Integrand size = 49, antiderivative size = 269 \[ \int \frac {\sqrt {b+a x} \sqrt {c+\sqrt {b+a x}}}{x \sqrt {d+\sqrt {c+\sqrt {b+a x}}}} \, dx=\frac {8}{15} \left (3 c+8 d^2\right ) \sqrt {d+\sqrt {c+\sqrt {b+a x}}}+\frac {8}{5} \sqrt {b+a x} \sqrt {d+\sqrt {c+\sqrt {b+a x}}}-\frac {32}{15} d \sqrt {c+\sqrt {b+a x}} \sqrt {d+\sqrt {c+\sqrt {b+a x}}}-b \text {RootSum}\left [b-c^2+2 c d^2-d^4-4 c d \text {$\#$1}^2+4 d^3 \text {$\#$1}^2+2 c \text {$\#$1}^4-6 d^2 \text {$\#$1}^4+4 d \text {$\#$1}^6-\text {$\#$1}^8\&,\frac {d \log \left (\sqrt {d+\sqrt {c+\sqrt {b+a x}}}-\text {$\#$1}\right )-\log \left (\sqrt {d+\sqrt {c+\sqrt {b+a x}}}-\text {$\#$1}\right ) \text {$\#$1}^2}{-c \text {$\#$1}+d^2 \text {$\#$1}-2 d \text {$\#$1}^3+\text {$\#$1}^5}\&\right ] \]
Time = 0.01 (sec) , antiderivative size = 223, normalized size of antiderivative = 0.83 \[ \int \frac {\sqrt {b+a x} \sqrt {c+\sqrt {b+a x}}}{x \sqrt {d+\sqrt {c+\sqrt {b+a x}}}} \, dx=\frac {8}{15} \sqrt {d+\sqrt {c+\sqrt {b+a x}}} \left (3 c+8 d^2+3 \sqrt {b+a x}-4 d \sqrt {c+\sqrt {b+a x}}\right )-b \text {RootSum}\left [b-c^2+2 c d^2-d^4-4 c d \text {$\#$1}^2+4 d^3 \text {$\#$1}^2+2 c \text {$\#$1}^4-6 d^2 \text {$\#$1}^4+4 d \text {$\#$1}^6-\text {$\#$1}^8\&,\frac {d \log \left (\sqrt {d+\sqrt {c+\sqrt {b+a x}}}-\text {$\#$1}\right )-\log \left (\sqrt {d+\sqrt {c+\sqrt {b+a x}}}-\text {$\#$1}\right ) \text {$\#$1}^2}{-c \text {$\#$1}+d^2 \text {$\#$1}-2 d \text {$\#$1}^3+\text {$\#$1}^5}\&\right ] \]
(8*Sqrt[d + Sqrt[c + Sqrt[b + a*x]]]*(3*c + 8*d^2 + 3*Sqrt[b + a*x] - 4*d* Sqrt[c + Sqrt[b + a*x]]))/15 - b*RootSum[b - c^2 + 2*c*d^2 - d^4 - 4*c*d*# 1^2 + 4*d^3*#1^2 + 2*c*#1^4 - 6*d^2*#1^4 + 4*d*#1^6 - #1^8 & , (d*Log[Sqrt [d + Sqrt[c + Sqrt[b + a*x]]] - #1] - Log[Sqrt[d + Sqrt[c + Sqrt[b + a*x]] ] - #1]*#1^2)/(-(c*#1) + d^2*#1 - 2*d*#1^3 + #1^5) & ]
Leaf count is larger than twice the leaf count of optimal. \(757\) vs. \(2(269)=538\).
Time = 5.00 (sec) , antiderivative size = 757, normalized size of antiderivative = 2.81, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.184, Rules used = {7267, 25, 7267, 2027, 2091, 7267, 25, 7291, 2009}
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.
| \(\displaystyle \int \frac {\sqrt {a x+b} \sqrt {\sqrt {a x+b}+c}}{x \sqrt {\sqrt {\sqrt {a x+b}+c}+d}} \, dx\) |
| \(\Big \downarrow \) 7267 |
| \(\displaystyle 2 \int \frac {(b+a x) \sqrt {c+\sqrt {b+a x}}}{a x \sqrt {d+\sqrt {c+\sqrt {b+a x}}}}d\sqrt {b+a x}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -2 \int -\frac {(b+a x) \sqrt {c+\sqrt {b+a x}}}{a x \sqrt {d+\sqrt {c+\sqrt {b+a x}}}}d\sqrt {b+a x}\) |
| \(\Big \downarrow \) 7267 |
| \(\displaystyle -4 \int \frac {\left (c \sqrt {c+\sqrt {b+a x}}-(b+a x)^{3/2}\right )^2}{\left (b-(-b+c-a x)^2\right ) \sqrt {d+\sqrt {c+\sqrt {b+a x}}}}d\sqrt {c+\sqrt {b+a x}}\) |
| \(\Big \downarrow \) 2027 |
| \(\displaystyle -4 \int \frac {(-b+c-a x)^2 (b+a x)}{\left (b-(-b+c-a x)^2\right ) \sqrt {d+\sqrt {c+\sqrt {b+a x}}}}d\sqrt {c+\sqrt {b+a x}}\) |
| \(\Big \downarrow \) 2091 |
| \(\displaystyle -4 \int \frac {(-b+c-a x)^2 (b+a x)}{\left (-c^2+2 (b+a x) c-(b+a x)^2+b\right ) \sqrt {d+\sqrt {c+\sqrt {b+a x}}}}d\sqrt {c+\sqrt {b+a x}}\) |
| \(\Big \downarrow \) 7267 |
| \(\displaystyle 8 \int -\frac {(-b+d-a x)^2 \left (c-(-b+d-a x)^2\right )^2}{b-\left (c-(-b+d-a x)^2\right )^2}d\sqrt {d+\sqrt {c+\sqrt {b+a x}}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -8 \int \frac {(-b+d-a x)^2 \left (c-(-b+d-a x)^2\right )^2}{b-\left (c-(-b+d-a x)^2\right )^2}d\sqrt {d+\sqrt {c+\sqrt {b+a x}}}\) |
| \(\Big \downarrow \) 7291 |
| \(\displaystyle -8 \int \left (\frac {b c-b \left (c-(-b+d-a x)^2\right )}{b-\left (c-(-b+d-a x)^2\right )^2}-(-b+d-a x)^2\right )d\sqrt {d+\sqrt {c+\sqrt {b+a x}}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle 8 \left (-\frac {\sqrt {b} \sqrt {\sqrt {b}+c} \arctan \left (\frac {\sqrt {\sqrt {\sqrt {a x+b}+c}+d}}{\sqrt {\sqrt {\sqrt {b}+c}-d}}\right )}{4 \sqrt {\sqrt {\sqrt {b}+c}-d}}+\frac {\sqrt {b} \left (\sqrt {b}-c\right ) \text {arctanh}\left (\frac {\sqrt {\sqrt {\sqrt {b}-c+d^2}+d}-\sqrt {2} \sqrt {\sqrt {\sqrt {a x+b}+c}+d}}{\sqrt {d-\sqrt {\sqrt {b}-c+d^2}}}\right )}{4 \sqrt {2} \sqrt {\sqrt {b}-c+d^2} \sqrt {d-\sqrt {\sqrt {b}-c+d^2}}}-\frac {\sqrt {b} \left (\sqrt {b}-c\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {\sqrt {\sqrt {a x+b}+c}+d}+\sqrt {\sqrt {\sqrt {b}-c+d^2}+d}}{\sqrt {d-\sqrt {\sqrt {b}-c+d^2}}}\right )}{4 \sqrt {2} \sqrt {\sqrt {b}-c+d^2} \sqrt {d-\sqrt {\sqrt {b}-c+d^2}}}-\frac {\sqrt {b} \sqrt {\sqrt {b}+c} \text {arctanh}\left (\frac {\sqrt {\sqrt {\sqrt {a x+b}+c}+d}}{\sqrt {\sqrt {\sqrt {b}+c}+d}}\right )}{4 \sqrt {\sqrt {\sqrt {b}+c}+d}}+d^2 \sqrt {\sqrt {\sqrt {a x+b}+c}+d}-\frac {\sqrt {b} \left (\sqrt {b}-c\right ) \log \left (-\sqrt {2} \sqrt {\sqrt {\sqrt {b}-c+d^2}+d} \sqrt {\sqrt {\sqrt {a x+b}+c}+d}+a x+\sqrt {\sqrt {b}-c+d^2}+b\right )}{8 \sqrt {2} \sqrt {\sqrt {b}-c+d^2} \sqrt {\sqrt {\sqrt {b}-c+d^2}+d}}+\frac {\sqrt {b} \left (\sqrt {b}-c\right ) \log \left (\sqrt {2} \sqrt {\sqrt {\sqrt {b}-c+d^2}+d} \sqrt {\sqrt {\sqrt {a x+b}+c}+d}+a x+\sqrt {\sqrt {b}-c+d^2}+b\right )}{8 \sqrt {2} \sqrt {\sqrt {b}-c+d^2} \sqrt {\sqrt {\sqrt {b}-c+d^2}+d}}-\frac {2}{3} d (a x+b)^{3/2}+\frac {1}{5} (a x+b)^{5/2}\right )\) |
8*((-2*d*(b + a*x)^(3/2))/3 + (b + a*x)^(5/2)/5 + d^2*Sqrt[d + Sqrt[c + Sq rt[b + a*x]]] - (Sqrt[b]*Sqrt[Sqrt[b] + c]*ArcTan[Sqrt[d + Sqrt[c + Sqrt[b + a*x]]]/Sqrt[Sqrt[Sqrt[b] + c] - d]])/(4*Sqrt[Sqrt[Sqrt[b] + c] - d]) - (Sqrt[b]*Sqrt[Sqrt[b] + c]*ArcTanh[Sqrt[d + Sqrt[c + Sqrt[b + a*x]]]/Sqrt[ Sqrt[Sqrt[b] + c] + d]])/(4*Sqrt[Sqrt[Sqrt[b] + c] + d]) + (Sqrt[b]*(Sqrt[ b] - c)*ArcTanh[(Sqrt[d + Sqrt[Sqrt[b] - c + d^2]] - Sqrt[2]*Sqrt[d + Sqrt [c + Sqrt[b + a*x]]])/Sqrt[d - Sqrt[Sqrt[b] - c + d^2]]])/(4*Sqrt[2]*Sqrt[ Sqrt[b] - c + d^2]*Sqrt[d - Sqrt[Sqrt[b] - c + d^2]]) - (Sqrt[b]*(Sqrt[b] - c)*ArcTanh[(Sqrt[d + Sqrt[Sqrt[b] - c + d^2]] + Sqrt[2]*Sqrt[d + Sqrt[c + Sqrt[b + a*x]]])/Sqrt[d - Sqrt[Sqrt[b] - c + d^2]]])/(4*Sqrt[2]*Sqrt[Sqr t[b] - c + d^2]*Sqrt[d - Sqrt[Sqrt[b] - c + d^2]]) - (Sqrt[b]*(Sqrt[b] - c )*Log[b + Sqrt[Sqrt[b] - c + d^2] + a*x - Sqrt[2]*Sqrt[d + Sqrt[Sqrt[b] - c + d^2]]*Sqrt[d + Sqrt[c + Sqrt[b + a*x]]]])/(8*Sqrt[2]*Sqrt[Sqrt[b] - c + d^2]*Sqrt[d + Sqrt[Sqrt[b] - c + d^2]]) + (Sqrt[b]*(Sqrt[b] - c)*Log[b + Sqrt[Sqrt[b] - c + d^2] + a*x + Sqrt[2]*Sqrt[d + Sqrt[Sqrt[b] - c + d^2]] *Sqrt[d + Sqrt[c + Sqrt[b + a*x]]]])/(8*Sqrt[2]*Sqrt[Sqrt[b] - c + d^2]*Sq rt[d + Sqrt[Sqrt[b] - c + d^2]]))
3.28.91.3.1 Defintions of rubi rules used
Int[(Fx_.)*((a_.)*(x_)^(r_.) + (b_.)*(x_)^(s_.))^(p_.), x_Symbol] :> Int[x^ (p*r)*(a + b*x^(s - r))^p*Fx, x] /; FreeQ[{a, b, r, s}, x] && IntegerQ[p] & & PosQ[s - r] && !(EqQ[p, 1] && EqQ[u, 1])
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]]
Int[(v_)/((a_) + (b_.)*(u_)^(n_.)), x_Symbol] :> Int[ExpandIntegrand[Polyno mialInSubst[v, u, x]/(a + b*x^n), x] /. x -> u, x] /; FreeQ[{a, b}, x] && I GtQ[n, 0] && PolynomialInQ[v, u, x]
Time = 0.20 (sec) , antiderivative size = 190, normalized size of antiderivative = 0.71
| method | result | size |
| derivativedivides | \(\frac {8 \left (d +\sqrt {c +\sqrt {a x +b}}\right )^{\frac {5}{2}}}{5}-\frac {16 d \left (d +\sqrt {c +\sqrt {a x +b}}\right )^{\frac {3}{2}}}{3}+8 \sqrt {d +\sqrt {c +\sqrt {a x +b}}}\, d^{2}+b \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{8}-4 d \,\textit {\_Z}^{6}+\left (6 d^{2}-2 c \right ) \textit {\_Z}^{4}+\left (-4 d^{3}+4 c d \right ) \textit {\_Z}^{2}+d^{4}-2 c \,d^{2}+c^{2}-b \right )}{\sum }\frac {\left (-\textit {\_R}^{4}+2 \textit {\_R}^{2} d -d^{2}\right ) \ln \left (\sqrt {d +\sqrt {c +\sqrt {a x +b}}}-\textit {\_R} \right )}{-\textit {\_R}^{7}+3 \textit {\_R}^{5} d -3 \textit {\_R}^{3} d^{2}+c \,\textit {\_R}^{3}+\textit {\_R} \,d^{3}-\textit {\_R} c d}\right )\) | \(190\) |
| default | \(\frac {8 \left (d +\sqrt {c +\sqrt {a x +b}}\right )^{\frac {5}{2}}}{5}-\frac {16 d \left (d +\sqrt {c +\sqrt {a x +b}}\right )^{\frac {3}{2}}}{3}+8 \sqrt {d +\sqrt {c +\sqrt {a x +b}}}\, d^{2}+b \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{8}-4 d \,\textit {\_Z}^{6}+\left (6 d^{2}-2 c \right ) \textit {\_Z}^{4}+\left (-4 d^{3}+4 c d \right ) \textit {\_Z}^{2}+d^{4}-2 c \,d^{2}+c^{2}-b \right )}{\sum }\frac {\left (-\textit {\_R}^{4}+2 \textit {\_R}^{2} d -d^{2}\right ) \ln \left (\sqrt {d +\sqrt {c +\sqrt {a x +b}}}-\textit {\_R} \right )}{-\textit {\_R}^{7}+3 \textit {\_R}^{5} d -3 \textit {\_R}^{3} d^{2}+c \,\textit {\_R}^{3}+\textit {\_R} \,d^{3}-\textit {\_R} c d}\right )\) | \(190\) |
int((a*x+b)^(1/2)*(c+(a*x+b)^(1/2))^(1/2)/x/(d+(c+(a*x+b)^(1/2))^(1/2))^(1 /2),x,method=_RETURNVERBOSE)
8/5*(d+(c+(a*x+b)^(1/2))^(1/2))^(5/2)-16/3*d*(d+(c+(a*x+b)^(1/2))^(1/2))^( 3/2)+8*(d+(c+(a*x+b)^(1/2))^(1/2))^(1/2)*d^2+b*sum((-_R^4+2*_R^2*d-d^2)/(- _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(_Z^8-4*d*_Z^6+(6*d^2-2*c)*_Z^4+(-4*d^3+4*c*d)*_Z^2 +d^4-2*c*d^2+c^2-b))
Timed out. \[ \int \frac {\sqrt {b+a x} \sqrt {c+\sqrt {b+a x}}}{x \sqrt {d+\sqrt {c+\sqrt {b+a x}}}} \, dx=\text {Timed out} \]
integrate((a*x+b)^(1/2)*(c+(a*x+b)^(1/2))^(1/2)/x/(d+(c+(a*x+b)^(1/2))^(1/ 2))^(1/2),x, algorithm="fricas")
Timed out. \[ \int \frac {\sqrt {b+a x} \sqrt {c+\sqrt {b+a x}}}{x \sqrt {d+\sqrt {c+\sqrt {b+a x}}}} \, dx=\text {Timed out} \]
Not integrable
Time = 1.35 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.14 \[ \int \frac {\sqrt {b+a x} \sqrt {c+\sqrt {b+a x}}}{x \sqrt {d+\sqrt {c+\sqrt {b+a x}}}} \, dx=\int { \frac {\sqrt {a x + b} \sqrt {c + \sqrt {a x + b}}}{\sqrt {d + \sqrt {c + \sqrt {a x + b}}} x} \,d x } \]
integrate((a*x+b)^(1/2)*(c+(a*x+b)^(1/2))^(1/2)/x/(d+(c+(a*x+b)^(1/2))^(1/ 2))^(1/2),x, algorithm="maxima")
Result contains higher order function than in optimal. Order 3 vs. order 1.
Time = 38.58 (sec) , antiderivative size = 1404, normalized size of antiderivative = 5.22 \[ \int \frac {\sqrt {b+a x} \sqrt {c+\sqrt {b+a x}}}{x \sqrt {d+\sqrt {c+\sqrt {b+a x}}}} \, dx=\text {Too large to display} \]
integrate((a*x+b)^(1/2)*(c+(a*x+b)^(1/2))^(1/2)/x/(d+(c+(a*x+b)^(1/2))^(1/ 2))^(1/2),x, algorithm="giac")
8/5*(d + sqrt(c + sqrt(a*x + b)))^(5/2) - 16/3*(d + sqrt(c + sqrt(a*x + b) ))^(3/2)*d + 8*sqrt(d + sqrt(c + sqrt(a*x + b)))*d^2 - (b*(d + sqrt(c + sq rt(b)))^2 - 2*b*(d + sqrt(c + sqrt(b)))*d + b*d^2)*log(sqrt(d + sqrt(c + s qrt(a*x + b))) + sqrt(d + sqrt(c + sqrt(b))))/((d + sqrt(c + sqrt(b)))^(7/ 2) - 3*(d + sqrt(c + sqrt(b)))^(5/2)*d + 3*(d + sqrt(c + sqrt(b)))^(3/2)*d ^2 - sqrt(d + sqrt(c + sqrt(b)))*d^3 - c*(d + sqrt(c + sqrt(b)))^(3/2) + c *sqrt(d + sqrt(c + sqrt(b)))*d) + (b*(d + sqrt(c + sqrt(b)))^2 - 2*b*(d + sqrt(c + sqrt(b)))*d + b*d^2)*log(sqrt(d + sqrt(c + sqrt(a*x + b))) - sqrt (d + sqrt(c + sqrt(b))))/((d + sqrt(c + sqrt(b)))^(7/2) - 3*(d + sqrt(c + sqrt(b)))^(5/2)*d + 3*(d + sqrt(c + sqrt(b)))^(3/2)*d^2 - sqrt(d + sqrt(c + sqrt(b)))*d^3 - c*(d + sqrt(c + sqrt(b)))^(3/2) + c*sqrt(d + sqrt(c + sq rt(b)))*d) - (b*(d - sqrt(c + sqrt(b)))^2 - 2*b*(d - sqrt(c + sqrt(b)))*d + b*d^2)*log(sqrt(d + sqrt(c + sqrt(a*x + b))) + sqrt(d - sqrt(c + sqrt(b) )))/((d - sqrt(c + sqrt(b)))^(7/2) - 3*(d - sqrt(c + sqrt(b)))^(5/2)*d + 3 *(d - sqrt(c + sqrt(b)))^(3/2)*d^2 - sqrt(d - sqrt(c + sqrt(b)))*d^3 - c*( d - sqrt(c + sqrt(b)))^(3/2) + c*sqrt(d - sqrt(c + sqrt(b)))*d) + (b*(d - sqrt(c + sqrt(b)))^2 - 2*b*(d - sqrt(c + sqrt(b)))*d + b*d^2)*log(sqrt(d + sqrt(c + sqrt(a*x + b))) - sqrt(d - sqrt(c + sqrt(b))))/((d - sqrt(c + sq rt(b)))^(7/2) - 3*(d - sqrt(c + sqrt(b)))^(5/2)*d + 3*(d - sqrt(c + sqrt(b )))^(3/2)*d^2 - sqrt(d - sqrt(c + sqrt(b)))*d^3 - c*(d - sqrt(c + sqrt(...
Not integrable
Time = 0.00 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.14 \[ \int \frac {\sqrt {b+a x} \sqrt {c+\sqrt {b+a x}}}{x \sqrt {d+\sqrt {c+\sqrt {b+a x}}}} \, dx=\int \frac {\sqrt {c+\sqrt {b+a\,x}}\,\sqrt {b+a\,x}}{x\,\sqrt {d+\sqrt {c+\sqrt {b+a\,x}}}} \,d x \]
int(((c + (b + a*x)^(1/2))^(1/2)*(b + a*x)^(1/2))/(x*(d + (c + (b + a*x)^( 1/2))^(1/2))^(1/2)),x)