Integrand size = 45, antiderivative size = 455 \[ \int \frac {x^2 \sqrt {b+a x}}{x^2-\sqrt {b+a x} \sqrt {c+\sqrt {b+a x}}} \, dx=\frac {2 (b+a x)^{3/2}}{3 a}+4 a \sqrt {c+\sqrt {b+a x}}-4 a \text {RootSum}\left [b^2-2 b c^2+c^4+a^2 c \text {$\#$1}+4 b c \text {$\#$1}^2-4 c^3 \text {$\#$1}^2-a^2 \text {$\#$1}^3-2 b \text {$\#$1}^4+6 c^2 \text {$\#$1}^4-4 c \text {$\#$1}^6+\text {$\#$1}^8\&,\frac {b^2 \log \left (\sqrt {c+\sqrt {b+a x}}-\text {$\#$1}\right )-2 b c^2 \log \left (\sqrt {c+\sqrt {b+a x}}-\text {$\#$1}\right )+c^4 \log \left (\sqrt {c+\sqrt {b+a x}}-\text {$\#$1}\right )+a^2 c \log \left (\sqrt {c+\sqrt {b+a x}}-\text {$\#$1}\right ) \text {$\#$1}+4 b c \log \left (\sqrt {c+\sqrt {b+a x}}-\text {$\#$1}\right ) \text {$\#$1}^2-3 c^3 \log \left (\sqrt {c+\sqrt {b+a x}}-\text {$\#$1}\right ) \text {$\#$1}^2-a^2 \log \left (\sqrt {c+\sqrt {b+a x}}-\text {$\#$1}\right ) \text {$\#$1}^3-2 b \log \left (\sqrt {c+\sqrt {b+a x}}-\text {$\#$1}\right ) \text {$\#$1}^4+3 c^2 \log \left (\sqrt {c+\sqrt {b+a x}}-\text {$\#$1}\right ) \text {$\#$1}^4-c \log \left (\sqrt {c+\sqrt {b+a x}}-\text {$\#$1}\right ) \text {$\#$1}^6}{a^2 c+8 b c \text {$\#$1}-8 c^3 \text {$\#$1}-3 a^2 \text {$\#$1}^2-8 b \text {$\#$1}^3+24 c^2 \text {$\#$1}^3-24 c \text {$\#$1}^5+8 \text {$\#$1}^7}\&\right ] \]
Time = 0.00 (sec) , antiderivative size = 459, normalized size of antiderivative = 1.01 \[ \int \frac {x^2 \sqrt {b+a x}}{x^2-\sqrt {b+a x} \sqrt {c+\sqrt {b+a x}}} \, dx=4 a \sqrt {c+\sqrt {b+a x}}+\frac {2 \left (c^3+(b+a x)^{3/2}\right )}{3 a}-4 a \text {RootSum}\left [b^2-2 b c^2+c^4+a^2 c \text {$\#$1}+4 b c \text {$\#$1}^2-4 c^3 \text {$\#$1}^2-a^2 \text {$\#$1}^3-2 b \text {$\#$1}^4+6 c^2 \text {$\#$1}^4-4 c \text {$\#$1}^6+\text {$\#$1}^8\&,\frac {b^2 \log \left (\sqrt {c+\sqrt {b+a x}}-\text {$\#$1}\right )-2 b c^2 \log \left (\sqrt {c+\sqrt {b+a x}}-\text {$\#$1}\right )+c^4 \log \left (\sqrt {c+\sqrt {b+a x}}-\text {$\#$1}\right )+a^2 c \log \left (\sqrt {c+\sqrt {b+a x}}-\text {$\#$1}\right ) \text {$\#$1}+4 b c \log \left (\sqrt {c+\sqrt {b+a x}}-\text {$\#$1}\right ) \text {$\#$1}^2-3 c^3 \log \left (\sqrt {c+\sqrt {b+a x}}-\text {$\#$1}\right ) \text {$\#$1}^2-a^2 \log \left (\sqrt {c+\sqrt {b+a x}}-\text {$\#$1}\right ) \text {$\#$1}^3-2 b \log \left (\sqrt {c+\sqrt {b+a x}}-\text {$\#$1}\right ) \text {$\#$1}^4+3 c^2 \log \left (\sqrt {c+\sqrt {b+a x}}-\text {$\#$1}\right ) \text {$\#$1}^4-c \log \left (\sqrt {c+\sqrt {b+a x}}-\text {$\#$1}\right ) \text {$\#$1}^6}{a^2 c+8 b c \text {$\#$1}-8 c^3 \text {$\#$1}-3 a^2 \text {$\#$1}^2-8 b \text {$\#$1}^3+24 c^2 \text {$\#$1}^3-24 c \text {$\#$1}^5+8 \text {$\#$1}^7}\&\right ] \]
4*a*Sqrt[c + Sqrt[b + a*x]] + (2*(c^3 + (b + a*x)^(3/2)))/(3*a) - 4*a*Root Sum[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^2*Log[Sqrt[c + Sqrt[b + a*x ]] - #1] - 2*b*c^2*Log[Sqrt[c + Sqrt[b + a*x]] - #1] + c^4*Log[Sqrt[c + Sq rt[b + a*x]] - #1] + a^2*c*Log[Sqrt[c + Sqrt[b + a*x]] - #1]*#1 + 4*b*c*Lo g[Sqrt[c + Sqrt[b + a*x]] - #1]*#1^2 - 3*c^3*Log[Sqrt[c + Sqrt[b + a*x]] - #1]*#1^2 - a^2*Log[Sqrt[c + Sqrt[b + a*x]] - #1]*#1^3 - 2*b*Log[Sqrt[c + Sqrt[b + a*x]] - #1]*#1^4 + 3*c^2*Log[Sqrt[c + Sqrt[b + a*x]] - #1]*#1^4 - c*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) & ]
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.
| \(\displaystyle \int \frac {x^2 \sqrt {a x+b}}{x^2-\sqrt {a x+b} \sqrt {\sqrt {a x+b}+c}} \, dx\) |
| \(\Big \downarrow \) 7267 |
| \(\displaystyle \frac {2 \int \frac {\left (b \sqrt {b+a x}-(b+a x)^{3/2}\right )^2}{-\sqrt {b+a x} \sqrt {c+\sqrt {b+a x}} a^2+b^2+(b+a x)^2-2 b (b+a x)}d\sqrt {b+a x}}{a}\) |
| \(\Big \downarrow \) 2027 |
| \(\displaystyle \frac {2 \int \frac {a^2 x^2 (b+a x)}{-\sqrt {b+a x} \sqrt {c+\sqrt {b+a x}} a^2+b^2+(b+a x)^2-2 b (b+a x)}d\sqrt {b+a x}}{a}\) |
| \(\Big \downarrow \) 7267 |
| \(\displaystyle \frac {4 \int \frac {(-b+c-a x)^2 \left (b-(-b+c-a x)^2\right )^2 \sqrt {c+\sqrt {b+a x}}}{(-b+c-a x)^4-2 b (-b+c-a x)^2+a^2 \sqrt {c+\sqrt {b+a x}} (-b+c-a x)+b^2}d\sqrt {c+\sqrt {b+a x}}}{a}\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \frac {4 \int \frac {(-b+c-a x)^2 \sqrt {c+\sqrt {b+a x}} \left (-c^2+2 (b+a x) c-(b+a x)^2+b\right )^2}{(-b+c-a x)^4-2 b (-b+c-a x)^2+a^2 \sqrt {c+\sqrt {b+a x}} (-b+c-a x)+b^2}d\sqrt {c+\sqrt {b+a x}}}{a}\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {4 \int \left ((b+a x)^{5/2}-2 c (b+a x)^{3/2}+a^2+c^2 \sqrt {c+\sqrt {b+a x}}-\frac {-(b+a x)^{3/2} a^4+c \sqrt {c+\sqrt {b+a x}} a^4-c (b+a x)^3 a^2+\left (b-c^2\right )^2 a^2-\left (2 b-3 c^2\right ) (b+a x)^2 a^2+c \left (4 b-3 c^2\right ) (b+a x) a^2}{(-b+c-a x)^4-2 b (-b+c-a x)^2+a^2 \sqrt {c+\sqrt {b+a x}} (-b+c-a x)+b^2}\right )d\sqrt {c+\sqrt {b+a x}}}{a}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {4 \left (-\int \frac {(b+a x)^{3/2}}{-(b+a x)^4+4 c (b+a x)^3+2 b \left (1-\frac {3 c^2}{b}\right ) (b+a x)^2+a^2 (b+a x)^{3/2}-4 b c \left (1-\frac {c^2}{b}\right ) (b+a x)-b^2 \left (\frac {c^4-2 b c^2}{b^2}+1\right )-a^2 c \sqrt {c+\sqrt {b+a x}}}d\sqrt {c+\sqrt {b+a x}} a^4-c \int \frac {\sqrt {c+\sqrt {b+a x}}}{(b+a x)^4-4 c (b+a x)^3-2 b \left (1-\frac {3 c^2}{b}\right ) (b+a x)^2-a^2 (b+a x)^{3/2}+4 b c \left (1-\frac {c^2}{b}\right ) (b+a x)+b^2 \left (\frac {c^4-2 b c^2}{b^2}+1\right )+a^2 c \sqrt {c+\sqrt {b+a x}}}d\sqrt {c+\sqrt {b+a x}} a^4-c \int \frac {(b+a x)^3}{-(b+a x)^4+4 c (b+a x)^3+2 b \left (1-\frac {3 c^2}{b}\right ) (b+a x)^2+a^2 (b+a x)^{3/2}-4 b c \left (1-\frac {c^2}{b}\right ) (b+a x)-b^2 \left (\frac {c^4-2 b c^2}{b^2}+1\right )-a^2 c \sqrt {c+\sqrt {b+a x}}}d\sqrt {c+\sqrt {b+a x}} a^2-\left (b-c^2\right )^2 \int \frac {1}{(b+a x)^4-4 c (b+a x)^3-2 b \left (1-\frac {3 c^2}{b}\right ) (b+a x)^2-a^2 (b+a x)^{3/2}+4 b c \left (1-\frac {c^2}{b}\right ) (b+a x)+b^2 \left (\frac {c^4-2 b c^2}{b^2}+1\right )+a^2 c \sqrt {c+\sqrt {b+a x}}}d\sqrt {c+\sqrt {b+a x}} a^2-c \left (4 b-3 c^2\right ) \int \frac {b+a x}{(b+a x)^4-4 c (b+a x)^3-2 b \left (1-\frac {3 c^2}{b}\right ) (b+a x)^2-a^2 (b+a x)^{3/2}+4 b c \left (1-\frac {c^2}{b}\right ) (b+a x)+b^2 \left (\frac {c^4-2 b c^2}{b^2}+1\right )+a^2 c \sqrt {c+\sqrt {b+a x}}}d\sqrt {c+\sqrt {b+a x}} a^2+\left (2 b-3 c^2\right ) \int \frac {(b+a x)^2}{(b+a x)^4-4 c (b+a x)^3-2 b \left (1-\frac {3 c^2}{b}\right ) (b+a x)^2-a^2 (b+a x)^{3/2}+4 b c \left (1-\frac {c^2}{b}\right ) (b+a x)+b^2 \left (\frac {c^4-2 b c^2}{b^2}+1\right )+a^2 c \sqrt {c+\sqrt {b+a x}}}d\sqrt {c+\sqrt {b+a x}} a^2+\sqrt {c+\sqrt {b+a x}} a^2+\frac {1}{6} (b+a x)^3-\frac {1}{2} c (b+a x)^2+\frac {1}{2} c^2 (b+a x)\right )}{a}\) |
3.31.44.3.1 Defintions of rubi rules used
Int[(Fx_.)*((a_.)*(x_)^(r_.) + (b_.)*(x_)^(s_.))^(p_.), x_Symbol] :> Int[x^ (p*r)*(a + b*x^(s - r))^p*Fx, x] /; FreeQ[{a, b, r, s}, x] && IntegerQ[p] & & PosQ[s - r] && !(EqQ[p, 1] && EqQ[u, 1])
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 = 0.00 (sec) , antiderivative size = 264, normalized size of antiderivative = 0.58
| method | result | size |
| derivativedivides | \(-\frac {2 \left (-\frac {\left (c +\sqrt {a x +b}\right )^{3}}{3}+c \left (c +\sqrt {a x +b}\right )^{2}-c^{2} \left (c +\sqrt {a x +b}\right )-2 \sqrt {c +\sqrt {a x +b}}\, a^{2}+2 a^{2} \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{8}-4 c \,\textit {\_Z}^{6}+\left (6 c^{2}-2 b \right ) \textit {\_Z}^{4}-a^{2} \textit {\_Z}^{3}+\left (-4 c^{3}+4 b c \right ) \textit {\_Z}^{2}+a^{2} c \textit {\_Z} +c^{4}-2 b \,c^{2}+b^{2}\right )}{\sum }\frac {\left (-c \,\textit {\_R}^{6}+\left (3 c^{2}-2 b \right ) \textit {\_R}^{4}-a^{2} \textit {\_R}^{3}+c \left (-3 c^{2}+4 b \right ) \textit {\_R}^{2}+a^{2} c \textit {\_R} +c^{4}-2 b \,c^{2}+b^{2}\right ) \ln \left (\sqrt {c +\sqrt {a x +b}}-\textit {\_R} \right )}{8 \textit {\_R}^{7}-24 \textit {\_R}^{5} c +24 \textit {\_R}^{3} c^{2}-8 \textit {\_R}^{3} b -3 \textit {\_R}^{2} a^{2}-8 \textit {\_R} \,c^{3}+8 \textit {\_R} b c +a^{2} c}\right )\right )}{a}\) | \(264\) |
| default | \(-\frac {2 \left (-\frac {\left (c +\sqrt {a x +b}\right )^{3}}{3}+c \left (c +\sqrt {a x +b}\right )^{2}-c^{2} \left (c +\sqrt {a x +b}\right )-2 \sqrt {c +\sqrt {a x +b}}\, a^{2}+2 a^{2} \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{8}-4 c \,\textit {\_Z}^{6}+\left (6 c^{2}-2 b \right ) \textit {\_Z}^{4}-a^{2} \textit {\_Z}^{3}+\left (-4 c^{3}+4 b c \right ) \textit {\_Z}^{2}+a^{2} c \textit {\_Z} +c^{4}-2 b \,c^{2}+b^{2}\right )}{\sum }\frac {\left (-c \,\textit {\_R}^{6}+\left (3 c^{2}-2 b \right ) \textit {\_R}^{4}-a^{2} \textit {\_R}^{3}+c \left (-3 c^{2}+4 b \right ) \textit {\_R}^{2}+a^{2} c \textit {\_R} +c^{4}-2 b \,c^{2}+b^{2}\right ) \ln \left (\sqrt {c +\sqrt {a x +b}}-\textit {\_R} \right )}{8 \textit {\_R}^{7}-24 \textit {\_R}^{5} c +24 \textit {\_R}^{3} c^{2}-8 \textit {\_R}^{3} b -3 \textit {\_R}^{2} a^{2}-8 \textit {\_R} \,c^{3}+8 \textit {\_R} b c +a^{2} c}\right )\right )}{a}\) | \(264\) |
-2/a*(-1/3*(c+(a*x+b)^(1/2))^3+c*(c+(a*x+b)^(1/2))^2-c^2*(c+(a*x+b)^(1/2)) -2*(c+(a*x+b)^(1/2))^(1/2)*a^2+2*a^2*sum((-c*_R^6+(3*c^2-2*b)*_R^4-a^2*_R^ 3+c*(-3*c^2+4*b)*_R^2+a^2*c*_R+c^4-2*b*c^2+b^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)))
Timed out. \[ \int \frac {x^2 \sqrt {b+a x}}{x^2-\sqrt {b+a x} \sqrt {c+\sqrt {b+a x}}} \, dx=\text {Timed out} \]
integrate(x^2*(a*x+b)^(1/2)/(x^2-(a*x+b)^(1/2)*(c+(a*x+b)^(1/2))^(1/2)),x, algorithm="fricas")
Timed out. \[ \int \frac {x^2 \sqrt {b+a x}}{x^2-\sqrt {b+a x} \sqrt {c+\sqrt {b+a x}}} \, dx=\text {Timed out} \]
Not integrable
Time = 0.39 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.09 \[ \int \frac {x^2 \sqrt {b+a x}}{x^2-\sqrt {b+a x} \sqrt {c+\sqrt {b+a x}}} \, dx=\int { \frac {\sqrt {a x + b} x^{2}}{x^{2} - \sqrt {a x + b} \sqrt {c + \sqrt {a x + b}}} \,d x } \]
integrate(x^2*(a*x+b)^(1/2)/(x^2-(a*x+b)^(1/2)*(c+(a*x+b)^(1/2))^(1/2)),x, algorithm="maxima")
Not integrable
Time = 2.79 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.09 \[ \int \frac {x^2 \sqrt {b+a x}}{x^2-\sqrt {b+a x} \sqrt {c+\sqrt {b+a x}}} \, dx=\int { \frac {\sqrt {a x + b} x^{2}}{x^{2} - \sqrt {a x + b} \sqrt {c + \sqrt {a x + b}}} \,d x } \]
Not integrable
Time = 0.00 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.09 \[ \int \frac {x^2 \sqrt {b+a x}}{x^2-\sqrt {b+a x} \sqrt {c+\sqrt {b+a x}}} \, dx=-\int \frac {x^2\,\sqrt {b+a\,x}}{\sqrt {c+\sqrt {b+a\,x}}\,\sqrt {b+a\,x}-x^2} \,d x \]