Integrand size = 20, antiderivative size = 58 \[ \int \frac {x^2 \left (c+d x^2\right )}{a+b x^2} \, dx=\frac {(b c-a d) x}{b^2}+\frac {d x^3}{3 b}-\frac {\sqrt {a} (b c-a d) \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{b^{5/2}} \]
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Time = 0.02 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {470, 327, 211} \[ \int \frac {x^2 \left (c+d x^2\right )}{a+b x^2} \, dx=-\frac {\sqrt {a} \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) (b c-a d)}{b^{5/2}}+\frac {x (b c-a d)}{b^2}+\frac {d x^3}{3 b} \]
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Rule 211
Rule 327
Rule 470
Rubi steps \begin{align*} \text {integral}& = \frac {d x^3}{3 b}-\frac {(-3 b c+3 a d) \int \frac {x^2}{a+b x^2} \, dx}{3 b} \\ & = \frac {(b c-a d) x}{b^2}+\frac {d x^3}{3 b}-\frac {(a (b c-a d)) \int \frac {1}{a+b x^2} \, dx}{b^2} \\ & = \frac {(b c-a d) x}{b^2}+\frac {d x^3}{3 b}-\frac {\sqrt {a} (b c-a d) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{b^{5/2}} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.98 \[ \int \frac {x^2 \left (c+d x^2\right )}{a+b x^2} \, dx=\frac {(b c-a d) x}{b^2}+\frac {d x^3}{3 b}+\frac {\sqrt {a} (-b c+a d) \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{b^{5/2}} \]
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Time = 2.71 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.88
method | result | size |
default | \(-\frac {-\frac {1}{3} b d \,x^{3}+a d x -b c x}{b^{2}}+\frac {\left (a d -b c \right ) a \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{b^{2} \sqrt {a b}}\) | \(51\) |
risch | \(\frac {d \,x^{3}}{3 b}-\frac {a d x}{b^{2}}+\frac {c x}{b}+\frac {\sqrt {-a b}\, \ln \left (-\sqrt {-a b}\, x +a \right ) a d}{2 b^{3}}-\frac {\sqrt {-a b}\, \ln \left (-\sqrt {-a b}\, x +a \right ) c}{2 b^{2}}-\frac {\sqrt {-a b}\, \ln \left (\sqrt {-a b}\, x +a \right ) a d}{2 b^{3}}+\frac {\sqrt {-a b}\, \ln \left (\sqrt {-a b}\, x +a \right ) c}{2 b^{2}}\) | \(121\) |
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Time = 0.30 (sec) , antiderivative size = 129, normalized size of antiderivative = 2.22 \[ \int \frac {x^2 \left (c+d x^2\right )}{a+b x^2} \, dx=\left [\frac {2 \, b d x^{3} - 3 \, {\left (b c - a d\right )} \sqrt {-\frac {a}{b}} \log \left (\frac {b x^{2} + 2 \, b x \sqrt {-\frac {a}{b}} - a}{b x^{2} + a}\right ) + 6 \, {\left (b c - a d\right )} x}{6 \, b^{2}}, \frac {b d x^{3} - 3 \, {\left (b c - a d\right )} \sqrt {\frac {a}{b}} \arctan \left (\frac {b x \sqrt {\frac {a}{b}}}{a}\right ) + 3 \, {\left (b c - a d\right )} x}{3 \, b^{2}}\right ] \]
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Time = 0.18 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.55 \[ \int \frac {x^2 \left (c+d x^2\right )}{a+b x^2} \, dx=x \left (- \frac {a d}{b^{2}} + \frac {c}{b}\right ) - \frac {\sqrt {- \frac {a}{b^{5}}} \left (a d - b c\right ) \log {\left (- b^{2} \sqrt {- \frac {a}{b^{5}}} + x \right )}}{2} + \frac {\sqrt {- \frac {a}{b^{5}}} \left (a d - b c\right ) \log {\left (b^{2} \sqrt {- \frac {a}{b^{5}}} + x \right )}}{2} + \frac {d x^{3}}{3 b} \]
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Time = 0.29 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.93 \[ \int \frac {x^2 \left (c+d x^2\right )}{a+b x^2} \, dx=-\frac {{\left (a b c - a^{2} d\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b} b^{2}} + \frac {b d x^{3} + 3 \, {\left (b c - a d\right )} x}{3 \, b^{2}} \]
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Time = 0.27 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.00 \[ \int \frac {x^2 \left (c+d x^2\right )}{a+b x^2} \, dx=-\frac {{\left (a b c - a^{2} d\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b} b^{2}} + \frac {b^{2} d x^{3} + 3 \, b^{2} c x - 3 \, a b d x}{3 \, b^{3}} \]
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Time = 5.22 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.21 \[ \int \frac {x^2 \left (c+d x^2\right )}{a+b x^2} \, dx=x\,\left (\frac {c}{b}-\frac {a\,d}{b^2}\right )+\frac {d\,x^3}{3\,b}+\frac {\sqrt {a}\,\mathrm {atan}\left (\frac {\sqrt {a}\,\sqrt {b}\,x\,\left (a\,d-b\,c\right )}{a^2\,d-a\,b\,c}\right )\,\left (a\,d-b\,c\right )}{b^{5/2}} \]
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