\(\int \frac {1}{\sqrt {d+e x} (a+b x+c x^2)} \, dx\) [536]

Optimal result

Integrand size = 22, antiderivative size = 199 \[ \int \frac {1}{\sqrt {d+e x} \left (a+b x+c x^2\right )} \, dx=-\frac {2 \sqrt {2} \sqrt {c} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}}\right )}{\sqrt {b^2-4 a c} \sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}}+\frac {2 \sqrt {2} \sqrt {c} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}}\right )}{\sqrt {b^2-4 a c} \sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \] Output:

-2*2^(1/2)*c^(1/2)*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)^(1/2)/(2*c*d-(b-(-4*a*c+b^2)^(1/2))*e) 
^(1/2)+2*2^(1/2)*c^(1/2)*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)^(1/2)/(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 = 0.66 (sec) , antiderivative size = 198, normalized size of antiderivative = 0.99 \[ \int \frac {1}{\sqrt {d+e x} \left (a+b x+c x^2\right )} \, dx=-\frac {2 i \sqrt {2} \sqrt {c} \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {-2 c d+b e-i \sqrt {-b^2+4 a c} e}}\right )}{\sqrt {-2 c d+\left (b-i \sqrt {-b^2+4 a c}\right ) e}}-\frac {\arctan \left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {-2 c d+b e+i \sqrt {-b^2+4 a c} e}}\right )}{\sqrt {-2 c d+\left (b+i \sqrt {-b^2+4 a c}\right ) e}}\right )}{\sqrt {-b^2+4 a c}} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.30 (sec) , antiderivative size = 207, normalized size of antiderivative = 1.04, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {1149, 1406, 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.

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

\(\Big \downarrow \) 1149

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

\(\Big \downarrow \) 1406

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

\(\Big \downarrow \) 221

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

Input:

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

Output:

2*e*(-((Sqrt[2]*Sqrt[c]*ArcTanh[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x])/Sqrt[2*c*d 
 - (b - Sqrt[b^2 - 4*a*c])*e]])/(Sqrt[b^2 - 4*a*c]*e*Sqrt[2*c*d - (b - Sqr 
t[b^2 - 4*a*c])*e])) + (Sqrt[2]*Sqrt[c]*ArcTanh[(Sqrt[2]*Sqrt[c]*Sqrt[d + 
e*x])/Sqrt[2*c*d - (b + Sqrt[b^2 - 4*a*c])*e]])/(Sqrt[b^2 - 4*a*c]*e*Sqrt[ 
2*c*d - (b + Sqrt[b^2 - 4*a*c])*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 1149

Int[1/(Sqrt[(d_.) + (e_.)*(x_)]*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)), x_Sym 
bol] :> Simp[2*e   Subst[Int[1/(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] /; FreeQ[{a, b, c, d, e}, x]
 

rule 1406

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

Maple [A] (verified)

Time = 1.16 (sec) , antiderivative size = 174, normalized size of antiderivative = 0.87

method	result	size
pseudoelliptic	\(\frac {2 c e \sqrt {2}\, \left (-\frac {\operatorname {arctanh}\left (\frac {\sqrt {e x +d}\, c \sqrt {2}}{\sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{\sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}-\frac {\arctan \left (\frac {\sqrt {e x +d}\, c \sqrt {2}}{\sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{\sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{\sqrt {-e^{2} \left (4 a c -b^{2}\right )}}\)	\(174\)
derivativedivides	\(8 e c \left (-\frac {\sqrt {2}\, \arctan \left (\frac {\sqrt {e x +d}\, c \sqrt {2}}{\sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{4 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, \sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}-\frac {\sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {e x +d}\, c \sqrt {2}}{\sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{4 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, \sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )\)	\(194\)
default	\(8 e c \left (-\frac {\sqrt {2}\, \arctan \left (\frac {\sqrt {e x +d}\, c \sqrt {2}}{\sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{4 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, \sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}-\frac {\sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {e x +d}\, c \sqrt {2}}{\sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{4 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, \sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )\)	\(194\)

Input:

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

Output:

2*c*e*2^(1/2)/(-e^2*(4*a*c-b^2))^(1/2)*(-1/((-b*e+2*c*d+(-e^2*(4*a*c-b^2)) 
^(1/2))*c)^(1/2)*arctanh((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/((b*e-2*c*d+(-e^2*(4*a*c-b^2))^(1/2))*c)^(1/2)*a 
rctan((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. 2641 vs. \(2 (159) = 318\).

Time = 0.11 (sec) , antiderivative size = 2641, normalized size of antiderivative = 13.27 \[ \int \frac {1}{\sqrt {d+e x} \left (a+b x+c x^2\right )} \, dx=\text {Too large to display} \] Input:

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

Output:

-1/2*sqrt(2)*sqrt((2*c*d - b*e + ((b^2*c - 4*a*c^2)*d^2 - (b^3 - 4*a*b*c)* 
d*e + (a*b^2 - 4*a^2*c)*e^2)*sqrt(e^2/((b^2*c^2 - 4*a*c^3)*d^4 - 2*(b^3*c 
- 4*a*b*c^2)*d^3*e + (b^4 - 2*a*b^2*c - 8*a^2*c^2)*d^2*e^2 - 2*(a*b^3 - 4* 
a^2*b*c)*d*e^3 + (a^2*b^2 - 4*a^3*c)*e^4)))/((b^2*c - 4*a*c^2)*d^2 - (b^3 
- 4*a*b*c)*d*e + (a*b^2 - 4*a^2*c)*e^2))*log(4*sqrt(e*x + d)*c*e + sqrt(2) 
*((b^2 - 4*a*c)*e^2 - (2*(b^2*c^2 - 4*a*c^3)*d^3 - 3*(b^3*c - 4*a*b*c^2)*d 
^2*e + (b^4 - 2*a*b^2*c - 8*a^2*c^2)*d*e^2 - (a*b^3 - 4*a^2*b*c)*e^3)*sqrt 
(e^2/((b^2*c^2 - 4*a*c^3)*d^4 - 2*(b^3*c - 4*a*b*c^2)*d^3*e + (b^4 - 2*a*b 
^2*c - 8*a^2*c^2)*d^2*e^2 - 2*(a*b^3 - 4*a^2*b*c)*d*e^3 + (a^2*b^2 - 4*a^3 
*c)*e^4)))*sqrt((2*c*d - b*e + ((b^2*c - 4*a*c^2)*d^2 - (b^3 - 4*a*b*c)*d* 
e + (a*b^2 - 4*a^2*c)*e^2)*sqrt(e^2/((b^2*c^2 - 4*a*c^3)*d^4 - 2*(b^3*c - 
4*a*b*c^2)*d^3*e + (b^4 - 2*a*b^2*c - 8*a^2*c^2)*d^2*e^2 - 2*(a*b^3 - 4*a^ 
2*b*c)*d*e^3 + (a^2*b^2 - 4*a^3*c)*e^4)))/((b^2*c - 4*a*c^2)*d^2 - (b^3 - 
4*a*b*c)*d*e + (a*b^2 - 4*a^2*c)*e^2))) + 1/2*sqrt(2)*sqrt((2*c*d - b*e + 
((b^2*c - 4*a*c^2)*d^2 - (b^3 - 4*a*b*c)*d*e + (a*b^2 - 4*a^2*c)*e^2)*sqrt 
(e^2/((b^2*c^2 - 4*a*c^3)*d^4 - 2*(b^3*c - 4*a*b*c^2)*d^3*e + (b^4 - 2*a*b 
^2*c - 8*a^2*c^2)*d^2*e^2 - 2*(a*b^3 - 4*a^2*b*c)*d*e^3 + (a^2*b^2 - 4*a^3 
*c)*e^4)))/((b^2*c - 4*a*c^2)*d^2 - (b^3 - 4*a*b*c)*d*e + (a*b^2 - 4*a^2*c 
)*e^2))*log(4*sqrt(e*x + d)*c*e - sqrt(2)*((b^2 - 4*a*c)*e^2 - (2*(b^2*c^2 
 - 4*a*c^3)*d^3 - 3*(b^3*c - 4*a*b*c^2)*d^2*e + (b^4 - 2*a*b^2*c - 8*a^...
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 414 vs. \(2 (159) = 318\).

Time = 0.34 (sec) , antiderivative size = 414, normalized size of antiderivative = 2.08 \[ \int \frac {1}{\sqrt {d+e x} \left (a+b x+c x^2\right )} \, dx=\frac {{\left (\sqrt {-4 \, c^{2} d + 2 \, {\left (b c - \sqrt {b^{2} - 4 \, a c} c\right )} e} \sqrt {b^{2} - 4 \, a c} e {\left | e \right |} - \sqrt {-4 \, c^{2} d + 2 \, {\left (b c - \sqrt {b^{2} - 4 \, a c} c\right )} e} {\left (2 \, c d e - b e^{2}\right )}\right )} \arctan \left (\frac {2 \, \sqrt {\frac {1}{2}} \sqrt {e x + d}}{\sqrt {-\frac {2 \, c d - b e + \sqrt {{\left (2 \, c d - b e\right )}^{2} - 4 \, {\left (c d^{2} - b d e + a e^{2}\right )} c}}{c}}}\right )}{2 \, {\left (\sqrt {b^{2} - 4 \, a c} c d^{2} - \sqrt {b^{2} - 4 \, a c} b d e + \sqrt {b^{2} - 4 \, a c} a e^{2}\right )} {\left | c \right |} {\left | e \right |}} + \frac {{\left (\sqrt {-4 \, c^{2} d + 2 \, {\left (b c + \sqrt {b^{2} - 4 \, a c} c\right )} e} \sqrt {b^{2} - 4 \, a c} e {\left | e \right |} + \sqrt {-4 \, c^{2} d + 2 \, {\left (b c + \sqrt {b^{2} - 4 \, a c} c\right )} e} {\left (2 \, c d e - b e^{2}\right )}\right )} \arctan \left (\frac {2 \, \sqrt {\frac {1}{2}} \sqrt {e x + d}}{\sqrt {-\frac {2 \, c d - b e - \sqrt {{\left (2 \, c d - b e\right )}^{2} - 4 \, {\left (c d^{2} - b d e + a e^{2}\right )} c}}{c}}}\right )}{2 \, {\left (\sqrt {b^{2} - 4 \, a c} c d^{2} - \sqrt {b^{2} - 4 \, a c} b d e + \sqrt {b^{2} - 4 \, a c} a e^{2}\right )} {\left | c \right |} {\left | e \right |}} \] Input:

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

Output:

1/2*(sqrt(-4*c^2*d + 2*(b*c - sqrt(b^2 - 4*a*c)*c)*e)*sqrt(b^2 - 4*a*c)*e* 
abs(e) - sqrt(-4*c^2*d + 2*(b*c - sqrt(b^2 - 4*a*c)*c)*e)*(2*c*d*e - b*e^2 
))*arctan(2*sqrt(1/2)*sqrt(e*x + d)/sqrt(-(2*c*d - b*e + sqrt((2*c*d - b*e 
)^2 - 4*(c*d^2 - b*d*e + a*e^2)*c))/c))/((sqrt(b^2 - 4*a*c)*c*d^2 - sqrt(b 
^2 - 4*a*c)*b*d*e + sqrt(b^2 - 4*a*c)*a*e^2)*abs(c)*abs(e)) + 1/2*(sqrt(-4 
*c^2*d + 2*(b*c + sqrt(b^2 - 4*a*c)*c)*e)*sqrt(b^2 - 4*a*c)*e*abs(e) + sqr 
t(-4*c^2*d + 2*(b*c + sqrt(b^2 - 4*a*c)*c)*e)*(2*c*d*e - b*e^2))*arctan(2* 
sqrt(1/2)*sqrt(e*x + d)/sqrt(-(2*c*d - b*e - sqrt((2*c*d - b*e)^2 - 4*(c*d 
^2 - b*d*e + a*e^2)*c))/c))/((sqrt(b^2 - 4*a*c)*c*d^2 - sqrt(b^2 - 4*a*c)* 
b*d*e + sqrt(b^2 - 4*a*c)*a*e^2)*abs(c)*abs(e))
 

Mupad [B] (verification not implemented)

Time = 0.66 (sec) , antiderivative size = 4449, normalized size of antiderivative = 22.36 \[ \int \frac {1}{\sqrt {d+e x} \left (a+b x+c x^2\right )} \, dx=\text {Too large to display} \] Input:

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.32 (sec) , antiderivative size = 2352, normalized size of antiderivative = 11.82 \[ \int \frac {1}{\sqrt {d+e x} \left (a+b x+c x^2\right )} \, dx =\text {Too large to display} \] Input:

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

Output:

(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*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))*b*e - 4*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))*c*d - 4*sqrt(c)*sqr 
t(2*sqrt(c)*sqrt(a*e**2 - b*d*e + c*d**2) + b*e - 2*c*d)*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*e**2 + 4* 
sqrt(c)*sqrt(2*sqrt(c)*sqrt(a*e**2 - b*d*e + c*d**2) + b*e - 2*c*d)*atan(( 
sqrt(2*sqrt(c)*sqrt(a*e**2 - b*d*e + c*d**2) - b*e + 2*c*d) - 2*sqrt(c)*sq 
rt(d + e*x))/sqrt(2*sqrt(c)*sqrt(a*e**2 - b*d*e + c*d**2) + b*e - 2*c*d))* 
b*d*e - 4*sqrt(c)*sqrt(2*sqrt(c)*sqrt(a*e**2 - b*d*e + c*d**2) + b*e - 2*c 
*d)*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))*c*d**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*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)*sqr 
t(a*e**2 - b*d*e + c*d**2) + b*e - 2*c*d))*b*e + 4*sqrt(2*sqrt(c)*sqrt(...