Integrand size = 25, antiderivative size = 174 \[ \int \frac {1}{x^2 \left (\sqrt {a+b x}+\sqrt {a+c x}\right )^2} \, dx=-\frac {2 a}{3 (b-c)^2 x^3}-\frac {b+c}{2 (b-c)^2 x^2}-\frac {(b+c) \sqrt {a+b x} \sqrt {a+c x}}{4 a^2 (b-c) x}-\frac {(b+c) \sqrt {a+b x} (a+c x)^{3/2}}{2 a^2 (b-c)^2 x^2}+\frac {2 (a+b x)^{3/2} (a+c x)^{3/2}}{3 a^2 (b-c)^2 x^3}+\frac {(b+c) \text {arctanh}\left (\frac {\sqrt {a+b x}}{\sqrt {a+c x}}\right )}{4 a^2} \]
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Time = 0.17 (sec) , antiderivative size = 174, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {6822, 98, 96, 95, 214} \[ \int \frac {1}{x^2 \left (\sqrt {a+b x}+\sqrt {a+c x}\right )^2} \, dx=\frac {(b+c) \text {arctanh}\left (\frac {\sqrt {a+b x}}{\sqrt {a+c x}}\right )}{4 a^2}+\frac {2 (a+b x)^{3/2} (a+c x)^{3/2}}{3 a^2 x^3 (b-c)^2}-\frac {(b+c) \sqrt {a+b x} (a+c x)^{3/2}}{2 a^2 x^2 (b-c)^2}-\frac {(b+c) \sqrt {a+b x} \sqrt {a+c x}}{4 a^2 x (b-c)}-\frac {2 a}{3 x^3 (b-c)^2}-\frac {b+c}{2 x^2 (b-c)^2} \]
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Rule 95
Rule 96
Rule 98
Rule 214
Rule 6822
Rubi steps \begin{align*} \text {integral}& = \frac {\int \left (\frac {2 a}{x^4}+\frac {b \left (1+\frac {c}{b}\right )}{x^3}-\frac {2 \sqrt {a+b x} \sqrt {a+c x}}{x^4}\right ) \, dx}{(b-c)^2} \\ & = -\frac {2 a}{3 (b-c)^2 x^3}-\frac {b+c}{2 (b-c)^2 x^2}-\frac {2 \int \frac {\sqrt {a+b x} \sqrt {a+c x}}{x^4} \, dx}{(b-c)^2} \\ & = -\frac {2 a}{3 (b-c)^2 x^3}-\frac {b+c}{2 (b-c)^2 x^2}+\frac {2 (a+b x)^{3/2} (a+c x)^{3/2}}{3 a^2 (b-c)^2 x^3}+\frac {(b+c) \int \frac {\sqrt {a+b x} \sqrt {a+c x}}{x^3} \, dx}{a (b-c)^2} \\ & = -\frac {2 a}{3 (b-c)^2 x^3}-\frac {b+c}{2 (b-c)^2 x^2}-\frac {(b+c) \sqrt {a+b x} (a+c x)^{3/2}}{2 a^2 (b-c)^2 x^2}+\frac {2 (a+b x)^{3/2} (a+c x)^{3/2}}{3 a^2 (b-c)^2 x^3}+\frac {(b+c) \int \frac {\sqrt {a+c x}}{x^2 \sqrt {a+b x}} \, dx}{4 a (b-c)} \\ & = -\frac {2 a}{3 (b-c)^2 x^3}-\frac {b+c}{2 (b-c)^2 x^2}-\frac {(b+c) \sqrt {a+b x} \sqrt {a+c x}}{4 a^2 (b-c) x}-\frac {(b+c) \sqrt {a+b x} (a+c x)^{3/2}}{2 a^2 (b-c)^2 x^2}+\frac {2 (a+b x)^{3/2} (a+c x)^{3/2}}{3 a^2 (b-c)^2 x^3}-\frac {(b+c) \int \frac {1}{x \sqrt {a+b x} \sqrt {a+c x}} \, dx}{8 a} \\ & = -\frac {2 a}{3 (b-c)^2 x^3}-\frac {b+c}{2 (b-c)^2 x^2}-\frac {(b+c) \sqrt {a+b x} \sqrt {a+c x}}{4 a^2 (b-c) x}-\frac {(b+c) \sqrt {a+b x} (a+c x)^{3/2}}{2 a^2 (b-c)^2 x^2}+\frac {2 (a+b x)^{3/2} (a+c x)^{3/2}}{3 a^2 (b-c)^2 x^3}-\frac {(b+c) \text {Subst}\left (\int \frac {1}{-a+a x^2} \, dx,x,\frac {\sqrt {a+b x}}{\sqrt {a+c x}}\right )}{4 a} \\ & = -\frac {2 a}{3 (b-c)^2 x^3}-\frac {b+c}{2 (b-c)^2 x^2}-\frac {(b+c) \sqrt {a+b x} \sqrt {a+c x}}{4 a^2 (b-c) x}-\frac {(b+c) \sqrt {a+b x} (a+c x)^{3/2}}{2 a^2 (b-c)^2 x^2}+\frac {2 (a+b x)^{3/2} (a+c x)^{3/2}}{3 a^2 (b-c)^2 x^3}+\frac {(b+c) \tanh ^{-1}\left (\frac {\sqrt {a+b x}}{\sqrt {a+c x}}\right )}{4 a^2} \\ \end{align*}
Time = 10.18 (sec) , antiderivative size = 153, normalized size of antiderivative = 0.88 \[ \int \frac {1}{x^2 \left (\sqrt {a+b x}+\sqrt {a+c x}\right )^2} \, dx=\frac {-8 a^3+2 a (b+c) x \sqrt {a+b x} \sqrt {a+c x}+\left (-3 b^2+2 b c-3 c^2\right ) x^2 \sqrt {a+b x} \sqrt {a+c x}+a^2 \left (-6 b x-6 c x+8 \sqrt {a+b x} \sqrt {a+c x}\right )+3 (b-c)^2 (b+c) x^3 \text {arctanh}\left (\frac {\sqrt {a+b x}}{\sqrt {a+c x}}\right )}{12 a^2 (b-c)^2 x^3} \]
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Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.04 (sec) , antiderivative size = 457, normalized size of antiderivative = 2.63
method | result | size |
default | \(-\frac {b}{2 x^{2} \left (b -c \right )^{2}}-\frac {c}{2 x^{2} \left (b -c \right )^{2}}-\frac {2 a}{3 \left (b -c \right )^{2} x^{3}}-\frac {\sqrt {b x +a}\, \sqrt {c x +a}\, \left (-3 \ln \left (\frac {a \left (2 \,\operatorname {csgn}\left (a \right ) \sqrt {b c \,x^{2}+a b x +a c x +a^{2}}+b x +c x +2 a \right )}{x}\right ) x^{3} b^{3}+3 \ln \left (\frac {a \left (2 \,\operatorname {csgn}\left (a \right ) \sqrt {b c \,x^{2}+a b x +a c x +a^{2}}+b x +c x +2 a \right )}{x}\right ) x^{3} b^{2} c +3 \ln \left (\frac {a \left (2 \,\operatorname {csgn}\left (a \right ) \sqrt {b c \,x^{2}+a b x +a c x +a^{2}}+b x +c x +2 a \right )}{x}\right ) x^{3} b \,c^{2}-3 \ln \left (\frac {a \left (2 \,\operatorname {csgn}\left (a \right ) \sqrt {b c \,x^{2}+a b x +a c x +a^{2}}+b x +c x +2 a \right )}{x}\right ) x^{3} c^{3}+6 \,\operatorname {csgn}\left (a \right ) \sqrt {b c \,x^{2}+a b x +a c x +a^{2}}\, x^{2} b^{2}-4 \,\operatorname {csgn}\left (a \right ) \sqrt {b c \,x^{2}+a b x +a c x +a^{2}}\, x^{2} b c +6 \,\operatorname {csgn}\left (a \right ) \sqrt {b c \,x^{2}+a b x +a c x +a^{2}}\, x^{2} c^{2}-4 \,\operatorname {csgn}\left (a \right ) a \sqrt {b c \,x^{2}+a b x +a c x +a^{2}}\, x b -4 \,\operatorname {csgn}\left (a \right ) a \sqrt {b c \,x^{2}+a b x +a c x +a^{2}}\, x c -16 \sqrt {b c \,x^{2}+a b x +a c x +a^{2}}\, a^{2} \operatorname {csgn}\left (a \right )\right ) \operatorname {csgn}\left (a \right )}{24 \left (b -c \right )^{2} a^{2} \sqrt {b c \,x^{2}+a b x +a c x +a^{2}}\, x^{3}}\) | \(457\) |
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Time = 0.34 (sec) , antiderivative size = 182, normalized size of antiderivative = 1.05 \[ \int \frac {1}{x^2 \left (\sqrt {a+b x}+\sqrt {a+c x}\right )^2} \, dx=-\frac {12 \, {\left (b^{3} - b^{2} c - b c^{2} + c^{3}\right )} x^{3} \log \left (-\frac {{\left (b + c\right )} x - 2 \, \sqrt {b x + a} \sqrt {c x + a} + 2 \, a}{x}\right ) + {\left (5 \, b^{3} + 3 \, b^{2} c + 3 \, b c^{2} + 5 \, c^{3}\right )} x^{3} + 64 \, a^{3} + 8 \, {\left ({\left (3 \, b^{2} - 2 \, b c + 3 \, c^{2}\right )} x^{2} - 8 \, a^{2} - 2 \, {\left (a b + a c\right )} x\right )} \sqrt {b x + a} \sqrt {c x + a} + 48 \, {\left (a^{2} b + a^{2} c\right )} x}{96 \, {\left (a^{2} b^{2} - 2 \, a^{2} b c + a^{2} c^{2}\right )} x^{3}} \]
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\[ \int \frac {1}{x^2 \left (\sqrt {a+b x}+\sqrt {a+c x}\right )^2} \, dx=\int \frac {1}{x^{2} \left (\sqrt {a + b x} + \sqrt {a + c x}\right )^{2}}\, dx \]
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\[ \int \frac {1}{x^2 \left (\sqrt {a+b x}+\sqrt {a+c x}\right )^2} \, dx=\int { \frac {1}{x^{2} {\left (\sqrt {b x + a} + \sqrt {c x + a}\right )}^{2}} \,d x } \]
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Leaf count of result is larger than twice the leaf count of optimal. 802 vs. \(2 (146) = 292\).
Time = 2.46 (sec) , antiderivative size = 802, normalized size of antiderivative = 4.61 \[ \int \frac {1}{x^2 \left (\sqrt {a+b x}+\sqrt {a+c x}\right )^2} \, dx=\frac {\sqrt {b c} {\left (b + c\right )} {\left | b \right |} \arctan \left (-\frac {a b^{2} + a b c - {\left (\sqrt {b c} \sqrt {b x + a} - \sqrt {a b^{2} + {\left (b x + a\right )} b c - a b c}\right )}^{2}}{2 \, \sqrt {-b c} a b}\right )}{4 \, \sqrt {-b c} a^{2} b} + \frac {3 \, {\left (b^{3} - b^{2} c - b c^{2} + c^{3}\right )} \sqrt {b c} {\left (\sqrt {b c} \sqrt {b x + a} - \sqrt {a b^{2} + {\left (b x + a\right )} b c - a b c}\right )}^{10} {\left | b \right |} - 3 \, {\left (5 \, b^{5} + 22 \, b^{3} c^{2} + 5 \, b c^{4}\right )} \sqrt {b c} {\left (\sqrt {b c} \sqrt {b x + a} - \sqrt {a b^{2} + {\left (b x + a\right )} b c - a b c}\right )}^{8} a {\left | b \right |} + 2 \, {\left (15 \, b^{7} - b^{6} c + 18 \, b^{5} c^{2} + 18 \, b^{4} c^{3} - b^{3} c^{4} + 15 \, b^{2} c^{5}\right )} \sqrt {b c} {\left (\sqrt {b c} \sqrt {b x + a} - \sqrt {a b^{2} + {\left (b x + a\right )} b c - a b c}\right )}^{6} a^{2} {\left | b \right |} - 6 \, {\left (5 \, b^{9} - 6 \, b^{8} c - 5 \, b^{7} c^{2} + 12 \, b^{6} c^{3} - 5 \, b^{5} c^{4} - 6 \, b^{4} c^{5} + 5 \, b^{3} c^{6}\right )} \sqrt {b c} {\left (\sqrt {b c} \sqrt {b x + a} - \sqrt {a b^{2} + {\left (b x + a\right )} b c - a b c}\right )}^{4} a^{3} {\left | b \right |} + 3 \, {\left (5 \, b^{11} - 17 \, b^{10} c + 21 \, b^{9} c^{2} - 9 \, b^{8} c^{3} - 9 \, b^{7} c^{4} + 21 \, b^{6} c^{5} - 17 \, b^{5} c^{6} + 5 \, b^{4} c^{7}\right )} \sqrt {b c} {\left (\sqrt {b c} \sqrt {b x + a} - \sqrt {a b^{2} + {\left (b x + a\right )} b c - a b c}\right )}^{2} a^{4} {\left | b \right |} - {\left (3 \, b^{13} - 20 \, b^{12} c + 60 \, b^{11} c^{2} - 108 \, b^{10} c^{3} + 130 \, b^{9} c^{4} - 108 \, b^{8} c^{5} + 60 \, b^{7} c^{6} - 20 \, b^{6} c^{7} + 3 \, b^{5} c^{8}\right )} \sqrt {b c} a^{5} {\left | b \right |}}{6 \, {\left ({\left (\sqrt {b c} \sqrt {b x + a} - \sqrt {a b^{2} + {\left (b x + a\right )} b c - a b c}\right )}^{4} - 2 \, {\left (b^{2} + b c\right )} {\left (\sqrt {b c} \sqrt {b x + a} - \sqrt {a b^{2} + {\left (b x + a\right )} b c - a b c}\right )}^{2} a + {\left (b^{4} - 2 \, b^{3} c + b^{2} c^{2}\right )} a^{2}\right )}^{3} {\left (b^{2} - 2 \, b c + c^{2}\right )} a} - \frac {3 \, {\left (b x + a\right )} b^{3} + a b^{3} + 3 \, {\left (b x + a\right )} b^{2} c - 3 \, a b^{2} c}{6 \, {\left (b^{2} - 2 \, b c + c^{2}\right )} b^{3} x^{3}} \]
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Time = 48.12 (sec) , antiderivative size = 1290, normalized size of antiderivative = 7.41 \[ \int \frac {1}{x^2 \left (\sqrt {a+b x}+\sqrt {a+c x}\right )^2} \, dx=\text {Too large to display} \]
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