Integrand size = 19, antiderivative size = 77 \[ \int x^2 \left (\frac {c}{a+b x^2}\right )^{3/2} \, dx=-\frac {c x \sqrt {\frac {c}{a+b x^2}}}{b}+\frac {\sqrt {a} c \sqrt {\frac {c}{a+b x^2}} \sqrt {1+\frac {b x^2}{a}} \text {arcsinh}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{b^{3/2}} \]
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Time = 0.03 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {1973, 294, 221} \[ \int x^2 \left (\frac {c}{a+b x^2}\right )^{3/2} \, dx=\frac {\sqrt {a} c \sqrt {\frac {b x^2}{a}+1} \text {arcsinh}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) \sqrt {\frac {c}{a+b x^2}}}{b^{3/2}}-\frac {c x \sqrt {\frac {c}{a+b x^2}}}{b} \]
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Rule 221
Rule 294
Rule 1973
Rubi steps \begin{align*} \text {integral}& = \frac {\left (c \sqrt {\frac {c}{a+b x^2}} \sqrt {1+\frac {b x^2}{a}}\right ) \int \frac {x^2}{\left (1+\frac {b x^2}{a}\right )^{3/2}} \, dx}{a} \\ & = -\frac {c x \sqrt {\frac {c}{a+b x^2}}}{b}+\frac {\left (c \sqrt {\frac {c}{a+b x^2}} \sqrt {1+\frac {b x^2}{a}}\right ) \int \frac {1}{\sqrt {1+\frac {b x^2}{a}}} \, dx}{b} \\ & = -\frac {c x \sqrt {\frac {c}{a+b x^2}}}{b}+\frac {\sqrt {a} c \sqrt {\frac {c}{a+b x^2}} \sqrt {1+\frac {b x^2}{a}} \sinh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{b^{3/2}} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.95 \[ \int x^2 \left (\frac {c}{a+b x^2}\right )^{3/2} \, dx=-\frac {c \sqrt {\frac {c}{a+b x^2}} \left (\sqrt {b} x+2 \sqrt {a+b x^2} \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a}-\sqrt {a+b x^2}}\right )\right )}{b^{3/2}} \]
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Time = 0.07 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.78
method | result | size |
default | \(-\frac {\left (\frac {c}{b \,x^{2}+a}\right )^{\frac {3}{2}} \left (b \,x^{2}+a \right ) \left (x \,b^{\frac {3}{2}}-\ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right ) b \sqrt {b \,x^{2}+a}\right )}{b^{\frac {5}{2}}}\) | \(60\) |
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Time = 0.29 (sec) , antiderivative size = 141, normalized size of antiderivative = 1.83 \[ \int x^2 \left (\frac {c}{a+b x^2}\right )^{3/2} \, dx=\left [-\frac {2 \, c x \sqrt {\frac {c}{b x^{2} + a}} - c \sqrt {\frac {c}{b}} \log \left (-2 \, b c x^{2} - a c - 2 \, {\left (b^{2} x^{3} + a b x\right )} \sqrt {\frac {c}{b x^{2} + a}} \sqrt {\frac {c}{b}}\right )}{2 \, b}, -\frac {c x \sqrt {\frac {c}{b x^{2} + a}} + c \sqrt {-\frac {c}{b}} \arctan \left (\frac {b x \sqrt {\frac {c}{b x^{2} + a}} \sqrt {-\frac {c}{b}}}{c}\right )}{b}\right ] \]
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\[ \int x^2 \left (\frac {c}{a+b x^2}\right )^{3/2} \, dx=\int x^{2} \left (\frac {c}{a + b x^{2}}\right )^{\frac {3}{2}}\, dx \]
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\[ \int x^2 \left (\frac {c}{a+b x^2}\right )^{3/2} \, dx=\int { x^{2} \left (\frac {c}{b x^{2} + a}\right )^{\frac {3}{2}} \,d x } \]
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Time = 0.34 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.92 \[ \int x^2 \left (\frac {c}{a+b x^2}\right )^{3/2} \, dx=-{\left (\frac {c x \mathrm {sgn}\left (b x^{2} + a\right )}{\sqrt {b c x^{2} + a c} b} + \frac {c \log \left ({\left | -\sqrt {b c} x + \sqrt {b c x^{2} + a c} \right |}\right ) \mathrm {sgn}\left (b x^{2} + a\right )}{\sqrt {b c} b}\right )} c \]
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Timed out. \[ \int x^2 \left (\frac {c}{a+b x^2}\right )^{3/2} \, dx=\int x^2\,{\left (\frac {c}{b\,x^2+a}\right )}^{3/2} \,d x \]
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