\(\int \frac {\sqrt {d+e x}}{(b x+c x^2)^2} \, dx\) [117]

Optimal result

Integrand size = 21, antiderivative size = 140 \[ \int \frac {\sqrt {d+e x}}{\left (b x+c x^2\right )^2} \, dx=-\frac {2 c \sqrt {d+e x}}{b^2 (b+c x)}-\frac {\sqrt {d+e x}}{b x (b+c x)}+\frac {(4 c d-b e) \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{b^3 \sqrt {d}}-\frac {\sqrt {c} (4 c d-3 b e) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt {c d-b e}}\right )}{b^3 \sqrt {c d-b e}} \] Output:

-2*c*(e*x+d)^(1/2)/b^2/(c*x+b)-(e*x+d)^(1/2)/b/x/(c*x+b)+(-b*e+4*c*d)*arct 
anh((e*x+d)^(1/2)/d^(1/2))/b^3/d^(1/2)-c^(1/2)*(-3*b*e+4*c*d)*arctanh(c^(1 
/2)*(e*x+d)^(1/2)/(-b*e+c*d)^(1/2))/b^3/(-b*e+c*d)^(1/2)
 

Mathematica [A] (verified)

Time = 0.66 (sec) , antiderivative size = 119, normalized size of antiderivative = 0.85 \[ \int \frac {\sqrt {d+e x}}{\left (b x+c x^2\right )^2} \, dx=\frac {-\frac {b (b+2 c x) \sqrt {d+e x}}{x (b+c x)}+\frac {\sqrt {c} (4 c d-3 b e) \arctan \left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt {-c d+b e}}\right )}{\sqrt {-c d+b e}}+\frac {(4 c d-b e) \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{\sqrt {d}}}{b^3} \] Input:

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

Output:

(-((b*(b + 2*c*x)*Sqrt[d + e*x])/(x*(b + c*x))) + (Sqrt[c]*(4*c*d - 3*b*e) 
*ArcTan[(Sqrt[c]*Sqrt[d + e*x])/Sqrt[-(c*d) + b*e]])/Sqrt[-(c*d) + b*e] + 
((4*c*d - b*e)*ArcTanh[Sqrt[d + e*x]/Sqrt[d]])/Sqrt[d])/b^3
 

Rubi [A] (verified)

Time = 0.62 (sec) , antiderivative size = 138, normalized size of antiderivative = 0.99, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {1163, 27, 1197, 27, 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.

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

\(\Big \downarrow \) 1163

\(\displaystyle \frac {\int -\frac {4 c d-b e+2 c e x}{2 \sqrt {d+e x} \left (c x^2+b x\right )}dx}{b^2}-\frac {(b+2 c x) \sqrt {d+e x}}{b^2 \left (b x+c x^2\right )}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {\int \frac {4 c d-b e+2 c e x}{\sqrt {d+e x} \left (c x^2+b x\right )}dx}{2 b^2}-\frac {(b+2 c x) \sqrt {d+e x}}{b^2 \left (b x+c x^2\right )}\)

\(\Big \downarrow \) 1197

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1480

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

\(\Big \downarrow \) 221

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

Input:

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

Output:

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

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 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 1163

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S 
ymbol] :> Simp[(d + e*x)^m*(b + 2*c*x)*((a + b*x + c*x^2)^(p + 1)/((p + 1)* 
(b^2 - 4*a*c))), x] - Simp[1/((p + 1)*(b^2 - 4*a*c))   Int[(d + e*x)^(m - 1 
)*(b*e*m + 2*c*d*(2*p + 3) + 2*c*e*(m + 2*p + 3)*x)*(a + b*x + c*x^2)^(p + 
1), x], x] /; FreeQ[{a, b, c, d, e}, x] && LtQ[p, -1] && GtQ[m, 0] && (LtQ[ 
m, 1] || (ILtQ[m + 2*p + 3, 0] && NeQ[m, 2])) && IntQuadraticQ[a, b, c, d, 
e, m, p, x]
 

rule 1197

Int[((f_.) + (g_.)*(x_))/(Sqrt[(d_.) + (e_.)*(x_)]*((a_.) + (b_.)*(x_) + (c 
_.)*(x_)^2)), x_Symbol] :> Simp[2   Subst[Int[(e*f - d*g + g*x^2)/(c*d^2 - 
b*d*e + a*e^2 - (2*c*d - b*e)*x^2 + c*x^4), x], x, Sqrt[d + e*x]], x] /; Fr 
eeQ[{a, b, 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 = 0.58 (sec) , antiderivative size = 133, normalized size of antiderivative = 0.95

method	result	size
pseudoelliptic	\(-\frac {-4 \left (c d -\frac {3 b e}{4}\right ) x \left (c x +b \right ) c \sqrt {d}\, \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {c \left (b e -c d \right )}}\right )+\sqrt {c \left (b e -c d \right )}\, \left (x \left (c x +b \right ) \left (b e -4 c d \right ) \operatorname {arctanh}\left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right )+\sqrt {d}\, \sqrt {e x +d}\, b \left (2 c x +b \right )\right )}{\sqrt {d}\, \sqrt {c \left (b e -c d \right )}\, b^{3} x \left (c x +b \right )}\)	\(133\)
derivativedivides	\(2 e^{3} \left (\frac {-\frac {b \sqrt {e x +d}}{2 x}-\frac {\left (b e -4 c d \right ) \operatorname {arctanh}\left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right )}{2 \sqrt {d}}}{b^{3} e^{3}}-\frac {c \left (\frac {b e \sqrt {e x +d}}{2 \left (e x +d \right ) c +2 b e -2 c d}+\frac {\left (3 b e -4 c d \right ) \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {c \left (b e -c d \right )}}\right )}{2 \sqrt {c \left (b e -c d \right )}}\right )}{b^{3} e^{3}}\right )\)	\(136\)
default	\(2 e^{3} \left (\frac {-\frac {b \sqrt {e x +d}}{2 x}-\frac {\left (b e -4 c d \right ) \operatorname {arctanh}\left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right )}{2 \sqrt {d}}}{b^{3} e^{3}}-\frac {c \left (\frac {b e \sqrt {e x +d}}{2 \left (e x +d \right ) c +2 b e -2 c d}+\frac {\left (3 b e -4 c d \right ) \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {c \left (b e -c d \right )}}\right )}{2 \sqrt {c \left (b e -c d \right )}}\right )}{b^{3} e^{3}}\right )\)	\(136\)
risch	\(-\frac {\sqrt {e x +d}}{b^{2} x}-\frac {e \left (-\frac {\left (-b e +4 c d \right ) \operatorname {arctanh}\left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right )}{b e \sqrt {d}}+\frac {2 c \left (\frac {b e \sqrt {e x +d}}{2 \left (e x +d \right ) c +2 b e -2 c d}+\frac {\left (3 b e -4 c d \right ) \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {c \left (b e -c d \right )}}\right )}{2 \sqrt {c \left (b e -c d \right )}}\right )}{b e}\right )}{b^{2}}\)	\(139\)

Input:

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

Output:

-1/d^(1/2)/(c*(b*e-c*d))^(1/2)*(-4*(c*d-3/4*b*e)*x*(c*x+b)*c*d^(1/2)*arcta 
n(c*(e*x+d)^(1/2)/(c*(b*e-c*d))^(1/2))+(c*(b*e-c*d))^(1/2)*(x*(c*x+b)*(b*e 
-4*c*d)*arctanh((e*x+d)^(1/2)/d^(1/2))+d^(1/2)*(e*x+d)^(1/2)*b*(2*c*x+b))) 
/b^3/x/(c*x+b)
 

Fricas [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 756, normalized size of antiderivative = 5.40 \[ \int \frac {\sqrt {d+e x}}{\left (b x+c x^2\right )^2} \, dx =\text {Too large to display} \] Input:

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

Output:

[-1/2*(((4*c^2*d^2 - 3*b*c*d*e)*x^2 + (4*b*c*d^2 - 3*b^2*d*e)*x)*sqrt(c/(c 
*d - b*e))*log((c*e*x + 2*c*d - b*e + 2*(c*d - b*e)*sqrt(e*x + d)*sqrt(c/( 
c*d - b*e)))/(c*x + b)) + ((4*c^2*d - b*c*e)*x^2 + (4*b*c*d - b^2*e)*x)*sq 
rt(d)*log((e*x - 2*sqrt(e*x + d)*sqrt(d) + 2*d)/x) + 2*(2*b*c*d*x + b^2*d) 
*sqrt(e*x + d))/(b^3*c*d*x^2 + b^4*d*x), 1/2*(2*((4*c^2*d^2 - 3*b*c*d*e)*x 
^2 + (4*b*c*d^2 - 3*b^2*d*e)*x)*sqrt(-c/(c*d - b*e))*arctan(sqrt(e*x + d)* 
sqrt(-c/(c*d - b*e))) - ((4*c^2*d - b*c*e)*x^2 + (4*b*c*d - b^2*e)*x)*sqrt 
(d)*log((e*x - 2*sqrt(e*x + d)*sqrt(d) + 2*d)/x) - 2*(2*b*c*d*x + b^2*d)*s 
qrt(e*x + d))/(b^3*c*d*x^2 + b^4*d*x), -1/2*(2*((4*c^2*d - b*c*e)*x^2 + (4 
*b*c*d - b^2*e)*x)*sqrt(-d)*arctan(sqrt(-d)/sqrt(e*x + d)) + ((4*c^2*d^2 - 
 3*b*c*d*e)*x^2 + (4*b*c*d^2 - 3*b^2*d*e)*x)*sqrt(c/(c*d - b*e))*log((c*e* 
x + 2*c*d - b*e + 2*(c*d - b*e)*sqrt(e*x + d)*sqrt(c/(c*d - b*e)))/(c*x + 
b)) + 2*(2*b*c*d*x + b^2*d)*sqrt(e*x + d))/(b^3*c*d*x^2 + b^4*d*x), (((4*c 
^2*d^2 - 3*b*c*d*e)*x^2 + (4*b*c*d^2 - 3*b^2*d*e)*x)*sqrt(-c/(c*d - b*e))* 
arctan(sqrt(e*x + d)*sqrt(-c/(c*d - b*e))) - ((4*c^2*d - b*c*e)*x^2 + (4*b 
*c*d - b^2*e)*x)*sqrt(-d)*arctan(sqrt(-d)/sqrt(e*x + d)) - (2*b*c*d*x + b^ 
2*d)*sqrt(e*x + d))/(b^3*c*d*x^2 + b^4*d*x)]
                                                                                    
                                                                                    
 

Sympy [F]

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

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

Output:

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

Maxima [F(-2)]

Exception generated. \[ \int \frac {\sqrt {d+e x}}{\left (b x+c x^2\right )^2} \, dx=\text {Exception raised: ValueError} \] Input:

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

Output:

Exception raised: ValueError >> Computation failed since Maxima requested 
additional constraints; using the 'assume' command before evaluation *may* 
 help (example of legal syntax is 'assume(b*e-c*d>0)', see `assume?` for m 
ore detail
 

Giac [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 166, normalized size of antiderivative = 1.19 \[ \int \frac {\sqrt {d+e x}}{\left (b x+c x^2\right )^2} \, dx=\frac {{\left (4 \, c^{2} d - 3 \, b c e\right )} \arctan \left (\frac {\sqrt {e x + d} c}{\sqrt {-c^{2} d + b c e}}\right )}{\sqrt {-c^{2} d + b c e} b^{3}} - \frac {{\left (4 \, c d - b e\right )} \arctan \left (\frac {\sqrt {e x + d}}{\sqrt {-d}}\right )}{b^{3} \sqrt {-d}} - \frac {2 \, {\left (e x + d\right )}^{\frac {3}{2}} c e - 2 \, \sqrt {e x + d} c d e + \sqrt {e x + d} b e^{2}}{{\left ({\left (e x + d\right )}^{2} c - 2 \, {\left (e x + d\right )} c d + c d^{2} + {\left (e x + d\right )} b e - b d e\right )} b^{2}} \] Input:

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

Output:

(4*c^2*d - 3*b*c*e)*arctan(sqrt(e*x + d)*c/sqrt(-c^2*d + b*c*e))/(sqrt(-c^ 
2*d + b*c*e)*b^3) - (4*c*d - b*e)*arctan(sqrt(e*x + d)/sqrt(-d))/(b^3*sqrt 
(-d)) - (2*(e*x + d)^(3/2)*c*e - 2*sqrt(e*x + d)*c*d*e + sqrt(e*x + d)*b*e 
^2)/(((e*x + d)^2*c - 2*(e*x + d)*c*d + c*d^2 + (e*x + d)*b*e - b*d*e)*b^2 
)
 

Mupad [B] (verification not implemented)

Time = 0.22 (sec) , antiderivative size = 1174, normalized size of antiderivative = 8.39 \[ \int \frac {\sqrt {d+e x}}{\left (b x+c x^2\right )^2} \, dx=\text {Too large to display} \] Input:

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

Output:

(atanh((2*c^2*e^6*(d + e*x)^(1/2))/(d^(3/2)*((8*c^3*e^5)/b - (2*c^2*e^6)/d 
)) - (8*c^3*e^5*(d + e*x)^(1/2))/(d^(1/2)*(8*c^3*e^5 - (2*b*c^2*e^6)/d)))* 
(b*e - 4*c*d))/(b^3*d^(1/2)) - ((2*c*e*(d + e*x)^(3/2))/b^2 + (e*(b*e - 2* 
c*d)*(d + e*x)^(1/2))/b^2)/((b*e - 2*c*d)*(d + e*x) + c*(d + e*x)^2 + c*d^ 
2 - b*d*e) + (atan(((((4*(d + e*x)^(1/2)*(5*b^2*c^3*e^4 + 16*c^5*d^2*e^2 - 
 16*b*c^4*d*e^3))/b^4 - ((-c*(b*e - c*d))^(1/2)*(3*b*e - 4*c*d)*((2*(2*b^7 
*c^2*e^4 - 4*b^6*c^3*d*e^3))/b^6 - (2*(2*b^7*c^2*e^3 - 4*b^6*c^3*d*e^2)*(- 
c*(b*e - c*d))^(1/2)*(3*b*e - 4*c*d)*(d + e*x)^(1/2))/(b^4*(b^4*e - b^3*c* 
d))))/(2*(b^4*e - b^3*c*d)))*(-c*(b*e - c*d))^(1/2)*(3*b*e - 4*c*d)*1i)/(2 
*(b^4*e - b^3*c*d)) + (((4*(d + e*x)^(1/2)*(5*b^2*c^3*e^4 + 16*c^5*d^2*e^2 
 - 16*b*c^4*d*e^3))/b^4 + ((-c*(b*e - c*d))^(1/2)*(3*b*e - 4*c*d)*((2*(2*b 
^7*c^2*e^4 - 4*b^6*c^3*d*e^3))/b^6 + (2*(2*b^7*c^2*e^3 - 4*b^6*c^3*d*e^2)* 
(-c*(b*e - c*d))^(1/2)*(3*b*e - 4*c*d)*(d + e*x)^(1/2))/(b^4*(b^4*e - b^3* 
c*d))))/(2*(b^4*e - b^3*c*d)))*(-c*(b*e - c*d))^(1/2)*(3*b*e - 4*c*d)*1i)/ 
(2*(b^4*e - b^3*c*d)))/((4*(3*b^2*c^3*e^5 + 16*c^5*d^2*e^3 - 16*b*c^4*d*e^ 
4))/b^6 - (((4*(d + e*x)^(1/2)*(5*b^2*c^3*e^4 + 16*c^5*d^2*e^2 - 16*b*c^4* 
d*e^3))/b^4 - ((-c*(b*e - c*d))^(1/2)*(3*b*e - 4*c*d)*((2*(2*b^7*c^2*e^4 - 
 4*b^6*c^3*d*e^3))/b^6 - (2*(2*b^7*c^2*e^3 - 4*b^6*c^3*d*e^2)*(-c*(b*e - c 
*d))^(1/2)*(3*b*e - 4*c*d)*(d + e*x)^(1/2))/(b^4*(b^4*e - b^3*c*d))))/(2*( 
b^4*e - b^3*c*d)))*(-c*(b*e - c*d))^(1/2)*(3*b*e - 4*c*d))/(2*(b^4*e - ...
 

Reduce [B] (verification not implemented)

Time = 0.25 (sec) , antiderivative size = 550, normalized size of antiderivative = 3.93 \[ \int \frac {\sqrt {d+e x}}{\left (b x+c x^2\right )^2} \, dx=\frac {-6 \sqrt {c}\, \sqrt {b e -c d}\, \mathit {atan} \left (\frac {\sqrt {e x +d}\, c}{\sqrt {c}\, \sqrt {b e -c d}}\right ) b^{2} d e x +8 \sqrt {c}\, \sqrt {b e -c d}\, \mathit {atan} \left (\frac {\sqrt {e x +d}\, c}{\sqrt {c}\, \sqrt {b e -c d}}\right ) b c \,d^{2} x -6 \sqrt {c}\, \sqrt {b e -c d}\, \mathit {atan} \left (\frac {\sqrt {e x +d}\, c}{\sqrt {c}\, \sqrt {b e -c d}}\right ) b c d e \,x^{2}+8 \sqrt {c}\, \sqrt {b e -c d}\, \mathit {atan} \left (\frac {\sqrt {e x +d}\, c}{\sqrt {c}\, \sqrt {b e -c d}}\right ) c^{2} d^{2} x^{2}-2 \sqrt {e x +d}\, b^{3} d e +2 \sqrt {e x +d}\, b^{2} c \,d^{2}-4 \sqrt {e x +d}\, b^{2} c d e x +4 \sqrt {e x +d}\, b \,c^{2} d^{2} x +\sqrt {d}\, \mathrm {log}\left (\sqrt {e x +d}-\sqrt {d}\right ) b^{3} e^{2} x -5 \sqrt {d}\, \mathrm {log}\left (\sqrt {e x +d}-\sqrt {d}\right ) b^{2} c d e x +\sqrt {d}\, \mathrm {log}\left (\sqrt {e x +d}-\sqrt {d}\right ) b^{2} c \,e^{2} x^{2}+4 \sqrt {d}\, \mathrm {log}\left (\sqrt {e x +d}-\sqrt {d}\right ) b \,c^{2} d^{2} x -5 \sqrt {d}\, \mathrm {log}\left (\sqrt {e x +d}-\sqrt {d}\right ) b \,c^{2} d e \,x^{2}+4 \sqrt {d}\, \mathrm {log}\left (\sqrt {e x +d}-\sqrt {d}\right ) c^{3} d^{2} x^{2}-\sqrt {d}\, \mathrm {log}\left (\sqrt {e x +d}+\sqrt {d}\right ) b^{3} e^{2} x +5 \sqrt {d}\, \mathrm {log}\left (\sqrt {e x +d}+\sqrt {d}\right ) b^{2} c d e x -\sqrt {d}\, \mathrm {log}\left (\sqrt {e x +d}+\sqrt {d}\right ) b^{2} c \,e^{2} x^{2}-4 \sqrt {d}\, \mathrm {log}\left (\sqrt {e x +d}+\sqrt {d}\right ) b \,c^{2} d^{2} x +5 \sqrt {d}\, \mathrm {log}\left (\sqrt {e x +d}+\sqrt {d}\right ) b \,c^{2} d e \,x^{2}-4 \sqrt {d}\, \mathrm {log}\left (\sqrt {e x +d}+\sqrt {d}\right ) c^{3} d^{2} x^{2}}{2 b^{3} d x \left (b c e x -c^{2} d x +b^{2} e -b c d \right )} \] Input:

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

Output:

( - 6*sqrt(c)*sqrt(b*e - c*d)*atan((sqrt(d + e*x)*c)/(sqrt(c)*sqrt(b*e - c 
*d)))*b**2*d*e*x + 8*sqrt(c)*sqrt(b*e - c*d)*atan((sqrt(d + e*x)*c)/(sqrt( 
c)*sqrt(b*e - c*d)))*b*c*d**2*x - 6*sqrt(c)*sqrt(b*e - c*d)*atan((sqrt(d + 
 e*x)*c)/(sqrt(c)*sqrt(b*e - c*d)))*b*c*d*e*x**2 + 8*sqrt(c)*sqrt(b*e - c* 
d)*atan((sqrt(d + e*x)*c)/(sqrt(c)*sqrt(b*e - c*d)))*c**2*d**2*x**2 - 2*sq 
rt(d + e*x)*b**3*d*e + 2*sqrt(d + e*x)*b**2*c*d**2 - 4*sqrt(d + e*x)*b**2* 
c*d*e*x + 4*sqrt(d + e*x)*b*c**2*d**2*x + sqrt(d)*log(sqrt(d + e*x) - sqrt 
(d))*b**3*e**2*x - 5*sqrt(d)*log(sqrt(d + e*x) - sqrt(d))*b**2*c*d*e*x + s 
qrt(d)*log(sqrt(d + e*x) - sqrt(d))*b**2*c*e**2*x**2 + 4*sqrt(d)*log(sqrt( 
d + e*x) - sqrt(d))*b*c**2*d**2*x - 5*sqrt(d)*log(sqrt(d + e*x) - sqrt(d)) 
*b*c**2*d*e*x**2 + 4*sqrt(d)*log(sqrt(d + e*x) - sqrt(d))*c**3*d**2*x**2 - 
 sqrt(d)*log(sqrt(d + e*x) + sqrt(d))*b**3*e**2*x + 5*sqrt(d)*log(sqrt(d + 
 e*x) + sqrt(d))*b**2*c*d*e*x - sqrt(d)*log(sqrt(d + e*x) + sqrt(d))*b**2* 
c*e**2*x**2 - 4*sqrt(d)*log(sqrt(d + e*x) + sqrt(d))*b*c**2*d**2*x + 5*sqr 
t(d)*log(sqrt(d + e*x) + sqrt(d))*b*c**2*d*e*x**2 - 4*sqrt(d)*log(sqrt(d + 
 e*x) + sqrt(d))*c**3*d**2*x**2)/(2*b**3*d*x*(b**2*e - b*c*d + b*c*e*x - c 
**2*d*x))