\(\int \frac {x^2}{x^2-\sqrt {b+a x} \sqrt {c+\sqrt {b+a x}}} \, dx\) [2614]

Optimal result

Integrand size = 36, antiderivative size = 228 \[ \int \frac {x^2}{x^2-\sqrt {b+a x} \sqrt {c+\sqrt {b+a x}}} \, dx=x+4 a \text {RootSum}\left [b^2-2 b c^2+c^4+a^2 c \text {$\#$1}+4 b c \text {$\#$1}^2-4 c^3 \text {$\#$1}^2-a^2 \text {$\#$1}^3-2 b \text {$\#$1}^4+6 c^2 \text {$\#$1}^4-4 c \text {$\#$1}^6+\text {$\#$1}^8\&,\frac {-c^2 \log \left (\sqrt {c+\sqrt {b+a x}}-\text {$\#$1}\right ) \text {$\#$1}^2+2 c \log \left (\sqrt {c+\sqrt {b+a x}}-\text {$\#$1}\right ) \text {$\#$1}^4-\log \left (\sqrt {c+\sqrt {b+a x}}-\text {$\#$1}\right ) \text {$\#$1}^6}{-a^2 c-8 b c \text {$\#$1}+8 c^3 \text {$\#$1}+3 a^2 \text {$\#$1}^2+8 b \text {$\#$1}^3-24 c^2 \text {$\#$1}^3+24 c \text {$\#$1}^5-8 \text {$\#$1}^7}\&\right ] \] Output:

Unintegrable
 

Mathematica [A] (verified)

Time = 0.45 (sec) , antiderivative size = 239, normalized size of antiderivative = 1.05 \[ \int \frac {x^2}{x^2-\sqrt {b+a x} \sqrt {c+\sqrt {b+a x}}} \, dx=\frac {b-c^2+a x}{a}-4 a \text {RootSum}\left [b^2-2 b c^2+c^4+a^2 c \text {$\#$1}+4 b c \text {$\#$1}^2-4 c^3 \text {$\#$1}^2-a^2 \text {$\#$1}^3-2 b \text {$\#$1}^4+6 c^2 \text {$\#$1}^4-4 c \text {$\#$1}^6+\text {$\#$1}^8\&,\frac {c^2 \log \left (\sqrt {c+\sqrt {b+a x}}-\text {$\#$1}\right ) \text {$\#$1}^2-2 c \log \left (\sqrt {c+\sqrt {b+a x}}-\text {$\#$1}\right ) \text {$\#$1}^4+\log \left (\sqrt {c+\sqrt {b+a x}}-\text {$\#$1}\right ) \text {$\#$1}^6}{-a^2 c-8 b c \text {$\#$1}+8 c^3 \text {$\#$1}+3 a^2 \text {$\#$1}^2+8 b \text {$\#$1}^3-24 c^2 \text {$\#$1}^3+24 c \text {$\#$1}^5-8 \text {$\#$1}^7}\&\right ] \] Input:

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

Output:

(b - c^2 + a*x)/a - 4*a*RootSum[b^2 - 2*b*c^2 + c^4 + a^2*c*#1 + 4*b*c*#1^ 
2 - 4*c^3*#1^2 - a^2*#1^3 - 2*b*#1^4 + 6*c^2*#1^4 - 4*c*#1^6 + #1^8 & , (c 
^2*Log[Sqrt[c + Sqrt[b + a*x]] - #1]*#1^2 - 2*c*Log[Sqrt[c + Sqrt[b + a*x] 
] - #1]*#1^4 + Log[Sqrt[c + Sqrt[b + a*x]] - #1]*#1^6)/(-(a^2*c) - 8*b*c*# 
1 + 8*c^3*#1 + 3*a^2*#1^2 + 8*b*#1^3 - 24*c^2*#1^3 + 24*c*#1^5 - 8*#1^7) & 
 ]
 

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.

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

\(\Big \downarrow \) 7267

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

\(\Big \downarrow \) 7267

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 7292

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

\(\Big \downarrow \) 7293

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

$Aborted
 

Maple [N/A] (verified)

Time = 0.08 (sec) , antiderivative size = 189, normalized size of antiderivative = 0.83

method	result	size
derivativedivides	\(-\frac {2 \left (-\frac {\left (c +\sqrt {a x +b}\right )^{2}}{2}+c \left (c +\sqrt {a x +b}\right )-2 a^{2} \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{8}-4 c \,\textit {\_Z}^{6}+\left (6 c^{2}-2 b \right ) \textit {\_Z}^{4}-a^{2} \textit {\_Z}^{3}+\left (-4 c^{3}+4 b c \right ) \textit {\_Z}^{2}+a^{2} c \textit {\_Z} +c^{4}-2 b \,c^{2}+b^{2}\right )}{\sum }\frac {\left (\textit {\_R}^{6}-2 \textit {\_R}^{4} c +\textit {\_R}^{2} c^{2}\right ) \ln \left (\sqrt {c +\sqrt {a x +b}}-\textit {\_R} \right )}{8 \textit {\_R}^{7}-24 \textit {\_R}^{5} c +24 \textit {\_R}^{3} c^{2}-8 \textit {\_R}^{3} b -3 \textit {\_R}^{2} a^{2}-8 \textit {\_R} \,c^{3}+8 \textit {\_R} b c +a^{2} c}\right )\right )}{a}\)	\(189\)
default	\(-\frac {2 \left (-\frac {\left (c +\sqrt {a x +b}\right )^{2}}{2}+c \left (c +\sqrt {a x +b}\right )-2 a^{2} \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{8}-4 c \,\textit {\_Z}^{6}+\left (6 c^{2}-2 b \right ) \textit {\_Z}^{4}-a^{2} \textit {\_Z}^{3}+\left (-4 c^{3}+4 b c \right ) \textit {\_Z}^{2}+a^{2} c \textit {\_Z} +c^{4}-2 b \,c^{2}+b^{2}\right )}{\sum }\frac {\left (\textit {\_R}^{6}-2 \textit {\_R}^{4} c +\textit {\_R}^{2} c^{2}\right ) \ln \left (\sqrt {c +\sqrt {a x +b}}-\textit {\_R} \right )}{8 \textit {\_R}^{7}-24 \textit {\_R}^{5} c +24 \textit {\_R}^{3} c^{2}-8 \textit {\_R}^{3} b -3 \textit {\_R}^{2} a^{2}-8 \textit {\_R} \,c^{3}+8 \textit {\_R} b c +a^{2} c}\right )\right )}{a}\)	\(189\)

Input:

int(x^2/(x^2-(a*x+b)^(1/2)*(c+(a*x+b)^(1/2))^(1/2)),x,method=_RETURNVERBOS 
E)
 

Output:

-2/a*(-1/2*(c+(a*x+b)^(1/2))^2+c*(c+(a*x+b)^(1/2))-2*a^2*sum((_R^6-2*_R^4* 
c+_R^2*c^2)/(8*_R^7-24*_R^5*c+24*_R^3*c^2-8*_R^3*b-3*_R^2*a^2-8*_R*c^3+8*_ 
R*b*c+a^2*c)*ln((c+(a*x+b)^(1/2))^(1/2)-_R),_R=RootOf(_Z^8-4*c*_Z^6+(6*c^2 
-2*b)*_Z^4-a^2*_Z^3+(-4*c^3+4*b*c)*_Z^2+a^2*c*_Z+c^4-2*b*c^2+b^2)))
 

Fricas [F(-1)]

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

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

Output:

Timed out
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [N/A]

Not integrable

Time = 0.19 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.14 \[ \int \frac {x^2}{x^2-\sqrt {b+a x} \sqrt {c+\sqrt {b+a x}}} \, dx=\int { \frac {x^{2}}{x^{2} - \sqrt {a x + b} \sqrt {c + \sqrt {a x + b}}} \,d x } \] Input:

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

Output:

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

Giac [N/A]

Not integrable

Time = 6.41 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.14 \[ \int \frac {x^2}{x^2-\sqrt {b+a x} \sqrt {c+\sqrt {b+a x}}} \, dx=\int { \frac {x^{2}}{x^{2} - \sqrt {a x + b} \sqrt {c + \sqrt {a x + b}}} \,d x } \] Input:

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

Output:

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

Mupad [N/A]

Not integrable

Time = 9.38 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.15 \[ \int \frac {x^2}{x^2-\sqrt {b+a x} \sqrt {c+\sqrt {b+a x}}} \, dx=-\int \frac {x^2}{\sqrt {c+\sqrt {b+a\,x}}\,\sqrt {b+a\,x}-x^2} \,d x \] Input:

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

Output:

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

Reduce [N/A]

Not integrable

Time = 200.03 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.14 \[ \int \frac {x^2}{x^2-\sqrt {b+a x} \sqrt {c+\sqrt {b+a x}}} \, dx=\int \frac {x^{2}}{x^{2}-\sqrt {a x +b}\, \sqrt {c +\sqrt {a x +b}}}d x \] Input:

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

Output:

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