Integrand size = 32, antiderivative size = 41 \[ \int \left (b x+c x^2\right ) \left (1+\left (d+\frac {b x^2}{2}+\frac {c x^3}{3}\right )^5\right ) \, dx=\frac {b x^2}{2}+\frac {c x^3}{3}+\frac {1}{6} \left (d+\frac {b x^2}{2}+\frac {c x^3}{3}\right )^6 \]
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Time = 0.03 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.031, Rules used = {1605} \[ \int \left (b x+c x^2\right ) \left (1+\left (d+\frac {b x^2}{2}+\frac {c x^3}{3}\right )^5\right ) \, dx=\frac {1}{6} \left (\frac {b x^2}{2}+\frac {c x^3}{3}+d\right )^6+\frac {b x^2}{2}+\frac {c x^3}{3} \]
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Rule 1605
Rubi steps \begin{align*} \text {integral}& = \text {Subst}\left (\int \left (1+x^5\right ) \, dx,x,d+\frac {b x^2}{2}+\frac {c x^3}{3}\right ) \\ & = \frac {b x^2}{2}+\frac {c x^3}{3}+\frac {1}{6} \left (d+\frac {b x^2}{2}+\frac {c x^3}{3}\right )^6 \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(146\) vs. \(2(41)=82\).
Time = 0.04 (sec) , antiderivative size = 146, normalized size of antiderivative = 3.56 \[ \int \left (b x+c x^2\right ) \left (1+\left (d+\frac {b x^2}{2}+\frac {c x^3}{3}\right )^5\right ) \, dx=\frac {x^2 (3 b+2 c x) \left (46656+46656 d^5+243 b^5 x^{10}+810 b^4 c x^{11}+1080 b^3 c^2 x^{12}+720 b^2 c^3 x^{13}+240 b c^4 x^{14}+32 c^5 x^{15}+19440 d^4 x^2 (3 b+2 c x)+4320 d^3 x^4 (3 b+2 c x)^2+540 d^2 x^6 (3 b+2 c x)^3+36 d x^8 (3 b+2 c x)^4\right )}{279936} \]
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Time = 0.08 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.80
method | result | size |
default | \(\frac {\left (d +\frac {1}{2} b \,x^{2}+\frac {1}{3} c \,x^{3}\right )^{6}}{6}+d +\frac {b \,x^{2}}{2}+\frac {c \,x^{3}}{3}\) | \(33\) |
norman | \(\left (\frac {1}{2} b \,d^{5}+\frac {1}{2} b \right ) x^{2}+\left (\frac {5}{324} b^{3} c^{3}+\frac {1}{243} c^{5} d \right ) x^{15}+\left (\frac {5}{12} b^{3} d^{3}+\frac {5}{18} c^{2} d^{4}\right ) x^{6}+\left (\frac {5}{288} b^{4} c^{2}+\frac {5}{162} b \,c^{4} d \right ) x^{14}+\left (\frac {5}{32} b^{4} d^{2}+\frac {5}{9} b \,c^{2} d^{3}\right ) x^{8}+\left (\frac {1}{96} b^{5} c +\frac {5}{54} b^{2} c^{3} d \right ) x^{13}+\left (\frac {1}{32} b^{5} d +\frac {5}{12} d^{2} c^{2} b^{2}\right ) x^{10}+\left (\frac {1}{3} c \,d^{5}+\frac {1}{3} c \right ) x^{3}+\left (\frac {5}{12} b^{3} c \,d^{2}+\frac {10}{81} c^{3} d^{3}\right ) x^{9}+\left (\frac {5}{48} d c \,b^{4}+\frac {5}{27} b \,c^{3} d^{2}\right ) x^{11}+\left (\frac {1}{384} b^{6}+\frac {5}{36} b^{3} c^{2} d +\frac {5}{162} d^{2} c^{4}\right ) x^{12}+\frac {c^{6} x^{18}}{4374}+\frac {b \,c^{5} x^{17}}{486}+\frac {5 b^{2} c^{4} x^{16}}{648}+\frac {5 x^{4} b^{2} d^{4}}{8}+\frac {5 b c \,d^{4} x^{5}}{6}+\frac {5 b^{2} c \,d^{3} x^{7}}{6}\) | \(285\) |
gosper | \(\frac {x^{2} \left (64 c^{6} x^{16}+576 b \,c^{5} x^{15}+2160 b^{2} c^{4} x^{14}+4320 b^{3} c^{3} x^{13}+1152 c^{5} d \,x^{13}+4860 b^{4} c^{2} x^{12}+8640 b \,c^{4} d \,x^{12}+2916 b^{5} c \,x^{11}+25920 b^{2} c^{3} d \,x^{11}+729 b^{6} x^{10}+38880 b^{3} c^{2} d \,x^{10}+8640 c^{4} d^{2} x^{10}+29160 b^{4} c d \,x^{9}+51840 b \,c^{3} d^{2} x^{9}+8748 b^{5} d \,x^{8}+116640 b^{2} c^{2} d^{2} x^{8}+116640 b^{3} c \,d^{2} x^{7}+34560 c^{3} d^{3} x^{7}+43740 b^{4} d^{2} x^{6}+155520 b \,c^{2} d^{3} x^{6}+233280 b^{2} c \,d^{3} x^{5}+116640 b^{3} d^{3} x^{4}+77760 c^{2} d^{4} x^{4}+233280 b c \,d^{4} x^{3}+174960 b^{2} d^{4} x^{2}+93312 c \,d^{5} x +139968 b \,d^{5}+93312 c x +139968 b \right )}{279936}\) | \(294\) |
risch | \(\frac {5}{162} x^{14} b \,c^{4} d +\frac {5}{54} x^{13} b^{2} c^{3} d +\frac {5}{12} x^{6} b^{3} d^{3}+\frac {1}{2} b \,d^{5} x^{2}+\frac {1}{384} b^{6} x^{12}+\frac {1}{3} c \,x^{3}+\frac {1}{2} b \,x^{2}+\frac {5}{162} x^{12} d^{2} c^{4}+\frac {10}{81} x^{9} c^{3} d^{3}+\frac {5}{18} x^{6} c^{2} d^{4}+\frac {1}{3} x^{3} c \,d^{5}+\frac {5}{6} b c \,d^{4} x^{5}+\frac {5}{6} b^{2} c \,d^{3} x^{7}+\frac {1}{96} b^{5} c \,x^{13}+\frac {5}{288} b^{4} c^{2} x^{14}+\frac {5}{324} b^{3} c^{3} x^{15}+\frac {5}{32} x^{8} b^{4} d^{2}+\frac {1}{243} c^{5} d \,x^{15}+\frac {5}{8} x^{4} b^{2} d^{4}+\frac {5}{648} b^{2} c^{4} x^{16}+\frac {1}{486} b \,c^{5} x^{17}+\frac {1}{32} b^{5} d \,x^{10}+\frac {5}{36} x^{12} b^{3} c^{2} d +\frac {5}{48} x^{11} d c \,b^{4}+\frac {5}{27} x^{11} b \,c^{3} d^{2}+\frac {5}{12} x^{10} d^{2} c^{2} b^{2}+\frac {5}{12} x^{9} b^{3} c \,d^{2}+\frac {5}{9} x^{8} b \,c^{2} d^{3}+\frac {1}{4374} c^{6} x^{18}\) | \(299\) |
parallelrisch | \(\frac {5}{162} x^{14} b \,c^{4} d +\frac {5}{54} x^{13} b^{2} c^{3} d +\frac {5}{12} x^{6} b^{3} d^{3}+\frac {1}{2} b \,d^{5} x^{2}+\frac {1}{384} b^{6} x^{12}+\frac {1}{3} c \,x^{3}+\frac {1}{2} b \,x^{2}+\frac {5}{162} x^{12} d^{2} c^{4}+\frac {10}{81} x^{9} c^{3} d^{3}+\frac {5}{18} x^{6} c^{2} d^{4}+\frac {1}{3} x^{3} c \,d^{5}+\frac {5}{6} b c \,d^{4} x^{5}+\frac {5}{6} b^{2} c \,d^{3} x^{7}+\frac {1}{96} b^{5} c \,x^{13}+\frac {5}{288} b^{4} c^{2} x^{14}+\frac {5}{324} b^{3} c^{3} x^{15}+\frac {5}{32} x^{8} b^{4} d^{2}+\frac {1}{243} c^{5} d \,x^{15}+\frac {5}{8} x^{4} b^{2} d^{4}+\frac {5}{648} b^{2} c^{4} x^{16}+\frac {1}{486} b \,c^{5} x^{17}+\frac {1}{32} b^{5} d \,x^{10}+\frac {5}{36} x^{12} b^{3} c^{2} d +\frac {5}{48} x^{11} d c \,b^{4}+\frac {5}{27} x^{11} b \,c^{3} d^{2}+\frac {5}{12} x^{10} d^{2} c^{2} b^{2}+\frac {5}{12} x^{9} b^{3} c \,d^{2}+\frac {5}{9} x^{8} b \,c^{2} d^{3}+\frac {1}{4374} c^{6} x^{18}\) | \(299\) |
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Leaf count of result is larger than twice the leaf count of optimal. 289 vs. \(2 (33) = 66\).
Time = 0.27 (sec) , antiderivative size = 289, normalized size of antiderivative = 7.05 \[ \int \left (b x+c x^2\right ) \left (1+\left (d+\frac {b x^2}{2}+\frac {c x^3}{3}\right )^5\right ) \, dx=\frac {1}{4374} \, c^{6} x^{18} + \frac {1}{486} \, b c^{5} x^{17} + \frac {5}{648} \, b^{2} c^{4} x^{16} + \frac {1}{972} \, {\left (15 \, b^{3} c^{3} + 4 \, c^{5} d\right )} x^{15} + \frac {5}{2592} \, {\left (9 \, b^{4} c^{2} + 16 \, b c^{4} d\right )} x^{14} + \frac {1}{864} \, {\left (9 \, b^{5} c + 80 \, b^{2} c^{3} d\right )} x^{13} + \frac {5}{6} \, b^{2} c d^{3} x^{7} + \frac {1}{10368} \, {\left (27 \, b^{6} + 1440 \, b^{3} c^{2} d + 320 \, c^{4} d^{2}\right )} x^{12} + \frac {5}{432} \, {\left (9 \, b^{4} c d + 16 \, b c^{3} d^{2}\right )} x^{11} + \frac {5}{6} \, b c d^{4} x^{5} + \frac {1}{96} \, {\left (3 \, b^{5} d + 40 \, b^{2} c^{2} d^{2}\right )} x^{10} + \frac {5}{8} \, b^{2} d^{4} x^{4} + \frac {5}{324} \, {\left (27 \, b^{3} c d^{2} + 8 \, c^{3} d^{3}\right )} x^{9} + \frac {5}{288} \, {\left (9 \, b^{4} d^{2} + 32 \, b c^{2} d^{3}\right )} x^{8} + \frac {5}{36} \, {\left (3 \, b^{3} d^{3} + 2 \, c^{2} d^{4}\right )} x^{6} + \frac {1}{3} \, {\left (c d^{5} + c\right )} x^{3} + \frac {1}{2} \, {\left (b d^{5} + b\right )} x^{2} \]
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Leaf count of result is larger than twice the leaf count of optimal. 321 vs. \(2 (31) = 62\).
Time = 0.06 (sec) , antiderivative size = 321, normalized size of antiderivative = 7.83 \[ \int \left (b x+c x^2\right ) \left (1+\left (d+\frac {b x^2}{2}+\frac {c x^3}{3}\right )^5\right ) \, dx=\frac {5 b^{2} c^{4} x^{16}}{648} + \frac {5 b^{2} c d^{3} x^{7}}{6} + \frac {5 b^{2} d^{4} x^{4}}{8} + \frac {b c^{5} x^{17}}{486} + \frac {5 b c d^{4} x^{5}}{6} + \frac {c^{6} x^{18}}{4374} + x^{15} \cdot \left (\frac {5 b^{3} c^{3}}{324} + \frac {c^{5} d}{243}\right ) + x^{14} \cdot \left (\frac {5 b^{4} c^{2}}{288} + \frac {5 b c^{4} d}{162}\right ) + x^{13} \left (\frac {b^{5} c}{96} + \frac {5 b^{2} c^{3} d}{54}\right ) + x^{12} \left (\frac {b^{6}}{384} + \frac {5 b^{3} c^{2} d}{36} + \frac {5 c^{4} d^{2}}{162}\right ) + x^{11} \cdot \left (\frac {5 b^{4} c d}{48} + \frac {5 b c^{3} d^{2}}{27}\right ) + x^{10} \left (\frac {b^{5} d}{32} + \frac {5 b^{2} c^{2} d^{2}}{12}\right ) + x^{9} \cdot \left (\frac {5 b^{3} c d^{2}}{12} + \frac {10 c^{3} d^{3}}{81}\right ) + x^{8} \cdot \left (\frac {5 b^{4} d^{2}}{32} + \frac {5 b c^{2} d^{3}}{9}\right ) + x^{6} \cdot \left (\frac {5 b^{3} d^{3}}{12} + \frac {5 c^{2} d^{4}}{18}\right ) + x^{3} \left (\frac {c d^{5}}{3} + \frac {c}{3}\right ) + x^{2} \left (\frac {b d^{5}}{2} + \frac {b}{2}\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. 289 vs. \(2 (33) = 66\).
Time = 0.20 (sec) , antiderivative size = 289, normalized size of antiderivative = 7.05 \[ \int \left (b x+c x^2\right ) \left (1+\left (d+\frac {b x^2}{2}+\frac {c x^3}{3}\right )^5\right ) \, dx=\frac {1}{4374} \, c^{6} x^{18} + \frac {1}{486} \, b c^{5} x^{17} + \frac {5}{648} \, b^{2} c^{4} x^{16} + \frac {1}{972} \, {\left (15 \, b^{3} c^{3} + 4 \, c^{5} d\right )} x^{15} + \frac {5}{2592} \, {\left (9 \, b^{4} c^{2} + 16 \, b c^{4} d\right )} x^{14} + \frac {1}{864} \, {\left (9 \, b^{5} c + 80 \, b^{2} c^{3} d\right )} x^{13} + \frac {5}{6} \, b^{2} c d^{3} x^{7} + \frac {1}{10368} \, {\left (27 \, b^{6} + 1440 \, b^{3} c^{2} d + 320 \, c^{4} d^{2}\right )} x^{12} + \frac {5}{432} \, {\left (9 \, b^{4} c d + 16 \, b c^{3} d^{2}\right )} x^{11} + \frac {5}{6} \, b c d^{4} x^{5} + \frac {1}{96} \, {\left (3 \, b^{5} d + 40 \, b^{2} c^{2} d^{2}\right )} x^{10} + \frac {5}{8} \, b^{2} d^{4} x^{4} + \frac {5}{324} \, {\left (27 \, b^{3} c d^{2} + 8 \, c^{3} d^{3}\right )} x^{9} + \frac {5}{288} \, {\left (9 \, b^{4} d^{2} + 32 \, b c^{2} d^{3}\right )} x^{8} + \frac {5}{36} \, {\left (3 \, b^{3} d^{3} + 2 \, c^{2} d^{4}\right )} x^{6} + \frac {1}{3} \, {\left (c d^{5} + c\right )} x^{3} + \frac {1}{2} \, {\left (b d^{5} + b\right )} x^{2} \]
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Leaf count of result is larger than twice the leaf count of optimal. 126 vs. \(2 (33) = 66\).
Time = 0.28 (sec) , antiderivative size = 126, normalized size of antiderivative = 3.07 \[ \int \left (b x+c x^2\right ) \left (1+\left (d+\frac {b x^2}{2}+\frac {c x^3}{3}\right )^5\right ) \, dx=\frac {1}{279936} \, {\left (2 \, c x^{3} + 3 \, b x^{2}\right )}^{6} + \frac {1}{7776} \, {\left (2 \, c x^{3} + 3 \, b x^{2}\right )}^{5} d + \frac {5}{2592} \, {\left (2 \, c x^{3} + 3 \, b x^{2}\right )}^{4} d^{2} + \frac {5}{324} \, {\left (2 \, c x^{3} + 3 \, b x^{2}\right )}^{3} d^{3} + \frac {5}{72} \, {\left (2 \, c x^{3} + 3 \, b x^{2}\right )}^{2} d^{4} + \frac {1}{6} \, {\left (2 \, c x^{3} + 3 \, b x^{2}\right )} d^{5} + \frac {1}{3} \, c x^{3} + \frac {1}{2} \, b x^{2} \]
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Time = 9.41 (sec) , antiderivative size = 273, normalized size of antiderivative = 6.66 \[ \int \left (b x+c x^2\right ) \left (1+\left (d+\frac {b x^2}{2}+\frac {c x^3}{3}\right )^5\right ) \, dx=x^{13}\,\left (\frac {b^5\,c}{96}+\frac {5\,d\,b^2\,c^3}{54}\right )+x^{14}\,\left (\frac {5\,b^4\,c^2}{288}+\frac {5\,d\,b\,c^4}{162}\right )+x^{12}\,\left (\frac {b^6}{384}+\frac {5\,b^3\,c^2\,d}{36}+\frac {5\,c^4\,d^2}{162}\right )+\frac {c^6\,x^{18}}{4374}+x^{15}\,\left (\frac {5\,b^3\,c^3}{324}+\frac {d\,c^5}{243}\right )+\frac {5\,d^3\,x^6\,\left (3\,b^3+2\,d\,c^2\right )}{36}+\frac {b\,c^5\,x^{17}}{486}+\frac {5\,b^2\,c^4\,x^{16}}{648}+\frac {b\,x^2\,\left (d^5+1\right )}{2}+\frac {5\,b^2\,d^4\,x^4}{8}+\frac {c\,x^3\,\left (d^5+1\right )}{3}+\frac {5\,b^2\,c\,d^3\,x^7}{6}+\frac {5\,b\,d^2\,x^8\,\left (9\,b^3+32\,d\,c^2\right )}{288}+\frac {b^2\,d\,x^{10}\,\left (3\,b^3+40\,d\,c^2\right )}{96}+\frac {5\,c\,d^2\,x^9\,\left (27\,b^3+8\,d\,c^2\right )}{324}+\frac {5\,b\,c\,d^4\,x^5}{6}+\frac {5\,b\,c\,d\,x^{11}\,\left (9\,b^3+16\,d\,c^2\right )}{432} \]
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