Integrand size = 40, antiderivative size = 394 \[ \int \frac {1}{x^2 (d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=-\frac {2 e (a e+c d x)}{3 d \left (c d^2-a e^2\right ) x \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}+\frac {2 \left (3 c^3 d^6+a c^2 d^4 e^2+9 a^2 c d^2 e^4-5 a^3 e^6+c d e \left (3 c^2 d^4+10 a c d^2 e^2-5 a^2 e^4\right ) x\right )}{3 a d^2 e \left (c d^2-a e^2\right )^3 x \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}-\frac {\left (9 c^3 d^6-9 a c^2 d^4 e^2+31 a^2 c d^2 e^4-15 a^3 e^6\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{3 a^2 d^3 e^2 \left (c d^2-a e^2\right )^3 x}+\frac {\left (3 c d^2+5 a e^2\right ) \text {arctanh}\left (\frac {2 a d e+\left (c d^2+a e^2\right ) x}{2 \sqrt {a} \sqrt {d} \sqrt {e} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\right )}{2 a^{5/2} d^{7/2} e^{5/2}} \]
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Time = 0.34 (sec) , antiderivative size = 394, 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 = {865, 836, 820, 738, 212} \[ \int \frac {1}{x^2 (d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=\frac {\left (5 a e^2+3 c d^2\right ) \text {arctanh}\left (\frac {x \left (a e^2+c d^2\right )+2 a d e}{2 \sqrt {a} \sqrt {d} \sqrt {e} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{2 a^{5/2} d^{7/2} e^{5/2}}+\frac {2 \left (-5 a^3 e^6+c d e x \left (-5 a^2 e^4+10 a c d^2 e^2+3 c^2 d^4\right )+9 a^2 c d^2 e^4+a c^2 d^4 e^2+3 c^3 d^6\right )}{3 a d^2 e x \left (c d^2-a e^2\right )^3 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}-\frac {\left (-15 a^3 e^6+31 a^2 c d^2 e^4-9 a c^2 d^4 e^2+9 c^3 d^6\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{3 a^2 d^3 e^2 x \left (c d^2-a e^2\right )^3}-\frac {2 e (a e+c d x)}{3 d x \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}} \]
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Rule 212
Rule 738
Rule 820
Rule 836
Rule 865
Rubi steps \begin{align*} \text {integral}& = \int \frac {a e+c d x}{x^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx \\ & = -\frac {2 e (a e+c d x)}{3 d \left (c d^2-a e^2\right ) x \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}-\frac {2 \int \frac {-\frac {1}{2} a e \left (3 c d^2-5 a e^2\right ) \left (c d^2-a e^2\right )+3 a c d e^2 \left (c d^2-a e^2\right ) x}{x^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx}{3 a d e \left (c d^2-a e^2\right )^2} \\ & = -\frac {2 e (a e+c d x)}{3 d \left (c d^2-a e^2\right ) x \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}+\frac {2 \left (3 c^3 d^6+a c^2 d^4 e^2+9 a^2 c d^2 e^4-5 a^3 e^6+c d e \left (3 c^2 d^4+10 a c d^2 e^2-5 a^2 e^4\right ) x\right )}{3 a d^2 e \left (c d^2-a e^2\right )^3 x \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {4 \int \frac {\frac {1}{4} a e \left (c d^2-a e^2\right ) \left (9 c^3 d^6-9 a c^2 d^4 e^2+31 a^2 c d^2 e^4-15 a^3 e^6\right )+\frac {1}{2} a c d e^2 \left (c d^2-a e^2\right ) \left (3 c^2 d^4+10 a c d^2 e^2-5 a^2 e^4\right ) x}{x^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{3 a^2 d^2 e^2 \left (c d^2-a e^2\right )^4} \\ & = -\frac {2 e (a e+c d x)}{3 d \left (c d^2-a e^2\right ) x \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}+\frac {2 \left (3 c^3 d^6+a c^2 d^4 e^2+9 a^2 c d^2 e^4-5 a^3 e^6+c d e \left (3 c^2 d^4+10 a c d^2 e^2-5 a^2 e^4\right ) x\right )}{3 a d^2 e \left (c d^2-a e^2\right )^3 x \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}-\frac {\left (9 c^3 d^6-9 a c^2 d^4 e^2+31 a^2 c d^2 e^4-15 a^3 e^6\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{3 a^2 d^3 e^2 \left (c d^2-a e^2\right )^3 x}-\frac {\left (3 c d^2+5 a e^2\right ) \int \frac {1}{x \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{2 a^2 d^3 e^2} \\ & = -\frac {2 e (a e+c d x)}{3 d \left (c d^2-a e^2\right ) x \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}+\frac {2 \left (3 c^3 d^6+a c^2 d^4 e^2+9 a^2 c d^2 e^4-5 a^3 e^6+c d e \left (3 c^2 d^4+10 a c d^2 e^2-5 a^2 e^4\right ) x\right )}{3 a d^2 e \left (c d^2-a e^2\right )^3 x \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}-\frac {\left (9 c^3 d^6-9 a c^2 d^4 e^2+31 a^2 c d^2 e^4-15 a^3 e^6\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{3 a^2 d^3 e^2 \left (c d^2-a e^2\right )^3 x}+\frac {\left (3 c d^2+5 a e^2\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 )}{a^2 d^3 e^2} \\ & = -\frac {2 e (a e+c d x)}{3 d \left (c d^2-a e^2\right ) x \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}+\frac {2 \left (3 c^3 d^6+a c^2 d^4 e^2+9 a^2 c d^2 e^4-5 a^3 e^6+c d e \left (3 c^2 d^4+10 a c d^2 e^2-5 a^2 e^4\right ) x\right )}{3 a d^2 e \left (c d^2-a e^2\right )^3 x \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}-\frac {\left (9 c^3 d^6-9 a c^2 d^4 e^2+31 a^2 c d^2 e^4-15 a^3 e^6\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{3 a^2 d^3 e^2 \left (c d^2-a e^2\right )^3 x}+\frac {\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 )}{2 a^{5/2} d^{7/2} e^{5/2}} \\ \end{align*}
Time = 0.53 (sec) , antiderivative size = 303, normalized size of antiderivative = 0.77 \[ \int \frac {1}{x^2 (d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=\frac {-\frac {\sqrt {a} \sqrt {d} \sqrt {e} (a e+c d x) \left (-9 c^4 d^7 x (d+e x)^2-3 a c^3 d^5 e (d-3 e x) (d+e x)^2+a^4 e^7 \left (3 d^2+20 d e x+15 e^2 x^2\right )+a^2 c^2 d^3 e^3 \left (9 d^3+9 d^2 e x-33 d e^2 x^2-31 e^3 x^3\right )-a^3 c d e^5 \left (9 d^3+39 d^2 e x+11 d e^2 x^2-15 e^3 x^3\right )\right )}{\left (-c d^2+a e^2\right )^3 x}+3 \left (3 c d^2+5 a e^2\right ) (a e+c d x)^{3/2} (d+e x)^{3/2} \text {arctanh}\left (\frac {\sqrt {a} \sqrt {e} \sqrt {d+e x}}{\sqrt {d} \sqrt {a e+c d x}}\right )}{3 a^{5/2} d^{7/2} e^{5/2} ((a e+c d x) (d+e x))^{3/2}} \]
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Time = 0.74 (sec) , antiderivative size = 716, normalized size of antiderivative = 1.82
method | result | size |
default | \(\frac {-\frac {1}{a d e x \sqrt {a d e +\left (e^{2} a +c \,d^{2}\right ) x +c d e \,x^{2}}}-\frac {3 \left (e^{2} a +c \,d^{2}\right ) \left (\frac {1}{a d e \sqrt {a d e +\left (e^{2} a +c \,d^{2}\right ) x +c d e \,x^{2}}}-\frac {\left (e^{2} a +c \,d^{2}\right ) \left (2 c d e x +e^{2} a +c \,d^{2}\right )}{a d e \left (4 a c \,d^{2} e^{2}-\left (e^{2} a +c \,d^{2}\right )^{2}\right ) \sqrt {a d e +\left (e^{2} a +c \,d^{2}\right ) x +c d e \,x^{2}}}-\frac {\ln \left (\frac {2 a d e +\left (e^{2} a +c \,d^{2}\right ) x +2 \sqrt {a d e}\, \sqrt {a d e +\left (e^{2} a +c \,d^{2}\right ) x +c d e \,x^{2}}}{x}\right )}{a d e \sqrt {a d e}}\right )}{2 a d e}-\frac {4 c \left (2 c d e x +e^{2} a +c \,d^{2}\right )}{a \left (4 a c \,d^{2} e^{2}-\left (e^{2} a +c \,d^{2}\right )^{2}\right ) \sqrt {a d e +\left (e^{2} a +c \,d^{2}\right ) x +c d e \,x^{2}}}}{d}-\frac {e \left (\frac {1}{a d e \sqrt {a d e +\left (e^{2} a +c \,d^{2}\right ) x +c d e \,x^{2}}}-\frac {\left (e^{2} a +c \,d^{2}\right ) \left (2 c d e x +e^{2} a +c \,d^{2}\right )}{a d e \left (4 a c \,d^{2} e^{2}-\left (e^{2} a +c \,d^{2}\right )^{2}\right ) \sqrt {a d e +\left (e^{2} a +c \,d^{2}\right ) x +c d e \,x^{2}}}-\frac {\ln \left (\frac {2 a d e +\left (e^{2} a +c \,d^{2}\right ) x +2 \sqrt {a d e}\, \sqrt {a d e +\left (e^{2} a +c \,d^{2}\right ) x +c d e \,x^{2}}}{x}\right )}{a d e \sqrt {a d e}}\right )}{d^{2}}+\frac {e \left (-\frac {2}{3 \left (e^{2} a -c \,d^{2}\right ) \left (x +\frac {d}{e}\right ) \sqrt {c d e \left (x +\frac {d}{e}\right )^{2}+\left (e^{2} a -c \,d^{2}\right ) \left (x +\frac {d}{e}\right )}}+\frac {8 c d e \left (2 c d e \left (x +\frac {d}{e}\right )+e^{2} a -c \,d^{2}\right )}{3 \left (e^{2} a -c \,d^{2}\right )^{3} \sqrt {c d e \left (x +\frac {d}{e}\right )^{2}+\left (e^{2} a -c \,d^{2}\right ) \left (x +\frac {d}{e}\right )}}\right )}{d^{2}}\) | \(716\) |
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Leaf count of result is larger than twice the leaf count of optimal. 896 vs. \(2 (364) = 728\).
Time = 11.47 (sec) , antiderivative size = 1812, normalized size of antiderivative = 4.60 \[ \int \frac {1}{x^2 (d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=\text {Too large to display} \]
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\[ \int \frac {1}{x^2 (d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=\int \frac {1}{x^{2} \left (\left (d + e x\right ) \left (a e + c d x\right )\right )^{\frac {3}{2}} \left (d + e x\right )}\, dx \]
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\[ \int \frac {1}{x^2 (d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=\int { \frac {1}{{\left (c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x\right )}^{\frac {3}{2}} {\left (e x + d\right )} x^{2}} \,d x } \]
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\[ \int \frac {1}{x^2 (d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=\int { \frac {1}{{\left (c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x\right )}^{\frac {3}{2}} {\left (e x + d\right )} x^{2}} \,d x } \]
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Timed out. \[ \int \frac {1}{x^2 (d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=\int \frac {1}{x^2\,\left (d+e\,x\right )\,{\left (c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e\right )}^{3/2}} \,d x \]
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