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

Optimal result

Integrand size = 25, antiderivative size = 152 \[ \int \frac {A+B x}{\sqrt {d+e x} \left (a-c x^2\right )} \, dx=\frac {\left (B-\frac {A \sqrt {c}}{\sqrt {a}}\right ) \text {arctanh}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {a} e}}\right )}{c^{3/4} \sqrt {\sqrt {c} d-\sqrt {a} e}}+\frac {\left (B+\frac {A \sqrt {c}}{\sqrt {a}}\right ) \text {arctanh}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d+\sqrt {a} e}}\right )}{c^{3/4} \sqrt {\sqrt {c} d+\sqrt {a} e}} \] Output:

(B-A*c^(1/2)/a^(1/2))*arctanh(c^(1/4)*(e*x+d)^(1/2)/(c^(1/2)*d-a^(1/2)*e)^ 
(1/2))/c^(3/4)/(c^(1/2)*d-a^(1/2)*e)^(1/2)+(B+A*c^(1/2)/a^(1/2))*arctanh(c 
^(1/4)*(e*x+d)^(1/2)/(c^(1/2)*d+a^(1/2)*e)^(1/2))/c^(3/4)/(c^(1/2)*d+a^(1/ 
2)*e)^(1/2)
 

Mathematica [A] (verified)

Time = 0.55 (sec) , antiderivative size = 188, normalized size of antiderivative = 1.24 \[ \int \frac {A+B x}{\sqrt {d+e x} \left (a-c x^2\right )} \, dx=\frac {\frac {\left (\sqrt {a} B+A \sqrt {c}\right ) \arctan \left (\frac {\sqrt {-c d-\sqrt {a} \sqrt {c} e} \sqrt {d+e x}}{\sqrt {c} d+\sqrt {a} e}\right )}{\sqrt {-c d-\sqrt {a} \sqrt {c} e}}+\frac {\left (\sqrt {a} B-A \sqrt {c}\right ) \arctan \left (\frac {\sqrt {-c d+\sqrt {a} \sqrt {c} e} \sqrt {d+e x}}{\sqrt {c} d-\sqrt {a} e}\right )}{\sqrt {-c d+\sqrt {a} \sqrt {c} e}}}{\sqrt {a} \sqrt {c}} \] Input:

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

Output:

(((Sqrt[a]*B + A*Sqrt[c])*ArcTan[(Sqrt[-(c*d) - Sqrt[a]*Sqrt[c]*e]*Sqrt[d 
+ e*x])/(Sqrt[c]*d + Sqrt[a]*e)])/Sqrt[-(c*d) - Sqrt[a]*Sqrt[c]*e] + ((Sqr 
t[a]*B - A*Sqrt[c])*ArcTan[(Sqrt[-(c*d) + Sqrt[a]*Sqrt[c]*e]*Sqrt[d + e*x] 
)/(Sqrt[c]*d - Sqrt[a]*e)])/Sqrt[-(c*d) + Sqrt[a]*Sqrt[c]*e])/(Sqrt[a]*Sqr 
t[c])
 

Rubi [A] (verified)

Time = 0.30 (sec) , antiderivative size = 160, normalized size of antiderivative = 1.05, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {654, 1480, 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.

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

\(\Big \downarrow \) 654

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

\(\Big \downarrow \) 1480

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

\(\Big \downarrow \) 221

\(\displaystyle 2 \left (\frac {\left (B-\frac {A \sqrt {c}}{\sqrt {a}}\right ) \text {arctanh}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {a} e}}\right )}{2 c^{3/4} \sqrt {\sqrt {c} d-\sqrt {a} e}}+\frac {\left (\frac {A \sqrt {c}}{\sqrt {a}}+B\right ) \text {arctanh}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {a} e+\sqrt {c} d}}\right )}{2 c^{3/4} \sqrt {\sqrt {a} e+\sqrt {c} d}}\right )\)

Input:

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

Output:

2*(((B - (A*Sqrt[c])/Sqrt[a])*ArcTanh[(c^(1/4)*Sqrt[d + e*x])/Sqrt[Sqrt[c] 
*d - Sqrt[a]*e]])/(2*c^(3/4)*Sqrt[Sqrt[c]*d - Sqrt[a]*e]) + ((B + (A*Sqrt[ 
c])/Sqrt[a])*ArcTanh[(c^(1/4)*Sqrt[d + e*x])/Sqrt[Sqrt[c]*d + Sqrt[a]*e]]) 
/(2*c^(3/4)*Sqrt[Sqrt[c]*d + Sqrt[a]*e]))
 

Defintions of rubi rules used

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 654

Int[((f_.) + (g_.)*(x_))/(Sqrt[(d_.) + (e_.)*(x_)]*((a_) + (c_.)*(x_)^2)), 
x_Symbol] :> Simp[2   Subst[Int[(e*f - d*g + g*x^2)/(c*d^2 + a*e^2 - 2*c*d* 
x^2 + c*x^4), x], x, Sqrt[d + e*x]], x] /; FreeQ[{a, c, d, e, f, g}, x]
 

rule 1480

Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] : 
> With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[(e/2 + (2*c*d - b*e)/(2*q))   Int[1/( 
b/2 - q/2 + c*x^2), x], x] + Simp[(e/2 - (2*c*d - b*e)/(2*q))   Int[1/(b/2 
+ q/2 + c*x^2), x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] 
 && NeQ[c*d^2 - a*e^2, 0] && PosQ[b^2 - 4*a*c]
 

Maple [A] (verified)

Time = 1.38 (sec) , antiderivative size = 133, normalized size of antiderivative = 0.88

method	result	size
pseudoelliptic	\(-\frac {-\frac {\left (A c e -B \sqrt {a c \,e^{2}}\right ) \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{\sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}-\frac {\left (A c e +B \sqrt {a c \,e^{2}}\right ) \operatorname {arctanh}\left (\frac {c \sqrt {e x +d}}{\sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{\sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}}{\sqrt {a c \,e^{2}}}\)	\(133\)
derivativedivides	\(-2 c \left (\frac {\left (-A c e +B \sqrt {a c \,e^{2}}\right ) \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{2 c \sqrt {a c \,e^{2}}\, \sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}-\frac {\left (A c e +B \sqrt {a c \,e^{2}}\right ) \operatorname {arctanh}\left (\frac {c \sqrt {e x +d}}{\sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{2 c \sqrt {a c \,e^{2}}\, \sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}\right )\)	\(148\)
default	\(2 c \left (\frac {\left (A c e -B \sqrt {a c \,e^{2}}\right ) \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{2 c \sqrt {a c \,e^{2}}\, \sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}-\frac {\left (-A c e -B \sqrt {a c \,e^{2}}\right ) \operatorname {arctanh}\left (\frac {c \sqrt {e x +d}}{\sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{2 c \sqrt {a c \,e^{2}}\, \sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}\right )\)	\(150\)

Input:

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

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2385 vs. \(2 (108) = 216\).

Time = 0.14 (sec) , antiderivative size = 2385, normalized size of antiderivative = 15.69 \[ \int \frac {A+B x}{\sqrt {d+e x} \left (a-c x^2\right )} \, dx=\text {Too large to display} \] Input:

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

Output:

1/2*sqrt(-(2*A*B*a*e - (B^2*a + A^2*c)*d + (a*c^2*d^2 - a^2*c*e^2)*sqrt((4 
*A^2*B^2*c^2*d^2 - 4*(A*B^3*a*c + A^3*B*c^2)*d*e + (B^4*a^2 + 2*A^2*B^2*a* 
c + A^4*c^2)*e^2)/(a*c^5*d^4 - 2*a^2*c^4*d^2*e^2 + a^3*c^3*e^4)))/(a*c^2*d 
^2 - a^2*c*e^2))*log((2*(A*B^3*a*c - A^3*B*c^2)*d - (B^4*a^2 - A^4*c^2)*e) 
*sqrt(e*x + d) + (2*A*B^2*a*c^2*d^2 - (B^3*a^2*c + 3*A^2*B*a*c^2)*d*e + (A 
*B^2*a^2*c + A^3*a*c^2)*e^2 + (A*a*c^4*d^3 - B*a^2*c^3*d^2*e - A*a^2*c^3*d 
*e^2 + B*a^3*c^2*e^3)*sqrt((4*A^2*B^2*c^2*d^2 - 4*(A*B^3*a*c + A^3*B*c^2)* 
d*e + (B^4*a^2 + 2*A^2*B^2*a*c + A^4*c^2)*e^2)/(a*c^5*d^4 - 2*a^2*c^4*d^2* 
e^2 + a^3*c^3*e^4)))*sqrt(-(2*A*B*a*e - (B^2*a + A^2*c)*d + (a*c^2*d^2 - a 
^2*c*e^2)*sqrt((4*A^2*B^2*c^2*d^2 - 4*(A*B^3*a*c + A^3*B*c^2)*d*e + (B^4*a 
^2 + 2*A^2*B^2*a*c + A^4*c^2)*e^2)/(a*c^5*d^4 - 2*a^2*c^4*d^2*e^2 + a^3*c^ 
3*e^4)))/(a*c^2*d^2 - a^2*c*e^2))) - 1/2*sqrt(-(2*A*B*a*e - (B^2*a + A^2*c 
)*d + (a*c^2*d^2 - a^2*c*e^2)*sqrt((4*A^2*B^2*c^2*d^2 - 4*(A*B^3*a*c + A^3 
*B*c^2)*d*e + (B^4*a^2 + 2*A^2*B^2*a*c + A^4*c^2)*e^2)/(a*c^5*d^4 - 2*a^2* 
c^4*d^2*e^2 + a^3*c^3*e^4)))/(a*c^2*d^2 - a^2*c*e^2))*log((2*(A*B^3*a*c - 
A^3*B*c^2)*d - (B^4*a^2 - A^4*c^2)*e)*sqrt(e*x + d) - (2*A*B^2*a*c^2*d^2 - 
 (B^3*a^2*c + 3*A^2*B*a*c^2)*d*e + (A*B^2*a^2*c + A^3*a*c^2)*e^2 + (A*a*c^ 
4*d^3 - B*a^2*c^3*d^2*e - A*a^2*c^3*d*e^2 + B*a^3*c^2*e^3)*sqrt((4*A^2*B^2 
*c^2*d^2 - 4*(A*B^3*a*c + A^3*B*c^2)*d*e + (B^4*a^2 + 2*A^2*B^2*a*c + A^4* 
c^2)*e^2)/(a*c^5*d^4 - 2*a^2*c^4*d^2*e^2 + a^3*c^3*e^4)))*sqrt(-(2*A*B*...
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 269 vs. \(2 (108) = 216\).

Time = 0.15 (sec) , antiderivative size = 269, normalized size of antiderivative = 1.77 \[ \int \frac {A+B x}{\sqrt {d+e x} \left (a-c x^2\right )} \, dx=-\frac {{\left (B a c d {\left | c \right |} {\left | e \right |} - A a c e {\left | c \right |} {\left | e \right |} + \sqrt {a c} A c d e {\left | c \right |} - \sqrt {a c} B a e^{2} {\left | c \right |}\right )} \arctan \left (\frac {\sqrt {e x + d}}{\sqrt {-\frac {c d + \sqrt {c^{2} d^{2} - {\left (c d^{2} - a e^{2}\right )} c}}{c}}}\right )}{{\left (a c^{2} d - \sqrt {a c} a c e\right )} \sqrt {-c^{2} d - \sqrt {a c} c e} {\left | e \right |}} - \frac {{\left (B a c d {\left | c \right |} {\left | e \right |} - A a c e {\left | c \right |} {\left | e \right |} - \sqrt {a c} A c d e {\left | c \right |} + \sqrt {a c} B a e^{2} {\left | c \right |}\right )} \arctan \left (\frac {\sqrt {e x + d}}{\sqrt {-\frac {c d - \sqrt {c^{2} d^{2} - {\left (c d^{2} - a e^{2}\right )} c}}{c}}}\right )}{{\left (a c^{2} d + \sqrt {a c} a c e\right )} \sqrt {-c^{2} d + \sqrt {a c} c e} {\left | e \right |}} \] Input:

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

Output:

-(B*a*c*d*abs(c)*abs(e) - A*a*c*e*abs(c)*abs(e) + sqrt(a*c)*A*c*d*e*abs(c) 
 - sqrt(a*c)*B*a*e^2*abs(c))*arctan(sqrt(e*x + d)/sqrt(-(c*d + sqrt(c^2*d^ 
2 - (c*d^2 - a*e^2)*c))/c))/((a*c^2*d - sqrt(a*c)*a*c*e)*sqrt(-c^2*d - sqr 
t(a*c)*c*e)*abs(e)) - (B*a*c*d*abs(c)*abs(e) - A*a*c*e*abs(c)*abs(e) - sqr 
t(a*c)*A*c*d*e*abs(c) + sqrt(a*c)*B*a*e^2*abs(c))*arctan(sqrt(e*x + d)/sqr 
t(-(c*d - sqrt(c^2*d^2 - (c*d^2 - a*e^2)*c))/c))/((a*c^2*d + sqrt(a*c)*a*c 
*e)*sqrt(-c^2*d + sqrt(a*c)*c*e)*abs(e))
 

Mupad [B] (verification not implemented)

Time = 6.93 (sec) , antiderivative size = 2065, normalized size of antiderivative = 13.59 \[ \int \frac {A+B x}{\sqrt {d+e x} \left (a-c x^2\right )} \, dx=\text {Too large to display} \] Input:

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

Output:

atan((a^2*c^5*d^3*((B^2*a*e*(a^3*c^3)^(1/2) + A^2*c*e*(a^3*c^3)^(1/2) + A^ 
2*a*c^3*d + B^2*a^2*c^2*d - 2*A*B*a^2*c^2*e - 2*A*B*c*d*(a^3*c^3)^(1/2))/( 
4*a^2*c^4*d^2 - 4*a^3*c^3*e^2))^(3/2)*(d + e*x)^(1/2)*8i + A^2*a^2*c^3*e^2 
*((B^2*a*e*(a^3*c^3)^(1/2) + A^2*c*e*(a^3*c^3)^(1/2) + A^2*a*c^3*d + B^2*a 
^2*c^2*d - 2*A*B*a^2*c^2*e - 2*A*B*c*d*(a^3*c^3)^(1/2))/(4*a^2*c^4*d^2 - 4 
*a^3*c^3*e^2))^(1/2)*(d + e*x)^(1/2)*2i - B^2*a^2*c^3*d^2*((B^2*a*e*(a^3*c 
^3)^(1/2) + A^2*c*e*(a^3*c^3)^(1/2) + A^2*a*c^3*d + B^2*a^2*c^2*d - 2*A*B* 
a^2*c^2*e - 2*A*B*c*d*(a^3*c^3)^(1/2))/(4*a^2*c^4*d^2 - 4*a^3*c^3*e^2))^(1 
/2)*(d + e*x)^(1/2)*2i + B^2*a^3*c^2*e^2*((B^2*a*e*(a^3*c^3)^(1/2) + A^2*c 
*e*(a^3*c^3)^(1/2) + A^2*a*c^3*d + B^2*a^2*c^2*d - 2*A*B*a^2*c^2*e - 2*A*B 
*c*d*(a^3*c^3)^(1/2))/(4*a^2*c^4*d^2 - 4*a^3*c^3*e^2))^(1/2)*(d + e*x)^(1/ 
2)*2i - A^2*a*c^4*d^2*((B^2*a*e*(a^3*c^3)^(1/2) + A^2*c*e*(a^3*c^3)^(1/2) 
+ A^2*a*c^3*d + B^2*a^2*c^2*d - 2*A*B*a^2*c^2*e - 2*A*B*c*d*(a^3*c^3)^(1/2 
))/(4*a^2*c^4*d^2 - 4*a^3*c^3*e^2))^(1/2)*(d + e*x)^(1/2)*2i - a^3*c^4*d*e 
^2*((B^2*a*e*(a^3*c^3)^(1/2) + A^2*c*e*(a^3*c^3)^(1/2) + A^2*a*c^3*d + B^2 
*a^2*c^2*d - 2*A*B*a^2*c^2*e - 2*A*B*c*d*(a^3*c^3)^(1/2))/(4*a^2*c^4*d^2 - 
 4*a^3*c^3*e^2))^(3/2)*(d + e*x)^(1/2)*8i)/(A^3*c*e^2*(a^3*c^3)^(1/2) - B^ 
3*a^3*c*e^2 - 2*A^2*B*a*c^3*d^2 - B^3*a*d*e*(a^3*c^3)^(1/2) - A^2*B*a^2*c^ 
2*e^2 + A*B^2*a*e^2*(a^3*c^3)^(1/2) + 2*A*B^2*c*d^2*(a^3*c^3)^(1/2) + A^3* 
a*c^3*d*e + 3*A*B^2*a^2*c^2*d*e - 3*A^2*B*c*d*e*(a^3*c^3)^(1/2)))*((B^2...
 

Reduce [B] (verification not implemented)

Time = 0.23 (sec) , antiderivative size = 516, normalized size of antiderivative = 3.39 \[ \int \frac {A+B x}{\sqrt {d+e x} \left (a-c x^2\right )} \, dx=\frac {-2 \sqrt {a}\, \sqrt {\sqrt {c}\, \sqrt {a}\, e -c d}\, \mathit {atan} \left (\frac {\sqrt {e x +d}\, c}{\sqrt {c}\, \sqrt {\sqrt {c}\, \sqrt {a}\, e -c d}}\right ) b e +2 \sqrt {a}\, \sqrt {\sqrt {c}\, \sqrt {a}\, e -c d}\, \mathit {atan} \left (\frac {\sqrt {e x +d}\, c}{\sqrt {c}\, \sqrt {\sqrt {c}\, \sqrt {a}\, e -c d}}\right ) c d +2 \sqrt {c}\, \sqrt {\sqrt {c}\, \sqrt {a}\, e -c d}\, \mathit {atan} \left (\frac {\sqrt {e x +d}\, c}{\sqrt {c}\, \sqrt {\sqrt {c}\, \sqrt {a}\, e -c d}}\right ) a e -2 \sqrt {c}\, \sqrt {\sqrt {c}\, \sqrt {a}\, e -c d}\, \mathit {atan} \left (\frac {\sqrt {e x +d}\, c}{\sqrt {c}\, \sqrt {\sqrt {c}\, \sqrt {a}\, e -c d}}\right ) b d -\sqrt {a}\, \sqrt {\sqrt {c}\, \sqrt {a}\, e +c d}\, \mathrm {log}\left (-\sqrt {\sqrt {c}\, \sqrt {a}\, e +c d}+\sqrt {c}\, \sqrt {e x +d}\right ) b e +\sqrt {a}\, \sqrt {\sqrt {c}\, \sqrt {a}\, e +c d}\, \mathrm {log}\left (-\sqrt {\sqrt {c}\, \sqrt {a}\, e +c d}+\sqrt {c}\, \sqrt {e x +d}\right ) c d +\sqrt {a}\, \sqrt {\sqrt {c}\, \sqrt {a}\, e +c d}\, \mathrm {log}\left (\sqrt {\sqrt {c}\, \sqrt {a}\, e +c d}+\sqrt {c}\, \sqrt {e x +d}\right ) b e -\sqrt {a}\, \sqrt {\sqrt {c}\, \sqrt {a}\, e +c d}\, \mathrm {log}\left (\sqrt {\sqrt {c}\, \sqrt {a}\, e +c d}+\sqrt {c}\, \sqrt {e x +d}\right ) c d -\sqrt {c}\, \sqrt {\sqrt {c}\, \sqrt {a}\, e +c d}\, \mathrm {log}\left (-\sqrt {\sqrt {c}\, \sqrt {a}\, e +c d}+\sqrt {c}\, \sqrt {e x +d}\right ) a e +\sqrt {c}\, \sqrt {\sqrt {c}\, \sqrt {a}\, e +c d}\, \mathrm {log}\left (-\sqrt {\sqrt {c}\, \sqrt {a}\, e +c d}+\sqrt {c}\, \sqrt {e x +d}\right ) b d +\sqrt {c}\, \sqrt {\sqrt {c}\, \sqrt {a}\, e +c d}\, \mathrm {log}\left (\sqrt {\sqrt {c}\, \sqrt {a}\, e +c d}+\sqrt {c}\, \sqrt {e x +d}\right ) a e -\sqrt {c}\, \sqrt {\sqrt {c}\, \sqrt {a}\, e +c d}\, \mathrm {log}\left (\sqrt {\sqrt {c}\, \sqrt {a}\, e +c d}+\sqrt {c}\, \sqrt {e x +d}\right ) b d}{2 c \left (a \,e^{2}-c \,d^{2}\right )} \] Input:

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

Output:

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