Integrand size = 19, antiderivative size = 54 \[ \int \frac {c+d x^2}{\left (a+b x^2\right )^{3/2}} \, dx=\frac {(b c-a d) x}{a b \sqrt {a+b x^2}}+\frac {d \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{b^{3/2}} \]
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Time = 0.01 (sec) , antiderivative size = 54, 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 = {393, 223, 212} \[ \int \frac {c+d x^2}{\left (a+b x^2\right )^{3/2}} \, dx=\frac {d \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{b^{3/2}}+\frac {x (b c-a d)}{a b \sqrt {a+b x^2}} \]
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Rule 212
Rule 223
Rule 393
Rubi steps \begin{align*} \text {integral}& = \frac {(b c-a d) x}{a b \sqrt {a+b x^2}}+\frac {d \int \frac {1}{\sqrt {a+b x^2}} \, dx}{b} \\ & = \frac {(b c-a d) x}{a b \sqrt {a+b x^2}}+\frac {d \text {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {x}{\sqrt {a+b x^2}}\right )}{b} \\ & = \frac {(b c-a d) x}{a b \sqrt {a+b x^2}}+\frac {d \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{b^{3/2}} \\ \end{align*}
Time = 0.08 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.07 \[ \int \frac {c+d x^2}{\left (a+b x^2\right )^{3/2}} \, dx=\frac {b c x-a d x}{a b \sqrt {a+b x^2}}-\frac {d \log \left (-\sqrt {b} x+\sqrt {a+b x^2}\right )}{b^{3/2}} \]
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Time = 2.31 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.02
method | result | size |
default | \(\frac {c x}{a \sqrt {b \,x^{2}+a}}+d \left (-\frac {x}{b \sqrt {b \,x^{2}+a}}+\frac {\ln \left (x \sqrt {b}+\sqrt {b \,x^{2}+a}\right )}{b^{\frac {3}{2}}}\right )\) | \(55\) |
pseudoelliptic | \(\frac {\operatorname {arctanh}\left (\frac {\sqrt {b \,x^{2}+a}}{x \sqrt {b}}\right ) a d \sqrt {b \,x^{2}+a}-a d x \sqrt {b}+b^{\frac {3}{2}} c x}{b^{\frac {3}{2}} \sqrt {b \,x^{2}+a}\, a}\) | \(61\) |
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Time = 0.27 (sec) , antiderivative size = 167, normalized size of antiderivative = 3.09 \[ \int \frac {c+d x^2}{\left (a+b x^2\right )^{3/2}} \, dx=\left [\frac {2 \, {\left (b^{2} c - a b d\right )} \sqrt {b x^{2} + a} x + {\left (a b d x^{2} + a^{2} d\right )} \sqrt {b} \log \left (-2 \, b x^{2} - 2 \, \sqrt {b x^{2} + a} \sqrt {b} x - a\right )}{2 \, {\left (a b^{3} x^{2} + a^{2} b^{2}\right )}}, \frac {{\left (b^{2} c - a b d\right )} \sqrt {b x^{2} + a} x - {\left (a b d x^{2} + a^{2} d\right )} \sqrt {-b} \arctan \left (\frac {\sqrt {-b} x}{\sqrt {b x^{2} + a}}\right )}{a b^{3} x^{2} + a^{2} b^{2}}\right ] \]
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Time = 2.12 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.11 \[ \int \frac {c+d x^2}{\left (a+b x^2\right )^{3/2}} \, dx=d \left (\frac {\operatorname {asinh}{\left (\frac {\sqrt {b} x}{\sqrt {a}} \right )}}{b^{\frac {3}{2}}} - \frac {x}{\sqrt {a} b \sqrt {1 + \frac {b x^{2}}{a}}}\right ) + \frac {c x}{a^{\frac {3}{2}} \sqrt {1 + \frac {b x^{2}}{a}}} \]
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Time = 0.19 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.85 \[ \int \frac {c+d x^2}{\left (a+b x^2\right )^{3/2}} \, dx=\frac {c x}{\sqrt {b x^{2} + a} a} - \frac {d x}{\sqrt {b x^{2} + a} b} + \frac {d \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{b^{\frac {3}{2}}} \]
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Time = 0.27 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.93 \[ \int \frac {c+d x^2}{\left (a+b x^2\right )^{3/2}} \, dx=-\frac {d \log \left ({\left | -\sqrt {b} x + \sqrt {b x^{2} + a} \right |}\right )}{b^{\frac {3}{2}}} + \frac {{\left (b c - a d\right )} x}{\sqrt {b x^{2} + a} a b} \]
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Time = 4.82 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.98 \[ \int \frac {c+d x^2}{\left (a+b x^2\right )^{3/2}} \, dx=\frac {d\,\ln \left (\sqrt {b}\,x+\sqrt {b\,x^2+a}\right )}{b^{3/2}}+\frac {c\,x}{a\,\sqrt {b\,x^2+a}}-\frac {d\,x}{b\,\sqrt {b\,x^2+a}} \]
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