Integrand size = 34, antiderivative size = 305 \[ \int \frac {x}{x^2-\sqrt {b+a x} \sqrt {c+\sqrt {b+a x}}} \, dx=-4 \text {RootSum}\left [b^2-2 b c^2+c^4+a^2 c \text {$\#$1}+4 b c \text {$\#$1}^2-4 c^3 \text {$\#$1}^2-a^2 \text {$\#$1}^3-2 b \text {$\#$1}^4+6 c^2 \text {$\#$1}^4-4 c \text {$\#$1}^6+\text {$\#$1}^8\&,\frac {-b c \log \left (\sqrt {c+\sqrt {b+a x}}-\text {$\#$1}\right ) \text {$\#$1}+c^3 \log \left (\sqrt {c+\sqrt {b+a x}}-\text {$\#$1}\right ) \text {$\#$1}+b \log \left (\sqrt {c+\sqrt {b+a x}}-\text {$\#$1}\right ) \text {$\#$1}^3-3 c^2 \log \left (\sqrt {c+\sqrt {b+a x}}-\text {$\#$1}\right ) \text {$\#$1}^3+3 c \log \left (\sqrt {c+\sqrt {b+a x}}-\text {$\#$1}\right ) \text {$\#$1}^5-\log \left (\sqrt {c+\sqrt {b+a x}}-\text {$\#$1}\right ) \text {$\#$1}^7}{a^2 c+8 b c \text {$\#$1}-8 c^3 \text {$\#$1}-3 a^2 \text {$\#$1}^2-8 b \text {$\#$1}^3+24 c^2 \text {$\#$1}^3-24 c \text {$\#$1}^5+8 \text {$\#$1}^7}\&\right ] \]
Time = 0.00 (sec) , antiderivative size = 305, normalized size of antiderivative = 1.00 \[ \int \frac {x}{x^2-\sqrt {b+a x} \sqrt {c+\sqrt {b+a x}}} \, dx=-4 \text {RootSum}\left [b^2-2 b c^2+c^4+a^2 c \text {$\#$1}+4 b c \text {$\#$1}^2-4 c^3 \text {$\#$1}^2-a^2 \text {$\#$1}^3-2 b \text {$\#$1}^4+6 c^2 \text {$\#$1}^4-4 c \text {$\#$1}^6+\text {$\#$1}^8\&,\frac {-b c \log \left (\sqrt {c+\sqrt {b+a x}}-\text {$\#$1}\right ) \text {$\#$1}+c^3 \log \left (\sqrt {c+\sqrt {b+a x}}-\text {$\#$1}\right ) \text {$\#$1}+b \log \left (\sqrt {c+\sqrt {b+a x}}-\text {$\#$1}\right ) \text {$\#$1}^3-3 c^2 \log \left (\sqrt {c+\sqrt {b+a x}}-\text {$\#$1}\right ) \text {$\#$1}^3+3 c \log \left (\sqrt {c+\sqrt {b+a x}}-\text {$\#$1}\right ) \text {$\#$1}^5-\log \left (\sqrt {c+\sqrt {b+a x}}-\text {$\#$1}\right ) \text {$\#$1}^7}{a^2 c+8 b c \text {$\#$1}-8 c^3 \text {$\#$1}-3 a^2 \text {$\#$1}^2-8 b \text {$\#$1}^3+24 c^2 \text {$\#$1}^3-24 c \text {$\#$1}^5+8 \text {$\#$1}^7}\&\right ] \]
-4*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 & , (-(b*c*Log[Sqrt[c + Sqr t[b + a*x]] - #1]*#1) + c^3*Log[Sqrt[c + Sqrt[b + a*x]] - #1]*#1 + b*Log[S qrt[c + Sqrt[b + a*x]] - #1]*#1^3 - 3*c^2*Log[Sqrt[c + Sqrt[b + a*x]] - #1 ]*#1^3 + 3*c*Log[Sqrt[c + Sqrt[b + a*x]] - #1]*#1^5 - Log[Sqrt[c + Sqrt[b + a*x]] - #1]*#1^7)/(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) & ]
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.
| \(\displaystyle \int \frac {x}{x^2-\sqrt {a x+b} \sqrt {\sqrt {a x+b}+c}} \, dx\) |
| \(\Big \downarrow \) 7267 |
| \(\displaystyle 2 \int \frac {a x \sqrt {b+a x}}{-\sqrt {b+a x} \sqrt {c+\sqrt {b+a x}} a^2+b^2+(b+a x)^2-2 b (b+a x)}d\sqrt {b+a x}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -2 \int -\frac {a x \sqrt {b+a x}}{-\sqrt {b+a x} \sqrt {c+\sqrt {b+a x}} a^2+b^2+(b+a x)^2-2 b (b+a x)}d\sqrt {b+a x}\) |
| \(\Big \downarrow \) 7267 |
| \(\displaystyle -4 \int -\frac {(-b+c-a x) \left (b-(-b+c-a x)^2\right ) \sqrt {c+\sqrt {b+a x}}}{(-b+c-a x)^4-2 b (-b+c-a x)^2+a^2 \sqrt {c+\sqrt {b+a x}} (-b+c-a x)+b^2}d\sqrt {c+\sqrt {b+a x}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle 4 \int \frac {(-b+c-a x) \left (b-(-b+c-a x)^2\right ) \sqrt {c+\sqrt {b+a x}}}{(-b+c-a x)^4-2 b (-b+c-a x)^2+a^2 \sqrt {c+\sqrt {b+a x}} (-b+c-a x)+b^2}d\sqrt {c+\sqrt {b+a x}}\) |
| \(\Big \downarrow \) 2525 |
| \(\displaystyle -4 \left (-\frac {1}{8} \int -\frac {a^2 (c-3 (b+a x))}{(-b+c-a x)^4-2 b (-b+c-a x)^2+a^2 \sqrt {c+\sqrt {b+a x}} (-b+c-a x)+b^2}d\sqrt {c+\sqrt {b+a x}}-\frac {1}{8} \log \left (a^2 \sqrt {\sqrt {a x+b}+c} (-a x-b+c)+(-a x-b+c)^4-2 b (-a x-b+c)^2+b^2\right )\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -4 \left (\frac {1}{8} \int \frac {a^2 (c-3 (b+a x))}{(-b+c-a x)^4-2 b (-b+c-a x)^2+a^2 \sqrt {c+\sqrt {b+a x}} (-b+c-a x)+b^2}d\sqrt {c+\sqrt {b+a x}}-\frac {1}{8} \log \left (a^2 \sqrt {\sqrt {a x+b}+c} (-a x-b+c)+(-a x-b+c)^4-2 b (-a x-b+c)^2+b^2\right )\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -4 \left (\frac {1}{8} a^2 \int \frac {c-3 (b+a x)}{(-b+c-a x)^4-2 b (-b+c-a x)^2+a^2 \sqrt {c+\sqrt {b+a x}} (-b+c-a x)+b^2}d\sqrt {c+\sqrt {b+a x}}-\frac {1}{8} \log \left (a^2 \sqrt {\sqrt {a x+b}+c} (-a x-b+c)+(-a x-b+c)^4-2 b (-a x-b+c)^2+b^2\right )\right )\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -4 \left (\frac {1}{8} a^2 \int \left (\frac {c}{(b+a x)^4-4 c (b+a x)^3-2 b \left (1-\frac {3 c^2}{b}\right ) (b+a x)^2-a^2 (b+a x)^{3/2}+4 b c \left (1-\frac {c^2}{b}\right ) (b+a x)+b^2 \left (\frac {c^4-2 b c^2}{b^2}+1\right )+a^2 c \sqrt {c+\sqrt {b+a x}}}+\frac {3 (b+a x)}{-(b+a x)^4+4 c (b+a x)^3+2 b \left (1-\frac {3 c^2}{b}\right ) (b+a x)^2+a^2 (b+a x)^{3/2}-4 b c \left (1-\frac {c^2}{b}\right ) (b+a x)-b^2 \left (\frac {c^4-2 b c^2}{b^2}+1\right )-a^2 c \sqrt {c+\sqrt {b+a x}}}\right )d\sqrt {c+\sqrt {b+a x}}-\frac {1}{8} \log \left (a^2 \sqrt {\sqrt {a x+b}+c} (-a x-b+c)+(-a x-b+c)^4-2 b (-a x-b+c)^2+b^2\right )\right )\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -4 \left (\frac {1}{8} a^2 \left (3 \int \frac {b+a x}{-(b+a x)^4+4 c (b+a x)^3+2 b \left (1-\frac {3 c^2}{b}\right ) (b+a x)^2+a^2 (b+a x)^{3/2}-4 b c \left (1-\frac {c^2}{b}\right ) (b+a x)-b^2 \left (\frac {c^4-2 b c^2}{b^2}+1\right )-a^2 c \sqrt {c+\sqrt {b+a x}}}d\sqrt {c+\sqrt {b+a x}}+c \int \frac {1}{(b+a x)^4-4 c (b+a x)^3-2 b \left (1-\frac {3 c^2}{b}\right ) (b+a x)^2-a^2 (b+a x)^{3/2}+4 b c \left (1-\frac {c^2}{b}\right ) (b+a x)+b^2 \left (\frac {c^4-2 b c^2}{b^2}+1\right )+a^2 c \sqrt {c+\sqrt {b+a x}}}d\sqrt {c+\sqrt {b+a x}}\right )-\frac {1}{8} \log \left (a^2 \sqrt {\sqrt {a x+b}+c} (-a x-b+c)+(-a x-b+c)^4-2 b (-a x-b+c)^2+b^2\right )\right )\) |
3.29.70.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[(Pm_)/(Qn_), x_Symbol] :> With[{m = Expon[Pm, x], n = Expon[Qn, x]}, Si mp[Coeff[Pm, x, m]*(Log[Qn]/(n*Coeff[Qn, x, n])), x] + Simp[1/(n*Coeff[Qn, x, n]) Int[ExpandToSum[n*Coeff[Qn, x, n]*Pm - Coeff[Pm, x, m]*D[Qn, x], x ]/Qn, x], x] /; EqQ[m, n - 1]] /; PolyQ[Pm, x] && PolyQ[Qn, x]
Int[u_, x_Symbol] :> With[{lst = SubstForFractionalPowerOfLinear[u, x]}, Si mp[lst[[2]]*lst[[4]] Subst[Int[lst[[1]], x], x, lst[[3]]^(1/lst[[2]])], x ] /; !FalseQ[lst] && SubstForFractionalPowerQ[u, lst[[3]], x]]
Time = 0.06 (sec) , antiderivative size = 172, normalized size of antiderivative = 0.56
| method | result | size |
| derivativedivides | \(4 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{8}-4 c \,\textit {\_Z}^{6}+\left (6 c^{2}-2 b \right ) \textit {\_Z}^{4}-a^{2} \textit {\_Z}^{3}+\left (-4 c^{3}+4 b c \right ) \textit {\_Z}^{2}+a^{2} c \textit {\_Z} +c^{4}-2 b \,c^{2}+b^{2}\right )}{\sum }\frac {\left (\textit {\_R}^{7}-3 c \,\textit {\_R}^{5}+\left (3 c^{2}-b \right ) \textit {\_R}^{3}+c \left (-c^{2}+b \right ) \textit {\_R} \right ) \ln \left (\sqrt {c +\sqrt {a x +b}}-\textit {\_R} \right )}{8 \textit {\_R}^{7}-24 c \,\textit {\_R}^{5}+24 \textit {\_R}^{3} c^{2}-8 \textit {\_R}^{3} b -3 \textit {\_R}^{2} a^{2}-8 \textit {\_R} \,c^{3}+8 \textit {\_R} b c +a^{2} c}\right )\) | \(172\) |
| default | \(4 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{8}-4 c \,\textit {\_Z}^{6}+\left (6 c^{2}-2 b \right ) \textit {\_Z}^{4}-a^{2} \textit {\_Z}^{3}+\left (-4 c^{3}+4 b c \right ) \textit {\_Z}^{2}+a^{2} c \textit {\_Z} +c^{4}-2 b \,c^{2}+b^{2}\right )}{\sum }\frac {\left (\textit {\_R}^{7}-3 c \,\textit {\_R}^{5}+\left (3 c^{2}-b \right ) \textit {\_R}^{3}+c \left (-c^{2}+b \right ) \textit {\_R} \right ) \ln \left (\sqrt {c +\sqrt {a x +b}}-\textit {\_R} \right )}{8 \textit {\_R}^{7}-24 c \,\textit {\_R}^{5}+24 \textit {\_R}^{3} c^{2}-8 \textit {\_R}^{3} b -3 \textit {\_R}^{2} a^{2}-8 \textit {\_R} \,c^{3}+8 \textit {\_R} b c +a^{2} c}\right )\) | \(172\) |
4*sum((_R^7-3*c*_R^5+(3*c^2-b)*_R^3+c*(-c^2+b)*_R)/(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))
Timed out. \[ \int \frac {x}{x^2-\sqrt {b+a x} \sqrt {c+\sqrt {b+a x}}} \, dx=\text {Timed out} \]
Timed out. \[ \int \frac {x}{x^2-\sqrt {b+a x} \sqrt {c+\sqrt {b+a x}}} \, dx=\text {Timed out} \]
Not integrable
Time = 0.28 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.10 \[ \int \frac {x}{x^2-\sqrt {b+a x} \sqrt {c+\sqrt {b+a x}}} \, dx=\int { \frac {x}{x^{2} - \sqrt {a x + b} \sqrt {c + \sqrt {a x + b}}} \,d x } \]
Not integrable
Time = 2.43 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.10 \[ \int \frac {x}{x^2-\sqrt {b+a x} \sqrt {c+\sqrt {b+a x}}} \, dx=\int { \frac {x}{x^{2} - \sqrt {a x + b} \sqrt {c + \sqrt {a x + b}}} \,d x } \]
Not integrable
Time = 0.00 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.11 \[ \int \frac {x}{x^2-\sqrt {b+a x} \sqrt {c+\sqrt {b+a x}}} \, dx=-\int \frac {x}{\sqrt {c+\sqrt {b+a\,x}}\,\sqrt {b+a\,x}-x^2} \,d x \]