Integrand size = 16, antiderivative size = 15 \[ \int (a+b x)^5 (a c+b c x) \, dx=\frac {c (a+b x)^7}{7 b} \]
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Time = 0.00 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {21, 32} \[ \int (a+b x)^5 (a c+b c x) \, dx=\frac {c (a+b x)^7}{7 b} \]
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Rule 21
Rule 32
Rubi steps \begin{align*} \text {integral}& = c \int (a+b x)^6 \, dx \\ & = \frac {c (a+b x)^7}{7 b} \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.00 \[ \int (a+b x)^5 (a c+b c x) \, dx=\frac {c (a+b x)^7}{7 b} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(66\) vs. \(2(13)=26\).
Time = 0.26 (sec) , antiderivative size = 67, normalized size of antiderivative = 4.47
method | result | size |
gosper | \(\frac {c x \left (b^{6} x^{6}+7 a \,x^{5} b^{5}+21 a^{2} x^{4} b^{4}+35 a^{3} x^{3} b^{3}+35 a^{4} x^{2} b^{2}+21 a^{5} x b +7 a^{6}\right )}{7}\) | \(67\) |
default | \(\frac {1}{7} b^{6} c \,x^{7}+a \,b^{5} c \,x^{6}+3 a^{2} b^{4} c \,x^{5}+5 a^{3} b^{3} c \,x^{4}+5 a^{4} b^{2} c \,x^{3}+3 a^{5} b c \,x^{2}+a^{6} c x\) | \(72\) |
norman | \(\frac {1}{7} b^{6} c \,x^{7}+a \,b^{5} c \,x^{6}+3 a^{2} b^{4} c \,x^{5}+5 a^{3} b^{3} c \,x^{4}+5 a^{4} b^{2} c \,x^{3}+3 a^{5} b c \,x^{2}+a^{6} c x\) | \(72\) |
parallelrisch | \(\frac {1}{7} b^{6} c \,x^{7}+a \,b^{5} c \,x^{6}+3 a^{2} b^{4} c \,x^{5}+5 a^{3} b^{3} c \,x^{4}+5 a^{4} b^{2} c \,x^{3}+3 a^{5} b c \,x^{2}+a^{6} c x\) | \(72\) |
risch | \(\frac {b^{6} c \,x^{7}}{7}+a \,b^{5} c \,x^{6}+3 a^{2} b^{4} c \,x^{5}+5 a^{3} b^{3} c \,x^{4}+5 a^{4} b^{2} c \,x^{3}+3 a^{5} b c \,x^{2}+a^{6} c x +\frac {c \,a^{7}}{7 b}\) | \(81\) |
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Leaf count of result is larger than twice the leaf count of optimal. 71 vs. \(2 (13) = 26\).
Time = 0.22 (sec) , antiderivative size = 71, normalized size of antiderivative = 4.73 \[ \int (a+b x)^5 (a c+b c x) \, dx=\frac {1}{7} \, b^{6} c x^{7} + a b^{5} c x^{6} + 3 \, a^{2} b^{4} c x^{5} + 5 \, a^{3} b^{3} c x^{4} + 5 \, a^{4} b^{2} c x^{3} + 3 \, a^{5} b c x^{2} + a^{6} c x \]
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Leaf count of result is larger than twice the leaf count of optimal. 78 vs. \(2 (10) = 20\).
Time = 0.05 (sec) , antiderivative size = 78, normalized size of antiderivative = 5.20 \[ \int (a+b x)^5 (a c+b c x) \, dx=a^{6} c x + 3 a^{5} b c x^{2} + 5 a^{4} b^{2} c x^{3} + 5 a^{3} b^{3} c x^{4} + 3 a^{2} b^{4} c x^{5} + a b^{5} c x^{6} + \frac {b^{6} c x^{7}}{7} \]
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Leaf count of result is larger than twice the leaf count of optimal. 71 vs. \(2 (13) = 26\).
Time = 0.20 (sec) , antiderivative size = 71, normalized size of antiderivative = 4.73 \[ \int (a+b x)^5 (a c+b c x) \, dx=\frac {1}{7} \, b^{6} c x^{7} + a b^{5} c x^{6} + 3 \, a^{2} b^{4} c x^{5} + 5 \, a^{3} b^{3} c x^{4} + 5 \, a^{4} b^{2} c x^{3} + 3 \, a^{5} b c x^{2} + a^{6} c x \]
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Leaf count of result is larger than twice the leaf count of optimal. 71 vs. \(2 (13) = 26\).
Time = 0.28 (sec) , antiderivative size = 71, normalized size of antiderivative = 4.73 \[ \int (a+b x)^5 (a c+b c x) \, dx=\frac {1}{7} \, b^{6} c x^{7} + a b^{5} c x^{6} + 3 \, a^{2} b^{4} c x^{5} + 5 \, a^{3} b^{3} c x^{4} + 5 \, a^{4} b^{2} c x^{3} + 3 \, a^{5} b c x^{2} + a^{6} c x \]
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Time = 0.04 (sec) , antiderivative size = 71, normalized size of antiderivative = 4.73 \[ \int (a+b x)^5 (a c+b c x) \, dx=c\,a^6\,x+3\,c\,a^5\,b\,x^2+5\,c\,a^4\,b^2\,x^3+5\,c\,a^3\,b^3\,x^4+3\,c\,a^2\,b^4\,x^5+c\,a\,b^5\,x^6+\frac {c\,b^6\,x^7}{7} \]
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