Integrand size = 27, antiderivative size = 183 \[ \int \frac {\sqrt {d^2-e^2 x^2}}{x^3 (d+e x)^4} \, dx=\frac {8 e^2 (d-e x)}{5 d \left (d^2-e^2 x^2\right )^{5/2}}+\frac {4 e^2 (10 d-13 e x)}{15 d^3 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {e^2 (135 d-164 e x)}{15 d^5 \sqrt {d^2-e^2 x^2}}-\frac {\sqrt {d^2-e^2 x^2}}{2 d^4 x^2}+\frac {4 e \sqrt {d^2-e^2 x^2}}{d^5 x}-\frac {19 e^2 \text {arctanh}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{2 d^5} \]
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Time = 0.25 (sec) , antiderivative size = 183, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.259, Rules used = {866, 1819, 1821, 821, 272, 65, 214} \[ \int \frac {\sqrt {d^2-e^2 x^2}}{x^3 (d+e x)^4} \, dx=-\frac {19 e^2 \text {arctanh}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{2 d^5}+\frac {8 e^2 (d-e x)}{5 d \left (d^2-e^2 x^2\right )^{5/2}}+\frac {e^2 (135 d-164 e x)}{15 d^5 \sqrt {d^2-e^2 x^2}}+\frac {4 e \sqrt {d^2-e^2 x^2}}{d^5 x}-\frac {\sqrt {d^2-e^2 x^2}}{2 d^4 x^2}+\frac {4 e^2 (10 d-13 e x)}{15 d^3 \left (d^2-e^2 x^2\right )^{3/2}} \]
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Rule 65
Rule 214
Rule 272
Rule 821
Rule 866
Rule 1819
Rule 1821
Rubi steps \begin{align*} \text {integral}& = \int \frac {(d-e x)^4}{x^3 \left (d^2-e^2 x^2\right )^{7/2}} \, dx \\ & = \frac {8 e^2 (d-e x)}{5 d \left (d^2-e^2 x^2\right )^{5/2}}-\frac {\int \frac {-5 d^4+20 d^3 e x-35 d^2 e^2 x^2+32 d e^3 x^3}{x^3 \left (d^2-e^2 x^2\right )^{5/2}} \, dx}{5 d^2} \\ & = \frac {8 e^2 (d-e x)}{5 d \left (d^2-e^2 x^2\right )^{5/2}}+\frac {4 e^2 (10 d-13 e x)}{15 d^3 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {\int \frac {15 d^4-60 d^3 e x+120 d^2 e^2 x^2-104 d e^3 x^3}{x^3 \left (d^2-e^2 x^2\right )^{3/2}} \, dx}{15 d^4} \\ & = \frac {8 e^2 (d-e x)}{5 d \left (d^2-e^2 x^2\right )^{5/2}}+\frac {4 e^2 (10 d-13 e x)}{15 d^3 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {e^2 (135 d-164 e x)}{15 d^5 \sqrt {d^2-e^2 x^2}}-\frac {\int \frac {-15 d^4+60 d^3 e x-135 d^2 e^2 x^2}{x^3 \sqrt {d^2-e^2 x^2}} \, dx}{15 d^6} \\ & = \frac {8 e^2 (d-e x)}{5 d \left (d^2-e^2 x^2\right )^{5/2}}+\frac {4 e^2 (10 d-13 e x)}{15 d^3 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {e^2 (135 d-164 e x)}{15 d^5 \sqrt {d^2-e^2 x^2}}-\frac {\sqrt {d^2-e^2 x^2}}{2 d^4 x^2}+\frac {\int \frac {-120 d^5 e+285 d^4 e^2 x}{x^2 \sqrt {d^2-e^2 x^2}} \, dx}{30 d^8} \\ & = \frac {8 e^2 (d-e x)}{5 d \left (d^2-e^2 x^2\right )^{5/2}}+\frac {4 e^2 (10 d-13 e x)}{15 d^3 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {e^2 (135 d-164 e x)}{15 d^5 \sqrt {d^2-e^2 x^2}}-\frac {\sqrt {d^2-e^2 x^2}}{2 d^4 x^2}+\frac {4 e \sqrt {d^2-e^2 x^2}}{d^5 x}+\frac {\left (19 e^2\right ) \int \frac {1}{x \sqrt {d^2-e^2 x^2}} \, dx}{2 d^4} \\ & = \frac {8 e^2 (d-e x)}{5 d \left (d^2-e^2 x^2\right )^{5/2}}+\frac {4 e^2 (10 d-13 e x)}{15 d^3 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {e^2 (135 d-164 e x)}{15 d^5 \sqrt {d^2-e^2 x^2}}-\frac {\sqrt {d^2-e^2 x^2}}{2 d^4 x^2}+\frac {4 e \sqrt {d^2-e^2 x^2}}{d^5 x}+\frac {\left (19 e^2\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {d^2-e^2 x}} \, dx,x,x^2\right )}{4 d^4} \\ & = \frac {8 e^2 (d-e x)}{5 d \left (d^2-e^2 x^2\right )^{5/2}}+\frac {4 e^2 (10 d-13 e x)}{15 d^3 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {e^2 (135 d-164 e x)}{15 d^5 \sqrt {d^2-e^2 x^2}}-\frac {\sqrt {d^2-e^2 x^2}}{2 d^4 x^2}+\frac {4 e \sqrt {d^2-e^2 x^2}}{d^5 x}-\frac {19 \text {Subst}\left (\int \frac {1}{\frac {d^2}{e^2}-\frac {x^2}{e^2}} \, dx,x,\sqrt {d^2-e^2 x^2}\right )}{2 d^4} \\ & = \frac {8 e^2 (d-e x)}{5 d \left (d^2-e^2 x^2\right )^{5/2}}+\frac {4 e^2 (10 d-13 e x)}{15 d^3 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {e^2 (135 d-164 e x)}{15 d^5 \sqrt {d^2-e^2 x^2}}-\frac {\sqrt {d^2-e^2 x^2}}{2 d^4 x^2}+\frac {4 e \sqrt {d^2-e^2 x^2}}{d^5 x}-\frac {19 e^2 \tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{2 d^5} \\ \end{align*}
Time = 0.50 (sec) , antiderivative size = 116, normalized size of antiderivative = 0.63 \[ \int \frac {\sqrt {d^2-e^2 x^2}}{x^3 (d+e x)^4} \, dx=\frac {\frac {\sqrt {d^2-e^2 x^2} \left (-15 d^4+75 d^3 e x+713 d^2 e^2 x^2+1059 d e^3 x^3+448 e^4 x^4\right )}{x^2 (d+e x)^3}+570 e^2 \text {arctanh}\left (\frac {\sqrt {-e^2} x-\sqrt {d^2-e^2 x^2}}{d}\right )}{30 d^5} \]
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Time = 0.45 (sec) , antiderivative size = 205, normalized size of antiderivative = 1.12
method | result | size |
risch | \(-\frac {\sqrt {-e^{2} x^{2}+d^{2}}\, \left (-8 e x +d \right )}{2 d^{5} x^{2}}-\frac {19 e^{2} \ln \left (\frac {2 d^{2}+2 \sqrt {d^{2}}\, \sqrt {-e^{2} x^{2}+d^{2}}}{x}\right )}{2 d^{4} \sqrt {d^{2}}}+\frac {164 e \sqrt {-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )}}{15 d^{5} \left (x +\frac {d}{e}\right )}+\frac {2 \sqrt {-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )}}{5 e \,d^{3} \left (x +\frac {d}{e}\right )^{3}}+\frac {29 \sqrt {-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )}}{15 d^{4} \left (x +\frac {d}{e}\right )^{2}}\) | \(205\) |
default | \(\frac {-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}{2 d^{2} x^{2}}-\frac {e^{2} \left (\sqrt {-e^{2} x^{2}+d^{2}}-\frac {d^{2} \ln \left (\frac {2 d^{2}+2 \sqrt {d^{2}}\, \sqrt {-e^{2} x^{2}+d^{2}}}{x}\right )}{\sqrt {d^{2}}}\right )}{2 d^{2}}}{d^{4}}-\frac {4 e \left (-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}{d^{2} x}-\frac {2 e^{2} \left (\frac {x \sqrt {-e^{2} x^{2}+d^{2}}}{2}+\frac {d^{2} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{2 \sqrt {e^{2}}}\right )}{d^{2}}\right )}{d^{5}}+\frac {10 e^{2} \left (\sqrt {-e^{2} x^{2}+d^{2}}-\frac {d^{2} \ln \left (\frac {2 d^{2}+2 \sqrt {d^{2}}\, \sqrt {-e^{2} x^{2}+d^{2}}}{x}\right )}{\sqrt {d^{2}}}\right )}{d^{6}}-\frac {-\frac {\left (-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {3}{2}}}{5 d e \left (x +\frac {d}{e}\right )^{4}}-\frac {\left (-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {3}{2}}}{15 d^{2} \left (x +\frac {d}{e}\right )^{3}}}{e \,d^{3}}-\frac {6 e \left (-\frac {\left (-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {3}{2}}}{d e \left (x +\frac {d}{e}\right )^{2}}-\frac {e \left (\sqrt {-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )}+\frac {d e \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )}}\right )}{\sqrt {e^{2}}}\right )}{d}\right )}{d^{5}}+\frac {\left (-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {3}{2}}}{e \,d^{5} \left (x +\frac {d}{e}\right )^{3}}-\frac {10 e^{2} \left (\sqrt {-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )}+\frac {d e \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )}}\right )}{\sqrt {e^{2}}}\right )}{d^{6}}\) | \(598\) |
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Time = 0.31 (sec) , antiderivative size = 202, normalized size of antiderivative = 1.10 \[ \int \frac {\sqrt {d^2-e^2 x^2}}{x^3 (d+e x)^4} \, dx=\frac {398 \, e^{5} x^{5} + 1194 \, d e^{4} x^{4} + 1194 \, d^{2} e^{3} x^{3} + 398 \, d^{3} e^{2} x^{2} + 285 \, {\left (e^{5} x^{5} + 3 \, d e^{4} x^{4} + 3 \, d^{2} e^{3} x^{3} + d^{3} e^{2} x^{2}\right )} \log \left (-\frac {d - \sqrt {-e^{2} x^{2} + d^{2}}}{x}\right ) + {\left (448 \, e^{4} x^{4} + 1059 \, d e^{3} x^{3} + 713 \, d^{2} e^{2} x^{2} + 75 \, d^{3} e x - 15 \, d^{4}\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{30 \, {\left (d^{5} e^{3} x^{5} + 3 \, d^{6} e^{2} x^{4} + 3 \, d^{7} e x^{3} + d^{8} x^{2}\right )}} \]
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\[ \int \frac {\sqrt {d^2-e^2 x^2}}{x^3 (d+e x)^4} \, dx=\int \frac {\sqrt {- \left (- d + e x\right ) \left (d + e x\right )}}{x^{3} \left (d + e x\right )^{4}}\, dx \]
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\[ \int \frac {\sqrt {d^2-e^2 x^2}}{x^3 (d+e x)^4} \, dx=\int { \frac {\sqrt {-e^{2} x^{2} + d^{2}}}{{\left (e x + d\right )}^{4} x^{3}} \,d x } \]
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Leaf count of result is larger than twice the leaf count of optimal. 382 vs. \(2 (162) = 324\).
Time = 0.31 (sec) , antiderivative size = 382, normalized size of antiderivative = 2.09 \[ \int \frac {\sqrt {d^2-e^2 x^2}}{x^3 (d+e x)^4} \, dx=-\frac {19 \, e^{3} \log \left (\frac {{\left | -2 \, d e - 2 \, \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |} \right |}}{2 \, e^{2} {\left | x \right |}}\right )}{2 \, d^{5} {\left | e \right |}} + \frac {{\left (15 \, e^{3} - \frac {165 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )} e}{x} - \frac {4234 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{2}}{e x^{2}} - \frac {14330 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{3}}{e^{3} x^{3}} - \frac {20965 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{4}}{e^{5} x^{4}} - \frac {14385 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{5}}{e^{7} x^{5}} - \frac {4080 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{6}}{e^{9} x^{6}}\right )} e^{4} x^{2}}{120 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{2} d^{5} {\left (\frac {d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}}{e^{2} x} + 1\right )}^{5} {\left | e \right |}} + \frac {\frac {16 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )} d^{5} e {\left | e \right |}}{x} - \frac {{\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{2} d^{5} {\left | e \right |}}{e x^{2}}}{8 \, d^{10} e^{2}} \]
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Timed out. \[ \int \frac {\sqrt {d^2-e^2 x^2}}{x^3 (d+e x)^4} \, dx=\int \frac {\sqrt {d^2-e^2\,x^2}}{x^3\,{\left (d+e\,x\right )}^4} \,d x \]
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