Integrand size = 20, antiderivative size = 89 \[ \int \frac {x^2 \left (A+B x^2\right )}{\left (a+b x^2\right )^3} \, dx=-\frac {(A b-a B) x}{4 b^2 \left (a+b x^2\right )^2}+\frac {(A b-5 a B) x}{8 a b^2 \left (a+b x^2\right )}+\frac {(A b+3 a B) \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{8 a^{3/2} b^{5/2}} \]
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Time = 0.04 (sec) , antiderivative size = 89, 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 = {466, 393, 211} \[ \int \frac {x^2 \left (A+B x^2\right )}{\left (a+b x^2\right )^3} \, dx=\frac {(3 a B+A b) \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{8 a^{3/2} b^{5/2}}+\frac {x (A b-5 a B)}{8 a b^2 \left (a+b x^2\right )}-\frac {x (A b-a B)}{4 b^2 \left (a+b x^2\right )^2} \]
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Rule 211
Rule 393
Rule 466
Rubi steps \begin{align*} \text {integral}& = -\frac {(A b-a B) x}{4 b^2 \left (a+b x^2\right )^2}-\frac {\int \frac {-A b+a B-4 b B x^2}{\left (a+b x^2\right )^2} \, dx}{4 b^2} \\ & = -\frac {(A b-a B) x}{4 b^2 \left (a+b x^2\right )^2}+\frac {(A b-5 a B) x}{8 a b^2 \left (a+b x^2\right )}+\frac {(A b+3 a B) \int \frac {1}{a+b x^2} \, dx}{8 a b^2} \\ & = -\frac {(A b-a B) x}{4 b^2 \left (a+b x^2\right )^2}+\frac {(A b-5 a B) x}{8 a b^2 \left (a+b x^2\right )}+\frac {(A b+3 a B) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{8 a^{3/2} b^{5/2}} \\ \end{align*}
Time = 0.05 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.93 \[ \int \frac {x^2 \left (A+B x^2\right )}{\left (a+b x^2\right )^3} \, dx=\frac {\frac {\sqrt {b} x \left (-3 a^2 B+A b^2 x^2-a b \left (A+5 B x^2\right )\right )}{a \left (a+b x^2\right )^2}+\frac {(A b+3 a B) \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{a^{3/2}}}{8 b^{5/2}} \]
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Time = 2.54 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.85
method | result | size |
default | \(\frac {\frac {\left (A b -5 B a \right ) x^{3}}{8 a b}-\frac {\left (A b +3 B a \right ) x}{8 b^{2}}}{\left (b \,x^{2}+a \right )^{2}}+\frac {\left (A b +3 B a \right ) \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{8 b^{2} a \sqrt {a b}}\) | \(76\) |
risch | \(\frac {\frac {\left (A b -5 B a \right ) x^{3}}{8 a b}-\frac {\left (A b +3 B a \right ) x}{8 b^{2}}}{\left (b \,x^{2}+a \right )^{2}}-\frac {\ln \left (b x +\sqrt {-a b}\right ) A}{16 \sqrt {-a b}\, b a}-\frac {3 \ln \left (b x +\sqrt {-a b}\right ) B}{16 \sqrt {-a b}\, b^{2}}+\frac {\ln \left (-b x +\sqrt {-a b}\right ) A}{16 \sqrt {-a b}\, b a}+\frac {3 \ln \left (-b x +\sqrt {-a b}\right ) B}{16 \sqrt {-a b}\, b^{2}}\) | \(146\) |
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Time = 0.30 (sec) , antiderivative size = 301, normalized size of antiderivative = 3.38 \[ \int \frac {x^2 \left (A+B x^2\right )}{\left (a+b x^2\right )^3} \, dx=\left [-\frac {2 \, {\left (5 \, B a^{2} b^{2} - A a b^{3}\right )} x^{3} + {\left ({\left (3 \, B a b^{2} + A b^{3}\right )} x^{4} + 3 \, B a^{3} + A a^{2} b + 2 \, {\left (3 \, B a^{2} b + A a b^{2}\right )} x^{2}\right )} \sqrt {-a b} \log \left (\frac {b x^{2} - 2 \, \sqrt {-a b} x - a}{b x^{2} + a}\right ) + 2 \, {\left (3 \, B a^{3} b + A a^{2} b^{2}\right )} x}{16 \, {\left (a^{2} b^{5} x^{4} + 2 \, a^{3} b^{4} x^{2} + a^{4} b^{3}\right )}}, -\frac {{\left (5 \, B a^{2} b^{2} - A a b^{3}\right )} x^{3} - {\left ({\left (3 \, B a b^{2} + A b^{3}\right )} x^{4} + 3 \, B a^{3} + A a^{2} b + 2 \, {\left (3 \, B a^{2} b + A a b^{2}\right )} x^{2}\right )} \sqrt {a b} \arctan \left (\frac {\sqrt {a b} x}{a}\right ) + {\left (3 \, B a^{3} b + A a^{2} b^{2}\right )} x}{8 \, {\left (a^{2} b^{5} x^{4} + 2 \, a^{3} b^{4} x^{2} + a^{4} b^{3}\right )}}\right ] \]
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Time = 0.40 (sec) , antiderivative size = 155, normalized size of antiderivative = 1.74 \[ \int \frac {x^2 \left (A+B x^2\right )}{\left (a+b x^2\right )^3} \, dx=- \frac {\sqrt {- \frac {1}{a^{3} b^{5}}} \left (A b + 3 B a\right ) \log {\left (- a^{2} b^{2} \sqrt {- \frac {1}{a^{3} b^{5}}} + x \right )}}{16} + \frac {\sqrt {- \frac {1}{a^{3} b^{5}}} \left (A b + 3 B a\right ) \log {\left (a^{2} b^{2} \sqrt {- \frac {1}{a^{3} b^{5}}} + x \right )}}{16} + \frac {x^{3} \left (A b^{2} - 5 B a b\right ) + x \left (- A a b - 3 B a^{2}\right )}{8 a^{3} b^{2} + 16 a^{2} b^{3} x^{2} + 8 a b^{4} x^{4}} \]
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Time = 0.29 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.03 \[ \int \frac {x^2 \left (A+B x^2\right )}{\left (a+b x^2\right )^3} \, dx=-\frac {{\left (5 \, B a b - A b^{2}\right )} x^{3} + {\left (3 \, B a^{2} + A a b\right )} x}{8 \, {\left (a b^{4} x^{4} + 2 \, a^{2} b^{3} x^{2} + a^{3} b^{2}\right )}} + \frac {{\left (3 \, B a + A b\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{8 \, \sqrt {a b} a b^{2}} \]
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Time = 0.28 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.88 \[ \int \frac {x^2 \left (A+B x^2\right )}{\left (a+b x^2\right )^3} \, dx=\frac {{\left (3 \, B a + A b\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{8 \, \sqrt {a b} a b^{2}} - \frac {5 \, B a b x^{3} - A b^{2} x^{3} + 3 \, B a^{2} x + A a b x}{8 \, {\left (b x^{2} + a\right )}^{2} a b^{2}} \]
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Time = 4.99 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.92 \[ \int \frac {x^2 \left (A+B x^2\right )}{\left (a+b x^2\right )^3} \, dx=\frac {\mathrm {atan}\left (\frac {\sqrt {b}\,x}{\sqrt {a}}\right )\,\left (A\,b+3\,B\,a\right )}{8\,a^{3/2}\,b^{5/2}}-\frac {\frac {x\,\left (A\,b+3\,B\,a\right )}{8\,b^2}-\frac {x^3\,\left (A\,b-5\,B\,a\right )}{8\,a\,b}}{a^2+2\,a\,b\,x^2+b^2\,x^4} \]
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