Integrand size = 19, antiderivative size = 94 \[ \int \frac {1}{(d+e x) \left (a+c x^2\right )^{3/2}} \, dx=\frac {a e+c d x}{a \left (c d^2+a e^2\right ) \sqrt {a+c x^2}}-\frac {e^2 \text {arctanh}\left (\frac {a e-c d x}{\sqrt {c d^2+a e^2} \sqrt {a+c x^2}}\right )}{\left (c d^2+a e^2\right )^{3/2}} \]
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Time = 0.03 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.211, Rules used = {755, 12, 739, 212} \[ \int \frac {1}{(d+e x) \left (a+c x^2\right )^{3/2}} \, dx=\frac {a e+c d x}{a \sqrt {a+c x^2} \left (a e^2+c d^2\right )}-\frac {e^2 \text {arctanh}\left (\frac {a e-c d x}{\sqrt {a+c x^2} \sqrt {a e^2+c d^2}}\right )}{\left (a e^2+c d^2\right )^{3/2}} \]
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Rule 12
Rule 212
Rule 739
Rule 755
Rubi steps \begin{align*} \text {integral}& = \frac {a e+c d x}{a \left (c d^2+a e^2\right ) \sqrt {a+c x^2}}+\frac {\int \frac {a e^2}{(d+e x) \sqrt {a+c x^2}} \, dx}{a \left (c d^2+a e^2\right )} \\ & = \frac {a e+c d x}{a \left (c d^2+a e^2\right ) \sqrt {a+c x^2}}+\frac {e^2 \int \frac {1}{(d+e x) \sqrt {a+c x^2}} \, dx}{c d^2+a e^2} \\ & = \frac {a e+c d x}{a \left (c d^2+a e^2\right ) \sqrt {a+c x^2}}-\frac {e^2 \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 {a e+c d x}{a \left (c d^2+a e^2\right ) \sqrt {a+c x^2}}-\frac {e^2 \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 = 0.05 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.11 \[ \int \frac {1}{(d+e x) \left (a+c x^2\right )^{3/2}} \, dx=\frac {a e+c d x}{a \left (c d^2+a e^2\right ) \sqrt {a+c x^2}}+\frac {2 e^2 \arctan \left (\frac {\sqrt {c} (d+e x)-e \sqrt {a+c x^2}}{\sqrt {-c d^2-a e^2}}\right )}{\left (-c d^2-a e^2\right )^{3/2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(314\) vs. \(2(86)=172\).
Time = 0.35 (sec) , antiderivative size = 315, normalized size of antiderivative = 3.35
method | result | size |
default | \(\frac {\frac {e^{2}}{\left (e^{2} a +c \,d^{2}\right ) \sqrt {\left (x +\frac {d}{e}\right )^{2} c -\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+\frac {e^{2} a +c \,d^{2}}{e^{2}}}}+\frac {2 e c d \left (2 c \left (x +\frac {d}{e}\right )-\frac {2 c d}{e}\right )}{\left (e^{2} a +c \,d^{2}\right ) \left (\frac {4 c \left (e^{2} a +c \,d^{2}\right )}{e^{2}}-\frac {4 c^{2} d^{2}}{e^{2}}\right ) \sqrt {\left (x +\frac {d}{e}\right )^{2} c -\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+\frac {e^{2} a +c \,d^{2}}{e^{2}}}}-\frac {e^{2} \ln \left (\frac {\frac {2 e^{2} a +2 c \,d^{2}}{e^{2}}-\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+2 \sqrt {\frac {e^{2} a +c \,d^{2}}{e^{2}}}\, \sqrt {\left (x +\frac {d}{e}\right )^{2} c -\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+\frac {e^{2} a +c \,d^{2}}{e^{2}}}}{x +\frac {d}{e}}\right )}{\left (e^{2} a +c \,d^{2}\right ) \sqrt {\frac {e^{2} a +c \,d^{2}}{e^{2}}}}}{e}\) | \(315\) |
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Leaf count of result is larger than twice the leaf count of optimal. 215 vs. \(2 (87) = 174\).
Time = 0.32 (sec) , antiderivative size = 456, normalized size of antiderivative = 4.85 \[ \int \frac {1}{(d+e x) \left (a+c x^2\right )^{3/2}} \, dx=\left [\frac {{\left (a c e^{2} x^{2} + a^{2} e^{2}\right )} \sqrt {c d^{2} + a e^{2}} \log \left (\frac {2 \, a c d e x - a c d^{2} - 2 \, a^{2} e^{2} - {\left (2 \, c^{2} d^{2} + a c e^{2}\right )} x^{2} - 2 \, \sqrt {c d^{2} + a e^{2}} {\left (c d x - a e\right )} \sqrt {c x^{2} + a}}{e^{2} x^{2} + 2 \, d e x + d^{2}}\right ) + 2 \, {\left (a c d^{2} e + a^{2} e^{3} + {\left (c^{2} d^{3} + a c d e^{2}\right )} x\right )} \sqrt {c x^{2} + a}}{2 \, {\left (a^{2} c^{2} d^{4} + 2 \, a^{3} c d^{2} e^{2} + a^{4} e^{4} + {\left (a c^{3} d^{4} + 2 \, a^{2} c^{2} d^{2} e^{2} + a^{3} c e^{4}\right )} x^{2}\right )}}, -\frac {{\left (a c e^{2} x^{2} + a^{2} e^{2}\right )} \sqrt {-c d^{2} - a e^{2}} \arctan \left (\frac {\sqrt {-c d^{2} - a e^{2}} {\left (c d x - a e\right )} \sqrt {c x^{2} + a}}{a c d^{2} + a^{2} e^{2} + {\left (c^{2} d^{2} + a c e^{2}\right )} x^{2}}\right ) - {\left (a c d^{2} e + a^{2} e^{3} + {\left (c^{2} d^{3} + a c d e^{2}\right )} x\right )} \sqrt {c x^{2} + a}}{a^{2} c^{2} d^{4} + 2 \, a^{3} c d^{2} e^{2} + a^{4} e^{4} + {\left (a c^{3} d^{4} + 2 \, a^{2} c^{2} d^{2} e^{2} + a^{3} c e^{4}\right )} x^{2}}\right ] \]
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\[ \int \frac {1}{(d+e x) \left (a+c x^2\right )^{3/2}} \, dx=\int \frac {1}{\left (a + c x^{2}\right )^{\frac {3}{2}} \left (d + e x\right )}\, dx \]
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Time = 0.20 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.39 \[ \int \frac {1}{(d+e x) \left (a+c x^2\right )^{3/2}} \, dx=\frac {c d x}{\sqrt {c x^{2} + a} a c d^{2} + \sqrt {c x^{2} + a} a^{2} e^{2}} + \frac {1}{\frac {\sqrt {c x^{2} + a} c d^{2}}{e} + \sqrt {c x^{2} + a} a e} + \frac {\operatorname {arsinh}\left (\frac {c d x}{e \sqrt {\frac {a c}{e^{2}}} {\left | e x + d \right |}} - \frac {a}{\sqrt {\frac {a c}{e^{2}}} {\left | e x + d \right |}}\right )}{{\left (a + \frac {c d^{2}}{e^{2}}\right )}^{\frac {3}{2}} e} \]
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Leaf count of result is larger than twice the leaf count of optimal. 180 vs. \(2 (87) = 174\).
Time = 0.28 (sec) , antiderivative size = 180, normalized size of antiderivative = 1.91 \[ \int \frac {1}{(d+e x) \left (a+c x^2\right )^{3/2}} \, dx=-\frac {2 \, e^{2} \arctan \left (\frac {{\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )} e + \sqrt {c} d}{\sqrt {-c d^{2} - a e^{2}}}\right )}{{\left (c d^{2} + a e^{2}\right )} \sqrt {-c d^{2} - a e^{2}}} + \frac {\frac {{\left (c^{2} d^{3} + a c d e^{2}\right )} x}{a c^{2} d^{4} + 2 \, a^{2} c d^{2} e^{2} + a^{3} e^{4}} + \frac {a c d^{2} e + a^{2} e^{3}}{a c^{2} d^{4} + 2 \, a^{2} c d^{2} e^{2} + a^{3} e^{4}}}{\sqrt {c x^{2} + a}} \]
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Timed out. \[ \int \frac {1}{(d+e x) \left (a+c x^2\right )^{3/2}} \, dx=\int \frac {1}{{\left (c\,x^2+a\right )}^{3/2}\,\left (d+e\,x\right )} \,d x \]
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