Integrand size = 40, antiderivative size = 229 \[ \int \frac {1}{x^2 (d+e x) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=-\frac {2 e (a e+c d x)}{d \left (c d^2-a e^2\right ) x \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}-\frac {\left (c d^2-3 a e^2\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{a d^2 e \left (c d^2-a e^2\right ) x}+\frac {\left (c d^2+3 a e^2\right ) \text {arctanh}\left (\frac {2 a d e+\left (c d^2+a e^2\right ) x}{2 \sqrt {a} \sqrt {d} \sqrt {e} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\right )}{2 a^{3/2} d^{5/2} e^{3/2}} \]
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Time = 0.18 (sec) , antiderivative size = 229, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {865, 836, 820, 738, 212} \[ \int \frac {1}{x^2 (d+e x) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=\frac {\left (3 a e^2+c d^2\right ) \text {arctanh}\left (\frac {x \left (a e^2+c d^2\right )+2 a d e}{2 \sqrt {a} \sqrt {d} \sqrt {e} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{2 a^{3/2} d^{5/2} e^{3/2}}-\frac {\left (c d^2-3 a e^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{a d^2 e x \left (c d^2-a e^2\right )}-\frac {2 e (a e+c d x)}{d x \left (c d^2-a e^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}} \]
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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 )^{3/2}} \, dx \\ & = -\frac {2 e (a e+c d x)}{d \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}}-\frac {2 \int \frac {-\frac {1}{2} a e \left (c d^2-3 a e^2\right ) \left (c d^2-a e^2\right )+a c d e^2 \left (c d^2-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}{a d e \left (c d^2-a e^2\right )^2} \\ & = -\frac {2 e (a e+c d x)}{d \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}}-\frac {\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}}{a d^2 e \left (c d^2-a e^2\right ) x}-\frac {1}{2} \left (\frac {c}{a e}+\frac {3 e}{d^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 \\ & = -\frac {2 e (a e+c d x)}{d \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}}-\frac {\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}}{a d^2 e \left (c d^2-a e^2\right ) x}-\left (-\frac {c}{a e}-\frac {3 e}{d^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 ) \\ & = -\frac {2 e (a e+c d x)}{d \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}}-\frac {\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}}{a d^2 e \left (c d^2-a e^2\right ) x}+\frac {\left (c d^2+3 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^{3/2} d^{5/2} e^{3/2}} \\ \end{align*}
Time = 0.26 (sec) , antiderivative size = 201, normalized size of antiderivative = 0.88 \[ \int \frac {1}{x^2 (d+e x) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=\frac {\sqrt {a} \sqrt {d} \sqrt {e} \left (-c^2 d^3 x (d+e x)+a^2 e^3 (d+3 e x)-a c d e \left (d^2-3 e^2 x^2\right )\right )+\left (c^2 d^4+2 a c d^2 e^2-3 a^2 e^4\right ) x \sqrt {a e+c d x} \sqrt {d+e x} \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a e+c d x}}{\sqrt {a} \sqrt {e} \sqrt {d+e x}}\right )}{a^{3/2} d^{5/2} e^{3/2} \left (c d^2-a e^2\right ) x \sqrt {(a e+c d x) (d+e x)}} \]
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Time = 0.69 (sec) , antiderivative size = 270, normalized size of antiderivative = 1.18
method | result | size |
default | \(\frac {-\frac {\sqrt {a d e +\left (e^{2} a +c \,d^{2}\right ) x +c d e \,x^{2}}}{a d e x}+\frac {\left (e^{2} a +c \,d^{2}\right ) \ln \left (\frac {2 a d e +\left (e^{2} a +c \,d^{2}\right ) x +2 \sqrt {a d e}\, \sqrt {a d e +\left (e^{2} a +c \,d^{2}\right ) x +c d e \,x^{2}}}{x}\right )}{2 a d e \sqrt {a d e}}}{d}+\frac {e \ln \left (\frac {2 a d e +\left (e^{2} a +c \,d^{2}\right ) x +2 \sqrt {a d e}\, \sqrt {a d e +\left (e^{2} a +c \,d^{2}\right ) x +c d e \,x^{2}}}{x}\right )}{d^{2} \sqrt {a d e}}-\frac {2 e \sqrt {c d e \left (x +\frac {d}{e}\right )^{2}+\left (e^{2} a -c \,d^{2}\right ) \left (x +\frac {d}{e}\right )}}{d^{2} \left (e^{2} a -c \,d^{2}\right ) \left (x +\frac {d}{e}\right )}\) | \(270\) |
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Time = 0.97 (sec) , antiderivative size = 610, normalized size of antiderivative = 2.66 \[ \int \frac {1}{x^2 (d+e x) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=\left [\frac {\sqrt {a d e} {\left ({\left (c^{2} d^{4} e + 2 \, a c d^{2} e^{3} - 3 \, a^{2} e^{5}\right )} x^{2} + {\left (c^{2} d^{5} + 2 \, a c d^{3} e^{2} - 3 \, a^{2} d e^{4}\right )} x\right )} \log \left (\frac {8 \, a^{2} d^{2} e^{2} + {\left (c^{2} d^{4} + 6 \, a c d^{2} e^{2} + a^{2} e^{4}\right )} x^{2} + 4 \, \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} {\left (2 \, a d e + {\left (c d^{2} + a e^{2}\right )} x\right )} \sqrt {a d e} + 8 \, {\left (a c d^{3} e + a^{2} d e^{3}\right )} x}{x^{2}}\right ) - 4 \, {\left (a c d^{4} e - a^{2} d^{2} e^{3} + {\left (a c d^{3} e^{2} - 3 \, a^{2} d e^{4}\right )} x\right )} \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x}}{4 \, {\left ({\left (a^{2} c d^{5} e^{3} - a^{3} d^{3} e^{5}\right )} x^{2} + {\left (a^{2} c d^{6} e^{2} - a^{3} d^{4} e^{4}\right )} x\right )}}, -\frac {\sqrt {-a d e} {\left ({\left (c^{2} d^{4} e + 2 \, a c d^{2} e^{3} - 3 \, a^{2} e^{5}\right )} x^{2} + {\left (c^{2} d^{5} + 2 \, a c d^{3} e^{2} - 3 \, a^{2} d e^{4}\right )} x\right )} \arctan \left (\frac {\sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} {\left (2 \, a d e + {\left (c d^{2} + a e^{2}\right )} x\right )} \sqrt {-a d e}}{2 \, {\left (a c d^{2} e^{2} x^{2} + a^{2} d^{2} e^{2} + {\left (a c d^{3} e + a^{2} d e^{3}\right )} x\right )}}\right ) + 2 \, {\left (a c d^{4} e - a^{2} d^{2} e^{3} + {\left (a c d^{3} e^{2} - 3 \, a^{2} d e^{4}\right )} x\right )} \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x}}{2 \, {\left ({\left (a^{2} c d^{5} e^{3} - a^{3} d^{3} e^{5}\right )} x^{2} + {\left (a^{2} c d^{6} e^{2} - a^{3} d^{4} e^{4}\right )} x\right )}}\right ] \]
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\[ \int \frac {1}{x^2 (d+e x) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=\int \frac {1}{x^{2} \sqrt {\left (d + e x\right ) \left (a e + c d x\right )} \left (d + e x\right )}\, dx \]
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\[ \int \frac {1}{x^2 (d+e x) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=\int { \frac {1}{\sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} {\left (e x + d\right )} x^{2}} \,d x } \]
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Exception generated. \[ \int \frac {1}{x^2 (d+e x) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int \frac {1}{x^2 (d+e x) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=\int \frac {1}{x^2\,\left (d+e\,x\right )\,\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}} \,d x \]
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