\(\int \frac {1}{(d+e x+f \sqrt {-a+b x+c x^2})^3} \, dx\) [40]

Optimal result

Integrand size = 25, antiderivative size = 896 \[ \int \frac {1}{\left (d+e x+f \sqrt {-a+b x+c x^2}\right )^3} \, dx=-\frac {b^3 d \left (2 e^2+\sqrt {c} e f+c f^2\right )+4 b c d \left (2 c d^2-4 a e^2+a \sqrt {c} e f+a c f^2\right )-2 b^2 \left (5 c d^2 e-a e^3+c^{3/2} d^2 f-a c e f^2\right )-8 a c \left (a e^3+c^{3/2} d^2 f-c e \left (d^2+a f^2\right )\right )-(2 c d-b e) \left (b^2 e f+4 a c e f-4 c^{3/2} \left (2 d^2+a f^2\right )+\sqrt {c} \left (8 b d e+8 a e^2-b^2 f^2\right )\right ) \left (\sqrt {c} x+\sqrt {-a+b x+c x^2}\right )}{\left (e+\sqrt {c} f\right )^2 \left (4 b d e+4 a e^2-b^2 f^2-4 c \left (d^2+a f^2\right )\right ) \left (b d+a \left (e-\sqrt {c} f\right )+\left (2 \sqrt {c} d+b f\right ) \left (\sqrt {c} x+\sqrt {-a+b x+c x^2}\right )+\left (e+\sqrt {c} f\right ) \left (\sqrt {c} x+\sqrt {-a+b x+c x^2}\right )^2\right )^2}-\frac {\frac {8 c \left (e+\sqrt {c} f\right ) \left (b d+a \left (e-\sqrt {c} f\right )\right ) \left (4 b d e+4 a e^2-b^2 f^2-4 c \left (d^2+a f^2\right )\right )+\left (2 \sqrt {c} d+b f\right ) \left (3 b^3 e^2 f+3 b^3 \sqrt {c} e f^2-2 b c f \left (7 b d e-2 a e^2-b^2 f^2\right )+8 c^{5/2} \left (2 d^3-a d f^2\right )-2 c^{3/2} \left (8 b d^2 e+8 a d e^2+b^2 d f^2-6 a b e f^2\right )-8 c^2 f \left (3 a d e-b \left (d^2+a f^2\right )\right )\right )}{\left (e+\sqrt {c} f\right )^2}-6 \left (b^2+4 a c\right ) (2 c d-b e) f \left (\sqrt {c} x+\sqrt {-a+b x+c x^2}\right )}{\left (4 b d e+4 a e^2-b^2 f^2-4 c \left (d^2+a f^2\right )\right )^2 \left (b d+a \left (e-\sqrt {c} f\right )+\left (2 \sqrt {c} d+b f\right ) \left (\sqrt {c} x+\sqrt {-a+b x+c x^2}\right )+\left (e+\sqrt {c} f\right ) \left (\sqrt {c} x+\sqrt {-a+b x+c x^2}\right )^2\right )}-\frac {12 \left (b^2+4 a c\right ) (2 c d-b e) f \text {arctanh}\left (\frac {b f+2 c f x+2 e \sqrt {-a+x (b+c x)}+2 \sqrt {c} \left (d+e x+f \sqrt {-a+x (b+c x)}\right )}{\sqrt {-4 b d e-4 a e^2+b^2 f^2+4 c \left (d^2+a f^2\right )}}\right )}{\left (-4 b d e-4 a e^2+b^2 f^2+4 c \left (d^2+a f^2\right )\right )^{5/2}} \] Output:

-(b^3*d*(2*e^2+c^(1/2)*e*f+c*f^2)+4*b*c*d*(2*c*d^2-4*a*e^2+a*c^(1/2)*e*f+a 
*c*f^2)-2*b^2*(5*c*d^2*e-a*e^3+c^(3/2)*d^2*f-a*c*e*f^2)-8*a*c*(a*e^3+c^(3/ 
2)*d^2*f-c*e*(a*f^2+d^2))-(-b*e+2*c*d)*(b^2*e*f+4*a*c*e*f-4*c^(3/2)*(a*f^2 
+2*d^2)+c^(1/2)*(-b^2*f^2+8*a*e^2+8*b*d*e))*(c^(1/2)*x+(c*x^2+b*x-a)^(1/2) 
))/(e+c^(1/2)*f)^2/(4*b*d*e+4*a*e^2-b^2*f^2-4*c*(a*f^2+d^2))/(b*d+a*(e-c^( 
1/2)*f)+(2*c^(1/2)*d+b*f)*(c^(1/2)*x+(c*x^2+b*x-a)^(1/2))+(e+c^(1/2)*f)*(c 
^(1/2)*x+(c*x^2+b*x-a)^(1/2))^2)^2-((8*c*(e+c^(1/2)*f)*(b*d+a*(e-c^(1/2)*f 
))*(4*b*d*e+4*a*e^2-b^2*f^2-4*c*(a*f^2+d^2))+(2*c^(1/2)*d+b*f)*(3*b^3*e^2* 
f+3*b^3*c^(1/2)*e*f^2-2*b*c*f*(-b^2*f^2-2*a*e^2+7*b*d*e)+8*c^(5/2)*(-a*d*f 
^2+2*d^3)-2*c^(3/2)*(-6*a*b*e*f^2+b^2*d*f^2+8*a*d*e^2+8*b*d^2*e)-8*c^2*f*( 
3*a*d*e-b*(a*f^2+d^2))))/(e+c^(1/2)*f)^2-6*(4*a*c+b^2)*(-b*e+2*c*d)*f*(c^( 
1/2)*x+(c*x^2+b*x-a)^(1/2)))/(4*b*d*e+4*a*e^2-b^2*f^2-4*c*(a*f^2+d^2))^2/( 
b*d+a*(e-c^(1/2)*f)+(2*c^(1/2)*d+b*f)*(c^(1/2)*x+(c*x^2+b*x-a)^(1/2))+(e+c 
^(1/2)*f)*(c^(1/2)*x+(c*x^2+b*x-a)^(1/2))^2)-12*(4*a*c+b^2)*(-b*e+2*c*d)*f 
*arctanh((b*f+2*c*f*x+2*e*(-a+x*(c*x+b))^(1/2)+2*c^(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))/(-4*b*d*e 
-4*a*e^2+b^2*f^2+4*c*(a*f^2+d^2))^(5/2)
 

Mathematica [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 75.18 (sec) , antiderivative size = 3628, normalized size of antiderivative = 4.05 \[ \int \frac {1}{\left (d+e x+f \sqrt {-a+b x+c x^2}\right )^3} \, dx=\text {Result too large to show} \] Input:

Integrate[(d + e*x + f*Sqrt[-a + b*x + c*x^2])^(-3),x]
 

Output:

(2*(-4*c^2*d^4*e*f^2 + 5*b*c*d^3*e^2*f^2 - b^2*d^2*e^3*f^2 + 2*a*c*d^2*e^3 
*f^2 + a*b*d*e^4*f^2 + 2*a^2*e^5*f^2 - b*c^2*d^3*f^4 - 6*a*c^2*d^2*e*f^4 + 
 3*a*b*c*d*e^2*f^4 - a*b^2*e^3*f^4 - 2*a^2*c*e^3*f^4 - 6*c^2*d^3*e^2*f^2*x 
 + 9*b*c*d^2*e^3*f^2*x - 3*b^2*d*e^4*f^2*x + 6*a*c*d*e^4*f^2*x - 3*a*b*e^5 
*f^2*x - 2*c^3*d^3*f^4*x + 3*b*c^2*d^2*e*f^4*x - 3*b^2*c*d*e^2*f^4*x - 6*a 
*c^2*d*e^2*f^4*x + b^3*e^3*f^4*x + 3*a*b*c*e^3*f^4*x))/((e^2 - c*f^2)^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) + (-8*c^2*d^4*e^3 + 16*b*c*d^3*e^4 - 8 
*b^2*d^2*e^5 + 16*a*c*d^2*e^5 - 16*a*b*d*e^6 - 8*a^2*e^7 - 24*c^3*d^4*e*f^ 
2 + 48*b*c^2*d^3*e^2*f^2 - 34*b^2*c*d^2*e^3*f^2 + 8*a*c^2*d^2*e^3*f^2 + 7* 
b^3*d*e^4*f^2 - 20*a*b*c*d*e^4*f^2 + 4*a*b^2*e^5*f^2 - 8*a^2*c*e^5*f^2 - 6 
*b^2*c^2*d^2*e*f^4 - 24*a*c^3*d^2*e*f^4 + 12*b^3*c*d*e^2*f^4 + 48*a*b*c^2* 
d*e^2*f^4 - 2*b^4*e^3*f^4 + 2*a*b^2*c*e^3*f^4 + 40*a^2*c^2*e^3*f^4 - 3*b^3 
*c^2*d*f^6 - 12*a*b*c^3*d*f^6 - 6*a*b^2*c^2*e*f^6 - 24*a^2*c^3*e*f^6 - 6*b 
^2*c*d*e^4*f^2*x - 24*a*c^2*d*e^4*f^2*x + 3*b^3*e^5*f^2*x + 12*a*b*c*e^5*f 
^2*x + 12*b^2*c^2*d*e^2*f^4*x + 48*a*c^3*d*e^2*f^4*x - 6*b^3*c*e^3*f^4*x - 
 24*a*b*c^2*e^3*f^4*x - 6*b^2*c^3*d*f^6*x - 24*a*c^4*d*f^6*x + 3*b^3*c^2*e 
*f^6*x + 12*a*b*c^3*e*f^6*x)/((e^2 - c*f^2)^2*(-4*c*d^2 + 4*b*d*e + 4*a*e^ 
2 - b^2*f^2 - 4*a*c*f^2)^2*(d^2 + a*f^2 + 2*d*e*x - b*f^2*x + e^2*x^2 - c* 
f^2*x^2)) + Sqrt[-a + b*x + c*x^2]*((2*(-2*c*d^3*e*f + 2*b*d^2*e^2*f + ...
 

Rubi [F]

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.

\(\displaystyle \int \frac {1}{\left (f \sqrt {-a+b x+c x^2}+d+e x\right )^3} \, dx\)

\(\Big \downarrow \) 7293

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

\(\Big \downarrow \) 7299

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

Input:

Int[(d + e*x + f*Sqrt[-a + b*x + c*x^2])^(-3),x]
 

Output:

$Aborted
 

Defintions of rubi rules used

rule 7293

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

rule 7299

Int[u_, x_] :> CannotIntegrate[u, x]
 

Maple [B] (warning: unable to verify)

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

Time = 114.04 (sec) , antiderivative size = 4527517, normalized size of antiderivative = 5053.03

method	result	size
default	\(\text {Expression too large to display}\)	\(4527517\)

Input:

int(1/(d+e*x+f*(c*x^2+b*x-a)^(1/2))^3,x,method=_RETURNVERBOSE)
 

Output:

result too large to display
 

Fricas [F(-1)]

Timed out. \[ \int \frac {1}{\left (d+e x+f \sqrt {-a+b x+c x^2}\right )^3} \, dx=\text {Timed out} \] Input:

integrate(1/(d+e*x+f*(c*x^2+b*x-a)^(1/2))^3,x, algorithm="fricas")
 

Output:

Timed out
 

Sympy [F]

\[ \int \frac {1}{\left (d+e x+f \sqrt {-a+b x+c x^2}\right )^3} \, dx=\int \frac {1}{\left (d + e x + f \sqrt {- a + b x + c x^{2}}\right )^{3}}\, dx \] Input:

integrate(1/(d+e*x+f*(c*x**2+b*x-a)**(1/2))**3,x)
 

Output:

Integral((d + e*x + f*sqrt(-a + b*x + c*x**2))**(-3), x)
 

Maxima [F]

\[ \int \frac {1}{\left (d+e x+f \sqrt {-a+b x+c x^2}\right )^3} \, dx=\int { \frac {1}{{\left (e x + \sqrt {c x^{2} + b x - a} f + d\right )}^{3}} \,d x } \] Input:

integrate(1/(d+e*x+f*(c*x^2+b*x-a)^(1/2))^3,x, algorithm="maxima")
 

Output:

integrate((e*x + sqrt(c*x^2 + b*x - a)*f + d)^(-3), x)
 

Giac [F(-2)]

Exception generated. \[ \int \frac {1}{\left (d+e x+f \sqrt {-a+b x+c x^2}\right )^3} \, dx=\text {Exception raised: TypeError} \] Input:

integrate(1/(d+e*x+f*(c*x^2+b*x-a)^(1/2))^3,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. \[ \int \frac {1}{\left (d+e x+f \sqrt {-a+b x+c x^2}\right )^3} \, dx=\int \frac {1}{{\left (d+f\,\sqrt {c\,x^2+b\,x-a}+e\,x\right )}^3} \,d x \] Input:

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

Output:

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

Reduce [F]

\[ \int \frac {1}{\left (d+e x+f \sqrt {-a+b x+c x^2}\right )^3} \, dx=\int \frac {1}{\left (d +e x +f \sqrt {c \,x^{2}+b x -a}\right )^{3}}d x \] Input:

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

Output:

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