Integrand size = 23, antiderivative size = 229 \[ \int \frac {\sqrt {a x+\sqrt {-b+a x}}}{x^2} \, dx=-\frac {\sqrt {a x+\sqrt {-b+a x}}}{x}-\frac {1}{4} a \text {RootSum}\left [b+b^2-4 b \text {$\#$1}+2 b \text {$\#$1}^2+\text {$\#$1}^4\&,\frac {-\log \left (\sqrt {-b+a x}-\sqrt {a x+\sqrt {-b+a x}}+\text {$\#$1}\right )+2 b \log \left (\sqrt {-b+a x}-\sqrt {a x+\sqrt {-b+a x}}+\text {$\#$1}\right )+2 \log \left (\sqrt {-b+a x}-\sqrt {a x+\sqrt {-b+a x}}+\text {$\#$1}\right ) \text {$\#$1}-2 \log \left (\sqrt {-b+a x}-\sqrt {a x+\sqrt {-b+a x}}+\text {$\#$1}\right ) \text {$\#$1}^2}{-b+b \text {$\#$1}+\text {$\#$1}^3}\&\right ] \]
Time = 0.32 (sec) , antiderivative size = 430, normalized size of antiderivative = 1.88 \[ \int \frac {\sqrt {a x+\sqrt {-b+a x}}}{x^2} \, dx=-\frac {\sqrt {a x+\sqrt {-b+a x}}}{x}-a \text {RootSum}\left [1-b+b^2+4 \text {$\#$1}+6 \text {$\#$1}^2+2 b \text {$\#$1}^2+4 \text {$\#$1}^3+\text {$\#$1}^4\&,\frac {b \log \left (-1-\sqrt {-b+a x}+\sqrt {a x+\sqrt {-b+a x}}-\text {$\#$1}\right )-\log \left (-1-\sqrt {-b+a x}+\sqrt {a x+\sqrt {-b+a x}}-\text {$\#$1}\right ) \text {$\#$1}^2}{1+3 \text {$\#$1}+b \text {$\#$1}+3 \text {$\#$1}^2+\text {$\#$1}^3}\&\right ]+a \text {RootSum}\left [1-8 b+16 b^2+4 \text {$\#$1}-16 b \text {$\#$1}+6 \text {$\#$1}^2+8 b \text {$\#$1}^2+4 \text {$\#$1}^3+\text {$\#$1}^4\&,\frac {-\log \left (-1-2 \sqrt {-b+a x}+2 \sqrt {a x+\sqrt {-b+a x}}-\text {$\#$1}\right )+4 b \log \left (-1-2 \sqrt {-b+a x}+2 \sqrt {a x+\sqrt {-b+a x}}-\text {$\#$1}\right )+4 \log \left (-1-2 \sqrt {-b+a x}+2 \sqrt {a x+\sqrt {-b+a x}}-\text {$\#$1}\right ) \text {$\#$1}-\log \left (-1-2 \sqrt {-b+a x}+2 \sqrt {a x+\sqrt {-b+a x}}-\text {$\#$1}\right ) \text {$\#$1}^2}{1-4 b+3 \text {$\#$1}+4 b \text {$\#$1}+3 \text {$\#$1}^2+\text {$\#$1}^3}\&\right ] \]
-(Sqrt[a*x + Sqrt[-b + a*x]]/x) - a*RootSum[1 - b + b^2 + 4*#1 + 6*#1^2 + 2*b*#1^2 + 4*#1^3 + #1^4 & , (b*Log[-1 - Sqrt[-b + a*x] + Sqrt[a*x + Sqrt[ -b + a*x]] - #1] - Log[-1 - Sqrt[-b + a*x] + Sqrt[a*x + Sqrt[-b + a*x]] - #1]*#1^2)/(1 + 3*#1 + b*#1 + 3*#1^2 + #1^3) & ] + a*RootSum[1 - 8*b + 16*b ^2 + 4*#1 - 16*b*#1 + 6*#1^2 + 8*b*#1^2 + 4*#1^3 + #1^4 & , (-Log[-1 - 2*S qrt[-b + a*x] + 2*Sqrt[a*x + Sqrt[-b + a*x]] - #1] + 4*b*Log[-1 - 2*Sqrt[- b + a*x] + 2*Sqrt[a*x + Sqrt[-b + a*x]] - #1] + 4*Log[-1 - 2*Sqrt[-b + a*x ] + 2*Sqrt[a*x + Sqrt[-b + a*x]] - #1]*#1 - Log[-1 - 2*Sqrt[-b + a*x] + 2* Sqrt[a*x + Sqrt[-b + a*x]] - #1]*#1^2)/(1 - 4*b + 3*#1 + 4*b*#1 + 3*#1^2 + #1^3) & ]
Time = 0.50 (sec) , antiderivative size = 177, normalized size of antiderivative = 0.77, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.391, Rules used = {7267, 1347, 27, 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.
| \(\displaystyle \int \frac {\sqrt {\sqrt {a x-b}+a x}}{x^2} \, dx\) |
| \(\Big \downarrow \) 7267 |
| \(\displaystyle 2 a \int \frac {\sqrt {a x-b} \sqrt {a x+\sqrt {a x-b}}}{a^2 x^2}d\sqrt {a x-b}\) |
| \(\Big \downarrow \) 1347 |
| \(\displaystyle 2 a \left (-\frac {\int -\frac {b \left (2 \sqrt {a x-b}+1\right )}{2 a x \sqrt {a x+\sqrt {a x-b}}}d\sqrt {a x-b}}{2 b}-\frac {\sqrt {\sqrt {a x-b}+a x}}{2 a x}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 2 a \left (\frac {1}{4} \int \frac {2 \sqrt {a x-b}+1}{a x \sqrt {a x+\sqrt {a x-b}}}d\sqrt {a x-b}-\frac {\sqrt {\sqrt {a x-b}+a x}}{2 a x}\right )\) |
| \(\Big \downarrow \) 1369 |
| \(\displaystyle 2 a \left (\frac {1}{4} \left (\frac {\int \frac {\left (1-2 \sqrt {b}\right ) \sqrt {b}-\left (1-2 \sqrt {b}\right ) \sqrt {a x-b}}{a x \sqrt {a x+\sqrt {a x-b}}}d\sqrt {a x-b}}{2 \sqrt {b}}-\frac {\int -\frac {\left (2 \sqrt {b}+1\right ) \left (\sqrt {b}+\sqrt {a x-b}\right )}{a x \sqrt {a x+\sqrt {a x-b}}}d\sqrt {a x-b}}{2 \sqrt {b}}\right )-\frac {\sqrt {\sqrt {a x-b}+a x}}{2 a x}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle 2 a \left (\frac {1}{4} \left (\frac {\int \frac {\left (2 \sqrt {b}+1\right ) \left (\sqrt {b}+\sqrt {a x-b}\right )}{a x \sqrt {a x+\sqrt {a x-b}}}d\sqrt {a x-b}}{2 \sqrt {b}}+\frac {\int \frac {\left (1-2 \sqrt {b}\right ) \sqrt {b}-\left (1-2 \sqrt {b}\right ) \sqrt {a x-b}}{a x \sqrt {a x+\sqrt {a x-b}}}d\sqrt {a x-b}}{2 \sqrt {b}}\right )-\frac {\sqrt {\sqrt {a x-b}+a x}}{2 a x}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 2 a \left (\frac {1}{4} \left (\frac {\left (2 \sqrt {b}+1\right ) \int \frac {\sqrt {b}+\sqrt {a x-b}}{a x \sqrt {a x+\sqrt {a x-b}}}d\sqrt {a x-b}}{2 \sqrt {b}}+\frac {\int \frac {\left (1-2 \sqrt {b}\right ) \sqrt {b}-\left (1-2 \sqrt {b}\right ) \sqrt {a x-b}}{a x \sqrt {a x+\sqrt {a x-b}}}d\sqrt {a x-b}}{2 \sqrt {b}}\right )-\frac {\sqrt {\sqrt {a x-b}+a x}}{2 a x}\right )\) |
| \(\Big \downarrow \) 1363 |
| \(\displaystyle 2 a \left (\frac {1}{4} \left (\left (1-2 \sqrt {b}\right )^2 b \int \frac {1}{b (a x-b)-2 \left (1-2 \sqrt {b}\right )^2 b^{5/2}}d\left (-\frac {\left (1-2 \sqrt {b}\right ) \sqrt {b} \left (\sqrt {b}+\sqrt {a x-b}\right )}{\sqrt {a x+\sqrt {a x-b}}}\right )-\left (2 \sqrt {b}+1\right ) b \int \frac {1}{2 b^{5/2}+(a x-b) b}d\frac {\sqrt {b} \left (\sqrt {b}-\sqrt {a x-b}\right )}{\sqrt {a x+\sqrt {a x-b}}}\right )-\frac {\sqrt {\sqrt {a x-b}+a x}}{2 a x}\right )\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle 2 a \left (\frac {1}{4} \left (\left (1-2 \sqrt {b}\right )^2 b \int \frac {1}{b (a x-b)-2 \left (1-2 \sqrt {b}\right )^2 b^{5/2}}d\left (-\frac {\left (1-2 \sqrt {b}\right ) \sqrt {b} \left (\sqrt {b}+\sqrt {a x-b}\right )}{\sqrt {a x+\sqrt {a x-b}}}\right )-\frac {\left (2 \sqrt {b}+1\right ) \arctan \left (\frac {\sqrt {b}-\sqrt {a x-b}}{\sqrt {2} \sqrt [4]{b} \sqrt {\sqrt {a x-b}+a x}}\right )}{\sqrt {2} b^{3/4}}\right )-\frac {\sqrt {\sqrt {a x-b}+a x}}{2 a x}\right )\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle 2 a \left (\frac {1}{4} \left (\frac {\left (1-2 \sqrt {b}\right ) \text {arctanh}\left (\frac {\sqrt {a x-b}+\sqrt {b}}{\sqrt {2} \sqrt [4]{b} \sqrt {\sqrt {a x-b}+a x}}\right )}{\sqrt {2} b^{3/4}}-\frac {\left (2 \sqrt {b}+1\right ) \arctan \left (\frac {\sqrt {b}-\sqrt {a x-b}}{\sqrt {2} \sqrt [4]{b} \sqrt {\sqrt {a x-b}+a x}}\right )}{\sqrt {2} b^{3/4}}\right )-\frac {\sqrt {\sqrt {a x-b}+a x}}{2 a x}\right )\) |
2*a*(-1/2*Sqrt[a*x + Sqrt[-b + a*x]]/(a*x) + (-(((1 + 2*Sqrt[b])*ArcTan[(S qrt[b] - Sqrt[-b + a*x])/(Sqrt[2]*b^(1/4)*Sqrt[a*x + Sqrt[-b + a*x]])])/(S qrt[2]*b^(3/4))) + ((1 - 2*Sqrt[b])*ArcTanh[(Sqrt[b] + Sqrt[-b + a*x])/(Sq rt[2]*b^(1/4)*Sqrt[a*x + Sqrt[-b + a*x]])])/(Sqrt[2]*b^(3/4)))/4)
3.27.20.3.1 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[(Rt[-a/b, 2]/a)*ArcTanh[x /Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
Int[((g_.) + (h_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_)*((d_) + (e_.)*(x_) + (f _.)*(x_)^2)^(q_), x_Symbol] :> Simp[(a*h - g*c*x)*(a + c*x^2)^(p + 1)*((d + e*x + f*x^2)^q/(2*a*c*(p + 1))), x] + Simp[2/(4*a*c*(p + 1)) Int[(a + c* x^2)^(p + 1)*(d + e*x + f*x^2)^(q - 1)*Simp[g*c*d*(2*p + 3) - a*(h*e*q) + ( g*c*e*(2*p + q + 3) - a*(2*h*f*q))*x + g*c*f*(2*p + 2*q + 3)*x^2, x], x], x ] /; FreeQ[{a, c, d, e, f, g, h}, x] && NeQ[e^2 - 4*d*f, 0] && LtQ[p, -1] & & GtQ[q, 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 = 0.22 (sec) , antiderivative size = 1056, normalized size of antiderivative = 4.61
| method | result | size |
| derivativedivides | \(\text {Expression too large to display}\) | \(1056\) |
| default | \(\text {Expression too large to display}\) | \(1056\) |
2*a*(1/4*(-b)^(1/2)/b*(1/(-b)^(1/2)/((a*x-b)^(1/2)+(-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)) ^(3/2)-1/2*(1-2*(-b)^(1/2))/(-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))))-2/(-b)^(1/2)*( 1/4*(2*(a*x-b)^(1/2)+1)*(((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)+1/8*(-4*(-b)^(1/2)-(1-2*(-b)^(1/ 2))^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))))-1/4*(-b)^(1/2)/b*(-1/(-b) ^(1/2)/((a*x-b)^(1/2)-(-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))^(3/2)+1/2*(1+2*(-b)^(1/2))/( -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...
Timed out. \[ \int \frac {\sqrt {a x+\sqrt {-b+a x}}}{x^2} \, dx=\text {Timed out} \]
Not integrable
Time = 0.76 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.08 \[ \int \frac {\sqrt {a x+\sqrt {-b+a x}}}{x^2} \, dx=\int \frac {\sqrt {a x + \sqrt {a x - b}}}{x^{2}}\, dx \]
Not integrable
Time = 0.26 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.09 \[ \int \frac {\sqrt {a x+\sqrt {-b+a x}}}{x^2} \, dx=\int { \frac {\sqrt {a x + \sqrt {a x - b}}}{x^{2}} \,d x } \]
Result contains higher order function than in optimal. Order 3 vs. order 1.
Time = 0.36 (sec) , antiderivative size = 240, normalized size of antiderivative = 1.05 \[ \int \frac {\sqrt {a x+\sqrt {-b+a x}}}{x^2} \, dx=\frac {2 \, a^{2} {\left (\sqrt {a x - b} - \sqrt {a x + \sqrt {a x - b}}\right )}^{3} + 2 \, a^{2} b {\left (\sqrt {a x - b} - \sqrt {a x + \sqrt {a x - b}}\right )} + 3 \, a^{2} {\left (\sqrt {a x - b} - \sqrt {a x + \sqrt {a x - b}}\right )}^{2} + a^{2} b + a^{2} {\left (\sqrt {a x - b} - \sqrt {a x + \sqrt {a x - b}}\right )}}{{\left ({\left (\sqrt {a x - b} - \sqrt {a x + \sqrt {a x - b}}\right )}^{4} + 2 \, b {\left (\sqrt {a x - b} - \sqrt {a x + \sqrt {a x - b}}\right )}^{2} + b^{2} + 4 \, b {\left (\sqrt {a x - b} - \sqrt {a x + \sqrt {a x - b}}\right )} + b\right )} a} \]
(2*a^2*(sqrt(a*x - b) - sqrt(a*x + sqrt(a*x - b)))^3 + 2*a^2*b*(sqrt(a*x - b) - sqrt(a*x + sqrt(a*x - b))) + 3*a^2*(sqrt(a*x - b) - sqrt(a*x + sqrt( a*x - b)))^2 + a^2*b + a^2*(sqrt(a*x - b) - sqrt(a*x + sqrt(a*x - b))))/(( (sqrt(a*x - b) - sqrt(a*x + sqrt(a*x - b)))^4 + 2*b*(sqrt(a*x - b) - sqrt( a*x + sqrt(a*x - b)))^2 + b^2 + 4*b*(sqrt(a*x - b) - sqrt(a*x + sqrt(a*x - b))) + b)*a)
Not integrable
Time = 6.54 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.09 \[ \int \frac {\sqrt {a x+\sqrt {-b+a x}}}{x^2} \, dx=\int \frac {\sqrt {a\,x+\sqrt {a\,x-b}}}{x^2} \,d x \]