Integrand size = 32, antiderivative size = 380 \[ \int \frac {x}{x-\sqrt {b+a x} \sqrt {c+\sqrt {b+a x}}} \, dx=x+\frac {4}{3} \left (3 a^2+c\right ) \sqrt {c+\sqrt {b+a x}}+\sqrt {b+a x} \left (2 a+\frac {4}{3} \sqrt {c+\sqrt {b+a x}}\right )-4 \text {RootSum}\left [b-c^2-a c \text {$\#$1}+2 c \text {$\#$1}^2+a \text {$\#$1}^3-\text {$\#$1}^4\&,\frac {-a^2 b \log \left (\sqrt {c+\sqrt {b+a x}}-\text {$\#$1}\right )+a^2 c^2 \log \left (\sqrt {c+\sqrt {b+a x}}-\text {$\#$1}\right )-a b \log \left (\sqrt {c+\sqrt {b+a x}}-\text {$\#$1}\right ) \text {$\#$1}+a^3 c \log \left (\sqrt {c+\sqrt {b+a x}}-\text {$\#$1}\right ) \text {$\#$1}+a c^2 \log \left (\sqrt {c+\sqrt {b+a x}}-\text {$\#$1}\right ) \text {$\#$1}-b \log \left (\sqrt {c+\sqrt {b+a x}}-\text {$\#$1}\right ) \text {$\#$1}^2-a^2 c \log \left (\sqrt {c+\sqrt {b+a x}}-\text {$\#$1}\right ) \text {$\#$1}^2-a^3 \log \left (\sqrt {c+\sqrt {b+a x}}-\text {$\#$1}\right ) \text {$\#$1}^3-a c \log \left (\sqrt {c+\sqrt {b+a x}}-\text {$\#$1}\right ) \text {$\#$1}^3}{a c-4 c \text {$\#$1}-3 a \text {$\#$1}^2+4 \text {$\#$1}^3}\&\right ] \]
Time = 0.00 (sec) , antiderivative size = 383, normalized size of antiderivative = 1.01 \[ \int \frac {x}{x-\sqrt {b+a x} \sqrt {c+\sqrt {b+a x}}} \, dx=\frac {b}{a}-\frac {c^2}{a}+x+2 a \left (c+\sqrt {b+a x}\right )+\frac {4}{3} \sqrt {c+\sqrt {b+a x}} \left (3 a^2+c+\sqrt {b+a x}\right )-4 \text {RootSum}\left [b-c^2-a c \text {$\#$1}+2 c \text {$\#$1}^2+a \text {$\#$1}^3-\text {$\#$1}^4\&,\frac {-a^2 b \log \left (\sqrt {c+\sqrt {b+a x}}-\text {$\#$1}\right )+a^2 c^2 \log \left (\sqrt {c+\sqrt {b+a x}}-\text {$\#$1}\right )-a b \log \left (\sqrt {c+\sqrt {b+a x}}-\text {$\#$1}\right ) \text {$\#$1}+a^3 c \log \left (\sqrt {c+\sqrt {b+a x}}-\text {$\#$1}\right ) \text {$\#$1}+a c^2 \log \left (\sqrt {c+\sqrt {b+a x}}-\text {$\#$1}\right ) \text {$\#$1}-b \log \left (\sqrt {c+\sqrt {b+a x}}-\text {$\#$1}\right ) \text {$\#$1}^2-a^2 c \log \left (\sqrt {c+\sqrt {b+a x}}-\text {$\#$1}\right ) \text {$\#$1}^2-a^3 \log \left (\sqrt {c+\sqrt {b+a x}}-\text {$\#$1}\right ) \text {$\#$1}^3-a c \log \left (\sqrt {c+\sqrt {b+a x}}-\text {$\#$1}\right ) \text {$\#$1}^3}{a c-4 c \text {$\#$1}-3 a \text {$\#$1}^2+4 \text {$\#$1}^3}\&\right ] \]
b/a - c^2/a + x + 2*a*(c + Sqrt[b + a*x]) + (4*Sqrt[c + Sqrt[b + a*x]]*(3* a^2 + c + Sqrt[b + a*x]))/3 - 4*RootSum[b - c^2 - a*c*#1 + 2*c*#1^2 + a*#1 ^3 - #1^4 & , (-(a^2*b*Log[Sqrt[c + Sqrt[b + a*x]] - #1]) + a^2*c^2*Log[Sq rt[c + Sqrt[b + a*x]] - #1] - a*b*Log[Sqrt[c + Sqrt[b + a*x]] - #1]*#1 + a ^3*c*Log[Sqrt[c + Sqrt[b + a*x]] - #1]*#1 + a*c^2*Log[Sqrt[c + Sqrt[b + a* x]] - #1]*#1 - b*Log[Sqrt[c + Sqrt[b + a*x]] - #1]*#1^2 - a^2*c*Log[Sqrt[c + Sqrt[b + a*x]] - #1]*#1^2 - a^3*Log[Sqrt[c + Sqrt[b + a*x]] - #1]*#1^3 - a*c*Log[Sqrt[c + Sqrt[b + a*x]] - #1]*#1^3)/(a*c - 4*c*#1 - 3*a*#1^2 + 4 *#1^3) & ]
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.
| \(\displaystyle \int \frac {x}{x-\sqrt {a x+b} \sqrt {\sqrt {a x+b}+c}} \, dx\) |
| \(\Big \downarrow \) 7267 |
| \(\displaystyle \frac {2 \int -\frac {a x \sqrt {b+a x}}{b-\sqrt {b+a x} \left (\sqrt {b+a x}-a \sqrt {c+\sqrt {b+a x}}\right )}d\sqrt {b+a x}}{a}\) |
| \(\Big \downarrow \) 7267 |
| \(\displaystyle \frac {4 \int -\frac {(-b+c-a x) \left (b-(-b+c-a x)^2\right ) \sqrt {c+\sqrt {b+a x}}}{b-(-b+c-a x) \left (c+\sqrt {c+\sqrt {b+a x}} \left (a-\sqrt {c+\sqrt {b+a x}}\right )\right )}d\sqrt {c+\sqrt {b+a x}}}{a}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {4 \int \frac {(-b+c-a x) \left (b-(-b+c-a x)^2\right ) \sqrt {c+\sqrt {b+a x}}}{b-(-b+c-a x) \left (c+\sqrt {c+\sqrt {b+a x}} \left (a-\sqrt {c+\sqrt {b+a x}}\right )\right )}d\sqrt {c+\sqrt {b+a x}}}{a}\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle -\frac {4 \int \frac {(-b+c-a x) \sqrt {c+\sqrt {b+a x}} \left (-c^2+2 (b+a x) c-(b+a x)^2+b\right )}{-c^2+2 (b+a x) c-a \sqrt {c+\sqrt {b+a x}} c-(b+a x)^2+a (b+a x)^{3/2}+b}d\sqrt {c+\sqrt {b+a x}}}{a}\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -\frac {4 \int \left (-a^3-(b+a x) a-(b+a x)^{3/2}+\frac {\left (b-c^2\right ) a^3+\left (a^2+c\right ) (b+a x)^{3/2} a^2+\left (b-c \left (a^2+c\right )\right ) \sqrt {c+\sqrt {b+a x}} a^2+\left (c a^2+b\right ) (b+a x) a}{-c^2+2 (b+a x) c-a \sqrt {c+\sqrt {b+a x}} c-(b+a x)^2+a (b+a x)^{3/2}+b}-\left (a^2-c\right ) \sqrt {c+\sqrt {b+a x}}\right )d\sqrt {c+\sqrt {b+a x}}}{a}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {4 \left (-a^2 b \int \frac {\sqrt {c+\sqrt {b+a x}}}{-c^2+2 (b+a x) c-a \sqrt {c+\sqrt {b+a x}} c-(b+a x)^2+a (b+a x)^{3/2}+b}d\sqrt {c+\sqrt {b+a x}}+\frac {1}{4} a \left (3 a^4+7 a^2 c+4 b\right ) \int \frac {b+a x}{c^2-2 (b+a x) c+a \sqrt {c+\sqrt {b+a x}} c+(b+a x)^2-a (b+a x)^{3/2}-b}d\sqrt {c+\sqrt {b+a x}}-\frac {1}{4} a^3 \left (4 b-c \left (a^2+5 c\right )\right ) \int \frac {1}{-c^2+2 (b+a x) c-a \sqrt {c+\sqrt {b+a x}} c-(b+a x)^2+a (b+a x)^{3/2}+b}d\sqrt {c+\sqrt {b+a x}}+a^3 \sqrt {\sqrt {a x+b}+c}+\frac {1}{4} a^2 \left (a^2+c\right ) \log \left (2 c (a x+b)-a c \sqrt {\sqrt {a x+b}+c}-(a x+b)^2+a (a x+b)^{3/2}+b-c^2\right )+\frac {1}{2} \left (a^2-c\right ) (a x+b)+\frac {1}{3} a (a x+b)^{3/2}+\frac {1}{4} (a x+b)^2\right )}{a}\) |
3.30.77.3.1 Defintions of rubi rules used
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 = 203, normalized size of antiderivative = 0.53
| method | result | size |
| derivativedivides | \(-\frac {2 \left (-2 a^{3} \sqrt {c +\sqrt {a x +b}}-a^{2} \left (c +\sqrt {a x +b}\right )-\frac {2 a \left (c +\sqrt {a x +b}\right )^{\frac {3}{2}}}{3}-\frac {\left (c +\sqrt {a x +b}\right )^{2}}{2}+c \left (c +\sqrt {a x +b}\right )+2 a \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{4}-a \,\textit {\_Z}^{3}-2 c \,\textit {\_Z}^{2}+a c \textit {\_Z} +c^{2}-b \right )}{\sum }\frac {\left (a \left (-a^{2}-c \right ) \textit {\_R}^{3}+\left (-a^{2} c -b \right ) \textit {\_R}^{2}+a \left (a^{2} c +c^{2}-b \right ) \textit {\_R} +a^{2} c^{2}-a^{2} b \right ) \ln \left (\sqrt {c +\sqrt {a x +b}}-\textit {\_R} \right )}{4 \textit {\_R}^{3}-3 \textit {\_R}^{2} a -4 \textit {\_R} c +a c}\right )\right )}{a}\) | \(203\) |
| default | \(\frac {4 a^{3} \sqrt {c +\sqrt {a x +b}}+2 a^{2} \left (c +\sqrt {a x +b}\right )+\frac {4 a \left (c +\sqrt {a x +b}\right )^{\frac {3}{2}}}{3}+\left (c +\sqrt {a x +b}\right )^{2}-2 c \left (c +\sqrt {a x +b}\right )-4 a \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{4}-a \,\textit {\_Z}^{3}-2 c \,\textit {\_Z}^{2}+a c \textit {\_Z} +c^{2}-b \right )}{\sum }\frac {\left (a \left (-a^{2}-c \right ) \textit {\_R}^{3}+\left (-a^{2} c -b \right ) \textit {\_R}^{2}+a \left (a^{2} c +c^{2}-b \right ) \textit {\_R} +a^{2} c^{2}-a^{2} b \right ) \ln \left (\sqrt {c +\sqrt {a x +b}}-\textit {\_R} \right )}{4 \textit {\_R}^{3}-3 \textit {\_R}^{2} a -4 \textit {\_R} c +a c}\right )}{a}\) | \(203\) |
-2/a*(-2*a^3*(c+(a*x+b)^(1/2))^(1/2)-a^2*(c+(a*x+b)^(1/2))-2/3*a*(c+(a*x+b )^(1/2))^(3/2)-1/2*(c+(a*x+b)^(1/2))^2+c*(c+(a*x+b)^(1/2))+2*a*sum((a*(-a^ 2-c)*_R^3+(-a^2*c-b)*_R^2+a*(a^2*c+c^2-b)*_R+a^2*c^2-a^2*b)/(4*_R^3-3*_R^2 *a-4*_R*c+a*c)*ln((c+(a*x+b)^(1/2))^(1/2)-_R),_R=RootOf(_Z^4-_Z^3*a-2*_Z^2 *c+_Z*a*c+c^2-b)))
Exception generated. \[ \int \frac {x}{x-\sqrt {b+a x} \sqrt {c+\sqrt {b+a x}}} \, dx=\text {Exception raised: AttributeError} \]
Timed out. \[ \int \frac {x}{x-\sqrt {b+a x} \sqrt {c+\sqrt {b+a x}}} \, dx=\text {Timed out} \]
Not integrable
Time = 0.31 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.08 \[ \int \frac {x}{x-\sqrt {b+a x} \sqrt {c+\sqrt {b+a x}}} \, dx=\int { -\frac {x}{\sqrt {a x + b} \sqrt {c + \sqrt {a x + b}} - x} \,d x } \]
Not integrable
Time = 0.57 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.08 \[ \int \frac {x}{x-\sqrt {b+a x} \sqrt {c+\sqrt {b+a x}}} \, dx=\int { -\frac {x}{\sqrt {a x + b} \sqrt {c + \sqrt {a x + b}} - x} \,d x } \]
Not integrable
Time = 0.00 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.07 \[ \int \frac {x}{x-\sqrt {b+a x} \sqrt {c+\sqrt {b+a x}}} \, dx=\int \frac {x}{x-\sqrt {c+\sqrt {b+a\,x}}\,\sqrt {b+a\,x}} \,d x \]