Integrand size = 31, antiderivative size = 306 \[ \int \frac {\sqrt {c+\sqrt {b+a x}}}{x-\sqrt {b+a x}} \, dx=4 \sqrt {c+\sqrt {b+a x}}+\frac {2 \left (\sqrt {2} a^2+2 \sqrt {2} b+\sqrt {2} a \sqrt {a^2+4 b}+\sqrt {2} a c+\sqrt {2} \sqrt {a^2+4 b} c\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c+\sqrt {b+a x}}}{\sqrt {-a-\sqrt {a^2+4 b}-2 c}}\right )}{\sqrt {a^2+4 b} \sqrt {-a-\sqrt {a^2+4 b}-2 c}}+\frac {2 \left (-\sqrt {2} a^2-2 \sqrt {2} b+\sqrt {2} a \sqrt {a^2+4 b}-\sqrt {2} a c+\sqrt {2} \sqrt {a^2+4 b} c\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c+\sqrt {b+a x}}}{\sqrt {-a+\sqrt {a^2+4 b}-2 c}}\right )}{\sqrt {a^2+4 b} \sqrt {-a+\sqrt {a^2+4 b}-2 c}} \]
4*(c+(a*x+b)^(1/2))^(1/2)+2*(2^(1/2)*a^2+2*2^(1/2)*b+2^(1/2)*a*(a^2+4*b)^( 1/2)+2^(1/2)*a*c+2^(1/2)*(a^2+4*b)^(1/2)*c)*arctan(2^(1/2)*(c+(a*x+b)^(1/2 ))^(1/2)/(-a-(a^2+4*b)^(1/2)-2*c)^(1/2))/(a^2+4*b)^(1/2)/(-a-(a^2+4*b)^(1/ 2)-2*c)^(1/2)+2*(-2^(1/2)*a^2-2*2^(1/2)*b+2^(1/2)*a*(a^2+4*b)^(1/2)-2^(1/2 )*a*c+2^(1/2)*(a^2+4*b)^(1/2)*c)*arctan(2^(1/2)*(c+(a*x+b)^(1/2))^(1/2)/(- a+(a^2+4*b)^(1/2)-2*c)^(1/2))/(a^2+4*b)^(1/2)/(-a+(a^2+4*b)^(1/2)-2*c)^(1/ 2)
Time = 0.52 (sec) , antiderivative size = 265, normalized size of antiderivative = 0.87 \[ \int \frac {\sqrt {c+\sqrt {b+a x}}}{x-\sqrt {b+a x}} \, dx=4 \sqrt {c+\sqrt {b+a x}}+\frac {2 \sqrt {2} \left (a^2+2 b+a \sqrt {a^2+4 b}+a c+\sqrt {a^2+4 b} c\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c+\sqrt {b+a x}}}{\sqrt {-a-\sqrt {a^2+4 b}-2 c}}\right )}{\sqrt {a^2+4 b} \sqrt {-a-\sqrt {a^2+4 b}-2 c}}+\frac {2 \sqrt {2} \left (-a^2-2 b+a \sqrt {a^2+4 b}-a c+\sqrt {a^2+4 b} c\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c+\sqrt {b+a x}}}{\sqrt {-a+\sqrt {a^2+4 b}-2 c}}\right )}{\sqrt {a^2+4 b} \sqrt {-a+\sqrt {a^2+4 b}-2 c}} \]
4*Sqrt[c + Sqrt[b + a*x]] + (2*Sqrt[2]*(a^2 + 2*b + a*Sqrt[a^2 + 4*b] + a* c + Sqrt[a^2 + 4*b]*c)*ArcTan[(Sqrt[2]*Sqrt[c + Sqrt[b + a*x]])/Sqrt[-a - Sqrt[a^2 + 4*b] - 2*c]])/(Sqrt[a^2 + 4*b]*Sqrt[-a - Sqrt[a^2 + 4*b] - 2*c] ) + (2*Sqrt[2]*(-a^2 - 2*b + a*Sqrt[a^2 + 4*b] - a*c + Sqrt[a^2 + 4*b]*c)* ArcTan[(Sqrt[2]*Sqrt[c + Sqrt[b + a*x]])/Sqrt[-a + Sqrt[a^2 + 4*b] - 2*c]] )/(Sqrt[a^2 + 4*b]*Sqrt[-a + Sqrt[a^2 + 4*b] - 2*c])
Time = 0.66 (sec) , antiderivative size = 214, normalized size of antiderivative = 0.70, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.194, Rules used = {7267, 1196, 25, 1197, 1480, 219}
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.
| \(\displaystyle \int \frac {\sqrt {\sqrt {a x+b}+c}}{x-\sqrt {a x+b}} \, dx\) |
| \(\Big \downarrow \) 7267 |
| \(\displaystyle -2 \int \frac {\sqrt {b+a x} \sqrt {c+\sqrt {b+a x}}}{a \sqrt {b+a x}-a x}d\sqrt {b+a x}\) |
| \(\Big \downarrow \) 1196 |
| \(\displaystyle -2 \left (-\int -\frac {b+(a+c) \sqrt {b+a x}}{\sqrt {c+\sqrt {b+a x}} \left (a \sqrt {b+a x}-a x\right )}d\sqrt {b+a x}-2 \sqrt {\sqrt {a x+b}+c}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -2 \left (\int \frac {b+(a+c) \sqrt {b+a x}}{\sqrt {c+\sqrt {b+a x}} \left (a \sqrt {b+a x}-a x\right )}d\sqrt {b+a x}-2 \sqrt {\sqrt {a x+b}+c}\right )\) |
| \(\Big \downarrow \) 1197 |
| \(\displaystyle -2 \left (2 \int \frac {b-c (a+c)+(a+c) (b+a x)}{-(b+a x)^2+(a+2 c) (b+a x)+b-c (a+c)}d\sqrt {c+\sqrt {b+a x}}-2 \sqrt {\sqrt {a x+b}+c}\right )\) |
| \(\Big \downarrow \) 1480 |
| \(\displaystyle -2 \left (2 \left (\frac {1}{2} \left (-\frac {a^2+a c+2 b}{\sqrt {a^2+4 b}}+a+c\right ) \int \frac {1}{-b+\frac {1}{2} \left (a+2 c-\sqrt {a^2+4 b}\right )-a x}d\sqrt {c+\sqrt {b+a x}}+\frac {1}{2} \left (\frac {a^2+a c+2 b}{\sqrt {a^2+4 b}}+a+c\right ) \int \frac {1}{-b+\frac {1}{2} \left (a+2 c+\sqrt {a^2+4 b}\right )-a x}d\sqrt {c+\sqrt {b+a x}}\right )-2 \sqrt {\sqrt {a x+b}+c}\right )\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle -2 \left (2 \left (\frac {\left (-\frac {a^2+a c+2 b}{\sqrt {a^2+4 b}}+a+c\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {\sqrt {a x+b}+c}}{\sqrt {-\sqrt {a^2+4 b}+a+2 c}}\right )}{\sqrt {2} \sqrt {-\sqrt {a^2+4 b}+a+2 c}}+\frac {\left (\frac {a^2+a c+2 b}{\sqrt {a^2+4 b}}+a+c\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {\sqrt {a x+b}+c}}{\sqrt {\sqrt {a^2+4 b}+a+2 c}}\right )}{\sqrt {2} \sqrt {\sqrt {a^2+4 b}+a+2 c}}\right )-2 \sqrt {\sqrt {a x+b}+c}\right )\) |
-2*(-2*Sqrt[c + Sqrt[b + a*x]] + 2*(((a + c - (a^2 + 2*b + a*c)/Sqrt[a^2 + 4*b])*ArcTanh[(Sqrt[2]*Sqrt[c + Sqrt[b + a*x]])/Sqrt[a - Sqrt[a^2 + 4*b] + 2*c]])/(Sqrt[2]*Sqrt[a - Sqrt[a^2 + 4*b] + 2*c]) + ((a + c + (a^2 + 2*b + a*c)/Sqrt[a^2 + 4*b])*ArcTanh[(Sqrt[2]*Sqrt[c + Sqrt[b + a*x]])/Sqrt[a + Sqrt[a^2 + 4*b] + 2*c]])/(Sqrt[2]*Sqrt[a + Sqrt[a^2 + 4*b] + 2*c])))
3.29.74.3.1 Defintions of rubi rules used
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[(((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_)))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[g*((d + e*x)^m/(c*m)), x] + Simp[1/c Int [(d + e*x)^(m - 1)*(Simp[c*d*f - a*e*g + (g*c*d - b*e*g + c*e*f)*x, x]/(a + b*x + c*x^2)), x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && FractionQ[m] & & GtQ[m, 0]
Int[((f_.) + (g_.)*(x_))/(Sqrt[(d_.) + (e_.)*(x_)]*((a_.) + (b_.)*(x_) + (c _.)*(x_)^2)), x_Symbol] :> Simp[2 Subst[Int[(e*f - d*g + g*x^2)/(c*d^2 - b*d*e + a*e^2 - (2*c*d - b*e)*x^2 + c*x^4), x], x, Sqrt[d + e*x]], x] /; Fr eeQ[{a, b, c, d, e, f, g}, x]
Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] : > With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[(e/2 + (2*c*d - b*e)/(2*q)) Int[1/( b/2 - q/2 + c*x^2), x], x] + Simp[(e/2 - (2*c*d - b*e)/(2*q)) Int[1/(b/2 + q/2 + c*x^2), x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - a*e^2, 0] && PosQ[b^2 - 4*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 = 0.13 (sec) , antiderivative size = 216, normalized size of antiderivative = 0.71
| method | result | size |
| derivativedivides | \(4 \sqrt {c +\sqrt {a x +b}}-\frac {4 \left (-a \sqrt {a^{2}+4 b}-c \sqrt {a^{2}+4 b}+a^{2}+a c +2 b \right ) \arctan \left (\frac {2 \sqrt {c +\sqrt {a x +b}}}{\sqrt {2 \sqrt {a^{2}+4 b}-2 a -4 c}}\right )}{\sqrt {a^{2}+4 b}\, \sqrt {2 \sqrt {a^{2}+4 b}-2 a -4 c}}-\frac {4 \left (-a \sqrt {a^{2}+4 b}-c \sqrt {a^{2}+4 b}-a^{2}-a c -2 b \right ) \arctan \left (\frac {2 \sqrt {c +\sqrt {a x +b}}}{\sqrt {-2 \sqrt {a^{2}+4 b}-2 a -4 c}}\right )}{\sqrt {a^{2}+4 b}\, \sqrt {-2 \sqrt {a^{2}+4 b}-2 a -4 c}}\) | \(216\) |
| default | \(4 \sqrt {c +\sqrt {a x +b}}-\frac {4 \left (-a \sqrt {a^{2}+4 b}-c \sqrt {a^{2}+4 b}+a^{2}+a c +2 b \right ) \arctan \left (\frac {2 \sqrt {c +\sqrt {a x +b}}}{\sqrt {2 \sqrt {a^{2}+4 b}-2 a -4 c}}\right )}{\sqrt {a^{2}+4 b}\, \sqrt {2 \sqrt {a^{2}+4 b}-2 a -4 c}}-\frac {4 \left (-a \sqrt {a^{2}+4 b}-c \sqrt {a^{2}+4 b}-a^{2}-a c -2 b \right ) \arctan \left (\frac {2 \sqrt {c +\sqrt {a x +b}}}{\sqrt {-2 \sqrt {a^{2}+4 b}-2 a -4 c}}\right )}{\sqrt {a^{2}+4 b}\, \sqrt {-2 \sqrt {a^{2}+4 b}-2 a -4 c}}\) | \(216\) |
4*(c+(a*x+b)^(1/2))^(1/2)-4*(-a*(a^2+4*b)^(1/2)-c*(a^2+4*b)^(1/2)+a^2+a*c+ 2*b)/(a^2+4*b)^(1/2)/(2*(a^2+4*b)^(1/2)-2*a-4*c)^(1/2)*arctan(2*(c+(a*x+b) ^(1/2))^(1/2)/(2*(a^2+4*b)^(1/2)-2*a-4*c)^(1/2))-4*(-a*(a^2+4*b)^(1/2)-c*( a^2+4*b)^(1/2)-a^2-a*c-2*b)/(a^2+4*b)^(1/2)/(-2*(a^2+4*b)^(1/2)-2*a-4*c)^( 1/2)*arctan(2*(c+(a*x+b)^(1/2))^(1/2)/(-2*(a^2+4*b)^(1/2)-2*a-4*c)^(1/2))
Leaf count of result is larger than twice the leaf count of optimal. 1106 vs. \(2 (242) = 484\).
Time = 0.31 (sec) , antiderivative size = 1106, normalized size of antiderivative = 3.61 \[ \int \frac {\sqrt {c+\sqrt {b+a x}}}{x-\sqrt {b+a x}} \, dx=\text {Too large to display} \]
-sqrt(2)*sqrt((a^3 + 3*a*b + (a^2 + 2*b)*c + (a^2 + 4*b)*sqrt((a^4 + a^2*c ^2 + 2*a^2*b + b^2 + 2*(a^3 + a*b)*c)/(a^2 + 4*b)))/(a^2 + 4*b))*log(8*sqr t(2)*(a^4 + 5*a^2*b + 4*b^2 + (a^3 + 4*a*b)*c - (a^3 + 4*a*b)*sqrt((a^4 + a^2*c^2 + 2*a^2*b + b^2 + 2*(a^3 + a*b)*c)/(a^2 + 4*b)))*sqrt((a^3 + 3*a*b + (a^2 + 2*b)*c + (a^2 + 4*b)*sqrt((a^4 + a^2*c^2 + 2*a^2*b + b^2 + 2*(a^ 3 + a*b)*c)/(a^2 + 4*b)))/(a^2 + 4*b)) + 32*(a^2*b + a*b*c + b^2)*sqrt(c + sqrt(a*x + b))) + sqrt(2)*sqrt((a^3 + 3*a*b + (a^2 + 2*b)*c + (a^2 + 4*b) *sqrt((a^4 + a^2*c^2 + 2*a^2*b + b^2 + 2*(a^3 + a*b)*c)/(a^2 + 4*b)))/(a^2 + 4*b))*log(-8*sqrt(2)*(a^4 + 5*a^2*b + 4*b^2 + (a^3 + 4*a*b)*c - (a^3 + 4*a*b)*sqrt((a^4 + a^2*c^2 + 2*a^2*b + b^2 + 2*(a^3 + a*b)*c)/(a^2 + 4*b)) )*sqrt((a^3 + 3*a*b + (a^2 + 2*b)*c + (a^2 + 4*b)*sqrt((a^4 + a^2*c^2 + 2* a^2*b + b^2 + 2*(a^3 + a*b)*c)/(a^2 + 4*b)))/(a^2 + 4*b)) + 32*(a^2*b + a* b*c + b^2)*sqrt(c + sqrt(a*x + b))) - sqrt(2)*sqrt((a^3 + 3*a*b + (a^2 + 2 *b)*c - (a^2 + 4*b)*sqrt((a^4 + a^2*c^2 + 2*a^2*b + b^2 + 2*(a^3 + a*b)*c) /(a^2 + 4*b)))/(a^2 + 4*b))*log(8*sqrt(2)*(a^4 + 5*a^2*b + 4*b^2 + (a^3 + 4*a*b)*c + (a^3 + 4*a*b)*sqrt((a^4 + a^2*c^2 + 2*a^2*b + b^2 + 2*(a^3 + a* b)*c)/(a^2 + 4*b)))*sqrt((a^3 + 3*a*b + (a^2 + 2*b)*c - (a^2 + 4*b)*sqrt(( a^4 + a^2*c^2 + 2*a^2*b + b^2 + 2*(a^3 + a*b)*c)/(a^2 + 4*b)))/(a^2 + 4*b) ) + 32*(a^2*b + a*b*c + b^2)*sqrt(c + sqrt(a*x + b))) + sqrt(2)*sqrt((a^3 + 3*a*b + (a^2 + 2*b)*c - (a^2 + 4*b)*sqrt((a^4 + a^2*c^2 + 2*a^2*b + b...
\[ \int \frac {\sqrt {c+\sqrt {b+a x}}}{x-\sqrt {b+a x}} \, dx=\int \frac {\sqrt {c + \sqrt {a x + b}}}{x - \sqrt {a x + b}}\, dx \]
\[ \int \frac {\sqrt {c+\sqrt {b+a x}}}{x-\sqrt {b+a x}} \, dx=\int { \frac {\sqrt {c + \sqrt {a x + b}}}{x - \sqrt {a x + b}} \,d x } \]
Time = 0.29 (sec) , antiderivative size = 210, normalized size of antiderivative = 0.69 \[ \int \frac {\sqrt {c+\sqrt {b+a x}}}{x-\sqrt {b+a x}} \, dx=-\frac {4 \, \sqrt {a^{2} + 4 \, b} \sqrt {-2 \, a - 4 \, c + 2 \, \sqrt {a^{2} + 4 \, b}} b \arctan \left (\frac {\sqrt {c + \sqrt {a x + b}}}{\sqrt {-\frac {1}{2} \, a - c + \frac {1}{2} \, \sqrt {{\left (a + 2 \, c\right )}^{2} - 4 \, a c - 4 \, c^{2} + 4 \, b}}}\right )}{a^{3} + 4 \, a b + {\left (a^{2} + 4 \, b\right )}^{\frac {3}{2}}} + \frac {4 \, \sqrt {a^{2} + 4 \, b} \sqrt {-2 \, a - 4 \, c - 2 \, \sqrt {a^{2} + 4 \, b}} b \arctan \left (\frac {\sqrt {c + \sqrt {a x + b}}}{\sqrt {-\frac {1}{2} \, a - c - \frac {1}{2} \, \sqrt {{\left (a + 2 \, c\right )}^{2} - 4 \, a c - 4 \, c^{2} + 4 \, b}}}\right )}{a^{3} + 4 \, a b - {\left (a^{2} + 4 \, b\right )}^{\frac {3}{2}}} + 4 \, \sqrt {c + \sqrt {a x + b}} \]
-4*sqrt(a^2 + 4*b)*sqrt(-2*a - 4*c + 2*sqrt(a^2 + 4*b))*b*arctan(sqrt(c + sqrt(a*x + b))/sqrt(-1/2*a - c + 1/2*sqrt((a + 2*c)^2 - 4*a*c - 4*c^2 + 4* b)))/(a^3 + 4*a*b + (a^2 + 4*b)^(3/2)) + 4*sqrt(a^2 + 4*b)*sqrt(-2*a - 4*c - 2*sqrt(a^2 + 4*b))*b*arctan(sqrt(c + sqrt(a*x + b))/sqrt(-1/2*a - c - 1 /2*sqrt((a + 2*c)^2 - 4*a*c - 4*c^2 + 4*b)))/(a^3 + 4*a*b - (a^2 + 4*b)^(3 /2)) + 4*sqrt(c + sqrt(a*x + b))
Timed out. \[ \int \frac {\sqrt {c+\sqrt {b+a x}}}{x-\sqrt {b+a x}} \, dx=\int \frac {\sqrt {c+\sqrt {b+a\,x}}}{x-\sqrt {b+a\,x}} \,d x \]