\(\int \frac {1}{d+e x+f \sqrt {a+b x+c x^2}} \, dx\) [27]

Optimal result

Integrand size = 23, antiderivative size = 361 \[ \int \frac {1}{d+e x+f \sqrt {a+b x+c x^2}} \, dx=\frac {2 \sqrt {c} f \text {arctanh}\left (\frac {\sqrt {a}-\sqrt {a+b x+c x^2}}{\sqrt {c} x}\right )}{e^2-c f^2}+\frac {2 (2 c d-b e) f \text {arctanh}\left (\frac {2 a f+b f x+2 d \sqrt {a+x (b+c x)}-2 \sqrt {a} \left (d+e x+f \sqrt {a+x (b+c x)}\right )}{\sqrt {-4 b d e+4 a e^2+b^2 f^2+4 c \left (d^2-a f^2\right )} x}\right )}{\left (e^2-c f^2\right ) \sqrt {-4 b d e+4 a e^2+b^2 f^2+4 c \left (d^2-a f^2\right )}}-\frac {e \log \left (c-\frac {\left (\sqrt {a}-\sqrt {a+b x+c x^2}\right )^2}{x^2}\right )}{e^2-c f^2}+\frac {e \log \left (b e-c \left (d+\sqrt {a} f\right )+\frac {\left (2 \sqrt {a} e-b f\right ) \left (\sqrt {a}-\sqrt {a+b x+c x^2}\right )}{x}+\frac {\left (d-\sqrt {a} f\right ) \left (\sqrt {a}-\sqrt {a+b x+c x^2}\right )^2}{x^2}\right )}{e^2-c f^2} \] Output:

2*c^(1/2)*f*arctanh((a^(1/2)-(c*x^2+b*x+a)^(1/2))/c^(1/2)/x)/(-c*f^2+e^2)+ 
2*(-b*e+2*c*d)*f*arctanh((2*a*f+b*f*x+2*d*(a+x*(c*x+b))^(1/2)-2*a^(1/2)*(d 
+e*x+f*(a+x*(c*x+b))^(1/2)))/(-4*b*d*e+4*a*e^2+b^2*f^2+4*c*(-a*f^2+d^2))^( 
1/2)/x)/(-c*f^2+e^2)/(-4*b*d*e+4*a*e^2+b^2*f^2+4*c*(-a*f^2+d^2))^(1/2)-e*l 
n(c-(a^(1/2)-(c*x^2+b*x+a)^(1/2))^2/x^2)/(-c*f^2+e^2)+e*ln(b*e-c*(d+a^(1/2 
)*f)+(2*a^(1/2)*e-b*f)*(a^(1/2)-(c*x^2+b*x+a)^(1/2))/x+(d-a^(1/2)*f)*(a^(1 
/2)-(c*x^2+b*x+a)^(1/2))^2/x^2)/(-c*f^2+e^2)
 

Mathematica [A] (verified)

Time = 3.17 (sec) , antiderivative size = 323, normalized size of antiderivative = 0.89 \[ \int \frac {1}{d+e x+f \sqrt {a+b x+c x^2}} \, dx=\frac {\frac {2 (2 c d-b e) f \arctan \left (\frac {\sqrt {-4 c d^2+4 b d e-4 a e^2-b^2 f^2+4 a c f^2} x}{2 a f+b f x+2 d \sqrt {a+x (b+c x)}-2 \sqrt {a} \left (d+e x+f \sqrt {a+x (b+c x)}\right )}\right )}{\sqrt {-4 c d^2+4 b d e-4 a e^2-b^2 f^2+4 a c f^2}}+2 \sqrt {c} f \text {arctanh}\left (\frac {\sqrt {c} x}{\sqrt {a}-\sqrt {a+x (b+c x)}}\right )+e \left (-\log \left (-2 a-b x+2 \sqrt {a} \sqrt {a+x (b+c x)}\right )+\log \left (-2 a^{3/2} f+2 a \left (d+e x+f \sqrt {a+x (b+c x)}\right )+b x \left (d+e x+f \sqrt {a+x (b+c x)}\right )-2 \sqrt {a} \left (b f x+c f x^2+(d+e x) \sqrt {a+x (b+c x)}\right )\right )\right )}{e^2-c f^2} \] Input:

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

Output:

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

Rubi [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(1087\) vs. \(2(361)=722\).

Time = 10.42 (sec) , antiderivative size = 1087, normalized size of antiderivative = 3.01, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {7293, 2009}

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.

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

\(\Big \downarrow \) 7293

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

((2*c*d - b*e)*f*ArcTanh[(2*d*e - b*f^2 + 2*(e^2 - c*f^2)*x)/(f*Sqrt[-4*b* 
d*e + 4*a*e^2 + b^2*f^2 + 4*c*(d^2 - a*f^2)])])/((e^2 - c*f^2)*Sqrt[-4*b*d 
*e + 4*a*e^2 + b^2*f^2 + 4*c*(d^2 - a*f^2)]) - (Sqrt[c]*f*ArcTanh[(b + 2*c 
*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/(e^2 - c*f^2) - (Sqrt[2*c*d^2*e^2 
- 2*b*d*e^3 + 2*a*e^4 + 2*c^2*d^2*f^2 - 2*b*c*d*e*f^2 + b^2*e^2*f^2 - 2*a* 
c*e^2*f^2 - e*(2*c*d - b*e)*f*Sqrt[-4*b*d*e + 4*a*e^2 + b^2*f^2 + 4*c*(d^2 
 - a*f^2)]]*ArcTanh[(2*b*d*e - 4*a*(e^2 - c*f^2) - b*f*(b*f + Sqrt[-4*b*d* 
e + 4*a*e^2 + b^2*f^2 + 4*c*(d^2 - a*f^2)]) + 2*(2*c*d*e - b*e^2 - c*f*Sqr 
t[-4*b*d*e + 4*a*e^2 + b^2*f^2 + 4*c*(d^2 - a*f^2)])*x)/(2*Sqrt[2]*Sqrt[2* 
c*d^2*e^2 - 2*b*d*e^3 + 2*a*e^4 + 2*c^2*d^2*f^2 - 2*b*c*d*e*f^2 + b^2*e^2* 
f^2 - 2*a*c*e^2*f^2 - e*(2*c*d - b*e)*f*Sqrt[-4*b*d*e + 4*a*e^2 + b^2*f^2 
+ 4*c*(d^2 - a*f^2)]]*Sqrt[a + b*x + c*x^2])])/(Sqrt[2]*(e^2 - c*f^2)*Sqrt 
[-4*b*d*e + 4*a*e^2 + b^2*f^2 + 4*c*(d^2 - a*f^2)]) + (Sqrt[2*c*d^2*e^2 - 
2*b*d*e^3 + 2*a*e^4 + 2*c^2*d^2*f^2 - 2*b*c*d*e*f^2 + b^2*e^2*f^2 - 2*a*c* 
e^2*f^2 + e*(2*c*d - b*e)*f*Sqrt[-4*b*d*e + 4*a*e^2 + b^2*f^2 + 4*c*(d^2 - 
 a*f^2)]]*ArcTanh[(2*b*d*e - 4*a*e^2 - b^2*f^2 + 4*a*c*f^2 + b*f*Sqrt[-4*b 
*d*e + 4*a*e^2 + b^2*f^2 + 4*c*(d^2 - a*f^2)] + 2*(2*c*d*e - b*e^2 + c*f*S 
qrt[-4*b*d*e + 4*a*e^2 + b^2*f^2 + 4*c*(d^2 - a*f^2)])*x)/(2*Sqrt[2]*Sqrt[ 
2*c*d^2*e^2 - 2*b*d*e^3 + 2*a*e^4 + 2*c^2*d^2*f^2 - 2*b*c*d*e*f^2 + b^2*e^ 
2*f^2 - 2*a*c*e^2*f^2 + e*(2*c*d - b*e)*f*Sqrt[-4*b*d*e + 4*a*e^2 + b^2...
 

Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 7293

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v] 
]
 

Maple [B] (warning: unable to verify)

Leaf count of result is larger than twice the leaf count of optimal. \(4878\) vs. \(2(325)=650\).

Time = 0.15 (sec) , antiderivative size = 4879, normalized size of antiderivative = 13.52

method	result	size
default	\(\text {Expression too large to display}\)	\(4879\)

Input:

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

Output:

f*(-2*(c*f^2-e^2)/(-f^2*(4*a*c*f^2-b^2*f^2-4*a*e^2+4*b*d*e-4*c*d^2))^(1/2) 
/(2*c*f^2-2*e^2)*(1/2*(4*(x+(b*f^2-2*d*e+(-f^2*(4*a*c*f^2-b^2*f^2-4*a*e^2+ 
4*b*d*e-4*c*d^2))^(1/2))/(2*c*f^2-2*e^2))^2*c-4*(b*e^2-2*c*d*e+(f^2*(-4*a* 
c*f^2+b^2*f^2+4*a*e^2-4*b*d*e+4*c*d^2))^(1/2)*c)/(c*f^2-e^2)*(x+(b*f^2-2*d 
*e+(-f^2*(4*a*c*f^2-b^2*f^2-4*a*e^2+4*b*d*e-4*c*d^2))^(1/2))/(2*c*f^2-2*e^ 
2))-2*(2*a*c*e^2*f^2-b^2*e^2*f^2+2*b*c*d*e*f^2-2*c^2*d^2*f^2-2*e^4*a+2*d*e 
^3*b-2*d^2*e^2*c-(f^2*(-4*a*c*f^2+b^2*f^2+4*a*e^2-4*b*d*e+4*c*d^2))^(1/2)* 
b*e^2+2*(f^2*(-4*a*c*f^2+b^2*f^2+4*a*e^2-4*b*d*e+4*c*d^2))^(1/2)*c*d*e)/(c 
*f^2-e^2)^2)^(1/2)-1/2*(b*e^2-2*c*d*e+(f^2*(-4*a*c*f^2+b^2*f^2+4*a*e^2-4*b 
*d*e+4*c*d^2))^(1/2)*c)/(c*f^2-e^2)*ln((-1/2*(b*e^2-2*c*d*e+(f^2*(-4*a*c*f 
^2+b^2*f^2+4*a*e^2-4*b*d*e+4*c*d^2))^(1/2)*c)/(c*f^2-e^2)+c*(x+(b*f^2-2*d* 
e+(-f^2*(4*a*c*f^2-b^2*f^2-4*a*e^2+4*b*d*e-4*c*d^2))^(1/2))/(2*c*f^2-2*e^2 
)))/c^(1/2)+((x+(b*f^2-2*d*e+(-f^2*(4*a*c*f^2-b^2*f^2-4*a*e^2+4*b*d*e-4*c* 
d^2))^(1/2))/(2*c*f^2-2*e^2))^2*c-(b*e^2-2*c*d*e+(f^2*(-4*a*c*f^2+b^2*f^2+ 
4*a*e^2-4*b*d*e+4*c*d^2))^(1/2)*c)/(c*f^2-e^2)*(x+(b*f^2-2*d*e+(-f^2*(4*a* 
c*f^2-b^2*f^2-4*a*e^2+4*b*d*e-4*c*d^2))^(1/2))/(2*c*f^2-2*e^2))-1/2*(2*a*c 
*e^2*f^2-b^2*e^2*f^2+2*b*c*d*e*f^2-2*c^2*d^2*f^2-2*e^4*a+2*d*e^3*b-2*d^2*e 
^2*c-(f^2*(-4*a*c*f^2+b^2*f^2+4*a*e^2-4*b*d*e+4*c*d^2))^(1/2)*b*e^2+2*(f^2 
*(-4*a*c*f^2+b^2*f^2+4*a*e^2-4*b*d*e+4*c*d^2))^(1/2)*c*d*e)/(c*f^2-e^2)^2) 
^(1/2))/c^(1/2)+(2*a*c*e^2*f^2-b^2*e^2*f^2+2*b*c*d*e*f^2-2*c^2*d^2*f^2-...
 

Fricas [F(-1)]

Timed out. \[ \int \frac {1}{d+e x+f \sqrt {a+b x+c x^2}} \, dx=\text {Timed out} \] Input:

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

Output:

Timed out
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F(-2)]

Exception generated. \[ \int \frac {1}{d+e x+f \sqrt {a+b x+c x^2}} \, dx=\text {Exception raised: TypeError} \] Input:

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

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Error: Bad Argument Type
                                                                                    
                                                                                    
 

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{d+e x+f \sqrt {a+b x+c x^2}} \, dx=\int \frac {1}{d+e\,x+f\,\sqrt {c\,x^2+b\,x+a}} \,d x \] Input:

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

Output:

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

Reduce [F]

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

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

Output:

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