\(\int \frac {1}{(d+e x+f \sqrt {-a+b x-c x^2})^2} \, dx\) [49]

Optimal result

Integrand size = 26, antiderivative size = 463 \[ \int \frac {1}{\left (d+e x+f \sqrt {-a+b x-c x^2}\right )^2} \, dx=\frac {4 \sqrt {b^2-4 a c} \left (2 c d+\left (b+\sqrt {b^2-4 a c}\right ) e-\frac {2 c \sqrt {b^2-4 a c} f \sqrt {-a+b x-c x^2}}{b-\sqrt {b^2-4 a c}-2 c x}\right )}{\left (4 b d e+4 a e^2-b^2 f^2+4 c \left (d^2+a f^2\right )\right ) \left (2 c d+\left (b+\sqrt {b^2-4 a c}\right ) e-\frac {4 c \sqrt {b^2-4 a c} f \sqrt {-a+b x-c x^2}}{b-\sqrt {b^2-4 a c}-2 c x}-\frac {4 c \left (2 c d+\left (b-\sqrt {b^2-4 a c}\right ) e\right ) \left (a-b x+c x^2\right )}{\left (b-\sqrt {b^2-4 a c}-2 c x\right )^2}\right )}+\frac {4 \left (b^2-4 a c\right ) f \arctan \left (\frac {\sqrt {b^2-4 a c} f-\frac {4 c d \sqrt {-a+b x-c x^2}}{b-\sqrt {b^2-4 a c}-2 c x}-\frac {2 \left (b-\sqrt {b^2-4 a c}\right ) e \sqrt {-a+b x-c x^2}}{b-\sqrt {b^2-4 a c}-2 c x}}{\sqrt {4 b d e+4 a e^2-b^2 f^2+4 c \left (d^2+a f^2\right )}}\right )}{\left (4 b d e+4 a e^2-b^2 f^2+4 c \left (d^2+a f^2\right )\right )^{3/2}} \] Output:

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

Mathematica [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 49.72 (sec) , antiderivative size = 2346, normalized size of antiderivative = 5.07 \[ \int \frac {1}{\left (d+e x+f \sqrt {-a+b x-c x^2}\right )^2} \, dx=\text {Result too large to show} \] Input:

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

Output:

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

Rubi [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(965\) vs. \(2(463)=926\).

Time = 4.31 (sec) , antiderivative size = 965, normalized size of antiderivative = 2.08, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, 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.

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

\(\Big \downarrow \) 7293

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

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

result has leaf size over 500,000. Avoiding possible recursion issues.

Time = 1.74 (sec) , antiderivative size = 609986, normalized size of antiderivative = 1317.46

method	result	size
default	\(\text {Expression too large to display}\)	\(609986\)

Input:

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

Output:

result too large to display
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2873 vs. \(2 (424) = 848\).

Time = 20.12 (sec) , antiderivative size = 6908, normalized size of antiderivative = 14.92 \[ \int \frac {1}{\left (d+e x+f \sqrt {-a+b x-c x^2}\right )^2} \, dx=\text {Too large to display} \] Input:

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

Output:

Too large to include
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F(-2)]

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

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

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

Output:

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

Reduce [F]

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

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

Output:

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