Integrand size = 13, antiderivative size = 38 \[ \int (a+b x)^4 (c+d x) \, dx=\frac {(b c-a d) (a+b x)^5}{5 b^2}+\frac {d (a+b x)^6}{6 b^2} \]
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Time = 0.01 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {45} \[ \int (a+b x)^4 (c+d x) \, dx=\frac {(a+b x)^5 (b c-a d)}{5 b^2}+\frac {d (a+b x)^6}{6 b^2} \]
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Rule 45
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {(b c-a d) (a+b x)^4}{b}+\frac {d (a+b x)^5}{b}\right ) \, dx \\ & = \frac {(b c-a d) (a+b x)^5}{5 b^2}+\frac {d (a+b x)^6}{6 b^2} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(84\) vs. \(2(38)=76\).
Time = 0.01 (sec) , antiderivative size = 84, normalized size of antiderivative = 2.21 \[ \int (a+b x)^4 (c+d x) \, dx=\frac {1}{30} x \left (15 a^4 (2 c+d x)+20 a^3 b x (3 c+2 d x)+15 a^2 b^2 x^2 (4 c+3 d x)+6 a b^3 x^3 (5 c+4 d x)+b^4 x^4 (6 c+5 d x)\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. \(93\) vs. \(2(34)=68\).
Time = 0.18 (sec) , antiderivative size = 94, normalized size of antiderivative = 2.47
method | result | size |
norman | \(\frac {b^{4} d \,x^{6}}{6}+\left (\frac {4}{5} a \,b^{3} d +\frac {1}{5} b^{4} c \right ) x^{5}+\left (\frac {3}{2} a^{2} b^{2} d +a \,b^{3} c \right ) x^{4}+\left (\frac {4}{3} a^{3} b d +2 a^{2} b^{2} c \right ) x^{3}+\left (\frac {1}{2} a^{4} d +2 a^{3} b c \right ) x^{2}+a^{4} c x\) | \(94\) |
default | \(\frac {b^{4} d \,x^{6}}{6}+\frac {\left (4 a \,b^{3} d +b^{4} c \right ) x^{5}}{5}+\frac {\left (6 a^{2} b^{2} d +4 a \,b^{3} c \right ) x^{4}}{4}+\frac {\left (4 a^{3} b d +6 a^{2} b^{2} c \right ) x^{3}}{3}+\frac {\left (a^{4} d +4 a^{3} b c \right ) x^{2}}{2}+a^{4} c x\) | \(97\) |
gosper | \(\frac {1}{6} b^{4} d \,x^{6}+\frac {4}{5} x^{5} a \,b^{3} d +\frac {1}{5} x^{5} b^{4} c +\frac {3}{2} x^{4} a^{2} b^{2} d +x^{4} a \,b^{3} c +\frac {4}{3} x^{3} a^{3} b d +2 x^{3} a^{2} b^{2} c +\frac {1}{2} x^{2} a^{4} d +2 x^{2} a^{3} b c +a^{4} c x\) | \(98\) |
risch | \(\frac {1}{6} b^{4} d \,x^{6}+\frac {4}{5} x^{5} a \,b^{3} d +\frac {1}{5} x^{5} b^{4} c +\frac {3}{2} x^{4} a^{2} b^{2} d +x^{4} a \,b^{3} c +\frac {4}{3} x^{3} a^{3} b d +2 x^{3} a^{2} b^{2} c +\frac {1}{2} x^{2} a^{4} d +2 x^{2} a^{3} b c +a^{4} c x\) | \(98\) |
parallelrisch | \(\frac {1}{6} b^{4} d \,x^{6}+\frac {4}{5} x^{5} a \,b^{3} d +\frac {1}{5} x^{5} b^{4} c +\frac {3}{2} x^{4} a^{2} b^{2} d +x^{4} a \,b^{3} c +\frac {4}{3} x^{3} a^{3} b d +2 x^{3} a^{2} b^{2} c +\frac {1}{2} x^{2} a^{4} d +2 x^{2} a^{3} b c +a^{4} c x\) | \(98\) |
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Leaf count of result is larger than twice the leaf count of optimal. 96 vs. \(2 (34) = 68\).
Time = 0.22 (sec) , antiderivative size = 96, normalized size of antiderivative = 2.53 \[ \int (a+b x)^4 (c+d x) \, dx=\frac {1}{6} \, b^{4} d x^{6} + a^{4} c x + \frac {1}{5} \, {\left (b^{4} c + 4 \, a b^{3} d\right )} x^{5} + \frac {1}{2} \, {\left (2 \, a b^{3} c + 3 \, a^{2} b^{2} d\right )} x^{4} + \frac {2}{3} \, {\left (3 \, a^{2} b^{2} c + 2 \, a^{3} b d\right )} x^{3} + \frac {1}{2} \, {\left (4 \, a^{3} b c + a^{4} d\right )} x^{2} \]
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Leaf count of result is larger than twice the leaf count of optimal. 100 vs. \(2 (32) = 64\).
Time = 0.03 (sec) , antiderivative size = 100, normalized size of antiderivative = 2.63 \[ \int (a+b x)^4 (c+d x) \, dx=a^{4} c x + \frac {b^{4} d x^{6}}{6} + x^{5} \cdot \left (\frac {4 a b^{3} d}{5} + \frac {b^{4} c}{5}\right ) + x^{4} \cdot \left (\frac {3 a^{2} b^{2} d}{2} + a b^{3} c\right ) + x^{3} \cdot \left (\frac {4 a^{3} b d}{3} + 2 a^{2} b^{2} c\right ) + x^{2} \left (\frac {a^{4} d}{2} + 2 a^{3} b c\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. 96 vs. \(2 (34) = 68\).
Time = 0.22 (sec) , antiderivative size = 96, normalized size of antiderivative = 2.53 \[ \int (a+b x)^4 (c+d x) \, dx=\frac {1}{6} \, b^{4} d x^{6} + a^{4} c x + \frac {1}{5} \, {\left (b^{4} c + 4 \, a b^{3} d\right )} x^{5} + \frac {1}{2} \, {\left (2 \, a b^{3} c + 3 \, a^{2} b^{2} d\right )} x^{4} + \frac {2}{3} \, {\left (3 \, a^{2} b^{2} c + 2 \, a^{3} b d\right )} x^{3} + \frac {1}{2} \, {\left (4 \, a^{3} b c + a^{4} d\right )} x^{2} \]
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Leaf count of result is larger than twice the leaf count of optimal. 97 vs. \(2 (34) = 68\).
Time = 0.28 (sec) , antiderivative size = 97, normalized size of antiderivative = 2.55 \[ \int (a+b x)^4 (c+d x) \, dx=\frac {1}{6} \, b^{4} d x^{6} + \frac {1}{5} \, b^{4} c x^{5} + \frac {4}{5} \, a b^{3} d x^{5} + a b^{3} c x^{4} + \frac {3}{2} \, a^{2} b^{2} d x^{4} + 2 \, a^{2} b^{2} c x^{3} + \frac {4}{3} \, a^{3} b d x^{3} + 2 \, a^{3} b c x^{2} + \frac {1}{2} \, a^{4} d x^{2} + a^{4} c x \]
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Time = 0.21 (sec) , antiderivative size = 88, normalized size of antiderivative = 2.32 \[ \int (a+b x)^4 (c+d x) \, dx=x^5\,\left (\frac {c\,b^4}{5}+\frac {4\,a\,d\,b^3}{5}\right )+x^2\,\left (\frac {d\,a^4}{2}+2\,b\,c\,a^3\right )+\frac {b^4\,d\,x^6}{6}+a^4\,c\,x+\frac {2\,a^2\,b\,x^3\,\left (2\,a\,d+3\,b\,c\right )}{3}+\frac {a\,b^2\,x^4\,\left (3\,a\,d+2\,b\,c\right )}{2} \]
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