Integrand size = 22, antiderivative size = 95 \[ \int \frac {x^2}{(d+e x) \left (a+c x^2\right )^{3/2}} \, dx=-\frac {a e+c d x}{c \left (c d^2+a e^2\right ) \sqrt {a+c x^2}}-\frac {d^2 \text {arctanh}\left (\frac {a e-c d x}{\sqrt {c d^2+a e^2} \sqrt {a+c x^2}}\right )}{\left (c d^2+a e^2\right )^{3/2}} \]
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Time = 0.08 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {1661, 12, 739, 212} \[ \int \frac {x^2}{(d+e x) \left (a+c x^2\right )^{3/2}} \, dx=-\frac {d^2 \text {arctanh}\left (\frac {a e-c d x}{\sqrt {a+c x^2} \sqrt {a e^2+c d^2}}\right )}{\left (a e^2+c d^2\right )^{3/2}}-\frac {a e+c d x}{c \sqrt {a+c x^2} \left (a e^2+c d^2\right )} \]
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Rule 12
Rule 212
Rule 739
Rule 1661
Rubi steps \begin{align*} \text {integral}& = -\frac {a e+c d x}{c \left (c d^2+a e^2\right ) \sqrt {a+c x^2}}+\frac {\int \frac {a c d^2}{\left (c d^2+a e^2\right ) (d+e x) \sqrt {a+c x^2}} \, dx}{a c} \\ & = -\frac {a e+c d x}{c \left (c d^2+a e^2\right ) \sqrt {a+c x^2}}+\frac {d^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}{c \left (c d^2+a e^2\right ) \sqrt {a+c x^2}}-\frac {d^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}{c \left (c d^2+a e^2\right ) \sqrt {a+c x^2}}-\frac {d^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 = 0.47 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.15 \[ \int \frac {x^2}{(d+e x) \left (a+c x^2\right )^{3/2}} \, dx=\frac {-a e-c d x}{c \left (c d^2+a e^2\right ) \sqrt {a+c x^2}}+\frac {2 d^2 \arctan \left (\frac {\sqrt {-c d^2-a e^2} x}{\sqrt {a} (d+e x)-d \sqrt {a+c x^2}}\right )}{\left (-c d^2-a e^2\right )^{3/2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(354\) vs. \(2(88)=176\).
Time = 0.42 (sec) , antiderivative size = 355, normalized size of antiderivative = 3.74
method | result | size |
default | \(-\frac {1}{e c \sqrt {c \,x^{2}+a}}-\frac {d x}{e^{2} a \sqrt {c \,x^{2}+a}}+\frac {d^{2} \left (\frac {e^{2}}{\left (e^{2} a +c \,d^{2}\right ) \sqrt {\left (x +\frac {d}{e}\right )^{2} c -\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+\frac {e^{2} a +c \,d^{2}}{e^{2}}}}+\frac {2 e c d \left (2 c \left (x +\frac {d}{e}\right )-\frac {2 c d}{e}\right )}{\left (e^{2} a +c \,d^{2}\right ) \left (\frac {4 c \left (e^{2} a +c \,d^{2}\right )}{e^{2}}-\frac {4 c^{2} d^{2}}{e^{2}}\right ) \sqrt {\left (x +\frac {d}{e}\right )^{2} c -\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+\frac {e^{2} a +c \,d^{2}}{e^{2}}}}-\frac {e^{2} \ln \left (\frac {\frac {2 e^{2} a +2 c \,d^{2}}{e^{2}}-\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+2 \sqrt {\frac {e^{2} a +c \,d^{2}}{e^{2}}}\, \sqrt {\left (x +\frac {d}{e}\right )^{2} c -\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+\frac {e^{2} a +c \,d^{2}}{e^{2}}}}{x +\frac {d}{e}}\right )}{\left (e^{2} a +c \,d^{2}\right ) \sqrt {\frac {e^{2} a +c \,d^{2}}{e^{2}}}}\right )}{e^{3}}\) | \(355\) |
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Leaf count of result is larger than twice the leaf count of optimal. 214 vs. \(2 (88) = 176\).
Time = 0.36 (sec) , antiderivative size = 455, normalized size of antiderivative = 4.79 \[ \int \frac {x^2}{(d+e x) \left (a+c x^2\right )^{3/2}} \, dx=\left [\frac {{\left (c^{2} d^{2} x^{2} + a c d^{2}\right )} \sqrt {c d^{2} + a e^{2}} \log \left (\frac {2 \, a c d e x - a c d^{2} - 2 \, a^{2} e^{2} - {\left (2 \, c^{2} d^{2} + a c e^{2}\right )} x^{2} - 2 \, \sqrt {c d^{2} + a e^{2}} {\left (c d x - a e\right )} \sqrt {c x^{2} + a}}{e^{2} x^{2} + 2 \, d e x + d^{2}}\right ) - 2 \, {\left (a c d^{2} e + a^{2} e^{3} + {\left (c^{2} d^{3} + a c d e^{2}\right )} x\right )} \sqrt {c x^{2} + a}}{2 \, {\left (a c^{3} d^{4} + 2 \, a^{2} c^{2} d^{2} e^{2} + a^{3} c e^{4} + {\left (c^{4} d^{4} + 2 \, a c^{3} d^{2} e^{2} + a^{2} c^{2} e^{4}\right )} x^{2}\right )}}, -\frac {{\left (c^{2} d^{2} x^{2} + a c d^{2}\right )} \sqrt {-c d^{2} - a e^{2}} \arctan \left (\frac {\sqrt {-c d^{2} - a e^{2}} {\left (c d x - a e\right )} \sqrt {c x^{2} + a}}{a c d^{2} + a^{2} e^{2} + {\left (c^{2} d^{2} + a c e^{2}\right )} x^{2}}\right ) + {\left (a c d^{2} e + a^{2} e^{3} + {\left (c^{2} d^{3} + a c d e^{2}\right )} x\right )} \sqrt {c x^{2} + a}}{a c^{3} d^{4} + 2 \, a^{2} c^{2} d^{2} e^{2} + a^{3} c e^{4} + {\left (c^{4} d^{4} + 2 \, a c^{3} d^{2} e^{2} + a^{2} c^{2} e^{4}\right )} x^{2}}\right ] \]
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\[ \int \frac {x^2}{(d+e x) \left (a+c x^2\right )^{3/2}} \, dx=\int \frac {x^{2}}{\left (a + c x^{2}\right )^{\frac {3}{2}} \left (d + e x\right )}\, dx \]
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Leaf count of result is larger than twice the leaf count of optimal. 179 vs. \(2 (88) = 176\).
Time = 0.22 (sec) , antiderivative size = 179, normalized size of antiderivative = 1.88 \[ \int \frac {x^2}{(d+e x) \left (a+c x^2\right )^{3/2}} \, dx=\frac {c d^{3} x}{\sqrt {c x^{2} + a} a c d^{2} e^{2} + \sqrt {c x^{2} + a} a^{2} e^{4}} + \frac {d^{2}}{\sqrt {c x^{2} + a} c d^{2} e + \sqrt {c x^{2} + a} a e^{3}} - \frac {d x}{\sqrt {c x^{2} + a} a e^{2}} + \frac {d^{2} \operatorname {arsinh}\left (\frac {c d x}{e \sqrt {\frac {a c}{e^{2}}} {\left | e x + d \right |}} - \frac {a}{\sqrt {\frac {a c}{e^{2}}} {\left | e x + d \right |}}\right )}{{\left (a + \frac {c d^{2}}{e^{2}}\right )}^{\frac {3}{2}} e^{3}} - \frac {1}{\sqrt {c x^{2} + a} c e} \]
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Leaf count of result is larger than twice the leaf count of optimal. 181 vs. \(2 (88) = 176\).
Time = 0.28 (sec) , antiderivative size = 181, normalized size of antiderivative = 1.91 \[ \int \frac {x^2}{(d+e x) \left (a+c x^2\right )^{3/2}} \, dx=-\frac {2 \, d^{2} \arctan \left (\frac {{\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )} e + \sqrt {c} d}{\sqrt {-c d^{2} - a e^{2}}}\right )}{{\left (c d^{2} + a e^{2}\right )} \sqrt {-c d^{2} - a e^{2}}} - \frac {\frac {{\left (c^{2} d^{3} + a c d e^{2}\right )} x}{c^{3} d^{4} + 2 \, a c^{2} d^{2} e^{2} + a^{2} c e^{4}} + \frac {a c d^{2} e + a^{2} e^{3}}{c^{3} d^{4} + 2 \, a c^{2} d^{2} e^{2} + a^{2} c e^{4}}}{\sqrt {c x^{2} + a}} \]
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Timed out. \[ \int \frac {x^2}{(d+e x) \left (a+c x^2\right )^{3/2}} \, dx=\int \frac {x^2}{{\left (c\,x^2+a\right )}^{3/2}\,\left (d+e\,x\right )} \,d x \]
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