Integrand size = 25, antiderivative size = 103 \[ \int \frac {1}{x \left (\sqrt {a+b x}+\sqrt {a+c x}\right )} \, dx=-\frac {\sqrt {a+b x}}{(b-c) x}+\frac {\sqrt {a+c x}}{(b-c) x}-\frac {b \text {arctanh}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{\sqrt {a} (b-c)}+\frac {c \text {arctanh}\left (\frac {\sqrt {a+c x}}{\sqrt {a}}\right )}{\sqrt {a} (b-c)} \]
-b*arctanh((b*x+a)^(1/2)/a^(1/2))/(b-c)/a^(1/2)+c*arctanh((c*x+a)^(1/2)/a^ (1/2))/(b-c)/a^(1/2)-(b*x+a)^(1/2)/(b-c)/x+(c*x+a)^(1/2)/(b-c)/x
Time = 10.17 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.31 \[ \int \frac {1}{x \left (\sqrt {a+b x}+\sqrt {a+c x}\right )} \, dx=\frac {-\frac {a}{\sqrt {a+b x}}-\frac {b x}{\sqrt {a+b x}}+\frac {a}{\sqrt {a+c x}}+\frac {c x}{\sqrt {a+c x}}-\frac {b x \sqrt {1+\frac {b x}{a}} \text {arctanh}\left (\sqrt {1+\frac {b x}{a}}\right )}{\sqrt {a+b x}}+\frac {c x \sqrt {1+\frac {c x}{a}} \text {arctanh}\left (\sqrt {1+\frac {c x}{a}}\right )}{\sqrt {a+c x}}}{b x-c x} \]
(-(a/Sqrt[a + b*x]) - (b*x)/Sqrt[a + b*x] + a/Sqrt[a + c*x] + (c*x)/Sqrt[a + c*x] - (b*x*Sqrt[1 + (b*x)/a]*ArcTanh[Sqrt[1 + (b*x)/a]])/Sqrt[a + b*x] + (c*x*Sqrt[1 + (c*x)/a]*ArcTanh[Sqrt[1 + (c*x)/a]])/Sqrt[a + c*x])/(b*x - c*x)
Time = 0.26 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.93, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {2528, 51, 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.
\(\displaystyle \int \frac {1}{x \left (\sqrt {a+b x}+\sqrt {a+c x}\right )} \, dx\) |
\(\Big \downarrow \) 2528 |
\(\displaystyle \frac {\int \frac {\sqrt {a+b x}}{x^2}dx}{b-c}-\frac {\int \frac {\sqrt {a+c x}}{x^2}dx}{b-c}\) |
\(\Big \downarrow \) 51 |
\(\displaystyle \frac {\frac {1}{2} b \int \frac {1}{x \sqrt {a+b x}}dx-\frac {\sqrt {a+b x}}{x}}{b-c}-\frac {\frac {1}{2} c \int \frac {1}{x \sqrt {a+c x}}dx-\frac {\sqrt {a+c x}}{x}}{b-c}\) |
\(\Big \downarrow \) 73 |
\(\displaystyle \frac {\int \frac {1}{\frac {a+b x}{b}-\frac {a}{b}}d\sqrt {a+b x}-\frac {\sqrt {a+b x}}{x}}{b-c}-\frac {\int \frac {1}{\frac {a+c x}{c}-\frac {a}{c}}d\sqrt {a+c x}-\frac {\sqrt {a+c x}}{x}}{b-c}\) |
\(\Big \downarrow \) 221 |
\(\displaystyle \frac {-\frac {b \text {arctanh}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{\sqrt {a}}-\frac {\sqrt {a+b x}}{x}}{b-c}-\frac {-\frac {c \text {arctanh}\left (\frac {\sqrt {a+c x}}{\sqrt {a}}\right )}{\sqrt {a}}-\frac {\sqrt {a+c x}}{x}}{b-c}\) |
(-(Sqrt[a + b*x]/x) - (b*ArcTanh[Sqrt[a + b*x]/Sqrt[a]])/Sqrt[a])/(b - c) - (-(Sqrt[a + c*x]/x) - (c*ArcTanh[Sqrt[a + c*x]/Sqrt[a]])/Sqrt[a])/(b - c )
3.5.30.3.1 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 + 1))), x] - Simp[d*(n/(b*(m + 1))) Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d, n}, x ] && ILtQ[m, -1] && FractionQ[n] && GtQ[n, 0]
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[c/(e*(b*c - a*d)) Int[(u*Sqrt[a + b*x])/x, x], x] - Si mp[a/(f*(b*c - a*d)) Int[(u*Sqrt[c + d*x])/x, x], x] /; FreeQ[{a, b, c, d , e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a*e^2 - c*f^2, 0]
Time = 0.04 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.85
method | result | size |
default | \(\frac {2 b \left (-\frac {\sqrt {b x +a}}{2 x b}-\frac {\operatorname {arctanh}\left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right )}{2 \sqrt {a}}\right )}{b -c}-\frac {2 c \left (-\frac {\sqrt {c x +a}}{2 c x}-\frac {\operatorname {arctanh}\left (\frac {\sqrt {c x +a}}{\sqrt {a}}\right )}{2 \sqrt {a}}\right )}{b -c}\) | \(88\) |
2/(b-c)*b*(-1/2*(b*x+a)^(1/2)/x/b-1/2*arctanh((b*x+a)^(1/2)/a^(1/2))/a^(1/ 2))-2/(b-c)*c*(-1/2*(c*x+a)^(1/2)/c/x-1/2/a^(1/2)*arctanh((c*x+a)^(1/2)/a^ (1/2)))
Time = 0.32 (sec) , antiderivative size = 182, normalized size of antiderivative = 1.77 \[ \int \frac {1}{x \left (\sqrt {a+b x}+\sqrt {a+c x}\right )} \, dx=\left [-\frac {\sqrt {a} b x \log \left (\frac {b x + 2 \, \sqrt {b x + a} \sqrt {a} + 2 \, a}{x}\right ) + \sqrt {a} c x \log \left (\frac {c x - 2 \, \sqrt {c x + a} \sqrt {a} + 2 \, a}{x}\right ) + 2 \, \sqrt {b x + a} a - 2 \, \sqrt {c x + a} a}{2 \, {\left (a b - a c\right )} x}, \frac {\sqrt {-a} b x \arctan \left (\frac {\sqrt {b x + a} \sqrt {-a}}{a}\right ) - \sqrt {-a} c x \arctan \left (\frac {\sqrt {c x + a} \sqrt {-a}}{a}\right ) - \sqrt {b x + a} a + \sqrt {c x + a} a}{{\left (a b - a c\right )} x}\right ] \]
[-1/2*(sqrt(a)*b*x*log((b*x + 2*sqrt(b*x + a)*sqrt(a) + 2*a)/x) + sqrt(a)* c*x*log((c*x - 2*sqrt(c*x + a)*sqrt(a) + 2*a)/x) + 2*sqrt(b*x + a)*a - 2*s qrt(c*x + a)*a)/((a*b - a*c)*x), (sqrt(-a)*b*x*arctan(sqrt(b*x + a)*sqrt(- a)/a) - sqrt(-a)*c*x*arctan(sqrt(c*x + a)*sqrt(-a)/a) - sqrt(b*x + a)*a + sqrt(c*x + a)*a)/((a*b - a*c)*x)]
\[ \int \frac {1}{x \left (\sqrt {a+b x}+\sqrt {a+c x}\right )} \, dx=\int \frac {1}{x \left (\sqrt {a + b x} + \sqrt {a + c x}\right )}\, dx \]
\[ \int \frac {1}{x \left (\sqrt {a+b x}+\sqrt {a+c x}\right )} \, dx=\int { \frac {1}{x {\left (\sqrt {b x + a} + \sqrt {c x + a}\right )}} \,d x } \]
Leaf count of result is larger than twice the leaf count of optimal. 1402 vs. \(2 (87) = 174\).
Time = 2.75 (sec) , antiderivative size = 1402, normalized size of antiderivative = 13.61 \[ \int \frac {1}{x \left (\sqrt {a+b x}+\sqrt {a+c x}\right )} \, dx=\text {Too large to display} \]
b*arctan(sqrt(b*x + a)/sqrt(-a))/(sqrt(-a)*(b - c)) - 2*((sqrt(b*c)*sqrt(b *x + a) - sqrt(a*b^2 + (b*x + a)*b*c - a*b*c))*a*b^2*c*abs(b) - (sqrt(b*c) *sqrt(b*x + a) - sqrt(a*b^2 + (b*x + a)*b*c - a*b*c))*a*b*c^2*abs(b) + (sq rt(b*c)*sqrt(b*x + a) - sqrt(a*b^2 + (b*x + a)*b*c - a*b*c))^3*c*abs(b))/( (a^2*b^4 - 2*a^2*b^3*c + a^2*b^2*c^2 - 2*(sqrt(b*c)*sqrt(b*x + a) - sqrt(a *b^2 + (b*x + a)*b*c - a*b*c))^2*a*b^2 - 2*(sqrt(b*c)*sqrt(b*x + a) - sqrt (a*b^2 + (b*x + a)*b*c - a*b*c))^2*a*b*c + (sqrt(b*c)*sqrt(b*x + a) - sqrt (a*b^2 + (b*x + a)*b*c - a*b*c))^4)*(b - c)) - sqrt(b*x + a)/((b - c)*x) + (2*(a*b^3*c^2 - a*b^2*c^3)*(a*b^2 - a*b*c)^2*sqrt(-a)*abs(b)*sgn(-2*b + 2 *c) + 2*(a*b^3*c - a*b^2*c^2)*(a*b^2 - a*b*c)^2*sqrt(-a*b*c)*abs(b) + (a^2 *b^5*c - 3*a^2*b^4*c^2 + 3*a^2*b^3*c^3 - a^2*b^2*c^4)*sqrt(-a*b*c)*abs(-a* b^2 + a*b*c)*abs(b)*sgn(-2*b + 2*c) + (a^2*b^6*c - 3*a^2*b^5*c^2 + 3*a^2*b ^4*c^3 - a^2*b^3*c^4)*sqrt(-a)*abs(-a*b^2 + a*b*c)*abs(b) + (a^3*b^7*c^2 - 2*a^3*b^6*c^3 + 2*a^3*b^4*c^5 - a^3*b^3*c^6)*sqrt(-a)*abs(b)*sgn(-2*b + 2 *c) + (a^3*b^7*c - 2*a^3*b^6*c^2 + 2*a^3*b^4*c^4 - a^3*b^3*c^5)*sqrt(-a*b* c)*abs(b))*arctan(-(sqrt(b*c)*sqrt(b*x + a) - sqrt(a*b^2 + (b*x + a)*b*c - a*b*c))/sqrt(-(a*b^3 - a*b*c^2 + sqrt((a*b^3 - a*b*c^2)^2 - (a^2*b^5 - 3* a^2*b^4*c + 3*a^2*b^3*c^2 - a^2*b^2*c^3)*(b - c)))/(b - c)))/((b^8 - 5*b^7 *c + 10*b^6*c^2 - 10*b^5*c^3 + 5*b^4*c^4 - b^3*c^5)*a^3*abs(-a*b^2 + a*b*c )) - (2*(a*b^3*c^2 - a*b^2*c^3)*(a*b^2 - a*b*c)^2*sqrt(-a)*abs(b)*sgn(-...
Time = 24.65 (sec) , antiderivative size = 1637, normalized size of antiderivative = 15.89 \[ \int \frac {1}{x \left (\sqrt {a+b x}+\sqrt {a+c x}\right )} \, dx=\text {Too large to display} \]
(2*a*b - 2*a*c + a*c*log(((a + b*x)^(1/2) - a^(1/2))/((a + c*x)^(1/2) - a^ (1/2))) - 2*a^(1/2)*b*(a + c*x)^(1/2) + 2*a^(1/2)*c*(a + b*x)^(1/2) + a*b* atan((b^3*(a + b*x)^(1/2)*1i - b^3*(a + c*x)^(1/2)*1i + c^3*(a + b*x)^(1/2 )*1i - a^(1/2)*c^3*1i + a^(1/2)*b*c^2*1i - b*c^2*(a + c*x)^(1/2)*1i)/(b^3* (a + b*x)^(1/2) - b^3*(a + c*x)^(1/2) - c^3*(a + b*x)^(1/2) + a^(1/2)*c^3 - a^(1/2)*b*c^2 + b*c^2*(a + c*x)^(1/2)))*2i - a*c*atan((b^3*(a + b*x)^(1/ 2)*1i - b^3*(a + c*x)^(1/2)*1i + c^3*(a + b*x)^(1/2)*1i - a^(1/2)*c^3*1i + a^(1/2)*b*c^2*1i - b*c^2*(a + c*x)^(1/2)*1i)/(b^3*(a + b*x)^(1/2) - b^3*( a + c*x)^(1/2) - c^3*(a + b*x)^(1/2) + a^(1/2)*c^3 - a^(1/2)*b*c^2 + b*c^2 *(a + c*x)^(1/2)))*2i + a*b*log(((a + b*x)^(1/2) - a^(1/2))/((a + c*x)^(1/ 2) - a^(1/2))) + b*atan((b^3*(a + b*x)^(1/2)*1i - b^3*(a + c*x)^(1/2)*1i + c^3*(a + b*x)^(1/2)*1i - a^(1/2)*c^3*1i + a^(1/2)*b*c^2*1i - b*c^2*(a + c *x)^(1/2)*1i)/(b^3*(a + b*x)^(1/2) - b^3*(a + c*x)^(1/2) - c^3*(a + b*x)^( 1/2) + a^(1/2)*c^3 - a^(1/2)*b*c^2 + b*c^2*(a + c*x)^(1/2)))*(a + b*x)^(1/ 2)*(a + c*x)^(1/2)*2i - c*atan((b^3*(a + b*x)^(1/2)*1i - b^3*(a + c*x)^(1/ 2)*1i + c^3*(a + b*x)^(1/2)*1i - a^(1/2)*c^3*1i + a^(1/2)*b*c^2*1i - b*c^2 *(a + c*x)^(1/2)*1i)/(b^3*(a + b*x)^(1/2) - b^3*(a + c*x)^(1/2) - c^3*(a + b*x)^(1/2) + a^(1/2)*c^3 - a^(1/2)*b*c^2 + b*c^2*(a + c*x)^(1/2)))*(a + b *x)^(1/2)*(a + c*x)^(1/2)*2i + b*log(((a + b*x)^(1/2) - a^(1/2))/((a + c*x )^(1/2) - a^(1/2)))*(a + b*x)^(1/2)*(a + c*x)^(1/2) + c*log(((a + b*x)^...