\(\int \frac {1}{a b c-(c+a b x)^2 \sqrt {c+b x+a x^2}} \, dx\) [3128]

Optimal result

Integrand size = 31, antiderivative size = 757 \[ \int \frac {1}{a b c-(c+a b x)^2 \sqrt {c+b x+a x^2}} \, dx=-2 \text {RootSum}\left [a b^4 c+\sqrt {a} b^2 c^3-2 a^{3/2} b^2 c^3+a^{5/2} b^2 c^3-6 a^{3/2} b^3 c \text {$\#$1}-b^3 c^2 \text {$\#$1}+2 a b^3 c^2 \text {$\#$1}-a^2 b^3 c^2 \text {$\#$1}-4 a b c^3 \text {$\#$1}+4 a^2 b c^3 \text {$\#$1}+12 a^2 b^2 c \text {$\#$1}^2+5 \sqrt {a} b^2 c^2 \text {$\#$1}^2-4 a^{3/2} b^2 c^2 \text {$\#$1}^2-a^{5/2} b^2 c^2 \text {$\#$1}^2+4 a^{3/2} c^3 \text {$\#$1}^2-8 a^{5/2} b c \text {$\#$1}^3-2 a b^3 c \text {$\#$1}^3+2 a^2 b^3 c \text {$\#$1}^3-8 a b c^2 \text {$\#$1}^3+6 a^{3/2} b^2 c \text {$\#$1}^4-a^{5/2} b^2 c \text {$\#$1}^4+4 a^{3/2} c^2 \text {$\#$1}^4-a^2 b^3 \text {$\#$1}^5-4 a^2 b c \text {$\#$1}^5+a^{5/2} b^2 \text {$\#$1}^6\&,\frac {-\sqrt {a} b c \log \left (-\sqrt {a} x+\sqrt {c+b x+a x^2}-\text {$\#$1}\right )+b^2 \log \left (-\sqrt {a} x+\sqrt {c+b x+a x^2}-\text {$\#$1}\right ) \text {$\#$1}+2 a c \log \left (-\sqrt {a} x+\sqrt {c+b x+a x^2}-\text {$\#$1}\right ) \text {$\#$1}-3 \sqrt {a} b \log \left (-\sqrt {a} x+\sqrt {c+b x+a x^2}-\text {$\#$1}\right ) \text {$\#$1}^2+2 a \log \left (-\sqrt {a} x+\sqrt {c+b x+a x^2}-\text {$\#$1}\right ) \text {$\#$1}^3}{6 a^{3/2} b^3 c+b^3 c^2-2 a b^3 c^2+a^2 b^3 c^2+4 a b c^3-4 a^2 b c^3-24 a^2 b^2 c \text {$\#$1}-10 \sqrt {a} b^2 c^2 \text {$\#$1}+8 a^{3/2} b^2 c^2 \text {$\#$1}+2 a^{5/2} b^2 c^2 \text {$\#$1}-8 a^{3/2} c^3 \text {$\#$1}+24 a^{5/2} b c \text {$\#$1}^2+6 a b^3 c \text {$\#$1}^2-6 a^2 b^3 c \text {$\#$1}^2+24 a b c^2 \text {$\#$1}^2-24 a^{3/2} b^2 c \text {$\#$1}^3+4 a^{5/2} b^2 c \text {$\#$1}^3-16 a^{3/2} c^2 \text {$\#$1}^3+5 a^2 b^3 \text {$\#$1}^4+20 a^2 b c \text {$\#$1}^4-6 a^{5/2} b^2 \text {$\#$1}^5}\&\right ] \] Output:

Unintegrable
 

Mathematica [A] (verified)

Time = 2.87 (sec) , antiderivative size = 734, normalized size of antiderivative = 0.97 \[ \int \frac {1}{a b c-(c+a b x)^2 \sqrt {c+b x+a x^2}} \, dx=-2 \text {RootSum}\left [-a^3 b^4 \sqrt {c}+a^4 b c+2 a^3 b^2 c^{3/2}-a^3 c^{5/2}+a^2 b^5 \text {$\#$1}-2 a^2 b^3 c \text {$\#$1}+4 a^3 b^3 c \text {$\#$1}+a^2 b c^2 \text {$\#$1}-4 a^3 b c^2 \text {$\#$1}-5 a^2 b^4 \sqrt {c} \text {$\#$1}^2-3 a^3 b c \text {$\#$1}^2+4 a^2 b^2 c^{3/2} \text {$\#$1}^2-4 a^3 b^2 c^{3/2} \text {$\#$1}^2+a^2 c^{5/2} \text {$\#$1}^2+2 a b^3 c \text {$\#$1}^3+8 a^2 b^3 c \text {$\#$1}^3-2 a b c^2 \text {$\#$1}^3+3 a^2 b c \text {$\#$1}^4-6 a b^2 c^{3/2} \text {$\#$1}^4-4 a^2 b^2 c^{3/2} \text {$\#$1}^4+a c^{5/2} \text {$\#$1}^4+b c^2 \text {$\#$1}^5+4 a b c^2 \text {$\#$1}^5-a b c \text {$\#$1}^6-c^{5/2} \text {$\#$1}^6\&,\frac {-a^2 \sqrt {c} \log (x)+a^2 \sqrt {c} \log \left (-\sqrt {c}+\sqrt {c+b x+a x^2}-x \text {$\#$1}\right )+a b \log (x) \text {$\#$1}-a b \log \left (-\sqrt {c}+\sqrt {c+b x+a x^2}-x \text {$\#$1}\right ) \text {$\#$1}-b \log (x) \text {$\#$1}^3+b \log \left (-\sqrt {c}+\sqrt {c+b x+a x^2}-x \text {$\#$1}\right ) \text {$\#$1}^3+\sqrt {c} \log (x) \text {$\#$1}^4-\sqrt {c} \log \left (-\sqrt {c}+\sqrt {c+b x+a x^2}-x \text {$\#$1}\right ) \text {$\#$1}^4}{-a^2 b^5+2 a^2 b^3 c-4 a^3 b^3 c-a^2 b c^2+4 a^3 b c^2+10 a^2 b^4 \sqrt {c} \text {$\#$1}+6 a^3 b c \text {$\#$1}-8 a^2 b^2 c^{3/2} \text {$\#$1}+8 a^3 b^2 c^{3/2} \text {$\#$1}-2 a^2 c^{5/2} \text {$\#$1}-6 a b^3 c \text {$\#$1}^2-24 a^2 b^3 c \text {$\#$1}^2+6 a b c^2 \text {$\#$1}^2-12 a^2 b c \text {$\#$1}^3+24 a b^2 c^{3/2} \text {$\#$1}^3+16 a^2 b^2 c^{3/2} \text {$\#$1}^3-4 a c^{5/2} \text {$\#$1}^3-5 b c^2 \text {$\#$1}^4-20 a b c^2 \text {$\#$1}^4+6 a b c \text {$\#$1}^5+6 c^{5/2} \text {$\#$1}^5}\&\right ] \] Input:

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

Output:

-2*RootSum[-(a^3*b^4*Sqrt[c]) + a^4*b*c + 2*a^3*b^2*c^(3/2) - a^3*c^(5/2) 
+ a^2*b^5*#1 - 2*a^2*b^3*c*#1 + 4*a^3*b^3*c*#1 + a^2*b*c^2*#1 - 4*a^3*b*c^ 
2*#1 - 5*a^2*b^4*Sqrt[c]*#1^2 - 3*a^3*b*c*#1^2 + 4*a^2*b^2*c^(3/2)*#1^2 - 
4*a^3*b^2*c^(3/2)*#1^2 + a^2*c^(5/2)*#1^2 + 2*a*b^3*c*#1^3 + 8*a^2*b^3*c*# 
1^3 - 2*a*b*c^2*#1^3 + 3*a^2*b*c*#1^4 - 6*a*b^2*c^(3/2)*#1^4 - 4*a^2*b^2*c 
^(3/2)*#1^4 + a*c^(5/2)*#1^4 + b*c^2*#1^5 + 4*a*b*c^2*#1^5 - a*b*c*#1^6 - 
c^(5/2)*#1^6 & , (-(a^2*Sqrt[c]*Log[x]) + a^2*Sqrt[c]*Log[-Sqrt[c] + Sqrt[ 
c + b*x + a*x^2] - x*#1] + a*b*Log[x]*#1 - a*b*Log[-Sqrt[c] + Sqrt[c + b*x 
 + a*x^2] - x*#1]*#1 - b*Log[x]*#1^3 + b*Log[-Sqrt[c] + Sqrt[c + b*x + a*x 
^2] - x*#1]*#1^3 + Sqrt[c]*Log[x]*#1^4 - Sqrt[c]*Log[-Sqrt[c] + Sqrt[c + b 
*x + a*x^2] - x*#1]*#1^4)/(-(a^2*b^5) + 2*a^2*b^3*c - 4*a^3*b^3*c - a^2*b* 
c^2 + 4*a^3*b*c^2 + 10*a^2*b^4*Sqrt[c]*#1 + 6*a^3*b*c*#1 - 8*a^2*b^2*c^(3/ 
2)*#1 + 8*a^3*b^2*c^(3/2)*#1 - 2*a^2*c^(5/2)*#1 - 6*a*b^3*c*#1^2 - 24*a^2* 
b^3*c*#1^2 + 6*a*b*c^2*#1^2 - 12*a^2*b*c*#1^3 + 24*a*b^2*c^(3/2)*#1^3 + 16 
*a^2*b^2*c^(3/2)*#1^3 - 4*a*c^(5/2)*#1^3 - 5*b*c^2*#1^4 - 20*a*b*c^2*#1^4 
+ 6*a*b*c*#1^5 + 6*c^(5/2)*#1^5) & ]
 

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.

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

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {c^2 \sqrt {a x^2+b x+c}}{-a^5 b^4 x^6-a^4 b^5 x^5 \left (\frac {4 c}{b^2}+1\right )-4 a^3 b^4 c x^4 \left (\frac {1}{4} \left (a+\frac {6 c}{b^2}\right )+1\right )+a^2 b^2 c^2 \left (1-\frac {c^3}{a^2 b^2}\right )-6 a^2 b^3 c^2 x^3 \left (\frac {2 \left (a b^2+c\right )}{3 b^2}+1\right )-4 a b^2 c^3 x^2 \left (\frac {1}{4} \left (6 a+\frac {c}{b^2}\right )+1\right )-(4 a+1) b c^4 x}+\frac {a b c}{-a^5 b^4 x^6-a^4 b^5 x^5 \left (\frac {4 c}{b^2}+1\right )-4 a^3 b^4 c x^4 \left (\frac {1}{4} \left (a+\frac {6 c}{b^2}\right )+1\right )+a^2 b^2 c^2 \left (1-\frac {c^3}{a^2 b^2}\right )-6 a^2 b^3 c^2 x^3 \left (\frac {2 \left (a b^2+c\right )}{3 b^2}+1\right )-4 a b^2 c^3 x^2 \left (\frac {1}{4} \left (6 a+\frac {c}{b^2}\right )+1\right )-(4 a+1) b c^4 x}+\frac {2 a b c x \sqrt {a x^2+b x+c}}{-a^5 b^4 x^6-a^4 b^5 x^5 \left (\frac {4 c}{b^2}+1\right )-4 a^3 b^4 c x^4 \left (\frac {1}{4} \left (a+\frac {6 c}{b^2}\right )+1\right )+a^2 b^2 c^2 \left (1-\frac {c^3}{a^2 b^2}\right )-6 a^2 b^3 c^2 x^3 \left (\frac {2 \left (a b^2+c\right )}{3 b^2}+1\right )-4 a b^2 c^3 x^2 \left (\frac {1}{4} \left (6 a+\frac {c}{b^2}\right )+1\right )-(4 a+1) b c^4 x}+\frac {a^2 b^2 x^2 \sqrt {a x^2+b x+c}}{-a^5 b^4 x^6-a^4 b^5 x^5 \left (\frac {4 c}{b^2}+1\right )-4 a^3 b^4 c x^4 \left (\frac {1}{4} \left (a+\frac {6 c}{b^2}\right )+1\right )+a^2 b^2 c^2 \left (1-\frac {c^3}{a^2 b^2}\right )-6 a^2 b^3 c^2 x^3 \left (\frac {2 \left (a b^2+c\right )}{3 b^2}+1\right )-4 a b^2 c^3 x^2 \left (\frac {1}{4} \left (6 a+\frac {c}{b^2}\right )+1\right )-(4 a+1) b c^4 x}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle a^2 b^2 \int \frac {x^2 \sqrt {a x^2+b x+c}}{-a^5 b^4 x^6-a^4 b^5 \left (\frac {4 c}{b^2}+1\right ) x^5-4 a^3 b^4 c \left (\frac {1}{4} \left (a+\frac {6 c}{b^2}\right )+1\right ) x^4-6 a^2 b^3 c^2 \left (\frac {2 \left (a b^2+c\right )}{3 b^2}+1\right ) x^3-4 a b^2 c^3 \left (\frac {1}{4} \left (6 a+\frac {c}{b^2}\right )+1\right ) x^2-(4 a+1) b c^4 x+a^2 b^2 c^2 \left (1-\frac {c^3}{a^2 b^2}\right )}dx+a b c \int \frac {1}{-a^5 b^4 x^6-a^4 b^5 \left (\frac {4 c}{b^2}+1\right ) x^5-4 a^3 b^4 c \left (\frac {1}{4} \left (a+\frac {6 c}{b^2}\right )+1\right ) x^4-6 a^2 b^3 c^2 \left (\frac {2 \left (a b^2+c\right )}{3 b^2}+1\right ) x^3-4 a b^2 c^3 \left (\frac {1}{4} \left (6 a+\frac {c}{b^2}\right )+1\right ) x^2-(4 a+1) b c^4 x+a^2 b^2 c^2 \left (1-\frac {c^3}{a^2 b^2}\right )}dx+2 a b c \int \frac {x \sqrt {a x^2+b x+c}}{-a^5 b^4 x^6-a^4 b^5 \left (\frac {4 c}{b^2}+1\right ) x^5-4 a^3 b^4 c \left (\frac {1}{4} \left (a+\frac {6 c}{b^2}\right )+1\right ) x^4-6 a^2 b^3 c^2 \left (\frac {2 \left (a b^2+c\right )}{3 b^2}+1\right ) x^3-4 a b^2 c^3 \left (\frac {1}{4} \left (6 a+\frac {c}{b^2}\right )+1\right ) x^2-(4 a+1) b c^4 x+a^2 b^2 c^2 \left (1-\frac {c^3}{a^2 b^2}\right )}dx+c^2 \int \frac {\sqrt {a x^2+b x+c}}{-a^5 b^4 x^6-a^4 b^5 \left (\frac {4 c}{b^2}+1\right ) x^5-4 a^3 b^4 c \left (\frac {1}{4} \left (a+\frac {6 c}{b^2}\right )+1\right ) x^4-6 a^2 b^3 c^2 \left (\frac {2 \left (a b^2+c\right )}{3 b^2}+1\right ) x^3-4 a b^2 c^3 \left (\frac {1}{4} \left (6 a+\frac {c}{b^2}\right )+1\right ) x^2-(4 a+1) b c^4 x+a^2 b^2 c^2 \left (1-\frac {c^3}{a^2 b^2}\right )}dx\)

Input:

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

Output:

$Aborted
 

Maple [F(-1)]

Timed out.

hanged

Input:

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

Output:

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

Fricas [F(-1)]

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

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

Output:

Timed out
 

Sympy [N/A]

Not integrable

Time = 3.22 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.09 \[ \int \frac {1}{a b c-(c+a b x)^2 \sqrt {c+b x+a x^2}} \, dx=- \int \frac {1}{a^{2} b^{2} x^{2} \sqrt {a x^{2} + b x + c} + 2 a b c x \sqrt {a x^{2} + b x + c} - a b c + c^{2} \sqrt {a x^{2} + b x + c}}\, dx \] Input:

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

Output:

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

Maxima [N/A]

Not integrable

Time = 0.07 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.04 \[ \int \frac {1}{a b c-(c+a b x)^2 \sqrt {c+b x+a x^2}} \, dx=\int { \frac {1}{a b c - {\left (a b x + c\right )}^{2} \sqrt {a x^{2} + b x + c}} \,d x } \] Input:

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

Output:

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

Giac [N/A]

Not integrable

Time = 2.54 (sec) , antiderivative size = 3, normalized size of antiderivative = 0.00 \[ \int \frac {1}{a b c-(c+a b x)^2 \sqrt {c+b x+a x^2}} \, dx=\int { \frac {1}{a b c - {\left (a b x + c\right )}^{2} \sqrt {a x^{2} + b x + c}} \,d x } \] Input:

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

Output:

sage0*x
 

Mupad [N/A]

Not integrable

Time = 13.82 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.04 \[ \int \frac {1}{a b c-(c+a b x)^2 \sqrt {c+b x+a x^2}} \, dx=\int -\frac {1}{{\left (c+a\,b\,x\right )}^2\,\sqrt {a\,x^2+b\,x+c}-a\,b\,c} \,d x \] Input:

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

Output:

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

Reduce [N/A]

Not integrable

Time = 200.03 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.04 \[ \int \frac {1}{a b c-(c+a b x)^2 \sqrt {c+b x+a x^2}} \, dx=\int \frac {1}{a b c -\left (a b x +c \right )^{2} \sqrt {a \,x^{2}+b x +c}}d x \] Input:

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

Output:

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