Integrand size = 40, antiderivative size = 286 \[ \int \frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{x^4 (d+e x)} \, dx=-\frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{3 d x^3}-\frac {\left (\frac {c}{a e}-\frac {5 e}{d^2}\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{12 x^2}+\frac {\left (3 c d^2-5 a e^2\right ) \left (c d^2+3 a e^2\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{24 a^2 d^3 e^2 x}-\frac {\left (c d^2-a e^2\right ) \left (c^2 d^4+2 a c d^2 e^2+5 a^2 e^4\right ) \text {arctanh}\left (\frac {2 a d e+\left (c d^2+a e^2\right ) x}{2 \sqrt {a} \sqrt {d} \sqrt {e} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\right )}{16 a^{5/2} d^{7/2} e^{5/2}} \]
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Time = 0.23 (sec) , antiderivative size = 286, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {863, 848, 820, 738, 212} \[ \int \frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{x^4 (d+e x)} \, dx=\frac {\left (3 c d^2-5 a e^2\right ) \left (3 a e^2+c d^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{24 a^2 d^3 e^2 x}-\frac {\left (c d^2-a e^2\right ) \left (5 a^2 e^4+2 a c d^2 e^2+c^2 d^4\right ) \text {arctanh}\left (\frac {x \left (a e^2+c d^2\right )+2 a d e}{2 \sqrt {a} \sqrt {d} \sqrt {e} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{16 a^{5/2} d^{7/2} e^{5/2}}-\frac {\left (\frac {c}{a e}-\frac {5 e}{d^2}\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{12 x^2}-\frac {\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{3 d x^3} \]
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Rule 212
Rule 738
Rule 820
Rule 848
Rule 863
Rubi steps \begin{align*} \text {integral}& = \int \frac {a e+c d x}{x^4 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx \\ & = -\frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{3 d x^3}-\frac {\int \frac {-\frac {1}{2} a e \left (c d^2-5 a e^2\right )+2 a c d e^2 x}{x^3 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{3 a d e} \\ & = -\frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{3 d x^3}-\frac {\left (\frac {c}{a e}-\frac {5 e}{d^2}\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{12 x^2}+\frac {\int \frac {-\frac {1}{4} a e \left (3 c d^2-5 a e^2\right ) \left (c d^2+3 a e^2\right )-\frac {1}{2} a c d e^2 \left (c d^2-5 a e^2\right ) x}{x^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{6 a^2 d^2 e^2} \\ & = -\frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{3 d x^3}-\frac {\left (\frac {c}{a e}-\frac {5 e}{d^2}\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{12 x^2}+\frac {\left (3 c d^2-5 a e^2\right ) \left (c d^2+3 a e^2\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{24 a^2 d^3 e^2 x}+\frac {\left (\left (c d^2-a e^2\right ) \left (c^2 d^4+2 a c d^2 e^2+5 a^2 e^4\right )\right ) \int \frac {1}{x \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{16 a^2 d^3 e^2} \\ & = -\frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{3 d x^3}-\frac {\left (\frac {c}{a e}-\frac {5 e}{d^2}\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{12 x^2}+\frac {\left (3 c d^2-5 a e^2\right ) \left (c d^2+3 a e^2\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{24 a^2 d^3 e^2 x}-\frac {\left (\left (c d^2-a e^2\right ) \left (c^2 d^4+2 a c d^2 e^2+5 a^2 e^4\right )\right ) \text {Subst}\left (\int \frac {1}{4 a d e-x^2} \, dx,x,\frac {2 a d e-\left (-c d^2-a e^2\right ) x}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\right )}{8 a^2 d^3 e^2} \\ & = -\frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{3 d x^3}-\frac {\left (\frac {c}{a e}-\frac {5 e}{d^2}\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{12 x^2}+\frac {\left (3 c d^2-5 a e^2\right ) \left (c d^2+3 a e^2\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{24 a^2 d^3 e^2 x}-\frac {\left (c d^2-a e^2\right ) \left (c^2 d^4+2 a c d^2 e^2+5 a^2 e^4\right ) \tanh ^{-1}\left (\frac {2 a d e+\left (c d^2+a e^2\right ) x}{2 \sqrt {a} \sqrt {d} \sqrt {e} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\right )}{16 a^{5/2} d^{7/2} e^{5/2}} \\ \end{align*}
Time = 10.17 (sec) , antiderivative size = 210, normalized size of antiderivative = 0.73 \[ \int \frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{x^4 (d+e x)} \, dx=\frac {\sqrt {(a e+c d x) (d+e x)} \left (\frac {\sqrt {a} \sqrt {d} \sqrt {e} \left (3 c^2 d^4 x^2-2 a c d^2 e x (d-2 e x)+a^2 e^2 \left (-8 d^2+10 d e x-15 e^2 x^2\right )\right )}{x^3}-\frac {3 \left (c^3 d^6+a c^2 d^4 e^2+3 a^2 c d^2 e^4-5 a^3 e^6\right ) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a e+c d x}}{\sqrt {a} \sqrt {e} \sqrt {d+e x}}\right )}{\sqrt {a e+c d x} \sqrt {d+e x}}\right )}{24 a^{5/2} d^{7/2} e^{5/2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(2058\) vs. \(2(256)=512\).
Time = 1.16 (sec) , antiderivative size = 2059, normalized size of antiderivative = 7.20
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Time = 1.23 (sec) , antiderivative size = 558, normalized size of antiderivative = 1.95 \[ \int \frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{x^4 (d+e x)} \, dx=\left [-\frac {3 \, {\left (c^{3} d^{6} + a c^{2} d^{4} e^{2} + 3 \, a^{2} c d^{2} e^{4} - 5 \, a^{3} e^{6}\right )} \sqrt {a d e} x^{3} \log \left (\frac {8 \, a^{2} d^{2} e^{2} + {\left (c^{2} d^{4} + 6 \, a c d^{2} e^{2} + a^{2} e^{4}\right )} x^{2} + 4 \, \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} {\left (2 \, a d e + {\left (c d^{2} + a e^{2}\right )} x\right )} \sqrt {a d e} + 8 \, {\left (a c d^{3} e + a^{2} d e^{3}\right )} x}{x^{2}}\right ) + 4 \, {\left (8 \, a^{3} d^{3} e^{3} - {\left (3 \, a c^{2} d^{5} e + 4 \, a^{2} c d^{3} e^{3} - 15 \, a^{3} d e^{5}\right )} x^{2} + 2 \, {\left (a^{2} c d^{4} e^{2} - 5 \, a^{3} d^{2} e^{4}\right )} x\right )} \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x}}{96 \, a^{3} d^{4} e^{3} x^{3}}, \frac {3 \, {\left (c^{3} d^{6} + a c^{2} d^{4} e^{2} + 3 \, a^{2} c d^{2} e^{4} - 5 \, a^{3} e^{6}\right )} \sqrt {-a d e} x^{3} \arctan \left (\frac {\sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} {\left (2 \, a d e + {\left (c d^{2} + a e^{2}\right )} x\right )} \sqrt {-a d e}}{2 \, {\left (a c d^{2} e^{2} x^{2} + a^{2} d^{2} e^{2} + {\left (a c d^{3} e + a^{2} d e^{3}\right )} x\right )}}\right ) - 2 \, {\left (8 \, a^{3} d^{3} e^{3} - {\left (3 \, a c^{2} d^{5} e + 4 \, a^{2} c d^{3} e^{3} - 15 \, a^{3} d e^{5}\right )} x^{2} + 2 \, {\left (a^{2} c d^{4} e^{2} - 5 \, a^{3} d^{2} e^{4}\right )} x\right )} \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x}}{48 \, a^{3} d^{4} e^{3} x^{3}}\right ] \]
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Timed out. \[ \int \frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{x^4 (d+e x)} \, dx=\text {Timed out} \]
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\[ \int \frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{x^4 (d+e x)} \, dx=\int { \frac {\sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x}}{{\left (e x + d\right )} x^{4}} \,d x } \]
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Leaf count of result is larger than twice the leaf count of optimal. 912 vs. \(2 (256) = 512\).
Time = 0.33 (sec) , antiderivative size = 912, normalized size of antiderivative = 3.19 \[ \int \frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{x^4 (d+e x)} \, dx=\frac {{\left (c^{3} d^{6} + a c^{2} d^{4} e^{2} + 3 \, a^{2} c d^{2} e^{4} - 5 \, a^{3} e^{6}\right )} \arctan \left (-\frac {\sqrt {c d e} x - \sqrt {c d e x^{2} + c d^{2} x + a e^{2} x + a d e}}{\sqrt {-a d e}}\right )}{8 \, \sqrt {-a d e} a^{2} d^{3} e^{2}} - \frac {3 \, {\left (\sqrt {c d e} x - \sqrt {c d e x^{2} + c d^{2} x + a e^{2} x + a d e}\right )} a^{2} c^{3} d^{8} e^{2} + 51 \, {\left (\sqrt {c d e} x - \sqrt {c d e x^{2} + c d^{2} x + a e^{2} x + a d e}\right )} a^{3} c^{2} d^{6} e^{4} + 105 \, {\left (\sqrt {c d e} x - \sqrt {c d e x^{2} + c d^{2} x + a e^{2} x + a d e}\right )} a^{4} c d^{4} e^{6} + 33 \, {\left (\sqrt {c d e} x - \sqrt {c d e x^{2} + c d^{2} x + a e^{2} x + a d e}\right )} a^{5} d^{2} e^{8} + 16 \, \sqrt {c d e} a^{4} c d^{5} e^{5} + 48 \, \sqrt {c d e} a^{5} d^{3} e^{7} + 8 \, {\left (\sqrt {c d e} x - \sqrt {c d e x^{2} + c d^{2} x + a e^{2} x + a d e}\right )}^{3} a c^{3} d^{7} e + 72 \, {\left (\sqrt {c d e} x - \sqrt {c d e x^{2} + c d^{2} x + a e^{2} x + a d e}\right )}^{3} a^{2} c^{2} d^{5} e^{3} + 24 \, {\left (\sqrt {c d e} x - \sqrt {c d e x^{2} + c d^{2} x + a e^{2} x + a d e}\right )}^{3} a^{3} c d^{3} e^{5} - 40 \, {\left (\sqrt {c d e} x - \sqrt {c d e x^{2} + c d^{2} x + a e^{2} x + a d e}\right )}^{3} a^{4} d e^{7} + 48 \, \sqrt {c d e} {\left (\sqrt {c d e} x - \sqrt {c d e x^{2} + c d^{2} x + a e^{2} x + a d e}\right )}^{2} a^{2} c^{2} d^{6} e^{2} + 144 \, \sqrt {c d e} {\left (\sqrt {c d e} x - \sqrt {c d e x^{2} + c d^{2} x + a e^{2} x + a d e}\right )}^{2} a^{3} c d^{4} e^{4} - 3 \, {\left (\sqrt {c d e} x - \sqrt {c d e x^{2} + c d^{2} x + a e^{2} x + a d e}\right )}^{5} c^{3} d^{6} - 3 \, {\left (\sqrt {c d e} x - \sqrt {c d e x^{2} + c d^{2} x + a e^{2} x + a d e}\right )}^{5} a c^{2} d^{4} e^{2} - 9 \, {\left (\sqrt {c d e} x - \sqrt {c d e x^{2} + c d^{2} x + a e^{2} x + a d e}\right )}^{5} a^{2} c d^{2} e^{4} + 15 \, {\left (\sqrt {c d e} x - \sqrt {c d e x^{2} + c d^{2} x + a e^{2} x + a d e}\right )}^{5} a^{3} e^{6}}{24 \, {\left (a d e - {\left (\sqrt {c d e} x - \sqrt {c d e x^{2} + c d^{2} x + a e^{2} x + a d e}\right )}^{2}\right )}^{3} a^{2} d^{3} e^{2}} \]
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Timed out. \[ \int \frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{x^4 (d+e x)} \, dx=\int \frac {\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}}{x^4\,\left (d+e\,x\right )} \,d x \]
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