Integrand size = 23, antiderivative size = 165 \[ \int \frac {x}{\left (\sqrt {a+b x}+\sqrt {c+b x}\right )^2} \, dx=\frac {(a+c) x^2}{2 (a-c)^2}+\frac {2 b x^3}{3 (a-c)^2}-\frac {(a+c) \sqrt {a+b x} \sqrt {c+b x}}{4 b^2 (a-c)}+\frac {(a+c) (a+b x)^{3/2} \sqrt {c+b x}}{2 b^2 (a-c)^2}-\frac {2 (a+b x)^{3/2} (c+b x)^{3/2}}{3 b^2 (a-c)^2}-\frac {(a+c) \text {arctanh}\left (\frac {\sqrt {a+b x}}{\sqrt {c+b x}}\right )}{4 b^2} \]
1/2*(a+c)*x^2/(a-c)^2+2/3*b*x^3/(a-c)^2-2/3*(b*x+a)^(3/2)*(b*x+c)^(3/2)/b^ 2/(a-c)^2-1/4*(a+c)*arctanh((b*x+a)^(1/2)/(b*x+c)^(1/2))/b^2+1/2*(a+c)*(b* x+a)^(3/2)*(b*x+c)^(1/2)/b^2/(a-c)^2-1/4*(a+c)*(b*x+a)^(1/2)*(b*x+c)^(1/2) /b^2/(a-c)
Time = 0.32 (sec) , antiderivative size = 133, normalized size of antiderivative = 0.81 \[ \int \frac {x}{\left (\sqrt {a+b x}+\sqrt {c+b x}\right )^2} \, dx=\frac {\frac {2 (c+b x) \left (-3 a c+c^2+3 a b x-b c x+4 b^2 x^2\right )}{(a-c)^2}+\frac {\sqrt {a+b x} \sqrt {c+b x} \left (3 a^2+3 c^2-2 b c x-8 b^2 x^2-2 a (c+b x)\right )}{(a-c)^2}+3 (a+c) \log \left (\sqrt {a+b x}-\sqrt {c+b x}\right )}{12 b^2} \]
((2*(c + b*x)*(-3*a*c + c^2 + 3*a*b*x - b*c*x + 4*b^2*x^2))/(a - c)^2 + (S qrt[a + b*x]*Sqrt[c + b*x]*(3*a^2 + 3*c^2 - 2*b*c*x - 8*b^2*x^2 - 2*a*(c + b*x)))/(a - c)^2 + 3*(a + c)*Log[Sqrt[a + b*x] - Sqrt[c + b*x]])/(12*b^2)
Time = 0.39 (sec) , antiderivative size = 151, normalized size of antiderivative = 0.92, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, 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.
| \(\displaystyle \int \frac {x}{\left (\sqrt {a+b x}+\sqrt {b x+c}\right )^2} \, dx\) |
| \(\Big \downarrow \) 7240 |
| \(\displaystyle \frac {\int \left (2 b x^2+(a+c) x-2 \sqrt {a+b x} \sqrt {c+b x} x\right )dx}{(a-c)^2}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {-\frac {\left (a^2-c^2\right ) \sqrt {a+b x} \sqrt {b x+c}}{4 b^2}-\frac {(a-c)^2 (a+c) \text {arctanh}\left (\frac {\sqrt {a+b x}}{\sqrt {b x+c}}\right )}{4 b^2}-\frac {2 (a+b x)^{3/2} (b x+c)^{3/2}}{3 b^2}+\frac {(a+c) (a+b x)^{3/2} \sqrt {b x+c}}{2 b^2}+\frac {1}{2} x^2 (a+c)+\frac {2 b x^3}{3}}{(a-c)^2}\) |
(((a + c)*x^2)/2 + (2*b*x^3)/3 - ((a^2 - c^2)*Sqrt[a + b*x]*Sqrt[c + b*x]) /(4*b^2) + ((a + c)*(a + b*x)^(3/2)*Sqrt[c + b*x])/(2*b^2) - (2*(a + b*x)^ (3/2)*(c + b*x)^(3/2))/(3*b^2) - ((a - c)^2*(a + c)*ArcTanh[Sqrt[a + b*x]/ Sqrt[c + b*x]])/(4*b^2))/(a - c)^2
3.5.7.3.1 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]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.04 (sec) , antiderivative size = 431, normalized size of antiderivative = 2.61
| method | result | size |
| default | \(\frac {x^{2} a}{2 \left (a -c \right )^{2}}+\frac {x^{2} c}{2 \left (a -c \right )^{2}}+\frac {2 b \,x^{3}}{3 \left (a -c \right )^{2}}-\frac {\sqrt {b x +a}\, \sqrt {b x +c}\, \left (16 \,\operatorname {csgn}\left (b \right ) x^{2} b^{2} \sqrt {b^{2} x^{2}+a b x +b c x +a c}+4 \,\operatorname {csgn}\left (b \right ) \sqrt {b^{2} x^{2}+a b x +b c x +a c}\, x a b +4 \,\operatorname {csgn}\left (b \right ) \sqrt {b^{2} x^{2}+a b x +b c x +a c}\, x b c -6 \,\operatorname {csgn}\left (b \right ) \sqrt {b^{2} x^{2}+a b x +b c x +a c}\, a^{2}+4 \,\operatorname {csgn}\left (b \right ) \sqrt {b^{2} x^{2}+a b x +b c x +a c}\, a c -6 \,\operatorname {csgn}\left (b \right ) \sqrt {b^{2} x^{2}+a b x +b c x +a c}\, c^{2}+3 \ln \left (\frac {\left (2 \sqrt {b^{2} x^{2}+a b x +b c x +a c}\, \operatorname {csgn}\left (b \right )+2 b x +a +c \right ) \operatorname {csgn}\left (b \right )}{2}\right ) a^{3}-3 \ln \left (\frac {\left (2 \sqrt {b^{2} x^{2}+a b x +b c x +a c}\, \operatorname {csgn}\left (b \right )+2 b x +a +c \right ) \operatorname {csgn}\left (b \right )}{2}\right ) a^{2} c -3 \ln \left (\frac {\left (2 \sqrt {b^{2} x^{2}+a b x +b c x +a c}\, \operatorname {csgn}\left (b \right )+2 b x +a +c \right ) \operatorname {csgn}\left (b \right )}{2}\right ) a \,c^{2}+3 \ln \left (\frac {\left (2 \sqrt {b^{2} x^{2}+a b x +b c x +a c}\, \operatorname {csgn}\left (b \right )+2 b x +a +c \right ) \operatorname {csgn}\left (b \right )}{2}\right ) c^{3}\right ) \operatorname {csgn}\left (b \right )}{24 \left (a -c \right )^{2} b^{2} \sqrt {b^{2} x^{2}+a b x +b c x +a c}}\) | \(431\) |
1/2*x^2/(a-c)^2*a+1/2*x^2/(a-c)^2*c+2/3*b*x^3/(a-c)^2-1/24/(a-c)^2*(b*x+a) ^(1/2)*(b*x+c)^(1/2)*(16*csgn(b)*x^2*b^2*(b^2*x^2+a*b*x+b*c*x+a*c)^(1/2)+4 *csgn(b)*(b^2*x^2+a*b*x+b*c*x+a*c)^(1/2)*x*a*b+4*csgn(b)*(b^2*x^2+a*b*x+b* c*x+a*c)^(1/2)*x*b*c-6*csgn(b)*(b^2*x^2+a*b*x+b*c*x+a*c)^(1/2)*a^2+4*csgn( b)*(b^2*x^2+a*b*x+b*c*x+a*c)^(1/2)*a*c-6*csgn(b)*(b^2*x^2+a*b*x+b*c*x+a*c) ^(1/2)*c^2+3*ln(1/2*(2*(b^2*x^2+a*b*x+b*c*x+a*c)^(1/2)*csgn(b)+2*b*x+a+c)* csgn(b))*a^3-3*ln(1/2*(2*(b^2*x^2+a*b*x+b*c*x+a*c)^(1/2)*csgn(b)+2*b*x+a+c )*csgn(b))*a^2*c-3*ln(1/2*(2*(b^2*x^2+a*b*x+b*c*x+a*c)^(1/2)*csgn(b)+2*b*x +a+c)*csgn(b))*a*c^2+3*ln(1/2*(2*(b^2*x^2+a*b*x+b*c*x+a*c)^(1/2)*csgn(b)+2 *b*x+a+c)*csgn(b))*c^3)*csgn(b)/b^2/(b^2*x^2+a*b*x+b*c*x+a*c)^(1/2)
Time = 0.32 (sec) , antiderivative size = 149, normalized size of antiderivative = 0.90 \[ \int \frac {x}{\left (\sqrt {a+b x}+\sqrt {c+b x}\right )^2} \, dx=\frac {16 \, b^{3} x^{3} + 12 \, {\left (a b^{2} + b^{2} c\right )} x^{2} - 2 \, {\left (8 \, b^{2} x^{2} - 3 \, a^{2} + 2 \, a c - 3 \, c^{2} + 2 \, {\left (a b + b c\right )} x\right )} \sqrt {b x + a} \sqrt {b x + c} + 3 \, {\left (a^{3} - a^{2} c - a c^{2} + c^{3}\right )} \log \left (-2 \, b x + 2 \, \sqrt {b x + a} \sqrt {b x + c} - a - c\right )}{24 \, {\left (a^{2} b^{2} - 2 \, a b^{2} c + b^{2} c^{2}\right )}} \]
1/24*(16*b^3*x^3 + 12*(a*b^2 + b^2*c)*x^2 - 2*(8*b^2*x^2 - 3*a^2 + 2*a*c - 3*c^2 + 2*(a*b + b*c)*x)*sqrt(b*x + a)*sqrt(b*x + c) + 3*(a^3 - a^2*c - a *c^2 + c^3)*log(-2*b*x + 2*sqrt(b*x + a)*sqrt(b*x + c) - a - c))/(a^2*b^2 - 2*a*b^2*c + b^2*c^2)
\[ \int \frac {x}{\left (\sqrt {a+b x}+\sqrt {c+b x}\right )^2} \, dx=\int \frac {x}{\left (\sqrt {a + b x} + \sqrt {b x + c}\right )^{2}}\, dx \]
\[ \int \frac {x}{\left (\sqrt {a+b x}+\sqrt {c+b x}\right )^2} \, dx=\int { \frac {x}{{\left (\sqrt {b x + a} + \sqrt {b x + c}\right )}^{2}} \,d x } \]
Leaf count of result is larger than twice the leaf count of optimal. 445 vs. \(2 (137) = 274\).
Time = 0.34 (sec) , antiderivative size = 445, normalized size of antiderivative = 2.70 \[ \int \frac {x}{\left (\sqrt {a+b x}+\sqrt {c+b x}\right )^2} \, dx=-\frac {{\left (2 \, {\left (b x + a\right )} {\left (\frac {4 \, {\left (a^{3} b^{2} - 3 \, a^{2} b^{2} c + 3 \, a b^{2} c^{2} - b^{2} c^{3}\right )} {\left (b x + a\right )}}{a^{5} b^{3} - 5 \, a^{4} b^{3} c + 10 \, a^{3} b^{3} c^{2} - 10 \, a^{2} b^{3} c^{3} + 5 \, a b^{3} c^{4} - b^{3} c^{5}} - \frac {7 \, a^{4} b^{2} - 22 \, a^{3} b^{2} c + 24 \, a^{2} b^{2} c^{2} - 10 \, a b^{2} c^{3} + b^{2} c^{4}}{a^{5} b^{3} - 5 \, a^{4} b^{3} c + 10 \, a^{3} b^{3} c^{2} - 10 \, a^{2} b^{3} c^{3} + 5 \, a b^{3} c^{4} - b^{3} c^{5}}\right )} + \frac {3 \, {\left (a^{5} b^{2} - 3 \, a^{4} b^{2} c + 2 \, a^{3} b^{2} c^{2} + 2 \, a^{2} b^{2} c^{3} - 3 \, a b^{2} c^{4} + b^{2} c^{5}\right )}}{a^{5} b^{3} - 5 \, a^{4} b^{3} c + 10 \, a^{3} b^{3} c^{2} - 10 \, a^{2} b^{3} c^{3} + 5 \, a b^{3} c^{4} - b^{3} c^{5}}\right )} \sqrt {b x + a} \sqrt {b x + c} - \frac {3 \, {\left (a + c\right )} \log \left ({\left | -\sqrt {b x + a} + \sqrt {b x + c} \right |}\right )}{b} - \frac {2 \, {\left (4 \, {\left (b x + a\right )}^{3} - 9 \, {\left (b x + a\right )}^{2} a + 6 \, {\left (b x + a\right )} a^{2} + 3 \, {\left (b x + a\right )}^{2} c - 6 \, {\left (b x + a\right )} a c\right )}}{a^{2} b - 2 \, a b c + b c^{2}}}{12 \, b} \]
-1/12*((2*(b*x + a)*(4*(a^3*b^2 - 3*a^2*b^2*c + 3*a*b^2*c^2 - b^2*c^3)*(b* x + a)/(a^5*b^3 - 5*a^4*b^3*c + 10*a^3*b^3*c^2 - 10*a^2*b^3*c^3 + 5*a*b^3* c^4 - b^3*c^5) - (7*a^4*b^2 - 22*a^3*b^2*c + 24*a^2*b^2*c^2 - 10*a*b^2*c^3 + b^2*c^4)/(a^5*b^3 - 5*a^4*b^3*c + 10*a^3*b^3*c^2 - 10*a^2*b^3*c^3 + 5*a *b^3*c^4 - b^3*c^5)) + 3*(a^5*b^2 - 3*a^4*b^2*c + 2*a^3*b^2*c^2 + 2*a^2*b^ 2*c^3 - 3*a*b^2*c^4 + b^2*c^5)/(a^5*b^3 - 5*a^4*b^3*c + 10*a^3*b^3*c^2 - 1 0*a^2*b^3*c^3 + 5*a*b^3*c^4 - b^3*c^5))*sqrt(b*x + a)*sqrt(b*x + c) - 3*(a + c)*log(abs(-sqrt(b*x + a) + sqrt(b*x + c)))/b - 2*(4*(b*x + a)^3 - 9*(b *x + a)^2*a + 6*(b*x + a)*a^2 + 3*(b*x + a)^2*c - 6*(b*x + a)*a*c)/(a^2*b - 2*a*b*c + b*c^2))/b
Time = 75.36 (sec) , antiderivative size = 1012, normalized size of antiderivative = 6.13 \[ \int \frac {x}{\left (\sqrt {a+b x}+\sqrt {c+b x}\right )^2} \, dx=\frac {\frac {\left (\sqrt {a+b\,x}-\sqrt {a}\right )\,\left (\frac {a}{2}+\frac {c}{2}\right )}{b^2\,\left (\sqrt {c+b\,x}-\sqrt {c}\right )}+\frac {{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^{11}\,\left (\frac {a}{2}+\frac {c}{2}\right )}{b^2\,{\left (\sqrt {c+b\,x}-\sqrt {c}\right )}^{11}}-\frac {{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^3\,\left (\frac {17\,a^3}{6}+\frac {101\,a^2\,c}{2}+\frac {101\,a\,c^2}{2}+\frac {17\,c^3}{6}\right )}{{\left (\sqrt {c+b\,x}-\sqrt {c}\right )}^3\,\left (a^2\,b^2-2\,a\,b^2\,c+b^2\,c^2\right )}-\frac {{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^9\,\left (\frac {17\,a^3}{6}+\frac {101\,a^2\,c}{2}+\frac {101\,a\,c^2}{2}+\frac {17\,c^3}{6}\right )}{{\left (\sqrt {c+b\,x}-\sqrt {c}\right )}^9\,\left (a^2\,b^2-2\,a\,b^2\,c+b^2\,c^2\right )}-\frac {{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^5\,\left (19\,a^3+269\,a^2\,c+269\,a\,c^2+19\,c^3\right )}{{\left (\sqrt {c+b\,x}-\sqrt {c}\right )}^5\,\left (a^2\,b^2-2\,a\,b^2\,c+b^2\,c^2\right )}-\frac {{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^7\,\left (19\,a^3+269\,a^2\,c+269\,a\,c^2+19\,c^3\right )}{{\left (\sqrt {c+b\,x}-\sqrt {c}\right )}^7\,\left (a^2\,b^2-2\,a\,b^2\,c+b^2\,c^2\right )}+\frac {16\,a^{3/2}\,c^{3/2}\,{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^2}{{\left (\sqrt {c+b\,x}-\sqrt {c}\right )}^2\,\left (a^2\,b^2-2\,a\,b^2\,c+b^2\,c^2\right )}+\frac {16\,a^{3/2}\,c^{3/2}\,{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^{10}}{{\left (\sqrt {c+b\,x}-\sqrt {c}\right )}^{10}\,\left (a^2\,b^2-2\,a\,b^2\,c+b^2\,c^2\right )}+\frac {\sqrt {a}\,\sqrt {c}\,{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^4\,\left (64\,a^2+192\,a\,c+64\,c^2\right )}{{\left (\sqrt {c+b\,x}-\sqrt {c}\right )}^4\,\left (a^2\,b^2-2\,a\,b^2\,c+b^2\,c^2\right )}+\frac {\sqrt {a}\,\sqrt {c}\,{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^8\,\left (64\,a^2+192\,a\,c+64\,c^2\right )}{{\left (\sqrt {c+b\,x}-\sqrt {c}\right )}^8\,\left (a^2\,b^2-2\,a\,b^2\,c+b^2\,c^2\right )}+\frac {\sqrt {a}\,\sqrt {c}\,{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^6\,\left (128\,a^2+\frac {1312\,a\,c}{3}+128\,c^2\right )}{{\left (\sqrt {c+b\,x}-\sqrt {c}\right )}^6\,\left (a^2\,b^2-2\,a\,b^2\,c+b^2\,c^2\right )}}{\frac {15\,{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^4}{{\left (\sqrt {c+b\,x}-\sqrt {c}\right )}^4}-\frac {6\,{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^2}{{\left (\sqrt {c+b\,x}-\sqrt {c}\right )}^2}-\frac {20\,{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^6}{{\left (\sqrt {c+b\,x}-\sqrt {c}\right )}^6}+\frac {15\,{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^8}{{\left (\sqrt {c+b\,x}-\sqrt {c}\right )}^8}-\frac {6\,{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^{10}}{{\left (\sqrt {c+b\,x}-\sqrt {c}\right )}^{10}}+\frac {{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^{12}}{{\left (\sqrt {c+b\,x}-\sqrt {c}\right )}^{12}}+1}-\frac {\mathrm {atanh}\left (\frac {\sqrt {a+b\,x}-\sqrt {a}}{\sqrt {c+b\,x}-\sqrt {c}}\right )\,\left (a+c\right )}{2\,b^2}+\frac {x^2\,\left (a+c\right )}{2\,{\left (a-c\right )}^2}+\frac {2\,b\,x^3}{3\,{\left (a-c\right )}^2} \]
((((a + b*x)^(1/2) - a^(1/2))*(a/2 + c/2))/(b^2*((c + b*x)^(1/2) - c^(1/2) )) + (((a + b*x)^(1/2) - a^(1/2))^11*(a/2 + c/2))/(b^2*((c + b*x)^(1/2) - c^(1/2))^11) - (((a + b*x)^(1/2) - a^(1/2))^3*((101*a*c^2)/2 + (101*a^2*c) /2 + (17*a^3)/6 + (17*c^3)/6))/(((c + b*x)^(1/2) - c^(1/2))^3*(a^2*b^2 + b ^2*c^2 - 2*a*b^2*c)) - (((a + b*x)^(1/2) - a^(1/2))^9*((101*a*c^2)/2 + (10 1*a^2*c)/2 + (17*a^3)/6 + (17*c^3)/6))/(((c + b*x)^(1/2) - c^(1/2))^9*(a^2 *b^2 + b^2*c^2 - 2*a*b^2*c)) - (((a + b*x)^(1/2) - a^(1/2))^5*(269*a*c^2 + 269*a^2*c + 19*a^3 + 19*c^3))/(((c + b*x)^(1/2) - c^(1/2))^5*(a^2*b^2 + b ^2*c^2 - 2*a*b^2*c)) - (((a + b*x)^(1/2) - a^(1/2))^7*(269*a*c^2 + 269*a^2 *c + 19*a^3 + 19*c^3))/(((c + b*x)^(1/2) - c^(1/2))^7*(a^2*b^2 + b^2*c^2 - 2*a*b^2*c)) + (16*a^(3/2)*c^(3/2)*((a + b*x)^(1/2) - a^(1/2))^2)/(((c + b *x)^(1/2) - c^(1/2))^2*(a^2*b^2 + b^2*c^2 - 2*a*b^2*c)) + (16*a^(3/2)*c^(3 /2)*((a + b*x)^(1/2) - a^(1/2))^10)/(((c + b*x)^(1/2) - c^(1/2))^10*(a^2*b ^2 + b^2*c^2 - 2*a*b^2*c)) + (a^(1/2)*c^(1/2)*((a + b*x)^(1/2) - a^(1/2))^ 4*(192*a*c + 64*a^2 + 64*c^2))/(((c + b*x)^(1/2) - c^(1/2))^4*(a^2*b^2 + b ^2*c^2 - 2*a*b^2*c)) + (a^(1/2)*c^(1/2)*((a + b*x)^(1/2) - a^(1/2))^8*(192 *a*c + 64*a^2 + 64*c^2))/(((c + b*x)^(1/2) - c^(1/2))^8*(a^2*b^2 + b^2*c^2 - 2*a*b^2*c)) + (a^(1/2)*c^(1/2)*((a + b*x)^(1/2) - a^(1/2))^6*((1312*a*c )/3 + 128*a^2 + 128*c^2))/(((c + b*x)^(1/2) - c^(1/2))^6*(a^2*b^2 + b^2*c^ 2 - 2*a*b^2*c)))/((15*((a + b*x)^(1/2) - a^(1/2))^4)/((c + b*x)^(1/2) -...