\(\int \frac {1}{(f+e x) \sqrt {d+\sqrt {c+\sqrt {b+a x}}}} \, dx\) [1760]

Optimal result

Integrand size = 29, antiderivative size = 118 \[ \int \frac {1}{(f+e x) \sqrt {d+\sqrt {c+\sqrt {b+a x}}}} \, dx=\frac {\text {RootSum}\left [b e-c^2 e+2 c d^2 e-d^4 e-a f-4 c d e \text {$\#$1}^2+4 d^3 e \text {$\#$1}^2+2 c e \text {$\#$1}^4-6 d^2 e \text {$\#$1}^4+4 d e \text {$\#$1}^6-e \text {$\#$1}^8\&,\frac {\log \left (\sqrt {d+\sqrt {c+\sqrt {b+a x}}}-\text {$\#$1}\right )}{\text {$\#$1}}\&\right ]}{e} \]

[Out]

Unintegrable

Rubi [F]

\[ \int \frac {1}{(f+e x) \sqrt {d+\sqrt {c+\sqrt {b+a x}}}} \, dx=\int \frac {1}{(f+e x) \sqrt {d+\sqrt {c+\sqrt {b+a x}}}} \, dx \]

[In]

Int[1/((f + e*x)*Sqrt[d + Sqrt[c + Sqrt[b + a*x]]]),x]

[Out]

-8*d*(c - d^2)*Defer[Subst][Defer[Int][(b*e*(1 - (c^2*e - 2*c*d^2*e + d^4*e + a*f)/(b*e)) - 4*c*d*(1 - d^2/c)*
e*x^2 + 2*c*(1 - (3*d^2)/c)*e*x^4 + 4*d*e*x^6 - e*x^8)^(-1), x], x, Sqrt[d + Sqrt[c + Sqrt[b + a*x]]]] + 8*(c
- 3*d^2)*Defer[Subst][Defer[Int][x^2/(b*e*(1 - (c^2*e - 2*c*d^2*e + d^4*e + a*f)/(b*e)) - 4*c*d*(1 - d^2/c)*e*
x^2 + 2*c*(1 - (3*d^2)/c)*e*x^4 + 4*d*e*x^6 - e*x^8), x], x, Sqrt[d + Sqrt[c + Sqrt[b + a*x]]]] + 24*d*Defer[S
ubst][Defer[Int][x^4/(b*e*(1 - (c^2*e - 2*c*d^2*e + d^4*e + a*f)/(b*e)) - 4*c*d*(1 - d^2/c)*e*x^2 + 2*c*(1 - (
3*d^2)/c)*e*x^4 + 4*d*e*x^6 - e*x^8), x], x, Sqrt[d + Sqrt[c + Sqrt[b + a*x]]]] + 8*Defer[Subst][Defer[Int][x^
6/(-(b*e*(1 - (c^2*e - 2*c*d^2*e + d^4*e + a*f)/(b*e))) + 4*c*d*(1 - d^2/c)*e*x^2 - 2*c*(1 - (3*d^2)/c)*e*x^4
- 4*d*e*x^6 + e*x^8), x], x, Sqrt[d + Sqrt[c + Sqrt[b + a*x]]]]

Rubi steps \begin{align*} \text {integral}& = 2 \text {Subst}\left (\int \frac {x}{\left (-b e+a f+e x^2\right ) \sqrt {d+\sqrt {c+x}}} \, dx,x,\sqrt {b+a x}\right ) \\ & = 4 \text {Subst}\left (\int \frac {x \left (-c+x^2\right )}{\sqrt {d+x} \left (-b e+a f+e \left (c-x^2\right )^2\right )} \, dx,x,\sqrt {c+\sqrt {b+a x}}\right ) \\ & = 8 \text {Subst}\left (\int \frac {\left (-d+x^2\right ) \left (-c+\left (d-x^2\right )^2\right )}{-b e+a f+e \left (c-\left (d-x^2\right )^2\right )^2} \, dx,x,\sqrt {d+\sqrt {c+\sqrt {b+a x}}}\right ) \\ & = 8 \text {Subst}\left (\int \frac {\left (d-x^2\right ) \left (-c+d^2-2 d x^2+x^4\right )}{b e \left (1-\frac {a f}{b e}\right )-e \left (c-\left (d-x^2\right )^2\right )^2} \, dx,x,\sqrt {d+\sqrt {c+\sqrt {b+a x}}}\right ) \\ & = 8 \text {Subst}\left (\int \left (\frac {\left (1-\frac {c}{d^2}\right ) d^3}{b e \left (1-\frac {c^2 e-2 c d^2 e+d^4 e+a f}{b e}\right )-4 c d \left (1-\frac {d^2}{c}\right ) e x^2+2 c \left (1-\frac {3 d^2}{c}\right ) e x^4+4 d e x^6-e x^8}+\frac {c \left (1-\frac {3 d^2}{c}\right ) x^2}{b e \left (1-\frac {c^2 e-2 c d^2 e+d^4 e+a f}{b e}\right )-4 c d \left (1-\frac {d^2}{c}\right ) e x^2+2 c \left (1-\frac {3 d^2}{c}\right ) e x^4+4 d e x^6-e x^8}+\frac {3 d x^4}{b e \left (1-\frac {c^2 e-2 c d^2 e+d^4 e+a f}{b e}\right )-4 c d \left (1-\frac {d^2}{c}\right ) e x^2+2 c \left (1-\frac {3 d^2}{c}\right ) e x^4+4 d e x^6-e x^8}+\frac {x^6}{-b e \left (1-\frac {c^2 e-2 c d^2 e+d^4 e+a f}{b e}\right )+4 c d \left (1-\frac {d^2}{c}\right ) e x^2-2 c \left (1-\frac {3 d^2}{c}\right ) e x^4-4 d e x^6+e x^8}\right ) \, dx,x,\sqrt {d+\sqrt {c+\sqrt {b+a x}}}\right ) \\ & = 8 \text {Subst}\left (\int \frac {x^6}{-b e \left (1-\frac {c^2 e-2 c d^2 e+d^4 e+a f}{b e}\right )+4 c d \left (1-\frac {d^2}{c}\right ) e x^2-2 c \left (1-\frac {3 d^2}{c}\right ) e x^4-4 d e x^6+e x^8} \, dx,x,\sqrt {d+\sqrt {c+\sqrt {b+a x}}}\right )+(24 d) \text {Subst}\left (\int \frac {x^4}{b e \left (1-\frac {c^2 e-2 c d^2 e+d^4 e+a f}{b e}\right )-4 c d \left (1-\frac {d^2}{c}\right ) e x^2+2 c \left (1-\frac {3 d^2}{c}\right ) e x^4+4 d e x^6-e x^8} \, dx,x,\sqrt {d+\sqrt {c+\sqrt {b+a x}}}\right )+\left (8 \left (c-3 d^2\right )\right ) \text {Subst}\left (\int \frac {x^2}{b e \left (1-\frac {c^2 e-2 c d^2 e+d^4 e+a f}{b e}\right )-4 c d \left (1-\frac {d^2}{c}\right ) e x^2+2 c \left (1-\frac {3 d^2}{c}\right ) e x^4+4 d e x^6-e x^8} \, dx,x,\sqrt {d+\sqrt {c+\sqrt {b+a x}}}\right )-\left (8 d \left (c-d^2\right )\right ) \text {Subst}\left (\int \frac {1}{b e \left (1-\frac {c^2 e-2 c d^2 e+d^4 e+a f}{b e}\right )-4 c d \left (1-\frac {d^2}{c}\right ) e x^2+2 c \left (1-\frac {3 d^2}{c}\right ) e x^4+4 d e x^6-e x^8} \, dx,x,\sqrt {d+\sqrt {c+\sqrt {b+a x}}}\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.27 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.00 \[ \int \frac {1}{(f+e x) \sqrt {d+\sqrt {c+\sqrt {b+a x}}}} \, dx=\frac {\text {RootSum}\left [b e-c^2 e+2 c d^2 e-d^4 e-a f-4 c d e \text {$\#$1}^2+4 d^3 e \text {$\#$1}^2+2 c e \text {$\#$1}^4-6 d^2 e \text {$\#$1}^4+4 d e \text {$\#$1}^6-e \text {$\#$1}^8\&,\frac {\log \left (\sqrt {d+\sqrt {c+\sqrt {b+a x}}}-\text {$\#$1}\right )}{\text {$\#$1}}\&\right ]}{e} \]

[In]

Integrate[1/((f + e*x)*Sqrt[d + Sqrt[c + Sqrt[b + a*x]]]),x]

[Out]

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

Maple [N/A] (verified)

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	\(-\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (e \,\textit {\_Z}^{8}-4 d e \,\textit {\_Z}^{6}+\left (6 d^{2} e -2 c e \right ) \textit {\_Z}^{4}+\left (-4 d^{3} e +4 c d e \right ) \textit {\_Z}^{2}+d^{4} e -2 c \,d^{2} e +c^{2} e +a f -b e \right )}{\sum }\frac {\left (-\textit {\_R}^{6}+3 d \,\textit {\_R}^{4}+\left (-3 d^{2}+c \right ) \textit {\_R}^{2}+d^{3}-c d \right ) \ln \left (\sqrt {d +\sqrt {c +\sqrt {a x +b}}}-\textit {\_R} \right )}{\textit {\_R}^{7}-3 \textit {\_R}^{5} d +3 \textit {\_R}^{3} d^{2}-c \,\textit {\_R}^{3}-\textit {\_R} \,d^{3}+\textit {\_R} c d}}{e}\)	\(165\)

[In]

int(1/(e*x+f)/(d+(c+(a*x+b)^(1/2))^(1/2))^(1/2),x,method=_RETURNVERBOSE)

[Out]

-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))

Fricas [F(-1)]

Timed out. \[ \int \frac {1}{(f+e x) \sqrt {d+\sqrt {c+\sqrt {b+a x}}}} \, dx=\text {Timed out} \]

[In]

integrate(1/(e*x+f)/(d+(c+(a*x+b)^(1/2))^(1/2))^(1/2),x, algorithm="fricas")

[Out]

Timed out

Sympy [F(-1)]

Timed out. \[ \int \frac {1}{(f+e x) \sqrt {d+\sqrt {c+\sqrt {b+a x}}}} \, dx=\text {Timed out} \]

[In]

integrate(1/(e*x+f)/(d+(c+(a*x+b)**(1/2))**(1/2))**(1/2),x)

[Out]

Timed out

Maxima [N/A]

Not integrable

Time = 1.34 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.21 \[ \int \frac {1}{(f+e x) \sqrt {d+\sqrt {c+\sqrt {b+a x}}}} \, dx=\int { \frac {1}{{\left (e x + f\right )} \sqrt {d + \sqrt {c + \sqrt {a x + b}}}} \,d x } \]

[In]

integrate(1/(e*x+f)/(d+(c+(a*x+b)^(1/2))^(1/2))^(1/2),x, algorithm="maxima")

[Out]

integrate(1/((e*x + f)*sqrt(d + sqrt(c + sqrt(a*x + b)))), x)

Giac [C] (verification not implemented)

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 \[ \int \frac {1}{(f+e x) \sqrt {d+\sqrt {c+\sqrt {b+a x}}}} \, dx=\text {Too large to display} \]

[In]

integrate(1/(e*x+f)/(d+(c+(a*x+b)^(1/2))^(1/2))^(1/2),x, algorithm="giac")

[Out]

-((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 + sqr
t(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 + sqr
t(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 + s
qrt((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 - sq
rt(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*sq
rt(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 + s
qrt(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 - sqr
t((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*sgn(sqrt(c + sqr
t(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 + sqr
t(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 - sqr
t((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 - sqr
t((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 - s
qrt((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 + sqr
t((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*s
gn(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))*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 + sqr
t((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 + sqr
t((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*sgn(sqrt(c + sqrt(a*x + b))) - c*(d - sqrt((c*e - s
qrt(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*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 - s
qrt(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 - s
qrt(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)

Mupad [N/A]

Not integrable

Time = 5.71 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.21 \[ \int \frac {1}{(f+e x) \sqrt {d+\sqrt {c+\sqrt {b+a x}}}} \, dx=\int \frac {1}{\left (f+e\,x\right )\,\sqrt {d+\sqrt {c+\sqrt {b+a\,x}}}} \,d x \]

[In]

int(1/((f + e*x)*(d + (c + (b + a*x)^(1/2))^(1/2))^(1/2)),x)

[Out]

int(1/((f + e*x)*(d + (c + (b + a*x)^(1/2))^(1/2))^(1/2)), x)