Integrand size = 29, antiderivative size = 144 \[ \int (e x)^m \left (a+b x^2\right )^2 \left (A+B x^2\right ) \left (c+d x^2\right ) \, dx=\frac {a^2 A c (e x)^{1+m}}{e (1+m)}+\frac {a (2 A b c+a B c+a A d) (e x)^{3+m}}{e^3 (3+m)}+\frac {(a B (2 b c+a d)+A b (b c+2 a d)) (e x)^{5+m}}{e^5 (5+m)}+\frac {b (b B c+A b d+2 a B d) (e x)^{7+m}}{e^7 (7+m)}+\frac {b^2 B d (e x)^{9+m}}{e^9 (9+m)} \]
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Time = 0.08 (sec) , antiderivative size = 144, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.034, Rules used = {584} \[ \int (e x)^m \left (a+b x^2\right )^2 \left (A+B x^2\right ) \left (c+d x^2\right ) \, dx=\frac {a^2 A c (e x)^{m+1}}{e (m+1)}+\frac {b (e x)^{m+7} (2 a B d+A b d+b B c)}{e^7 (m+7)}+\frac {(e x)^{m+5} (A b (2 a d+b c)+a B (a d+2 b c))}{e^5 (m+5)}+\frac {a (e x)^{m+3} (a A d+a B c+2 A b c)}{e^3 (m+3)}+\frac {b^2 B d (e x)^{m+9}}{e^9 (m+9)} \]
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Rule 584
Rubi steps \begin{align*} \text {integral}& = \int \left (a^2 A c (e x)^m+\frac {a (2 A b c+a B c+a A d) (e x)^{2+m}}{e^2}+\frac {(a B (2 b c+a d)+A b (b c+2 a d)) (e x)^{4+m}}{e^4}+\frac {b (b B c+A b d+2 a B d) (e x)^{6+m}}{e^6}+\frac {b^2 B d (e x)^{8+m}}{e^8}\right ) \, dx \\ & = \frac {a^2 A c (e x)^{1+m}}{e (1+m)}+\frac {a (2 A b c+a B c+a A d) (e x)^{3+m}}{e^3 (3+m)}+\frac {(a B (2 b c+a d)+A b (b c+2 a d)) (e x)^{5+m}}{e^5 (5+m)}+\frac {b (b B c+A b d+2 a B d) (e x)^{7+m}}{e^7 (7+m)}+\frac {b^2 B d (e x)^{9+m}}{e^9 (9+m)} \\ \end{align*}
Time = 0.13 (sec) , antiderivative size = 113, normalized size of antiderivative = 0.78 \[ \int (e x)^m \left (a+b x^2\right )^2 \left (A+B x^2\right ) \left (c+d x^2\right ) \, dx=x (e x)^m \left (\frac {a^2 A c}{1+m}+\frac {a (2 A b c+a B c+a A d) x^2}{3+m}+\frac {(a B (2 b c+a d)+A b (b c+2 a d)) x^4}{5+m}+\frac {b (b B c+A b d+2 a B d) x^6}{7+m}+\frac {b^2 B d x^8}{9+m}\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. \(710\) vs. \(2(144)=288\).
Time = 3.38 (sec) , antiderivative size = 711, normalized size of antiderivative = 4.94
method | result | size |
gosper | \(\frac {x \left (B \,b^{2} d \,m^{4} x^{8}+16 B \,b^{2} d \,m^{3} x^{8}+A \,b^{2} d \,m^{4} x^{6}+2 B a b d \,m^{4} x^{6}+B \,b^{2} c \,m^{4} x^{6}+86 B \,b^{2} d \,m^{2} x^{8}+18 A \,b^{2} d \,m^{3} x^{6}+36 B a b d \,m^{3} x^{6}+18 B \,b^{2} c \,m^{3} x^{6}+176 m \,x^{8} b^{2} B d +2 A a b d \,m^{4} x^{4}+A \,b^{2} c \,m^{4} x^{4}+104 A \,b^{2} d \,m^{2} x^{6}+B \,a^{2} d \,m^{4} x^{4}+2 B a b c \,m^{4} x^{4}+208 B a b d \,m^{2} x^{6}+104 B \,b^{2} c \,m^{2} x^{6}+105 b^{2} B d \,x^{8}+40 A a b d \,m^{3} x^{4}+20 A \,b^{2} c \,m^{3} x^{4}+222 A \,b^{2} d \,x^{6} m +20 B \,a^{2} d \,m^{3} x^{4}+40 B a b c \,m^{3} x^{4}+444 B a b d \,x^{6} m +222 B \,b^{2} c \,x^{6} m +A \,a^{2} d \,m^{4} x^{2}+2 A a b c \,m^{4} x^{2}+260 A a b d \,m^{2} x^{4}+130 A \,b^{2} c \,m^{2} x^{4}+135 A \,b^{2} d \,x^{6}+B \,a^{2} c \,m^{4} x^{2}+130 B \,a^{2} d \,m^{2} x^{4}+260 B a b c \,m^{2} x^{4}+270 B a b d \,x^{6}+135 B \,b^{2} c \,x^{6}+22 A \,a^{2} d \,m^{3} x^{2}+44 A a b c \,m^{3} x^{2}+600 A a b d \,x^{4} m +300 A \,b^{2} c \,x^{4} m +22 B \,a^{2} c \,m^{3} x^{2}+300 B \,a^{2} d \,x^{4} m +600 B a b c \,x^{4} m +A \,a^{2} c \,m^{4}+164 A \,a^{2} d \,m^{2} x^{2}+328 A a b c \,m^{2} x^{2}+378 A a b d \,x^{4}+189 A \,b^{2} c \,x^{4}+164 B \,a^{2} c \,m^{2} x^{2}+189 B \,a^{2} d \,x^{4}+378 B a b c \,x^{4}+24 A \,a^{2} c \,m^{3}+458 A \,a^{2} d \,x^{2} m +916 A a b c \,x^{2} m +458 B \,a^{2} c \,x^{2} m +206 A \,a^{2} c \,m^{2}+315 A \,a^{2} d \,x^{2}+630 A a b c \,x^{2}+315 B \,a^{2} c \,x^{2}+744 A \,a^{2} c m +945 A \,a^{2} c \right ) \left (e x \right )^{m}}{\left (9+m \right ) \left (7+m \right ) \left (5+m \right ) \left (3+m \right ) \left (1+m \right )}\) | \(711\) |
risch | \(\frac {x \left (B \,b^{2} d \,m^{4} x^{8}+16 B \,b^{2} d \,m^{3} x^{8}+A \,b^{2} d \,m^{4} x^{6}+2 B a b d \,m^{4} x^{6}+B \,b^{2} c \,m^{4} x^{6}+86 B \,b^{2} d \,m^{2} x^{8}+18 A \,b^{2} d \,m^{3} x^{6}+36 B a b d \,m^{3} x^{6}+18 B \,b^{2} c \,m^{3} x^{6}+176 m \,x^{8} b^{2} B d +2 A a b d \,m^{4} x^{4}+A \,b^{2} c \,m^{4} x^{4}+104 A \,b^{2} d \,m^{2} x^{6}+B \,a^{2} d \,m^{4} x^{4}+2 B a b c \,m^{4} x^{4}+208 B a b d \,m^{2} x^{6}+104 B \,b^{2} c \,m^{2} x^{6}+105 b^{2} B d \,x^{8}+40 A a b d \,m^{3} x^{4}+20 A \,b^{2} c \,m^{3} x^{4}+222 A \,b^{2} d \,x^{6} m +20 B \,a^{2} d \,m^{3} x^{4}+40 B a b c \,m^{3} x^{4}+444 B a b d \,x^{6} m +222 B \,b^{2} c \,x^{6} m +A \,a^{2} d \,m^{4} x^{2}+2 A a b c \,m^{4} x^{2}+260 A a b d \,m^{2} x^{4}+130 A \,b^{2} c \,m^{2} x^{4}+135 A \,b^{2} d \,x^{6}+B \,a^{2} c \,m^{4} x^{2}+130 B \,a^{2} d \,m^{2} x^{4}+260 B a b c \,m^{2} x^{4}+270 B a b d \,x^{6}+135 B \,b^{2} c \,x^{6}+22 A \,a^{2} d \,m^{3} x^{2}+44 A a b c \,m^{3} x^{2}+600 A a b d \,x^{4} m +300 A \,b^{2} c \,x^{4} m +22 B \,a^{2} c \,m^{3} x^{2}+300 B \,a^{2} d \,x^{4} m +600 B a b c \,x^{4} m +A \,a^{2} c \,m^{4}+164 A \,a^{2} d \,m^{2} x^{2}+328 A a b c \,m^{2} x^{2}+378 A a b d \,x^{4}+189 A \,b^{2} c \,x^{4}+164 B \,a^{2} c \,m^{2} x^{2}+189 B \,a^{2} d \,x^{4}+378 B a b c \,x^{4}+24 A \,a^{2} c \,m^{3}+458 A \,a^{2} d \,x^{2} m +916 A a b c \,x^{2} m +458 B \,a^{2} c \,x^{2} m +206 A \,a^{2} c \,m^{2}+315 A \,a^{2} d \,x^{2}+630 A a b c \,x^{2}+315 B \,a^{2} c \,x^{2}+744 A \,a^{2} c m +945 A \,a^{2} c \right ) \left (e x \right )^{m}}{\left (9+m \right ) \left (7+m \right ) \left (5+m \right ) \left (3+m \right ) \left (1+m \right )}\) | \(711\) |
parallelrisch | \(\text {Expression too large to display}\) | \(1010\) |
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Leaf count of result is larger than twice the leaf count of optimal. 532 vs. \(2 (144) = 288\).
Time = 0.28 (sec) , antiderivative size = 532, normalized size of antiderivative = 3.69 \[ \int (e x)^m \left (a+b x^2\right )^2 \left (A+B x^2\right ) \left (c+d x^2\right ) \, dx=\frac {{\left ({\left (B b^{2} d m^{4} + 16 \, B b^{2} d m^{3} + 86 \, B b^{2} d m^{2} + 176 \, B b^{2} d m + 105 \, B b^{2} d\right )} x^{9} + {\left ({\left (B b^{2} c + {\left (2 \, B a b + A b^{2}\right )} d\right )} m^{4} + 135 \, B b^{2} c + 18 \, {\left (B b^{2} c + {\left (2 \, B a b + A b^{2}\right )} d\right )} m^{3} + 104 \, {\left (B b^{2} c + {\left (2 \, B a b + A b^{2}\right )} d\right )} m^{2} + 135 \, {\left (2 \, B a b + A b^{2}\right )} d + 222 \, {\left (B b^{2} c + {\left (2 \, B a b + A b^{2}\right )} d\right )} m\right )} x^{7} + {\left ({\left ({\left (2 \, B a b + A b^{2}\right )} c + {\left (B a^{2} + 2 \, A a b\right )} d\right )} m^{4} + 20 \, {\left ({\left (2 \, B a b + A b^{2}\right )} c + {\left (B a^{2} + 2 \, A a b\right )} d\right )} m^{3} + 130 \, {\left ({\left (2 \, B a b + A b^{2}\right )} c + {\left (B a^{2} + 2 \, A a b\right )} d\right )} m^{2} + 189 \, {\left (2 \, B a b + A b^{2}\right )} c + 189 \, {\left (B a^{2} + 2 \, A a b\right )} d + 300 \, {\left ({\left (2 \, B a b + A b^{2}\right )} c + {\left (B a^{2} + 2 \, A a b\right )} d\right )} m\right )} x^{5} + {\left ({\left (A a^{2} d + {\left (B a^{2} + 2 \, A a b\right )} c\right )} m^{4} + 315 \, A a^{2} d + 22 \, {\left (A a^{2} d + {\left (B a^{2} + 2 \, A a b\right )} c\right )} m^{3} + 164 \, {\left (A a^{2} d + {\left (B a^{2} + 2 \, A a b\right )} c\right )} m^{2} + 315 \, {\left (B a^{2} + 2 \, A a b\right )} c + 458 \, {\left (A a^{2} d + {\left (B a^{2} + 2 \, A a b\right )} c\right )} m\right )} x^{3} + {\left (A a^{2} c m^{4} + 24 \, A a^{2} c m^{3} + 206 \, A a^{2} c m^{2} + 744 \, A a^{2} c m + 945 \, A a^{2} c\right )} x\right )} \left (e x\right )^{m}}{m^{5} + 25 \, m^{4} + 230 \, m^{3} + 950 \, m^{2} + 1689 \, m + 945} \]
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Leaf count of result is larger than twice the leaf count of optimal. 3271 vs. \(2 (139) = 278\).
Time = 0.66 (sec) , antiderivative size = 3271, normalized size of antiderivative = 22.72 \[ \int (e x)^m \left (a+b x^2\right )^2 \left (A+B x^2\right ) \left (c+d x^2\right ) \, dx=\text {Too large to display} \]
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Time = 0.23 (sec) , antiderivative size = 242, normalized size of antiderivative = 1.68 \[ \int (e x)^m \left (a+b x^2\right )^2 \left (A+B x^2\right ) \left (c+d x^2\right ) \, dx=\frac {B b^{2} d e^{m} x^{9} x^{m}}{m + 9} + \frac {B b^{2} c e^{m} x^{7} x^{m}}{m + 7} + \frac {2 \, B a b d e^{m} x^{7} x^{m}}{m + 7} + \frac {A b^{2} d e^{m} x^{7} x^{m}}{m + 7} + \frac {2 \, B a b c e^{m} x^{5} x^{m}}{m + 5} + \frac {A b^{2} c e^{m} x^{5} x^{m}}{m + 5} + \frac {B a^{2} d e^{m} x^{5} x^{m}}{m + 5} + \frac {2 \, A a b d e^{m} x^{5} x^{m}}{m + 5} + \frac {B a^{2} c e^{m} x^{3} x^{m}}{m + 3} + \frac {2 \, A a b c e^{m} x^{3} x^{m}}{m + 3} + \frac {A a^{2} d e^{m} x^{3} x^{m}}{m + 3} + \frac {\left (e x\right )^{m + 1} A a^{2} c}{e {\left (m + 1\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 1009 vs. \(2 (144) = 288\).
Time = 0.30 (sec) , antiderivative size = 1009, normalized size of antiderivative = 7.01 \[ \int (e x)^m \left (a+b x^2\right )^2 \left (A+B x^2\right ) \left (c+d x^2\right ) \, dx=\frac {\left (e x\right )^{m} B b^{2} d m^{4} x^{9} + 16 \, \left (e x\right )^{m} B b^{2} d m^{3} x^{9} + \left (e x\right )^{m} B b^{2} c m^{4} x^{7} + 2 \, \left (e x\right )^{m} B a b d m^{4} x^{7} + \left (e x\right )^{m} A b^{2} d m^{4} x^{7} + 86 \, \left (e x\right )^{m} B b^{2} d m^{2} x^{9} + 18 \, \left (e x\right )^{m} B b^{2} c m^{3} x^{7} + 36 \, \left (e x\right )^{m} B a b d m^{3} x^{7} + 18 \, \left (e x\right )^{m} A b^{2} d m^{3} x^{7} + 176 \, \left (e x\right )^{m} B b^{2} d m x^{9} + 2 \, \left (e x\right )^{m} B a b c m^{4} x^{5} + \left (e x\right )^{m} A b^{2} c m^{4} x^{5} + \left (e x\right )^{m} B a^{2} d m^{4} x^{5} + 2 \, \left (e x\right )^{m} A a b d m^{4} x^{5} + 104 \, \left (e x\right )^{m} B b^{2} c m^{2} x^{7} + 208 \, \left (e x\right )^{m} B a b d m^{2} x^{7} + 104 \, \left (e x\right )^{m} A b^{2} d m^{2} x^{7} + 105 \, \left (e x\right )^{m} B b^{2} d x^{9} + 40 \, \left (e x\right )^{m} B a b c m^{3} x^{5} + 20 \, \left (e x\right )^{m} A b^{2} c m^{3} x^{5} + 20 \, \left (e x\right )^{m} B a^{2} d m^{3} x^{5} + 40 \, \left (e x\right )^{m} A a b d m^{3} x^{5} + 222 \, \left (e x\right )^{m} B b^{2} c m x^{7} + 444 \, \left (e x\right )^{m} B a b d m x^{7} + 222 \, \left (e x\right )^{m} A b^{2} d m x^{7} + \left (e x\right )^{m} B a^{2} c m^{4} x^{3} + 2 \, \left (e x\right )^{m} A a b c m^{4} x^{3} + \left (e x\right )^{m} A a^{2} d m^{4} x^{3} + 260 \, \left (e x\right )^{m} B a b c m^{2} x^{5} + 130 \, \left (e x\right )^{m} A b^{2} c m^{2} x^{5} + 130 \, \left (e x\right )^{m} B a^{2} d m^{2} x^{5} + 260 \, \left (e x\right )^{m} A a b d m^{2} x^{5} + 135 \, \left (e x\right )^{m} B b^{2} c x^{7} + 270 \, \left (e x\right )^{m} B a b d x^{7} + 135 \, \left (e x\right )^{m} A b^{2} d x^{7} + 22 \, \left (e x\right )^{m} B a^{2} c m^{3} x^{3} + 44 \, \left (e x\right )^{m} A a b c m^{3} x^{3} + 22 \, \left (e x\right )^{m} A a^{2} d m^{3} x^{3} + 600 \, \left (e x\right )^{m} B a b c m x^{5} + 300 \, \left (e x\right )^{m} A b^{2} c m x^{5} + 300 \, \left (e x\right )^{m} B a^{2} d m x^{5} + 600 \, \left (e x\right )^{m} A a b d m x^{5} + \left (e x\right )^{m} A a^{2} c m^{4} x + 164 \, \left (e x\right )^{m} B a^{2} c m^{2} x^{3} + 328 \, \left (e x\right )^{m} A a b c m^{2} x^{3} + 164 \, \left (e x\right )^{m} A a^{2} d m^{2} x^{3} + 378 \, \left (e x\right )^{m} B a b c x^{5} + 189 \, \left (e x\right )^{m} A b^{2} c x^{5} + 189 \, \left (e x\right )^{m} B a^{2} d x^{5} + 378 \, \left (e x\right )^{m} A a b d x^{5} + 24 \, \left (e x\right )^{m} A a^{2} c m^{3} x + 458 \, \left (e x\right )^{m} B a^{2} c m x^{3} + 916 \, \left (e x\right )^{m} A a b c m x^{3} + 458 \, \left (e x\right )^{m} A a^{2} d m x^{3} + 206 \, \left (e x\right )^{m} A a^{2} c m^{2} x + 315 \, \left (e x\right )^{m} B a^{2} c x^{3} + 630 \, \left (e x\right )^{m} A a b c x^{3} + 315 \, \left (e x\right )^{m} A a^{2} d x^{3} + 744 \, \left (e x\right )^{m} A a^{2} c m x + 945 \, \left (e x\right )^{m} A a^{2} c x}{m^{5} + 25 \, m^{4} + 230 \, m^{3} + 950 \, m^{2} + 1689 \, m + 945} \]
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Time = 5.63 (sec) , antiderivative size = 305, normalized size of antiderivative = 2.12 \[ \int (e x)^m \left (a+b x^2\right )^2 \left (A+B x^2\right ) \left (c+d x^2\right ) \, dx={\left (e\,x\right )}^m\,\left (\frac {x^5\,\left (A\,b^2\,c+B\,a^2\,d+2\,A\,a\,b\,d+2\,B\,a\,b\,c\right )\,\left (m^4+20\,m^3+130\,m^2+300\,m+189\right )}{m^5+25\,m^4+230\,m^3+950\,m^2+1689\,m+945}+\frac {a\,x^3\,\left (A\,a\,d+2\,A\,b\,c+B\,a\,c\right )\,\left (m^4+22\,m^3+164\,m^2+458\,m+315\right )}{m^5+25\,m^4+230\,m^3+950\,m^2+1689\,m+945}+\frac {b\,x^7\,\left (A\,b\,d+2\,B\,a\,d+B\,b\,c\right )\,\left (m^4+18\,m^3+104\,m^2+222\,m+135\right )}{m^5+25\,m^4+230\,m^3+950\,m^2+1689\,m+945}+\frac {A\,a^2\,c\,x\,\left (m^4+24\,m^3+206\,m^2+744\,m+945\right )}{m^5+25\,m^4+230\,m^3+950\,m^2+1689\,m+945}+\frac {B\,b^2\,d\,x^9\,\left (m^4+16\,m^3+86\,m^2+176\,m+105\right )}{m^5+25\,m^4+230\,m^3+950\,m^2+1689\,m+945}\right ) \]
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