Integrand size = 40, antiderivative size = 295 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{x^5 (d+e x)} \, dx=\frac {\left (c d^2-a e^2\right ) \left (3 c d^2+5 a e^2\right ) \left (2 a d e+\left (c d^2+a e^2\right ) x\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{64 a^2 d^3 e^2 x^2}-\frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{4 d x^4}-\frac {\left (\frac {3 c}{a e}-\frac {5 e}{d^2}\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{24 x^3}-\frac {\left (c d^2-a e^2\right )^3 \left (3 c d^2+5 a e^2\right ) \text {arctanh}\left (\frac {2 a d e+\left (c d^2+a e^2\right ) x}{2 \sqrt {a} \sqrt {d} \sqrt {e} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\right )}{128 a^{5/2} d^{7/2} e^{5/2}} \]
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Time = 0.23 (sec) , antiderivative size = 295, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {863, 848, 820, 734, 738, 212} \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{x^5 (d+e x)} \, dx=-\frac {\left (5 a e^2+3 c d^2\right ) \left (c d^2-a e^2\right )^3 \text {arctanh}\left (\frac {x \left (a e^2+c d^2\right )+2 a d e}{2 \sqrt {a} \sqrt {d} \sqrt {e} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{128 a^{5/2} d^{7/2} e^{5/2}}+\frac {\left (5 a e^2+3 c d^2\right ) \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+2 a d e\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{64 a^2 d^3 e^2 x^2}-\frac {\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{4 d x^4}-\frac {\left (\frac {3 c}{a e}-\frac {5 e}{d^2}\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{24 x^3} \]
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Rule 212
Rule 734
Rule 738
Rule 820
Rule 848
Rule 863
Rubi steps \begin{align*} \text {integral}& = \int \frac {(a e+c d x) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{x^5} \, dx \\ & = -\frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{4 d x^4}-\frac {\int \frac {\left (-\frac {1}{2} a e \left (3 c d^2-5 a e^2\right )+a c d e^2 x\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{x^4} \, dx}{4 a d e} \\ & = -\frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{4 d x^4}-\frac {\left (\frac {3 c}{a e}-\frac {5 e}{d^2}\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{24 x^3}-\frac {\left (\frac {3 c^2 d^2}{a}+2 c e^2-\frac {5 a e^4}{d^2}\right ) \int \frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{x^3} \, dx}{16 e} \\ & = \frac {\left (c d^2-a e^2\right ) \left (3 c d^2+5 a e^2\right ) \left (2 a d e+\left (c d^2+a e^2\right ) x\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{64 a^2 d^3 e^2 x^2}-\frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{4 d x^4}-\frac {\left (\frac {3 c}{a e}-\frac {5 e}{d^2}\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{24 x^3}+\frac {\left (\left (c d^2-a e^2\right )^3 \left (3 c d^2+5 a e^2\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}{128 a^2 d^3 e^2} \\ & = \frac {\left (c d^2-a e^2\right ) \left (3 c d^2+5 a e^2\right ) \left (2 a d e+\left (c d^2+a e^2\right ) x\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{64 a^2 d^3 e^2 x^2}-\frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{4 d x^4}-\frac {\left (\frac {3 c}{a e}-\frac {5 e}{d^2}\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{24 x^3}-\frac {\left (\left (c d^2-a e^2\right )^3 \left (3 c d^2+5 a e^2\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 )}{64 a^2 d^3 e^2} \\ & = \frac {\left (c d^2-a e^2\right ) \left (3 c d^2+5 a e^2\right ) \left (2 a d e+\left (c d^2+a e^2\right ) x\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{64 a^2 d^3 e^2 x^2}-\frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{4 d x^4}-\frac {\left (\frac {3 c}{a e}-\frac {5 e}{d^2}\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{24 x^3}-\frac {\left (c d^2-a e^2\right )^3 \left (3 c d^2+5 a e^2\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 )}{128 a^{5/2} d^{7/2} e^{5/2}} \\ \end{align*}
Time = 0.53 (sec) , antiderivative size = 247, normalized size of antiderivative = 0.84 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{x^5 (d+e x)} \, dx=\frac {\sqrt {(a e+c d x) (d+e x)} \left (-\frac {\sqrt {a} \sqrt {d} \sqrt {e} \left (-9 c^3 d^6 x^3+3 a c^2 d^4 e x^2 (2 d+3 e x)+a^2 c d^2 e^2 x \left (72 d^2+20 d e x-31 e^2 x^2\right )+a^3 e^3 \left (48 d^3+8 d^2 e x-10 d e^2 x^2+15 e^3 x^3\right )\right )}{x^4}-\frac {3 \left (c d^2-a e^2\right )^3 \left (3 c d^2+5 a e^2\right ) \text {arctanh}\left (\frac {\sqrt {a} \sqrt {e} \sqrt {d+e x}}{\sqrt {d} \sqrt {a e+c d x}}\right )}{\sqrt {a e+c d x} \sqrt {d+e x}}\right )}{192 a^{5/2} d^{7/2} e^{5/2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(7420\) vs. \(2(265)=530\).
Time = 1.06 (sec) , antiderivative size = 7421, normalized size of antiderivative = 25.16
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Time = 4.94 (sec) , antiderivative size = 704, normalized size of antiderivative = 2.39 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{x^5 (d+e x)} \, dx=\left [-\frac {3 \, {\left (3 \, c^{4} d^{8} - 4 \, a c^{3} d^{6} e^{2} - 6 \, a^{2} c^{2} d^{4} e^{4} + 12 \, a^{3} c d^{2} e^{6} - 5 \, a^{4} e^{8}\right )} \sqrt {a d e} x^{4} \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 (48 \, a^{4} d^{4} e^{4} - {\left (9 \, a c^{3} d^{7} e - 9 \, a^{2} c^{2} d^{5} e^{3} + 31 \, a^{3} c d^{3} e^{5} - 15 \, a^{4} d e^{7}\right )} x^{3} + 2 \, {\left (3 \, a^{2} c^{2} d^{6} e^{2} + 10 \, a^{3} c d^{4} e^{4} - 5 \, a^{4} d^{2} e^{6}\right )} x^{2} + 8 \, {\left (9 \, a^{3} c d^{5} e^{3} + a^{4} d^{3} e^{5}\right )} x\right )} \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x}}{768 \, a^{3} d^{4} e^{3} x^{4}}, \frac {3 \, {\left (3 \, c^{4} d^{8} - 4 \, a c^{3} d^{6} e^{2} - 6 \, a^{2} c^{2} d^{4} e^{4} + 12 \, a^{3} c d^{2} e^{6} - 5 \, a^{4} e^{8}\right )} \sqrt {-a d e} x^{4} \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 (48 \, a^{4} d^{4} e^{4} - {\left (9 \, a c^{3} d^{7} e - 9 \, a^{2} c^{2} d^{5} e^{3} + 31 \, a^{3} c d^{3} e^{5} - 15 \, a^{4} d e^{7}\right )} x^{3} + 2 \, {\left (3 \, a^{2} c^{2} d^{6} e^{2} + 10 \, a^{3} c d^{4} e^{4} - 5 \, a^{4} d^{2} e^{6}\right )} x^{2} + 8 \, {\left (9 \, a^{3} c d^{5} e^{3} + a^{4} d^{3} e^{5}\right )} x\right )} \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x}}{384 \, a^{3} d^{4} e^{3} x^{4}}\right ] \]
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Timed out. \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{x^5 (d+e x)} \, dx=\text {Timed out} \]
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\[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{x^5 (d+e x)} \, dx=\int { \frac {{\left (c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x\right )}^{\frac {3}{2}}}{{\left (e x + d\right )} x^{5}} \,d x } \]
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Leaf count of result is larger than twice the leaf count of optimal. 1618 vs. \(2 (265) = 530\).
Time = 0.37 (sec) , antiderivative size = 1618, normalized size of antiderivative = 5.48 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{x^5 (d+e x)} \, dx=\text {Too large to display} \]
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Timed out. \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{x^5 (d+e x)} \, dx=\int \frac {{\left (c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e\right )}^{3/2}}{x^5\,\left (d+e\,x\right )} \,d x \]
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