Integrand size = 22, antiderivative size = 268 \[ \int \frac {1}{x^3 (d+e x)^2 \sqrt {a+c x^2}} \, dx=-\frac {\sqrt {a+c x^2}}{2 a d^2 x^2}+\frac {2 e \sqrt {a+c x^2}}{a d^3 x}+\frac {e^4 \sqrt {a+c x^2}}{d^3 \left (c d^2+a e^2\right ) (d+e x)}+\frac {c e^3 \text {arctanh}\left (\frac {a e-c d x}{\sqrt {c d^2+a e^2} \sqrt {a+c x^2}}\right )}{d^2 \left (c d^2+a e^2\right )^{3/2}}+\frac {3 e^3 \text {arctanh}\left (\frac {a e-c d x}{\sqrt {c d^2+a e^2} \sqrt {a+c x^2}}\right )}{d^4 \sqrt {c d^2+a e^2}}+\frac {c \text {arctanh}\left (\frac {\sqrt {a+c x^2}}{\sqrt {a}}\right )}{2 a^{3/2} d^2}-\frac {3 e^2 \text {arctanh}\left (\frac {\sqrt {a+c x^2}}{\sqrt {a}}\right )}{\sqrt {a} d^4} \]
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Time = 0.14 (sec) , antiderivative size = 268, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.409, Rules used = {975, 272, 44, 65, 214, 270, 745, 739, 212} \[ \int \frac {1}{x^3 (d+e x)^2 \sqrt {a+c x^2}} \, dx=\frac {c \text {arctanh}\left (\frac {\sqrt {a+c x^2}}{\sqrt {a}}\right )}{2 a^{3/2} d^2}-\frac {3 e^2 \text {arctanh}\left (\frac {\sqrt {a+c x^2}}{\sqrt {a}}\right )}{\sqrt {a} d^4}+\frac {c e^3 \text {arctanh}\left (\frac {a e-c d x}{\sqrt {a+c x^2} \sqrt {a e^2+c d^2}}\right )}{d^2 \left (a e^2+c d^2\right )^{3/2}}+\frac {3 e^3 \text {arctanh}\left (\frac {a e-c d x}{\sqrt {a+c x^2} \sqrt {a e^2+c d^2}}\right )}{d^4 \sqrt {a e^2+c d^2}}+\frac {2 e \sqrt {a+c x^2}}{a d^3 x}-\frac {\sqrt {a+c x^2}}{2 a d^2 x^2}+\frac {e^4 \sqrt {a+c x^2}}{d^3 (d+e x) \left (a e^2+c d^2\right )} \]
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Rule 44
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^3 \sqrt {a+c x^2}}-\frac {2 e}{d^3 x^2 \sqrt {a+c x^2}}+\frac {3 e^2}{d^4 x \sqrt {a+c x^2}}-\frac {e^3}{d^3 (d+e x)^2 \sqrt {a+c x^2}}-\frac {3 e^3}{d^4 (d+e x) \sqrt {a+c x^2}}\right ) \, dx \\ & = \frac {\int \frac {1}{x^3 \sqrt {a+c x^2}} \, dx}{d^2}-\frac {(2 e) \int \frac {1}{x^2 \sqrt {a+c x^2}} \, dx}{d^3}+\frac {\left (3 e^2\right ) \int \frac {1}{x \sqrt {a+c x^2}} \, dx}{d^4}-\frac {\left (3 e^3\right ) \int \frac {1}{(d+e x) \sqrt {a+c x^2}} \, dx}{d^4}-\frac {e^3 \int \frac {1}{(d+e x)^2 \sqrt {a+c x^2}} \, dx}{d^3} \\ & = \frac {2 e \sqrt {a+c x^2}}{a d^3 x}+\frac {e^4 \sqrt {a+c x^2}}{d^3 \left (c d^2+a e^2\right ) (d+e x)}+\frac {\text {Subst}\left (\int \frac {1}{x^2 \sqrt {a+c x}} \, dx,x,x^2\right )}{2 d^2}+\frac {\left (3 e^2\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {a+c x}} \, dx,x,x^2\right )}{2 d^4}+\frac {\left (3 e^3\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^4}-\frac {\left (c e^3\right ) \int \frac {1}{(d+e x) \sqrt {a+c x^2}} \, dx}{d^2 \left (c d^2+a e^2\right )} \\ & = -\frac {\sqrt {a+c x^2}}{2 a d^2 x^2}+\frac {2 e \sqrt {a+c x^2}}{a d^3 x}+\frac {e^4 \sqrt {a+c x^2}}{d^3 \left (c d^2+a e^2\right ) (d+e x)}+\frac {3 e^3 \tanh ^{-1}\left (\frac {a e-c d x}{\sqrt {c d^2+a e^2} \sqrt {a+c x^2}}\right )}{d^4 \sqrt {c d^2+a e^2}}-\frac {c \text {Subst}\left (\int \frac {1}{x \sqrt {a+c x}} \, dx,x,x^2\right )}{4 a d^2}+\frac {\left (3 e^2\right ) \text {Subst}\left (\int \frac {1}{-\frac {a}{c}+\frac {x^2}{c}} \, dx,x,\sqrt {a+c x^2}\right )}{c d^4}+\frac {\left (c e^3\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^2 \left (c d^2+a e^2\right )} \\ & = -\frac {\sqrt {a+c x^2}}{2 a d^2 x^2}+\frac {2 e \sqrt {a+c x^2}}{a d^3 x}+\frac {e^4 \sqrt {a+c x^2}}{d^3 \left (c d^2+a e^2\right ) (d+e x)}+\frac {c e^3 \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 {3 e^3 \tanh ^{-1}\left (\frac {a e-c d x}{\sqrt {c d^2+a e^2} \sqrt {a+c x^2}}\right )}{d^4 \sqrt {c d^2+a e^2}}-\frac {3 e^2 \tanh ^{-1}\left (\frac {\sqrt {a+c x^2}}{\sqrt {a}}\right )}{\sqrt {a} d^4}-\frac {\text {Subst}\left (\int \frac {1}{-\frac {a}{c}+\frac {x^2}{c}} \, dx,x,\sqrt {a+c x^2}\right )}{2 a d^2} \\ & = -\frac {\sqrt {a+c x^2}}{2 a d^2 x^2}+\frac {2 e \sqrt {a+c x^2}}{a d^3 x}+\frac {e^4 \sqrt {a+c x^2}}{d^3 \left (c d^2+a e^2\right ) (d+e x)}+\frac {c e^3 \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 {3 e^3 \tanh ^{-1}\left (\frac {a e-c d x}{\sqrt {c d^2+a e^2} \sqrt {a+c x^2}}\right )}{d^4 \sqrt {c d^2+a e^2}}+\frac {c \tanh ^{-1}\left (\frac {\sqrt {a+c x^2}}{\sqrt {a}}\right )}{2 a^{3/2} d^2}-\frac {3 e^2 \tanh ^{-1}\left (\frac {\sqrt {a+c x^2}}{\sqrt {a}}\right )}{\sqrt {a} d^4} \\ \end{align*}
Time = 1.13 (sec) , antiderivative size = 224, normalized size of antiderivative = 0.84 \[ \int \frac {1}{x^3 (d+e x)^2 \sqrt {a+c x^2}} \, dx=\frac {\frac {d \sqrt {a+c x^2} \left (c d^2 \left (-d^2+3 d e x+4 e^2 x^2\right )+a e^2 \left (-d^2+3 d e x+6 e^2 x^2\right )\right )}{a \left (c d^2+a e^2\right ) x^2 (d+e x)}-\frac {4 e^3 \left (4 c d^2+3 a e^2\right ) \arctan \left (\frac {\sqrt {c} (d+e x)-e \sqrt {a+c x^2}}{\sqrt {-c d^2-a e^2}}\right )}{\left (-c d^2-a e^2\right )^{3/2}}+\frac {2 \left (-c d^2+6 a e^2\right ) \text {arctanh}\left (\frac {\sqrt {c} x-\sqrt {a+c x^2}}{\sqrt {a}}\right )}{a^{3/2}}}{2 d^4} \]
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Time = 0.42 (sec) , antiderivative size = 424, normalized size of antiderivative = 1.58
method | result | size |
risch | \(-\frac {\sqrt {c \,x^{2}+a}\, \left (-4 e x +d \right )}{2 a \,d^{3} x^{2}}-\frac {2 a e \left (-\frac {e^{2} \sqrt {\left (x +\frac {d}{e}\right )^{2} c -\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+\frac {e^{2} a +c \,d^{2}}{e^{2}}}}{\left (e^{2} a +c \,d^{2}\right ) \left (x +\frac {d}{e}\right )}-\frac {e c d \ln \left (\frac {\frac {2 e^{2} a +2 c \,d^{2}}{e^{2}}-\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+2 \sqrt {\frac {e^{2} a +c \,d^{2}}{e^{2}}}\, \sqrt {\left (x +\frac {d}{e}\right )^{2} c -\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+\frac {e^{2} a +c \,d^{2}}{e^{2}}}}{x +\frac {d}{e}}\right )}{\left (e^{2} a +c \,d^{2}\right ) \sqrt {\frac {e^{2} a +c \,d^{2}}{e^{2}}}}\right )-\frac {6 e^{2} a \ln \left (\frac {\frac {2 e^{2} a +2 c \,d^{2}}{e^{2}}-\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+2 \sqrt {\frac {e^{2} a +c \,d^{2}}{e^{2}}}\, \sqrt {\left (x +\frac {d}{e}\right )^{2} c -\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+\frac {e^{2} a +c \,d^{2}}{e^{2}}}}{x +\frac {d}{e}}\right )}{d \sqrt {\frac {e^{2} a +c \,d^{2}}{e^{2}}}}-\frac {\left (-6 e^{2} a +c \,d^{2}\right ) \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {c \,x^{2}+a}}{x}\right )}{d \sqrt {a}}}{2 d^{3} a}\) | \(424\) |
default | \(\frac {-\frac {\sqrt {c \,x^{2}+a}}{2 a \,x^{2}}+\frac {c \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {c \,x^{2}+a}}{x}\right )}{2 a^{\frac {3}{2}}}}{d^{2}}-\frac {3 e^{2} \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {c \,x^{2}+a}}{x}\right )}{d^{4} \sqrt {a}}+\frac {2 e \sqrt {c \,x^{2}+a}}{a \,d^{3} x}-\frac {e \left (-\frac {e^{2} \sqrt {\left (x +\frac {d}{e}\right )^{2} c -\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+\frac {e^{2} a +c \,d^{2}}{e^{2}}}}{\left (e^{2} a +c \,d^{2}\right ) \left (x +\frac {d}{e}\right )}-\frac {e c d \ln \left (\frac {\frac {2 e^{2} a +2 c \,d^{2}}{e^{2}}-\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+2 \sqrt {\frac {e^{2} a +c \,d^{2}}{e^{2}}}\, \sqrt {\left (x +\frac {d}{e}\right )^{2} c -\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+\frac {e^{2} a +c \,d^{2}}{e^{2}}}}{x +\frac {d}{e}}\right )}{\left (e^{2} a +c \,d^{2}\right ) \sqrt {\frac {e^{2} a +c \,d^{2}}{e^{2}}}}\right )}{d^{3}}+\frac {3 e^{2} \ln \left (\frac {\frac {2 e^{2} a +2 c \,d^{2}}{e^{2}}-\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+2 \sqrt {\frac {e^{2} a +c \,d^{2}}{e^{2}}}\, \sqrt {\left (x +\frac {d}{e}\right )^{2} c -\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+\frac {e^{2} a +c \,d^{2}}{e^{2}}}}{x +\frac {d}{e}}\right )}{d^{4} \sqrt {\frac {e^{2} a +c \,d^{2}}{e^{2}}}}\) | \(453\) |
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Time = 1.02 (sec) , antiderivative size = 1867, normalized size of antiderivative = 6.97 \[ \int \frac {1}{x^3 (d+e x)^2 \sqrt {a+c x^2}} \, dx=\text {Too large to display} \]
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\[ \int \frac {1}{x^3 (d+e x)^2 \sqrt {a+c x^2}} \, dx=\int \frac {1}{x^{3} \sqrt {a + c x^{2}} \left (d + e x\right )^{2}}\, dx \]
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\[ \int \frac {1}{x^3 (d+e x)^2 \sqrt {a+c x^2}} \, dx=\int { \frac {1}{\sqrt {c x^{2} + a} {\left (e x + d\right )}^{2} x^{3}} \,d x } \]
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\[ \int \frac {1}{x^3 (d+e x)^2 \sqrt {a+c x^2}} \, dx=\int { \frac {1}{\sqrt {c x^{2} + a} {\left (e x + d\right )}^{2} x^{3}} \,d x } \]
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Timed out. \[ \int \frac {1}{x^3 (d+e x)^2 \sqrt {a+c x^2}} \, dx=\int \frac {1}{x^3\,\sqrt {c\,x^2+a}\,{\left (d+e\,x\right )}^2} \,d x \]
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