Integrand size = 40, antiderivative size = 407 \[ \int \frac {\sqrt {b+a x} \sqrt {c+\sqrt {b+a x}}}{x-\sqrt {b+a x}} \, dx=\frac {4}{3} (3 a+c) \sqrt {c+\sqrt {b+a x}}+\frac {4}{3} \sqrt {b+a x} \sqrt {c+\sqrt {b+a x}}+\frac {2 \left (\sqrt {2} a^3+3 \sqrt {2} a b+\sqrt {2} a^2 \sqrt {a^2+4 b}+\sqrt {2} b \sqrt {a^2+4 b}+\sqrt {2} a^2 c+2 \sqrt {2} b c+\sqrt {2} a \sqrt {a^2+4 b} c\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c+\sqrt {b+a x}}}{\sqrt {-a-\sqrt {a^2+4 b}-2 c}}\right )}{\sqrt {a^2+4 b} \sqrt {-a-\sqrt {a^2+4 b}-2 c}}+\frac {2 \left (-\sqrt {2} a^3-3 \sqrt {2} a b+\sqrt {2} a^2 \sqrt {a^2+4 b}+\sqrt {2} b \sqrt {a^2+4 b}-\sqrt {2} a^2 c-2 \sqrt {2} b c+\sqrt {2} a \sqrt {a^2+4 b} c\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c+\sqrt {b+a x}}}{\sqrt {-a+\sqrt {a^2+4 b}-2 c}}\right )}{\sqrt {a^2+4 b} \sqrt {-a+\sqrt {a^2+4 b}-2 c}} \]
4/3*(3*a+c)*(c+(a*x+b)^(1/2))^(1/2)+4/3*(a*x+b)^(1/2)*(c+(a*x+b)^(1/2))^(1 /2)+2*(2^(1/2)*a^3+3*2^(1/2)*a*b+2^(1/2)*a^2*(a^2+4*b)^(1/2)+2^(1/2)*b*(a^ 2+4*b)^(1/2)+2^(1/2)*a^2*c+2*2^(1/2)*b*c+2^(1/2)*a*(a^2+4*b)^(1/2)*c)*arct an(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^3-3*2^(1/2)*a*b+2^(1 /2)*a^2*(a^2+4*b)^(1/2)+2^(1/2)*b*(a^2+4*b)^(1/2)-2^(1/2)*a^2*c-2*2^(1/2)* b*c+2^(1/2)*a*(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 = 0.67 (sec) , antiderivative size = 324, normalized size of antiderivative = 0.80 \[ \int \frac {\sqrt {b+a x} \sqrt {c+\sqrt {b+a x}}}{x-\sqrt {b+a x}} \, dx=\frac {4}{3} \sqrt {c+\sqrt {b+a x}} \left (3 a+c+\sqrt {b+a x}\right )+\frac {2 \sqrt {2} \left (a^3+a^2 \left (\sqrt {a^2+4 b}+c\right )+b \left (\sqrt {a^2+4 b}+2 c\right )+a \left (3 b+\sqrt {a^2+4 b} c\right )\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c+\sqrt {b+a x}}}{\sqrt {-a-\sqrt {a^2+4 b}-2 c}}\right )}{\sqrt {a^2+4 b} \sqrt {-a-\sqrt {a^2+4 b}-2 c}}+\frac {2 \sqrt {2} \left (-a^3+b \left (\sqrt {a^2+4 b}-2 c\right )+a^2 \left (\sqrt {a^2+4 b}-c\right )+a \left (-3 b+\sqrt {a^2+4 b} c\right )\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c+\sqrt {b+a x}}}{\sqrt {-a+\sqrt {a^2+4 b}-2 c}}\right )}{\sqrt {a^2+4 b} \sqrt {-a+\sqrt {a^2+4 b}-2 c}} \]
(4*Sqrt[c + Sqrt[b + a*x]]*(3*a + c + Sqrt[b + a*x]))/3 + (2*Sqrt[2]*(a^3 + a^2*(Sqrt[a^2 + 4*b] + c) + b*(Sqrt[a^2 + 4*b] + 2*c) + a*(3*b + 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*Sqr t[2]*(-a^3 + b*(Sqrt[a^2 + 4*b] - 2*c) + a^2*(Sqrt[a^2 + 4*b] - c) + a*(-3 *b + 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 = 2.72 (sec) , antiderivative size = 249, normalized size of antiderivative = 0.61, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.075, Rules used = {7267, 1199, 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.
| \(\displaystyle \int \frac {\sqrt {a x+b} \sqrt {\sqrt {a x+b}+c}}{x-\sqrt {a x+b}} \, dx\) |
| \(\Big \downarrow \) 7267 |
| \(\displaystyle -2 \int \frac {(b+a x) \sqrt {c+\sqrt {b+a x}}}{a \sqrt {b+a x}-a x}d\sqrt {b+a x}\) |
| \(\Big \downarrow \) 1199 |
| \(\displaystyle -4 \int \left (-x a-a-b+\frac {a (b-c (a+c))+\left (a^2+c a+b\right ) (b+a x)}{-(b+a x)^2+(a+2 c) (b+a x)+b-c (a+c)}\right )d\sqrt {c+\sqrt {b+a x}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -4 \left (\frac {\left (a^2-\frac {a^3+a^2 c+3 a b+2 b c}{\sqrt {a^2+4 b}}+a c+b\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {\sqrt {a x+b}+c}}{\sqrt {-\sqrt {a^2+4 b}+a+2 c}}\right )}{\sqrt {2} \sqrt {-\sqrt {a^2+4 b}+a+2 c}}+\frac {\left (a^2+\frac {a^3+a^2 c+3 a b+2 b c}{\sqrt {a^2+4 b}}+a c+b\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {\sqrt {a x+b}+c}}{\sqrt {\sqrt {a^2+4 b}+a+2 c}}\right )}{\sqrt {2} \sqrt {\sqrt {a^2+4 b}+a+2 c}}-a \sqrt {\sqrt {a x+b}+c}-\frac {1}{3} (a x+b)^{3/2}\right )\) |
-4*(-1/3*(b + a*x)^(3/2) - a*Sqrt[c + Sqrt[b + a*x]] + ((a^2 + b + a*c - ( a^3 + 3*a*b + a^2*c + 2*b*c)/Sqrt[a^2 + 4*b])*ArcTanh[(Sqrt[2]*Sqrt[c + Sq rt[b + a*x]])/Sqrt[a - Sqrt[a^2 + 4*b] + 2*c]])/(Sqrt[2]*Sqrt[a - Sqrt[a^2 + 4*b] + 2*c]) + ((a^2 + b + a*c + (a^3 + 3*a*b + a^2*c + 2*b*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]))
3.31.11.3.1 Defintions of rubi rules used
Int[(((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_))/((a_.) + (b_.)*(x _) + (c_.)*(x_)^2), x_Symbol] :> With[{q = Denominator[m]}, Simp[q/e Subs t[Int[ExpandIntegrand[x^(q*(m + 1) - 1)*(((e*f - d*g)/e + g*(x^q/e))^n/((c* d^2 - b*d*e + a*e^2)/e^2 - (2*c*d - b*e)*(x^q/e^2) + c*(x^(2*q)/e^2))), x], x], x, (d + e*x)^(1/q)], x]] /; FreeQ[{a, b, c, d, e, f, g}, x] && Integer Q[n] && FractionQ[m]
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 = 0.10 (sec) , antiderivative size = 274, normalized size of antiderivative = 0.67
| method | result | size |
| derivativedivides | \(\frac {4 \left (c +\sqrt {a x +b}\right )^{\frac {3}{2}}}{3}+4 a \sqrt {c +\sqrt {a x +b}}-\frac {4 \left (-a^{2} \sqrt {a^{2}+4 b}-a c \sqrt {a^{2}+4 b}+a^{3}+a^{2} c -b \sqrt {a^{2}+4 b}+3 a b +2 b c \right ) \arctan \left (\frac {2 \sqrt {c +\sqrt {a x +b}}}{\sqrt {2 \sqrt {a^{2}+4 b}-2 a -4 c}}\right )}{\sqrt {a^{2}+4 b}\, \sqrt {2 \sqrt {a^{2}+4 b}-2 a -4 c}}-\frac {4 \left (-a^{2} \sqrt {a^{2}+4 b}-a c \sqrt {a^{2}+4 b}-a^{3}-a^{2} c -b \sqrt {a^{2}+4 b}-3 a b -2 b c \right ) \arctan \left (\frac {2 \sqrt {c +\sqrt {a x +b}}}{\sqrt {-2 \sqrt {a^{2}+4 b}-2 a -4 c}}\right )}{\sqrt {a^{2}+4 b}\, \sqrt {-2 \sqrt {a^{2}+4 b}-2 a -4 c}}\) | \(274\) |
| default | \(\frac {4 \left (c +\sqrt {a x +b}\right )^{\frac {3}{2}}}{3}+4 a \sqrt {c +\sqrt {a x +b}}-\frac {4 \left (-a^{2} \sqrt {a^{2}+4 b}-a c \sqrt {a^{2}+4 b}+a^{3}+a^{2} c -b \sqrt {a^{2}+4 b}+3 a b +2 b c \right ) \arctan \left (\frac {2 \sqrt {c +\sqrt {a x +b}}}{\sqrt {2 \sqrt {a^{2}+4 b}-2 a -4 c}}\right )}{\sqrt {a^{2}+4 b}\, \sqrt {2 \sqrt {a^{2}+4 b}-2 a -4 c}}-\frac {4 \left (-a^{2} \sqrt {a^{2}+4 b}-a c \sqrt {a^{2}+4 b}-a^{3}-a^{2} c -b \sqrt {a^{2}+4 b}-3 a b -2 b c \right ) \arctan \left (\frac {2 \sqrt {c +\sqrt {a x +b}}}{\sqrt {-2 \sqrt {a^{2}+4 b}-2 a -4 c}}\right )}{\sqrt {a^{2}+4 b}\, \sqrt {-2 \sqrt {a^{2}+4 b}-2 a -4 c}}\) | \(274\) |
4/3*(c+(a*x+b)^(1/2))^(3/2)+4*a*(c+(a*x+b)^(1/2))^(1/2)-4*(-a^2*(a^2+4*b)^ (1/2)-a*c*(a^2+4*b)^(1/2)+a^3+a^2*c-b*(a^2+4*b)^(1/2)+3*a*b+2*b*c)/(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^2*(a^2+4*b)^(1/2)-a*c*(a^2+4*b) ^(1/2)-a^3-a^2*c-b*(a^2+4*b)^(1/2)-3*a*b-2*b*c)/(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. 2014 vs. \(2 (322) = 644\).
Time = 0.32 (sec) , antiderivative size = 2014, normalized size of antiderivative = 4.95 \[ \int \frac {\sqrt {b+a x} \sqrt {c+\sqrt {b+a x}}}{x-\sqrt {b+a x}} \, dx=\text {Too large to display} \]
sqrt(2)*sqrt((a^5 + 5*a^3*b + 5*a*b^2 + (a^4 + 4*a^2*b + 2*b^2)*c + (a^2 + 4*b)*sqrt((a^8 + 6*a^6*b + 11*a^4*b^2 + 6*a^2*b^3 + b^4 + (a^6 + 4*a^4*b + 4*a^2*b^2)*c^2 + 2*(a^7 + 5*a^5*b + 7*a^3*b^2 + 2*a*b^3)*c)/(a^2 + 4*b)) )/(a^2 + 4*b))*log(8*sqrt(2)*(a^7 + 7*a^5*b + 13*a^3*b^2 + 4*a*b^3 + (a^6 + 6*a^4*b + 8*a^2*b^2)*c - (a^4 + 6*a^2*b + 8*b^2)*sqrt((a^8 + 6*a^6*b + 1 1*a^4*b^2 + 6*a^2*b^3 + b^4 + (a^6 + 4*a^4*b + 4*a^2*b^2)*c^2 + 2*(a^7 + 5 *a^5*b + 7*a^3*b^2 + 2*a*b^3)*c)/(a^2 + 4*b)))*sqrt((a^5 + 5*a^3*b + 5*a*b ^2 + (a^4 + 4*a^2*b + 2*b^2)*c + (a^2 + 4*b)*sqrt((a^8 + 6*a^6*b + 11*a^4* b^2 + 6*a^2*b^3 + b^4 + (a^6 + 4*a^4*b + 4*a^2*b^2)*c^2 + 2*(a^7 + 5*a^5*b + 7*a^3*b^2 + 2*a*b^3)*c)/(a^2 + 4*b)))/(a^2 + 4*b)) + 32*(a^4*b^2 + 3*a^ 2*b^3 + b^4 + (a^3*b^2 + 2*a*b^3)*c)*sqrt(c + sqrt(a*x + b))) - sqrt(2)*sq rt((a^5 + 5*a^3*b + 5*a*b^2 + (a^4 + 4*a^2*b + 2*b^2)*c + (a^2 + 4*b)*sqrt ((a^8 + 6*a^6*b + 11*a^4*b^2 + 6*a^2*b^3 + b^4 + (a^6 + 4*a^4*b + 4*a^2*b^ 2)*c^2 + 2*(a^7 + 5*a^5*b + 7*a^3*b^2 + 2*a*b^3)*c)/(a^2 + 4*b)))/(a^2 + 4 *b))*log(-8*sqrt(2)*(a^7 + 7*a^5*b + 13*a^3*b^2 + 4*a*b^3 + (a^6 + 6*a^4*b + 8*a^2*b^2)*c - (a^4 + 6*a^2*b + 8*b^2)*sqrt((a^8 + 6*a^6*b + 11*a^4*b^2 + 6*a^2*b^3 + b^4 + (a^6 + 4*a^4*b + 4*a^2*b^2)*c^2 + 2*(a^7 + 5*a^5*b + 7*a^3*b^2 + 2*a*b^3)*c)/(a^2 + 4*b)))*sqrt((a^5 + 5*a^3*b + 5*a*b^2 + (a^4 + 4*a^2*b + 2*b^2)*c + (a^2 + 4*b)*sqrt((a^8 + 6*a^6*b + 11*a^4*b^2 + 6*a ^2*b^3 + b^4 + (a^6 + 4*a^4*b + 4*a^2*b^2)*c^2 + 2*(a^7 + 5*a^5*b + 7*a...
\[ \int \frac {\sqrt {b+a x} \sqrt {c+\sqrt {b+a x}}}{x-\sqrt {b+a x}} \, dx=\int \frac {\sqrt {c + \sqrt {a x + b}} \sqrt {a x + b}}{x - \sqrt {a x + b}}\, dx \]
\[ \int \frac {\sqrt {b+a x} \sqrt {c+\sqrt {b+a x}}}{x-\sqrt {b+a x}} \, dx=\int { \frac {\sqrt {a x + b} \sqrt {c + \sqrt {a x + b}}}{x - \sqrt {a x + b}} \,d x } \]
Time = 0.32 (sec) , antiderivative size = 259, normalized size of antiderivative = 0.64 \[ \int \frac {\sqrt {b+a x} \sqrt {c+\sqrt {b+a x}}}{x-\sqrt {b+a x}} \, dx=\frac {4 \, \sqrt {a^{2} + 4 \, b} \sqrt {-2 \, a - 4 \, c + 2 \, \sqrt {a^{2} + 4 \, b}} b^{2} \arctan \left (\frac {\sqrt {c + \sqrt {a x + b}}}{\sqrt {-\frac {1}{2} \, a - c + \frac {1}{2} \, \sqrt {{\left (a + 2 \, c\right )}^{2} - 4 \, a c - 4 \, c^{2} + 4 \, b}}}\right )}{a^{4} + 6 \, a^{2} b + 8 \, b^{2} + {\left (a^{3} + 4 \, a b\right )} \sqrt {a^{2} + 4 \, b}} - \frac {4 \, \sqrt {a^{2} + 4 \, b} \sqrt {-2 \, a - 4 \, c - 2 \, \sqrt {a^{2} + 4 \, b}} b^{2} \arctan \left (\frac {\sqrt {c + \sqrt {a x + b}}}{\sqrt {-\frac {1}{2} \, a - c - \frac {1}{2} \, \sqrt {{\left (a + 2 \, c\right )}^{2} - 4 \, a c - 4 \, c^{2} + 4 \, b}}}\right )}{a^{4} + 6 \, a^{2} b + 8 \, b^{2} - {\left (a^{3} + 4 \, a b\right )} \sqrt {a^{2} + 4 \, b}} + 4 \, a \sqrt {c + \sqrt {a x + b}} + \frac {4}{3} \, {\left (c + \sqrt {a x + b}\right )}^{\frac {3}{2}} \]
4*sqrt(a^2 + 4*b)*sqrt(-2*a - 4*c + 2*sqrt(a^2 + 4*b))*b^2*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^4 + 6*a^2*b + 8*b^2 + (a^3 + 4*a*b)*sqrt(a^2 + 4*b)) - 4*sqrt(a^2 + 4*b)*sqrt(-2*a - 4*c - 2*sqrt(a^2 + 4*b))*b^2*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^4 + 6*a^2*b + 8*b^2 - (a^3 + 4*a*b)*sqrt(a^2 + 4*b)) + 4*a*sqrt(c + sqrt(a* x + b)) + 4/3*(c + sqrt(a*x + b))^(3/2)
Timed out. \[ \int \frac {\sqrt {b+a x} \sqrt {c+\sqrt {b+a x}}}{x-\sqrt {b+a x}} \, dx=\int \frac {\sqrt {c+\sqrt {b+a\,x}}\,\sqrt {b+a\,x}}{x-\sqrt {b+a\,x}} \,d x \]