Integrand size = 20, antiderivative size = 90 \[ \int \frac {x}{(d+e x)^2 \sqrt {a+c x^2}} \, dx=\frac {d \sqrt {a+c x^2}}{\left (c d^2+a e^2\right ) (d+e x)}-\frac {a e \text {arctanh}\left (\frac {a e-c d x}{\sqrt {c d^2+a e^2} \sqrt {a+c x^2}}\right )}{\left (c d^2+a e^2\right )^{3/2}} \]
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Time = 0.03 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {821, 739, 212} \[ \int \frac {x}{(d+e x)^2 \sqrt {a+c x^2}} \, dx=\frac {d \sqrt {a+c x^2}}{(d+e x) \left (a e^2+c d^2\right )}-\frac {a e \text {arctanh}\left (\frac {a e-c d x}{\sqrt {a+c x^2} \sqrt {a e^2+c d^2}}\right )}{\left (a e^2+c d^2\right )^{3/2}} \]
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Rule 212
Rule 739
Rule 821
Rubi steps \begin{align*} \text {integral}& = \frac {d \sqrt {a+c x^2}}{\left (c d^2+a e^2\right ) (d+e x)}+\frac {(a e) \int \frac {1}{(d+e x) \sqrt {a+c x^2}} \, dx}{c d^2+a e^2} \\ & = \frac {d \sqrt {a+c x^2}}{\left (c d^2+a e^2\right ) (d+e x)}-\frac {(a e) \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 )}{c d^2+a e^2} \\ & = \frac {d \sqrt {a+c x^2}}{\left (c d^2+a e^2\right ) (d+e x)}-\frac {a e \tanh ^{-1}\left (\frac {a e-c d x}{\sqrt {c d^2+a e^2} \sqrt {a+c x^2}}\right )}{\left (c d^2+a e^2\right )^{3/2}} \\ \end{align*}
Time = 0.40 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.11 \[ \int \frac {x}{(d+e x)^2 \sqrt {a+c x^2}} \, dx=\frac {d \sqrt {a+c x^2}}{\left (c d^2+a e^2\right ) (d+e x)}+\frac {2 a e \arctan \left (\frac {\sqrt {c} (d+e x)-e \sqrt {a+c x^2}}{\sqrt {-c d^2-a e^2}}\right )}{\left (-c d^2-a e^2\right )^{3/2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(343\) vs. \(2(82)=164\).
Time = 0.42 (sec) , antiderivative size = 344, normalized size of antiderivative = 3.82
method | result | size |
default | \(-\frac {\ln \left (\frac {\frac {2 e^{2} a +2 c \,d^{2}}{e^{2}}-\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+2 \sqrt {\frac {e^{2} a +c \,d^{2}}{e^{2}}}\, \sqrt {\left (x +\frac {d}{e}\right )^{2} c -\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+\frac {e^{2} a +c \,d^{2}}{e^{2}}}}{x +\frac {d}{e}}\right )}{e^{2} \sqrt {\frac {e^{2} a +c \,d^{2}}{e^{2}}}}-\frac {d \left (-\frac {e^{2} \sqrt {\left (x +\frac {d}{e}\right )^{2} c -\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+\frac {e^{2} a +c \,d^{2}}{e^{2}}}}{\left (e^{2} a +c \,d^{2}\right ) \left (x +\frac {d}{e}\right )}-\frac {e c d \ln \left (\frac {\frac {2 e^{2} a +2 c \,d^{2}}{e^{2}}-\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+2 \sqrt {\frac {e^{2} a +c \,d^{2}}{e^{2}}}\, \sqrt {\left (x +\frac {d}{e}\right )^{2} c -\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+\frac {e^{2} a +c \,d^{2}}{e^{2}}}}{x +\frac {d}{e}}\right )}{\left (e^{2} a +c \,d^{2}\right ) \sqrt {\frac {e^{2} a +c \,d^{2}}{e^{2}}}}\right )}{e^{3}}\) | \(344\) |
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Leaf count of result is larger than twice the leaf count of optimal. 178 vs. \(2 (83) = 166\).
Time = 0.45 (sec) , antiderivative size = 382, normalized size of antiderivative = 4.24 \[ \int \frac {x}{(d+e x)^2 \sqrt {a+c x^2}} \, dx=\left [\frac {{\left (a e^{2} x + a d e\right )} \sqrt {c d^{2} + a e^{2}} \log \left (\frac {2 \, a c d e x - a c d^{2} - 2 \, a^{2} e^{2} - {\left (2 \, c^{2} d^{2} + a c e^{2}\right )} x^{2} - 2 \, \sqrt {c d^{2} + a e^{2}} {\left (c d x - a e\right )} \sqrt {c x^{2} + a}}{e^{2} x^{2} + 2 \, d e x + d^{2}}\right ) + 2 \, {\left (c d^{3} + a d e^{2}\right )} \sqrt {c x^{2} + a}}{2 \, {\left (c^{2} d^{5} + 2 \, a c d^{3} e^{2} + a^{2} d e^{4} + {\left (c^{2} d^{4} e + 2 \, a c d^{2} e^{3} + a^{2} e^{5}\right )} x\right )}}, -\frac {{\left (a e^{2} x + a d e\right )} \sqrt {-c d^{2} - a e^{2}} \arctan \left (\frac {\sqrt {-c d^{2} - a e^{2}} {\left (c d x - a e\right )} \sqrt {c x^{2} + a}}{a c d^{2} + a^{2} e^{2} + {\left (c^{2} d^{2} + a c e^{2}\right )} x^{2}}\right ) - {\left (c d^{3} + a d e^{2}\right )} \sqrt {c x^{2} + a}}{c^{2} d^{5} + 2 \, a c d^{3} e^{2} + a^{2} d e^{4} + {\left (c^{2} d^{4} e + 2 \, a c d^{2} e^{3} + a^{2} e^{5}\right )} x}\right ] \]
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\[ \int \frac {x}{(d+e x)^2 \sqrt {a+c x^2}} \, dx=\int \frac {x}{\sqrt {a + c x^{2}} \left (d + e x\right )^{2}}\, dx \]
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Exception generated. \[ \int \frac {x}{(d+e x)^2 \sqrt {a+c x^2}} \, dx=\text {Exception raised: ValueError} \]
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\[ \int \frac {x}{(d+e x)^2 \sqrt {a+c x^2}} \, dx=\int { \frac {x}{\sqrt {c x^{2} + a} {\left (e x + d\right )}^{2}} \,d x } \]
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Timed out. \[ \int \frac {x}{(d+e x)^2 \sqrt {a+c x^2}} \, dx=\int \frac {x}{\sqrt {c\,x^2+a}\,{\left (d+e\,x\right )}^2} \,d x \]
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