Integrand size = 29, antiderivative size = 118 \[ \int \frac {1}{(f+e x) \sqrt {d+\sqrt {c+\sqrt {b+a x}}}} \, dx=\frac {\text {RootSum}\left [b e-c^2 e+2 c d^2 e-d^4 e-a f-4 c d e \text {$\#$1}^2+4 d^3 e \text {$\#$1}^2+2 c e \text {$\#$1}^4-6 d^2 e \text {$\#$1}^4+4 d e \text {$\#$1}^6-e \text {$\#$1}^8\&,\frac {\log \left (\sqrt {d+\sqrt {c+\sqrt {b+a x}}}-\text {$\#$1}\right )}{\text {$\#$1}}\&\right ]}{e} \]
Time = 0.27 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.00 \[ \int \frac {1}{(f+e x) \sqrt {d+\sqrt {c+\sqrt {b+a x}}}} \, dx=\frac {\text {RootSum}\left [b e-c^2 e+2 c d^2 e-d^4 e-a f-4 c d e \text {$\#$1}^2+4 d^3 e \text {$\#$1}^2+2 c e \text {$\#$1}^4-6 d^2 e \text {$\#$1}^4+4 d e \text {$\#$1}^6-e \text {$\#$1}^8\&,\frac {\log \left (\sqrt {d+\sqrt {c+\sqrt {b+a x}}}-\text {$\#$1}\right )}{\text {$\#$1}}\&\right ]}{e} \]
RootSum[b*e - c^2*e + 2*c*d^2*e - d^4*e - a*f - 4*c*d*e*#1^2 + 4*d^3*e*#1^ 2 + 2*c*e*#1^4 - 6*d^2*e*#1^4 + 4*d*e*#1^6 - e*#1^8 & , Log[Sqrt[d + Sqrt[ c + Sqrt[b + a*x]]] - #1]/#1 & ]/e
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.
| \(\displaystyle \int \frac {1}{(e x+f) \sqrt {\sqrt {\sqrt {a x+b}+c}+d}} \, dx\) |
| \(\Big \downarrow \) 7267 |
| \(\displaystyle 2 \int -\frac {\sqrt {b+a x}}{(b e-(b+a x) e-a f) \sqrt {d+\sqrt {c+\sqrt {b+a x}}}}d\sqrt {b+a x}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -2 \int \frac {\sqrt {b+a x}}{(b e-(b+a x) e-a f) \sqrt {d+\sqrt {c+\sqrt {b+a x}}}}d\sqrt {b+a x}\) |
| \(\Big \downarrow \) 7267 |
| \(\displaystyle -4 \int -\frac {(-b+c-a x) \sqrt {c+\sqrt {b+a x}}}{\left (-e (-b+c-a x)^2+b e-a f\right ) \sqrt {d+\sqrt {c+\sqrt {b+a x}}}}d\sqrt {c+\sqrt {b+a x}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle 4 \int \frac {(-b+c-a x) \sqrt {c+\sqrt {b+a x}}}{\left (-e (-b+c-a x)^2+b e-a f\right ) \sqrt {d+\sqrt {c+\sqrt {b+a x}}}}d\sqrt {c+\sqrt {b+a x}}\) |
| \(\Big \downarrow \) 2091 |
| \(\displaystyle 4 \int \frac {(-b+c-a x) \sqrt {c+\sqrt {b+a x}}}{\left (-e c^2+2 e (b+a x) c-e (b+a x)^2+b e-a f\right ) \sqrt {d+\sqrt {c+\sqrt {b+a x}}}}d\sqrt {c+\sqrt {b+a x}}\) |
| \(\Big \downarrow \) 7267 |
| \(\displaystyle 8 \int -\frac {(-b+d-a x) \left (c-(-b+d-a x)^2\right )}{-e (-b+d-a x)^4+2 c e (-b+d-a x)^2-c^2 e+b e-a f}d\sqrt {d+\sqrt {c+\sqrt {b+a x}}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -8 \int \frac {(-b+d-a x) \left (c-(-b+d-a x)^2\right )}{-e (-b+d-a x)^4+2 c e (-b+d-a x)^2-c^2 e+b e-a f}d\sqrt {d+\sqrt {c+\sqrt {b+a x}}}\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle -8 \int \frac {(-b+d-a x) \left (-d^2+2 (b+a x) d-(b+a x)^2+c\right )}{-e (-b+d-a x)^4+2 c e (-b+d-a x)^2+b e \left (1-\frac {e c^2+a f}{b e}\right )}d\sqrt {d+\sqrt {c+\sqrt {b+a x}}}\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -8 \int \left (\frac {(b+a x)^3}{-e (b+a x)^4+4 d e (b+a x)^3+2 c \left (1-\frac {3 d^2}{c}\right ) e (b+a x)^2-4 c d \left (1-\frac {d^2}{c}\right ) e (b+a x)+b e \left (1-\frac {e d^4-2 c e d^2+c^2 e+a f}{b e}\right )}+\frac {3 d (b+a x)^2}{e (b+a x)^4-4 d e (b+a x)^3-2 c \left (1-\frac {3 d^2}{c}\right ) e (b+a x)^2+4 c d \left (1-\frac {d^2}{c}\right ) e (b+a x)-b e \left (1-\frac {e d^4-2 c e d^2+c^2 e+a f}{b e}\right )}+\frac {3 \left (1-\frac {c}{3 d^2}\right ) d^2 (b+a x)}{-e (b+a x)^4+4 d e (b+a x)^3+2 c \left (1-\frac {3 d^2}{c}\right ) e (b+a x)^2-4 c d \left (1-\frac {d^2}{c}\right ) e (b+a x)+b e \left (1-\frac {e d^4-2 c e d^2+c^2 e+a f}{b e}\right )}+\frac {c d \left (1-\frac {d^2}{c}\right )}{-e (b+a x)^4+4 d e (b+a x)^3+2 c \left (1-\frac {3 d^2}{c}\right ) e (b+a x)^2-4 c d \left (1-\frac {d^2}{c}\right ) e (b+a x)+b e \left (1-\frac {e d^4-2 c e d^2+c^2 e+a f}{b e}\right )}\right )d\sqrt {d+\sqrt {c+\sqrt {b+a x}}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle 8 \left (-d \left (c-d^2\right ) \int \frac {1}{-e (b+a x)^4+4 d e (b+a x)^3+2 c \left (1-\frac {3 d^2}{c}\right ) e (b+a x)^2-4 c d \left (1-\frac {d^2}{c}\right ) e (b+a x)+b e \left (1-\frac {e d^4-2 c e d^2+c^2 e+a f}{b e}\right )}d\sqrt {d+\sqrt {c+\sqrt {b+a x}}}+\left (c-3 d^2\right ) \int \frac {b+a x}{-e (b+a x)^4+4 d e (b+a x)^3+2 c \left (1-\frac {3 d^2}{c}\right ) e (b+a x)^2-4 c d \left (1-\frac {d^2}{c}\right ) e (b+a x)+b e \left (1-\frac {e d^4-2 c e d^2+c^2 e+a f}{b e}\right )}d\sqrt {d+\sqrt {c+\sqrt {b+a x}}}-\int \frac {(b+a x)^3}{-e (b+a x)^4+4 d e (b+a x)^3+2 c \left (1-\frac {3 d^2}{c}\right ) e (b+a x)^2-4 c d \left (1-\frac {d^2}{c}\right ) e (b+a x)+b e \left (1-\frac {e d^4-2 c e d^2+c^2 e+a f}{b e}\right )}d\sqrt {d+\sqrt {c+\sqrt {b+a x}}}-3 d \int \frac {(b+a x)^2}{e (b+a x)^4-4 d e (b+a x)^3-2 c \left (1-\frac {3 d^2}{c}\right ) e (b+a x)^2+4 c d \left (1-\frac {d^2}{c}\right ) e (b+a x)-b e \left (1-\frac {e d^4-2 c e d^2+c^2 e+a f}{b e}\right )}d\sqrt {d+\sqrt {c+\sqrt {b+a x}}}\right )\) |
3.18.60.3.1 Defintions of rubi rules used
Int[(Px_)*(u_)^(p_.)*(z_)^(q_.), x_Symbol] :> Int[Px*ExpandToSum[z, x]^q*Ex pandToSum[u, x]^p, x] /; FreeQ[{p, q}, x] && PolyQ[Px, x] && BinomialQ[z, x ] && TrinomialQ[u, x] && !(BinomialMatchQ[z, x] && TrinomialMatchQ[u, 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 = 0.58 (sec) , antiderivative size = 165, normalized size of antiderivative = 1.40
| method | result | size |
| derivativedivides | \(-\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (e \,\textit {\_Z}^{8}-4 d e \,\textit {\_Z}^{6}+\left (6 d^{2} e -2 c e \right ) \textit {\_Z}^{4}+\left (-4 d^{3} e +4 c d e \right ) \textit {\_Z}^{2}+d^{4} e -2 c \,d^{2} e +c^{2} e +a f -b e \right )}{\sum }\frac {\left (-\textit {\_R}^{6}+3 d \,\textit {\_R}^{4}+\left (-3 d^{2}+c \right ) \textit {\_R}^{2}+d^{3}-c d \right ) \ln \left (\sqrt {d +\sqrt {c +\sqrt {a x +b}}}-\textit {\_R} \right )}{\textit {\_R}^{7}-3 \textit {\_R}^{5} d +3 \textit {\_R}^{3} d^{2}-c \,\textit {\_R}^{3}-\textit {\_R} \,d^{3}+\textit {\_R} c d}}{e}\) | \(165\) |
| default | \(-\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (e \,\textit {\_Z}^{8}-4 d e \,\textit {\_Z}^{6}+\left (6 d^{2} e -2 c e \right ) \textit {\_Z}^{4}+\left (-4 d^{3} e +4 c d e \right ) \textit {\_Z}^{2}+d^{4} e -2 c \,d^{2} e +c^{2} e +a f -b e \right )}{\sum }\frac {\left (-\textit {\_R}^{6}+3 d \,\textit {\_R}^{4}+\left (-3 d^{2}+c \right ) \textit {\_R}^{2}+d^{3}-c d \right ) \ln \left (\sqrt {d +\sqrt {c +\sqrt {a x +b}}}-\textit {\_R} \right )}{\textit {\_R}^{7}-3 \textit {\_R}^{5} d +3 \textit {\_R}^{3} d^{2}-c \,\textit {\_R}^{3}-\textit {\_R} \,d^{3}+\textit {\_R} c d}}{e}\) | \(165\) |
-1/e*sum((-_R^6+3*d*_R^4+(-3*d^2+c)*_R^2+d^3-c*d)/(_R^7-3*_R^5*d+3*_R^3*d^ 2-_R^3*c-_R*d^3+_R*c*d)*ln((d+(c+(a*x+b)^(1/2))^(1/2))^(1/2)-_R),_R=RootOf (e*_Z^8-4*d*e*_Z^6+(6*d^2*e-2*c*e)*_Z^4+(-4*d^3*e+4*c*d*e)*_Z^2+d^4*e-2*c* d^2*e+c^2*e+a*f-b*e))
Timed out. \[ \int \frac {1}{(f+e x) \sqrt {d+\sqrt {c+\sqrt {b+a x}}}} \, dx=\text {Timed out} \]
Timed out. \[ \int \frac {1}{(f+e x) \sqrt {d+\sqrt {c+\sqrt {b+a x}}}} \, dx=\text {Timed out} \]
Not integrable
Time = 1.34 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.21 \[ \int \frac {1}{(f+e x) \sqrt {d+\sqrt {c+\sqrt {b+a x}}}} \, dx=\int { \frac {1}{{\left (e x + f\right )} \sqrt {d + \sqrt {c + \sqrt {a x + b}}}} \,d x } \]
Result contains higher order function than in optimal. Order 3 vs. order 1.
Time = 16.79 (sec) , antiderivative size = 3653, normalized size of antiderivative = 30.96 \[ \int \frac {1}{(f+e x) \sqrt {d+\sqrt {c+\sqrt {b+a x}}}} \, dx=\text {Too large to display} \]
-((d + sqrt((c*e + sqrt(b*e^2 - a*e*f))/e))^3*sgn(sqrt(c + sqrt(a*x + b))) - 3*(d + sqrt((c*e + sqrt(b*e^2 - a*e*f))/e))^2*d*sgn(sqrt(c + sqrt(a*x + b))) + 3*(d + sqrt((c*e + sqrt(b*e^2 - a*e*f))/e))*d^2*sgn(sqrt(c + sqrt( a*x + b))) - d^3*sgn(sqrt(c + sqrt(a*x + b))) - c*(d + sqrt((c*e + sqrt(b* e^2 - a*e*f))/e))*sgn(sqrt(c + sqrt(a*x + b))) + c*d*sgn(sqrt(c + sqrt(a*x + b))))*log(sqrt(d + sqrt(c + sqrt(a*x + b))) + sqrt(d + sqrt((c*e + sqrt (b*e^2 - a*e*f))/e)))/((d + sqrt((c*e + sqrt(b*e^2 - a*e*f))/e))^(7/2)*e - 3*(d + sqrt((c*e + sqrt(b*e^2 - a*e*f))/e))^(5/2)*d*e + 3*(d + sqrt((c*e + sqrt(b*e^2 - a*e*f))/e))^(3/2)*d^2*e - sqrt(d + sqrt((c*e + sqrt(b*e^2 - a*e*f))/e))*d^3*e - c*(d + sqrt((c*e + sqrt(b*e^2 - a*e*f))/e))^(3/2)*e + c*sqrt(d + sqrt((c*e + sqrt(b*e^2 - a*e*f))/e))*d*e) + ((d + sqrt((c*e + sqrt(b*e^2 - a*e*f))/e))^3*sgn(sqrt(c + sqrt(a*x + b))) - 3*(d + sqrt((c*e + sqrt(b*e^2 - a*e*f))/e))^2*d*sgn(sqrt(c + sqrt(a*x + b))) + 3*(d + sqrt ((c*e + sqrt(b*e^2 - a*e*f))/e))*d^2*sgn(sqrt(c + sqrt(a*x + b))) - d^3*sg n(sqrt(c + sqrt(a*x + b))) - c*(d + sqrt((c*e + sqrt(b*e^2 - a*e*f))/e))*s gn(sqrt(c + sqrt(a*x + b))) + c*d*sgn(sqrt(c + sqrt(a*x + b))))*log(sqrt(d + sqrt(c + sqrt(a*x + b))) - sqrt(d + sqrt((c*e + sqrt(b*e^2 - a*e*f))/e) ))/((d + sqrt((c*e + sqrt(b*e^2 - a*e*f))/e))^(7/2)*e - 3*(d + sqrt((c*e + sqrt(b*e^2 - a*e*f))/e))^(5/2)*d*e + 3*(d + sqrt((c*e + sqrt(b*e^2 - a*e* f))/e))^(3/2)*d^2*e - sqrt(d + sqrt((c*e + sqrt(b*e^2 - a*e*f))/e))*d^3...
Not integrable
Time = 5.71 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.21 \[ \int \frac {1}{(f+e x) \sqrt {d+\sqrt {c+\sqrt {b+a x}}}} \, dx=\int \frac {1}{\left (f+e\,x\right )\,\sqrt {d+\sqrt {c+\sqrt {b+a\,x}}}} \,d x \]