Integrand size = 20, antiderivative size = 60 \[ \int \frac {x^3 \left (A+B x^2\right )}{\left (a+b x^2\right )^2} \, dx=\frac {B x^2}{2 b^2}+\frac {a (A b-a B)}{2 b^3 \left (a+b x^2\right )}+\frac {(A b-2 a B) \log \left (a+b x^2\right )}{2 b^3} \]
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Time = 0.05 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {457, 78} \[ \int \frac {x^3 \left (A+B x^2\right )}{\left (a+b x^2\right )^2} \, dx=\frac {a (A b-a B)}{2 b^3 \left (a+b x^2\right )}+\frac {(A b-2 a B) \log \left (a+b x^2\right )}{2 b^3}+\frac {B x^2}{2 b^2} \]
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Rule 78
Rule 457
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \text {Subst}\left (\int \frac {x (A+B x)}{(a+b x)^2} \, dx,x,x^2\right ) \\ & = \frac {1}{2} \text {Subst}\left (\int \left (\frac {B}{b^2}+\frac {a (-A b+a B)}{b^2 (a+b x)^2}+\frac {A b-2 a B}{b^2 (a+b x)}\right ) \, dx,x,x^2\right ) \\ & = \frac {B x^2}{2 b^2}+\frac {a (A b-a B)}{2 b^3 \left (a+b x^2\right )}+\frac {(A b-2 a B) \log \left (a+b x^2\right )}{2 b^3} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.83 \[ \int \frac {x^3 \left (A+B x^2\right )}{\left (a+b x^2\right )^2} \, dx=\frac {b B x^2+\frac {a (A b-a B)}{a+b x^2}+(A b-2 a B) \log \left (a+b x^2\right )}{2 b^3} \]
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Time = 2.54 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.95
method | result | size |
norman | \(\frac {\frac {B \,x^{4}}{2 b}+\frac {a \left (A b -2 B a \right )}{2 b^{3}}}{b \,x^{2}+a}+\frac {\left (A b -2 B a \right ) \ln \left (b \,x^{2}+a \right )}{2 b^{3}}\) | \(57\) |
default | \(\frac {B \,x^{2}}{2 b^{2}}+\frac {\frac {\left (A b -2 B a \right ) \ln \left (b \,x^{2}+a \right )}{b}+\frac {a \left (A b -B a \right )}{b \left (b \,x^{2}+a \right )}}{2 b^{2}}\) | \(59\) |
risch | \(\frac {B \,x^{2}}{2 b^{2}}+\frac {a A}{2 b^{2} \left (b \,x^{2}+a \right )}-\frac {a^{2} B}{2 b^{3} \left (b \,x^{2}+a \right )}+\frac {\ln \left (b \,x^{2}+a \right ) A}{2 b^{2}}-\frac {\ln \left (b \,x^{2}+a \right ) B a}{b^{3}}\) | \(74\) |
parallelrisch | \(\frac {b^{2} B \,x^{4}+A \ln \left (b \,x^{2}+a \right ) x^{2} b^{2}-2 B \ln \left (b \,x^{2}+a \right ) x^{2} a b +A \ln \left (b \,x^{2}+a \right ) a b -2 B \ln \left (b \,x^{2}+a \right ) a^{2}+a b A -2 a^{2} B}{2 b^{3} \left (b \,x^{2}+a \right )}\) | \(92\) |
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Time = 0.27 (sec) , antiderivative size = 81, normalized size of antiderivative = 1.35 \[ \int \frac {x^3 \left (A+B x^2\right )}{\left (a+b x^2\right )^2} \, dx=\frac {B b^{2} x^{4} + B a b x^{2} - B a^{2} + A a b - {\left (2 \, B a^{2} - A a b + {\left (2 \, B a b - A b^{2}\right )} x^{2}\right )} \log \left (b x^{2} + a\right )}{2 \, {\left (b^{4} x^{2} + a b^{3}\right )}} \]
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Time = 0.32 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.93 \[ \int \frac {x^3 \left (A+B x^2\right )}{\left (a+b x^2\right )^2} \, dx=\frac {B x^{2}}{2 b^{2}} + \frac {A a b - B a^{2}}{2 a b^{3} + 2 b^{4} x^{2}} - \frac {\left (- A b + 2 B a\right ) \log {\left (a + b x^{2} \right )}}{2 b^{3}} \]
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Time = 0.20 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.00 \[ \int \frac {x^3 \left (A+B x^2\right )}{\left (a+b x^2\right )^2} \, dx=\frac {B x^{2}}{2 \, b^{2}} - \frac {B a^{2} - A a b}{2 \, {\left (b^{4} x^{2} + a b^{3}\right )}} - \frac {{\left (2 \, B a - A b\right )} \log \left (b x^{2} + a\right )}{2 \, b^{3}} \]
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Time = 0.27 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.52 \[ \int \frac {x^3 \left (A+B x^2\right )}{\left (a+b x^2\right )^2} \, dx=\frac {\frac {{\left (b x^{2} + a\right )} B}{b^{2}} + \frac {{\left (2 \, B a - A b\right )} \log \left (\frac {{\left | b x^{2} + a \right |}}{{\left (b x^{2} + a\right )}^{2} {\left | b \right |}}\right )}{b^{2}} - \frac {\frac {B a^{2} b}{b x^{2} + a} - \frac {A a b^{2}}{b x^{2} + a}}{b^{3}}}{2 \, b} \]
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Time = 5.05 (sec) , antiderivative size = 62, normalized size of antiderivative = 1.03 \[ \int \frac {x^3 \left (A+B x^2\right )}{\left (a+b x^2\right )^2} \, dx=\frac {B\,x^2}{2\,b^2}+\frac {\ln \left (b\,x^2+a\right )\,\left (A\,b-2\,B\,a\right )}{2\,b^3}-\frac {B\,a^2-A\,a\,b}{2\,b\,\left (b^3\,x^2+a\,b^2\right )} \]
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