Integrand size = 22, antiderivative size = 109 \[ \int x^m \left (a+b x^2\right )^2 \left (c+d x^2\right )^2 \, dx=\frac {a^2 c^2 x^{1+m}}{1+m}+\frac {2 a c (b c+a d) x^{3+m}}{3+m}+\frac {\left (b^2 c^2+4 a b c d+a^2 d^2\right ) x^{5+m}}{5+m}+\frac {2 b d (b c+a d) x^{7+m}}{7+m}+\frac {b^2 d^2 x^{9+m}}{9+m} \]
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Time = 0.04 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {459} \[ \int x^m \left (a+b x^2\right )^2 \left (c+d x^2\right )^2 \, dx=\frac {x^{m+5} \left (a^2 d^2+4 a b c d+b^2 c^2\right )}{m+5}+\frac {a^2 c^2 x^{m+1}}{m+1}+\frac {2 a c x^{m+3} (a d+b c)}{m+3}+\frac {2 b d x^{m+7} (a d+b c)}{m+7}+\frac {b^2 d^2 x^{m+9}}{m+9} \]
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Rule 459
Rubi steps \begin{align*} \text {integral}& = \int \left (a^2 c^2 x^m+2 a c (b c+a d) x^{2+m}+\left (b^2 c^2+4 a b c d+a^2 d^2\right ) x^{4+m}+2 b d (b c+a d) x^{6+m}+b^2 d^2 x^{8+m}\right ) \, dx \\ & = \frac {a^2 c^2 x^{1+m}}{1+m}+\frac {2 a c (b c+a d) x^{3+m}}{3+m}+\frac {\left (b^2 c^2+4 a b c d+a^2 d^2\right ) x^{5+m}}{5+m}+\frac {2 b d (b c+a d) x^{7+m}}{7+m}+\frac {b^2 d^2 x^{9+m}}{9+m} \\ \end{align*}
Time = 0.10 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.93 \[ \int x^m \left (a+b x^2\right )^2 \left (c+d x^2\right )^2 \, dx=x^m \left (\frac {a^2 c^2 x}{1+m}+\frac {2 a c (b c+a d) x^3}{3+m}+\frac {\left (b^2 c^2+4 a b c d+a^2 d^2\right ) x^5}{5+m}+\frac {2 b d (b c+a d) x^7}{7+m}+\frac {b^2 d^2 x^9}{9+m}\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. \(567\) vs. \(2(109)=218\).
Time = 2.73 (sec) , antiderivative size = 568, normalized size of antiderivative = 5.21
method | result | size |
risch | \(\frac {x \left (b^{2} d^{2} m^{4} x^{8}+16 b^{2} d^{2} m^{3} x^{8}+2 a b \,d^{2} m^{4} x^{6}+2 b^{2} c d \,m^{4} x^{6}+86 b^{2} d^{2} m^{2} x^{8}+36 a b \,d^{2} m^{3} x^{6}+36 b^{2} c d \,m^{3} x^{6}+176 m \,x^{8} b^{2} d^{2}+a^{2} d^{2} m^{4} x^{4}+4 a b c d \,m^{4} x^{4}+208 a b \,d^{2} m^{2} x^{6}+b^{2} c^{2} m^{4} x^{4}+208 b^{2} c d \,m^{2} x^{6}+105 b^{2} d^{2} x^{8}+20 a^{2} d^{2} m^{3} x^{4}+80 a b c d \,m^{3} x^{4}+444 a b \,d^{2} x^{6} m +20 b^{2} c^{2} m^{3} x^{4}+444 b^{2} c d \,x^{6} m +2 a^{2} c d \,m^{4} x^{2}+130 a^{2} d^{2} m^{2} x^{4}+2 a b \,c^{2} m^{4} x^{2}+520 a b c d \,m^{2} x^{4}+270 a b \,d^{2} x^{6}+130 b^{2} c^{2} m^{2} x^{4}+270 b^{2} c d \,x^{6}+44 a^{2} c d \,m^{3} x^{2}+300 a^{2} d^{2} x^{4} m +44 a b \,c^{2} m^{3} x^{2}+1200 x^{4} a b c d m +300 b^{2} c^{2} x^{4} m +a^{2} c^{2} m^{4}+328 a^{2} c d \,m^{2} x^{2}+189 a^{2} d^{2} x^{4}+328 a b \,c^{2} m^{2} x^{2}+756 x^{4} a b c d +189 b^{2} c^{2} x^{4}+24 a^{2} c^{2} m^{3}+916 a^{2} c d \,x^{2} m +916 a b \,c^{2} x^{2} m +206 a^{2} c^{2} m^{2}+630 a^{2} c d \,x^{2}+630 a b \,c^{2} x^{2}+744 a^{2} c^{2} m +945 a^{2} c^{2}\right ) x^{m}}{\left (9+m \right ) \left (7+m \right ) \left (5+m \right ) \left (3+m \right ) \left (1+m \right )}\) | \(568\) |
gosper | \(\frac {x^{1+m} \left (b^{2} d^{2} m^{4} x^{8}+16 b^{2} d^{2} m^{3} x^{8}+2 a b \,d^{2} m^{4} x^{6}+2 b^{2} c d \,m^{4} x^{6}+86 b^{2} d^{2} m^{2} x^{8}+36 a b \,d^{2} m^{3} x^{6}+36 b^{2} c d \,m^{3} x^{6}+176 m \,x^{8} b^{2} d^{2}+a^{2} d^{2} m^{4} x^{4}+4 a b c d \,m^{4} x^{4}+208 a b \,d^{2} m^{2} x^{6}+b^{2} c^{2} m^{4} x^{4}+208 b^{2} c d \,m^{2} x^{6}+105 b^{2} d^{2} x^{8}+20 a^{2} d^{2} m^{3} x^{4}+80 a b c d \,m^{3} x^{4}+444 a b \,d^{2} x^{6} m +20 b^{2} c^{2} m^{3} x^{4}+444 b^{2} c d \,x^{6} m +2 a^{2} c d \,m^{4} x^{2}+130 a^{2} d^{2} m^{2} x^{4}+2 a b \,c^{2} m^{4} x^{2}+520 a b c d \,m^{2} x^{4}+270 a b \,d^{2} x^{6}+130 b^{2} c^{2} m^{2} x^{4}+270 b^{2} c d \,x^{6}+44 a^{2} c d \,m^{3} x^{2}+300 a^{2} d^{2} x^{4} m +44 a b \,c^{2} m^{3} x^{2}+1200 x^{4} a b c d m +300 b^{2} c^{2} x^{4} m +a^{2} c^{2} m^{4}+328 a^{2} c d \,m^{2} x^{2}+189 a^{2} d^{2} x^{4}+328 a b \,c^{2} m^{2} x^{2}+756 x^{4} a b c d +189 b^{2} c^{2} x^{4}+24 a^{2} c^{2} m^{3}+916 a^{2} c d \,x^{2} m +916 a b \,c^{2} x^{2} m +206 a^{2} c^{2} m^{2}+630 a^{2} c d \,x^{2}+630 a b \,c^{2} x^{2}+744 a^{2} c^{2} m +945 a^{2} c^{2}\right )}{\left (1+m \right ) \left (3+m \right ) \left (5+m \right ) \left (7+m \right ) \left (9+m \right )}\) | \(569\) |
parallelrisch | \(\frac {300 x^{5} x^{m} a^{2} d^{2} m +300 x^{5} x^{m} b^{2} c^{2} m +x \,x^{m} a^{2} c^{2} m^{4}+24 x \,x^{m} a^{2} c^{2} m^{3}+630 x^{3} x^{m} a^{2} c d +630 x^{3} x^{m} a b \,c^{2}+2 x^{7} x^{m} a b \,d^{2} m^{4}+2 x^{7} x^{m} b^{2} c d \,m^{4}+36 x^{7} x^{m} a b \,d^{2} m^{3}+36 x^{7} x^{m} b^{2} c d \,m^{3}+208 x^{7} x^{m} a b \,d^{2} m^{2}+208 x^{7} x^{m} b^{2} c d \,m^{2}+444 x^{7} x^{m} a b \,d^{2} m +x^{9} x^{m} b^{2} d^{2} m^{4}+16 x^{9} x^{m} b^{2} d^{2} m^{3}+86 x^{9} x^{m} b^{2} d^{2} m^{2}+176 x^{9} x^{m} b^{2} d^{2} m +x^{5} x^{m} a^{2} d^{2} m^{4}+x^{5} x^{m} b^{2} c^{2} m^{4}+20 x^{5} x^{m} a^{2} d^{2} m^{3}+20 x^{5} x^{m} b^{2} c^{2} m^{3}+270 x^{7} x^{m} a b \,d^{2}+270 x^{7} x^{m} b^{2} c d +130 x^{5} x^{m} a^{2} d^{2} m^{2}+130 x^{5} x^{m} b^{2} c^{2} m^{2}+2 x^{3} x^{m} a b \,c^{2} m^{4}+756 x^{5} x^{m} a b c d +328 x^{3} x^{m} a^{2} c d \,m^{2}+328 x^{3} x^{m} a b \,c^{2} m^{2}+44 x^{3} x^{m} a^{2} c d \,m^{3}+916 x^{3} x^{m} a^{2} c d m +916 x^{3} x^{m} a b \,c^{2} m +444 x^{7} x^{m} b^{2} c d m +2 x^{3} x^{m} a^{2} c d \,m^{4}+44 x^{3} x^{m} a b \,c^{2} m^{3}+105 x^{9} x^{m} b^{2} d^{2}+189 x^{5} x^{m} a^{2} d^{2}+189 x^{5} x^{m} b^{2} c^{2}+945 x \,x^{m} a^{2} c^{2}+206 x \,x^{m} a^{2} c^{2} m^{2}+744 x \,x^{m} a^{2} c^{2} m +520 x^{5} x^{m} a b c d \,m^{2}+1200 x^{5} x^{m} a b c d m +4 x^{5} x^{m} a b c d \,m^{4}+80 x^{5} x^{m} a b c d \,m^{3}}{\left (9+m \right ) \left (7+m \right ) \left (5+m \right ) \left (3+m \right ) \left (1+m \right )}\) | \(704\) |
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Leaf count of result is larger than twice the leaf count of optimal. 442 vs. \(2 (109) = 218\).
Time = 0.26 (sec) , antiderivative size = 442, normalized size of antiderivative = 4.06 \[ \int x^m \left (a+b x^2\right )^2 \left (c+d x^2\right )^2 \, dx=\frac {{\left ({\left (b^{2} d^{2} m^{4} + 16 \, b^{2} d^{2} m^{3} + 86 \, b^{2} d^{2} m^{2} + 176 \, b^{2} d^{2} m + 105 \, b^{2} d^{2}\right )} x^{9} + 2 \, {\left ({\left (b^{2} c d + a b d^{2}\right )} m^{4} + 135 \, b^{2} c d + 135 \, a b d^{2} + 18 \, {\left (b^{2} c d + a b d^{2}\right )} m^{3} + 104 \, {\left (b^{2} c d + a b d^{2}\right )} m^{2} + 222 \, {\left (b^{2} c d + a b d^{2}\right )} m\right )} x^{7} + {\left ({\left (b^{2} c^{2} + 4 \, a b c d + a^{2} d^{2}\right )} m^{4} + 189 \, b^{2} c^{2} + 756 \, a b c d + 189 \, a^{2} d^{2} + 20 \, {\left (b^{2} c^{2} + 4 \, a b c d + a^{2} d^{2}\right )} m^{3} + 130 \, {\left (b^{2} c^{2} + 4 \, a b c d + a^{2} d^{2}\right )} m^{2} + 300 \, {\left (b^{2} c^{2} + 4 \, a b c d + a^{2} d^{2}\right )} m\right )} x^{5} + 2 \, {\left ({\left (a b c^{2} + a^{2} c d\right )} m^{4} + 315 \, a b c^{2} + 315 \, a^{2} c d + 22 \, {\left (a b c^{2} + a^{2} c d\right )} m^{3} + 164 \, {\left (a b c^{2} + a^{2} c d\right )} m^{2} + 458 \, {\left (a b c^{2} + a^{2} c d\right )} m\right )} x^{3} + {\left (a^{2} c^{2} m^{4} + 24 \, a^{2} c^{2} m^{3} + 206 \, a^{2} c^{2} m^{2} + 744 \, a^{2} c^{2} m + 945 \, a^{2} c^{2}\right )} x\right )} x^{m}}{m^{5} + 25 \, m^{4} + 230 \, m^{3} + 950 \, m^{2} + 1689 \, m + 945} \]
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Leaf count of result is larger than twice the leaf count of optimal. 2363 vs. \(2 (100) = 200\).
Time = 0.67 (sec) , antiderivative size = 2363, normalized size of antiderivative = 21.68 \[ \int x^m \left (a+b x^2\right )^2 \left (c+d x^2\right )^2 \, dx=\text {Too large to display} \]
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Time = 0.20 (sec) , antiderivative size = 153, normalized size of antiderivative = 1.40 \[ \int x^m \left (a+b x^2\right )^2 \left (c+d x^2\right )^2 \, dx=\frac {b^{2} d^{2} x^{m + 9}}{m + 9} + \frac {2 \, b^{2} c d x^{m + 7}}{m + 7} + \frac {2 \, a b d^{2} x^{m + 7}}{m + 7} + \frac {b^{2} c^{2} x^{m + 5}}{m + 5} + \frac {4 \, a b c d x^{m + 5}}{m + 5} + \frac {a^{2} d^{2} x^{m + 5}}{m + 5} + \frac {2 \, a b c^{2} x^{m + 3}}{m + 3} + \frac {2 \, a^{2} c d x^{m + 3}}{m + 3} + \frac {a^{2} c^{2} x^{m + 1}}{m + 1} \]
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Leaf count of result is larger than twice the leaf count of optimal. 703 vs. \(2 (109) = 218\).
Time = 0.34 (sec) , antiderivative size = 703, normalized size of antiderivative = 6.45 \[ \int x^m \left (a+b x^2\right )^2 \left (c+d x^2\right )^2 \, dx=\frac {b^{2} d^{2} m^{4} x^{9} x^{m} + 16 \, b^{2} d^{2} m^{3} x^{9} x^{m} + 2 \, b^{2} c d m^{4} x^{7} x^{m} + 2 \, a b d^{2} m^{4} x^{7} x^{m} + 86 \, b^{2} d^{2} m^{2} x^{9} x^{m} + 36 \, b^{2} c d m^{3} x^{7} x^{m} + 36 \, a b d^{2} m^{3} x^{7} x^{m} + 176 \, b^{2} d^{2} m x^{9} x^{m} + b^{2} c^{2} m^{4} x^{5} x^{m} + 4 \, a b c d m^{4} x^{5} x^{m} + a^{2} d^{2} m^{4} x^{5} x^{m} + 208 \, b^{2} c d m^{2} x^{7} x^{m} + 208 \, a b d^{2} m^{2} x^{7} x^{m} + 105 \, b^{2} d^{2} x^{9} x^{m} + 20 \, b^{2} c^{2} m^{3} x^{5} x^{m} + 80 \, a b c d m^{3} x^{5} x^{m} + 20 \, a^{2} d^{2} m^{3} x^{5} x^{m} + 444 \, b^{2} c d m x^{7} x^{m} + 444 \, a b d^{2} m x^{7} x^{m} + 2 \, a b c^{2} m^{4} x^{3} x^{m} + 2 \, a^{2} c d m^{4} x^{3} x^{m} + 130 \, b^{2} c^{2} m^{2} x^{5} x^{m} + 520 \, a b c d m^{2} x^{5} x^{m} + 130 \, a^{2} d^{2} m^{2} x^{5} x^{m} + 270 \, b^{2} c d x^{7} x^{m} + 270 \, a b d^{2} x^{7} x^{m} + 44 \, a b c^{2} m^{3} x^{3} x^{m} + 44 \, a^{2} c d m^{3} x^{3} x^{m} + 300 \, b^{2} c^{2} m x^{5} x^{m} + 1200 \, a b c d m x^{5} x^{m} + 300 \, a^{2} d^{2} m x^{5} x^{m} + a^{2} c^{2} m^{4} x x^{m} + 328 \, a b c^{2} m^{2} x^{3} x^{m} + 328 \, a^{2} c d m^{2} x^{3} x^{m} + 189 \, b^{2} c^{2} x^{5} x^{m} + 756 \, a b c d x^{5} x^{m} + 189 \, a^{2} d^{2} x^{5} x^{m} + 24 \, a^{2} c^{2} m^{3} x x^{m} + 916 \, a b c^{2} m x^{3} x^{m} + 916 \, a^{2} c d m x^{3} x^{m} + 206 \, a^{2} c^{2} m^{2} x x^{m} + 630 \, a b c^{2} x^{3} x^{m} + 630 \, a^{2} c d x^{3} x^{m} + 744 \, a^{2} c^{2} m x x^{m} + 945 \, a^{2} c^{2} x x^{m}}{m^{5} + 25 \, m^{4} + 230 \, m^{3} + 950 \, m^{2} + 1689 \, m + 945} \]
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Time = 5.15 (sec) , antiderivative size = 302, normalized size of antiderivative = 2.77 \[ \int x^m \left (a+b x^2\right )^2 \left (c+d x^2\right )^2 \, dx=\frac {x^m\,x^5\,\left (a^2\,d^2+4\,a\,b\,c\,d+b^2\,c^2\right )\,\left (m^4+20\,m^3+130\,m^2+300\,m+189\right )}{m^5+25\,m^4+230\,m^3+950\,m^2+1689\,m+945}+\frac {b^2\,d^2\,x^m\,x^9\,\left (m^4+16\,m^3+86\,m^2+176\,m+105\right )}{m^5+25\,m^4+230\,m^3+950\,m^2+1689\,m+945}+\frac {a^2\,c^2\,x\,x^m\,\left (m^4+24\,m^3+206\,m^2+744\,m+945\right )}{m^5+25\,m^4+230\,m^3+950\,m^2+1689\,m+945}+\frac {2\,a\,c\,x^m\,x^3\,\left (a\,d+b\,c\right )\,\left (m^4+22\,m^3+164\,m^2+458\,m+315\right )}{m^5+25\,m^4+230\,m^3+950\,m^2+1689\,m+945}+\frac {2\,b\,d\,x^m\,x^7\,\left (a\,d+b\,c\right )\,\left (m^4+18\,m^3+104\,m^2+222\,m+135\right )}{m^5+25\,m^4+230\,m^3+950\,m^2+1689\,m+945} \]
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