\(\int \frac {1}{\sqrt {d+e x} (b x+c x^2)^2} \, dx\) [118]

Optimal result

Integrand size = 21, antiderivative size = 164 \[ \int \frac {1}{\sqrt {d+e x} \left (b x+c x^2\right )^2} \, dx=-\frac {c (2 c d-b e) \sqrt {d+e x}}{b^2 d (c d-b e) (b+c x)}-\frac {\sqrt {d+e x}}{b d x (b+c x)}+\frac {(4 c d+b e) \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{b^3 d^{3/2}}-\frac {c^{3/2} (4 c d-5 b e) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt {c d-b e}}\right )}{b^3 (c d-b e)^{3/2}} \] Output:

-c*(-b*e+2*c*d)*(e*x+d)^(1/2)/b^2/d/(-b*e+c*d)/(c*x+b)-(e*x+d)^(1/2)/b/d/x 
/(c*x+b)+(b*e+4*c*d)*arctanh((e*x+d)^(1/2)/d^(1/2))/b^3/d^(3/2)-c^(3/2)*(- 
5*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 
)^(3/2)
 

Mathematica [A] (verified)

Time = 0.77 (sec) , antiderivative size = 149, normalized size of antiderivative = 0.91 \[ \int \frac {1}{\sqrt {d+e x} \left (b x+c x^2\right )^2} \, dx=\frac {-\frac {b \sqrt {d+e x} \left (-b c d+b^2 e-2 c^2 d x+b c e x\right )}{d (-c d+b e) x (b+c x)}-\frac {c^{3/2} (4 c d-5 b e) \arctan \left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt {-c d+b e}}\right )}{(-c d+b e)^{3/2}}+\frac {(4 c d+b e) \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{d^{3/2}}}{b^3} \] Input:

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

Output:

(-((b*Sqrt[d + e*x]*(-(b*c*d) + b^2*e - 2*c^2*d*x + b*c*e*x))/(d*(-(c*d) + 
 b*e)*x*(b + c*x))) - (c^(3/2)*(4*c*d - 5*b*e)*ArcTan[(Sqrt[c]*Sqrt[d + e* 
x])/Sqrt[-(c*d) + b*e]])/(-(c*d) + b*e)^(3/2) + ((4*c*d + b*e)*ArcTanh[Sqr 
t[d + e*x]/Sqrt[d]])/d^(3/2))/b^3
 

Rubi [A] (verified)

Time = 0.72 (sec) , antiderivative size = 189, normalized size of antiderivative = 1.15, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {1165, 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.

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

\(\Big \downarrow \) 1165

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1197

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1480

\(\displaystyle -\frac {e \left (\frac {c (c d-b e) (b e+4 c d) \int \frac {1}{c (d+e x)-c d}d\sqrt {d+e x}}{b e}-\frac {c^2 d (4 c d-5 b e) \int \frac {1}{-c d+b e+c (d+e x)}d\sqrt {d+e x}}{b e}\right )}{b^2 d (c d-b e)}-\frac {\sqrt {d+e x} (c x (2 c d-b e)+b (c d-b e))}{b^2 d \left (b x+c x^2\right ) (c d-b e)}\)

\(\Big \downarrow \) 221

\(\displaystyle -\frac {e \left (\frac {c^{3/2} d (4 c d-5 b e) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt {c d-b e}}\right )}{b e \sqrt {c d-b e}}-\frac {\text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right ) (c d-b e) (b e+4 c d)}{b \sqrt {d} e}\right )}{b^2 d (c d-b e)}-\frac {\sqrt {d+e x} (c x (2 c d-b e)+b (c d-b e))}{b^2 d \left (b x+c x^2\right ) (c d-b e)}\)

Input:

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

Output:

-((Sqrt[d + e*x]*(b*(c*d - b*e) + c*(2*c*d - b*e)*x))/(b^2*d*(c*d - b*e)*( 
b*x + c*x^2))) - (e*(-(((c*d - b*e)*(4*c*d + b*e)*ArcTanh[Sqrt[d + e*x]/Sq 
rt[d]])/(b*Sqrt[d]*e)) + (c^(3/2)*d*(4*c*d - 5*b*e)*ArcTanh[(Sqrt[c]*Sqrt[ 
d + e*x])/Sqrt[c*d - b*e]])/(b*e*Sqrt[c*d - b*e])))/(b^2*d*(c*d - b*e))
 

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 1165

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S 
ymbol] :> Simp[(d + e*x)^(m + 1)*(b*c*d - b^2*e + 2*a*c*e + c*(2*c*d - b*e) 
*x)*((a + b*x + c*x^2)^(p + 1)/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^ 
2))), x] + Simp[1/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2))   Int[(d 
+ e*x)^m*Simp[b*c*d*e*(2*p - m + 2) + b^2*e^2*(m + p + 2) - 2*c^2*d^2*(2*p 
+ 3) - 2*a*c*e^2*(m + 2*p + 3) - c*e*(2*c*d - b*e)*(m + 2*p + 4)*x, x]*(a + 
 b*x + c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, m}, x] && LtQ[p, -1] 
 && 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 = 0.60 (sec) , antiderivative size = 160, normalized size of antiderivative = 0.98

method	result	size
derivativedivides	\(2 e^{3} \left (\frac {c^{2} \left (\frac {b e \sqrt {e x +d}}{2 \left (b e -c d \right ) \left (\left (e x +d \right ) c +b e -c d \right )}+\frac {\left (5 b e -4 c d \right ) \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {c \left (b e -c d \right )}}\right )}{2 \left (b e -c d \right ) \sqrt {c \left (b e -c d \right )}}\right )}{b^{3} e^{3}}+\frac {-\frac {b \sqrt {e x +d}}{2 d x}+\frac {\left (b e +4 c d \right ) \operatorname {arctanh}\left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right )}{2 d^{\frac {3}{2}}}}{b^{3} e^{3}}\right )\)	\(160\)
default	\(2 e^{3} \left (\frac {c^{2} \left (\frac {b e \sqrt {e x +d}}{2 \left (b e -c d \right ) \left (\left (e x +d \right ) c +b e -c d \right )}+\frac {\left (5 b e -4 c d \right ) \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {c \left (b e -c d \right )}}\right )}{2 \left (b e -c d \right ) \sqrt {c \left (b e -c d \right )}}\right )}{b^{3} e^{3}}+\frac {-\frac {b \sqrt {e x +d}}{2 d x}+\frac {\left (b e +4 c d \right ) \operatorname {arctanh}\left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right )}{2 d^{\frac {3}{2}}}}{b^{3} e^{3}}\right )\)	\(160\)
risch	\(-\frac {\sqrt {e x +d}}{d \,b^{2} x}-\frac {e \left (-\frac {\left (b e +4 c d \right ) \operatorname {arctanh}\left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right )}{b e \sqrt {d}}-\frac {2 d \,c^{2} \left (\frac {b e \sqrt {e x +d}}{2 \left (b e -c d \right ) \left (\left (e x +d \right ) c +b e -c d \right )}+\frac {\left (5 b e -4 c d \right ) \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {c \left (b e -c d \right )}}\right )}{2 \left (b e -c d \right ) \sqrt {c \left (b e -c d \right )}}\right )}{b e}\right )}{b^{2} d}\)	\(167\)
pseudoelliptic	\(-\frac {4 \left (\left (c d -\frac {5 b e}{4}\right ) x \left (c x +b \right ) c^{2} d^{\frac {5}{2}} \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {c \left (b e -c d \right )}}\right )-\frac {\sqrt {c \left (b e -c d \right )}\, \left (d x \left (b e +4 c d \right ) \left (b e -c d \right ) \left (c x +b \right ) \operatorname {arctanh}\left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right )+\sqrt {e x +d}\, b \,d^{\frac {3}{2}} \left (2 c^{2} d x +b \left (-e x +d \right ) c -e \,b^{2}\right )\right )}{4}\right )}{\sqrt {c \left (b e -c d \right )}\, d^{\frac {5}{2}} x \,b^{3} \left (b e -c d \right ) \left (c x +b \right )}\)	\(171\)

Input:

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

Output:

2*e^3*(c^2/b^3/e^3*(1/2*b*e/(b*e-c*d)*(e*x+d)^(1/2)/((e*x+d)*c+b*e-c*d)+1/ 
2*(5*b*e-4*c*d)/(b*e-c*d)/(c*(b*e-c*d))^(1/2)*arctan(c*(e*x+d)^(1/2)/(c*(b 
*e-c*d))^(1/2)))+1/b^3/e^3*(-1/2*b/d*(e*x+d)^(1/2)/x+1/2*(b*e+4*c*d)/d^(3/ 
2)*arctanh((e*x+d)^(1/2)/d^(1/2))))
 

Fricas [A] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 1120, normalized size of antiderivative = 6.83 \[ \int \frac {1}{\sqrt {d+e x} \left (b x+c x^2\right )^2} \, dx =\text {Too large to display} \] Input:

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F(-2)]

Exception generated. \[ \int \frac {1}{\sqrt {d+e x} \left (b x+c x^2\right )^2} \, dx=\text {Exception raised: ValueError} \] Input:

integrate(1/(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 = 0.19 (sec) , antiderivative size = 236, normalized size of antiderivative = 1.44 \[ \int \frac {1}{\sqrt {d+e x} \left (b x+c x^2\right )^2} \, dx=\frac {{\left (4 \, c^{3} d - 5 \, b c^{2} e\right )} \arctan \left (\frac {\sqrt {e x + d} c}{\sqrt {-c^{2} d + b c e}}\right )}{{\left (b^{3} c d - b^{4} e\right )} \sqrt {-c^{2} d + b c e}} - \frac {2 \, {\left (e x + d\right )}^{\frac {3}{2}} c^{2} d e - 2 \, \sqrt {e x + d} c^{2} d^{2} e - {\left (e x + d\right )}^{\frac {3}{2}} b c e^{2} + 2 \, \sqrt {e x + d} b c d e^{2} - \sqrt {e x + d} b^{2} e^{3}}{{\left (b^{2} c d^{2} - b^{3} d e\right )} {\left ({\left (e x + d\right )}^{2} c - 2 \, {\left (e x + d\right )} c d + c d^{2} + {\left (e x + d\right )} b e - b d e\right )}} - \frac {{\left (4 \, c d + b e\right )} \arctan \left (\frac {\sqrt {e x + d}}{\sqrt {-d}}\right )}{b^{3} \sqrt {-d} d} \] Input:

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

Output:

(4*c^3*d - 5*b*c^2*e)*arctan(sqrt(e*x + d)*c/sqrt(-c^2*d + b*c*e))/((b^3*c 
*d - b^4*e)*sqrt(-c^2*d + b*c*e)) - (2*(e*x + d)^(3/2)*c^2*d*e - 2*sqrt(e* 
x + d)*c^2*d^2*e - (e*x + d)^(3/2)*b*c*e^2 + 2*sqrt(e*x + d)*b*c*d*e^2 - s 
qrt(e*x + d)*b^2*e^3)/((b^2*c*d^2 - b^3*d*e)*((e*x + d)^2*c - 2*(e*x + d)* 
c*d + c*d^2 + (e*x + d)*b*e - b*d*e)) - (4*c*d + b*e)*arctan(sqrt(e*x + d) 
/sqrt(-d))/(b^3*sqrt(-d)*d)
 

Mupad [B] (verification not implemented)

Time = 5.90 (sec) , antiderivative size = 3784, normalized size of antiderivative = 23.07 \[ \int \frac {1}{\sqrt {d+e x} \left (b x+c x^2\right )^2} \, dx=\text {Too large to display} \] Input:

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

Output:

(((d + e*x)^(1/2)*(b^2*e^3 + 2*c^2*d^2*e - 2*b*c*d*e^2))/(b^2*(c*d^2 - b*d 
*e)) + (c*(b*e^2 - 2*c*d*e)*(d + e*x)^(3/2))/(b^2*(c*d^2 - b*d*e)))/((b*e 
- 2*c*d)*(d + e*x) + c*(d + e*x)^2 + c*d^2 - b*d*e) + (atan((((-c^3*(b*e - 
 c*d)^3)^(1/2)*((2*(d + e*x)^(1/2)*(b^4*c^3*e^6 + 32*c^7*d^4*e^2 - 64*b*c^ 
6*d^3*e^3 + 6*b^3*c^4*d*e^5 + 26*b^2*c^5*d^2*e^4))/(b^4*c^2*d^4 + b^6*d^2* 
e^2 - 2*b^5*c*d^3*e) - ((-c^3*(b*e - c*d)^3)^(1/2)*(5*b*e - 4*c*d)*((4*b^9 
*c^2*d*e^6 + 8*b^6*c^5*d^4*e^3 - 16*b^7*c^4*d^3*e^4 + 4*b^8*c^3*d^2*e^5)/( 
b^6*c^2*d^4 + b^8*d^2*e^2 - 2*b^7*c*d^3*e) + ((-c^3*(b*e - c*d)^3)^(1/2)*( 
5*b*e - 4*c*d)*(d + e*x)^(1/2)*(8*b^6*c^5*d^5*e^2 - 20*b^7*c^4*d^4*e^3 + 1 
6*b^8*c^3*d^3*e^4 - 4*b^9*c^2*d^2*e^5))/((b^4*c^2*d^4 + b^6*d^2*e^2 - 2*b^ 
5*c*d^3*e)*(b^6*e^3 - b^3*c^3*d^3 + 3*b^4*c^2*d^2*e - 3*b^5*c*d*e^2))))/(2 
*(b^6*e^3 - b^3*c^3*d^3 + 3*b^4*c^2*d^2*e - 3*b^5*c*d*e^2)))*(5*b*e - 4*c* 
d)*1i)/(2*(b^6*e^3 - b^3*c^3*d^3 + 3*b^4*c^2*d^2*e - 3*b^5*c*d*e^2)) + ((- 
c^3*(b*e - c*d)^3)^(1/2)*((2*(d + e*x)^(1/2)*(b^4*c^3*e^6 + 32*c^7*d^4*e^2 
 - 64*b*c^6*d^3*e^3 + 6*b^3*c^4*d*e^5 + 26*b^2*c^5*d^2*e^4))/(b^4*c^2*d^4 
+ b^6*d^2*e^2 - 2*b^5*c*d^3*e) + ((-c^3*(b*e - c*d)^3)^(1/2)*(5*b*e - 4*c* 
d)*((4*b^9*c^2*d*e^6 + 8*b^6*c^5*d^4*e^3 - 16*b^7*c^4*d^3*e^4 + 4*b^8*c^3* 
d^2*e^5)/(b^6*c^2*d^4 + b^8*d^2*e^2 - 2*b^7*c*d^3*e) - ((-c^3*(b*e - c*d)^ 
3)^(1/2)*(5*b*e - 4*c*d)*(d + e*x)^(1/2)*(8*b^6*c^5*d^5*e^2 - 20*b^7*c^4*d 
^4*e^3 + 16*b^8*c^3*d^3*e^4 - 4*b^9*c^2*d^2*e^5))/((b^4*c^2*d^4 + b^6*d...
 

Reduce [B] (verification not implemented)

Time = 0.24 (sec) , antiderivative size = 743, normalized size of antiderivative = 4.53 \[ \int \frac {1}{\sqrt {d+e x} \left (b x+c x^2\right )^2} \, dx =\text {Too large to display} \] Input:

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

Output:

(10*sqrt(c)*sqrt(b*e - c*d)*atan((sqrt(d + e*x)*c)/(sqrt(c)*sqrt(b*e - c*d 
)))*b**2*c*d**2*e*x - 8*sqrt(c)*sqrt(b*e - c*d)*atan((sqrt(d + e*x)*c)/(sq 
rt(c)*sqrt(b*e - c*d)))*b*c**2*d**3*x + 10*sqrt(c)*sqrt(b*e - c*d)*atan((s 
qrt(d + e*x)*c)/(sqrt(c)*sqrt(b*e - c*d)))*b*c**2*d**2*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**3*d** 
3*x**2 - 2*sqrt(d + e*x)*b**4*d*e**2 + 4*sqrt(d + e*x)*b**3*c*d**2*e - 2*s 
qrt(d + e*x)*b**3*c*d*e**2*x - 2*sqrt(d + e*x)*b**2*c**2*d**3 + 6*sqrt(d + 
 e*x)*b**2*c**2*d**2*e*x - 4*sqrt(d + e*x)*b*c**3*d**3*x - sqrt(d)*log(sqr 
t(d + e*x) - sqrt(d))*b**4*e**3*x - 2*sqrt(d)*log(sqrt(d + e*x) - sqrt(d)) 
*b**3*c*d*e**2*x - sqrt(d)*log(sqrt(d + e*x) - sqrt(d))*b**3*c*e**3*x**2 + 
 7*sqrt(d)*log(sqrt(d + e*x) - sqrt(d))*b**2*c**2*d**2*e*x - 2*sqrt(d)*log 
(sqrt(d + e*x) - sqrt(d))*b**2*c**2*d*e**2*x**2 - 4*sqrt(d)*log(sqrt(d + e 
*x) - sqrt(d))*b*c**3*d**3*x + 7*sqrt(d)*log(sqrt(d + e*x) - sqrt(d))*b*c* 
*3*d**2*e*x**2 - 4*sqrt(d)*log(sqrt(d + e*x) - sqrt(d))*c**4*d**3*x**2 + s 
qrt(d)*log(sqrt(d + e*x) + sqrt(d))*b**4*e**3*x + 2*sqrt(d)*log(sqrt(d + e 
*x) + sqrt(d))*b**3*c*d*e**2*x + sqrt(d)*log(sqrt(d + e*x) + sqrt(d))*b**3 
*c*e**3*x**2 - 7*sqrt(d)*log(sqrt(d + e*x) + sqrt(d))*b**2*c**2*d**2*e*x + 
 2*sqrt(d)*log(sqrt(d + e*x) + sqrt(d))*b**2*c**2*d*e**2*x**2 + 4*sqrt(d)* 
log(sqrt(d + e*x) + sqrt(d))*b*c**3*d**3*x - 7*sqrt(d)*log(sqrt(d + e*x) + 
 sqrt(d))*b*c**3*d**2*e*x**2 + 4*sqrt(d)*log(sqrt(d + e*x) + sqrt(d))*c...