Integrand size = 42, antiderivative size = 586 \[ \int \frac {\sqrt {a x+\sqrt {-b+a x}}}{\sqrt {-b+a x} \left (1+a^2 x^2\right )} \, dx=-\frac {\text {RootSum}\left [1+b^2+2 b^3+b^4-8 \text {$\#$1}-8 b^2 \text {$\#$1}-8 b^3 \text {$\#$1}+24 \text {$\#$1}^2+20 b^2 \text {$\#$1}^2+4 b^3 \text {$\#$1}^2-32 \text {$\#$1}^3-16 b^2 \text {$\#$1}^3+16 \text {$\#$1}^4+2 b \text {$\#$1}^4+6 b^2 \text {$\#$1}^4-8 b \text {$\#$1}^5+4 b \text {$\#$1}^6+\text {$\#$1}^8\&,\frac {-b^2 \log \left (\sqrt {-b+a x}-\sqrt {a x+\sqrt {-b+a x}}+\text {$\#$1}\right )+2 b \log \left (\sqrt {-b+a x}-\sqrt {a x+\sqrt {-b+a x}}+\text {$\#$1}\right ) \text {$\#$1}+2 b^2 \log \left (\sqrt {-b+a x}-\sqrt {a x+\sqrt {-b+a x}}+\text {$\#$1}\right ) \text {$\#$1}-\log \left (\sqrt {-b+a x}-\sqrt {a x+\sqrt {-b+a x}}+\text {$\#$1}\right ) \text {$\#$1}^2-6 b \log \left (\sqrt {-b+a x}-\sqrt {a x+\sqrt {-b+a x}}+\text {$\#$1}\right ) \text {$\#$1}^2+4 \log \left (\sqrt {-b+a x}-\sqrt {a x+\sqrt {-b+a x}}+\text {$\#$1}\right ) \text {$\#$1}^3+4 b \log \left (\sqrt {-b+a x}-\sqrt {a x+\sqrt {-b+a x}}+\text {$\#$1}\right ) \text {$\#$1}^3-5 \log \left (\sqrt {-b+a x}-\sqrt {a x+\sqrt {-b+a x}}+\text {$\#$1}\right ) \text {$\#$1}^4+2 \log \left (\sqrt {-b+a x}-\sqrt {a x+\sqrt {-b+a x}}+\text {$\#$1}\right ) \text {$\#$1}^5}{-1-b^2-b^3+6 \text {$\#$1}+5 b^2 \text {$\#$1}+b^3 \text {$\#$1}-12 \text {$\#$1}^2-6 b^2 \text {$\#$1}^2+8 \text {$\#$1}^3+b \text {$\#$1}^3+3 b^2 \text {$\#$1}^3-5 b \text {$\#$1}^4+3 b \text {$\#$1}^5+\text {$\#$1}^7}\&\right ]}{2 a} \]
Time = 0.51 (sec) , antiderivative size = 604, normalized size of antiderivative = 1.03 \[ \int \frac {\sqrt {a x+\sqrt {-b+a x}}}{\sqrt {-b+a x} \left (1+a^2 x^2\right )} \, dx=-\frac {\text {RootSum}\left [1+b^2+2 b^3+b^4-8 \text {$\#$1}-8 b^2 \text {$\#$1}-8 b^3 \text {$\#$1}+24 \text {$\#$1}^2+20 b^2 \text {$\#$1}^2+4 b^3 \text {$\#$1}^2-32 \text {$\#$1}^3-16 b^2 \text {$\#$1}^3+16 \text {$\#$1}^4+2 b \text {$\#$1}^4+6 b^2 \text {$\#$1}^4-8 b \text {$\#$1}^5+4 b \text {$\#$1}^6+\text {$\#$1}^8\&,\frac {-b^2 \log \left (-\sqrt {-b+a x}+\sqrt {a x+\sqrt {-b+a x}}-\text {$\#$1}\right )+2 b \log \left (-\sqrt {-b+a x}+\sqrt {a x+\sqrt {-b+a x}}-\text {$\#$1}\right ) \text {$\#$1}+2 b^2 \log \left (-\sqrt {-b+a x}+\sqrt {a x+\sqrt {-b+a x}}-\text {$\#$1}\right ) \text {$\#$1}-\log \left (-\sqrt {-b+a x}+\sqrt {a x+\sqrt {-b+a x}}-\text {$\#$1}\right ) \text {$\#$1}^2-6 b \log \left (-\sqrt {-b+a x}+\sqrt {a x+\sqrt {-b+a x}}-\text {$\#$1}\right ) \text {$\#$1}^2+4 \log \left (-\sqrt {-b+a x}+\sqrt {a x+\sqrt {-b+a x}}-\text {$\#$1}\right ) \text {$\#$1}^3+4 b \log \left (-\sqrt {-b+a x}+\sqrt {a x+\sqrt {-b+a x}}-\text {$\#$1}\right ) \text {$\#$1}^3-5 \log \left (-\sqrt {-b+a x}+\sqrt {a x+\sqrt {-b+a x}}-\text {$\#$1}\right ) \text {$\#$1}^4+2 \log \left (-\sqrt {-b+a x}+\sqrt {a x+\sqrt {-b+a x}}-\text {$\#$1}\right ) \text {$\#$1}^5}{-1-b^2-b^3+6 \text {$\#$1}+5 b^2 \text {$\#$1}+b^3 \text {$\#$1}-12 \text {$\#$1}^2-6 b^2 \text {$\#$1}^2+8 \text {$\#$1}^3+b \text {$\#$1}^3+3 b^2 \text {$\#$1}^3-5 b \text {$\#$1}^4+3 b \text {$\#$1}^5+\text {$\#$1}^7}\&\right ]}{2 a} \]
-1/2*RootSum[1 + b^2 + 2*b^3 + b^4 - 8*#1 - 8*b^2*#1 - 8*b^3*#1 + 24*#1^2 + 20*b^2*#1^2 + 4*b^3*#1^2 - 32*#1^3 - 16*b^2*#1^3 + 16*#1^4 + 2*b*#1^4 + 6*b^2*#1^4 - 8*b*#1^5 + 4*b*#1^6 + #1^8 & , (-(b^2*Log[-Sqrt[-b + a*x] + S qrt[a*x + Sqrt[-b + a*x]] - #1]) + 2*b*Log[-Sqrt[-b + a*x] + Sqrt[a*x + Sq rt[-b + a*x]] - #1]*#1 + 2*b^2*Log[-Sqrt[-b + a*x] + Sqrt[a*x + Sqrt[-b + a*x]] - #1]*#1 - Log[-Sqrt[-b + a*x] + Sqrt[a*x + Sqrt[-b + a*x]] - #1]*#1 ^2 - 6*b*Log[-Sqrt[-b + a*x] + Sqrt[a*x + Sqrt[-b + a*x]] - #1]*#1^2 + 4*L og[-Sqrt[-b + a*x] + Sqrt[a*x + Sqrt[-b + a*x]] - #1]*#1^3 + 4*b*Log[-Sqrt [-b + a*x] + Sqrt[a*x + Sqrt[-b + a*x]] - #1]*#1^3 - 5*Log[-Sqrt[-b + a*x] + Sqrt[a*x + Sqrt[-b + a*x]] - #1]*#1^4 + 2*Log[-Sqrt[-b + a*x] + Sqrt[a* x + Sqrt[-b + a*x]] - #1]*#1^5)/(-1 - b^2 - b^3 + 6*#1 + 5*b^2*#1 + b^3*#1 - 12*#1^2 - 6*b^2*#1^2 + 8*#1^3 + b*#1^3 + 3*b^2*#1^3 - 5*b*#1^4 + 3*b*#1 ^5 + #1^7) & ]/a
Result contains complex when optimal does not.
Time = 1.69 (sec) , antiderivative size = 587, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {7267, 7279, 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.
| \(\displaystyle \int \frac {\sqrt {\sqrt {a x-b}+a x}}{\left (a^2 x^2+1\right ) \sqrt {a x-b}} \, dx\) |
| \(\Big \downarrow \) 7267 |
| \(\displaystyle \frac {2 \int \frac {\sqrt {a x+\sqrt {a x-b}}}{b^2+2 (a x-b) b+(a x-b)^2+1}d\sqrt {a x-b}}{a}\) |
| \(\Big \downarrow \) 7279 |
| \(\displaystyle \frac {2 \int \left (\frac {i \sqrt {a x+\sqrt {a x-b}}}{2 b+2 (a x-b)+2 i}-\frac {i \sqrt {a x+\sqrt {a x-b}}}{2 b+2 (a x-b)-2 i}\right )d\sqrt {a x-b}}{a}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {2 \left (-\frac {\sqrt {-i b+\sqrt {b-(1-i)}+(1+i)} \arctan \left (\frac {\left (b+i \sqrt {b-(1-i)}-(1-i)\right ) \sqrt {a x-b}+\sqrt {b-(1-i)} (b+i)}{\sqrt {2} \sqrt {1-i b} \sqrt {-i b+\sqrt {b-(1-i)}+(1+i)} \sqrt [4]{b-(1-i)} \sqrt {\sqrt {a x-b}+a x}}\right )}{2 \sqrt {2} \sqrt {1-i b} \sqrt [4]{b-(1-i)}}+\frac {i \sqrt {i-\sqrt {-b+i}} \text {arctanh}\left (\frac {\left (1-2 \sqrt {-b+i}\right ) \left (-\sqrt {a x-b}\right )-2 b+\sqrt {-b+i}}{2 \sqrt {i-\sqrt {-b+i}} \sqrt {\sqrt {a x-b}+a x}}\right )}{4 \sqrt {-b+i}}+\frac {i \sqrt {\sqrt {-b+i}+i} \text {arctanh}\left (\frac {\left (1+2 \sqrt {-b+i}\right ) \sqrt {a x-b}+2 b+\sqrt {-b+i}}{2 \sqrt {\sqrt {-b+i}+i} \sqrt {\sqrt {a x-b}+a x}}\right )}{4 \sqrt {-b+i}}+\frac {i \sqrt {b-i \sqrt {b-(1-i)}-(1-i)} \text {arctanh}\left (\frac {\left (-b+i \sqrt {b-(1-i)}+(1-i)\right ) \sqrt {a x-b}+\sqrt {b-(1-i)} (b+i)}{\sqrt {2} \sqrt {-b-i} \sqrt [4]{b-(1-i)} \sqrt {b-i \sqrt {b-(1-i)}-(1-i)} \sqrt {\sqrt {a x-b}+a x}}\right )}{2 \sqrt {2} \sqrt {-b-i} \sqrt [4]{b-(1-i)}}\right )}{a}\) |
(2*(-1/2*(Sqrt[(1 + I) + Sqrt[(-1 + I) + b] - I*b]*ArcTan[(Sqrt[(-1 + I) + b]*(I + b) + ((-1 + I) + I*Sqrt[(-1 + I) + b] + b)*Sqrt[-b + a*x])/(Sqrt[ 2]*Sqrt[1 - I*b]*Sqrt[(1 + I) + Sqrt[(-1 + I) + b] - I*b]*((-1 + I) + b)^( 1/4)*Sqrt[a*x + Sqrt[-b + a*x]])])/(Sqrt[2]*Sqrt[1 - I*b]*((-1 + I) + b)^( 1/4)) + ((I/4)*Sqrt[I - Sqrt[I - b]]*ArcTanh[(Sqrt[I - b] - 2*b - (1 - 2*S qrt[I - b])*Sqrt[-b + a*x])/(2*Sqrt[I - Sqrt[I - b]]*Sqrt[a*x + Sqrt[-b + a*x]])])/Sqrt[I - b] + ((I/4)*Sqrt[I + Sqrt[I - b]]*ArcTanh[(Sqrt[I - b] + 2*b + (1 + 2*Sqrt[I - b])*Sqrt[-b + a*x])/(2*Sqrt[I + Sqrt[I - b]]*Sqrt[a *x + Sqrt[-b + a*x]])])/Sqrt[I - b] + ((I/2)*Sqrt[(-1 + I) - I*Sqrt[(-1 + I) + b] + b]*ArcTanh[(Sqrt[(-1 + I) + b]*(I + b) + ((1 - I) + I*Sqrt[(-1 + I) + b] - b)*Sqrt[-b + a*x])/(Sqrt[2]*Sqrt[-I - b]*((-1 + I) + b)^(1/4)*S qrt[(-1 + I) - I*Sqrt[(-1 + I) + b] + b]*Sqrt[a*x + Sqrt[-b + a*x]])])/(Sq rt[2]*Sqrt[-I - b]*((-1 + I) + b)^(1/4))))/a
3.32.3.3.1 Defintions of rubi rules used
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[(u_)/((a_.) + (b_.)*(x_)^(n_.) + (c_.)*(x_)^(n2_.)), x_Symbol] :> With[ {v = RationalFunctionExpand[u/(a + b*x^n + c*x^(2*n)), x]}, Int[v, x] /; Su mQ[v]] /; FreeQ[{a, b, c}, x] && EqQ[n2, 2*n] && IGtQ[n, 0]
Time = 0.37 (sec) , antiderivative size = 239, normalized size of antiderivative = 0.41
| method | result | size |
| derivativedivides | \(-\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{8}+4 b \,\textit {\_Z}^{6}-8 b \,\textit {\_Z}^{5}+\left (6 b^{2}+2 b +16\right ) \textit {\_Z}^{4}+\left (-16 b^{2}-32\right ) \textit {\_Z}^{3}+\left (4 b^{3}+20 b^{2}+24\right ) \textit {\_Z}^{2}+\left (-8 b^{3}-8 b^{2}-8\right ) \textit {\_Z} +b^{4}+2 b^{3}+b^{2}+1\right )}{\sum }\frac {\left (2 \textit {\_R}^{5}-5 \textit {\_R}^{4}+4 \left (1+b \right ) \textit {\_R}^{3}+\left (-6 b -1\right ) \textit {\_R}^{2}+2 b \left (1+b \right ) \textit {\_R} -b^{2}\right ) \ln \left (\sqrt {a x +\sqrt {a x -b}}-\sqrt {a x -b}-\textit {\_R} \right )}{\textit {\_R}^{7}+3 \textit {\_R}^{5} b -5 \textit {\_R}^{4} b +3 \textit {\_R}^{3} b^{2}+\textit {\_R}^{3} b -6 \textit {\_R}^{2} b^{2}+\textit {\_R} \,b^{3}+8 \textit {\_R}^{3}+5 \textit {\_R} \,b^{2}-b^{3}-12 \textit {\_R}^{2}-b^{2}+6 \textit {\_R} -1}}{2 a}\) | \(239\) |
| default | \(-\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{8}+4 b \,\textit {\_Z}^{6}-8 b \,\textit {\_Z}^{5}+\left (6 b^{2}+2 b +16\right ) \textit {\_Z}^{4}+\left (-16 b^{2}-32\right ) \textit {\_Z}^{3}+\left (4 b^{3}+20 b^{2}+24\right ) \textit {\_Z}^{2}+\left (-8 b^{3}-8 b^{2}-8\right ) \textit {\_Z} +b^{4}+2 b^{3}+b^{2}+1\right )}{\sum }\frac {\left (2 \textit {\_R}^{5}-5 \textit {\_R}^{4}+4 \left (1+b \right ) \textit {\_R}^{3}+\left (-6 b -1\right ) \textit {\_R}^{2}+2 b \left (1+b \right ) \textit {\_R} -b^{2}\right ) \ln \left (\sqrt {a x +\sqrt {a x -b}}-\sqrt {a x -b}-\textit {\_R} \right )}{\textit {\_R}^{7}+3 \textit {\_R}^{5} b -5 \textit {\_R}^{4} b +3 \textit {\_R}^{3} b^{2}+\textit {\_R}^{3} b -6 \textit {\_R}^{2} b^{2}+\textit {\_R} \,b^{3}+8 \textit {\_R}^{3}+5 \textit {\_R} \,b^{2}-b^{3}-12 \textit {\_R}^{2}-b^{2}+6 \textit {\_R} -1}}{2 a}\) | \(239\) |
-1/2/a*sum((2*_R^5-5*_R^4+4*(1+b)*_R^3+(-6*b-1)*_R^2+2*b*(1+b)*_R-b^2)/(_R ^7+3*_R^5*b-5*_R^4*b+3*_R^3*b^2+_R^3*b-6*_R^2*b^2+_R*b^3+8*_R^3+5*_R*b^2-b ^3-12*_R^2-b^2+6*_R-1)*ln((a*x+(a*x-b)^(1/2))^(1/2)-(a*x-b)^(1/2)-_R),_R=R ootOf(_Z^8+4*b*_Z^6-8*b*_Z^5+(6*b^2+2*b+16)*_Z^4+(-16*b^2-32)*_Z^3+(4*b^3+ 20*b^2+24)*_Z^2+(-8*b^3-8*b^2-8)*_Z+b^4+2*b^3+b^2+1))
Timed out. \[ \int \frac {\sqrt {a x+\sqrt {-b+a x}}}{\sqrt {-b+a x} \left (1+a^2 x^2\right )} \, dx=\text {Timed out} \]
Not integrable
Time = 2.27 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.05 \[ \int \frac {\sqrt {a x+\sqrt {-b+a x}}}{\sqrt {-b+a x} \left (1+a^2 x^2\right )} \, dx=\int \frac {\sqrt {a x + \sqrt {a x - b}}}{\sqrt {a x - b} \left (a^{2} x^{2} + 1\right )}\, dx \]
Not integrable
Time = 0.26 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.06 \[ \int \frac {\sqrt {a x+\sqrt {-b+a x}}}{\sqrt {-b+a x} \left (1+a^2 x^2\right )} \, dx=\int { \frac {\sqrt {a x + \sqrt {a x - b}}}{{\left (a^{2} x^{2} + 1\right )} \sqrt {a x - b}} \,d x } \]
Timed out. \[ \int \frac {\sqrt {a x+\sqrt {-b+a x}}}{\sqrt {-b+a x} \left (1+a^2 x^2\right )} \, dx=\text {Timed out} \]
Not integrable
Time = 8.89 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.06 \[ \int \frac {\sqrt {a x+\sqrt {-b+a x}}}{\sqrt {-b+a x} \left (1+a^2 x^2\right )} \, dx=\int \frac {\sqrt {a\,x+\sqrt {a\,x-b}}}{\left (a^2\,x^2+1\right )\,\sqrt {a\,x-b}} \,d x \]