Integrand size = 27, antiderivative size = 137 \[ \int \frac {\left (d^2-e^2 x^2\right )^{5/2}}{x^4 (d+e x)^4} \, dx=-\frac {8 e^3 (d-e x)}{d^2 \sqrt {d^2-e^2 x^2}}-\frac {\sqrt {d^2-e^2 x^2}}{3 x^3}+\frac {2 e \sqrt {d^2-e^2 x^2}}{d x^2}-\frac {23 e^2 \sqrt {d^2-e^2 x^2}}{3 d^2 x}+\frac {10 e^3 \text {arctanh}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{d^2} \]
[Out]
Time = 0.20 (sec) , antiderivative size = 137, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.259, Rules used = {866, 1819, 1821, 821, 272, 65, 214} \[ \int \frac {\left (d^2-e^2 x^2\right )^{5/2}}{x^4 (d+e x)^4} \, dx=\frac {10 e^3 \text {arctanh}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{d^2}-\frac {23 e^2 \sqrt {d^2-e^2 x^2}}{3 d^2 x}+\frac {2 e \sqrt {d^2-e^2 x^2}}{d x^2}-\frac {\sqrt {d^2-e^2 x^2}}{3 x^3}-\frac {8 e^3 (d-e x)}{d^2 \sqrt {d^2-e^2 x^2}} \]
[In]
[Out]
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^4 \left (d^2-e^2 x^2\right )^{3/2}} \, dx \\ & = -\frac {8 e^3 (d-e x)}{d^2 \sqrt {d^2-e^2 x^2}}-\frac {\int \frac {-d^4+4 d^3 e x-7 d^2 e^2 x^2+8 d e^3 x^3}{x^4 \sqrt {d^2-e^2 x^2}} \, dx}{d^2} \\ & = -\frac {8 e^3 (d-e x)}{d^2 \sqrt {d^2-e^2 x^2}}-\frac {\sqrt {d^2-e^2 x^2}}{3 x^3}+\frac {\int \frac {-12 d^5 e+23 d^4 e^2 x-24 d^3 e^3 x^2}{x^3 \sqrt {d^2-e^2 x^2}} \, dx}{3 d^4} \\ & = -\frac {8 e^3 (d-e x)}{d^2 \sqrt {d^2-e^2 x^2}}-\frac {\sqrt {d^2-e^2 x^2}}{3 x^3}+\frac {2 e \sqrt {d^2-e^2 x^2}}{d x^2}-\frac {\int \frac {-46 d^6 e^2+60 d^5 e^3 x}{x^2 \sqrt {d^2-e^2 x^2}} \, dx}{6 d^6} \\ & = -\frac {8 e^3 (d-e x)}{d^2 \sqrt {d^2-e^2 x^2}}-\frac {\sqrt {d^2-e^2 x^2}}{3 x^3}+\frac {2 e \sqrt {d^2-e^2 x^2}}{d x^2}-\frac {23 e^2 \sqrt {d^2-e^2 x^2}}{3 d^2 x}-\frac {\left (10 e^3\right ) \int \frac {1}{x \sqrt {d^2-e^2 x^2}} \, dx}{d} \\ & = -\frac {8 e^3 (d-e x)}{d^2 \sqrt {d^2-e^2 x^2}}-\frac {\sqrt {d^2-e^2 x^2}}{3 x^3}+\frac {2 e \sqrt {d^2-e^2 x^2}}{d x^2}-\frac {23 e^2 \sqrt {d^2-e^2 x^2}}{3 d^2 x}-\frac {\left (5 e^3\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {d^2-e^2 x}} \, dx,x,x^2\right )}{d} \\ & = -\frac {8 e^3 (d-e x)}{d^2 \sqrt {d^2-e^2 x^2}}-\frac {\sqrt {d^2-e^2 x^2}}{3 x^3}+\frac {2 e \sqrt {d^2-e^2 x^2}}{d x^2}-\frac {23 e^2 \sqrt {d^2-e^2 x^2}}{3 d^2 x}+\frac {(10 e) \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 )}{d} \\ & = -\frac {8 e^3 (d-e x)}{d^2 \sqrt {d^2-e^2 x^2}}-\frac {\sqrt {d^2-e^2 x^2}}{3 x^3}+\frac {2 e \sqrt {d^2-e^2 x^2}}{d x^2}-\frac {23 e^2 \sqrt {d^2-e^2 x^2}}{3 d^2 x}+\frac {10 e^3 \tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{d^2} \\ \end{align*}
Time = 0.39 (sec) , antiderivative size = 117, normalized size of antiderivative = 0.85 \[ \int \frac {\left (d^2-e^2 x^2\right )^{5/2}}{x^4 (d+e x)^4} \, dx=-\frac {\frac {d \sqrt {d^2-e^2 x^2} \left (d^3-5 d^2 e x+17 d e^2 x^2+47 e^3 x^3\right )}{x^3 (d+e x)}-30 \sqrt {d^2} e^3 \log (x)+30 \sqrt {d^2} e^3 \log \left (\sqrt {d^2}-\sqrt {d^2-e^2 x^2}\right )}{3 d^3} \]
[In]
[Out]
Time = 0.70 (sec) , antiderivative size = 131, normalized size of antiderivative = 0.96
method | result | size |
risch | \(-\frac {\sqrt {-e^{2} x^{2}+d^{2}}\, \left (23 e^{2} x^{2}-6 d e x +d^{2}\right )}{3 x^{3} d^{2}}+\frac {10 e^{3} \ln \left (\frac {2 d^{2}+2 \sqrt {d^{2}}\, \sqrt {-e^{2} x^{2}+d^{2}}}{x}\right )}{d \sqrt {d^{2}}}-\frac {8 e^{2} \sqrt {-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )}}{d^{2} \left (x +\frac {d}{e}\right )}\) | \(131\) |
default | \(\text {Expression too large to display}\) | \(1626\) |
[In]
[Out]
none
Time = 0.30 (sec) , antiderivative size = 123, normalized size of antiderivative = 0.90 \[ \int \frac {\left (d^2-e^2 x^2\right )^{5/2}}{x^4 (d+e x)^4} \, dx=-\frac {24 \, e^{4} x^{4} + 24 \, d e^{3} x^{3} + 30 \, {\left (e^{4} x^{4} + d e^{3} x^{3}\right )} \log \left (-\frac {d - \sqrt {-e^{2} x^{2} + d^{2}}}{x}\right ) + {\left (47 \, e^{3} x^{3} + 17 \, d e^{2} x^{2} - 5 \, d^{2} e x + d^{3}\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{3 \, {\left (d^{2} e x^{4} + d^{3} x^{3}\right )}} \]
[In]
[Out]
\[ \int \frac {\left (d^2-e^2 x^2\right )^{5/2}}{x^4 (d+e x)^4} \, dx=\int \frac {\left (- \left (- d + e x\right ) \left (d + e x\right )\right )^{\frac {5}{2}}}{x^{4} \left (d + e x\right )^{4}}\, dx \]
[In]
[Out]
\[ \int \frac {\left (d^2-e^2 x^2\right )^{5/2}}{x^4 (d+e x)^4} \, dx=\int { \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}}}{{\left (e x + d\right )}^{4} x^{4}} \,d x } \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 318 vs. \(2 (124) = 248\).
Time = 0.31 (sec) , antiderivative size = 318, normalized size of antiderivative = 2.32 \[ \int \frac {\left (d^2-e^2 x^2\right )^{5/2}}{x^4 (d+e x)^4} \, dx=\frac {{\left (e^{4} - \frac {11 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )} e^{2}}{x} + \frac {81 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{2}}{x^{2}} + \frac {477 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{3}}{e^{2} x^{3}}\right )} e^{6} x^{3}}{24 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{3} d^{2} {\left (\frac {d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}}{e^{2} x} + 1\right )} {\left | e \right |}} + \frac {10 \, e^{4} \log \left (\frac {{\left | -2 \, d e - 2 \, \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |} \right |}}{2 \, e^{2} {\left | x \right |}}\right )}{d^{2} {\left | e \right |}} - \frac {\frac {93 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )} d^{4} e^{4}}{x} - \frac {12 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{2} d^{4} e^{2}}{x^{2}} + \frac {{\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{3} d^{4}}{x^{3}}}{24 \, d^{6} e^{2} {\left | e \right |}} \]
[In]
[Out]
Timed out. \[ \int \frac {\left (d^2-e^2 x^2\right )^{5/2}}{x^4 (d+e x)^4} \, dx=\int \frac {{\left (d^2-e^2\,x^2\right )}^{5/2}}{x^4\,{\left (d+e\,x\right )}^4} \,d x \]
[In]
[Out]