\(\int \frac {\sqrt {d+e x}}{(a+b x+c x^2)^2} \, dx\) [542]

Optimal result

Integrand size = 22, antiderivative size = 287 \[ \int \frac {\sqrt {d+e x}}{\left (a+b x+c x^2\right )^2} \, dx=-\frac {(b+2 c x) \sqrt {d+e x}}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}+\frac {\sqrt {2} \sqrt {c} \left (4 c d-\left (2 b-\sqrt {b^2-4 a c}\right ) e\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}}\right )}{\left (b^2-4 a c\right )^{3/2} \sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}}-\frac {\sqrt {2} \sqrt {c} \left (4 c d-\left (2 b+\sqrt {b^2-4 a c}\right ) e\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}}\right )}{\left (b^2-4 a c\right )^{3/2} \sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \] Output:

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

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 1.85 (sec) , antiderivative size = 314, normalized size of antiderivative = 1.09 \[ \int \frac {\sqrt {d+e x}}{\left (a+b x+c x^2\right )^2} \, dx=-\frac {\frac {(b+2 c x) \sqrt {d+e x}}{a+x (b+c x)}+\frac {\sqrt {2} \sqrt {c} \left (-4 i c d+\left (2 i b+\sqrt {-b^2+4 a c}\right ) e\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {-2 c d+b e-i \sqrt {-b^2+4 a c} e}}\right )}{\sqrt {-b^2+4 a c} \sqrt {-2 c d+\left (b-i \sqrt {-b^2+4 a c}\right ) e}}+\frac {\sqrt {2} \sqrt {c} \left (4 i c d+\left (-2 i b+\sqrt {-b^2+4 a c}\right ) e\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {-2 c d+b e+i \sqrt {-b^2+4 a c} e}}\right )}{\sqrt {-b^2+4 a c} \sqrt {-2 c d+\left (b+i \sqrt {-b^2+4 a c}\right ) e}}}{b^2-4 a c} \] Input:

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

Output:

-((((b + 2*c*x)*Sqrt[d + e*x])/(a + x*(b + c*x)) + (Sqrt[2]*Sqrt[c]*((-4*I 
)*c*d + ((2*I)*b + Sqrt[-b^2 + 4*a*c])*e)*ArcTan[(Sqrt[2]*Sqrt[c]*Sqrt[d + 
 e*x])/Sqrt[-2*c*d + b*e - I*Sqrt[-b^2 + 4*a*c]*e]])/(Sqrt[-b^2 + 4*a*c]*S 
qrt[-2*c*d + (b - I*Sqrt[-b^2 + 4*a*c])*e]) + (Sqrt[2]*Sqrt[c]*((4*I)*c*d 
+ ((-2*I)*b + Sqrt[-b^2 + 4*a*c])*e)*ArcTan[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x] 
)/Sqrt[-2*c*d + b*e + I*Sqrt[-b^2 + 4*a*c]*e]])/(Sqrt[-b^2 + 4*a*c]*Sqrt[- 
2*c*d + (b + I*Sqrt[-b^2 + 4*a*c])*e]))/(b^2 - 4*a*c))
 

Rubi [A] (verified)

Time = 0.54 (sec) , antiderivative size = 283, normalized size of antiderivative = 0.99, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, 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.

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

\(\Big \downarrow \) 1163

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1197

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1480

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

\(\Big \downarrow \) 221

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

Input:

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

Output:

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

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 = 1.71 (sec) , antiderivative size = 419, normalized size of antiderivative = 1.46

method	result	size
derivativedivides	\(32 e^{3} c^{2} \left (\frac {-\frac {\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}\, \sqrt {e x +d}}{8 c \left (-e x -\frac {b e}{2 c}-\frac {\sqrt {\left (-4 a c +b^{2}\right ) e^{2}}}{2 c}\right )}+\frac {\left (b e -2 c d +\frac {\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}{2}\right ) \sqrt {2}\, \arctan \left (\frac {\sqrt {e x +d}\, c \sqrt {2}}{\sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{4 \sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}}{4 c \,e^{2} \left (4 a c -b^{2}\right ) \sqrt {-e^{2} \left (4 a c -b^{2}\right )}}+\frac {-\frac {\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}\, \sqrt {e x +d}}{8 c \left (-e x -\frac {b e}{2 c}+\frac {\sqrt {\left (-4 a c +b^{2}\right ) e^{2}}}{2 c}\right )}-\frac {\left (-b e +2 c d +\frac {\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}{2}\right ) \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {e x +d}\, c \sqrt {2}}{\sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{4 \sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}}{4 c \,e^{2} \left (4 a c -b^{2}\right ) \sqrt {-e^{2} \left (4 a c -b^{2}\right )}}\right )\)	\(419\)
default	\(32 e^{3} c^{2} \left (\frac {-\frac {\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}\, \sqrt {e x +d}}{8 c \left (-e x -\frac {b e}{2 c}-\frac {\sqrt {\left (-4 a c +b^{2}\right ) e^{2}}}{2 c}\right )}+\frac {\left (b e -2 c d +\frac {\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}{2}\right ) \sqrt {2}\, \arctan \left (\frac {\sqrt {e x +d}\, c \sqrt {2}}{\sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{4 \sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}}{4 c \,e^{2} \left (4 a c -b^{2}\right ) \sqrt {-e^{2} \left (4 a c -b^{2}\right )}}+\frac {-\frac {\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}\, \sqrt {e x +d}}{8 c \left (-e x -\frac {b e}{2 c}+\frac {\sqrt {\left (-4 a c +b^{2}\right ) e^{2}}}{2 c}\right )}-\frac {\left (-b e +2 c d +\frac {\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}}{2}\right ) \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {e x +d}\, c \sqrt {2}}{\sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{4 \sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}}{4 c \,e^{2} \left (4 a c -b^{2}\right ) \sqrt {-e^{2} \left (4 a c -b^{2}\right )}}\right )\)	\(419\)
pseudoelliptic	\(\frac {e \left (-\sqrt {\left (b e -2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) c}\, \left (-e^{2} \left (2 c x +b \right )^{2} \sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}-16 c^{3} d \,e^{2} x^{2}-16 \left (-\frac {b e \,x^{2}}{2}+d \left (b x +a \right )\right ) e^{2} c^{2}+8 b \,e^{3} \left (b x +a \right ) c +\left (-e^{2} \left (4 a c -b^{2}\right )\right )^{\frac {3}{2}}\right ) \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {e x +d}\, c \sqrt {2}}{\sqrt {\left (-b e +2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) c}}\right )+\sqrt {\left (-b e +2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) c}\, \left (\left (-e^{2} \left (2 c x +b \right )^{2} \sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}+16 c^{3} d \,e^{2} x^{2}+16 \left (-\frac {b e \,x^{2}}{2}+d \left (b x +a \right )\right ) e^{2} c^{2}-8 b \,e^{3} \left (b x +a \right ) c +\left (-e^{2} \left (4 a c -b^{2}\right )\right )^{\frac {3}{2}}\right ) \sqrt {2}\, \arctan \left (\frac {\sqrt {e x +d}\, c \sqrt {2}}{\sqrt {\left (b e -2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) c}}\right )-4 \sqrt {\left (b e -2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) c}\, \sqrt {e x +d}\, e \left (2 c x +b \right ) \sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right )\right ) c}{4 \sqrt {\left (b e -2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) c}\, \sqrt {\left (-b e +2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) c}\, \sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\, \left (\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}-e \left (2 c x +b \right )\right ) \left (\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}+e \left (2 c x +b \right )\right ) \left (a c -\frac {b^{2}}{4}\right )}\)	\(544\)

Input:

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

Output:

32*e^3*c^2*(1/4/c/e^2/(4*a*c-b^2)/(-e^2*(4*a*c-b^2))^(1/2)*(-1/8/c*(-4*a*c 
*e^2+b^2*e^2)^(1/2)*(e*x+d)^(1/2)/(-e*x-1/2*b/c*e-1/2/c*((-4*a*c+b^2)*e^2) 
^(1/2))+1/4*(b*e-2*c*d+1/2*(-4*a*c*e^2+b^2*e^2)^(1/2))*2^(1/2)/((b*e-2*c*d 
+(-e^2*(4*a*c-b^2))^(1/2))*c)^(1/2)*arctan((e*x+d)^(1/2)*c*2^(1/2)/((b*e-2 
*c*d+(-e^2*(4*a*c-b^2))^(1/2))*c)^(1/2)))+1/4/c/e^2/(4*a*c-b^2)/(-e^2*(4*a 
*c-b^2))^(1/2)*(-1/8/c*(-4*a*c*e^2+b^2*e^2)^(1/2)*(e*x+d)^(1/2)/(-e*x-1/2* 
b/c*e+1/2/c*((-4*a*c+b^2)*e^2)^(1/2))-1/4*(-b*e+2*c*d+1/2*(-4*a*c*e^2+b^2* 
e^2)^(1/2))*2^(1/2)/((-b*e+2*c*d+(-e^2*(4*a*c-b^2))^(1/2))*c)^(1/2)*arctan 
h((e*x+d)^(1/2)*c*2^(1/2)/((-b*e+2*c*d+(-e^2*(4*a*c-b^2))^(1/2))*c)^(1/2)) 
))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 6393 vs. \(2 (241) = 482\).

Time = 0.21 (sec) , antiderivative size = 6393, normalized size of antiderivative = 22.28 \[ \int \frac {\sqrt {d+e x}}{\left (a+b x+c x^2\right )^2} \, dx=\text {Too large to display} \] Input:

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

Output:

Too large to include
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1005 vs. \(2 (241) = 482\).

Time = 0.54 (sec) , antiderivative size = 1005, normalized size of antiderivative = 3.50 \[ \int \frac {\sqrt {d+e x}}{\left (a+b x+c x^2\right )^2} \, dx =\text {Too large to display} \] Input:

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

Output:

-(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 + a*e^2)*(b 
^2 - 4*a*c)) - 1/4*(sqrt(-4*c^2*d + 2*(b*c - sqrt(b^2 - 4*a*c)*c)*e)*(b^2* 
e - 4*a*c*e)^2*e + (2*sqrt(b^2 - 4*a*c)*c*d*e - sqrt(b^2 - 4*a*c)*b*e^2)*s 
qrt(-4*c^2*d + 2*(b*c - sqrt(b^2 - 4*a*c)*c)*e)*abs(b^2*e - 4*a*c*e) - 2*( 
4*(b^2*c^2 - 4*a*c^3)*d^2*e - 4*(b^3*c - 4*a*b*c^2)*d*e^2 + (b^4 - 4*a*b^2 
*c)*e^3)*sqrt(-4*c^2*d + 2*(b*c - sqrt(b^2 - 4*a*c)*c)*e))*arctan(2*sqrt(1 
/2)*sqrt(e*x + d)/sqrt(-(2*b^2*c*d - 8*a*c^2*d - b^3*e + 4*a*b*c*e + sqrt( 
(2*b^2*c*d - 8*a*c^2*d - b^3*e + 4*a*b*c*e)^2 - 4*(b^2*c*d^2 - 4*a*c^2*d^2 
 - b^3*d*e + 4*a*b*c*d*e + a*b^2*e^2 - 4*a^2*c*e^2)*(b^2*c - 4*a*c^2)))/(b 
^2*c - 4*a*c^2)))/(((b^2*c - 4*a*c^2)*sqrt(b^2 - 4*a*c)*d^2 - (b^3 - 4*a*b 
*c)*sqrt(b^2 - 4*a*c)*d*e + (a*b^2 - 4*a^2*c)*sqrt(b^2 - 4*a*c)*e^2)*abs(b 
^2*e - 4*a*c*e)*abs(c)) + 1/4*(sqrt(-4*c^2*d + 2*(b*c + sqrt(b^2 - 4*a*c)* 
c)*e)*(b^2*e - 4*a*c*e)^2*e - (2*sqrt(b^2 - 4*a*c)*c*d*e - sqrt(b^2 - 4*a* 
c)*b*e^2)*sqrt(-4*c^2*d + 2*(b*c + sqrt(b^2 - 4*a*c)*c)*e)*abs(b^2*e - 4*a 
*c*e) - 2*(4*(b^2*c^2 - 4*a*c^3)*d^2*e - 4*(b^3*c - 4*a*b*c^2)*d*e^2 + (b^ 
4 - 4*a*b^2*c)*e^3)*sqrt(-4*c^2*d + 2*(b*c + sqrt(b^2 - 4*a*c)*c)*e))*arct 
an(2*sqrt(1/2)*sqrt(e*x + d)/sqrt(-(2*b^2*c*d - 8*a*c^2*d - b^3*e + 4*a*b* 
c*e - sqrt((2*b^2*c*d - 8*a*c^2*d - b^3*e + 4*a*b*c*e)^2 - 4*(b^2*c*d^2 - 
4*a*c^2*d^2 - b^3*d*e + 4*a*b*c*d*e + a*b^2*e^2 - 4*a^2*c*e^2)*(b^2*c -...
 

Mupad [B] (verification not implemented)

Time = 9.74 (sec) , antiderivative size = 5740, normalized size of antiderivative = 20.00 \[ \int \frac {\sqrt {d+e x}}{\left (a+b x+c x^2\right )^2} \, dx=\text {Too large to display} \] Input:

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

Output:

((2*c*e*(d + e*x)^(3/2))/(4*a*c - b^2) + (e*(b*e - 2*c*d)*(d + e*x)^(1/2)) 
/(4*a*c - b^2))/((b*e - 2*c*d)*(d + e*x) + c*(d + e*x)^2 + a*e^2 + c*d^2 - 
 b*d*e) - log(((4*c^2*e^3*(b*e - 2*c*d) - 8*c^2*e^2*(4*a*c - b^2)*(b*e - 2 
*c*d)*(d + e*x)^(1/2)*(((e^3*(-(4*a*c - b^2)^9)^(1/2))/8 - (b^9*e^3)/8 - 2 
56*a^3*c^6*d^3 + 4*b^6*c^3*d^3 - 48*a*b^4*c^4*d^3 + 96*a^4*b*c^4*e^3 - 192 
*a^4*c^5*d*e^2 - 6*b^7*c^2*d^2*e + 192*a^2*b^2*c^5*d^3 + 12*a^2*b^5*c^2*e^ 
3 - 64*a^3*b^3*c^3*e^3 + (9*b^8*c*d*e^2)/4 + 72*a*b^5*c^3*d^2*e - 24*a*b^6 
*c^2*d*e^2 + 384*a^3*b*c^5*d^2*e - 288*a^2*b^3*c^4*d^2*e + 72*a^2*b^4*c^3* 
d*e^2)/((4*a*c - b^2)^6*(a*e^2 + c*d^2 - b*d*e)))^(1/2))*(((e^3*(-(4*a*c - 
 b^2)^9)^(1/2))/8 - (b^9*e^3)/8 - 256*a^3*c^6*d^3 + 4*b^6*c^3*d^3 - 48*a*b 
^4*c^4*d^3 + 96*a^4*b*c^4*e^3 - 192*a^4*c^5*d*e^2 - 6*b^7*c^2*d^2*e + 192* 
a^2*b^2*c^5*d^3 + 12*a^2*b^5*c^2*e^3 - 64*a^3*b^3*c^3*e^3 + (9*b^8*c*d*e^2 
)/4 + 72*a*b^5*c^3*d^2*e - 24*a*b^6*c^2*d*e^2 + 384*a^3*b*c^5*d^2*e - 288* 
a^2*b^3*c^4*d^2*e + 72*a^2*b^4*c^3*d*e^2)/((4*a*c - b^2)^6*(a*e^2 + c*d^2 
- b*d*e)))^(1/2) + (4*c^3*e^2*(d + e*x)^(1/2)*(5*b^2*e^2 + 16*c^2*d^2 - 4* 
a*c*e^2 - 16*b*c*d*e))/(4*a*c - b^2)^2)*(((e^3*(-(4*a*c - b^2)^9)^(1/2))/8 
 - (b^9*e^3)/8 - 256*a^3*c^6*d^3 + 4*b^6*c^3*d^3 - 48*a*b^4*c^4*d^3 + 96*a 
^4*b*c^4*e^3 - 192*a^4*c^5*d*e^2 - 6*b^7*c^2*d^2*e + 192*a^2*b^2*c^5*d^3 + 
 12*a^2*b^5*c^2*e^3 - 64*a^3*b^3*c^3*e^3 + (9*b^8*c*d*e^2)/4 + 72*a*b^5*c^ 
3*d^2*e - 24*a*b^6*c^2*d*e^2 + 384*a^3*b*c^5*d^2*e - 288*a^2*b^3*c^4*d^...
 

Reduce [B] (verification not implemented)

Time = 10.43 (sec) , antiderivative size = 13560, normalized size of antiderivative = 47.25 \[ \int \frac {\sqrt {d+e x}}{\left (a+b x+c x^2\right )^2} \, dx =\text {Too large to display} \] Input:

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

Output:

( - 8*sqrt(2*sqrt(c)*sqrt(a*e**2 - b*d*e + c*d**2) + b*e - 2*c*d)*sqrt(a*e 
**2 - b*d*e + c*d**2)*atan((sqrt(2*sqrt(c)*sqrt(a*e**2 - b*d*e + c*d**2) - 
 b*e + 2*c*d) - 2*sqrt(c)*sqrt(d + e*x))/sqrt(2*sqrt(c)*sqrt(a*e**2 - b*d* 
e + c*d**2) + b*e - 2*c*d))*a**2*c*e**2 - 2*sqrt(2*sqrt(c)*sqrt(a*e**2 - b 
*d*e + c*d**2) + b*e - 2*c*d)*sqrt(a*e**2 - b*d*e + c*d**2)*atan((sqrt(2*s 
qrt(c)*sqrt(a*e**2 - b*d*e + c*d**2) - b*e + 2*c*d) - 2*sqrt(c)*sqrt(d + e 
*x))/sqrt(2*sqrt(c)*sqrt(a*e**2 - b*d*e + c*d**2) + b*e - 2*c*d))*a*b**2*e 
**2 + 16*sqrt(2*sqrt(c)*sqrt(a*e**2 - b*d*e + c*d**2) + b*e - 2*c*d)*sqrt( 
a*e**2 - b*d*e + c*d**2)*atan((sqrt(2*sqrt(c)*sqrt(a*e**2 - b*d*e + c*d**2 
) - b*e + 2*c*d) - 2*sqrt(c)*sqrt(d + e*x))/sqrt(2*sqrt(c)*sqrt(a*e**2 - b 
*d*e + c*d**2) + b*e - 2*c*d))*a*b*c*d*e - 8*sqrt(2*sqrt(c)*sqrt(a*e**2 - 
b*d*e + c*d**2) + b*e - 2*c*d)*sqrt(a*e**2 - b*d*e + c*d**2)*atan((sqrt(2* 
sqrt(c)*sqrt(a*e**2 - b*d*e + c*d**2) - b*e + 2*c*d) - 2*sqrt(c)*sqrt(d + 
e*x))/sqrt(2*sqrt(c)*sqrt(a*e**2 - b*d*e + c*d**2) + b*e - 2*c*d))*a*b*c*e 
**2*x - 16*sqrt(2*sqrt(c)*sqrt(a*e**2 - b*d*e + c*d**2) + b*e - 2*c*d)*sqr 
t(a*e**2 - b*d*e + c*d**2)*atan((sqrt(2*sqrt(c)*sqrt(a*e**2 - b*d*e + c*d* 
*2) - b*e + 2*c*d) - 2*sqrt(c)*sqrt(d + e*x))/sqrt(2*sqrt(c)*sqrt(a*e**2 - 
 b*d*e + c*d**2) + b*e - 2*c*d))*a*c**2*d**2 - 8*sqrt(2*sqrt(c)*sqrt(a*e** 
2 - b*d*e + c*d**2) + b*e - 2*c*d)*sqrt(a*e**2 - b*d*e + c*d**2)*atan((sqr 
t(2*sqrt(c)*sqrt(a*e**2 - b*d*e + c*d**2) - b*e + 2*c*d) - 2*sqrt(c)*sq...