Integrand size = 26, antiderivative size = 65 \[ \int (d+e x)^2 \left (a^2+2 a b x+b^2 x^2\right )^2 \, dx=\frac {(b d-a e)^2 (a+b x)^5}{5 b^3}+\frac {e (b d-a e) (a+b x)^6}{3 b^3}+\frac {e^2 (a+b x)^7}{7 b^3} \]
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Time = 0.06 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {27, 45} \[ \int (d+e x)^2 \left (a^2+2 a b x+b^2 x^2\right )^2 \, dx=\frac {e (a+b x)^6 (b d-a e)}{3 b^3}+\frac {(a+b x)^5 (b d-a e)^2}{5 b^3}+\frac {e^2 (a+b x)^7}{7 b^3} \]
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Rule 27
Rule 45
Rubi steps \begin{align*} \text {integral}& = \int (a+b x)^4 (d+e x)^2 \, dx \\ & = \int \left (\frac {(b d-a e)^2 (a+b x)^4}{b^2}+\frac {2 e (b d-a e) (a+b x)^5}{b^2}+\frac {e^2 (a+b x)^6}{b^2}\right ) \, dx \\ & = \frac {(b d-a e)^2 (a+b x)^5}{5 b^3}+\frac {e (b d-a e) (a+b x)^6}{3 b^3}+\frac {e^2 (a+b x)^7}{7 b^3} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(148\) vs. \(2(65)=130\).
Time = 0.02 (sec) , antiderivative size = 148, normalized size of antiderivative = 2.28 \[ \int (d+e x)^2 \left (a^2+2 a b x+b^2 x^2\right )^2 \, dx=a^4 d^2 x+a^3 d (2 b d+a e) x^2+\frac {1}{3} a^2 \left (6 b^2 d^2+8 a b d e+a^2 e^2\right ) x^3+a b \left (b^2 d^2+3 a b d e+a^2 e^2\right ) x^4+\frac {1}{5} b^2 \left (b^2 d^2+8 a b d e+6 a^2 e^2\right ) x^5+\frac {1}{3} b^3 e (b d+2 a e) x^6+\frac {1}{7} b^4 e^2 x^7 \]
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Leaf count of result is larger than twice the leaf count of optimal. \(156\) vs. \(2(59)=118\).
Time = 2.26 (sec) , antiderivative size = 157, normalized size of antiderivative = 2.42
method | result | size |
norman | \(\frac {e^{2} b^{4} x^{7}}{7}+\left (\frac {2}{3} e^{2} a \,b^{3}+\frac {1}{3} b^{4} d e \right ) x^{6}+\left (\frac {6}{5} a^{2} b^{2} e^{2}+\frac {8}{5} a \,b^{3} d e +\frac {1}{5} b^{4} d^{2}\right ) x^{5}+\left (e^{2} a^{3} b +3 d e \,a^{2} b^{2}+d^{2} a \,b^{3}\right ) x^{4}+\left (\frac {1}{3} a^{4} e^{2}+\frac {8}{3} d e \,a^{3} b +2 b^{2} d^{2} a^{2}\right ) x^{3}+\left (d e \,a^{4}+2 a^{3} b \,d^{2}\right ) x^{2}+a^{4} d^{2} x\) | \(157\) |
default | \(\frac {e^{2} b^{4} x^{7}}{7}+\frac {\left (4 e^{2} a \,b^{3}+2 b^{4} d e \right ) x^{6}}{6}+\frac {\left (6 a^{2} b^{2} e^{2}+8 a \,b^{3} d e +b^{4} d^{2}\right ) x^{5}}{5}+\frac {\left (4 e^{2} a^{3} b +12 d e \,a^{2} b^{2}+4 d^{2} a \,b^{3}\right ) x^{4}}{4}+\frac {\left (a^{4} e^{2}+8 d e \,a^{3} b +6 b^{2} d^{2} a^{2}\right ) x^{3}}{3}+\frac {\left (2 d e \,a^{4}+4 a^{3} b \,d^{2}\right ) x^{2}}{2}+a^{4} d^{2} x\) | \(163\) |
risch | \(\frac {1}{7} e^{2} b^{4} x^{7}+\frac {2}{3} x^{6} e^{2} a \,b^{3}+\frac {1}{3} x^{6} b^{4} d e +\frac {6}{5} x^{5} a^{2} b^{2} e^{2}+\frac {8}{5} x^{5} a \,b^{3} d e +\frac {1}{5} d^{2} x^{5} b^{4}+a^{3} b \,e^{2} x^{4}+3 a^{2} b^{2} d e \,x^{4}+a \,b^{3} d^{2} x^{4}+\frac {1}{3} x^{3} a^{4} e^{2}+\frac {8}{3} x^{3} d e \,a^{3} b +2 a^{2} b^{2} d^{2} x^{3}+d e \,a^{4} x^{2}+2 a^{3} b \,d^{2} x^{2}+a^{4} d^{2} x\) | \(171\) |
parallelrisch | \(\frac {1}{7} e^{2} b^{4} x^{7}+\frac {2}{3} x^{6} e^{2} a \,b^{3}+\frac {1}{3} x^{6} b^{4} d e +\frac {6}{5} x^{5} a^{2} b^{2} e^{2}+\frac {8}{5} x^{5} a \,b^{3} d e +\frac {1}{5} d^{2} x^{5} b^{4}+a^{3} b \,e^{2} x^{4}+3 a^{2} b^{2} d e \,x^{4}+a \,b^{3} d^{2} x^{4}+\frac {1}{3} x^{3} a^{4} e^{2}+\frac {8}{3} x^{3} d e \,a^{3} b +2 a^{2} b^{2} d^{2} x^{3}+d e \,a^{4} x^{2}+2 a^{3} b \,d^{2} x^{2}+a^{4} d^{2} x\) | \(171\) |
gosper | \(\frac {x \left (15 e^{2} b^{4} x^{6}+70 x^{5} e^{2} a \,b^{3}+35 x^{5} b^{4} d e +126 x^{4} a^{2} b^{2} e^{2}+168 x^{4} a \,b^{3} d e +21 x^{4} b^{4} d^{2}+105 a^{3} b \,e^{2} x^{3}+315 a^{2} b^{2} d e \,x^{3}+105 a \,b^{3} d^{2} x^{3}+35 x^{2} a^{4} e^{2}+280 x^{2} d e \,a^{3} b +210 x^{2} b^{2} d^{2} a^{2}+105 a^{4} d e x +210 a^{3} b \,d^{2} x +105 a^{4} d^{2}\right )}{105}\) | \(173\) |
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Leaf count of result is larger than twice the leaf count of optimal. 156 vs. \(2 (59) = 118\).
Time = 0.26 (sec) , antiderivative size = 156, normalized size of antiderivative = 2.40 \[ \int (d+e x)^2 \left (a^2+2 a b x+b^2 x^2\right )^2 \, dx=\frac {1}{7} \, b^{4} e^{2} x^{7} + a^{4} d^{2} x + \frac {1}{3} \, {\left (b^{4} d e + 2 \, a b^{3} e^{2}\right )} x^{6} + \frac {1}{5} \, {\left (b^{4} d^{2} + 8 \, a b^{3} d e + 6 \, a^{2} b^{2} e^{2}\right )} x^{5} + {\left (a b^{3} d^{2} + 3 \, a^{2} b^{2} d e + a^{3} b e^{2}\right )} x^{4} + \frac {1}{3} \, {\left (6 \, a^{2} b^{2} d^{2} + 8 \, a^{3} b d e + a^{4} e^{2}\right )} x^{3} + {\left (2 \, a^{3} b d^{2} + a^{4} d e\right )} x^{2} \]
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Leaf count of result is larger than twice the leaf count of optimal. 168 vs. \(2 (54) = 108\).
Time = 0.04 (sec) , antiderivative size = 168, normalized size of antiderivative = 2.58 \[ \int (d+e x)^2 \left (a^2+2 a b x+b^2 x^2\right )^2 \, dx=a^{4} d^{2} x + \frac {b^{4} e^{2} x^{7}}{7} + x^{6} \cdot \left (\frac {2 a b^{3} e^{2}}{3} + \frac {b^{4} d e}{3}\right ) + x^{5} \cdot \left (\frac {6 a^{2} b^{2} e^{2}}{5} + \frac {8 a b^{3} d e}{5} + \frac {b^{4} d^{2}}{5}\right ) + x^{4} \left (a^{3} b e^{2} + 3 a^{2} b^{2} d e + a b^{3} d^{2}\right ) + x^{3} \left (\frac {a^{4} e^{2}}{3} + \frac {8 a^{3} b d e}{3} + 2 a^{2} b^{2} d^{2}\right ) + x^{2} \left (a^{4} d e + 2 a^{3} b d^{2}\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. 156 vs. \(2 (59) = 118\).
Time = 0.19 (sec) , antiderivative size = 156, normalized size of antiderivative = 2.40 \[ \int (d+e x)^2 \left (a^2+2 a b x+b^2 x^2\right )^2 \, dx=\frac {1}{7} \, b^{4} e^{2} x^{7} + a^{4} d^{2} x + \frac {1}{3} \, {\left (b^{4} d e + 2 \, a b^{3} e^{2}\right )} x^{6} + \frac {1}{5} \, {\left (b^{4} d^{2} + 8 \, a b^{3} d e + 6 \, a^{2} b^{2} e^{2}\right )} x^{5} + {\left (a b^{3} d^{2} + 3 \, a^{2} b^{2} d e + a^{3} b e^{2}\right )} x^{4} + \frac {1}{3} \, {\left (6 \, a^{2} b^{2} d^{2} + 8 \, a^{3} b d e + a^{4} e^{2}\right )} x^{3} + {\left (2 \, a^{3} b d^{2} + a^{4} d e\right )} x^{2} \]
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Leaf count of result is larger than twice the leaf count of optimal. 170 vs. \(2 (59) = 118\).
Time = 0.26 (sec) , antiderivative size = 170, normalized size of antiderivative = 2.62 \[ \int (d+e x)^2 \left (a^2+2 a b x+b^2 x^2\right )^2 \, dx=\frac {1}{7} \, b^{4} e^{2} x^{7} + \frac {1}{3} \, b^{4} d e x^{6} + \frac {2}{3} \, a b^{3} e^{2} x^{6} + \frac {1}{5} \, b^{4} d^{2} x^{5} + \frac {8}{5} \, a b^{3} d e x^{5} + \frac {6}{5} \, a^{2} b^{2} e^{2} x^{5} + a b^{3} d^{2} x^{4} + 3 \, a^{2} b^{2} d e x^{4} + a^{3} b e^{2} x^{4} + 2 \, a^{2} b^{2} d^{2} x^{3} + \frac {8}{3} \, a^{3} b d e x^{3} + \frac {1}{3} \, a^{4} e^{2} x^{3} + 2 \, a^{3} b d^{2} x^{2} + a^{4} d e x^{2} + a^{4} d^{2} x \]
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Time = 9.65 (sec) , antiderivative size = 144, normalized size of antiderivative = 2.22 \[ \int (d+e x)^2 \left (a^2+2 a b x+b^2 x^2\right )^2 \, dx=x^3\,\left (\frac {a^4\,e^2}{3}+\frac {8\,a^3\,b\,d\,e}{3}+2\,a^2\,b^2\,d^2\right )+x^5\,\left (\frac {6\,a^2\,b^2\,e^2}{5}+\frac {8\,a\,b^3\,d\,e}{5}+\frac {b^4\,d^2}{5}\right )+a^4\,d^2\,x+\frac {b^4\,e^2\,x^7}{7}+a^3\,d\,x^2\,\left (a\,e+2\,b\,d\right )+\frac {b^3\,e\,x^6\,\left (2\,a\,e+b\,d\right )}{3}+a\,b\,x^4\,\left (a^2\,e^2+3\,a\,b\,d\,e+b^2\,d^2\right ) \]
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