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

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

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Rubi [F]  time = 3.07, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {1}{(f+e x) \sqrt {d+\sqrt {c+\sqrt {b+a x}}}} \, dx \end {gather*}

Verification is not applicable to the result.

[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*} \int \frac {1}{(f+e x) \sqrt {d+\sqrt {c+\sqrt {b+a x}}}} \, dx &=2 \operatorname {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 \operatorname {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 \operatorname {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 \operatorname {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 \operatorname {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 \operatorname {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) \operatorname {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 ) \operatorname {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 ) \operatorname {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*}

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Mathematica [B]  time = 24.33, size = 388, normalized size = 3.29 \begin {gather*} \text {RootSum}\left [\text {$\#$1}^8 a f-\text {$\#$1}^8 b e+\text {$\#$1}^8 c^2 e-2 \text {$\#$1}^8 c d^2 e+\text {$\#$1}^8 d^4 e+4 \text {$\#$1}^6 c d e-4 \text {$\#$1}^6 d^3 e-2 \text {$\#$1}^4 c e+6 \text {$\#$1}^4 d^2 e-4 \text {$\#$1}^2 d e+e\&,\frac {\text {$\#$1}^6 d^3 \log \left (\frac {1}{\sqrt {\sqrt {\sqrt {a x+b}+c}+d}}-\text {$\#$1}\right )-\text {$\#$1}^6 c d \log \left (\frac {1}{\sqrt {\sqrt {\sqrt {a x+b}+c}+d}}-\text {$\#$1}\right )-3 \text {$\#$1}^4 d^2 \log \left (\frac {1}{\sqrt {\sqrt {\sqrt {a x+b}+c}+d}}-\text {$\#$1}\right )+\text {$\#$1}^4 c \log \left (\frac {1}{\sqrt {\sqrt {\sqrt {a x+b}+c}+d}}-\text {$\#$1}\right )+3 \text {$\#$1}^2 d \log \left (\frac {1}{\sqrt {\sqrt {\sqrt {a x+b}+c}+d}}-\text {$\#$1}\right )-\log \left (\frac {1}{\sqrt {\sqrt {\sqrt {a x+b}+c}+d}}-\text {$\#$1}\right )}{\text {$\#$1}^7 a f-\text {$\#$1}^7 b e+\text {$\#$1}^7 c^2 e-2 \text {$\#$1}^7 c d^2 e+\text {$\#$1}^7 d^4 e+3 \text {$\#$1}^5 c d e-3 \text {$\#$1}^5 d^3 e-\text {$\#$1}^3 c e+3 \text {$\#$1}^3 d^2 e-\text {$\#$1} d e}\&\right ] \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

RootSum[e - 4*d*e*#1^2 - 2*c*e*#1^4 + 6*d^2*e*#1^4 + 4*c*d*e*#1^6 - 4*d^3*e*#1^6 - b*e*#1^8 + c^2*e*#1^8 - 2*c
*d^2*e*#1^8 + d^4*e*#1^8 + a*f*#1^8 & , (-Log[1/Sqrt[d + Sqrt[c + Sqrt[b + a*x]]] - #1] + 3*d*Log[1/Sqrt[d + S
qrt[c + Sqrt[b + a*x]]] - #1]*#1^2 + c*Log[1/Sqrt[d + Sqrt[c + Sqrt[b + a*x]]] - #1]*#1^4 - 3*d^2*Log[1/Sqrt[d
+ Sqrt[c + Sqrt[b + a*x]]] - #1]*#1^4 - c*d*Log[1/Sqrt[d + Sqrt[c + Sqrt[b + a*x]]] - #1]*#1^6 + d^3*Log[1/Sq
rt[d + Sqrt[c + Sqrt[b + a*x]]] - #1]*#1^6)/(-(d*e*#1) - c*e*#1^3 + 3*d^2*e*#1^3 + 3*c*d*e*#1^5 - 3*d^3*e*#1^5
- b*e*#1^7 + c^2*e*#1^7 - 2*c*d^2*e*#1^7 + d^4*e*#1^7 + a*f*#1^7) & ]

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IntegrateAlgebraic [A]  time = 0.40, size = 118, normalized size = 1.00 \begin {gather*} \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} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[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

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fricas [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,x):;OUTPUT:Warn
ing, integration of abs or sign assumes constant sign by intervals (correct if the argument is real):Check [ab
s(t_nostep^2-d)]Unable to divide, perhaps due to rounding error%%%{-1073741824,[2,10,18,5,4,2,0,0]%%%}+%%%{214
7483648,[2,10,18,3,5,2,0,0]%%%}+%%%{1073741824,[2,10,18,3,3,3,0,0]%%%}+%%%{-1073741824,[2,10,18,1,6,2,0,0]%%%}
+%%%{-1073741824,[2,10,18,1,4,3,0,0]%%%}+%%%{2147483648,[2,10,17,5,4,1,1,1]%%%}+%%%{-4294967296,[2,10,17,3,5,1
,1,1]%%%}+%%%{-3221225472,[2,10,17,3,3,2,1,1]%%%}+%%%{2147483648,[2,10,17,1,6,1,1,1]%%%}+%%%{3221225472,[2,10,
17,1,4,2,1,1]%%%}+%%%{-1073741824,[2,10,16,5,4,0,2,2]%%%}+%%%{2147483648,[2,10,16,3,5,0,2,2]%%%}+%%%{322122547
2,[2,10,16,3,3,1,2,2]%%%}+%%%{-1073741824,[2,10,16,1,6,0,2,2]%%%}+%%%{-3221225472,[2,10,16,1,4,1,2,2]%%%}+%%%{
-1073741824,[2,10,15,3,3,0,3,3]%%%}+%%%{1073741824,[2,10,15,1,4,0,3,3]%%%}+%%%{33554432,[2,8,16,4,3,1,0,0]%%%}
+%%%{-50331648,[2,8,16,2,4,1,0,0]%%%}+%%%{16777216,[2,8,16,0,5,1,0,0]%%%}+%%%{-33554432,[2,8,15,4,3,0,1,1]%%%}
+%%%{50331648,[2,8,15,2,4,0,1,1]%%%}+%%%{-16777216,[2,8,15,0,5,0,1,1]%%%}+%%%{-262144,[2,6,14,3,2,0,0,0]%%%}+%
%%{262144,[2,6,14,1,3,0,0,0]%%%}+%%%{-134217728,[1,9,17,4,3,2,0,0]%%%}+%%%{402653184,[1,9,17,2,4,2,0,0]%%%}+%%
%{-268435456,[1,9,17,0,5,2,0,0]%%%}+%%%{268435456,[1,9,16,4,3,1,1,1]%%%}+%%%{-805306368,[1,9,16,2,4,1,1,1]%%%}
+%%%{536870912,[1,9,16,0,5,1,1,1]%%%}+%%%{-134217728,[1,9,15,4,3,0,2,2]%%%}+%%%{402653184,[1,9,15,2,4,0,2,2]%%
%}+%%%{-268435456,[1,9,15,0,5,0,2,2]%%%}+%%%{2097152,[1,7,15,3,2,1,0,0]%%%}+%%%{-2097152,[1,7,15,1,3,1,0,0]%%%
}+%%%{-2097152,[1,7,14,3,2,0,1,1]%%%}+%%%{2097152,[1,7,14,1,3,0,1,1]%%%}+%%%{1073741824,[0,10,18,6,4,2,0,0]%%%
}+%%%{-3221225472,[0,10,18,4,5,2,0,0]%%%}+%%%{3221225472,[0,10,18,2,6,2,0,0]%%%}+%%%{-1073741824,[0,10,18,2,4,
3,0,0]%%%}+%%%{-1073741824,[0,10,18,0,7,2,0,0]%%%}+%%%{1073741824,[0,10,18,0,5,3,0,0]%%%}+%%%{-2147483648,[0,1
0,17,6,4,1,1,1]%%%}+%%%{6442450944,[0,10,17,4,5,1,1,1]%%%}+%%%{-6442450944,[0,10,17,2,6,1,1,1]%%%}+%%%{3221225
472,[0,10,17,2,4,2,1,1]%%%}+%%%{2147483648,[0,10,17,0,7,1,1,1]%%%}+%%%{-3221225472,[0,10,17,0,5,2,1,1]%%%}+%%%
{1073741824,[0,10,16,6,4,0,2,2]%%%}+%%%{-3221225472,[0,10,16,4,5,0,2,2]%%%}+%%%{3221225472,[0,10,16,2,6,0,2,2]
%%%}+%%%{-3221225472,[0,10,16,2,4,1,2,2]%%%}+%%%{-1073741824,[0,10,16,0,7,0,2,2]%%%}+%%%{3221225472,[0,10,16,0
,5,1,2,2]%%%}+%%%{1073741824,[0,10,15,2,4,0,3,3]%%%}+%%%{-1073741824,[0,10,15,0,5,0,3,3]%%%}+%%%{-33554432,[0,
8,16,5,3,1,0,0]%%%}+%%%{67108864,[0,8,16,3,4,1,0,0]%%%}+%%%{-33554432,[0,8,16,1,5,1,0,0]%%%}+%%%{33554432,[0,8
,15,5,3,0,1,1]%%%}+%%%{-67108864,[0,8,15,3,4,0,1,1]%%%}+%%%{33554432,[0,8,15,1,5,0,1,1]%%%}+%%%{262144,[0,6,14
,4,2,0,0,0]%%%}+%%%{-262144,[0,6,14,2,3,0,0,0]%%%} / %%%{4096,[0,4,7,2,2,0,0,0]%%%}+%%%{-4096,[0,4,7,0,3,0,0,0
]%%%} Error: Bad Argument Value

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maple [B]  time = 0.60, size = 165, normalized size = 1.40

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

Verification of antiderivative is not currently implemented for this CAS.

[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*e*d*_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))

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{{\left (e x + f\right )} \sqrt {d + \sqrt {c + \sqrt {a x + b}}}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {1}{\left (f+e\,x\right )\,\sqrt {d+\sqrt {c+\sqrt {b+a\,x}}}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\sqrt {d + \sqrt {c + \sqrt {a x + b}}} \left (e x + f\right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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