Integrand size = 20, antiderivative size = 54 \[ \int \frac {x^3 \left (A+B x^2\right )}{a+b x^2} \, dx=\frac {(A b-a B) x^2}{2 b^2}+\frac {B x^4}{4 b}-\frac {a (A b-a B) \log \left (a+b x^2\right )}{2 b^3} \]
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Time = 0.05 (sec) , antiderivative size = 54, 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} \[ \int \frac {x^3 \left (A+B x^2\right )}{a+b x^2} \, dx=-\frac {a (A b-a B) \log \left (a+b x^2\right )}{2 b^3}+\frac {x^2 (A b-a B)}{2 b^2}+\frac {B x^4}{4 b} \]
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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} \, dx,x,x^2\right ) \\ & = \frac {1}{2} \text {Subst}\left (\int \left (\frac {A b-a B}{b^2}+\frac {B x}{b}+\frac {a (-A b+a B)}{b^2 (a+b x)}\right ) \, dx,x,x^2\right ) \\ & = \frac {(A b-a B) x^2}{2 b^2}+\frac {B x^4}{4 b}-\frac {a (A b-a B) \log \left (a+b x^2\right )}{2 b^3} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.87 \[ \int \frac {x^3 \left (A+B x^2\right )}{a+b x^2} \, dx=\frac {b x^2 \left (2 A b-2 a B+b B x^2\right )+2 a (-A b+a B) \log \left (a+b x^2\right )}{4 b^3} \]
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Time = 2.52 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.91
method | result | size |
norman | \(\frac {\left (A b -B a \right ) x^{2}}{2 b^{2}}+\frac {B \,x^{4}}{4 b}-\frac {a \left (A b -B a \right ) \ln \left (b \,x^{2}+a \right )}{2 b^{3}}\) | \(49\) |
default | \(\frac {\frac {1}{2} b B \,x^{4}+A b \,x^{2}-B a \,x^{2}}{2 b^{2}}-\frac {a \left (A b -B a \right ) \ln \left (b \,x^{2}+a \right )}{2 b^{3}}\) | \(50\) |
parallelrisch | \(-\frac {-b^{2} B \,x^{4}-2 A \,b^{2} x^{2}+2 B a b \,x^{2}+2 A \ln \left (b \,x^{2}+a \right ) a b -2 B \ln \left (b \,x^{2}+a \right ) a^{2}}{4 b^{3}}\) | \(60\) |
risch | \(\frac {B \,x^{4}}{4 b}+\frac {A \,x^{2}}{2 b}-\frac {B a \,x^{2}}{2 b^{2}}+\frac {A^{2}}{4 b B}-\frac {A a}{2 b^{2}}+\frac {B \,a^{2}}{4 b^{3}}-\frac {a \ln \left (b \,x^{2}+a \right ) A}{2 b^{2}}+\frac {a^{2} \ln \left (b \,x^{2}+a \right ) B}{2 b^{3}}\) | \(89\) |
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Time = 0.25 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.94 \[ \int \frac {x^3 \left (A+B x^2\right )}{a+b x^2} \, dx=\frac {B b^{2} x^{4} - 2 \, {\left (B a b - A b^{2}\right )} x^{2} + 2 \, {\left (B a^{2} - A a b\right )} \log \left (b x^{2} + a\right )}{4 \, b^{3}} \]
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Time = 0.15 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.85 \[ \int \frac {x^3 \left (A+B x^2\right )}{a+b x^2} \, dx=\frac {B x^{4}}{4 b} + \frac {a \left (- A b + B a\right ) \log {\left (a + b x^{2} \right )}}{2 b^{3}} + x^{2} \left (\frac {A}{2 b} - \frac {B a}{2 b^{2}}\right ) \]
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Time = 0.19 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.93 \[ \int \frac {x^3 \left (A+B x^2\right )}{a+b x^2} \, dx=\frac {B b x^{4} - 2 \, {\left (B a - A b\right )} x^{2}}{4 \, b^{2}} + \frac {{\left (B a^{2} - A a b\right )} \log \left (b x^{2} + a\right )}{2 \, b^{3}} \]
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Time = 0.29 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.96 \[ \int \frac {x^3 \left (A+B x^2\right )}{a+b x^2} \, dx=\frac {B b x^{4} - 2 \, B a x^{2} + 2 \, A b x^{2}}{4 \, b^{2}} + \frac {{\left (B a^{2} - A a b\right )} \log \left ({\left | b x^{2} + a \right |}\right )}{2 \, b^{3}} \]
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Time = 4.91 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.96 \[ \int \frac {x^3 \left (A+B x^2\right )}{a+b x^2} \, dx=x^2\,\left (\frac {A}{2\,b}-\frac {B\,a}{2\,b^2}\right )+\frac {\ln \left (b\,x^2+a\right )\,\left (B\,a^2-A\,a\,b\right )}{2\,b^3}+\frac {B\,x^4}{4\,b} \]
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