Integrand size = 22, antiderivative size = 212 \[ \int \frac {1}{x^2 (d+e x)^2 \sqrt {a+c x^2}} \, dx=-\frac {\sqrt {a+c x^2}}{a d^2 x}-\frac {e^3 \sqrt {a+c x^2}}{d^2 \left (c d^2+a e^2\right ) (d+e x)}-\frac {c e^2 \text {arctanh}\left (\frac {a e-c d x}{\sqrt {c d^2+a e^2} \sqrt {a+c x^2}}\right )}{d \left (c d^2+a e^2\right )^{3/2}}-\frac {2 e^2 \text {arctanh}\left (\frac {a e-c d x}{\sqrt {c d^2+a e^2} \sqrt {a+c x^2}}\right )}{d^3 \sqrt {c d^2+a e^2}}+\frac {2 e \text {arctanh}\left (\frac {\sqrt {a+c x^2}}{\sqrt {a}}\right )}{\sqrt {a} d^3} \]
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Time = 0.11 (sec) , antiderivative size = 212, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {975, 270, 272, 65, 214, 745, 739, 212} \[ \int \frac {1}{x^2 (d+e x)^2 \sqrt {a+c x^2}} \, dx=\frac {2 e \text {arctanh}\left (\frac {\sqrt {a+c x^2}}{\sqrt {a}}\right )}{\sqrt {a} d^3}-\frac {c e^2 \text {arctanh}\left (\frac {a e-c d x}{\sqrt {a+c x^2} \sqrt {a e^2+c d^2}}\right )}{d \left (a e^2+c d^2\right )^{3/2}}-\frac {2 e^2 \text {arctanh}\left (\frac {a e-c d x}{\sqrt {a+c x^2} \sqrt {a e^2+c d^2}}\right )}{d^3 \sqrt {a e^2+c d^2}}-\frac {e^3 \sqrt {a+c x^2}}{d^2 (d+e x) \left (a e^2+c d^2\right )}-\frac {\sqrt {a+c x^2}}{a d^2 x} \]
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Rule 65
Rule 212
Rule 214
Rule 270
Rule 272
Rule 739
Rule 745
Rule 975
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {1}{d^2 x^2 \sqrt {a+c x^2}}-\frac {2 e}{d^3 x \sqrt {a+c x^2}}+\frac {e^2}{d^2 (d+e x)^2 \sqrt {a+c x^2}}+\frac {2 e^2}{d^3 (d+e x) \sqrt {a+c x^2}}\right ) \, dx \\ & = \frac {\int \frac {1}{x^2 \sqrt {a+c x^2}} \, dx}{d^2}-\frac {(2 e) \int \frac {1}{x \sqrt {a+c x^2}} \, dx}{d^3}+\frac {\left (2 e^2\right ) \int \frac {1}{(d+e x) \sqrt {a+c x^2}} \, dx}{d^3}+\frac {e^2 \int \frac {1}{(d+e x)^2 \sqrt {a+c x^2}} \, dx}{d^2} \\ & = -\frac {\sqrt {a+c x^2}}{a d^2 x}-\frac {e^3 \sqrt {a+c x^2}}{d^2 \left (c d^2+a e^2\right ) (d+e x)}-\frac {e \text {Subst}\left (\int \frac {1}{x \sqrt {a+c x}} \, dx,x,x^2\right )}{d^3}-\frac {\left (2 e^2\right ) \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^3}+\frac {\left (c e^2\right ) \int \frac {1}{(d+e x) \sqrt {a+c x^2}} \, dx}{d \left (c d^2+a e^2\right )} \\ & = -\frac {\sqrt {a+c x^2}}{a d^2 x}-\frac {e^3 \sqrt {a+c x^2}}{d^2 \left (c d^2+a e^2\right ) (d+e x)}-\frac {2 e^2 \tanh ^{-1}\left (\frac {a e-c d x}{\sqrt {c d^2+a e^2} \sqrt {a+c x^2}}\right )}{d^3 \sqrt {c d^2+a e^2}}-\frac {(2 e) \text {Subst}\left (\int \frac {1}{-\frac {a}{c}+\frac {x^2}{c}} \, dx,x,\sqrt {a+c x^2}\right )}{c d^3}-\frac {\left (c e^2\right ) \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 \left (c d^2+a e^2\right )} \\ & = -\frac {\sqrt {a+c x^2}}{a d^2 x}-\frac {e^3 \sqrt {a+c x^2}}{d^2 \left (c d^2+a e^2\right ) (d+e x)}-\frac {c e^2 \tanh ^{-1}\left (\frac {a e-c d x}{\sqrt {c d^2+a e^2} \sqrt {a+c x^2}}\right )}{d \left (c d^2+a e^2\right )^{3/2}}-\frac {2 e^2 \tanh ^{-1}\left (\frac {a e-c d x}{\sqrt {c d^2+a e^2} \sqrt {a+c x^2}}\right )}{d^3 \sqrt {c d^2+a e^2}}+\frac {2 e \tanh ^{-1}\left (\frac {\sqrt {a+c x^2}}{\sqrt {a}}\right )}{\sqrt {a} d^3} \\ \end{align*}
Time = 0.91 (sec) , antiderivative size = 183, normalized size of antiderivative = 0.86 \[ \int \frac {1}{x^2 (d+e x)^2 \sqrt {a+c x^2}} \, dx=\frac {-\frac {d \sqrt {a+c x^2} \left (c d^2 (d+e x)+a e^2 (d+2 e x)\right )}{a \left (c d^2+a e^2\right ) x (d+e x)}+\frac {2 e^2 \left (3 c d^2+2 a e^2\right ) \arctan \left (\frac {\sqrt {c} (d+e x)-e \sqrt {a+c x^2}}{\sqrt {-c d^2-a e^2}}\right )}{\left (-c d^2-a e^2\right )^{3/2}}-\frac {4 e \text {arctanh}\left (\frac {\sqrt {c} x-\sqrt {a+c x^2}}{\sqrt {a}}\right )}{\sqrt {a}}}{d^3} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(394\) vs. \(2(190)=380\).
Time = 0.39 (sec) , antiderivative size = 395, normalized size of antiderivative = 1.86
method | result | size |
default | \(-\frac {\sqrt {c \,x^{2}+a}}{a \,d^{2} x}+\frac {2 e \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {c \,x^{2}+a}}{x}\right )}{d^{3} \sqrt {a}}+\frac {-\frac {e^{2} \sqrt {\left (x +\frac {d}{e}\right )^{2} c -\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+\frac {e^{2} a +c \,d^{2}}{e^{2}}}}{\left (e^{2} a +c \,d^{2}\right ) \left (x +\frac {d}{e}\right )}-\frac {e c d \ln \left (\frac {\frac {2 e^{2} a +2 c \,d^{2}}{e^{2}}-\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+2 \sqrt {\frac {e^{2} a +c \,d^{2}}{e^{2}}}\, \sqrt {\left (x +\frac {d}{e}\right )^{2} c -\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+\frac {e^{2} a +c \,d^{2}}{e^{2}}}}{x +\frac {d}{e}}\right )}{\left (e^{2} a +c \,d^{2}\right ) \sqrt {\frac {e^{2} a +c \,d^{2}}{e^{2}}}}}{d^{2}}-\frac {2 e \ln \left (\frac {\frac {2 e^{2} a +2 c \,d^{2}}{e^{2}}-\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+2 \sqrt {\frac {e^{2} a +c \,d^{2}}{e^{2}}}\, \sqrt {\left (x +\frac {d}{e}\right )^{2} c -\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+\frac {e^{2} a +c \,d^{2}}{e^{2}}}}{x +\frac {d}{e}}\right )}{d^{3} \sqrt {\frac {e^{2} a +c \,d^{2}}{e^{2}}}}\) | \(395\) |
risch | \(-\frac {\sqrt {c \,x^{2}+a}}{a \,d^{2} x}+\frac {2 e \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {c \,x^{2}+a}}{x}\right )}{d^{3} \sqrt {a}}-\frac {e^{2} \sqrt {\left (x +\frac {d}{e}\right )^{2} c -\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+\frac {e^{2} a +c \,d^{2}}{e^{2}}}}{d^{2} \left (e^{2} a +c \,d^{2}\right ) \left (x +\frac {d}{e}\right )}-\frac {e c \ln \left (\frac {\frac {2 e^{2} a +2 c \,d^{2}}{e^{2}}-\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+2 \sqrt {\frac {e^{2} a +c \,d^{2}}{e^{2}}}\, \sqrt {\left (x +\frac {d}{e}\right )^{2} c -\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+\frac {e^{2} a +c \,d^{2}}{e^{2}}}}{x +\frac {d}{e}}\right )}{d \left (e^{2} a +c \,d^{2}\right ) \sqrt {\frac {e^{2} a +c \,d^{2}}{e^{2}}}}-\frac {2 e \ln \left (\frac {\frac {2 e^{2} a +2 c \,d^{2}}{e^{2}}-\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+2 \sqrt {\frac {e^{2} a +c \,d^{2}}{e^{2}}}\, \sqrt {\left (x +\frac {d}{e}\right )^{2} c -\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+\frac {e^{2} a +c \,d^{2}}{e^{2}}}}{x +\frac {d}{e}}\right )}{d^{3} \sqrt {\frac {e^{2} a +c \,d^{2}}{e^{2}}}}\) | \(395\) |
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Time = 0.67 (sec) , antiderivative size = 1512, normalized size of antiderivative = 7.13 \[ \int \frac {1}{x^2 (d+e x)^2 \sqrt {a+c x^2}} \, dx=\text {Too large to display} \]
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\[ \int \frac {1}{x^2 (d+e x)^2 \sqrt {a+c x^2}} \, dx=\int \frac {1}{x^{2} \sqrt {a + c x^{2}} \left (d + e x\right )^{2}}\, dx \]
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\[ \int \frac {1}{x^2 (d+e x)^2 \sqrt {a+c x^2}} \, dx=\int { \frac {1}{\sqrt {c x^{2} + a} {\left (e x + d\right )}^{2} x^{2}} \,d x } \]
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\[ \int \frac {1}{x^2 (d+e x)^2 \sqrt {a+c x^2}} \, dx=\int { \frac {1}{\sqrt {c x^{2} + a} {\left (e x + d\right )}^{2} x^{2}} \,d x } \]
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Timed out. \[ \int \frac {1}{x^2 (d+e x)^2 \sqrt {a+c x^2}} \, dx=\int \frac {1}{x^2\,\sqrt {c\,x^2+a}\,{\left (d+e\,x\right )}^2} \,d x \]
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