Integrand size = 20, antiderivative size = 91 \[ \int \frac {1}{(a+a x)^{7/2} (c-c x)^{7/2}} \, dx=\frac {x}{5 a c (a+a x)^{5/2} (c-c x)^{5/2}}+\frac {4 x}{15 a^2 c^2 (a+a x)^{3/2} (c-c x)^{3/2}}+\frac {8 x}{15 a^3 c^3 \sqrt {a+a x} \sqrt {c-c x}} \]
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Time = 0.01 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {40, 39} \[ \int \frac {1}{(a+a x)^{7/2} (c-c x)^{7/2}} \, dx=\frac {8 x}{15 a^3 c^3 \sqrt {a x+a} \sqrt {c-c x}}+\frac {4 x}{15 a^2 c^2 (a x+a)^{3/2} (c-c x)^{3/2}}+\frac {x}{5 a c (a x+a)^{5/2} (c-c x)^{5/2}} \]
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Rule 39
Rule 40
Rubi steps \begin{align*} \text {integral}& = \frac {x}{5 a c (a+a x)^{5/2} (c-c x)^{5/2}}+\frac {4 \int \frac {1}{(a+a x)^{5/2} (c-c x)^{5/2}} \, dx}{5 a c} \\ & = \frac {x}{5 a c (a+a x)^{5/2} (c-c x)^{5/2}}+\frac {4 x}{15 a^2 c^2 (a+a x)^{3/2} (c-c x)^{3/2}}+\frac {8 \int \frac {1}{(a+a x)^{3/2} (c-c x)^{3/2}} \, dx}{15 a^2 c^2} \\ & = \frac {x}{5 a c (a+a x)^{5/2} (c-c x)^{5/2}}+\frac {4 x}{15 a^2 c^2 (a+a x)^{3/2} (c-c x)^{3/2}}+\frac {8 x}{15 a^3 c^3 \sqrt {a+a x} \sqrt {c-c x}} \\ \end{align*}
Time = 0.20 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.54 \[ \int \frac {1}{(a+a x)^{7/2} (c-c x)^{7/2}} \, dx=\frac {x \left (15-20 x^2+8 x^4\right )}{15 a^3 c^3 \sqrt {a (1+x)} \sqrt {c-c x} \left (-1+x^2\right )^2} \]
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Time = 0.18 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.41
method | result | size |
gosper | \(-\frac {\left (-1+x \right ) \left (1+x \right ) x \left (8 x^{4}-20 x^{2}+15\right )}{15 \left (a x +a \right )^{\frac {7}{2}} \left (-c x +c \right )^{\frac {7}{2}}}\) | \(37\) |
default | \(-\frac {1}{5 a c \left (a x +a \right )^{\frac {5}{2}} \left (-c x +c \right )^{\frac {5}{2}}}+\frac {-\frac {1}{3 a c \left (a x +a \right )^{\frac {3}{2}} \left (-c x +c \right )^{\frac {5}{2}}}+\frac {-\frac {4}{3 a c \sqrt {a x +a}\, \left (-c x +c \right )^{\frac {5}{2}}}+\frac {4 \left (\frac {3 \sqrt {a x +a}}{5 a c \left (-c x +c \right )^{\frac {5}{2}}}+\frac {3 \left (\frac {2 \sqrt {a x +a}}{15 a c \left (-c x +c \right )^{\frac {3}{2}}}+\frac {2 \sqrt {a x +a}}{15 a \,c^{2} \sqrt {-c x +c}}\right )}{c}\right )}{3 a}}{a}}{a}\) | \(163\) |
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Time = 0.23 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.81 \[ \int \frac {1}{(a+a x)^{7/2} (c-c x)^{7/2}} \, dx=-\frac {{\left (8 \, x^{5} - 20 \, x^{3} + 15 \, x\right )} \sqrt {a x + a} \sqrt {-c x + c}}{15 \, {\left (a^{4} c^{4} x^{6} - 3 \, a^{4} c^{4} x^{4} + 3 \, a^{4} c^{4} x^{2} - a^{4} c^{4}\right )}} \]
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Result contains complex when optimal does not.
Time = 50.31 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.93 \[ \int \frac {1}{(a+a x)^{7/2} (c-c x)^{7/2}} \, dx=- \frac {2 i {G_{6, 6}^{5, 3}\left (\begin {matrix} \frac {7}{4}, \frac {9}{4}, 1 & \frac {1}{2}, \frac {7}{2}, 4 \\\frac {7}{4}, \frac {9}{4}, 3, \frac {7}{2}, 4 & 0 \end {matrix} \middle | {\frac {1}{x^{2}}} \right )}}{15 \pi ^{\frac {3}{2}} a^{\frac {7}{2}} c^{\frac {7}{2}}} + \frac {2 {G_{6, 6}^{2, 6}\left (\begin {matrix} - \frac {1}{2}, 0, \frac {1}{2}, \frac {5}{4}, \frac {7}{4}, 1 & \\\frac {5}{4}, \frac {7}{4} & - \frac {1}{2}, 0, 3, 0 \end {matrix} \middle | {\frac {e^{- 2 i \pi }}{x^{2}}} \right )}}{15 \pi ^{\frac {3}{2}} a^{\frac {7}{2}} c^{\frac {7}{2}}} \]
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Time = 0.21 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.74 \[ \int \frac {1}{(a+a x)^{7/2} (c-c x)^{7/2}} \, dx=\frac {x}{5 \, {\left (-a c x^{2} + a c\right )}^{\frac {5}{2}} a c} + \frac {4 \, x}{15 \, {\left (-a c x^{2} + a c\right )}^{\frac {3}{2}} a^{2} c^{2}} + \frac {8 \, x}{15 \, \sqrt {-a c x^{2} + a c} a^{3} c^{3}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 333 vs. \(2 (73) = 146\).
Time = 0.36 (sec) , antiderivative size = 333, normalized size of antiderivative = 3.66 \[ \int \frac {1}{(a+a x)^{7/2} (c-c x)^{7/2}} \, dx=-\frac {\sqrt {-{\left (a x + a\right )} a c + 2 \, a^{2} c} \sqrt {a x + a} {\left ({\left (a x + a\right )} {\left (\frac {64 \, {\left (a x + a\right )}}{c {\left | a \right |}} - \frac {275 \, a}{c {\left | a \right |}}\right )} + \frac {300 \, a^{2}}{c {\left | a \right |}}\right )}}{240 \, {\left ({\left (a x + a\right )} a c - 2 \, a^{2} c\right )}^{3}} + \frac {1024 \, a^{8} c^{4} - 2200 \, {\left (\sqrt {-a c} \sqrt {a x + a} - \sqrt {-{\left (a x + a\right )} a c + 2 \, a^{2} c}\right )}^{2} a^{6} c^{3} + 1660 \, {\left (\sqrt {-a c} \sqrt {a x + a} - \sqrt {-{\left (a x + a\right )} a c + 2 \, a^{2} c}\right )}^{4} a^{4} c^{2} - 450 \, {\left (\sqrt {-a c} \sqrt {a x + a} - \sqrt {-{\left (a x + a\right )} a c + 2 \, a^{2} c}\right )}^{6} a^{2} c + 45 \, {\left (\sqrt {-a c} \sqrt {a x + a} - \sqrt {-{\left (a x + a\right )} a c + 2 \, a^{2} c}\right )}^{8}}{60 \, {\left (2 \, a^{2} c - {\left (\sqrt {-a c} \sqrt {a x + a} - \sqrt {-{\left (a x + a\right )} a c + 2 \, a^{2} c}\right )}^{2}\right )}^{5} \sqrt {-a c} c^{2} {\left | a \right |}} \]
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Time = 0.49 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.55 \[ \int \frac {1}{(a+a x)^{7/2} (c-c x)^{7/2}} \, dx=\frac {x\,\left (8\,x^4-20\,x^2+15\right )}{15\,a^3\,\sqrt {a+a\,x}\,{\left (c-c\,x\right )}^{5/2}\,\left (c+3\,c\,x-x\,\left (c-c\,x\right )\right )} \]
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