Integrand size = 25, antiderivative size = 97 \[ \int \frac {1}{x \left (\sqrt {a+b x}+\sqrt {c+b x}\right )} \, dx=\frac {2 \sqrt {a+b x}}{a-c}-\frac {2 \sqrt {c+b x}}{a-c}-\frac {2 \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{a-c}+\frac {2 \sqrt {c} \text {arctanh}\left (\frac {\sqrt {c+b x}}{\sqrt {c}}\right )}{a-c} \]
-2*arctanh((b*x+a)^(1/2)/a^(1/2))*a^(1/2)/(a-c)+2*arctanh((b*x+c)^(1/2)/c^ (1/2))*c^(1/2)/(a-c)+2*(b*x+a)^(1/2)/(a-c)-2*(b*x+c)^(1/2)/(a-c)
Time = 0.50 (sec) , antiderivative size = 160, normalized size of antiderivative = 1.65 \[ \int \frac {1}{x \left (\sqrt {a+b x}+\sqrt {c+b x}\right )} \, dx=\frac {2 \left (\sqrt {a+b x}-\sqrt {c+b x}-\sqrt {-\left (\sqrt {a}-\sqrt {c}\right )^2} \arctan \left (\frac {\sqrt {a+b x}-\sqrt {c+b x}}{\sqrt {-\left (\sqrt {a}-\sqrt {c}\right )^2}}\right )-\sqrt {-\left (\sqrt {a}+\sqrt {c}\right )^2} \arctan \left (\frac {\sqrt {a+b x}-\sqrt {c+b x}}{\sqrt {-\left (\sqrt {a}+\sqrt {c}\right )^2}}\right )\right )}{a-c} \]
(2*(Sqrt[a + b*x] - Sqrt[c + b*x] - Sqrt[-(Sqrt[a] - Sqrt[c])^2]*ArcTan[(S qrt[a + b*x] - Sqrt[c + b*x])/Sqrt[-(Sqrt[a] - Sqrt[c])^2]] - Sqrt[-(Sqrt[ a] + Sqrt[c])^2]*ArcTan[(Sqrt[a + b*x] - Sqrt[c + b*x])/Sqrt[-(Sqrt[a] + S qrt[c])^2]]))/(a - c)
Time = 0.26 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.91, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {2529, 60, 73, 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.
| \(\displaystyle \int \frac {1}{x \left (\sqrt {a+b x}+\sqrt {b x+c}\right )} \, dx\) |
| \(\Big \downarrow \) 2529 |
| \(\displaystyle \frac {\int \frac {\sqrt {a+b x}}{x}dx}{a-c}-\frac {\int \frac {\sqrt {c+b x}}{x}dx}{a-c}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {a \int \frac {1}{x \sqrt {a+b x}}dx+2 \sqrt {a+b x}}{a-c}-\frac {c \int \frac {1}{x \sqrt {c+b x}}dx+2 \sqrt {b x+c}}{a-c}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {\frac {2 a \int \frac {1}{\frac {a+b x}{b}-\frac {a}{b}}d\sqrt {a+b x}}{b}+2 \sqrt {a+b x}}{a-c}-\frac {\frac {2 c \int \frac {1}{\frac {c+b x}{b}-\frac {c}{b}}d\sqrt {c+b x}}{b}+2 \sqrt {b x+c}}{a-c}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {2 \sqrt {a+b x}-2 \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{a-c}-\frac {2 \sqrt {b x+c}-2 \sqrt {c} \text {arctanh}\left (\frac {\sqrt {b x+c}}{\sqrt {c}}\right )}{a-c}\) |
(2*Sqrt[a + b*x] - 2*Sqrt[a]*ArcTanh[Sqrt[a + b*x]/Sqrt[a]])/(a - c) - (2* Sqrt[c + b*x] - 2*Sqrt[c]*ArcTanh[Sqrt[c + b*x]/Sqrt[c]])/(a - c)
3.5.4.3.1 Defintions of rubi rules used
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^n/(b*(m + n + 1))), x] + Simp[n*((b*c - a*d)/( b*(m + n + 1))) Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d}, x] && GtQ[n, 0] && NeQ[m + n + 1, 0] && !(IGtQ[m, 0] && ( !Integer Q[n] || (GtQ[m, 0] && LtQ[m - n, 0]))) && !ILtQ[m + n + 2, 0] && IntLinear Q[a, b, c, d, m, n, x]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ {p = Denominator[m]}, Simp[p/b Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL inearQ[a, b, c, d, m, n, x]
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[(u_)/((e_.)*Sqrt[(a_.) + (b_.)*(x_)] + (f_.)*Sqrt[(c_.) + (d_.)*(x_)]), x_Symbol] :> Simp[-d/(e*(b*c - a*d)) Int[u*Sqrt[a + b*x], x], x] + Simp[ b/(f*(b*c - a*d)) Int[u*Sqrt[c + d*x], x], x] /; FreeQ[{a, b, c, d, e, f} , x] && NeQ[b*c - a*d, 0] && EqQ[b*e^2 - d*f^2, 0]
Time = 0.02 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.75
| method | result | size |
| default | \(\frac {2 \sqrt {b x +a}-2 \sqrt {a}\, \operatorname {arctanh}\left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right )}{a -c}-\frac {2 \sqrt {b x +c}-2 \sqrt {c}\, \operatorname {arctanh}\left (\frac {\sqrt {b x +c}}{\sqrt {c}}\right )}{a -c}\) | \(73\) |
1/(a-c)*(2*(b*x+a)^(1/2)-2*a^(1/2)*arctanh((b*x+a)^(1/2)/a^(1/2)))-1/(a-c) *(2*(b*x+c)^(1/2)-2*c^(1/2)*arctanh((b*x+c)^(1/2)/c^(1/2)))
Time = 0.32 (sec) , antiderivative size = 318, normalized size of antiderivative = 3.28 \[ \int \frac {1}{x \left (\sqrt {a+b x}+\sqrt {c+b x}\right )} \, dx=\left [-\frac {\sqrt {a} \log \left (\frac {b x + 2 \, \sqrt {b x + a} \sqrt {a} + 2 \, a}{x}\right ) + \sqrt {c} \log \left (\frac {b x - 2 \, \sqrt {b x + c} \sqrt {c} + 2 \, c}{x}\right ) - 2 \, \sqrt {b x + a} + 2 \, \sqrt {b x + c}}{a - c}, -\frac {2 \, \sqrt {-c} \arctan \left (\frac {\sqrt {b x + c} \sqrt {-c}}{c}\right ) + \sqrt {a} \log \left (\frac {b x + 2 \, \sqrt {b x + a} \sqrt {a} + 2 \, a}{x}\right ) - 2 \, \sqrt {b x + a} + 2 \, \sqrt {b x + c}}{a - c}, \frac {2 \, \sqrt {-a} \arctan \left (\frac {\sqrt {b x + a} \sqrt {-a}}{a}\right ) - \sqrt {c} \log \left (\frac {b x - 2 \, \sqrt {b x + c} \sqrt {c} + 2 \, c}{x}\right ) + 2 \, \sqrt {b x + a} - 2 \, \sqrt {b x + c}}{a - c}, \frac {2 \, {\left (\sqrt {-a} \arctan \left (\frac {\sqrt {b x + a} \sqrt {-a}}{a}\right ) - \sqrt {-c} \arctan \left (\frac {\sqrt {b x + c} \sqrt {-c}}{c}\right ) + \sqrt {b x + a} - \sqrt {b x + c}\right )}}{a - c}\right ] \]
[-(sqrt(a)*log((b*x + 2*sqrt(b*x + a)*sqrt(a) + 2*a)/x) + sqrt(c)*log((b*x - 2*sqrt(b*x + c)*sqrt(c) + 2*c)/x) - 2*sqrt(b*x + a) + 2*sqrt(b*x + c))/ (a - c), -(2*sqrt(-c)*arctan(sqrt(b*x + c)*sqrt(-c)/c) + sqrt(a)*log((b*x + 2*sqrt(b*x + a)*sqrt(a) + 2*a)/x) - 2*sqrt(b*x + a) + 2*sqrt(b*x + c))/( a - c), (2*sqrt(-a)*arctan(sqrt(b*x + a)*sqrt(-a)/a) - sqrt(c)*log((b*x - 2*sqrt(b*x + c)*sqrt(c) + 2*c)/x) + 2*sqrt(b*x + a) - 2*sqrt(b*x + c))/(a - c), 2*(sqrt(-a)*arctan(sqrt(b*x + a)*sqrt(-a)/a) - sqrt(-c)*arctan(sqrt( b*x + c)*sqrt(-c)/c) + sqrt(b*x + a) - sqrt(b*x + c))/(a - c)]
\[ \int \frac {1}{x \left (\sqrt {a+b x}+\sqrt {c+b x}\right )} \, dx=\int \frac {1}{x \left (\sqrt {a + b x} + \sqrt {b x + c}\right )}\, dx \]
\[ \int \frac {1}{x \left (\sqrt {a+b x}+\sqrt {c+b x}\right )} \, dx=\int { \frac {1}{x {\left (\sqrt {b x + a} + \sqrt {b x + c}\right )}} \,d x } \]
Leaf count of result is larger than twice the leaf count of optimal. 1015 vs. \(2 (81) = 162\).
Time = 0.51 (sec) , antiderivative size = 1015, normalized size of antiderivative = 10.46 \[ \int \frac {1}{x \left (\sqrt {a+b x}+\sqrt {c+b x}\right )} \, dx=\frac {2 \, a \arctan \left (\frac {\sqrt {b x + a}}{\sqrt {-a}}\right )}{\sqrt {-a} {\left (a - c\right )}} - \frac {2 \, {\left (a^{4} c - a^{3} c^{2} - a^{2} c^{3} + a c^{4} + 2 \, {\left (a c^{2} + \sqrt {a c} c^{2}\right )} {\left (a - c\right )}^{2} \mathrm {sgn}\left (-a + c\right ) - 2 \, {\left (a c^{2} + \sqrt {a c} a c\right )} {\left (a - c\right )}^{2} + {\left (a^{2} c^{2} - 2 \, a c^{3} + c^{4} + {\left (a^{2} c - 2 \, a c^{2} + c^{3}\right )} \sqrt {a c}\right )} {\left | -a + c \right |} \mathrm {sgn}\left (-a + c\right ) - {\left (a^{3} c - 2 \, a^{2} c^{2} + a c^{3} + {\left (a^{2} c - 2 \, a c^{2} + c^{3}\right )} \sqrt {a c}\right )} {\left | -a + c \right |} - {\left (a^{4} c - a^{3} c^{2} - a^{2} c^{3} + a c^{4} - {\left (a^{3} c - a^{2} c^{2} - a c^{3} + c^{4}\right )} \sqrt {a c}\right )} \mathrm {sgn}\left (-a + c\right ) + {\left (a^{4} - a^{3} c - a^{2} c^{2} + a c^{3}\right )} \sqrt {a c}\right )} \arctan \left (-\frac {\sqrt {b x + a} - \sqrt {b x + c}}{\sqrt {-\frac {a^{2} - c^{2} + \sqrt {{\left (a^{2} - c^{2}\right )}^{2} - {\left (a^{3} - 3 \, a^{2} c + 3 \, a c^{2} - c^{3}\right )} {\left (a - c\right )}}}{a - c}}}\right )}{{\left (\sqrt {-a} a^{4} - a^{4} \sqrt {-c} - 4 \, \sqrt {-a} a^{3} c + 4 \, a^{3} \sqrt {-c} c + 6 \, \sqrt {-a} a^{2} c^{2} - 6 \, a^{2} \sqrt {-c} c^{2} - 4 \, \sqrt {-a} a c^{3} + 4 \, a \sqrt {-c} c^{3} + \sqrt {-a} c^{4} - \sqrt {-c} c^{4}\right )} {\left | -a + c \right |}} + \frac {2 \, {\left (a^{4} c - a^{3} c^{2} - a^{2} c^{3} + a c^{4} - 2 \, {\left (a c^{2} - \sqrt {a c} c^{2}\right )} {\left (a - c\right )}^{2} \mathrm {sgn}\left (-a + c\right ) - 2 \, {\left (a c^{2} + \sqrt {a c} a c\right )} {\left (a - c\right )}^{2} + {\left (a^{2} c^{2} - 2 \, a c^{3} + c^{4} - {\left (a^{2} c - 2 \, a c^{2} + c^{3}\right )} \sqrt {a c}\right )} {\left | -a + c \right |} \mathrm {sgn}\left (-a + c\right ) + {\left (a^{3} c - 2 \, a^{2} c^{2} + a c^{3} + {\left (a^{2} c - 2 \, a c^{2} + c^{3}\right )} \sqrt {a c}\right )} {\left | -a + c \right |} + {\left (a^{4} c - a^{3} c^{2} - a^{2} c^{3} + a c^{4} + {\left (a^{3} c - a^{2} c^{2} - a c^{3} + c^{4}\right )} \sqrt {a c}\right )} \mathrm {sgn}\left (-a + c\right ) + {\left (a^{4} - a^{3} c - a^{2} c^{2} + a c^{3}\right )} \sqrt {a c}\right )} \arctan \left (-\frac {\sqrt {b x + a} - \sqrt {b x + c}}{\sqrt {-\frac {a^{2} - c^{2} - \sqrt {{\left (a^{2} - c^{2}\right )}^{2} - {\left (a^{3} - 3 \, a^{2} c + 3 \, a c^{2} - c^{3}\right )} {\left (a - c\right )}}}{a - c}}}\right )}{{\left (\sqrt {-a} a^{4} - a^{4} \sqrt {-c} - 4 \, \sqrt {-a} a^{3} c + 4 \, a^{3} \sqrt {-c} c + 6 \, \sqrt {-a} a^{2} c^{2} - 6 \, a^{2} \sqrt {-c} c^{2} - 4 \, \sqrt {-a} a c^{3} + 4 \, a \sqrt {-c} c^{3} + \sqrt {-a} c^{4} - \sqrt {-c} c^{4}\right )} {\left | -a + c \right |}} + \frac {2 \, \sqrt {b x + a}}{a - c} - \frac {2 \, \sqrt {b x + c}}{a - c} \]
2*a*arctan(sqrt(b*x + a)/sqrt(-a))/(sqrt(-a)*(a - c)) - 2*(a^4*c - a^3*c^2 - a^2*c^3 + a*c^4 + 2*(a*c^2 + sqrt(a*c)*c^2)*(a - c)^2*sgn(-a + c) - 2*( a*c^2 + sqrt(a*c)*a*c)*(a - c)^2 + (a^2*c^2 - 2*a*c^3 + c^4 + (a^2*c - 2*a *c^2 + c^3)*sqrt(a*c))*abs(-a + c)*sgn(-a + c) - (a^3*c - 2*a^2*c^2 + a*c^ 3 + (a^2*c - 2*a*c^2 + c^3)*sqrt(a*c))*abs(-a + c) - (a^4*c - a^3*c^2 - a^ 2*c^3 + a*c^4 - (a^3*c - a^2*c^2 - a*c^3 + c^4)*sqrt(a*c))*sgn(-a + c) + ( a^4 - a^3*c - a^2*c^2 + a*c^3)*sqrt(a*c))*arctan(-(sqrt(b*x + a) - sqrt(b* x + c))/sqrt(-(a^2 - c^2 + sqrt((a^2 - c^2)^2 - (a^3 - 3*a^2*c + 3*a*c^2 - c^3)*(a - c)))/(a - c)))/((sqrt(-a)*a^4 - a^4*sqrt(-c) - 4*sqrt(-a)*a^3*c + 4*a^3*sqrt(-c)*c + 6*sqrt(-a)*a^2*c^2 - 6*a^2*sqrt(-c)*c^2 - 4*sqrt(-a) *a*c^3 + 4*a*sqrt(-c)*c^3 + sqrt(-a)*c^4 - sqrt(-c)*c^4)*abs(-a + c)) + 2* (a^4*c - a^3*c^2 - a^2*c^3 + a*c^4 - 2*(a*c^2 - sqrt(a*c)*c^2)*(a - c)^2*s gn(-a + c) - 2*(a*c^2 + sqrt(a*c)*a*c)*(a - c)^2 + (a^2*c^2 - 2*a*c^3 + c^ 4 - (a^2*c - 2*a*c^2 + c^3)*sqrt(a*c))*abs(-a + c)*sgn(-a + c) + (a^3*c - 2*a^2*c^2 + a*c^3 + (a^2*c - 2*a*c^2 + c^3)*sqrt(a*c))*abs(-a + c) + (a^4* c - a^3*c^2 - a^2*c^3 + a*c^4 + (a^3*c - a^2*c^2 - a*c^3 + c^4)*sqrt(a*c)) *sgn(-a + c) + (a^4 - a^3*c - a^2*c^2 + a*c^3)*sqrt(a*c))*arctan(-(sqrt(b* x + a) - sqrt(b*x + c))/sqrt(-(a^2 - c^2 - sqrt((a^2 - c^2)^2 - (a^3 - 3*a ^2*c + 3*a*c^2 - c^3)*(a - c)))/(a - c)))/((sqrt(-a)*a^4 - a^4*sqrt(-c) - 4*sqrt(-a)*a^3*c + 4*a^3*sqrt(-c)*c + 6*sqrt(-a)*a^2*c^2 - 6*a^2*sqrt(-...
Time = 32.29 (sec) , antiderivative size = 2983, normalized size of antiderivative = 30.75 \[ \int \frac {1}{x \left (\sqrt {a+b x}+\sqrt {c+b x}\right )} \, dx=\text {Too large to display} \]
(atan((a^2*c^(5/2)*(a*c^3 + a^3*c - 2*a^2*c^2)^(1/2)*2i - a^3*c^(3/2)*(a*c ^3 + a^3*c - 2*a^2*c^2)^(1/2)*2i - a^(7/2)*c*(a*c^3 + a^3*c - 2*a^2*c^2)^( 1/2)*2i + a^(5/2)*c^2*(a*c^3 + a^3*c - 2*a^2*c^2)^(1/2)*2i + a*c^3*(a + b* x)^(1/2)*(a*c^3 + a^3*c - 2*a^2*c^2)^(1/2)*2i + a^3*c*(a + b*x)^(1/2)*(a*c ^3 + a^3*c - 2*a^2*c^2)^(1/2)*2i + a^(3/2)*c^(5/2)*(a + b*x)^(1/2)*(a*c^3 + a^3*c - 2*a^2*c^2)^(1/2)*2i + a^(5/2)*c^(3/2)*(a + b*x)^(1/2)*(a*c^3 + a ^3*c - 2*a^2*c^2)^(1/2)*2i - a^2*c^2*(c + b*x)^(1/2)*(a*c^3 + a^3*c - 2*a^ 2*c^2)^(1/2)*4i - a^(3/2)*c^(5/2)*(c + b*x)^(1/2)*(a*c^3 + a^3*c - 2*a^2*c ^2)^(1/2)*2i - a^(5/2)*c^(3/2)*(c + b*x)^(1/2)*(a*c^3 + a^3*c - 2*a^2*c^2) ^(1/2)*2i)/(2*a^5*c^(3/2) - 4*a^4*c^(5/2) + 2*a^(5/2)*c^4 + 2*a^3*c^(7/2) - 4*a^(7/2)*c^3 + 2*a^(9/2)*c^2 - 2*a^2*c^4*(a + b*x)^(1/2) + 4*a^3*c^3*(a + b*x)^(1/2) - 2*a^4*c^2*(a + b*x)^(1/2) - 2*a^(3/2)*c^(9/2)*(a + b*x)^(1 /2) + 2*a^(5/2)*c^(7/2)*(a + b*x)^(1/2) + 2*a^(7/2)*c^(5/2)*(a + b*x)^(1/2 ) - 2*a^(9/2)*c^(3/2)*(a + b*x)^(1/2) + 2*a^2*c^4*(c + b*x)^(1/2) - 4*a^3* c^3*(c + b*x)^(1/2) + 2*a^4*c^2*(c + b*x)^(1/2)))*(a + b*x)^(1/2)*(a*c^3 + a^3*c - 2*a^2*c^2)^(1/2)*2i - 4*a^(3/2)*c - 8*a*c^(3/2) + atan((a^2*c^(5/ 2)*(a*c^3 + a^3*c - 2*a^2*c^2)^(1/2)*2i - a^3*c^(3/2)*(a*c^3 + a^3*c - 2*a ^2*c^2)^(1/2)*2i - a^(7/2)*c*(a*c^3 + a^3*c - 2*a^2*c^2)^(1/2)*2i + a^(5/2 )*c^2*(a*c^3 + a^3*c - 2*a^2*c^2)^(1/2)*2i + a*c^3*(a + b*x)^(1/2)*(a*c^3 + a^3*c - 2*a^2*c^2)^(1/2)*2i + a^3*c*(a + b*x)^(1/2)*(a*c^3 + a^3*c - ...