\(\int \frac {f+e x}{(h+g x) \sqrt {d+\sqrt {c+\sqrt {b+a x}}}} \, dx\) [2893]

Optimal result

Integrand size = 34, antiderivative size = 316 \[ \int \frac {f+e x}{(h+g x) \sqrt {d+\sqrt {c+\sqrt {b+a x}}}} \, dx=\frac {32 \left (13 c d-12 d^3\right ) e \sqrt {d+\sqrt {c+\sqrt {b+a x}}}}{105 a g}-\frac {32 \left (5 c-6 d^2\right ) e \sqrt {c+\sqrt {b+a x}} \sqrt {d+\sqrt {c+\sqrt {b+a x}}}}{105 a g}+\sqrt {b+a x} \left (-\frac {48 d e \sqrt {d+\sqrt {c+\sqrt {b+a x}}}}{35 a g}+\frac {8 e \sqrt {c+\sqrt {b+a x}} \sqrt {d+\sqrt {c+\sqrt {b+a x}}}}{7 a g}\right )+\frac {(f g-e h) \text {RootSum}\left [b g-c^2 g+2 c d^2 g-d^4 g-a h-4 c d g \text {$\#$1}^2+4 d^3 g \text {$\#$1}^2+2 c g \text {$\#$1}^4-6 d^2 g \text {$\#$1}^4+4 d g \text {$\#$1}^6-g \text {$\#$1}^8\&,\frac {\log \left (\sqrt {d+\sqrt {c+\sqrt {b+a x}}}-\text {$\#$1}\right )}{\text {$\#$1}}\&\right ]}{g^2} \]

[Out]

Unintegrable

Rubi [F]

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

[In]

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

[Out]

(8*d*(c - d^2)*e*Sqrt[d + Sqrt[c + Sqrt[b + a*x]]])/(a*g) - (8*(c - 3*d^2)*e*(d + Sqrt[c + Sqrt[b + a*x]])^(3/
2))/(3*a*g) - (24*d*e*(d + Sqrt[c + Sqrt[b + a*x]])^(5/2))/(5*a*g) + (8*e*(d + Sqrt[c + Sqrt[b + a*x]])^(7/2))
/(7*a*g) - (8*d*(c - d^2)*(f*g - e*h)*Defer[Subst][Defer[Int][(b*g*(1 - (c^2*g - 2*c*d^2*g + d^4*g + a*h)/(b*g
)) - 4*c*d*(1 - d^2/c)*g*x^2 + 2*c*(1 - (3*d^2)/c)*g*x^4 + 4*d*g*x^6 - g*x^8)^(-1), x], x, Sqrt[d + Sqrt[c + S
qrt[b + a*x]]]])/g + (8*(c - 3*d^2)*(f*g - e*h)*Defer[Subst][Defer[Int][x^2/(b*g*(1 - (c^2*g - 2*c*d^2*g + d^4
*g + a*h)/(b*g)) - 4*c*d*(1 - d^2/c)*g*x^2 + 2*c*(1 - (3*d^2)/c)*g*x^4 + 4*d*g*x^6 - g*x^8), x], x, Sqrt[d + S
qrt[c + Sqrt[b + a*x]]]])/g - (8*(f*g - e*h)*Defer[Subst][Defer[Int][x^6/(b*g*(1 - (c^2*g - 2*c*d^2*g + d^4*g
+ a*h)/(b*g)) - 4*c*d*(1 - d^2/c)*g*x^2 + 2*c*(1 - (3*d^2)/c)*g*x^4 + 4*d*g*x^6 - g*x^8), x], x, Sqrt[d + Sqrt
[c + Sqrt[b + a*x]]]])/g - (24*d*(f*g - e*h)*Defer[Subst][Defer[Int][x^4/(-(b*g*(1 - (c^2*g - 2*c*d^2*g + d^4*
g + a*h)/(b*g))) + 4*c*d*(1 - d^2/c)*g*x^2 - 2*c*(1 - (3*d^2)/c)*g*x^4 - 4*d*g*x^6 + g*x^8), x], x, Sqrt[d + S
qrt[c + Sqrt[b + a*x]]]])/g

Rubi steps \begin{align*} \text {integral}& = \frac {2 \text {Subst}\left (\int \frac {x \left (-b e+a f+e x^2\right )}{\left (-b g+a h+g x^2\right ) \sqrt {d+\sqrt {c+x}}} \, dx,x,\sqrt {b+a x}\right )}{a} \\ & = \frac {4 \text {Subst}\left (\int \frac {x \left (-c+x^2\right ) \left (-b e+a f+e \left (c-x^2\right )^2\right )}{\sqrt {d+x} \left (-b g+a h+g \left (c-x^2\right )^2\right )} \, dx,x,\sqrt {c+\sqrt {b+a x}}\right )}{a} \\ & = \frac {8 \text {Subst}\left (\int \frac {\left (-d+x^2\right ) \left (-c+\left (d-x^2\right )^2\right ) \left (-b e+a f+e \left (c-\left (d-x^2\right )^2\right )^2\right )}{-b g+a h+g \left (c-\left (d-x^2\right )^2\right )^2} \, dx,x,\sqrt {d+\sqrt {c+\sqrt {b+a x}}}\right )}{a} \\ & = \frac {8 \text {Subst}\left (\int \frac {\left (d-x^2\right ) \left (c-d^2+2 d x^2-x^4\right ) \left (b e \left (1-\frac {a f}{b e}\right )-e \left (c-\left (d-x^2\right )^2\right )^2\right )}{b g \left (1-\frac {a h}{b g}\right )-g \left (c-\left (d-x^2\right )^2\right )^2} \, dx,x,\sqrt {d+\sqrt {c+\sqrt {b+a x}}}\right )}{a} \\ & = \frac {8 \text {Subst}\left (\int \left (\frac {d \left (c-d^2\right ) e}{g}-\frac {\left (c-3 d^2\right ) e x^2}{g}-\frac {3 d e x^4}{g}+\frac {e x^6}{g}-\frac {a d \left (c-d^2\right ) (f g-e h)-a \left (c-3 d^2\right ) (f g-e h) x^2-3 a d (f g-e h) x^4+a (f g-e h) x^6}{g \left (b g \left (1-\frac {a h}{b g}\right )-g \left (c-\left (d-x^2\right )^2\right )^2\right )}\right ) \, dx,x,\sqrt {d+\sqrt {c+\sqrt {b+a x}}}\right )}{a} \\ & = \frac {8 d \left (c-d^2\right ) e \sqrt {d+\sqrt {c+\sqrt {b+a x}}}}{a g}-\frac {8 \left (c-3 d^2\right ) e \left (d+\sqrt {c+\sqrt {b+a x}}\right )^{3/2}}{3 a g}-\frac {24 d e \left (d+\sqrt {c+\sqrt {b+a x}}\right )^{5/2}}{5 a g}+\frac {8 e \left (d+\sqrt {c+\sqrt {b+a x}}\right )^{7/2}}{7 a g}-\frac {8 \text {Subst}\left (\int \frac {a d \left (c-d^2\right ) (f g-e h)-a \left (c-3 d^2\right ) (f g-e h) x^2-3 a d (f g-e h) x^4+a (f g-e h) x^6}{b g \left (1-\frac {a h}{b g}\right )-g \left (c-\left (d-x^2\right )^2\right )^2} \, dx,x,\sqrt {d+\sqrt {c+\sqrt {b+a x}}}\right )}{a g} \\ & = \frac {8 d \left (c-d^2\right ) e \sqrt {d+\sqrt {c+\sqrt {b+a x}}}}{a g}-\frac {8 \left (c-3 d^2\right ) e \left (d+\sqrt {c+\sqrt {b+a x}}\right )^{3/2}}{3 a g}-\frac {24 d e \left (d+\sqrt {c+\sqrt {b+a x}}\right )^{5/2}}{5 a g}+\frac {8 e \left (d+\sqrt {c+\sqrt {b+a x}}\right )^{7/2}}{7 a g}-\frac {8 \text {Subst}\left (\int \frac {a (f g-e h) \left (d-x^2\right ) \left (c-d^2+2 d x^2-x^4\right )}{b g \left (1-\frac {a h}{b g}\right )-g \left (c-\left (d-x^2\right )^2\right )^2} \, dx,x,\sqrt {d+\sqrt {c+\sqrt {b+a x}}}\right )}{a g} \\ & = \frac {8 d \left (c-d^2\right ) e \sqrt {d+\sqrt {c+\sqrt {b+a x}}}}{a g}-\frac {8 \left (c-3 d^2\right ) e \left (d+\sqrt {c+\sqrt {b+a x}}\right )^{3/2}}{3 a g}-\frac {24 d e \left (d+\sqrt {c+\sqrt {b+a x}}\right )^{5/2}}{5 a g}+\frac {8 e \left (d+\sqrt {c+\sqrt {b+a x}}\right )^{7/2}}{7 a g}-\frac {(8 (f g-e h)) \text {Subst}\left (\int \frac {\left (d-x^2\right ) \left (c-d^2+2 d x^2-x^4\right )}{b g \left (1-\frac {a h}{b g}\right )-g \left (c-\left (d-x^2\right )^2\right )^2} \, dx,x,\sqrt {d+\sqrt {c+\sqrt {b+a x}}}\right )}{g} \\ & = \frac {8 d \left (c-d^2\right ) e \sqrt {d+\sqrt {c+\sqrt {b+a x}}}}{a g}-\frac {8 \left (c-3 d^2\right ) e \left (d+\sqrt {c+\sqrt {b+a x}}\right )^{3/2}}{3 a g}-\frac {24 d e \left (d+\sqrt {c+\sqrt {b+a x}}\right )^{5/2}}{5 a g}+\frac {8 e \left (d+\sqrt {c+\sqrt {b+a x}}\right )^{7/2}}{7 a g}-\frac {(8 (f g-e h)) \text {Subst}\left (\int \left (\frac {c d \left (1-\frac {d^2}{c}\right )}{b g \left (1-\frac {c^2 g-2 c d^2 g+d^4 g+a h}{b g}\right )-4 c d \left (1-\frac {d^2}{c}\right ) g x^2+2 c \left (1-\frac {3 d^2}{c}\right ) g x^4+4 d g x^6-g x^8}+\frac {3 \left (1-\frac {c}{3 d^2}\right ) d^2 x^2}{b g \left (1-\frac {c^2 g-2 c d^2 g+d^4 g+a h}{b g}\right )-4 c d \left (1-\frac {d^2}{c}\right ) g x^2+2 c \left (1-\frac {3 d^2}{c}\right ) g x^4+4 d g x^6-g x^8}+\frac {x^6}{b g \left (1-\frac {c^2 g-2 c d^2 g+d^4 g+a h}{b g}\right )-4 c d \left (1-\frac {d^2}{c}\right ) g x^2+2 c \left (1-\frac {3 d^2}{c}\right ) g x^4+4 d g x^6-g x^8}+\frac {3 d x^4}{-b g \left (1-\frac {c^2 g-2 c d^2 g+d^4 g+a h}{b g}\right )+4 c d \left (1-\frac {d^2}{c}\right ) g x^2-2 c \left (1-\frac {3 d^2}{c}\right ) g x^4-4 d g x^6+g x^8}\right ) \, dx,x,\sqrt {d+\sqrt {c+\sqrt {b+a x}}}\right )}{g} \\ & = \frac {8 d \left (c-d^2\right ) e \sqrt {d+\sqrt {c+\sqrt {b+a x}}}}{a g}-\frac {8 \left (c-3 d^2\right ) e \left (d+\sqrt {c+\sqrt {b+a x}}\right )^{3/2}}{3 a g}-\frac {24 d e \left (d+\sqrt {c+\sqrt {b+a x}}\right )^{5/2}}{5 a g}+\frac {8 e \left (d+\sqrt {c+\sqrt {b+a x}}\right )^{7/2}}{7 a g}-\frac {(8 (f g-e h)) \text {Subst}\left (\int \frac {x^6}{b g \left (1-\frac {c^2 g-2 c d^2 g+d^4 g+a h}{b g}\right )-4 c d \left (1-\frac {d^2}{c}\right ) g x^2+2 c \left (1-\frac {3 d^2}{c}\right ) g x^4+4 d g x^6-g x^8} \, dx,x,\sqrt {d+\sqrt {c+\sqrt {b+a x}}}\right )}{g}-\frac {(24 d (f g-e h)) \text {Subst}\left (\int \frac {x^4}{-b g \left (1-\frac {c^2 g-2 c d^2 g+d^4 g+a h}{b g}\right )+4 c d \left (1-\frac {d^2}{c}\right ) g x^2-2 c \left (1-\frac {3 d^2}{c}\right ) g x^4-4 d g x^6+g x^8} \, dx,x,\sqrt {d+\sqrt {c+\sqrt {b+a x}}}\right )}{g}+\frac {\left (8 \left (c-3 d^2\right ) (f g-e h)\right ) \text {Subst}\left (\int \frac {x^2}{b g \left (1-\frac {c^2 g-2 c d^2 g+d^4 g+a h}{b g}\right )-4 c d \left (1-\frac {d^2}{c}\right ) g x^2+2 c \left (1-\frac {3 d^2}{c}\right ) g x^4+4 d g x^6-g x^8} \, dx,x,\sqrt {d+\sqrt {c+\sqrt {b+a x}}}\right )}{g}-\frac {\left (8 d \left (c-d^2\right ) (f g-e h)\right ) \text {Subst}\left (\int \frac {1}{b g \left (1-\frac {c^2 g-2 c d^2 g+d^4 g+a h}{b g}\right )-4 c d \left (1-\frac {d^2}{c}\right ) g x^2+2 c \left (1-\frac {3 d^2}{c}\right ) g x^4+4 d g x^6-g x^8} \, dx,x,\sqrt {d+\sqrt {c+\sqrt {b+a x}}}\right )}{g} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.50 (sec) , antiderivative size = 247, normalized size of antiderivative = 0.78 \[ \int \frac {f+e x}{(h+g x) \sqrt {d+\sqrt {c+\sqrt {b+a x}}}} \, dx=-\frac {8 e \left (20 c-24 d^2-15 \sqrt {b+a x}\right ) \sqrt {c+\sqrt {b+a x}} \sqrt {d+\sqrt {c+\sqrt {b+a x}}}}{105 a g}-\frac {16 d e \left (-26 c+24 d^2+9 \sqrt {b+a x}\right ) \sqrt {d+\sqrt {c+\sqrt {b+a x}}}}{105 a g}+\frac {(f g-e h) \text {RootSum}\left [b g-c^2 g+2 c d^2 g-d^4 g-a h-4 c d g \text {$\#$1}^2+4 d^3 g \text {$\#$1}^2+2 c g \text {$\#$1}^4-6 d^2 g \text {$\#$1}^4+4 d g \text {$\#$1}^6-g \text {$\#$1}^8\&,\frac {\log \left (\sqrt {d+\sqrt {c+\sqrt {b+a x}}}-\text {$\#$1}\right )}{\text {$\#$1}}\&\right ]}{g^2} \]

[In]

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

[Out]

(-8*e*(20*c - 24*d^2 - 15*Sqrt[b + a*x])*Sqrt[c + Sqrt[b + a*x]]*Sqrt[d + Sqrt[c + Sqrt[b + a*x]]])/(105*a*g)
- (16*d*e*(-26*c + 24*d^2 + 9*Sqrt[b + a*x])*Sqrt[d + Sqrt[c + Sqrt[b + a*x]]])/(105*a*g) + ((f*g - e*h)*RootS
um[b*g - c^2*g + 2*c*d^2*g - d^4*g - a*h - 4*c*d*g*#1^2 + 4*d^3*g*#1^2 + 2*c*g*#1^4 - 6*d^2*g*#1^4 + 4*d*g*#1^
6 - g*#1^8 & , Log[Sqrt[d + Sqrt[c + Sqrt[b + a*x]]] - #1]/#1 & ])/g^2

Maple [N/A] (verified)

Time = 0.58 (sec) , antiderivative size = 313, normalized size of antiderivative = 0.99

method	result	size
derivativedivides	\(\frac {\frac {8 e \left (\frac {\left (d +\sqrt {c +\sqrt {a x +b}}\right )^{\frac {7}{2}}}{7}-\frac {3 \left (d +\sqrt {c +\sqrt {a x +b}}\right )^{\frac {5}{2}} d}{5}+\frac {\left (3 d^{2}-c \right ) \left (d +\sqrt {c +\sqrt {a x +b}}\right )^{\frac {3}{2}}}{3}+\sqrt {d +\sqrt {c +\sqrt {a x +b}}}\, d \left (-d^{2}+c \right )\right )}{g}-\frac {a \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (g \,\textit {\_Z}^{8}-4 d g \,\textit {\_Z}^{6}+\left (6 d^{2} g -2 c g \right ) \textit {\_Z}^{4}+\left (-4 d^{3} g +4 c d g \right ) \textit {\_Z}^{2}+d^{4} g -2 c \,d^{2} g +c^{2} g +h a -b g \right )}{\sum }\frac {\left (\left (e h -f g \right ) \textit {\_R}^{6}+3 d \left (-e h +f g \right ) \textit {\_R}^{4}+\left (3 d^{2} e h -3 d^{2} f g -c e h +c f g \right ) \textit {\_R}^{2}-d^{3} e h +d^{3} f g +c d e h -c d f g \right ) \ln \left (\sqrt {d +\sqrt {c +\sqrt {a x +b}}}-\textit {\_R} \right )}{\textit {\_R}^{7}-3 \textit {\_R}^{5} d +3 \textit {\_R}^{3} d^{2}-\textit {\_R}^{3} c -\textit {\_R} \,d^{3}+\textit {\_R} c d}\right )}{g^{2}}}{a}\)	\(313\)
default	\(\frac {\frac {8 e \left (\frac {\left (d +\sqrt {c +\sqrt {a x +b}}\right )^{\frac {7}{2}}}{7}-\frac {3 \left (d +\sqrt {c +\sqrt {a x +b}}\right )^{\frac {5}{2}} d}{5}+\frac {\left (3 d^{2}-c \right ) \left (d +\sqrt {c +\sqrt {a x +b}}\right )^{\frac {3}{2}}}{3}+\sqrt {d +\sqrt {c +\sqrt {a x +b}}}\, d \left (-d^{2}+c \right )\right )}{g}-\frac {a \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (g \,\textit {\_Z}^{8}-4 d g \,\textit {\_Z}^{6}+\left (6 d^{2} g -2 c g \right ) \textit {\_Z}^{4}+\left (-4 d^{3} g +4 c d g \right ) \textit {\_Z}^{2}+d^{4} g -2 c \,d^{2} g +c^{2} g +h a -b g \right )}{\sum }\frac {\left (\left (e h -f g \right ) \textit {\_R}^{6}+3 d \left (-e h +f g \right ) \textit {\_R}^{4}+\left (3 d^{2} e h -3 d^{2} f g -c e h +c f g \right ) \textit {\_R}^{2}-d^{3} e h +d^{3} f g +c d e h -c d f g \right ) \ln \left (\sqrt {d +\sqrt {c +\sqrt {a x +b}}}-\textit {\_R} \right )}{\textit {\_R}^{7}-3 \textit {\_R}^{5} d +3 \textit {\_R}^{3} d^{2}-\textit {\_R}^{3} c -\textit {\_R} \,d^{3}+\textit {\_R} c d}\right )}{g^{2}}}{a}\)	\(313\)

[In]

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

[Out]

2/a*(4*e/g*(1/7*(d+(c+(a*x+b)^(1/2))^(1/2))^(7/2)-3/5*(d+(c+(a*x+b)^(1/2))^(1/2))^(5/2)*d+1/3*(3*d^2-c)*(d+(c+
(a*x+b)^(1/2))^(1/2))^(3/2)+(d+(c+(a*x+b)^(1/2))^(1/2))^(1/2)*d*(-d^2+c))-1/2*a/g^2*sum(((e*h-f*g)*_R^6+3*d*(-
e*h+f*g)*_R^4+(3*d^2*e*h-3*d^2*f*g-c*e*h+c*f*g)*_R^2-d^3*e*h+d^3*f*g+c*d*e*h-c*d*f*g)/(_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(g*_Z^8-4*d*g*_Z^6+(6*d^2*g-2*c*g)*_
Z^4+(-4*d^3*g+4*c*d*g)*_Z^2+d^4*g-2*c*d^2*g+c^2*g+h*a-b*g)))

Fricas [F(-1)]

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

[In]

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

[Out]

Timed out

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [N/A]

Not integrable

Time = 1.28 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.09 \[ \int \frac {f+e x}{(h+g x) \sqrt {d+\sqrt {c+\sqrt {b+a x}}}} \, dx=\int { \frac {e x + f}{{\left (g x + h\right )} \sqrt {d + \sqrt {c + \sqrt {a x + b}}}} \,d x } \]

[In]

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

[Out]

integrate((e*x + f)/((g*x + h)*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 = 34.17 (sec) , antiderivative size = 5825, normalized size of antiderivative = 18.43 \[ \int \frac {f+e x}{(h+g x) \sqrt {d+\sqrt {c+\sqrt {b+a x}}}} \, dx=\text {Too large to display} \]

[In]

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

[Out]

8/105*(15*a^6*(d + sqrt(c + sqrt(a*x + b)))^(7/2)*e*g^6*sgn(sqrt(c + sqrt(a*x + b))) - 63*a^6*(d + sqrt(c + sq
rt(a*x + b)))^(5/2)*d*e*g^6*sgn(sqrt(c + sqrt(a*x + b))) + 105*a^6*(d + sqrt(c + sqrt(a*x + b)))^(3/2)*d^2*e*g
^6*sgn(sqrt(c + sqrt(a*x + b))) - 105*a^6*sqrt(d + sqrt(c + sqrt(a*x + b)))*d^3*e*g^6*sgn(sqrt(c + sqrt(a*x +
b))) - 35*a^6*c*(d + sqrt(c + sqrt(a*x + b)))^(3/2)*e*g^6*sgn(sqrt(c + sqrt(a*x + b))) + 105*a^6*c*sqrt(d + sq
rt(c + sqrt(a*x + b)))*d*e*g^6*sgn(sqrt(c + sqrt(a*x + b))))/(a^7*g^7) - ((a^8*(d + sqrt((c*g + sqrt(b*g^2 - a
*g*h))/g))^3*f*g^7*sgn(sqrt(c + sqrt(a*x + b))) - 3*a^8*(d + sqrt((c*g + sqrt(b*g^2 - a*g*h))/g))^2*d*f*g^7*sg
n(sqrt(c + sqrt(a*x + b))) + 3*a^8*(d + sqrt((c*g + sqrt(b*g^2 - a*g*h))/g))*d^2*f*g^7*sgn(sqrt(c + sqrt(a*x +
b))) - a^8*d^3*f*g^7*sgn(sqrt(c + sqrt(a*x + b))) - a^8*(d + sqrt((c*g + sqrt(b*g^2 - a*g*h))/g))^3*e*g^6*h*s
gn(sqrt(c + sqrt(a*x + b))) + 3*a^8*(d + sqrt((c*g + sqrt(b*g^2 - a*g*h))/g))^2*d*e*g^6*h*sgn(sqrt(c + sqrt(a*
x + b))) - 3*a^8*(d + sqrt((c*g + sqrt(b*g^2 - a*g*h))/g))*d^2*e*g^6*h*sgn(sqrt(c + sqrt(a*x + b))) + a^8*d^3*
e*g^6*h*sgn(sqrt(c + sqrt(a*x + b))) - a^8*c*(d + sqrt((c*g + sqrt(b*g^2 - a*g*h))/g))*f*g^7*sgn(sqrt(c + sqrt
(a*x + b))) + a^8*c*d*f*g^7*sgn(sqrt(c + sqrt(a*x + b))) + a^8*c*(d + sqrt((c*g + sqrt(b*g^2 - a*g*h))/g))*e*g
^6*h*sgn(sqrt(c + sqrt(a*x + b))) - a^8*c*d*e*g^6*h*sgn(sqrt(c + sqrt(a*x + b))))*log(sqrt(d + sqrt(c + sqrt(a
*x + b))) + sqrt(d + sqrt((c*g + sqrt(b*g^2 - a*g*h))/g)))/((d + sqrt((c*g + sqrt(b*g^2 - a*g*h))/g))^(7/2)*g
- 3*(d + sqrt((c*g + sqrt(b*g^2 - a*g*h))/g))^(5/2)*d*g + (3*d^2*g - c*g)*(d + sqrt((c*g + sqrt(b*g^2 - a*g*h)
)/g))^(3/2) - (d^3*g - c*d*g)*sqrt(d + sqrt((c*g + sqrt(b*g^2 - a*g*h))/g))) - (a^8*(d + sqrt((c*g + sqrt(b*g^
2 - a*g*h))/g))^3*f*g^7*sgn(sqrt(c + sqrt(a*x + b))) - 3*a^8*(d + sqrt((c*g + sqrt(b*g^2 - a*g*h))/g))^2*d*f*g
^7*sgn(sqrt(c + sqrt(a*x + b))) + 3*a^8*(d + sqrt((c*g + sqrt(b*g^2 - a*g*h))/g))*d^2*f*g^7*sgn(sqrt(c + sqrt(
a*x + b))) - a^8*d^3*f*g^7*sgn(sqrt(c + sqrt(a*x + b))) - a^8*(d + sqrt((c*g + sqrt(b*g^2 - a*g*h))/g))^3*e*g^
6*h*sgn(sqrt(c + sqrt(a*x + b))) + 3*a^8*(d + sqrt((c*g + sqrt(b*g^2 - a*g*h))/g))^2*d*e*g^6*h*sgn(sqrt(c + sq
rt(a*x + b))) - 3*a^8*(d + sqrt((c*g + sqrt(b*g^2 - a*g*h))/g))*d^2*e*g^6*h*sgn(sqrt(c + sqrt(a*x + b))) + a^8
*d^3*e*g^6*h*sgn(sqrt(c + sqrt(a*x + b))) - a^8*c*(d + sqrt((c*g + sqrt(b*g^2 - a*g*h))/g))*f*g^7*sgn(sqrt(c +
sqrt(a*x + b))) + a^8*c*d*f*g^7*sgn(sqrt(c + sqrt(a*x + b))) + a^8*c*(d + sqrt((c*g + sqrt(b*g^2 - a*g*h))/g)
)*e*g^6*h*sgn(sqrt(c + sqrt(a*x + b))) - a^8*c*d*e*g^6*h*sgn(sqrt(c + sqrt(a*x + b))))*log(sqrt(d + sqrt(c + s
qrt(a*x + b))) - sqrt(d + sqrt((c*g + sqrt(b*g^2 - a*g*h))/g)))/((d + sqrt((c*g + sqrt(b*g^2 - a*g*h))/g))^(7/
2)*g - 3*(d + sqrt((c*g + sqrt(b*g^2 - a*g*h))/g))^(5/2)*d*g + (3*d^2*g - c*g)*(d + sqrt((c*g + sqrt(b*g^2 - a
*g*h))/g))^(3/2) - (d^3*g - c*d*g)*sqrt(d + sqrt((c*g + sqrt(b*g^2 - a*g*h))/g))) + (a^8*(d - sqrt((c*g + sqrt
(b*g^2 - a*g*h))/g))^3*f*g^7*sgn(sqrt(c + sqrt(a*x + b))) - 3*a^8*(d - sqrt((c*g + sqrt(b*g^2 - a*g*h))/g))^2*
d*f*g^7*sgn(sqrt(c + sqrt(a*x + b))) + 3*a^8*(d - sqrt((c*g + sqrt(b*g^2 - a*g*h))/g))*d^2*f*g^7*sgn(sqrt(c +
sqrt(a*x + b))) - a^8*d^3*f*g^7*sgn(sqrt(c + sqrt(a*x + b))) - a^8*(d - sqrt((c*g + sqrt(b*g^2 - a*g*h))/g))^3
*e*g^6*h*sgn(sqrt(c + sqrt(a*x + b))) + 3*a^8*(d - sqrt((c*g + sqrt(b*g^2 - a*g*h))/g))^2*d*e*g^6*h*sgn(sqrt(c
+ sqrt(a*x + b))) - 3*a^8*(d - sqrt((c*g + sqrt(b*g^2 - a*g*h))/g))*d^2*e*g^6*h*sgn(sqrt(c + sqrt(a*x + b)))
+ a^8*d^3*e*g^6*h*sgn(sqrt(c + sqrt(a*x + b))) - a^8*c*(d - sqrt((c*g + sqrt(b*g^2 - a*g*h))/g))*f*g^7*sgn(sqr
t(c + sqrt(a*x + b))) + a^8*c*d*f*g^7*sgn(sqrt(c + sqrt(a*x + b))) + a^8*c*(d - sqrt((c*g + sqrt(b*g^2 - a*g*h
))/g))*e*g^6*h*sgn(sqrt(c + sqrt(a*x + b))) - a^8*c*d*e*g^6*h*sgn(sqrt(c + sqrt(a*x + b))))*log(sqrt(d + sqrt(
c + sqrt(a*x + b))) + sqrt(d - sqrt((c*g + sqrt(b*g^2 - a*g*h))/g)))/((d - sqrt((c*g + sqrt(b*g^2 - a*g*h))/g)
)^(7/2)*g - 3*(d - sqrt((c*g + sqrt(b*g^2 - a*g*h))/g))^(5/2)*d*g + (3*d^2*g - c*g)*(d - sqrt((c*g + sqrt(b*g^
2 - a*g*h))/g))^(3/2) - (d^3*g - c*d*g)*sqrt(d - sqrt((c*g + sqrt(b*g^2 - a*g*h))/g))) - (a^8*(d - sqrt((c*g +
sqrt(b*g^2 - a*g*h))/g))^3*f*g^7*sgn(sqrt(c + sqrt(a*x + b))) - 3*a^8*(d - sqrt((c*g + sqrt(b*g^2 - a*g*h))/g
))^2*d*f*g^7*sgn(sqrt(c + sqrt(a*x + b))) + 3*a^8*(d - sqrt((c*g + sqrt(b*g^2 - a*g*h))/g))*d^2*f*g^7*sgn(sqrt
(c + sqrt(a*x + b))) - a^8*d^3*f*g^7*sgn(sqrt(c + sqrt(a*x + b))) - a^8*(d - sqrt((c*g + sqrt(b*g^2 - a*g*h))/
g))^3*e*g^6*h*sgn(sqrt(c + sqrt(a*x + b))) + 3*a^8*(d - sqrt((c*g + sqrt(b*g^2 - a*g*h))/g))^2*d*e*g^6*h*sgn(s
qrt(c + sqrt(a*x + b))) - 3*a^8*(d - sqrt((c*g + sqrt(b*g^2 - a*g*h))/g))*d^2*e*g^6*h*sgn(sqrt(c + sqrt(a*x +
b))) + a^8*d^3*e*g^6*h*sgn(sqrt(c + sqrt(a*x + b))) - a^8*c*(d - sqrt((c*g + sqrt(b*g^2 - a*g*h))/g))*f*g^7*sg
n(sqrt(c + sqrt(a*x + b))) + a^8*c*d*f*g^7*sgn(sqrt(c + sqrt(a*x + b))) + a^8*c*(d - sqrt((c*g + sqrt(b*g^2 -
a*g*h))/g))*e*g^6*h*sgn(sqrt(c + sqrt(a*x + b))) - a^8*c*d*e*g^6*h*sgn(sqrt(c + sqrt(a*x + b))))*log(sqrt(d +
sqrt(c + sqrt(a*x + b))) - sqrt(d - sqrt((c*g + sqrt(b*g^2 - a*g*h))/g)))/((d - sqrt((c*g + sqrt(b*g^2 - a*g*h
))/g))^(7/2)*g - 3*(d - sqrt((c*g + sqrt(b*g^2 - a*g*h))/g))^(5/2)*d*g + (3*d^2*g - c*g)*(d - sqrt((c*g + sqrt
(b*g^2 - a*g*h))/g))^(3/2) - (d^3*g - c*d*g)*sqrt(d - sqrt((c*g + sqrt(b*g^2 - a*g*h))/g))) + (a^8*(d + sqrt((
c*g - sqrt(b*g^2 - a*g*h))/g))^3*f*g^7*sgn(sqrt(c + sqrt(a*x + b))) - 3*a^8*(d + sqrt((c*g - sqrt(b*g^2 - a*g*
h))/g))^2*d*f*g^7*sgn(sqrt(c + sqrt(a*x + b))) + 3*a^8*(d + sqrt((c*g - sqrt(b*g^2 - a*g*h))/g))*d^2*f*g^7*sgn
(sqrt(c + sqrt(a*x + b))) - a^8*d^3*f*g^7*sgn(sqrt(c + sqrt(a*x + b))) - a^8*(d + sqrt((c*g - sqrt(b*g^2 - a*g
*h))/g))^3*e*g^6*h*sgn(sqrt(c + sqrt(a*x + b))) + 3*a^8*(d + sqrt((c*g - sqrt(b*g^2 - a*g*h))/g))^2*d*e*g^6*h*
sgn(sqrt(c + sqrt(a*x + b))) - 3*a^8*(d + sqrt((c*g - sqrt(b*g^2 - a*g*h))/g))*d^2*e*g^6*h*sgn(sqrt(c + sqrt(a
*x + b))) + a^8*d^3*e*g^6*h*sgn(sqrt(c + sqrt(a*x + b))) - a^8*c*(d + sqrt((c*g - sqrt(b*g^2 - a*g*h))/g))*f*g
^7*sgn(sqrt(c + sqrt(a*x + b))) + a^8*c*d*f*g^7*sgn(sqrt(c + sqrt(a*x + b))) + a^8*c*(d + sqrt((c*g - sqrt(b*g
^2 - a*g*h))/g))*e*g^6*h*sgn(sqrt(c + sqrt(a*x + b))) - a^8*c*d*e*g^6*h*sgn(sqrt(c + sqrt(a*x + b))))*log(sqrt
(d + sqrt(c + sqrt(a*x + b))) + sqrt(d + sqrt((c*g - sqrt(b*g^2 - a*g*h))/g)))/((d + sqrt((c*g - sqrt(b*g^2 -
a*g*h))/g))^(7/2)*g - 3*(d + sqrt((c*g - sqrt(b*g^2 - a*g*h))/g))^(5/2)*d*g + (3*d^2*g - c*g)*(d + sqrt((c*g -
sqrt(b*g^2 - a*g*h))/g))^(3/2) - (d^3*g - c*d*g)*sqrt(d + sqrt((c*g - sqrt(b*g^2 - a*g*h))/g))) - (a^8*(d + s
qrt((c*g - sqrt(b*g^2 - a*g*h))/g))^3*f*g^7*sgn(sqrt(c + sqrt(a*x + b))) - 3*a^8*(d + sqrt((c*g - sqrt(b*g^2 -
a*g*h))/g))^2*d*f*g^7*sgn(sqrt(c + sqrt(a*x + b))) + 3*a^8*(d + sqrt((c*g - sqrt(b*g^2 - a*g*h))/g))*d^2*f*g^
7*sgn(sqrt(c + sqrt(a*x + b))) - a^8*d^3*f*g^7*sgn(sqrt(c + sqrt(a*x + b))) - a^8*(d + sqrt((c*g - sqrt(b*g^2
- a*g*h))/g))^3*e*g^6*h*sgn(sqrt(c + sqrt(a*x + b))) + 3*a^8*(d + sqrt((c*g - sqrt(b*g^2 - a*g*h))/g))^2*d*e*g
^6*h*sgn(sqrt(c + sqrt(a*x + b))) - 3*a^8*(d + sqrt((c*g - sqrt(b*g^2 - a*g*h))/g))*d^2*e*g^6*h*sgn(sqrt(c + s
qrt(a*x + b))) + a^8*d^3*e*g^6*h*sgn(sqrt(c + sqrt(a*x + b))) - a^8*c*(d + sqrt((c*g - sqrt(b*g^2 - a*g*h))/g)
)*f*g^7*sgn(sqrt(c + sqrt(a*x + b))) + a^8*c*d*f*g^7*sgn(sqrt(c + sqrt(a*x + b))) + a^8*c*(d + sqrt((c*g - sqr
t(b*g^2 - a*g*h))/g))*e*g^6*h*sgn(sqrt(c + sqrt(a*x + b))) - a^8*c*d*e*g^6*h*sgn(sqrt(c + sqrt(a*x + b))))*log
(sqrt(d + sqrt(c + sqrt(a*x + b))) - sqrt(d + sqrt((c*g - sqrt(b*g^2 - a*g*h))/g)))/((d + sqrt((c*g - sqrt(b*g
^2 - a*g*h))/g))^(7/2)*g - 3*(d + sqrt((c*g - sqrt(b*g^2 - a*g*h))/g))^(5/2)*d*g + (3*d^2*g - c*g)*(d + sqrt((
c*g - sqrt(b*g^2 - a*g*h))/g))^(3/2) - (d^3*g - c*d*g)*sqrt(d + sqrt((c*g - sqrt(b*g^2 - a*g*h))/g))) + (a^8*(
d - sqrt((c*g - sqrt(b*g^2 - a*g*h))/g))^3*f*g^7*sgn(sqrt(c + sqrt(a*x + b))) - 3*a^8*(d - sqrt((c*g - sqrt(b*
g^2 - a*g*h))/g))^2*d*f*g^7*sgn(sqrt(c + sqrt(a*x + b))) + 3*a^8*(d - sqrt((c*g - sqrt(b*g^2 - a*g*h))/g))*d^2
*f*g^7*sgn(sqrt(c + sqrt(a*x + b))) - a^8*d^3*f*g^7*sgn(sqrt(c + sqrt(a*x + b))) - a^8*(d - sqrt((c*g - sqrt(b
*g^2 - a*g*h))/g))^3*e*g^6*h*sgn(sqrt(c + sqrt(a*x + b))) + 3*a^8*(d - sqrt((c*g - sqrt(b*g^2 - a*g*h))/g))^2*
d*e*g^6*h*sgn(sqrt(c + sqrt(a*x + b))) - 3*a^8*(d - sqrt((c*g - sqrt(b*g^2 - a*g*h))/g))*d^2*e*g^6*h*sgn(sqrt(
c + sqrt(a*x + b))) + a^8*d^3*e*g^6*h*sgn(sqrt(c + sqrt(a*x + b))) - a^8*c*(d - sqrt((c*g - sqrt(b*g^2 - a*g*h
))/g))*f*g^7*sgn(sqrt(c + sqrt(a*x + b))) + a^8*c*d*f*g^7*sgn(sqrt(c + sqrt(a*x + b))) + a^8*c*(d - sqrt((c*g
- sqrt(b*g^2 - a*g*h))/g))*e*g^6*h*sgn(sqrt(c + sqrt(a*x + b))) - a^8*c*d*e*g^6*h*sgn(sqrt(c + sqrt(a*x + b)))
)*log(sqrt(d + sqrt(c + sqrt(a*x + b))) + sqrt(d - sqrt((c*g - sqrt(b*g^2 - a*g*h))/g)))/((d - sqrt((c*g - sqr
t(b*g^2 - a*g*h))/g))^(7/2)*g - 3*(d - sqrt((c*g - sqrt(b*g^2 - a*g*h))/g))^(5/2)*d*g + (3*d^2*g - c*g)*(d - s
qrt((c*g - sqrt(b*g^2 - a*g*h))/g))^(3/2) - (d^3*g - c*d*g)*sqrt(d - sqrt((c*g - sqrt(b*g^2 - a*g*h))/g))) - (
a^8*(d - sqrt((c*g - sqrt(b*g^2 - a*g*h))/g))^3*f*g^7*sgn(sqrt(c + sqrt(a*x + b))) - 3*a^8*(d - sqrt((c*g - sq
rt(b*g^2 - a*g*h))/g))^2*d*f*g^7*sgn(sqrt(c + sqrt(a*x + b))) + 3*a^8*(d - sqrt((c*g - sqrt(b*g^2 - a*g*h))/g)
)*d^2*f*g^7*sgn(sqrt(c + sqrt(a*x + b))) - a^8*d^3*f*g^7*sgn(sqrt(c + sqrt(a*x + b))) - a^8*(d - sqrt((c*g - s
qrt(b*g^2 - a*g*h))/g))^3*e*g^6*h*sgn(sqrt(c + sqrt(a*x + b))) + 3*a^8*(d - sqrt((c*g - sqrt(b*g^2 - a*g*h))/g
))^2*d*e*g^6*h*sgn(sqrt(c + sqrt(a*x + b))) - 3*a^8*(d - sqrt((c*g - sqrt(b*g^2 - a*g*h))/g))*d^2*e*g^6*h*sgn(
sqrt(c + sqrt(a*x + b))) + a^8*d^3*e*g^6*h*sgn(sqrt(c + sqrt(a*x + b))) - a^8*c*(d - sqrt((c*g - sqrt(b*g^2 -
a*g*h))/g))*f*g^7*sgn(sqrt(c + sqrt(a*x + b))) + a^8*c*d*f*g^7*sgn(sqrt(c + sqrt(a*x + b))) + a^8*c*(d - sqrt(
(c*g - sqrt(b*g^2 - a*g*h))/g))*e*g^6*h*sgn(sqrt(c + sqrt(a*x + b))) - a^8*c*d*e*g^6*h*sgn(sqrt(c + sqrt(a*x +
b))))*log(sqrt(d + sqrt(c + sqrt(a*x + b))) - sqrt(d - sqrt((c*g - sqrt(b*g^2 - a*g*h))/g)))/((d - sqrt((c*g
- sqrt(b*g^2 - a*g*h))/g))^(7/2)*g - 3*(d - sqrt((c*g - sqrt(b*g^2 - a*g*h))/g))^(5/2)*d*g + (3*d^2*g - c*g)*(
d - sqrt((c*g - sqrt(b*g^2 - a*g*h))/g))^(3/2) - (d^3*g - c*d*g)*sqrt(d - sqrt((c*g - sqrt(b*g^2 - a*g*h))/g))
))/(a^8*g^7)

Mupad [N/A]

Not integrable

Time = 8.87 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.09 \[ \int \frac {f+e x}{(h+g x) \sqrt {d+\sqrt {c+\sqrt {b+a x}}}} \, dx=\int \frac {f+e\,x}{\left (h+g\,x\right )\,\sqrt {d+\sqrt {c+\sqrt {b+a\,x}}}} \,d x \]

[In]

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

[Out]

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