Integrand size = 23, antiderivative size = 100 \[ \int \frac {1}{(a+b x)^{7/2} (a c-b c x)^{7/2}} \, dx=\frac {x}{5 a^2 c (a+b x)^{5/2} (a c-b c x)^{5/2}}+\frac {4 x}{15 a^4 c^2 (a+b x)^{3/2} (a c-b c x)^{3/2}}+\frac {8 x}{15 a^6 c^3 \sqrt {a+b x} \sqrt {a c-b c x}} \]
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Time = 0.01 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {40, 39} \[ \int \frac {1}{(a+b x)^{7/2} (a c-b c x)^{7/2}} \, dx=\frac {8 x}{15 a^6 c^3 \sqrt {a+b x} \sqrt {a c-b c x}}+\frac {4 x}{15 a^4 c^2 (a+b x)^{3/2} (a c-b c x)^{3/2}}+\frac {x}{5 a^2 c (a+b x)^{5/2} (a c-b c x)^{5/2}} \]
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Rule 39
Rule 40
Rubi steps \begin{align*} \text {integral}& = \frac {x}{5 a^2 c (a+b x)^{5/2} (a c-b c x)^{5/2}}+\frac {4 \int \frac {1}{(a+b x)^{5/2} (a c-b c x)^{5/2}} \, dx}{5 a^2 c} \\ & = \frac {x}{5 a^2 c (a+b x)^{5/2} (a c-b c x)^{5/2}}+\frac {4 x}{15 a^4 c^2 (a+b x)^{3/2} (a c-b c x)^{3/2}}+\frac {8 \int \frac {1}{(a+b x)^{3/2} (a c-b c x)^{3/2}} \, dx}{15 a^4 c^2} \\ & = \frac {x}{5 a^2 c (a+b x)^{5/2} (a c-b c x)^{5/2}}+\frac {4 x}{15 a^4 c^2 (a+b x)^{3/2} (a c-b c x)^{3/2}}+\frac {8 x}{15 a^6 c^3 \sqrt {a+b x} \sqrt {a c-b c x}} \\ \end{align*}
Time = 0.13 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.57 \[ \int \frac {1}{(a+b x)^{7/2} (a c-b c x)^{7/2}} \, dx=\frac {15 a^4 x-20 a^2 b^2 x^3+8 b^4 x^5}{15 a^6 c (c (a-b x))^{5/2} (a+b x)^{5/2}} \]
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Time = 0.37 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.56
method | result | size |
gosper | \(\frac {\left (-b x +a \right ) x \left (8 b^{4} x^{4}-20 a^{2} b^{2} x^{2}+15 a^{4}\right )}{15 \left (b x +a \right )^{\frac {5}{2}} a^{6} \left (-b c x +a c \right )^{\frac {7}{2}}}\) | \(56\) |
default | \(-\frac {1}{5 b a c \left (b x +a \right )^{\frac {5}{2}} \left (-b c x +a c \right )^{\frac {5}{2}}}+\frac {-\frac {1}{3 b a c \left (b x +a \right )^{\frac {3}{2}} \left (-b c x +a c \right )^{\frac {5}{2}}}+\frac {-\frac {4}{3 b a c \sqrt {b x +a}\, \left (-b c x +a c \right )^{\frac {5}{2}}}+\frac {4 \left (\frac {3 \sqrt {b x +a}}{5 b a c \left (-b c x +a c \right )^{\frac {5}{2}}}+\frac {3 \left (\frac {2 \sqrt {b x +a}}{15 b a c \left (-b c x +a c \right )^{\frac {3}{2}}}+\frac {2 \sqrt {b x +a}}{15 b \,a^{2} c^{2} \sqrt {-b c x +a c}}\right )}{a c}\right )}{3 a}}{a}}{a}\) | \(202\) |
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Time = 0.26 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.98 \[ \int \frac {1}{(a+b x)^{7/2} (a c-b c x)^{7/2}} \, dx=-\frac {{\left (8 \, b^{4} x^{5} - 20 \, a^{2} b^{2} x^{3} + 15 \, a^{4} x\right )} \sqrt {-b c x + a c} \sqrt {b x + a}}{15 \, {\left (a^{6} b^{6} c^{4} x^{6} - 3 \, a^{8} b^{4} c^{4} x^{4} + 3 \, a^{10} b^{2} c^{4} x^{2} - a^{12} c^{4}\right )}} \]
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Result contains complex when optimal does not.
Time = 58.00 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.97 \[ \int \frac {1}{(a+b x)^{7/2} (a c-b c x)^{7/2}} \, dx=- \frac {2 i {G_{6, 6}^{5, 3}\left (\begin {matrix} \frac {7}{4}, \frac {9}{4}, 1 & \frac {1}{2}, \frac {7}{2}, 4 \\\frac {7}{4}, \frac {9}{4}, 3, \frac {7}{2}, 4 & 0 \end {matrix} \middle | {\frac {a^{2}}{b^{2} x^{2}}} \right )}}{15 \pi ^{\frac {3}{2}} a^{6} b c^{\frac {7}{2}}} + \frac {2 {G_{6, 6}^{2, 6}\left (\begin {matrix} - \frac {1}{2}, 0, \frac {1}{2}, \frac {5}{4}, \frac {7}{4}, 1 & \\\frac {5}{4}, \frac {7}{4} & - \frac {1}{2}, 0, 3, 0 \end {matrix} \middle | {\frac {a^{2} e^{- 2 i \pi }}{b^{2} x^{2}}} \right )}}{15 \pi ^{\frac {3}{2}} a^{6} b c^{\frac {7}{2}}} \]
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Time = 0.20 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.79 \[ \int \frac {1}{(a+b x)^{7/2} (a c-b c x)^{7/2}} \, dx=\frac {x}{5 \, {\left (-b^{2} c x^{2} + a^{2} c\right )}^{\frac {5}{2}} a^{2} c} + \frac {4 \, x}{15 \, {\left (-b^{2} c x^{2} + a^{2} c\right )}^{\frac {3}{2}} a^{4} c^{2}} + \frac {8 \, x}{15 \, \sqrt {-b^{2} c x^{2} + a^{2} c} a^{6} c^{3}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 296 vs. \(2 (82) = 164\).
Time = 0.37 (sec) , antiderivative size = 296, normalized size of antiderivative = 2.96 \[ \int \frac {1}{(a+b x)^{7/2} (a c-b c x)^{7/2}} \, dx=-\frac {\frac {\sqrt {-{\left (b x + a\right )} c + 2 \, a c} \sqrt {b x + a} {\left ({\left (b x + a\right )} {\left (\frac {64 \, {\left (b x + a\right )}}{a^{6} c} - \frac {275}{a^{5} c}\right )} + \frac {300}{a^{4} c}\right )}}{{\left ({\left (b x + a\right )} c - 2 \, a c\right )}^{3}} + \frac {4 \, {\left (45 \, {\left (\sqrt {b x + a} \sqrt {-c} - \sqrt {-{\left (b x + a\right )} c + 2 \, a c}\right )}^{8} - 450 \, a {\left (\sqrt {b x + a} \sqrt {-c} - \sqrt {-{\left (b x + a\right )} c + 2 \, a c}\right )}^{6} c + 1660 \, a^{2} {\left (\sqrt {b x + a} \sqrt {-c} - \sqrt {-{\left (b x + a\right )} c + 2 \, a c}\right )}^{4} c^{2} - 2200 \, a^{3} {\left (\sqrt {b x + a} \sqrt {-c} - \sqrt {-{\left (b x + a\right )} c + 2 \, a c}\right )}^{2} c^{3} + 1024 \, a^{4} c^{4}\right )}}{{\left ({\left (\sqrt {b x + a} \sqrt {-c} - \sqrt {-{\left (b x + a\right )} c + 2 \, a c}\right )}^{2} - 2 \, a c\right )}^{5} a^{5} \sqrt {-c} c^{2}}}{240 \, b} \]
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Time = 0.72 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.11 \[ \int \frac {1}{(a+b x)^{7/2} (a c-b c x)^{7/2}} \, dx=\frac {15\,a^4\,x\,\sqrt {a\,c-b\,c\,x}+8\,b^4\,x^5\,\sqrt {a\,c-b\,c\,x}-20\,a^2\,b^2\,x^3\,\sqrt {a\,c-b\,c\,x}}{{\left (a\,c-b\,c\,x\right )}^3\,\left (60\,a^8\,c-\left (a\,c-b\,c\,x\right )\,\left (45\,a^7+15\,b\,x\,a^6\right )\right )\,\sqrt {a+b\,x}} \]
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