Integrand size = 22, antiderivative size = 194 \[ \int \frac {1}{x^2 (d+e x) \left (a+c x^2\right )^{3/2}} \, dx=-\frac {e}{a d^2 \sqrt {a+c x^2}}-\frac {1}{a d x \sqrt {a+c x^2}}-\frac {2 c x}{a^2 d \sqrt {a+c x^2}}+\frac {e^2 (a e+c d x)}{a d^2 \left (c d^2+a e^2\right ) \sqrt {a+c x^2}}-\frac {e^4 \text {arctanh}\left (\frac {a e-c d x}{\sqrt {c d^2+a e^2} \sqrt {a+c x^2}}\right )}{d^2 \left (c d^2+a e^2\right )^{3/2}}+\frac {e \text {arctanh}\left (\frac {\sqrt {a+c x^2}}{\sqrt {a}}\right )}{a^{3/2} d^2} \]
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Time = 0.11 (sec) , antiderivative size = 194, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {975, 277, 197, 272, 53, 65, 214, 755, 12, 739, 212} \[ \int \frac {1}{x^2 (d+e x) \left (a+c x^2\right )^{3/2}} \, dx=\frac {e \text {arctanh}\left (\frac {\sqrt {a+c x^2}}{\sqrt {a}}\right )}{a^{3/2} d^2}-\frac {2 c x}{a^2 d \sqrt {a+c x^2}}-\frac {e^4 \text {arctanh}\left (\frac {a e-c d x}{\sqrt {a+c x^2} \sqrt {a e^2+c d^2}}\right )}{d^2 \left (a e^2+c d^2\right )^{3/2}}+\frac {e^2 (a e+c d x)}{a d^2 \sqrt {a+c x^2} \left (a e^2+c d^2\right )}-\frac {e}{a d^2 \sqrt {a+c x^2}}-\frac {1}{a d x \sqrt {a+c x^2}} \]
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Rule 12
Rule 53
Rule 65
Rule 197
Rule 212
Rule 214
Rule 272
Rule 277
Rule 739
Rule 755
Rule 975
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {1}{d x^2 \left (a+c x^2\right )^{3/2}}-\frac {e}{d^2 x \left (a+c x^2\right )^{3/2}}+\frac {e^2}{d^2 (d+e x) \left (a+c x^2\right )^{3/2}}\right ) \, dx \\ & = \frac {\int \frac {1}{x^2 \left (a+c x^2\right )^{3/2}} \, dx}{d}-\frac {e \int \frac {1}{x \left (a+c x^2\right )^{3/2}} \, dx}{d^2}+\frac {e^2 \int \frac {1}{(d+e x) \left (a+c x^2\right )^{3/2}} \, dx}{d^2} \\ & = -\frac {1}{a d x \sqrt {a+c x^2}}+\frac {e^2 (a e+c d x)}{a d^2 \left (c d^2+a e^2\right ) \sqrt {a+c x^2}}-\frac {(2 c) \int \frac {1}{\left (a+c x^2\right )^{3/2}} \, dx}{a d}-\frac {e \text {Subst}\left (\int \frac {1}{x (a+c x)^{3/2}} \, dx,x,x^2\right )}{2 d^2}+\frac {e^2 \int \frac {a e^2}{(d+e x) \sqrt {a+c x^2}} \, dx}{a d^2 \left (c d^2+a e^2\right )} \\ & = -\frac {e}{a d^2 \sqrt {a+c x^2}}-\frac {1}{a d x \sqrt {a+c x^2}}-\frac {2 c x}{a^2 d \sqrt {a+c x^2}}+\frac {e^2 (a e+c d x)}{a d^2 \left (c d^2+a e^2\right ) \sqrt {a+c x^2}}-\frac {e \text {Subst}\left (\int \frac {1}{x \sqrt {a+c x}} \, dx,x,x^2\right )}{2 a d^2}+\frac {e^4 \int \frac {1}{(d+e x) \sqrt {a+c x^2}} \, dx}{d^2 \left (c d^2+a e^2\right )} \\ & = -\frac {e}{a d^2 \sqrt {a+c x^2}}-\frac {1}{a d x \sqrt {a+c x^2}}-\frac {2 c x}{a^2 d \sqrt {a+c x^2}}+\frac {e^2 (a e+c d x)}{a d^2 \left (c d^2+a e^2\right ) \sqrt {a+c x^2}}-\frac {e \text {Subst}\left (\int \frac {1}{-\frac {a}{c}+\frac {x^2}{c}} \, dx,x,\sqrt {a+c x^2}\right )}{a c d^2}-\frac {e^4 \text {Subst}\left (\int \frac {1}{c d^2+a e^2-x^2} \, dx,x,\frac {a e-c d x}{\sqrt {a+c x^2}}\right )}{d^2 \left (c d^2+a e^2\right )} \\ & = -\frac {e}{a d^2 \sqrt {a+c x^2}}-\frac {1}{a d x \sqrt {a+c x^2}}-\frac {2 c x}{a^2 d \sqrt {a+c x^2}}+\frac {e^2 (a e+c d x)}{a d^2 \left (c d^2+a e^2\right ) \sqrt {a+c x^2}}-\frac {e^4 \tanh ^{-1}\left (\frac {a e-c d x}{\sqrt {c d^2+a e^2} \sqrt {a+c x^2}}\right )}{d^2 \left (c d^2+a e^2\right )^{3/2}}+\frac {e \tanh ^{-1}\left (\frac {\sqrt {a+c x^2}}{\sqrt {a}}\right )}{a^{3/2} d^2} \\ \end{align*}
Time = 0.69 (sec) , antiderivative size = 178, normalized size of antiderivative = 0.92 \[ \int \frac {1}{x^2 (d+e x) \left (a+c x^2\right )^{3/2}} \, dx=\frac {-\frac {d \left (a^2 e^2+2 c^2 d^2 x^2+a c \left (d^2+d e x+e^2 x^2\right )\right )}{a^2 \left (c d^2+a e^2\right ) x \sqrt {a+c x^2}}+\frac {2 e^4 \arctan \left (\frac {\sqrt {c} (d+e x)-e \sqrt {a+c x^2}}{\sqrt {-c d^2-a e^2}}\right )}{\left (-c d^2-a e^2\right )^{3/2}}-\frac {2 e \text {arctanh}\left (\frac {\sqrt {c} x-\sqrt {a+c x^2}}{\sqrt {a}}\right )}{a^{3/2}}}{d^2} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(360\) vs. \(2(174)=348\).
Time = 0.39 (sec) , antiderivative size = 361, normalized size of antiderivative = 1.86
method | result | size |
risch | \(-\frac {\sqrt {c \,x^{2}+a}}{a^{2} d x}+\frac {e \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {c \,x^{2}+a}}{x}\right )}{a^{\frac {3}{2}} d^{2}}-\frac {c \sqrt {\left (x -\frac {\sqrt {-a c}}{c}\right )^{2} c +2 \sqrt {-a c}\, \left (x -\frac {\sqrt {-a c}}{c}\right )}}{2 a^{2} \left (e \sqrt {-a c}+c d \right ) \left (x -\frac {\sqrt {-a c}}{c}\right )}+\frac {c \sqrt {\left (x +\frac {\sqrt {-a c}}{c}\right )^{2} c -2 \sqrt {-a c}\, \left (x +\frac {\sqrt {-a c}}{c}\right )}}{2 a^{2} \left (e \sqrt {-a c}-c d \right ) \left (x +\frac {\sqrt {-a c}}{c}\right )}+\frac {c \,e^{3} \ln \left (\frac {\frac {2 e^{2} a +2 c \,d^{2}}{e^{2}}-\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+2 \sqrt {\frac {e^{2} a +c \,d^{2}}{e^{2}}}\, \sqrt {\left (x +\frac {d}{e}\right )^{2} c -\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+\frac {e^{2} a +c \,d^{2}}{e^{2}}}}{x +\frac {d}{e}}\right )}{d^{2} \left (e \sqrt {-a c}+c d \right ) \left (e \sqrt {-a c}-c d \right ) \sqrt {\frac {e^{2} a +c \,d^{2}}{e^{2}}}}\) | \(361\) |
default | \(\frac {-\frac {1}{a x \sqrt {c \,x^{2}+a}}-\frac {2 c x}{a^{2} \sqrt {c \,x^{2}+a}}}{d}-\frac {e \left (\frac {1}{a \sqrt {c \,x^{2}+a}}-\frac {\ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {c \,x^{2}+a}}{x}\right )}{a^{\frac {3}{2}}}\right )}{d^{2}}+\frac {e \left (\frac {e^{2}}{\left (e^{2} a +c \,d^{2}\right ) \sqrt {\left (x +\frac {d}{e}\right )^{2} c -\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+\frac {e^{2} a +c \,d^{2}}{e^{2}}}}+\frac {2 e c d \left (2 c \left (x +\frac {d}{e}\right )-\frac {2 c d}{e}\right )}{\left (e^{2} a +c \,d^{2}\right ) \left (\frac {4 c \left (e^{2} a +c \,d^{2}\right )}{e^{2}}-\frac {4 c^{2} d^{2}}{e^{2}}\right ) \sqrt {\left (x +\frac {d}{e}\right )^{2} c -\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+\frac {e^{2} a +c \,d^{2}}{e^{2}}}}-\frac {e^{2} \ln \left (\frac {\frac {2 e^{2} a +2 c \,d^{2}}{e^{2}}-\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+2 \sqrt {\frac {e^{2} a +c \,d^{2}}{e^{2}}}\, \sqrt {\left (x +\frac {d}{e}\right )^{2} c -\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+\frac {e^{2} a +c \,d^{2}}{e^{2}}}}{x +\frac {d}{e}}\right )}{\left (e^{2} a +c \,d^{2}\right ) \sqrt {\frac {e^{2} a +c \,d^{2}}{e^{2}}}}\right )}{d^{2}}\) | \(403\) |
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Leaf count of result is larger than twice the leaf count of optimal. 369 vs. \(2 (175) = 350\).
Time = 0.50 (sec) , antiderivative size = 1556, normalized size of antiderivative = 8.02 \[ \int \frac {1}{x^2 (d+e x) \left (a+c x^2\right )^{3/2}} \, dx=\text {Too large to display} \]
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\[ \int \frac {1}{x^2 (d+e x) \left (a+c x^2\right )^{3/2}} \, dx=\int \frac {1}{x^{2} \left (a + c x^{2}\right )^{\frac {3}{2}} \left (d + e x\right )}\, dx \]
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\[ \int \frac {1}{x^2 (d+e x) \left (a+c x^2\right )^{3/2}} \, dx=\int { \frac {1}{{\left (c x^{2} + a\right )}^{\frac {3}{2}} {\left (e x + d\right )} x^{2}} \,d x } \]
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Time = 0.28 (sec) , antiderivative size = 273, normalized size of antiderivative = 1.41 \[ \int \frac {1}{x^2 (d+e x) \left (a+c x^2\right )^{3/2}} \, dx=-\frac {2 \, e^{4} \arctan \left (\frac {{\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )} e + \sqrt {c} d}{\sqrt {-c d^{2} - a e^{2}}}\right )}{{\left (c d^{4} + a d^{2} e^{2}\right )} \sqrt {-c d^{2} - a e^{2}}} - \frac {\frac {{\left (a c^{3} d^{3} + a^{2} c^{2} d e^{2}\right )} x}{a^{3} c^{2} d^{4} + 2 \, a^{4} c d^{2} e^{2} + a^{5} e^{4}} + \frac {a^{2} c^{2} d^{2} e + a^{3} c e^{3}}{a^{3} c^{2} d^{4} + 2 \, a^{4} c d^{2} e^{2} + a^{5} e^{4}}}{\sqrt {c x^{2} + a}} - \frac {2 \, e \arctan \left (-\frac {\sqrt {c} x - \sqrt {c x^{2} + a}}{\sqrt {-a}}\right )}{\sqrt {-a} a d^{2}} + \frac {2 \, \sqrt {c}}{{\left ({\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{2} - a\right )} a d} \]
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Timed out. \[ \int \frac {1}{x^2 (d+e x) \left (a+c x^2\right )^{3/2}} \, dx=\int \frac {1}{x^2\,{\left (c\,x^2+a\right )}^{3/2}\,\left (d+e\,x\right )} \,d x \]
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