Integrand size = 20, antiderivative size = 104 \[ \int \frac {x^7 \left (A+B x^2\right )}{\left (a+b x^2\right )^2} \, dx=-\frac {a (2 A b-3 a B) x^2}{2 b^4}+\frac {(A b-2 a B) x^4}{4 b^3}+\frac {B x^6}{6 b^2}+\frac {a^3 (A b-a B)}{2 b^5 \left (a+b x^2\right )}+\frac {a^2 (3 A b-4 a B) \log \left (a+b x^2\right )}{2 b^5} \]
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Time = 0.08 (sec) , antiderivative size = 104, 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} \[ \int \frac {x^7 \left (A+B x^2\right )}{\left (a+b x^2\right )^2} \, dx=\frac {a^3 (A b-a B)}{2 b^5 \left (a+b x^2\right )}+\frac {a^2 (3 A b-4 a B) \log \left (a+b x^2\right )}{2 b^5}-\frac {a x^2 (2 A b-3 a B)}{2 b^4}+\frac {x^4 (A b-2 a B)}{4 b^3}+\frac {B x^6}{6 b^2} \]
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Rule 78
Rule 457
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \text {Subst}\left (\int \frac {x^3 (A+B x)}{(a+b x)^2} \, dx,x,x^2\right ) \\ & = \frac {1}{2} \text {Subst}\left (\int \left (\frac {a (-2 A b+3 a B)}{b^4}+\frac {(A b-2 a B) x}{b^3}+\frac {B x^2}{b^2}+\frac {a^3 (-A b+a B)}{b^4 (a+b x)^2}-\frac {a^2 (-3 A b+4 a B)}{b^4 (a+b x)}\right ) \, dx,x,x^2\right ) \\ & = -\frac {a (2 A b-3 a B) x^2}{2 b^4}+\frac {(A b-2 a B) x^4}{4 b^3}+\frac {B x^6}{6 b^2}+\frac {a^3 (A b-a B)}{2 b^5 \left (a+b x^2\right )}+\frac {a^2 (3 A b-4 a B) \log \left (a+b x^2\right )}{2 b^5} \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.89 \[ \int \frac {x^7 \left (A+B x^2\right )}{\left (a+b x^2\right )^2} \, dx=\frac {6 a b (-2 A b+3 a B) x^2+3 b^2 (A b-2 a B) x^4+2 b^3 B x^6+\frac {6 a^3 (A b-a B)}{a+b x^2}+6 a^2 (3 A b-4 a B) \log \left (a+b x^2\right )}{12 b^5} \]
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Time = 2.60 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.98
method | result | size |
norman | \(\frac {\frac {B \,x^{8}}{6 b}+\frac {a \left (3 a^{2} b A -4 a^{3} B \right )}{2 b^{5}}+\frac {\left (3 A b -4 B a \right ) x^{6}}{12 b^{2}}-\frac {a \left (3 A b -4 B a \right ) x^{4}}{4 b^{3}}}{b \,x^{2}+a}+\frac {a^{2} \left (3 A b -4 B a \right ) \ln \left (b \,x^{2}+a \right )}{2 b^{5}}\) | \(102\) |
default | \(-\frac {-\frac {b^{2} B \,x^{6}}{6}+\frac {\left (-b^{2} A +2 a b B \right ) x^{4}}{4}+\frac {\left (2 a b A -3 a^{2} B \right ) x^{2}}{2}}{b^{4}}+\frac {a^{2} \left (\frac {\left (3 A b -4 B a \right ) \ln \left (b \,x^{2}+a \right )}{b}+\frac {a \left (A b -B a \right )}{b \left (b \,x^{2}+a \right )}\right )}{2 b^{4}}\) | \(103\) |
risch | \(\frac {B \,x^{6}}{6 b^{2}}+\frac {A \,x^{4}}{4 b^{2}}-\frac {B a \,x^{4}}{2 b^{3}}-\frac {a A \,x^{2}}{b^{3}}+\frac {3 a^{2} B \,x^{2}}{2 b^{4}}+\frac {a^{3} A}{2 b^{4} \left (b \,x^{2}+a \right )}-\frac {a^{4} B}{2 b^{5} \left (b \,x^{2}+a \right )}+\frac {3 a^{2} \ln \left (b \,x^{2}+a \right ) A}{2 b^{4}}-\frac {2 a^{3} \ln \left (b \,x^{2}+a \right ) B}{b^{5}}\) | \(122\) |
parallelrisch | \(\frac {2 B \,x^{8} b^{4}+3 A \,x^{6} b^{4}-4 B \,x^{6} a \,b^{3}-9 A \,x^{4} a \,b^{3}+12 B \,x^{4} a^{2} b^{2}+18 A \ln \left (b \,x^{2}+a \right ) x^{2} a^{2} b^{2}-24 B \ln \left (b \,x^{2}+a \right ) x^{2} a^{3} b +18 A \ln \left (b \,x^{2}+a \right ) a^{3} b -24 B \ln \left (b \,x^{2}+a \right ) a^{4}+18 A \,a^{3} b -24 B \,a^{4}}{12 b^{5} \left (b \,x^{2}+a \right )}\) | \(146\) |
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Time = 0.28 (sec) , antiderivative size = 148, normalized size of antiderivative = 1.42 \[ \int \frac {x^7 \left (A+B x^2\right )}{\left (a+b x^2\right )^2} \, dx=\frac {2 \, B b^{4} x^{8} - {\left (4 \, B a b^{3} - 3 \, A b^{4}\right )} x^{6} - 6 \, B a^{4} + 6 \, A a^{3} b + 3 \, {\left (4 \, B a^{2} b^{2} - 3 \, A a b^{3}\right )} x^{4} + 6 \, {\left (3 \, B a^{3} b - 2 \, A a^{2} b^{2}\right )} x^{2} - 6 \, {\left (4 \, B a^{4} - 3 \, A a^{3} b + {\left (4 \, B a^{3} b - 3 \, A a^{2} b^{2}\right )} x^{2}\right )} \log \left (b x^{2} + a\right )}{12 \, {\left (b^{6} x^{2} + a b^{5}\right )}} \]
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Time = 0.42 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.00 \[ \int \frac {x^7 \left (A+B x^2\right )}{\left (a+b x^2\right )^2} \, dx=\frac {B x^{6}}{6 b^{2}} - \frac {a^{2} \left (- 3 A b + 4 B a\right ) \log {\left (a + b x^{2} \right )}}{2 b^{5}} + x^{4} \left (\frac {A}{4 b^{2}} - \frac {B a}{2 b^{3}}\right ) + x^{2} \left (- \frac {A a}{b^{3}} + \frac {3 B a^{2}}{2 b^{4}}\right ) + \frac {A a^{3} b - B a^{4}}{2 a b^{5} + 2 b^{6} x^{2}} \]
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Time = 0.20 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.03 \[ \int \frac {x^7 \left (A+B x^2\right )}{\left (a+b x^2\right )^2} \, dx=-\frac {B a^{4} - A a^{3} b}{2 \, {\left (b^{6} x^{2} + a b^{5}\right )}} + \frac {2 \, B b^{2} x^{6} - 3 \, {\left (2 \, B a b - A b^{2}\right )} x^{4} + 6 \, {\left (3 \, B a^{2} - 2 \, A a b\right )} x^{2}}{12 \, b^{4}} - \frac {{\left (4 \, B a^{3} - 3 \, A a^{2} b\right )} \log \left (b x^{2} + a\right )}{2 \, b^{5}} \]
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Time = 0.28 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.30 \[ \int \frac {x^7 \left (A+B x^2\right )}{\left (a+b x^2\right )^2} \, dx=-\frac {{\left (4 \, B a^{3} - 3 \, A a^{2} b\right )} \log \left ({\left | b x^{2} + a \right |}\right )}{2 \, b^{5}} + \frac {2 \, B b^{4} x^{6} - 6 \, B a b^{3} x^{4} + 3 \, A b^{4} x^{4} + 18 \, B a^{2} b^{2} x^{2} - 12 \, A a b^{3} x^{2}}{12 \, b^{6}} + \frac {4 \, B a^{3} b x^{2} - 3 \, A a^{2} b^{2} x^{2} + 3 \, B a^{4} - 2 \, A a^{3} b}{2 \, {\left (b x^{2} + a\right )} b^{5}} \]
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Time = 4.97 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.16 \[ \int \frac {x^7 \left (A+B x^2\right )}{\left (a+b x^2\right )^2} \, dx=x^4\,\left (\frac {A}{4\,b^2}-\frac {B\,a}{2\,b^3}\right )-x^2\,\left (\frac {a\,\left (\frac {A}{b^2}-\frac {2\,B\,a}{b^3}\right )}{b}+\frac {B\,a^2}{2\,b^4}\right )+\frac {B\,x^6}{6\,b^2}-\frac {\ln \left (b\,x^2+a\right )\,\left (4\,B\,a^3-3\,A\,a^2\,b\right )}{2\,b^5}-\frac {B\,a^4-A\,a^3\,b}{2\,b\,\left (b^5\,x^2+a\,b^4\right )} \]
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