Integrand size = 34, antiderivative size = 286 \[ \int \frac {\sqrt {a x+\sqrt {-b+a x}}}{x \sqrt {-b+a x}} \, dx=-2 \log \left (1+2 \sqrt {-b+a x}-2 \sqrt {a x+\sqrt {-b+a x}}\right )-2 \text {RootSum}\left [1-8 b+16 b^2+4 \text {$\#$1}-16 b \text {$\#$1}+6 \text {$\#$1}^2+8 b \text {$\#$1}^2+4 \text {$\#$1}^3+\text {$\#$1}^4\&,\frac {-\log \left (1+2 \sqrt {-b+a x}-2 \sqrt {a x+\sqrt {-b+a x}}+\text {$\#$1}\right )+4 b \log \left (1+2 \sqrt {-b+a x}-2 \sqrt {a x+\sqrt {-b+a x}}+\text {$\#$1}\right )-2 \log \left (1+2 \sqrt {-b+a x}-2 \sqrt {a x+\sqrt {-b+a x}}+\text {$\#$1}\right ) \text {$\#$1}-\log \left (1+2 \sqrt {-b+a x}-2 \sqrt {a x+\sqrt {-b+a x}}+\text {$\#$1}\right ) \text {$\#$1}^2}{1-4 b+3 \text {$\#$1}+4 b \text {$\#$1}+3 \text {$\#$1}^2+\text {$\#$1}^3}\&\right ] \]
Time = 0.15 (sec) , antiderivative size = 166, normalized size of antiderivative = 0.58 \[ \int \frac {\sqrt {a x+\sqrt {-b+a x}}}{x \sqrt {-b+a x}} \, dx=-2 \log \left (-1-2 \sqrt {-b+a x}+2 \sqrt {a x+\sqrt {-b+a x}}\right )-\text {RootSum}\left [b+b^2-4 b \text {$\#$1}+2 b \text {$\#$1}^2+\text {$\#$1}^4\&,\frac {b \log \left (-\sqrt {-b+a x}+\sqrt {a x+\sqrt {-b+a x}}-\text {$\#$1}\right )-\log \left (-\sqrt {-b+a x}+\sqrt {a x+\sqrt {-b+a x}}-\text {$\#$1}\right ) \text {$\#$1}^2}{-b+b \text {$\#$1}+\text {$\#$1}^3}\&\right ] \]
-2*Log[-1 - 2*Sqrt[-b + a*x] + 2*Sqrt[a*x + Sqrt[-b + a*x]]] - RootSum[b + b^2 - 4*b*#1 + 2*b*#1^2 + #1^4 & , (b*Log[-Sqrt[-b + a*x] + Sqrt[a*x + Sq rt[-b + a*x]] - #1] - Log[-Sqrt[-b + a*x] + Sqrt[a*x + Sqrt[-b + a*x]] - # 1]*#1^2)/(-b + b*#1 + #1^3) & ]
Time = 0.80 (sec) , antiderivative size = 164, normalized size of antiderivative = 0.57, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.324, Rules used = {7267, 1321, 25, 1092, 219, 1369, 25, 27, 1363, 218, 221}
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.
| \(\displaystyle \int \frac {\sqrt {\sqrt {a x-b}+a x}}{x \sqrt {a x-b}} \, dx\) |
| \(\Big \downarrow \) 7267 |
| \(\displaystyle 2 \int \frac {\sqrt {a x+\sqrt {a x-b}}}{a x}d\sqrt {a x-b}\) |
| \(\Big \downarrow \) 1321 |
| \(\displaystyle 2 \left (\int \frac {1}{\sqrt {a x+\sqrt {a x-b}}}d\sqrt {a x-b}-\int -\frac {\sqrt {a x-b}}{a x \sqrt {a x+\sqrt {a x-b}}}d\sqrt {a x-b}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle 2 \left (\int \frac {1}{\sqrt {a x+\sqrt {a x-b}}}d\sqrt {a x-b}+\int \frac {\sqrt {a x-b}}{a x \sqrt {a x+\sqrt {a x-b}}}d\sqrt {a x-b}\right )\) |
| \(\Big \downarrow \) 1092 |
| \(\displaystyle 2 \left (2 \int \frac {1}{b-a x+4}d\frac {2 \sqrt {a x-b}+1}{\sqrt {a x+\sqrt {a x-b}}}+\int \frac {\sqrt {a x-b}}{a x \sqrt {a x+\sqrt {a x-b}}}d\sqrt {a x-b}\right )\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle 2 \left (\int \frac {\sqrt {a x-b}}{a x \sqrt {a x+\sqrt {a x-b}}}d\sqrt {a x-b}+\text {arctanh}\left (\frac {2 \sqrt {a x-b}+1}{2 \sqrt {\sqrt {a x-b}+a x}}\right )\right )\) |
| \(\Big \downarrow \) 1369 |
| \(\displaystyle 2 \left (\frac {\int -\frac {\sqrt {b} \left (\sqrt {b}-\sqrt {a x-b}\right )}{a x \sqrt {a x+\sqrt {a x-b}}}d\sqrt {a x-b}}{2 \sqrt {b}}-\frac {\int -\frac {\sqrt {b} \left (\sqrt {b}+\sqrt {a x-b}\right )}{a x \sqrt {a x+\sqrt {a x-b}}}d\sqrt {a x-b}}{2 \sqrt {b}}+\text {arctanh}\left (\frac {2 \sqrt {a x-b}+1}{2 \sqrt {\sqrt {a x-b}+a x}}\right )\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle 2 \left (-\frac {\int \frac {\sqrt {b} \left (\sqrt {b}-\sqrt {a x-b}\right )}{a x \sqrt {a x+\sqrt {a x-b}}}d\sqrt {a x-b}}{2 \sqrt {b}}+\frac {\int \frac {\sqrt {b} \left (\sqrt {b}+\sqrt {a x-b}\right )}{a x \sqrt {a x+\sqrt {a x-b}}}d\sqrt {a x-b}}{2 \sqrt {b}}+\text {arctanh}\left (\frac {2 \sqrt {a x-b}+1}{2 \sqrt {\sqrt {a x-b}+a x}}\right )\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 2 \left (-\frac {1}{2} \int \frac {\sqrt {b}-\sqrt {a x-b}}{a x \sqrt {a x+\sqrt {a x-b}}}d\sqrt {a x-b}+\frac {1}{2} \int \frac {\sqrt {b}+\sqrt {a x-b}}{a x \sqrt {a x+\sqrt {a x-b}}}d\sqrt {a x-b}+\text {arctanh}\left (\frac {2 \sqrt {a x-b}+1}{2 \sqrt {\sqrt {a x-b}+a x}}\right )\right )\) |
| \(\Big \downarrow \) 1363 |
| \(\displaystyle 2 \left (b^{3/2} \left (-\int \frac {1}{b (a x-b)-2 b^{5/2}}d\left (-\frac {\sqrt {b} \left (\sqrt {b}+\sqrt {a x-b}\right )}{\sqrt {a x+\sqrt {a x-b}}}\right )\right )-b^{3/2} \int \frac {1}{2 b^{5/2}+(a x-b) b}d\frac {\sqrt {b} \left (\sqrt {b}-\sqrt {a x-b}\right )}{\sqrt {a x+\sqrt {a x-b}}}+\text {arctanh}\left (\frac {2 \sqrt {a x-b}+1}{2 \sqrt {\sqrt {a x-b}+a x}}\right )\right )\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle 2 \left (b^{3/2} \left (-\int \frac {1}{b (a x-b)-2 b^{5/2}}d\left (-\frac {\sqrt {b} \left (\sqrt {b}+\sqrt {a x-b}\right )}{\sqrt {a x+\sqrt {a x-b}}}\right )\right )-\frac {\arctan \left (\frac {\sqrt {b}-\sqrt {a x-b}}{\sqrt {2} \sqrt [4]{b} \sqrt {\sqrt {a x-b}+a x}}\right )}{\sqrt {2} \sqrt [4]{b}}+\text {arctanh}\left (\frac {2 \sqrt {a x-b}+1}{2 \sqrt {\sqrt {a x-b}+a x}}\right )\right )\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle 2 \left (-\frac {\arctan \left (\frac {\sqrt {b}-\sqrt {a x-b}}{\sqrt {2} \sqrt [4]{b} \sqrt {\sqrt {a x-b}+a x}}\right )}{\sqrt {2} \sqrt [4]{b}}-\frac {\text {arctanh}\left (\frac {\sqrt {a x-b}+\sqrt {b}}{\sqrt {2} \sqrt [4]{b} \sqrt {\sqrt {a x-b}+a x}}\right )}{\sqrt {2} \sqrt [4]{b}}+\text {arctanh}\left (\frac {2 \sqrt {a x-b}+1}{2 \sqrt {\sqrt {a x-b}+a x}}\right )\right )\) |
2*(-(ArcTan[(Sqrt[b] - Sqrt[-b + a*x])/(Sqrt[2]*b^(1/4)*Sqrt[a*x + Sqrt[-b + a*x]])]/(Sqrt[2]*b^(1/4))) - ArcTanh[(Sqrt[b] + Sqrt[-b + a*x])/(Sqrt[2 ]*b^(1/4)*Sqrt[a*x + Sqrt[-b + a*x]])]/(Sqrt[2]*b^(1/4)) + ArcTanh[(1 + 2* Sqrt[-b + a*x])/(2*Sqrt[a*x + Sqrt[-b + a*x]])])
3.29.27.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/R t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[a/b]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x /Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
Int[1/Sqrt[(a_) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Simp[2 Subst[I nt[1/(4*c - x^2), x], x, (b + 2*c*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a , b, c}, x]
Int[Sqrt[(a_) + (b_.)*(x_) + (c_.)*(x_)^2]/((d_) + (f_.)*(x_)^2), x_Symbol] :> Simp[c/f Int[1/Sqrt[a + b*x + c*x^2], x], x] - Simp[1/f Int[(c*d - a*f - b*f*x)/(Sqrt[a + b*x + c*x^2]*(d + f*x^2)), x], x] /; FreeQ[{a, b, c, d, f}, x] && NeQ[b^2 - 4*a*c, 0]
Int[((g_) + (h_.)*(x_))/(((a_) + (c_.)*(x_)^2)*Sqrt[(d_.) + (e_.)*(x_) + (f _.)*(x_)^2]), x_Symbol] :> Simp[-2*a*g*h Subst[Int[1/Simp[2*a^2*g*h*c + a *e*x^2, x], x], x, Simp[a*h - g*c*x, x]/Sqrt[d + e*x + f*x^2]], x] /; FreeQ [{a, c, d, e, f, g, h}, x] && EqQ[a*h^2*e + 2*g*h*(c*d - a*f) - g^2*c*e, 0]
Int[((g_.) + (h_.)*(x_))/(((a_) + (c_.)*(x_)^2)*Sqrt[(d_.) + (e_.)*(x_) + ( f_.)*(x_)^2]), x_Symbol] :> With[{q = Rt[(c*d - a*f)^2 + a*c*e^2, 2]}, Simp [1/(2*q) Int[Simp[(-a)*h*e - g*(c*d - a*f - q) + (h*(c*d - a*f + q) - g*c *e)*x, x]/((a + c*x^2)*Sqrt[d + e*x + f*x^2]), x], x] - Simp[1/(2*q) Int[ Simp[(-a)*h*e - g*(c*d - a*f + q) + (h*(c*d - a*f - q) - g*c*e)*x, x]/((a + c*x^2)*Sqrt[d + e*x + f*x^2]), x], x]] /; FreeQ[{a, c, d, e, f, g, h}, x] && NeQ[e^2 - 4*d*f, 0] && NegQ[(-a)*c]
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 = 0.36 (sec) , antiderivative size = 529, normalized size of antiderivative = 1.85
| method | result | size |
| derivativedivides | \(-\frac {\sqrt {\left (\sqrt {a x -b}+\sqrt {-b}\right )^{2}+\left (1-2 \sqrt {-b}\right ) \left (\sqrt {a x -b}+\sqrt {-b}\right )-\sqrt {-b}}+\frac {\left (1-2 \sqrt {-b}\right ) \ln \left (\frac {1}{2}+\sqrt {a x -b}+\sqrt {\left (\sqrt {a x -b}+\sqrt {-b}\right )^{2}+\left (1-2 \sqrt {-b}\right ) \left (\sqrt {a x -b}+\sqrt {-b}\right )-\sqrt {-b}}\right )}{2}+\frac {\sqrt {-b}\, \ln \left (\frac {-2 \sqrt {-b}+\left (1-2 \sqrt {-b}\right ) \left (\sqrt {a x -b}+\sqrt {-b}\right )+2 \sqrt {-\sqrt {-b}}\, \sqrt {\left (\sqrt {a x -b}+\sqrt {-b}\right )^{2}+\left (1-2 \sqrt {-b}\right ) \left (\sqrt {a x -b}+\sqrt {-b}\right )-\sqrt {-b}}}{\sqrt {a x -b}+\sqrt {-b}}\right )}{\sqrt {-\sqrt {-b}}}}{\sqrt {-b}}+\frac {\sqrt {\left (\sqrt {a x -b}-\sqrt {-b}\right )^{2}+\left (1+2 \sqrt {-b}\right ) \left (\sqrt {a x -b}-\sqrt {-b}\right )+\sqrt {-b}}+\frac {\left (1+2 \sqrt {-b}\right ) \ln \left (\frac {1}{2}+\sqrt {a x -b}+\sqrt {\left (\sqrt {a x -b}-\sqrt {-b}\right )^{2}+\left (1+2 \sqrt {-b}\right ) \left (\sqrt {a x -b}-\sqrt {-b}\right )+\sqrt {-b}}\right )}{2}-\left (-b \right )^{\frac {1}{4}} \ln \left (\frac {2 \sqrt {-b}+\left (1+2 \sqrt {-b}\right ) \left (\sqrt {a x -b}-\sqrt {-b}\right )+2 \left (-b \right )^{\frac {1}{4}} \sqrt {\left (\sqrt {a x -b}-\sqrt {-b}\right )^{2}+\left (1+2 \sqrt {-b}\right ) \left (\sqrt {a x -b}-\sqrt {-b}\right )+\sqrt {-b}}}{\sqrt {a x -b}-\sqrt {-b}}\right )}{\sqrt {-b}}\) | \(529\) |
| default | \(-\frac {\sqrt {\left (\sqrt {a x -b}+\sqrt {-b}\right )^{2}+\left (1-2 \sqrt {-b}\right ) \left (\sqrt {a x -b}+\sqrt {-b}\right )-\sqrt {-b}}+\frac {\left (1-2 \sqrt {-b}\right ) \ln \left (\frac {1}{2}+\sqrt {a x -b}+\sqrt {\left (\sqrt {a x -b}+\sqrt {-b}\right )^{2}+\left (1-2 \sqrt {-b}\right ) \left (\sqrt {a x -b}+\sqrt {-b}\right )-\sqrt {-b}}\right )}{2}+\frac {\sqrt {-b}\, \ln \left (\frac {-2 \sqrt {-b}+\left (1-2 \sqrt {-b}\right ) \left (\sqrt {a x -b}+\sqrt {-b}\right )+2 \sqrt {-\sqrt {-b}}\, \sqrt {\left (\sqrt {a x -b}+\sqrt {-b}\right )^{2}+\left (1-2 \sqrt {-b}\right ) \left (\sqrt {a x -b}+\sqrt {-b}\right )-\sqrt {-b}}}{\sqrt {a x -b}+\sqrt {-b}}\right )}{\sqrt {-\sqrt {-b}}}}{\sqrt {-b}}+\frac {\sqrt {\left (\sqrt {a x -b}-\sqrt {-b}\right )^{2}+\left (1+2 \sqrt {-b}\right ) \left (\sqrt {a x -b}-\sqrt {-b}\right )+\sqrt {-b}}+\frac {\left (1+2 \sqrt {-b}\right ) \ln \left (\frac {1}{2}+\sqrt {a x -b}+\sqrt {\left (\sqrt {a x -b}-\sqrt {-b}\right )^{2}+\left (1+2 \sqrt {-b}\right ) \left (\sqrt {a x -b}-\sqrt {-b}\right )+\sqrt {-b}}\right )}{2}-\left (-b \right )^{\frac {1}{4}} \ln \left (\frac {2 \sqrt {-b}+\left (1+2 \sqrt {-b}\right ) \left (\sqrt {a x -b}-\sqrt {-b}\right )+2 \left (-b \right )^{\frac {1}{4}} \sqrt {\left (\sqrt {a x -b}-\sqrt {-b}\right )^{2}+\left (1+2 \sqrt {-b}\right ) \left (\sqrt {a x -b}-\sqrt {-b}\right )+\sqrt {-b}}}{\sqrt {a x -b}-\sqrt {-b}}\right )}{\sqrt {-b}}\) | \(529\) |
-1/(-b)^(1/2)*((((a*x-b)^(1/2)+(-b)^(1/2))^2+(1-2*(-b)^(1/2))*((a*x-b)^(1/ 2)+(-b)^(1/2))-(-b)^(1/2))^(1/2)+1/2*(1-2*(-b)^(1/2))*ln(1/2+(a*x-b)^(1/2) +(((a*x-b)^(1/2)+(-b)^(1/2))^2+(1-2*(-b)^(1/2))*((a*x-b)^(1/2)+(-b)^(1/2)) -(-b)^(1/2))^(1/2))+(-b)^(1/2)/(-(-b)^(1/2))^(1/2)*ln((-2*(-b)^(1/2)+(1-2* (-b)^(1/2))*((a*x-b)^(1/2)+(-b)^(1/2))+2*(-(-b)^(1/2))^(1/2)*(((a*x-b)^(1/ 2)+(-b)^(1/2))^2+(1-2*(-b)^(1/2))*((a*x-b)^(1/2)+(-b)^(1/2))-(-b)^(1/2))^( 1/2))/((a*x-b)^(1/2)+(-b)^(1/2))))+1/(-b)^(1/2)*((((a*x-b)^(1/2)-(-b)^(1/2 ))^2+(1+2*(-b)^(1/2))*((a*x-b)^(1/2)-(-b)^(1/2))+(-b)^(1/2))^(1/2)+1/2*(1+ 2*(-b)^(1/2))*ln(1/2+(a*x-b)^(1/2)+(((a*x-b)^(1/2)-(-b)^(1/2))^2+(1+2*(-b) ^(1/2))*((a*x-b)^(1/2)-(-b)^(1/2))+(-b)^(1/2))^(1/2))-(-b)^(1/4)*ln((2*(-b )^(1/2)+(1+2*(-b)^(1/2))*((a*x-b)^(1/2)-(-b)^(1/2))+2*(-b)^(1/4)*(((a*x-b) ^(1/2)-(-b)^(1/2))^2+(1+2*(-b)^(1/2))*((a*x-b)^(1/2)-(-b)^(1/2))+(-b)^(1/2 ))^(1/2))/((a*x-b)^(1/2)-(-b)^(1/2))))
Timed out. \[ \int \frac {\sqrt {a x+\sqrt {-b+a x}}}{x \sqrt {-b+a x}} \, dx=\text {Timed out} \]
Not integrable
Time = 0.71 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.09 \[ \int \frac {\sqrt {a x+\sqrt {-b+a x}}}{x \sqrt {-b+a x}} \, dx=\int \frac {\sqrt {a x + \sqrt {a x - b}}}{x \sqrt {a x - b}}\, dx \]
Not integrable
Time = 0.29 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.10 \[ \int \frac {\sqrt {a x+\sqrt {-b+a x}}}{x \sqrt {-b+a x}} \, dx=\int { \frac {\sqrt {a x + \sqrt {a x - b}}}{\sqrt {a x - b} x} \,d x } \]
Timed out. \[ \int \frac {\sqrt {a x+\sqrt {-b+a x}}}{x \sqrt {-b+a x}} \, dx=\text {Timed out} \]
Not integrable
Time = 7.03 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.10 \[ \int \frac {\sqrt {a x+\sqrt {-b+a x}}}{x \sqrt {-b+a x}} \, dx=\int \frac {\sqrt {a\,x+\sqrt {a\,x-b}}}{x\,\sqrt {a\,x-b}} \,d x \]