Integrand size = 29, antiderivative size = 118 \[ \int \frac {1}{(f+e x) \sqrt {d+\sqrt {c+\sqrt {b+a x}}}} \, dx=\frac {\text {RootSum}\left [b e-c^2 e+2 c d^2 e-d^4 e-a f-4 c d e \text {$\#$1}^2+4 d^3 e \text {$\#$1}^2+2 c e \text {$\#$1}^4-6 d^2 e \text {$\#$1}^4+4 d e \text {$\#$1}^6-e \text {$\#$1}^8\&,\frac {\log \left (\sqrt {d+\sqrt {c+\sqrt {b+a x}}}-\text {$\#$1}\right )}{\text {$\#$1}}\&\right ]}{e} \] Output:
Unintegrable
Time = 0.27 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.00 \[ \int \frac {1}{(f+e x) \sqrt {d+\sqrt {c+\sqrt {b+a x}}}} \, dx=\frac {\text {RootSum}\left [b e-c^2 e+2 c d^2 e-d^4 e-a f-4 c d e \text {$\#$1}^2+4 d^3 e \text {$\#$1}^2+2 c e \text {$\#$1}^4-6 d^2 e \text {$\#$1}^4+4 d e \text {$\#$1}^6-e \text {$\#$1}^8\&,\frac {\log \left (\sqrt {d+\sqrt {c+\sqrt {b+a x}}}-\text {$\#$1}\right )}{\text {$\#$1}}\&\right ]}{e} \] Input:
Integrate[1/((f + e*x)*Sqrt[d + Sqrt[c + Sqrt[b + a*x]]]),x]
Output:
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 |
\(\displaystyle 8 \left (-d \left (c-d^2\right ) \int \frac {1}{-e (b+a x)^4+4 d e (b+a x)^3+2 c \left (1-\frac {3 d^2}{c}\right ) e (b+a x)^2-4 c d \left (1-\frac {d^2}{c}\right ) e (b+a x)+b e \left (1-\frac {e d^4-2 c e d^2+c^2 e+a f}{b e}\right )}d\sqrt {d+\sqrt {c+\sqrt {b+a x}}}+\left (c-3 d^2\right ) \int \frac {b+a x}{-e (b+a x)^4+4 d e (b+a x)^3+2 c \left (1-\frac {3 d^2}{c}\right ) e (b+a x)^2-4 c d \left (1-\frac {d^2}{c}\right ) e (b+a x)+b e \left (1-\frac {e d^4-2 c e d^2+c^2 e+a f}{b e}\right )}d\sqrt {d+\sqrt {c+\sqrt {b+a x}}}-\int \frac {(b+a x)^3}{-e (b+a x)^4+4 d e (b+a x)^3+2 c \left (1-\frac {3 d^2}{c}\right ) e (b+a x)^2-4 c d \left (1-\frac {d^2}{c}\right ) e (b+a x)+b e \left (1-\frac {e d^4-2 c e d^2+c^2 e+a f}{b e}\right )}d\sqrt {d+\sqrt {c+\sqrt {b+a x}}}-3 d \int \frac {(b+a x)^2}{e (b+a x)^4-4 d e (b+a x)^3-2 c \left (1-\frac {3 d^2}{c}\right ) e (b+a x)^2+4 c d \left (1-\frac {d^2}{c}\right ) e (b+a x)-b e \left (1-\frac {e d^4-2 c e d^2+c^2 e+a f}{b e}\right )}d\sqrt {d+\sqrt {c+\sqrt {b+a x}}}\right )\) |
Input:
Int[1/((f + e*x)*Sqrt[d + Sqrt[c + Sqrt[b + a*x]]]),x]
Output:
$Aborted
Time = 0.84 (sec) , antiderivative size = 165, normalized size of antiderivative = 1.40
method | result | size |
derivativedivides | \(-\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (e \,\textit {\_Z}^{8}-4 d e \,\textit {\_Z}^{6}+\left (6 d^{2} e -2 c e \right ) \textit {\_Z}^{4}+\left (-4 d^{3} e +4 c d e \right ) \textit {\_Z}^{2}+d^{4} e -2 c \,d^{2} e +c^{2} e +a f -b e \right )}{\sum }\frac {\left (-\textit {\_R}^{6}+3 d \,\textit {\_R}^{4}+\left (-3 d^{2}+c \right ) \textit {\_R}^{2}+d^{3}-d c \right ) \ln \left (\sqrt {d +\sqrt {c +\sqrt {a x +b}}}-\textit {\_R} \right )}{\textit {\_R}^{7}-3 \textit {\_R}^{5} d +3 \textit {\_R}^{3} d^{2}-\textit {\_R}^{3} c -\textit {\_R} \,d^{3}+\textit {\_R} c d}}{e}\) | \(165\) |
default | \(-\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (e \,\textit {\_Z}^{8}-4 d e \,\textit {\_Z}^{6}+\left (6 d^{2} e -2 c e \right ) \textit {\_Z}^{4}+\left (-4 d^{3} e +4 c d e \right ) \textit {\_Z}^{2}+d^{4} e -2 c \,d^{2} e +c^{2} e +a f -b e \right )}{\sum }\frac {\left (-\textit {\_R}^{6}+3 d \,\textit {\_R}^{4}+\left (-3 d^{2}+c \right ) \textit {\_R}^{2}+d^{3}-d c \right ) \ln \left (\sqrt {d +\sqrt {c +\sqrt {a x +b}}}-\textit {\_R} \right )}{\textit {\_R}^{7}-3 \textit {\_R}^{5} d +3 \textit {\_R}^{3} d^{2}-\textit {\_R}^{3} c -\textit {\_R} \,d^{3}+\textit {\_R} c d}}{e}\) | \(165\) |
Input:
int(1/(e*x+f)/(d+(c+(a*x+b)^(1/2))^(1/2))^(1/2),x,method=_RETURNVERBOSE)
Output:
-1/e*sum((-_R^6+3*d*_R^4+(-3*d^2+c)*_R^2+d^3-d*c)/(_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. \[ \int \frac {1}{(f+e x) \sqrt {d+\sqrt {c+\sqrt {b+a x}}}} \, dx=\text {Timed out} \] Input:
integrate(1/(e*x+f)/(d+(c+(a*x+b)^(1/2))^(1/2))^(1/2),x, algorithm="fricas ")
Output:
Timed out
Timed out. \[ \int \frac {1}{(f+e x) \sqrt {d+\sqrt {c+\sqrt {b+a x}}}} \, dx=\text {Timed out} \] Input:
integrate(1/(e*x+f)/(d+(c+(a*x+b)**(1/2))**(1/2))**(1/2),x)
Output:
Timed out
Not integrable
Time = 1.87 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.21 \[ \int \frac {1}{(f+e x) \sqrt {d+\sqrt {c+\sqrt {b+a x}}}} \, dx=\int { \frac {1}{{\left (e x + f\right )} \sqrt {d + \sqrt {c + \sqrt {a x + b}}}} \,d x } \] Input:
integrate(1/(e*x+f)/(d+(c+(a*x+b)^(1/2))^(1/2))^(1/2),x, algorithm="maxima ")
Output:
integrate(1/((e*x + f)*sqrt(d + sqrt(c + sqrt(a*x + b)))), x)
Result contains higher order function than in optimal. Order 3 vs. order 1.
Time = 24.86 (sec) , antiderivative size = 3653, normalized size of antiderivative = 30.96 \[ \int \frac {1}{(f+e x) \sqrt {d+\sqrt {c+\sqrt {b+a x}}}} \, dx=\text {Too large to display} \] Input:
integrate(1/(e*x+f)/(d+(c+(a*x+b)^(1/2))^(1/2))^(1/2),x, algorithm="giac")
Output:
-((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 = 8.36 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.21 \[ \int \frac {1}{(f+e x) \sqrt {d+\sqrt {c+\sqrt {b+a x}}}} \, dx=\int \frac {1}{\left (f+e\,x\right )\,\sqrt {d+\sqrt {c+\sqrt {b+a\,x}}}} \,d x \] Input:
int(1/((f + e*x)*(d + (c + (b + a*x)^(1/2))^(1/2))^(1/2)),x)
Output:
int(1/((f + e*x)*(d + (c + (b + a*x)^(1/2))^(1/2))^(1/2)), x)
Not integrable
Time = 1.72 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.04 \[ \int \frac {1}{(f+e x) \sqrt {d+\sqrt {c+\sqrt {b+a x}}}} \, dx=-\left (\int \frac {\sqrt {\sqrt {\sqrt {a x +b}+c}+d}}{\sqrt {a x +b}\, e x +\sqrt {a x +b}\, f +c e x +c f -d^{2} e x -d^{2} f}d x \right ) d +\int \frac {\sqrt {\sqrt {a x +b}+c}\, \sqrt {\sqrt {\sqrt {a x +b}+c}+d}}{\sqrt {a x +b}\, e x +\sqrt {a x +b}\, f +c e x +c f -d^{2} e x -d^{2} f}d x \] Input:
int(1/(e*x+f)/(d+(c+(a*x+b)^(1/2))^(1/2))^(1/2),x)
Output:
- int(sqrt(sqrt(sqrt(a*x + b) + c) + d)/(sqrt(a*x + b)*e*x + sqrt(a*x + b )*f + c*e*x + c*f - d**2*e*x - d**2*f),x)*d + int((sqrt(sqrt(a*x + b) + c) *sqrt(sqrt(sqrt(a*x + b) + c) + d))/(sqrt(a*x + b)*e*x + sqrt(a*x + b)*f + c*e*x + c*f - d**2*e*x - d**2*f),x)