Integrand size = 21, antiderivative size = 64 \[ \int \frac {1}{\left (\sqrt {a+b x}+\sqrt {c+b x}\right )^3} \, dx=\frac {(a-c)^2}{10 b \left (\sqrt {a+b x}+\sqrt {c+b x}\right )^5}-\frac {1}{2 b \left (\sqrt {a+b x}+\sqrt {c+b x}\right )} \]
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Leaf count is larger than twice the leaf count of optimal. \(151\) vs. \(2(64)=128\).
Time = 0.07 (sec) , antiderivative size = 151, normalized size of antiderivative = 2.36, number of steps used = 6, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {6821, 45} \[ \int \frac {1}{\left (\sqrt {a+b x}+\sqrt {c+b x}\right )^3} \, dx=\frac {8 (a+b x)^{5/2}}{5 b (a-c)^3}+\frac {2 (a+3 c) (a+b x)^{3/2}}{3 b (a-c)^3}-\frac {8 a (a+b x)^{3/2}}{3 b (a-c)^3}-\frac {8 (b x+c)^{5/2}}{5 b (a-c)^3}+\frac {8 c (b x+c)^{3/2}}{3 b (a-c)^3}-\frac {2 (3 a+c) (b x+c)^{3/2}}{3 b (a-c)^3} \]
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Rule 45
Rule 6821
Rubi steps \begin{align*} \text {integral}& = \frac {\int \left (a \left (1+\frac {3 c}{a}\right ) \sqrt {a+b x}+4 b x \sqrt {a+b x}-3 a \left (1+\frac {c}{3 a}\right ) \sqrt {c+b x}-4 b x \sqrt {c+b x}\right ) \, dx}{(a-c)^3} \\ & = \frac {2 (a+3 c) (a+b x)^{3/2}}{3 b (a-c)^3}-\frac {2 (3 a+c) (c+b x)^{3/2}}{3 b (a-c)^3}+\frac {(4 b) \int x \sqrt {a+b x} \, dx}{(a-c)^3}-\frac {(4 b) \int x \sqrt {c+b x} \, dx}{(a-c)^3} \\ & = \frac {2 (a+3 c) (a+b x)^{3/2}}{3 b (a-c)^3}-\frac {2 (3 a+c) (c+b x)^{3/2}}{3 b (a-c)^3}+\frac {(4 b) \int \left (-\frac {a \sqrt {a+b x}}{b}+\frac {(a+b x)^{3/2}}{b}\right ) \, dx}{(a-c)^3}-\frac {(4 b) \int \left (-\frac {c \sqrt {c+b x}}{b}+\frac {(c+b x)^{3/2}}{b}\right ) \, dx}{(a-c)^3} \\ & = -\frac {8 a (a+b x)^{3/2}}{3 b (a-c)^3}+\frac {2 (a+3 c) (a+b x)^{3/2}}{3 b (a-c)^3}+\frac {8 (a+b x)^{5/2}}{5 b (a-c)^3}+\frac {8 c (c+b x)^{3/2}}{3 b (a-c)^3}-\frac {2 (3 a+c) (c+b x)^{3/2}}{3 b (a-c)^3}-\frac {8 (c+b x)^{5/2}}{5 b (a-c)^3} \\ \end{align*}
Time = 0.48 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.86 \[ \int \frac {1}{\left (\sqrt {a+b x}+\sqrt {c+b x}\right )^3} \, dx=\frac {2 \left ((-5 a+c-4 b x) (c+b x)^{3/2}+(a+b x)^{3/2} (-a+5 c+4 b x)\right )}{5 b (a-c)^3} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(145\) vs. \(2(52)=104\).
Time = 0.06 (sec) , antiderivative size = 146, normalized size of antiderivative = 2.28
method | result | size |
default | \(\frac {2 a \left (b x +a \right )^{\frac {3}{2}}}{3 \left (a -c \right )^{3} b}+\frac {2 c \left (b x +a \right )^{\frac {3}{2}}}{\left (a -c \right )^{3} b}-\frac {2 a \left (b x +c \right )^{\frac {3}{2}}}{\left (a -c \right )^{3} b}-\frac {2 c \left (b x +c \right )^{\frac {3}{2}}}{3 \left (a -c \right )^{3} b}+\frac {\frac {8 \left (b x +a \right )^{\frac {5}{2}}}{5}-\frac {8 \left (b x +a \right )^{\frac {3}{2}} a}{3}}{\left (a -c \right )^{3} b}-\frac {8 \left (\frac {\left (b x +c \right )^{\frac {5}{2}}}{5}-\frac {c \left (b x +c \right )^{\frac {3}{2}}}{3}\right )}{\left (a -c \right )^{3} b}\) | \(146\) |
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Leaf count of result is larger than twice the leaf count of optimal. 106 vs. \(2 (52) = 104\).
Time = 0.29 (sec) , antiderivative size = 106, normalized size of antiderivative = 1.66 \[ \int \frac {1}{\left (\sqrt {a+b x}+\sqrt {c+b x}\right )^3} \, dx=\frac {2 \, {\left ({\left (4 \, b^{2} x^{2} - a^{2} + 5 \, a c + {\left (3 \, a b + 5 \, b c\right )} x\right )} \sqrt {b x + a} - {\left (4 \, b^{2} x^{2} + 5 \, a c - c^{2} + {\left (5 \, a b + 3 \, b c\right )} x\right )} \sqrt {b x + c}\right )}}{5 \, {\left (a^{3} b - 3 \, a^{2} b c + 3 \, a b c^{2} - b c^{3}\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 384 vs. \(2 (48) = 96\).
Time = 0.73 (sec) , antiderivative size = 384, normalized size of antiderivative = 6.00 \[ \int \frac {1}{\left (\sqrt {a+b x}+\sqrt {c+b x}\right )^3} \, dx=\begin {cases} - \frac {2 a}{5 a b \sqrt {a + b x} + 15 a b \sqrt {b x + c} + 20 b^{2} x \sqrt {a + b x} + 20 b^{2} x \sqrt {b x + c} + 15 b c \sqrt {a + b x} + 5 b c \sqrt {b x + c}} - \frac {4 b x}{5 a b \sqrt {a + b x} + 15 a b \sqrt {b x + c} + 20 b^{2} x \sqrt {a + b x} + 20 b^{2} x \sqrt {b x + c} + 15 b c \sqrt {a + b x} + 5 b c \sqrt {b x + c}} - \frac {2 c}{5 a b \sqrt {a + b x} + 15 a b \sqrt {b x + c} + 20 b^{2} x \sqrt {a + b x} + 20 b^{2} x \sqrt {b x + c} + 15 b c \sqrt {a + b x} + 5 b c \sqrt {b x + c}} - \frac {6 \sqrt {a + b x} \sqrt {b x + c}}{5 a b \sqrt {a + b x} + 15 a b \sqrt {b x + c} + 20 b^{2} x \sqrt {a + b x} + 20 b^{2} x \sqrt {b x + c} + 15 b c \sqrt {a + b x} + 5 b c \sqrt {b x + c}} & \text {for}\: b \neq 0 \\\frac {x}{\left (\sqrt {a} + \sqrt {c}\right )^{3}} & \text {otherwise} \end {cases} \]
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\[ \int \frac {1}{\left (\sqrt {a+b x}+\sqrt {c+b x}\right )^3} \, dx=\int { \frac {1}{{\left (\sqrt {b x + a} + \sqrt {b x + c}\right )}^{3}} \,d x } \]
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Leaf count of result is larger than twice the leaf count of optimal. 427 vs. \(2 (52) = 104\).
Time = 0.35 (sec) , antiderivative size = 427, normalized size of antiderivative = 6.67 \[ \int \frac {1}{\left (\sqrt {a+b x}+\sqrt {c+b x}\right )^3} \, dx=-\frac {2}{5} \, {\left ({\left (b x + a\right )} {\left (\frac {4 \, {\left (a^{3} b^{2} - 3 \, a^{2} b^{2} c + 3 \, a b^{2} c^{2} - b^{2} c^{3}\right )} {\left (b x + a\right )}}{a^{6} b^{3} - 6 \, a^{5} b^{3} c + 15 \, a^{4} b^{3} c^{2} - 20 \, a^{3} b^{3} c^{3} + 15 \, a^{2} b^{3} c^{4} - 6 \, a b^{3} c^{5} + b^{3} c^{6}} - \frac {3 \, {\left (a^{4} b^{2} - 4 \, a^{3} b^{2} c + 6 \, a^{2} b^{2} c^{2} - 4 \, a b^{2} c^{3} + b^{2} c^{4}\right )}}{a^{6} b^{3} - 6 \, a^{5} b^{3} c + 15 \, a^{4} b^{3} c^{2} - 20 \, a^{3} b^{3} c^{3} + 15 \, a^{2} b^{3} c^{4} - 6 \, a b^{3} c^{5} + b^{3} c^{6}}\right )} - \frac {a^{5} b^{2} - 5 \, a^{4} b^{2} c + 10 \, a^{3} b^{2} c^{2} - 10 \, a^{2} b^{2} c^{3} + 5 \, a b^{2} c^{4} - b^{2} c^{5}}{a^{6} b^{3} - 6 \, a^{5} b^{3} c + 15 \, a^{4} b^{3} c^{2} - 20 \, a^{3} b^{3} c^{3} + 15 \, a^{2} b^{3} c^{4} - 6 \, a b^{3} c^{5} + b^{3} c^{6}}\right )} \sqrt {b x + c} + \frac {2 \, {\left (4 \, {\left (b x + a\right )}^{\frac {5}{2}} - 5 \, {\left (b x + a\right )}^{\frac {3}{2}} a + 5 \, {\left (b x + a\right )}^{\frac {3}{2}} c\right )}}{5 \, {\left (a^{3} b - 3 \, a^{2} b c + 3 \, a b c^{2} - b c^{3}\right )}} \]
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Time = 16.64 (sec) , antiderivative size = 252, normalized size of antiderivative = 3.94 \[ \int \frac {1}{\left (\sqrt {a+b x}+\sqrt {c+b x}\right )^3} \, dx=\frac {\left (\frac {2\,\left (a^2+3\,c\,a\right )}{{\left (a-c\right )}^3}+\frac {2\,a\,\left (\frac {32\,a\,b}{5\,{\left (a-c\right )}^3}-\frac {2\,b\,\left (5\,a+3\,c\right )}{{\left (a-c\right )}^3}\right )}{3\,b}\right )\,\sqrt {a+b\,x}}{b}-\frac {\left (\frac {2\,c\,\left (3\,a+c\right )}{{\left (a-c\right )}^3}+\frac {2\,c\,\left (\frac {32\,b\,c}{5\,{\left (a-c\right )}^3}-\frac {2\,b\,\left (3\,a+5\,c\right )}{{\left (a-c\right )}^3}\right )}{3\,b}\right )\,\sqrt {c+b\,x}}{b}+\frac {8\,b\,x^2\,\sqrt {a+b\,x}}{5\,{\left (a-c\right )}^3}-\frac {8\,b\,x^2\,\sqrt {c+b\,x}}{5\,{\left (a-c\right )}^3}-\frac {x\,\left (\frac {32\,a\,b}{5\,{\left (a-c\right )}^3}-\frac {2\,b\,\left (5\,a+3\,c\right )}{{\left (a-c\right )}^3}\right )\,\sqrt {a+b\,x}}{3\,b}+\frac {x\,\left (\frac {32\,b\,c}{5\,{\left (a-c\right )}^3}-\frac {2\,b\,\left (3\,a+5\,c\right )}{{\left (a-c\right )}^3}\right )\,\sqrt {c+b\,x}}{3\,b} \]
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