d x}}\, dx \] Input:

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

Output:

-Integral(1/(-a*x*sqrt(c + d*x) + b*x**3*sqrt(c + d*x)), x)
 

Maxima [F]

\[ \int \frac {1}{x \sqrt {c+d x} \left (a-b x^2\right )} \, dx=\int { -\frac {1}{{\left (b x^{2} - a\right )} \sqrt {d x + c} x} \,d x } \] Input:

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

Output:

-integrate(1/((b*x^2 - a)*sqrt(d*x + c)*x), x)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 246 vs. \(2 (113) = 226\).

Time = 0.15 (sec) , antiderivative size = 246, normalized size of antiderivative = 1.59 \[ \int \frac {1}{x \sqrt {c+d x} \left (a-b x^2\right )} \, dx=-\frac {{\left ({\left | a \right |} {\left | b \right |} {\left | d \right |} - \sqrt {a b} c {\left | b \right |}\right )} \arctan \left (\frac {\sqrt {d x + c}}{\sqrt {-\frac {a b c + \sqrt {a^{2} b^{2} c^{2} - {\left (a b c^{2} - a^{2} d^{2}\right )} a b}}{a b}}}\right )}{\sqrt {-b^{2} c - \sqrt {a b} b d} {\left (a^{2} d - \sqrt {a b} a c\right )}} - \frac {{\left ({\left | a \right |} {\left | b \right |} {\left | d \right |} + \sqrt {a b} c {\left | b \right |}\right )} \arctan \left (\frac {\sqrt {d x + c}}{\sqrt {-\frac {a b c - \sqrt {a^{2} b^{2} c^{2} - {\left (a b c^{2} - a^{2} d^{2}\right )} a b}}{a b}}}\right )}{\sqrt {-b^{2} c + \sqrt {a b} b d} {\left (a^{2} d + \sqrt {a b} a c\right )}} + \frac {2 \, \arctan \left (\frac {\sqrt {d x + c}}{\sqrt {-c}}\right )}{a \sqrt {-c}} \] Input:

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

Output:

-(abs(a)*abs(b)*abs(d) - sqrt(a*b)*c*abs(b))*arctan(sqrt(d*x + c)/sqrt(-(a 
*b*c + sqrt(a^2*b^2*c^2 - (a*b*c^2 - a^2*d^2)*a*b))/(a*b)))/(sqrt(-b^2*c - 
 sqrt(a*b)*b*d)*(a^2*d - sqrt(a*b)*a*c)) - (abs(a)*abs(b)*abs(d) + sqrt(a* 
b)*c*abs(b))*arctan(sqrt(d*x + c)/sqrt(-(a*b*c - sqrt(a^2*b^2*c^2 - (a*b*c 
^2 - a^2*d^2)*a*b))/(a*b)))/(sqrt(-b^2*c + sqrt(a*b)*b*d)*(a^2*d + sqrt(a* 
b)*a*c)) + 2*arctan(sqrt(d*x + c)/sqrt(-c))/(a*sqrt(-c))
 

Mupad [B] (verification not implemented)

Time = 9.05 (sec) , antiderivative size = 2691, normalized size of antiderivative = 17.36 \[ \int \frac {1}{x \sqrt {c+d x} \left (a-b x^2\right )} \, dx=\text {Too large to display} \] Input:

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

Output:

- atan(((((((512*a^5*b^4*d^10 - 768*a^4*b^5*c^2*d^8)*(c + d*x)^(1/2)*(-(d* 
(a^5*b)^(1/2) + a^2*b*c)/(4*(a^5*d^2 - a^4*b*c^2)))^(1/2) - 512*a^4*b^4*d^ 
10 + 384*a^3*b^5*c^2*d^8)*(-(d*(a^5*b)^(1/2) + a^2*b*c)/(4*(a^5*d^2 - a^4* 
b*c^2)))^(1/2) + 576*a^2*b^5*c*d^8*(c + d*x)^(1/2))*(-(d*(a^5*b)^(1/2) + a 
^2*b*c)/(4*(a^5*d^2 - a^4*b*c^2)))^(1/2) - 96*a*b^5*c*d^8)*(-(d*(a^5*b)^(1 
/2) + a^2*b*c)/(4*(a^5*d^2 - a^4*b*c^2)))^(1/2) - 96*b^5*d^8*(c + d*x)^(1/ 
2))*(-(d*(a^5*b)^(1/2) + a^2*b*c)/(4*(a^5*d^2 - a^4*b*c^2)))^(1/2)*1i + (( 
((512*a^4*b^4*d^10 + (512*a^5*b^4*d^10 - 768*a^4*b^5*c^2*d^8)*(c + d*x)^(1 
/2)*(-(d*(a^5*b)^(1/2) + a^2*b*c)/(4*(a^5*d^2 - a^4*b*c^2)))^(1/2) - 384*a 
^3*b^5*c^2*d^8)*(-(d*(a^5*b)^(1/2) + a^2*b*c)/(4*(a^5*d^2 - a^4*b*c^2)))^( 
1/2) + 576*a^2*b^5*c*d^8*(c + d*x)^(1/2))*(-(d*(a^5*b)^(1/2) + a^2*b*c)/(4 
*(a^5*d^2 - a^4*b*c^2)))^(1/2) + 96*a*b^5*c*d^8)*(-(d*(a^5*b)^(1/2) + a^2* 
b*c)/(4*(a^5*d^2 - a^4*b*c^2)))^(1/2) - 96*b^5*d^8*(c + d*x)^(1/2))*(-(d*( 
a^5*b)^(1/2) + a^2*b*c)/(4*(a^5*d^2 - a^4*b*c^2)))^(1/2)*1i)/((((((512*a^5 
*b^4*d^10 - 768*a^4*b^5*c^2*d^8)*(c + d*x)^(1/2)*(-(d*(a^5*b)^(1/2) + a^2* 
b*c)/(4*(a^5*d^2 - a^4*b*c^2)))^(1/2) - 512*a^4*b^4*d^10 + 384*a^3*b^5*c^2 
*d^8)*(-(d*(a^5*b)^(1/2) + a^2*b*c)/(4*(a^5*d^2 - a^4*b*c^2)))^(1/2) + 576 
*a^2*b^5*c*d^8*(c + d*x)^(1/2))*(-(d*(a^5*b)^(1/2) + a^2*b*c)/(4*(a^5*d^2 
- a^4*b*c^2)))^(1/2) - 96*a*b^5*c*d^8)*(-(d*(a^5*b)^(1/2) + a^2*b*c)/(4*(a 
^5*d^2 - a^4*b*c^2)))^(1/2) - 96*b^5*d^8*(c + d*x)^(1/2))*(-(d*(a^5*b)^...
 

Reduce [B] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 350, normalized size of antiderivative = 2.26 \[ \int \frac {1}{x \sqrt {c+d x} \left (a-b x^2\right )} \, dx=\frac {-2 \sqrt {a}\, \sqrt {\sqrt {b}\, \sqrt {a}\, d -b c}\, \mathit {atan} \left (\frac {\sqrt {d x +c}\, b}{\sqrt {b}\, \sqrt {\sqrt {b}\, \sqrt {a}\, d -b c}}\right ) c d -2 \sqrt {b}\, \sqrt {\sqrt {b}\, \sqrt {a}\, d -b c}\, \mathit {atan} \left (\frac {\sqrt {d x +c}\, b}{\sqrt {b}\, \sqrt {\sqrt {b}\, \sqrt {a}\, d -b c}}\right ) c^{2}-\sqrt {a}\, \sqrt {\sqrt {b}\, \sqrt {a}\, d +b c}\, \mathrm {log}\left (-\sqrt {\sqrt {b}\, \sqrt {a}\, d +b c}+\sqrt {b}\, \sqrt {d x +c}\right ) c d +\sqrt {a}\, \sqrt {\sqrt {b}\, \sqrt {a}\, d +b c}\, \mathrm {log}\left (\sqrt {\sqrt {b}\, \sqrt {a}\, d +b c}+\sqrt {b}\, \sqrt {d x +c}\right ) c d +\sqrt {b}\, \sqrt {\sqrt {b}\, \sqrt {a}\, d +b c}\, \mathrm {log}\left (-\sqrt {\sqrt {b}\, \sqrt {a}\, d +b c}+\sqrt {b}\, \sqrt {d x +c}\right ) c^{2}-\sqrt {b}\, \sqrt {\sqrt {b}\, \sqrt {a}\, d +b c}\, \mathrm {log}\left (\sqrt {\sqrt {b}\, \sqrt {a}\, d +b c}+\sqrt {b}\, \sqrt {d x +c}\right ) c^{2}+2 \sqrt {c}\, \mathrm {log}\left (\sqrt {d x +c}-\sqrt {c}\right ) a \,d^{2}-2 \sqrt {c}\, \mathrm {log}\left (\sqrt {d x +c}-\sqrt {c}\right ) b \,c^{2}-2 \sqrt {c}\, \mathrm {log}\left (\sqrt {d x +c}+\sqrt {c}\right ) a \,d^{2}+2 \sqrt {c}\, \mathrm {log}\left (\sqrt {d x +c}+\sqrt {c}\right ) b \,c^{2}}{2 a c \left (a \,d^{2}-b \,c^{2}\right )} \] Input:

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

Output:

( - 2*sqrt(a)*sqrt(sqrt(b)*sqrt(a)*d - b*c)*atan((sqrt(c + d*x)*b)/(sqrt(b 
)*sqrt(sqrt(b)*sqrt(a)*d - b*c)))*c*d - 2*sqrt(b)*sqrt(sqrt(b)*sqrt(a)*d - 
 b*c)*atan((sqrt(c + d*x)*b)/(sqrt(b)*sqrt(sqrt(b)*sqrt(a)*d - b*c)))*c**2 
 - sqrt(a)*sqrt(sqrt(b)*sqrt(a)*d + b*c)*log( - sqrt(sqrt(b)*sqrt(a)*d + b 
*c) + sqrt(b)*sqrt(c + d*x))*c*d + sqrt(a)*sqrt(sqrt(b)*sqrt(a)*d + b*c)*l 
og(sqrt(sqrt(b)*sqrt(a)*d + b*c) + sqrt(b)*sqrt(c + d*x))*c*d + sqrt(b)*sq 
rt(sqrt(b)*sqrt(a)*d + b*c)*log( - sqrt(sqrt(b)*sqrt(a)*d + b*c) + sqrt(b) 
*sqrt(c + d*x))*c**2 - sqrt(b)*sqrt(sqrt(b)*sqrt(a)*d + b*c)*log(sqrt(sqrt 
(b)*sqrt(a)*d + b*c) + sqrt(b)*sqrt(c + d*x))*c**2 + 2*sqrt(c)*log(sqrt(c 
+ d*x) - sqrt(c))*a*d**2 - 2*sqrt(c)*log(sqrt(c + d*x) - sqrt(c))*b*c**2 - 
 2*sqrt(c)*log(sqrt(c + d*x) + sqrt(c))*a*d**2 + 2*sqrt(c)*log(sqrt(c + d* 
x) + sqrt(c))*b*c**2)/(2*a*c*(a*d**2 - b*c**2))