Integrand size = 20, antiderivative size = 66 \[ \int \frac {x^2 (A+B x)}{\left (a+b x^2\right )^{3/2}} \, dx=-\frac {x (A+B x)}{b \sqrt {a+b x^2}}+\frac {2 B \sqrt {a+b x^2}}{b^2}+\frac {A \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{b^{3/2}} \]
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Time = 0.03 (sec) , antiderivative size = 66, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {833, 655, 223, 212} \[ \int \frac {x^2 (A+B x)}{\left (a+b x^2\right )^{3/2}} \, dx=\frac {A \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{b^{3/2}}-\frac {x (A+B x)}{b \sqrt {a+b x^2}}+\frac {2 B \sqrt {a+b x^2}}{b^2} \]
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Rule 212
Rule 223
Rule 655
Rule 833
Rubi steps \begin{align*} \text {integral}& = -\frac {x (A+B x)}{b \sqrt {a+b x^2}}+\frac {\int \frac {a A+2 a B x}{\sqrt {a+b x^2}} \, dx}{a b} \\ & = -\frac {x (A+B x)}{b \sqrt {a+b x^2}}+\frac {2 B \sqrt {a+b x^2}}{b^2}+\frac {A \int \frac {1}{\sqrt {a+b x^2}} \, dx}{b} \\ & = -\frac {x (A+B x)}{b \sqrt {a+b x^2}}+\frac {2 B \sqrt {a+b x^2}}{b^2}+\frac {A \text {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {x}{\sqrt {a+b x^2}}\right )}{b} \\ & = -\frac {x (A+B x)}{b \sqrt {a+b x^2}}+\frac {2 B \sqrt {a+b x^2}}{b^2}+\frac {A \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{b^{3/2}} \\ \end{align*}
Time = 0.27 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.05 \[ \int \frac {x^2 (A+B x)}{\left (a+b x^2\right )^{3/2}} \, dx=\frac {2 a B-A b x+b B x^2}{b^2 \sqrt {a+b x^2}}+\frac {2 A \text {arctanh}\left (\frac {\sqrt {b} x}{-\sqrt {a}+\sqrt {a+b x^2}}\right )}{b^{3/2}} \]
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Time = 3.53 (sec) , antiderivative size = 68, normalized size of antiderivative = 1.03
method | result | size |
risch | \(\frac {B \sqrt {b \,x^{2}+a}}{b^{2}}-\frac {A x}{b \sqrt {b \,x^{2}+a}}+\frac {A \ln \left (x \sqrt {b}+\sqrt {b \,x^{2}+a}\right )}{b^{\frac {3}{2}}}+\frac {a B}{b^{2} \sqrt {b \,x^{2}+a}}\) | \(68\) |
default | \(B \left (\frac {x^{2}}{\sqrt {b \,x^{2}+a}\, b}+\frac {2 a}{b^{2} \sqrt {b \,x^{2}+a}}\right )+A \left (-\frac {x}{b \sqrt {b \,x^{2}+a}}+\frac {\ln \left (x \sqrt {b}+\sqrt {b \,x^{2}+a}\right )}{b^{\frac {3}{2}}}\right )\) | \(74\) |
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Time = 0.28 (sec) , antiderivative size = 164, normalized size of antiderivative = 2.48 \[ \int \frac {x^2 (A+B x)}{\left (a+b x^2\right )^{3/2}} \, dx=\left [\frac {{\left (A b x^{2} + A a\right )} \sqrt {b} \log \left (-2 \, b x^{2} - 2 \, \sqrt {b x^{2} + a} \sqrt {b} x - a\right ) + 2 \, {\left (B b x^{2} - A b x + 2 \, B a\right )} \sqrt {b x^{2} + a}}{2 \, {\left (b^{3} x^{2} + a b^{2}\right )}}, -\frac {{\left (A b x^{2} + A a\right )} \sqrt {-b} \arctan \left (\frac {\sqrt {-b} x}{\sqrt {b x^{2} + a}}\right ) - {\left (B b x^{2} - A b x + 2 \, B a\right )} \sqrt {b x^{2} + a}}{b^{3} x^{2} + a b^{2}}\right ] \]
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Time = 2.71 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.26 \[ \int \frac {x^2 (A+B x)}{\left (a+b x^2\right )^{3/2}} \, dx=A \left (\frac {\operatorname {asinh}{\left (\frac {\sqrt {b} x}{\sqrt {a}} \right )}}{b^{\frac {3}{2}}} - \frac {x}{\sqrt {a} b \sqrt {1 + \frac {b x^{2}}{a}}}\right ) + B \left (\begin {cases} \frac {2 a}{b^{2} \sqrt {a + b x^{2}}} + \frac {x^{2}}{b \sqrt {a + b x^{2}}} & \text {for}\: b \neq 0 \\\frac {x^{4}}{4 a^{\frac {3}{2}}} & \text {otherwise} \end {cases}\right ) \]
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Time = 0.20 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.97 \[ \int \frac {x^2 (A+B x)}{\left (a+b x^2\right )^{3/2}} \, dx=\frac {B x^{2}}{\sqrt {b x^{2} + a} b} - \frac {A x}{\sqrt {b x^{2} + a} b} + \frac {A \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{b^{\frac {3}{2}}} + \frac {2 \, B a}{\sqrt {b x^{2} + a} b^{2}} \]
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Time = 0.30 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.88 \[ \int \frac {x^2 (A+B x)}{\left (a+b x^2\right )^{3/2}} \, dx=\frac {{\left (\frac {B x}{b} - \frac {A}{b}\right )} x + \frac {2 \, B a}{b^{2}}}{\sqrt {b x^{2} + a}} - \frac {A \log \left ({\left | -\sqrt {b} x + \sqrt {b x^{2} + a} \right |}\right )}{b^{\frac {3}{2}}} \]
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Time = 6.06 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.92 \[ \int \frac {x^2 (A+B x)}{\left (a+b x^2\right )^{3/2}} \, dx=\frac {A\,\ln \left (\sqrt {b}\,x+\sqrt {b\,x^2+a}\right )}{b^{3/2}}-\frac {A\,x}{b\,\sqrt {b\,x^2+a}}+\frac {B\,\left (b\,x^2+2\,a\right )}{b^2\,\sqrt {b\,x^2+a}} \]
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