Integrand size = 25, antiderivative size = 162 \[ \int \frac {1}{x^2 \left (\sqrt {a+b x}+\sqrt {c+b x}\right )^3} \, dx=\frac {8 b \sqrt {a+b x}}{(a-c)^3}-\frac {(a+3 c) \sqrt {a+b x}}{(a-c)^3 x}-\frac {8 b \sqrt {c+b x}}{(a-c)^3}+\frac {(3 a+c) \sqrt {c+b x}}{(a-c)^3 x}-\frac {3 b (3 a+c) \text {arctanh}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{\sqrt {a} (a-c)^3}-\frac {3 b (a+3 c) \text {arctanh}\left (\frac {\sqrt {c+b x}}{\sqrt {c}}\right )}{\sqrt {c} (-a+c)^3} \]
-3*b*(3*a+c)*arctanh((b*x+a)^(1/2)/a^(1/2))/(a-c)^3/a^(1/2)-3*b*(a+3*c)*ar ctanh((b*x+c)^(1/2)/c^(1/2))/(-a+c)^3/c^(1/2)+8*b*(b*x+a)^(1/2)/(a-c)^3-(a +3*c)*(b*x+a)^(1/2)/(a-c)^3/x-8*b*(b*x+c)^(1/2)/(a-c)^3+(3*a+c)*(b*x+c)^(1 /2)/(a-c)^3/x
Time = 10.46 (sec) , antiderivative size = 187, normalized size of antiderivative = 1.15 \[ \int \frac {1}{x^2 \left (\sqrt {a+b x}+\sqrt {c+b x}\right )^3} \, dx=\frac {b \left (8 \sqrt {a+b x}-8 \sqrt {c+b x}-8 \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )+8 \sqrt {c} \text {arctanh}\left (\frac {\sqrt {c+b x}}{\sqrt {c}}\right )-\frac {(a+3 c) \left (a+b x+b x \sqrt {1+\frac {b x}{a}} \text {arctanh}\left (\sqrt {1+\frac {b x}{a}}\right )\right )}{b x \sqrt {a+b x}}+\frac {(3 a+c) \left (c+b x+b x \sqrt {1+\frac {b x}{c}} \text {arctanh}\left (\sqrt {1+\frac {b x}{c}}\right )\right )}{b x \sqrt {c+b x}}\right )}{(a-c)^3} \]
(b*(8*Sqrt[a + b*x] - 8*Sqrt[c + b*x] - 8*Sqrt[a]*ArcTanh[Sqrt[a + b*x]/Sq rt[a]] + 8*Sqrt[c]*ArcTanh[Sqrt[c + b*x]/Sqrt[c]] - ((a + 3*c)*(a + b*x + b*x*Sqrt[1 + (b*x)/a]*ArcTanh[Sqrt[1 + (b*x)/a]]))/(b*x*Sqrt[a + b*x]) + ( (3*a + c)*(c + b*x + b*x*Sqrt[1 + (b*x)/c]*ArcTanh[Sqrt[1 + (b*x)/c]]))/(b *x*Sqrt[c + b*x])))/(a - c)^3
Time = 0.45 (sec) , antiderivative size = 175, normalized size of antiderivative = 1.08, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {7240, 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.
| \(\displaystyle \int \frac {1}{x^2 \left (\sqrt {a+b x}+\sqrt {b x+c}\right )^3} \, dx\) |
| \(\Big \downarrow \) 7240 |
| \(\displaystyle \frac {\int \left (\frac {4 \sqrt {a+b x} b}{x}-\frac {4 \sqrt {c+b x} b}{x}+\frac {(a+3 c) \sqrt {a+b x}}{x^2}-\frac {(3 a+c) \sqrt {c+b x}}{x^2}\right )dx}{(a-c)^3}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {-\frac {b (a+3 c) \text {arctanh}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{\sqrt {a}}+\frac {b (3 a+c) \text {arctanh}\left (\frac {\sqrt {b x+c}}{\sqrt {c}}\right )}{\sqrt {c}}-8 \sqrt {a} b \text {arctanh}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )-\frac {(a+3 c) \sqrt {a+b x}}{x}+\frac {(3 a+c) \sqrt {b x+c}}{x}+8 b \sqrt {a+b x}+8 b \sqrt {c} \text {arctanh}\left (\frac {\sqrt {b x+c}}{\sqrt {c}}\right )-8 b \sqrt {b x+c}}{(a-c)^3}\) |
(8*b*Sqrt[a + b*x] - ((a + 3*c)*Sqrt[a + b*x])/x - 8*b*Sqrt[c + b*x] + ((3 *a + c)*Sqrt[c + b*x])/x - 8*Sqrt[a]*b*ArcTanh[Sqrt[a + b*x]/Sqrt[a]] - (b *(a + 3*c)*ArcTanh[Sqrt[a + b*x]/Sqrt[a]])/Sqrt[a] + 8*b*Sqrt[c]*ArcTanh[S qrt[c + b*x]/Sqrt[c]] + (b*(3*a + c)*ArcTanh[Sqrt[c + b*x]/Sqrt[c]])/Sqrt[ c])/(a - c)^3
3.5.15.3.1 Defintions of rubi rules used
Int[(u_.)*((e_.)*Sqrt[(a_.) + (b_.)*(x_)^(n_.)] + (f_.)*Sqrt[(c_.) + (d_.)* (x_)^(n_.)])^(m_), x_Symbol] :> Simp[(a*e^2 - c*f^2)^m Int[ExpandIntegran d[u/(e*Sqrt[a + b*x^n] - f*Sqrt[c + d*x^n])^m, x], x], x] /; FreeQ[{a, b, c , d, e, f, n}, x] && ILtQ[m, 0] && EqQ[b*e^2 - d*f^2, 0]
Time = 0.04 (sec) , antiderivative size = 252, normalized size of antiderivative = 1.56
| method | result | size |
| default | \(\frac {2 a b \left (-\frac {\sqrt {b x +a}}{2 x b}-\frac {\operatorname {arctanh}\left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right )}{2 \sqrt {a}}\right )}{\left (a -c \right )^{3}}+\frac {6 c b \left (-\frac {\sqrt {b x +a}}{2 x b}-\frac {\operatorname {arctanh}\left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right )}{2 \sqrt {a}}\right )}{\left (a -c \right )^{3}}-\frac {6 a b \left (-\frac {\sqrt {b x +c}}{2 x b}-\frac {\operatorname {arctanh}\left (\frac {\sqrt {b x +c}}{\sqrt {c}}\right )}{2 \sqrt {c}}\right )}{\left (a -c \right )^{3}}-\frac {2 c b \left (-\frac {\sqrt {b x +c}}{2 x b}-\frac {\operatorname {arctanh}\left (\frac {\sqrt {b x +c}}{\sqrt {c}}\right )}{2 \sqrt {c}}\right )}{\left (a -c \right )^{3}}+\frac {4 b \left (2 \sqrt {b x +a}-2 \sqrt {a}\, \operatorname {arctanh}\left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right )\right )}{\left (a -c \right )^{3}}-\frac {4 b \left (2 \sqrt {b x +c}-2 \sqrt {c}\, \operatorname {arctanh}\left (\frac {\sqrt {b x +c}}{\sqrt {c}}\right )\right )}{\left (a -c \right )^{3}}\) | \(252\) |
2/(a-c)^3*a*b*(-1/2*(b*x+a)^(1/2)/x/b-1/2*arctanh((b*x+a)^(1/2)/a^(1/2))/a ^(1/2))+6/(a-c)^3*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))-6/(a-c)^3*a*b*(-1/2*(b*x+c)^(1/2)/x/b-1/2/c^(1/2)*arctanh( (b*x+c)^(1/2)/c^(1/2)))-2/(a-c)^3*c*b*(-1/2*(b*x+c)^(1/2)/x/b-1/2/c^(1/2)* arctanh((b*x+c)^(1/2)/c^(1/2)))+4/(a-c)^3*b*(2*(b*x+a)^(1/2)-2*a^(1/2)*arc tanh((b*x+a)^(1/2)/a^(1/2)))-4/(a-c)^3*b*(2*(b*x+c)^(1/2)-2*c^(1/2)*arctan h((b*x+c)^(1/2)/c^(1/2)))
Time = 0.31 (sec) , antiderivative size = 675, normalized size of antiderivative = 4.17 \[ \int \frac {1}{x^2 \left (\sqrt {a+b x}+\sqrt {c+b x}\right )^3} \, dx=\left [-\frac {3 \, {\left (3 \, a b c + b c^{2}\right )} \sqrt {a} x \log \left (\frac {b x + 2 \, \sqrt {b x + a} \sqrt {a} + 2 \, a}{x}\right ) + 3 \, {\left (a^{2} b + 3 \, a b c\right )} \sqrt {c} x \log \left (\frac {b x - 2 \, \sqrt {b x + c} \sqrt {c} + 2 \, c}{x}\right ) - 2 \, {\left (8 \, a b c x - a^{2} c - 3 \, a c^{2}\right )} \sqrt {b x + a} + 2 \, {\left (8 \, a b c x - 3 \, a^{2} c - a c^{2}\right )} \sqrt {b x + c}}{2 \, {\left (a^{4} c - 3 \, a^{3} c^{2} + 3 \, a^{2} c^{3} - a c^{4}\right )} x}, -\frac {6 \, {\left (a^{2} b + 3 \, a b c\right )} \sqrt {-c} x \arctan \left (\frac {\sqrt {b x + c} \sqrt {-c}}{c}\right ) + 3 \, {\left (3 \, a b c + b c^{2}\right )} \sqrt {a} x \log \left (\frac {b x + 2 \, \sqrt {b x + a} \sqrt {a} + 2 \, a}{x}\right ) - 2 \, {\left (8 \, a b c x - a^{2} c - 3 \, a c^{2}\right )} \sqrt {b x + a} + 2 \, {\left (8 \, a b c x - 3 \, a^{2} c - a c^{2}\right )} \sqrt {b x + c}}{2 \, {\left (a^{4} c - 3 \, a^{3} c^{2} + 3 \, a^{2} c^{3} - a c^{4}\right )} x}, \frac {6 \, {\left (3 \, a b c + b c^{2}\right )} \sqrt {-a} x \arctan \left (\frac {\sqrt {b x + a} \sqrt {-a}}{a}\right ) - 3 \, {\left (a^{2} b + 3 \, a b c\right )} \sqrt {c} x \log \left (\frac {b x - 2 \, \sqrt {b x + c} \sqrt {c} + 2 \, c}{x}\right ) + 2 \, {\left (8 \, a b c x - a^{2} c - 3 \, a c^{2}\right )} \sqrt {b x + a} - 2 \, {\left (8 \, a b c x - 3 \, a^{2} c - a c^{2}\right )} \sqrt {b x + c}}{2 \, {\left (a^{4} c - 3 \, a^{3} c^{2} + 3 \, a^{2} c^{3} - a c^{4}\right )} x}, \frac {3 \, {\left (3 \, a b c + b c^{2}\right )} \sqrt {-a} x \arctan \left (\frac {\sqrt {b x + a} \sqrt {-a}}{a}\right ) - 3 \, {\left (a^{2} b + 3 \, a b c\right )} \sqrt {-c} x \arctan \left (\frac {\sqrt {b x + c} \sqrt {-c}}{c}\right ) + {\left (8 \, a b c x - a^{2} c - 3 \, a c^{2}\right )} \sqrt {b x + a} - {\left (8 \, a b c x - 3 \, a^{2} c - a c^{2}\right )} \sqrt {b x + c}}{{\left (a^{4} c - 3 \, a^{3} c^{2} + 3 \, a^{2} c^{3} - a c^{4}\right )} x}\right ] \]
[-1/2*(3*(3*a*b*c + b*c^2)*sqrt(a)*x*log((b*x + 2*sqrt(b*x + a)*sqrt(a) + 2*a)/x) + 3*(a^2*b + 3*a*b*c)*sqrt(c)*x*log((b*x - 2*sqrt(b*x + c)*sqrt(c) + 2*c)/x) - 2*(8*a*b*c*x - a^2*c - 3*a*c^2)*sqrt(b*x + a) + 2*(8*a*b*c*x - 3*a^2*c - a*c^2)*sqrt(b*x + c))/((a^4*c - 3*a^3*c^2 + 3*a^2*c^3 - a*c^4) *x), -1/2*(6*(a^2*b + 3*a*b*c)*sqrt(-c)*x*arctan(sqrt(b*x + c)*sqrt(-c)/c) + 3*(3*a*b*c + b*c^2)*sqrt(a)*x*log((b*x + 2*sqrt(b*x + a)*sqrt(a) + 2*a) /x) - 2*(8*a*b*c*x - a^2*c - 3*a*c^2)*sqrt(b*x + a) + 2*(8*a*b*c*x - 3*a^2 *c - a*c^2)*sqrt(b*x + c))/((a^4*c - 3*a^3*c^2 + 3*a^2*c^3 - a*c^4)*x), 1/ 2*(6*(3*a*b*c + b*c^2)*sqrt(-a)*x*arctan(sqrt(b*x + a)*sqrt(-a)/a) - 3*(a^ 2*b + 3*a*b*c)*sqrt(c)*x*log((b*x - 2*sqrt(b*x + c)*sqrt(c) + 2*c)/x) + 2* (8*a*b*c*x - a^2*c - 3*a*c^2)*sqrt(b*x + a) - 2*(8*a*b*c*x - 3*a^2*c - a*c ^2)*sqrt(b*x + c))/((a^4*c - 3*a^3*c^2 + 3*a^2*c^3 - a*c^4)*x), (3*(3*a*b* c + b*c^2)*sqrt(-a)*x*arctan(sqrt(b*x + a)*sqrt(-a)/a) - 3*(a^2*b + 3*a*b* c)*sqrt(-c)*x*arctan(sqrt(b*x + c)*sqrt(-c)/c) + (8*a*b*c*x - a^2*c - 3*a* c^2)*sqrt(b*x + a) - (8*a*b*c*x - 3*a^2*c - a*c^2)*sqrt(b*x + c))/((a^4*c - 3*a^3*c^2 + 3*a^2*c^3 - a*c^4)*x)]
\[ \int \frac {1}{x^2 \left (\sqrt {a+b x}+\sqrt {c+b x}\right )^3} \, dx=\int \frac {1}{x^{2} \left (\sqrt {a + b x} + \sqrt {b x + c}\right )^{3}}\, dx \]
\[ \int \frac {1}{x^2 \left (\sqrt {a+b x}+\sqrt {c+b x}\right )^3} \, dx=\int { \frac {1}{x^{2} {\left (\sqrt {b x + a} + \sqrt {b x + c}\right )}^{3}} \,d x } \]
Leaf count of result is larger than twice the leaf count of optimal. 2594 vs. \(2 (142) = 284\).
Time = 7.56 (sec) , antiderivative size = 2594, normalized size of antiderivative = 16.01 \[ \int \frac {1}{x^2 \left (\sqrt {a+b x}+\sqrt {c+b x}\right )^3} \, dx=\text {Too large to display} \]
8*sqrt(b*x + a)*b/(a^3 - 3*a^2*c + 3*a*c^2 - c^3) - 8*sqrt(b*x + c)*b/(a^3 - 3*a^2*c + 3*a*c^2 - c^3) + 3*(3*a*b + b*c)*arctan(sqrt(b*x + a)/sqrt(-a ))/((a^3 - 3*a^2*c + 3*a*c^2 - c^3)*sqrt(-a)) - 3*(2*(a^2*c^2 + 3*a*c^3 + (a*c^2 + 3*c^3)*sqrt(a*c))*(a^3 - 3*a^2*c + 3*a*c^2 - c^3)^2*b*sgn(-2*a^3 + 6*a^2*c - 6*a*c^2 + 2*c^3) - 2*(a^2*c^2 + 3*a*c^3 + (a^2*c + 3*a*c^2)*sq rt(a*c))*(a^3 - 3*a^2*c + 3*a*c^2 - c^3)^2*b + (a^5*c^2 - a^4*c^3 - 6*a^3* c^4 + 14*a^2*c^5 - 11*a*c^6 + 3*c^7 + (a^5*c - a^4*c^2 - 6*a^3*c^3 + 14*a^ 2*c^4 - 11*a*c^5 + 3*c^6)*sqrt(a*c))*b*abs(-a^3 + 3*a^2*c - 3*a*c^2 + c^3) *sgn(-2*a^3 + 6*a^2*c - 6*a*c^2 + 2*c^3) - (a^6*c - a^5*c^2 - 6*a^4*c^3 + 14*a^3*c^4 - 11*a^2*c^5 + 3*a*c^6 + (a^5*c - a^4*c^2 - 6*a^3*c^3 + 14*a^2* c^4 - 11*a*c^5 + 3*c^6)*sqrt(a*c))*b*abs(-a^3 + 3*a^2*c - 3*a*c^2 + c^3) - (a^9*c - 2*a^8*c^2 - 6*a^7*c^3 + 22*a^6*c^4 - 20*a^5*c^5 - 6*a^4*c^6 + 22 *a^3*c^7 - 14*a^2*c^8 + 3*a*c^9 + (a^8*c - 2*a^7*c^2 - 6*a^6*c^3 + 22*a^5* c^4 - 20*a^4*c^5 - 6*a^3*c^6 + 22*a^2*c^7 - 14*a*c^8 + 3*c^9)*sqrt(a*c))*b *sgn(-2*a^3 + 6*a^2*c - 6*a*c^2 + 2*c^3) + (a^9*c - 2*a^8*c^2 - 6*a^7*c^3 + 22*a^6*c^4 - 20*a^5*c^5 - 6*a^4*c^6 + 22*a^3*c^7 - 14*a^2*c^8 + 3*a*c^9 + (a^9 - 2*a^8*c - 6*a^7*c^2 + 22*a^6*c^3 - 20*a^5*c^4 - 6*a^4*c^5 + 22*a^ 3*c^6 - 14*a^2*c^7 + 3*a*c^8)*sqrt(a*c))*b)*arctan(-(sqrt(b*x + a) - sqrt( b*x + c))/sqrt(-(a^4 - 2*a^3*c + 2*a*c^3 - c^4 + sqrt((a^4 - 2*a^3*c + 2*a *c^3 - c^4)^2 - (a^5 - 5*a^4*c + 10*a^3*c^2 - 10*a^2*c^3 + 5*a*c^4 - c^...
Time = 51.54 (sec) , antiderivative size = 4681, normalized size of antiderivative = 28.90 \[ \int \frac {1}{x^2 \left (\sqrt {a+b x}+\sqrt {c+b x}\right )^3} \, dx=\text {Too large to display} \]
(b*atan(((b*((a^(1/2)*c^(3/2) - 2*a*c + a^(3/2)*c^(1/2))*(2*a*c + a^(1/2)* c^(3/2) + a^(3/2)*c^(1/2)))^(1/2)*((9*a^6*b*c^(7/2) - 9*a^(7/2)*b*c^6 - 24 *a^5*b*c^(9/2) + 24*a^(9/2)*b*c^5 + 18*a^4*b*c^(11/2) - 18*a^(11/2)*b*c^4 - 3*a^2*b*c^(15/2) + 3*a^(15/2)*b*c^2)/(a^3*c^9 - 6*a^4*c^8 + 15*a^5*c^7 - 20*a^6*c^6 + 15*a^7*c^5 - 6*a^8*c^4 + a^9*c^3) + (((a + b*x)^(1/2) - a^(1 /2))*(6*a^(3/2)*b*c^8 - 6*a^8*b*c^(3/2) + 36*a^6*b*c^(7/2) - 36*a^(7/2)*b* c^6 - 48*a^5*b*c^(9/2) + 48*a^(9/2)*b*c^5 + 18*a^4*b*c^(11/2) - 18*a^(11/2 )*b*c^4))/(2*((c + b*x)^(1/2) - c^(1/2))*(a^3*c^9 - 6*a^4*c^8 + 15*a^5*c^7 - 20*a^6*c^6 + 15*a^7*c^5 - 6*a^8*c^4 + a^9*c^3)) - (3*b*((a^(5/2)*c^(19/ 2) - 5*a^(7/2)*c^(17/2) + 9*a^(9/2)*c^(15/2) - 5*a^(11/2)*c^(13/2) - 5*a^( 13/2)*c^(11/2) + 9*a^(15/2)*c^(9/2) - 5*a^(17/2)*c^(7/2) + a^(19/2)*c^(5/2 ))/(a^3*c^9 - 6*a^4*c^8 + 15*a^5*c^7 - 20*a^6*c^6 + 15*a^7*c^5 - 6*a^8*c^4 + a^9*c^3) - (((a + b*x)^(1/2) - a^(1/2))*(4*a^2*c^10 - 28*a^3*c^9 + 88*a ^4*c^8 - 164*a^5*c^7 + 200*a^6*c^6 - 164*a^7*c^5 + 88*a^8*c^4 - 28*a^9*c^3 + 4*a^10*c^2))/(2*((c + b*x)^(1/2) - c^(1/2))*(a^3*c^9 - 6*a^4*c^8 + 15*a ^5*c^7 - 20*a^6*c^6 + 15*a^7*c^5 - 6*a^8*c^4 + a^9*c^3)))*((a^(1/2)*c^(3/2 ) - 2*a*c + a^(3/2)*c^(1/2))*(2*a*c + a^(1/2)*c^(3/2) + a^(3/2)*c^(1/2)))^ (1/2)*(a*c^(7/2) + a^(7/2)*c - 3*a^3*c^(3/2) - 3*a^(3/2)*c^3 + 2*a^2*c^(5/ 2) + 2*a^(5/2)*c^2))/(2*(a^2*c^7 - 5*a^3*c^6 + 10*a^4*c^5 - 10*a^5*c^4 + 5 *a^6*c^3 - a^7*c^2)))*(a*c^(7/2) + a^(7/2)*c - 3*a^3*c^(3/2) - 3*a^(3/2...