Integrand size = 20, antiderivative size = 82 \[ \int \frac {x^5 \left (A+B x^2\right )}{\left (a+b x^2\right )^2} \, dx=\frac {(A b-2 a B) x^2}{2 b^3}+\frac {B x^4}{4 b^2}-\frac {a^2 (A b-a B)}{2 b^4 \left (a+b x^2\right )}-\frac {a (2 A b-3 a B) \log \left (a+b x^2\right )}{2 b^4} \]
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Time = 0.06 (sec) , antiderivative size = 82, 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} \[ \int \frac {x^5 \left (A+B x^2\right )}{\left (a+b x^2\right )^2} \, dx=-\frac {a^2 (A b-a B)}{2 b^4 \left (a+b x^2\right )}-\frac {a (2 A b-3 a B) \log \left (a+b x^2\right )}{2 b^4}+\frac {x^2 (A b-2 a B)}{2 b^3}+\frac {B x^4}{4 b^2} \]
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Rule 78
Rule 457
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \text {Subst}\left (\int \frac {x^2 (A+B x)}{(a+b x)^2} \, dx,x,x^2\right ) \\ & = \frac {1}{2} \text {Subst}\left (\int \left (\frac {A b-2 a B}{b^3}+\frac {B x}{b^2}-\frac {a^2 (-A b+a B)}{b^3 (a+b x)^2}+\frac {a (-2 A b+3 a B)}{b^3 (a+b x)}\right ) \, dx,x,x^2\right ) \\ & = \frac {(A b-2 a B) x^2}{2 b^3}+\frac {B x^4}{4 b^2}-\frac {a^2 (A b-a B)}{2 b^4 \left (a+b x^2\right )}-\frac {a (2 A b-3 a B) \log \left (a+b x^2\right )}{2 b^4} \\ \end{align*}
Time = 0.05 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.88 \[ \int \frac {x^5 \left (A+B x^2\right )}{\left (a+b x^2\right )^2} \, dx=\frac {2 b (A b-2 a B) x^2+b^2 B x^4+\frac {2 a^2 (-A b+a B)}{a+b x^2}+2 a (-2 A b+3 a B) \log \left (a+b x^2\right )}{4 b^4} \]
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Time = 2.50 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.93
method | result | size |
default | \(\frac {\left (b B \,x^{2}+A b -2 B a \right )^{2}}{4 b^{4} B}-\frac {a \left (\frac {\left (2 A b -3 B a \right ) \ln \left (b \,x^{2}+a \right )}{b}+\frac {a \left (A b -B a \right )}{b \left (b \,x^{2}+a \right )}\right )}{2 b^{3}}\) | \(76\) |
norman | \(\frac {\frac {B \,x^{6}}{4 b}-\frac {a \left (2 a b A -3 a^{2} B \right )}{2 b^{4}}+\frac {\left (2 A b -3 B a \right ) x^{4}}{4 b^{2}}}{b \,x^{2}+a}-\frac {a \left (2 A b -3 B a \right ) \ln \left (b \,x^{2}+a \right )}{2 b^{4}}\) | \(80\) |
parallelrisch | \(-\frac {-b^{3} B \,x^{6}-2 A \,x^{4} b^{3}+3 B \,x^{4} a \,b^{2}+4 A \ln \left (b \,x^{2}+a \right ) x^{2} a \,b^{2}-6 B \ln \left (b \,x^{2}+a \right ) x^{2} a^{2} b +4 A \ln \left (b \,x^{2}+a \right ) a^{2} b -6 B \ln \left (b \,x^{2}+a \right ) a^{3}+4 a^{2} b A -6 a^{3} B}{4 b^{4} \left (b \,x^{2}+a \right )}\) | \(122\) |
risch | \(\frac {B \,x^{4}}{4 b^{2}}+\frac {A \,x^{2}}{2 b^{2}}-\frac {B a \,x^{2}}{b^{3}}+\frac {A^{2}}{4 b^{2} B}-\frac {A a}{b^{3}}+\frac {B \,a^{2}}{b^{4}}-\frac {a^{2} A}{2 b^{3} \left (b \,x^{2}+a \right )}+\frac {a^{3} B}{2 b^{4} \left (b \,x^{2}+a \right )}-\frac {a \ln \left (b \,x^{2}+a \right ) A}{b^{3}}+\frac {3 a^{2} \ln \left (b \,x^{2}+a \right ) B}{2 b^{4}}\) | \(124\) |
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Time = 0.32 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.48 \[ \int \frac {x^5 \left (A+B x^2\right )}{\left (a+b x^2\right )^2} \, dx=\frac {B b^{3} x^{6} - {\left (3 \, B a b^{2} - 2 \, A b^{3}\right )} x^{4} + 2 \, B a^{3} - 2 \, A a^{2} b - 2 \, {\left (2 \, B a^{2} b - A a b^{2}\right )} x^{2} + 2 \, {\left (3 \, B a^{3} - 2 \, A a^{2} b + {\left (3 \, B a^{2} b - 2 \, A a b^{2}\right )} x^{2}\right )} \log \left (b x^{2} + a\right )}{4 \, {\left (b^{5} x^{2} + a b^{4}\right )}} \]
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Time = 0.39 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.95 \[ \int \frac {x^5 \left (A+B x^2\right )}{\left (a+b x^2\right )^2} \, dx=\frac {B x^{4}}{4 b^{2}} + \frac {a \left (- 2 A b + 3 B a\right ) \log {\left (a + b x^{2} \right )}}{2 b^{4}} + x^{2} \left (\frac {A}{2 b^{2}} - \frac {B a}{b^{3}}\right ) + \frac {- A a^{2} b + B a^{3}}{2 a b^{4} + 2 b^{5} x^{2}} \]
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Time = 0.19 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.00 \[ \int \frac {x^5 \left (A+B x^2\right )}{\left (a+b x^2\right )^2} \, dx=\frac {B a^{3} - A a^{2} b}{2 \, {\left (b^{5} x^{2} + a b^{4}\right )}} + \frac {B b x^{4} - 2 \, {\left (2 \, B a - A b\right )} x^{2}}{4 \, b^{3}} + \frac {{\left (3 \, B a^{2} - 2 \, A a b\right )} \log \left (b x^{2} + a\right )}{2 \, b^{4}} \]
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Time = 0.28 (sec) , antiderivative size = 106, normalized size of antiderivative = 1.29 \[ \int \frac {x^5 \left (A+B x^2\right )}{\left (a+b x^2\right )^2} \, dx=\frac {{\left (3 \, B a^{2} - 2 \, A a b\right )} \log \left ({\left | b x^{2} + a \right |}\right )}{2 \, b^{4}} + \frac {B b^{2} x^{4} - 4 \, B a b x^{2} + 2 \, A b^{2} x^{2}}{4 \, b^{4}} - \frac {3 \, B a^{2} b x^{2} - 2 \, A a b^{2} x^{2} + 2 \, B a^{3} - A a^{2} b}{2 \, {\left (b x^{2} + a\right )} b^{4}} \]
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Time = 0.07 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.05 \[ \int \frac {x^5 \left (A+B x^2\right )}{\left (a+b x^2\right )^2} \, dx=x^2\,\left (\frac {A}{2\,b^2}-\frac {B\,a}{b^3}\right )+\frac {\ln \left (b\,x^2+a\right )\,\left (3\,B\,a^2-2\,A\,a\,b\right )}{2\,b^4}+\frac {B\,x^4}{4\,b^2}+\frac {B\,a^3-A\,a^2\,b}{2\,b\,\left (b^4\,x^2+a\,b^3\right )} \]
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