Integrand size = 34, antiderivative size = 316 \[ \int \frac {f+e x}{(h+g x) \sqrt {d+\sqrt {c+\sqrt {b+a x}}}} \, dx=\frac {32 \left (13 c d-12 d^3\right ) e \sqrt {d+\sqrt {c+\sqrt {b+a x}}}}{105 a g}-\frac {32 \left (5 c-6 d^2\right ) e \sqrt {c+\sqrt {b+a x}} \sqrt {d+\sqrt {c+\sqrt {b+a x}}}}{105 a g}+\sqrt {b+a x} \left (-\frac {48 d e \sqrt {d+\sqrt {c+\sqrt {b+a x}}}}{35 a g}+\frac {8 e \sqrt {c+\sqrt {b+a x}} \sqrt {d+\sqrt {c+\sqrt {b+a x}}}}{7 a g}\right )+\frac {(f g-e h) \text {RootSum}\left [b g-c^2 g+2 c d^2 g-d^4 g-a h-4 c d g \text {$\#$1}^2+4 d^3 g \text {$\#$1}^2+2 c g \text {$\#$1}^4-6 d^2 g \text {$\#$1}^4+4 d g \text {$\#$1}^6-g \text {$\#$1}^8\&,\frac {\log \left (\sqrt {d+\sqrt {c+\sqrt {b+a x}}}-\text {$\#$1}\right )}{\text {$\#$1}}\&\right ]}{g^2} \]
Time = 0.50 (sec) , antiderivative size = 247, normalized size of antiderivative = 0.78 \[ \int \frac {f+e x}{(h+g x) \sqrt {d+\sqrt {c+\sqrt {b+a x}}}} \, dx=-\frac {8 e \left (20 c-24 d^2-15 \sqrt {b+a x}\right ) \sqrt {c+\sqrt {b+a x}} \sqrt {d+\sqrt {c+\sqrt {b+a x}}}}{105 a g}-\frac {16 d e \left (-26 c+24 d^2+9 \sqrt {b+a x}\right ) \sqrt {d+\sqrt {c+\sqrt {b+a x}}}}{105 a g}+\frac {(f g-e h) \text {RootSum}\left [b g-c^2 g+2 c d^2 g-d^4 g-a h-4 c d g \text {$\#$1}^2+4 d^3 g \text {$\#$1}^2+2 c g \text {$\#$1}^4-6 d^2 g \text {$\#$1}^4+4 d g \text {$\#$1}^6-g \text {$\#$1}^8\&,\frac {\log \left (\sqrt {d+\sqrt {c+\sqrt {b+a x}}}-\text {$\#$1}\right )}{\text {$\#$1}}\&\right ]}{g^2} \]
(-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)*RootSum [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 + Sqr t[b + a*x]]] - #1]/#1 & ])/g^2
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.
| \(\displaystyle \int \frac {e x+f}{(g x+h) \sqrt {\sqrt {\sqrt {a x+b}+c}+d}} \, dx\) |
| \(\Big \downarrow \) 7267 |
| \(\displaystyle \frac {2 \int \frac {\sqrt {b+a x} (b e-(b+a x) e-a f)}{(b g-(b+a x) g-a h) \sqrt {d+\sqrt {c+\sqrt {b+a x}}}}d\sqrt {b+a x}}{a}\) |
| \(\Big \downarrow \) 7267 |
| \(\displaystyle \frac {4 \int -\frac {(-b+c-a x) \left (-e (-b+c-a x)^2+b e-a f\right ) \sqrt {c+\sqrt {b+a x}}}{\left (-g (-b+c-a x)^2+b g-a h\right ) \sqrt {d+\sqrt {c+\sqrt {b+a x}}}}d\sqrt {c+\sqrt {b+a x}}}{a}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {4 \int \frac {(-b+c-a x) \left (-e (-b+c-a x)^2+b e-a f\right ) \sqrt {c+\sqrt {b+a x}}}{\left (-g (-b+c-a x)^2+b g-a h\right ) \sqrt {d+\sqrt {c+\sqrt {b+a x}}}}d\sqrt {c+\sqrt {b+a x}}}{a}\) |
| \(\Big \downarrow \) 2091 |
| \(\displaystyle -\frac {4 \int \frac {(-b+c-a x) \left (-e (-b+c-a x)^2+b e-a f\right ) \sqrt {c+\sqrt {b+a x}}}{\left (-g c^2+2 g (b+a x) c-g (b+a x)^2+b g-a h\right ) \sqrt {d+\sqrt {c+\sqrt {b+a x}}}}d\sqrt {c+\sqrt {b+a x}}}{a}\) |
| \(\Big \downarrow \) 7267 |
| \(\displaystyle -\frac {8 \int -\frac {(-b+d-a x) \left (c-(-b+d-a x)^2\right ) \left (-e \left (c-(-b+d-a x)^2\right )^2+b e-a f\right )}{-g (-b+d-a x)^4+2 c g (-b+d-a x)^2-c^2 g+b g-a h}d\sqrt {d+\sqrt {c+\sqrt {b+a x}}}}{a}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {8 \int \frac {(-b+d-a x) \left (c-(-b+d-a x)^2\right ) \left (-e \left (c-(-b+d-a x)^2\right )^2+b e-a f\right )}{-g (-b+d-a x)^4+2 c g (-b+d-a x)^2-c^2 g+b g-a h}d\sqrt {d+\sqrt {c+\sqrt {b+a x}}}}{a}\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \frac {8 \int \frac {(-b+d-a x) \left (-d^2+2 (b+a x) d-(b+a x)^2+c\right ) \left (b e \left (1-\frac {a f}{b e}\right )-e \left (c-(-b+d-a x)^2\right )^2\right )}{-g (-b+d-a x)^4+2 c g (-b+d-a x)^2+b g \left (1-\frac {g c^2+a h}{b g}\right )}d\sqrt {d+\sqrt {c+\sqrt {b+a x}}}}{a}\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {8 \int \left (\frac {e (b+a x)^3}{g}-\frac {3 d e (b+a x)^2}{g}-\frac {\left (c-3 d^2\right ) e (b+a x)}{g}-\frac {a (f g-e h) (b+a x)^3-3 a d (f g-e h) (b+a x)^2-a \left (c-3 d^2\right ) (f g-e h) (b+a x)+a d \left (c-d^2\right ) (f g-e h)}{g \left (-g (-b+d-a x)^4+2 c g (-b+d-a x)^2+b g \left (1-\frac {g c^2+a h}{b g}\right )\right )}+\frac {d \left (c-d^2\right ) e}{g}\right )d\sqrt {d+\sqrt {c+\sqrt {b+a x}}}}{a}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {8 \left (\frac {a d \left (c-d^2\right ) (f g-e h) \int \frac {1}{-g (b+a x)^4+4 d g (b+a x)^3+2 c \left (1-\frac {3 d^2}{c}\right ) g (b+a x)^2-4 c d \left (1-\frac {d^2}{c}\right ) g (b+a x)+b g \left (1-\frac {g d^4-2 c g d^2+c^2 g+a h}{b g}\right )}d\sqrt {d+\sqrt {c+\sqrt {b+a x}}}}{g}-\frac {a \left (c-3 d^2\right ) (f g-e h) \int \frac {b+a x}{-g (b+a x)^4+4 d g (b+a x)^3+2 c \left (1-\frac {3 d^2}{c}\right ) g (b+a x)^2-4 c d \left (1-\frac {d^2}{c}\right ) g (b+a x)+b g \left (1-\frac {g d^4-2 c g d^2+c^2 g+a h}{b g}\right )}d\sqrt {d+\sqrt {c+\sqrt {b+a x}}}}{g}+\frac {a (f g-e h) \int \frac {(b+a x)^3}{-g (b+a x)^4+4 d g (b+a x)^3+2 c \left (1-\frac {3 d^2}{c}\right ) g (b+a x)^2-4 c d \left (1-\frac {d^2}{c}\right ) g (b+a x)+b g \left (1-\frac {g d^4-2 c g d^2+c^2 g+a h}{b g}\right )}d\sqrt {d+\sqrt {c+\sqrt {b+a x}}}}{g}+\frac {3 a d (f g-e h) \int \frac {(b+a x)^2}{g (b+a x)^4-4 d g (b+a x)^3-2 c \left (1-\frac {3 d^2}{c}\right ) g (b+a x)^2+4 c d \left (1-\frac {d^2}{c}\right ) g (b+a x)-b g \left (1-\frac {g d^4-2 c g d^2+c^2 g+a h}{b g}\right )}d\sqrt {d+\sqrt {c+\sqrt {b+a x}}}}{g}+\frac {e \left (c-3 d^2\right ) (a x+b)^{3/2}}{3 g}-\frac {d e \left (c-d^2\right ) \sqrt {\sqrt {\sqrt {a x+b}+c}+d}}{g}+\frac {3 d e (a x+b)^{5/2}}{5 g}-\frac {e (a x+b)^{7/2}}{7 g}\right )}{a}\) |
3.29.93.3.1 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 = 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 | \(\frac {\frac {8 e \left (\frac {\left (d +\sqrt {c +\sqrt {a x +b}}\right )^{\frac {7}{2}}}{7}-\frac {3 \left (d +\sqrt {c +\sqrt {a x +b}}\right )^{\frac {5}{2}} d}{5}+\frac {\left (3 d^{2}-c \right ) \left (d +\sqrt {c +\sqrt {a x +b}}\right )^{\frac {3}{2}}}{3}+\sqrt {d +\sqrt {c +\sqrt {a x +b}}}\, d \left (-d^{2}+c \right )\right )}{g}-\frac {a \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (g \,\textit {\_Z}^{8}-4 d g \,\textit {\_Z}^{6}+\left (6 d^{2} g -2 c g \right ) \textit {\_Z}^{4}+\left (-4 d^{3} g +4 c d g \right ) \textit {\_Z}^{2}+d^{4} g -2 c \,d^{2} g +c^{2} g +h a -b g \right )}{\sum }\frac {\left (\left (e h -f g \right ) \textit {\_R}^{6}+3 d \left (-e h +f g \right ) \textit {\_R}^{4}+\left (3 d^{2} e h -3 d^{2} f g -c e h +c f g \right ) \textit {\_R}^{2}-d^{3} e h +d^{3} f g +c d e h -c d f g \right ) \ln \left (\sqrt {d +\sqrt {c +\sqrt {a x +b}}}-\textit {\_R} \right )}{\textit {\_R}^{7}-3 \textit {\_R}^{5} d +3 \textit {\_R}^{3} d^{2}-\textit {\_R}^{3} c -\textit {\_R} \,d^{3}+\textit {\_R} c d}\right )}{g^{2}}}{a}\) | \(313\) |
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)))
Timed out. \[ \int \frac {f+e x}{(h+g x) \sqrt {d+\sqrt {c+\sqrt {b+a x}}}} \, dx=\text {Timed out} \]
Timed out. \[ \int \frac {f+e x}{(h+g x) \sqrt {d+\sqrt {c+\sqrt {b+a x}}}} \, dx=\text {Timed out} \]
Not integrable
Time = 1.28 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.09 \[ \int \frac {f+e x}{(h+g x) \sqrt {d+\sqrt {c+\sqrt {b+a x}}}} \, dx=\int { \frac {e x + f}{{\left (g x + h\right )} \sqrt {d + \sqrt {c + \sqrt {a x + b}}}} \,d x } \]
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 \[ \int \frac {f+e x}{(h+g x) \sqrt {d+\sqrt {c+\sqrt {b+a x}}}} \, dx=\text {Too large to display} \]
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 + sqrt(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 + sqrt(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*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*sg n(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(sqr t(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...
Not integrable
Time = 8.87 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.09 \[ \int \frac {f+e x}{(h+g x) \sqrt {d+\sqrt {c+\sqrt {b+a x}}}} \, dx=\int \frac {f+e\,x}{\left (h+g\,x\right )\,\sqrt {d+\sqrt {c+\sqrt {b+a\,x}}}} \,d x \]