Integrand size = 20, antiderivative size = 96 \[ \int x^m \left (a+b x^2\right )^3 \left (A+B x^2\right ) \, dx=\frac {a^3 A x^{1+m}}{1+m}+\frac {a^2 (3 A b+a B) x^{3+m}}{3+m}+\frac {3 a b (A b+a B) x^{5+m}}{5+m}+\frac {b^2 (A b+3 a B) x^{7+m}}{7+m}+\frac {b^3 B x^{9+m}}{9+m} \]
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Time = 0.05 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {459} \[ \int x^m \left (a+b x^2\right )^3 \left (A+B x^2\right ) \, dx=\frac {a^3 A x^{m+1}}{m+1}+\frac {a^2 x^{m+3} (a B+3 A b)}{m+3}+\frac {b^2 x^{m+7} (3 a B+A b)}{m+7}+\frac {3 a b x^{m+5} (a B+A b)}{m+5}+\frac {b^3 B x^{m+9}}{m+9} \]
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Rule 459
Rubi steps \begin{align*} \text {integral}& = \int \left (a^3 A x^m+a^2 (3 A b+a B) x^{2+m}+3 a b (A b+a B) x^{4+m}+b^2 (A b+3 a B) x^{6+m}+b^3 B x^{8+m}\right ) \, dx \\ & = \frac {a^3 A x^{1+m}}{1+m}+\frac {a^2 (3 A b+a B) x^{3+m}}{3+m}+\frac {3 a b (A b+a B) x^{5+m}}{5+m}+\frac {b^2 (A b+3 a B) x^{7+m}}{7+m}+\frac {b^3 B x^{9+m}}{9+m} \\ \end{align*}
Time = 0.13 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.93 \[ \int x^m \left (a+b x^2\right )^3 \left (A+B x^2\right ) \, dx=x^{1+m} \left (\frac {a^3 A}{1+m}+\frac {a^2 (3 A b+a B) x^2}{3+m}+\frac {3 a b (A b+a B) x^4}{5+m}+\frac {b^2 (A b+3 a B) x^6}{7+m}+\frac {b^3 B x^8}{9+m}\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. \(472\) vs. \(2(96)=192\).
Time = 2.75 (sec) , antiderivative size = 473, normalized size of antiderivative = 4.93
method | result | size |
risch | \(\frac {x \left (B \,b^{3} m^{4} x^{8}+16 B \,b^{3} m^{3} x^{8}+A \,b^{3} m^{4} x^{6}+3 B a \,b^{2} m^{4} x^{6}+86 B \,b^{3} m^{2} x^{8}+18 A \,b^{3} m^{3} x^{6}+54 B a \,b^{2} m^{3} x^{6}+176 m \,x^{8} B \,b^{3}+3 A a \,b^{2} m^{4} x^{4}+104 A \,b^{3} m^{2} x^{6}+3 B \,a^{2} b \,m^{4} x^{4}+312 B a \,b^{2} m^{2} x^{6}+105 b^{3} B \,x^{8}+60 A a \,b^{2} m^{3} x^{4}+222 A \,x^{6} b^{3} m +60 B \,a^{2} b \,m^{3} x^{4}+666 B \,x^{6} a \,b^{2} m +3 A \,a^{2} b \,m^{4} x^{2}+390 A a \,b^{2} m^{2} x^{4}+135 A \,x^{6} b^{3}+B \,a^{3} m^{4} x^{2}+390 B \,a^{2} b \,m^{2} x^{4}+405 B \,x^{6} a \,b^{2}+66 A \,a^{2} b \,m^{3} x^{2}+900 A a \,b^{2} x^{4} m +22 B \,a^{3} m^{3} x^{2}+900 B \,a^{2} b \,x^{4} m +A \,a^{3} m^{4}+492 A \,a^{2} b \,m^{2} x^{2}+567 A a \,b^{2} x^{4}+164 B \,a^{3} m^{2} x^{2}+567 B \,a^{2} b \,x^{4}+24 A \,a^{3} m^{3}+1374 A \,a^{2} b \,x^{2} m +458 B \,a^{3} x^{2} m +206 A \,a^{3} m^{2}+945 A \,a^{2} b \,x^{2}+315 B \,a^{3} x^{2}+744 a^{3} A m +945 a^{3} A \right ) x^{m}}{\left (9+m \right ) \left (7+m \right ) \left (5+m \right ) \left (3+m \right ) \left (1+m \right )}\) | \(473\) |
gosper | \(\frac {x^{1+m} \left (B \,b^{3} m^{4} x^{8}+16 B \,b^{3} m^{3} x^{8}+A \,b^{3} m^{4} x^{6}+3 B a \,b^{2} m^{4} x^{6}+86 B \,b^{3} m^{2} x^{8}+18 A \,b^{3} m^{3} x^{6}+54 B a \,b^{2} m^{3} x^{6}+176 m \,x^{8} B \,b^{3}+3 A a \,b^{2} m^{4} x^{4}+104 A \,b^{3} m^{2} x^{6}+3 B \,a^{2} b \,m^{4} x^{4}+312 B a \,b^{2} m^{2} x^{6}+105 b^{3} B \,x^{8}+60 A a \,b^{2} m^{3} x^{4}+222 A \,x^{6} b^{3} m +60 B \,a^{2} b \,m^{3} x^{4}+666 B \,x^{6} a \,b^{2} m +3 A \,a^{2} b \,m^{4} x^{2}+390 A a \,b^{2} m^{2} x^{4}+135 A \,x^{6} b^{3}+B \,a^{3} m^{4} x^{2}+390 B \,a^{2} b \,m^{2} x^{4}+405 B \,x^{6} a \,b^{2}+66 A \,a^{2} b \,m^{3} x^{2}+900 A a \,b^{2} x^{4} m +22 B \,a^{3} m^{3} x^{2}+900 B \,a^{2} b \,x^{4} m +A \,a^{3} m^{4}+492 A \,a^{2} b \,m^{2} x^{2}+567 A a \,b^{2} x^{4}+164 B \,a^{3} m^{2} x^{2}+567 B \,a^{2} b \,x^{4}+24 A \,a^{3} m^{3}+1374 A \,a^{2} b \,x^{2} m +458 B \,a^{3} x^{2} m +206 A \,a^{3} m^{2}+945 A \,a^{2} b \,x^{2}+315 B \,a^{3} x^{2}+744 a^{3} A m +945 a^{3} A \right )}{\left (1+m \right ) \left (3+m \right ) \left (5+m \right ) \left (7+m \right ) \left (9+m \right )}\) | \(474\) |
parallelrisch | \(\frac {312 B \,x^{7} x^{m} a \,b^{2} m^{2}+3 B \,x^{5} x^{m} a^{2} b \,m^{4}+60 B \,x^{5} x^{m} a^{2} b \,m^{3}+492 A \,x^{3} x^{m} a^{2} b \,m^{2}+1374 A \,x^{3} x^{m} a^{2} b m +3 B \,x^{7} x^{m} a \,b^{2} m^{4}+54 B \,x^{7} x^{m} a \,b^{2} m^{3}+3 A \,x^{5} x^{m} a \,b^{2} m^{4}+60 A \,x^{5} x^{m} a \,b^{2} m^{3}+666 B \,x^{7} x^{m} a \,b^{2} m +105 B \,x^{9} x^{m} b^{3}+135 A \,x^{7} x^{m} b^{3}+315 B \,x^{3} x^{m} a^{3}+945 A x \,x^{m} a^{3}+222 A \,x^{7} x^{m} b^{3} m +405 B \,x^{7} x^{m} a \,b^{2}+B \,x^{3} x^{m} a^{3} m^{4}+22 B \,x^{3} x^{m} a^{3} m^{3}+567 A \,x^{5} x^{m} a \,b^{2}+A x \,x^{m} a^{3} m^{4}+567 B \,x^{5} x^{m} a^{2} b +164 B \,x^{3} x^{m} a^{3} m^{2}+24 A x \,x^{m} a^{3} m^{3}+458 B \,x^{3} x^{m} a^{3} m +945 A \,x^{3} x^{m} a^{2} b +206 A x \,x^{m} a^{3} m^{2}+744 A x \,x^{m} a^{3} m +104 A \,x^{7} x^{m} b^{3} m^{2}+B \,x^{9} x^{m} b^{3} m^{4}+16 B \,x^{9} x^{m} b^{3} m^{3}+A \,x^{7} x^{m} b^{3} m^{4}+86 B \,x^{9} x^{m} b^{3} m^{2}+18 A \,x^{7} x^{m} b^{3} m^{3}+176 B \,x^{9} x^{m} b^{3} m +390 A \,x^{5} x^{m} a \,b^{2} m^{2}+3 A \,x^{3} x^{m} a^{2} b \,m^{4}+390 B \,x^{5} x^{m} a^{2} b \,m^{2}+900 A \,x^{5} x^{m} a \,b^{2} m +66 A \,x^{3} x^{m} a^{2} b \,m^{3}+900 B \,x^{5} x^{m} a^{2} b m}{\left (9+m \right ) \left (7+m \right ) \left (5+m \right ) \left (3+m \right ) \left (1+m \right )}\) | \(594\) |
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Leaf count of result is larger than twice the leaf count of optimal. 379 vs. \(2 (96) = 192\).
Time = 0.24 (sec) , antiderivative size = 379, normalized size of antiderivative = 3.95 \[ \int x^m \left (a+b x^2\right )^3 \left (A+B x^2\right ) \, dx=\frac {{\left ({\left (B b^{3} m^{4} + 16 \, B b^{3} m^{3} + 86 \, B b^{3} m^{2} + 176 \, B b^{3} m + 105 \, B b^{3}\right )} x^{9} + {\left ({\left (3 \, B a b^{2} + A b^{3}\right )} m^{4} + 405 \, B a b^{2} + 135 \, A b^{3} + 18 \, {\left (3 \, B a b^{2} + A b^{3}\right )} m^{3} + 104 \, {\left (3 \, B a b^{2} + A b^{3}\right )} m^{2} + 222 \, {\left (3 \, B a b^{2} + A b^{3}\right )} m\right )} x^{7} + 3 \, {\left ({\left (B a^{2} b + A a b^{2}\right )} m^{4} + 189 \, B a^{2} b + 189 \, A a b^{2} + 20 \, {\left (B a^{2} b + A a b^{2}\right )} m^{3} + 130 \, {\left (B a^{2} b + A a b^{2}\right )} m^{2} + 300 \, {\left (B a^{2} b + A a b^{2}\right )} m\right )} x^{5} + {\left ({\left (B a^{3} + 3 \, A a^{2} b\right )} m^{4} + 315 \, B a^{3} + 945 \, A a^{2} b + 22 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} m^{3} + 164 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} m^{2} + 458 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} m\right )} x^{3} + {\left (A a^{3} m^{4} + 24 \, A a^{3} m^{3} + 206 \, A a^{3} m^{2} + 744 \, A a^{3} m + 945 \, A a^{3}\right )} x\right )} x^{m}}{m^{5} + 25 \, m^{4} + 230 \, m^{3} + 950 \, m^{2} + 1689 \, m + 945} \]
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Leaf count of result is larger than twice the leaf count of optimal. 2069 vs. \(2 (87) = 174\).
Time = 0.65 (sec) , antiderivative size = 2069, normalized size of antiderivative = 21.55 \[ \int x^m \left (a+b x^2\right )^3 \left (A+B x^2\right ) \, dx=\text {Too large to display} \]
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Time = 0.21 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.34 \[ \int x^m \left (a+b x^2\right )^3 \left (A+B x^2\right ) \, dx=\frac {B b^{3} x^{m + 9}}{m + 9} + \frac {3 \, B a b^{2} x^{m + 7}}{m + 7} + \frac {A b^{3} x^{m + 7}}{m + 7} + \frac {3 \, B a^{2} b x^{m + 5}}{m + 5} + \frac {3 \, A a b^{2} x^{m + 5}}{m + 5} + \frac {B a^{3} x^{m + 3}}{m + 3} + \frac {3 \, A a^{2} b x^{m + 3}}{m + 3} + \frac {A a^{3} x^{m + 1}}{m + 1} \]
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Leaf count of result is larger than twice the leaf count of optimal. 593 vs. \(2 (96) = 192\).
Time = 0.36 (sec) , antiderivative size = 593, normalized size of antiderivative = 6.18 \[ \int x^m \left (a+b x^2\right )^3 \left (A+B x^2\right ) \, dx=\frac {B b^{3} m^{4} x^{9} x^{m} + 16 \, B b^{3} m^{3} x^{9} x^{m} + 3 \, B a b^{2} m^{4} x^{7} x^{m} + A b^{3} m^{4} x^{7} x^{m} + 86 \, B b^{3} m^{2} x^{9} x^{m} + 54 \, B a b^{2} m^{3} x^{7} x^{m} + 18 \, A b^{3} m^{3} x^{7} x^{m} + 176 \, B b^{3} m x^{9} x^{m} + 3 \, B a^{2} b m^{4} x^{5} x^{m} + 3 \, A a b^{2} m^{4} x^{5} x^{m} + 312 \, B a b^{2} m^{2} x^{7} x^{m} + 104 \, A b^{3} m^{2} x^{7} x^{m} + 105 \, B b^{3} x^{9} x^{m} + 60 \, B a^{2} b m^{3} x^{5} x^{m} + 60 \, A a b^{2} m^{3} x^{5} x^{m} + 666 \, B a b^{2} m x^{7} x^{m} + 222 \, A b^{3} m x^{7} x^{m} + B a^{3} m^{4} x^{3} x^{m} + 3 \, A a^{2} b m^{4} x^{3} x^{m} + 390 \, B a^{2} b m^{2} x^{5} x^{m} + 390 \, A a b^{2} m^{2} x^{5} x^{m} + 405 \, B a b^{2} x^{7} x^{m} + 135 \, A b^{3} x^{7} x^{m} + 22 \, B a^{3} m^{3} x^{3} x^{m} + 66 \, A a^{2} b m^{3} x^{3} x^{m} + 900 \, B a^{2} b m x^{5} x^{m} + 900 \, A a b^{2} m x^{5} x^{m} + A a^{3} m^{4} x x^{m} + 164 \, B a^{3} m^{2} x^{3} x^{m} + 492 \, A a^{2} b m^{2} x^{3} x^{m} + 567 \, B a^{2} b x^{5} x^{m} + 567 \, A a b^{2} x^{5} x^{m} + 24 \, A a^{3} m^{3} x x^{m} + 458 \, B a^{3} m x^{3} x^{m} + 1374 \, A a^{2} b m x^{3} x^{m} + 206 \, A a^{3} m^{2} x x^{m} + 315 \, B a^{3} x^{3} x^{m} + 945 \, A a^{2} b x^{3} x^{m} + 744 \, A a^{3} m x x^{m} + 945 \, A a^{3} x x^{m}}{m^{5} + 25 \, m^{4} + 230 \, m^{3} + 950 \, m^{2} + 1689 \, m + 945} \]
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Time = 5.13 (sec) , antiderivative size = 289, normalized size of antiderivative = 3.01 \[ \int x^m \left (a+b x^2\right )^3 \left (A+B x^2\right ) \, dx=\frac {A\,a^3\,x\,x^m\,\left (m^4+24\,m^3+206\,m^2+744\,m+945\right )}{m^5+25\,m^4+230\,m^3+950\,m^2+1689\,m+945}+\frac {B\,b^3\,x^m\,x^9\,\left (m^4+16\,m^3+86\,m^2+176\,m+105\right )}{m^5+25\,m^4+230\,m^3+950\,m^2+1689\,m+945}+\frac {a^2\,x^m\,x^3\,\left (3\,A\,b+B\,a\right )\,\left (m^4+22\,m^3+164\,m^2+458\,m+315\right )}{m^5+25\,m^4+230\,m^3+950\,m^2+1689\,m+945}+\frac {b^2\,x^m\,x^7\,\left (A\,b+3\,B\,a\right )\,\left (m^4+18\,m^3+104\,m^2+222\,m+135\right )}{m^5+25\,m^4+230\,m^3+950\,m^2+1689\,m+945}+\frac {3\,a\,b\,x^m\,x^5\,\left (A\,b+B\,a\right )\,\left (m^4+20\,m^3+130\,m^2+300\,m+189\right )}{m^5+25\,m^4+230\,m^3+950\,m^2+1689\,m+945} \]
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