Integrand size = 20, antiderivative size = 71 \[ \int x^m \left (a+b x^2\right )^2 \left (c+d x^2\right ) \, dx=\frac {a^2 c x^{1+m}}{1+m}+\frac {a (2 b c+a d) x^{3+m}}{3+m}+\frac {b (b c+2 a d) x^{5+m}}{5+m}+\frac {b^2 d x^{7+m}}{7+m} \]
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Time = 0.03 (sec) , antiderivative size = 71, 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} \[ \int x^m \left (a+b x^2\right )^2 \left (c+d x^2\right ) \, dx=\frac {a^2 c x^{m+1}}{m+1}+\frac {a x^{m+3} (a d+2 b c)}{m+3}+\frac {b x^{m+5} (2 a d+b c)}{m+5}+\frac {b^2 d x^{m+7}}{m+7} \]
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Rule 459
Rubi steps \begin{align*} \text {integral}& = \int \left (a^2 c x^m+a (2 b c+a d) x^{2+m}+b (b c+2 a d) x^{4+m}+b^2 d x^{6+m}\right ) \, dx \\ & = \frac {a^2 c x^{1+m}}{1+m}+\frac {a (2 b c+a d) x^{3+m}}{3+m}+\frac {b (b c+2 a d) x^{5+m}}{5+m}+\frac {b^2 d x^{7+m}}{7+m} \\ \end{align*}
Time = 0.07 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.93 \[ \int x^m \left (a+b x^2\right )^2 \left (c+d x^2\right ) \, dx=x^{1+m} \left (\frac {a^2 c}{1+m}+\frac {a (2 b c+a d) x^2}{3+m}+\frac {b (b c+2 a d) x^4}{5+m}+\frac {b^2 d x^6}{7+m}\right ) \]
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Time = 2.63 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.15
method | result | size |
norman | \(\frac {a \left (a d +2 b c \right ) x^{3} {\mathrm e}^{m \ln \left (x \right )}}{3+m}+\frac {a^{2} c x \,{\mathrm e}^{m \ln \left (x \right )}}{1+m}+\frac {b \left (2 a d +b c \right ) x^{5} {\mathrm e}^{m \ln \left (x \right )}}{5+m}+\frac {b^{2} d \,x^{7} {\mathrm e}^{m \ln \left (x \right )}}{7+m}\) | \(82\) |
risch | \(\frac {x \left (b^{2} d \,m^{3} x^{6}+9 b^{2} d \,m^{2} x^{6}+2 a b d \,m^{3} x^{4}+b^{2} c \,m^{3} x^{4}+23 m \,x^{6} b^{2} d +22 a b d \,m^{2} x^{4}+11 b^{2} c \,m^{2} x^{4}+15 b^{2} d \,x^{6}+a^{2} d \,m^{3} x^{2}+2 a b c \,m^{3} x^{2}+62 a b d \,x^{4} m +31 b^{2} c \,x^{4} m +13 a^{2} d \,m^{2} x^{2}+26 a b c \,m^{2} x^{2}+42 a b d \,x^{4}+21 b^{2} c \,x^{4}+a^{2} c \,m^{3}+47 a^{2} d \,x^{2} m +94 a b c \,x^{2} m +15 a^{2} c \,m^{2}+35 a^{2} d \,x^{2}+70 a b c \,x^{2}+71 a^{2} c m +105 a^{2} c \right ) x^{m}}{\left (7+m \right ) \left (5+m \right ) \left (3+m \right ) \left (1+m \right )}\) | \(261\) |
gosper | \(\frac {x^{1+m} \left (b^{2} d \,m^{3} x^{6}+9 b^{2} d \,m^{2} x^{6}+2 a b d \,m^{3} x^{4}+b^{2} c \,m^{3} x^{4}+23 m \,x^{6} b^{2} d +22 a b d \,m^{2} x^{4}+11 b^{2} c \,m^{2} x^{4}+15 b^{2} d \,x^{6}+a^{2} d \,m^{3} x^{2}+2 a b c \,m^{3} x^{2}+62 a b d \,x^{4} m +31 b^{2} c \,x^{4} m +13 a^{2} d \,m^{2} x^{2}+26 a b c \,m^{2} x^{2}+42 a b d \,x^{4}+21 b^{2} c \,x^{4}+a^{2} c \,m^{3}+47 a^{2} d \,x^{2} m +94 a b c \,x^{2} m +15 a^{2} c \,m^{2}+35 a^{2} d \,x^{2}+70 a b c \,x^{2}+71 a^{2} c m +105 a^{2} c \right )}{\left (1+m \right ) \left (3+m \right ) \left (5+m \right ) \left (7+m \right )}\) | \(262\) |
parallelrisch | \(\frac {15 x^{7} x^{m} b^{2} d +70 x^{3} x^{m} a b c +15 x \,x^{m} a^{2} c \,m^{2}+71 x \,x^{m} a^{2} c m +2 x^{3} x^{m} a b c \,m^{3}+21 x^{5} x^{m} b^{2} c +35 x^{3} x^{m} a^{2} d +105 x \,x^{m} a^{2} c +26 x^{3} x^{m} a b c \,m^{2}+94 x^{3} x^{m} a b c m +x^{7} x^{m} b^{2} d \,m^{3}+9 x^{7} x^{m} b^{2} d \,m^{2}+23 x^{7} x^{m} b^{2} d m +x^{5} x^{m} b^{2} c \,m^{3}+11 x^{5} x^{m} b^{2} c \,m^{2}+31 x^{5} x^{m} b^{2} c m +x^{3} x^{m} a^{2} d \,m^{3}+42 x^{5} x^{m} a b d +13 x^{3} x^{m} a^{2} d \,m^{2}+47 x^{3} x^{m} a^{2} d m +x \,x^{m} a^{2} c \,m^{3}+2 x^{5} x^{m} a b d \,m^{3}+22 x^{5} x^{m} a b d \,m^{2}+62 x^{5} x^{m} a b d m}{\left (7+m \right ) \left (5+m \right ) \left (3+m \right ) \left (1+m \right )}\) | \(333\) |
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Leaf count of result is larger than twice the leaf count of optimal. 215 vs. \(2 (71) = 142\).
Time = 0.27 (sec) , antiderivative size = 215, normalized size of antiderivative = 3.03 \[ \int x^m \left (a+b x^2\right )^2 \left (c+d x^2\right ) \, dx=\frac {{\left ({\left (b^{2} d m^{3} + 9 \, b^{2} d m^{2} + 23 \, b^{2} d m + 15 \, b^{2} d\right )} x^{7} + {\left ({\left (b^{2} c + 2 \, a b d\right )} m^{3} + 21 \, b^{2} c + 42 \, a b d + 11 \, {\left (b^{2} c + 2 \, a b d\right )} m^{2} + 31 \, {\left (b^{2} c + 2 \, a b d\right )} m\right )} x^{5} + {\left ({\left (2 \, a b c + a^{2} d\right )} m^{3} + 70 \, a b c + 35 \, a^{2} d + 13 \, {\left (2 \, a b c + a^{2} d\right )} m^{2} + 47 \, {\left (2 \, a b c + a^{2} d\right )} m\right )} x^{3} + {\left (a^{2} c m^{3} + 15 \, a^{2} c m^{2} + 71 \, a^{2} c m + 105 \, a^{2} c\right )} x\right )} x^{m}}{m^{4} + 16 \, m^{3} + 86 \, m^{2} + 176 \, m + 105} \]
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Leaf count of result is larger than twice the leaf count of optimal. 1044 vs. \(2 (63) = 126\).
Time = 0.51 (sec) , antiderivative size = 1044, normalized size of antiderivative = 14.70 \[ \int x^m \left (a+b x^2\right )^2 \left (c+d x^2\right ) \, dx=\begin {cases} - \frac {a^{2} c}{6 x^{6}} - \frac {a^{2} d}{4 x^{4}} - \frac {a b c}{2 x^{4}} - \frac {a b d}{x^{2}} - \frac {b^{2} c}{2 x^{2}} + b^{2} d \log {\left (x \right )} & \text {for}\: m = -7 \\- \frac {a^{2} c}{4 x^{4}} - \frac {a^{2} d}{2 x^{2}} - \frac {a b c}{x^{2}} + 2 a b d \log {\left (x \right )} + b^{2} c \log {\left (x \right )} + \frac {b^{2} d x^{2}}{2} & \text {for}\: m = -5 \\- \frac {a^{2} c}{2 x^{2}} + a^{2} d \log {\left (x \right )} + 2 a b c \log {\left (x \right )} + a b d x^{2} + \frac {b^{2} c x^{2}}{2} + \frac {b^{2} d x^{4}}{4} & \text {for}\: m = -3 \\a^{2} c \log {\left (x \right )} + \frac {a^{2} d x^{2}}{2} + a b c x^{2} + \frac {a b d x^{4}}{2} + \frac {b^{2} c x^{4}}{4} + \frac {b^{2} d x^{6}}{6} & \text {for}\: m = -1 \\\frac {a^{2} c m^{3} x x^{m}}{m^{4} + 16 m^{3} + 86 m^{2} + 176 m + 105} + \frac {15 a^{2} c m^{2} x x^{m}}{m^{4} + 16 m^{3} + 86 m^{2} + 176 m + 105} + \frac {71 a^{2} c m x x^{m}}{m^{4} + 16 m^{3} + 86 m^{2} + 176 m + 105} + \frac {105 a^{2} c x x^{m}}{m^{4} + 16 m^{3} + 86 m^{2} + 176 m + 105} + \frac {a^{2} d m^{3} x^{3} x^{m}}{m^{4} + 16 m^{3} + 86 m^{2} + 176 m + 105} + \frac {13 a^{2} d m^{2} x^{3} x^{m}}{m^{4} + 16 m^{3} + 86 m^{2} + 176 m + 105} + \frac {47 a^{2} d m x^{3} x^{m}}{m^{4} + 16 m^{3} + 86 m^{2} + 176 m + 105} + \frac {35 a^{2} d x^{3} x^{m}}{m^{4} + 16 m^{3} + 86 m^{2} + 176 m + 105} + \frac {2 a b c m^{3} x^{3} x^{m}}{m^{4} + 16 m^{3} + 86 m^{2} + 176 m + 105} + \frac {26 a b c m^{2} x^{3} x^{m}}{m^{4} + 16 m^{3} + 86 m^{2} + 176 m + 105} + \frac {94 a b c m x^{3} x^{m}}{m^{4} + 16 m^{3} + 86 m^{2} + 176 m + 105} + \frac {70 a b c x^{3} x^{m}}{m^{4} + 16 m^{3} + 86 m^{2} + 176 m + 105} + \frac {2 a b d m^{3} x^{5} x^{m}}{m^{4} + 16 m^{3} + 86 m^{2} + 176 m + 105} + \frac {22 a b d m^{2} x^{5} x^{m}}{m^{4} + 16 m^{3} + 86 m^{2} + 176 m + 105} + \frac {62 a b d m x^{5} x^{m}}{m^{4} + 16 m^{3} + 86 m^{2} + 176 m + 105} + \frac {42 a b d x^{5} x^{m}}{m^{4} + 16 m^{3} + 86 m^{2} + 176 m + 105} + \frac {b^{2} c m^{3} x^{5} x^{m}}{m^{4} + 16 m^{3} + 86 m^{2} + 176 m + 105} + \frac {11 b^{2} c m^{2} x^{5} x^{m}}{m^{4} + 16 m^{3} + 86 m^{2} + 176 m + 105} + \frac {31 b^{2} c m x^{5} x^{m}}{m^{4} + 16 m^{3} + 86 m^{2} + 176 m + 105} + \frac {21 b^{2} c x^{5} x^{m}}{m^{4} + 16 m^{3} + 86 m^{2} + 176 m + 105} + \frac {b^{2} d m^{3} x^{7} x^{m}}{m^{4} + 16 m^{3} + 86 m^{2} + 176 m + 105} + \frac {9 b^{2} d m^{2} x^{7} x^{m}}{m^{4} + 16 m^{3} + 86 m^{2} + 176 m + 105} + \frac {23 b^{2} d m x^{7} x^{m}}{m^{4} + 16 m^{3} + 86 m^{2} + 176 m + 105} + \frac {15 b^{2} d x^{7} x^{m}}{m^{4} + 16 m^{3} + 86 m^{2} + 176 m + 105} & \text {otherwise} \end {cases} \]
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Time = 0.20 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.28 \[ \int x^m \left (a+b x^2\right )^2 \left (c+d x^2\right ) \, dx=\frac {b^{2} d x^{m + 7}}{m + 7} + \frac {b^{2} c x^{m + 5}}{m + 5} + \frac {2 \, a b d x^{m + 5}}{m + 5} + \frac {2 \, a b c x^{m + 3}}{m + 3} + \frac {a^{2} d x^{m + 3}}{m + 3} + \frac {a^{2} c x^{m + 1}}{m + 1} \]
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Leaf count of result is larger than twice the leaf count of optimal. 332 vs. \(2 (71) = 142\).
Time = 0.34 (sec) , antiderivative size = 332, normalized size of antiderivative = 4.68 \[ \int x^m \left (a+b x^2\right )^2 \left (c+d x^2\right ) \, dx=\frac {b^{2} d m^{3} x^{7} x^{m} + 9 \, b^{2} d m^{2} x^{7} x^{m} + b^{2} c m^{3} x^{5} x^{m} + 2 \, a b d m^{3} x^{5} x^{m} + 23 \, b^{2} d m x^{7} x^{m} + 11 \, b^{2} c m^{2} x^{5} x^{m} + 22 \, a b d m^{2} x^{5} x^{m} + 15 \, b^{2} d x^{7} x^{m} + 2 \, a b c m^{3} x^{3} x^{m} + a^{2} d m^{3} x^{3} x^{m} + 31 \, b^{2} c m x^{5} x^{m} + 62 \, a b d m x^{5} x^{m} + 26 \, a b c m^{2} x^{3} x^{m} + 13 \, a^{2} d m^{2} x^{3} x^{m} + 21 \, b^{2} c x^{5} x^{m} + 42 \, a b d x^{5} x^{m} + a^{2} c m^{3} x x^{m} + 94 \, a b c m x^{3} x^{m} + 47 \, a^{2} d m x^{3} x^{m} + 15 \, a^{2} c m^{2} x x^{m} + 70 \, a b c x^{3} x^{m} + 35 \, a^{2} d x^{3} x^{m} + 71 \, a^{2} c m x x^{m} + 105 \, a^{2} c x x^{m}}{m^{4} + 16 \, m^{3} + 86 \, m^{2} + 176 \, m + 105} \]
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Time = 4.98 (sec) , antiderivative size = 177, normalized size of antiderivative = 2.49 \[ \int x^m \left (a+b x^2\right )^2 \left (c+d x^2\right ) \, dx=x^m\,\left (\frac {a\,x^3\,\left (a\,d+2\,b\,c\right )\,\left (m^3+13\,m^2+47\,m+35\right )}{m^4+16\,m^3+86\,m^2+176\,m+105}+\frac {b\,x^5\,\left (2\,a\,d+b\,c\right )\,\left (m^3+11\,m^2+31\,m+21\right )}{m^4+16\,m^3+86\,m^2+176\,m+105}+\frac {b^2\,d\,x^7\,\left (m^3+9\,m^2+23\,m+15\right )}{m^4+16\,m^3+86\,m^2+176\,m+105}+\frac {a^2\,c\,x\,\left (m^3+15\,m^2+71\,m+105\right )}{m^4+16\,m^3+86\,m^2+176\,m+105}\right ) \]
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