Integrand size = 31, antiderivative size = 34 \[ \int \left (b x+c x^2\right ) \left (1+\left (\frac {b x^2}{2}+\frac {c x^3}{3}\right )^5\right ) \, dx=\frac {b x^2}{2}+\frac {c x^3}{3}+\frac {x^{12} (3 b+2 c x)^6}{279936} \]
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Time = 0.02 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.032, Rules used = {1605} \[ \int \left (b x+c x^2\right ) \left (1+\left (\frac {b x^2}{2}+\frac {c x^3}{3}\right )^5\right ) \, dx=\frac {x^{12} (3 b+2 c x)^6}{279936}+\frac {b x^2}{2}+\frac {c x^3}{3} \]
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Rule 1605
Rubi steps \begin{align*} \text {integral}& = \text {Subst}\left (\int \left (1+x^5\right ) \, dx,x,\frac {b x^2}{2}+\frac {c x^3}{3}\right ) \\ & = \frac {b x^2}{2}+\frac {c x^3}{3}+\frac {x^{12} (3 b+2 c x)^6}{279936} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(98\) vs. \(2(34)=68\).
Time = 0.01 (sec) , antiderivative size = 98, normalized size of antiderivative = 2.88 \[ \int \left (b x+c x^2\right ) \left (1+\left (\frac {b x^2}{2}+\frac {c x^3}{3}\right )^5\right ) \, dx=\frac {b x^2}{2}+\frac {c x^3}{3}+\frac {b^6 x^{12}}{384}+\frac {1}{96} b^5 c x^{13}+\frac {5}{288} b^4 c^2 x^{14}+\frac {5}{324} b^3 c^3 x^{15}+\frac {5}{648} b^2 c^4 x^{16}+\frac {1}{486} b c^5 x^{17}+\frac {c^6 x^{18}}{4374} \]
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Time = 0.73 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.91
method | result | size |
default | \(\frac {\left (\frac {1}{2} b \,x^{2}+\frac {1}{3} c \,x^{3}\right )^{6}}{6}+\frac {b \,x^{2}}{2}+\frac {c \,x^{3}}{3}\) | \(31\) |
gosper | \(\frac {x^{2} \left (64 c^{6} x^{16}+576 b \,c^{5} x^{15}+2160 b^{2} c^{4} x^{14}+4320 b^{3} c^{3} x^{13}+4860 b^{4} c^{2} x^{12}+2916 b^{5} c \,x^{11}+729 b^{6} x^{10}+93312 c x +139968 b \right )}{279936}\) | \(81\) |
norman | \(\frac {1}{2} b \,x^{2}+\frac {1}{384} b^{6} x^{12}+\frac {1}{3} c \,x^{3}+\frac {1}{4374} c^{6} x^{18}+\frac {1}{486} b \,c^{5} x^{17}+\frac {5}{648} b^{2} c^{4} x^{16}+\frac {5}{324} b^{3} c^{3} x^{15}+\frac {5}{288} b^{4} c^{2} x^{14}+\frac {1}{96} b^{5} c \,x^{13}\) | \(81\) |
risch | \(\frac {1}{2} b \,x^{2}+\frac {1}{384} b^{6} x^{12}+\frac {1}{3} c \,x^{3}+\frac {1}{4374} c^{6} x^{18}+\frac {1}{486} b \,c^{5} x^{17}+\frac {5}{648} b^{2} c^{4} x^{16}+\frac {5}{324} b^{3} c^{3} x^{15}+\frac {5}{288} b^{4} c^{2} x^{14}+\frac {1}{96} b^{5} c \,x^{13}\) | \(81\) |
parallelrisch | \(\frac {1}{2} b \,x^{2}+\frac {1}{384} b^{6} x^{12}+\frac {1}{3} c \,x^{3}+\frac {1}{4374} c^{6} x^{18}+\frac {1}{486} b \,c^{5} x^{17}+\frac {5}{648} b^{2} c^{4} x^{16}+\frac {5}{324} b^{3} c^{3} x^{15}+\frac {5}{288} b^{4} c^{2} x^{14}+\frac {1}{96} b^{5} c \,x^{13}\) | \(81\) |
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Leaf count of result is larger than twice the leaf count of optimal. 80 vs. \(2 (28) = 56\).
Time = 0.27 (sec) , antiderivative size = 80, normalized size of antiderivative = 2.35 \[ \int \left (b x+c x^2\right ) \left (1+\left (\frac {b x^2}{2}+\frac {c x^3}{3}\right )^5\right ) \, dx=\frac {1}{4374} \, c^{6} x^{18} + \frac {1}{486} \, b c^{5} x^{17} + \frac {5}{648} \, b^{2} c^{4} x^{16} + \frac {5}{324} \, b^{3} c^{3} x^{15} + \frac {5}{288} \, b^{4} c^{2} x^{14} + \frac {1}{96} \, b^{5} c x^{13} + \frac {1}{384} \, b^{6} x^{12} + \frac {1}{3} \, c x^{3} + \frac {1}{2} \, b x^{2} \]
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Leaf count of result is larger than twice the leaf count of optimal. 90 vs. \(2 (27) = 54\).
Time = 0.04 (sec) , antiderivative size = 90, normalized size of antiderivative = 2.65 \[ \int \left (b x+c x^2\right ) \left (1+\left (\frac {b x^2}{2}+\frac {c x^3}{3}\right )^5\right ) \, dx=\frac {b^{6} x^{12}}{384} + \frac {b^{5} c x^{13}}{96} + \frac {5 b^{4} c^{2} x^{14}}{288} + \frac {5 b^{3} c^{3} x^{15}}{324} + \frac {5 b^{2} c^{4} x^{16}}{648} + \frac {b c^{5} x^{17}}{486} + \frac {b x^{2}}{2} + \frac {c^{6} x^{18}}{4374} + \frac {c x^{3}}{3} \]
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Leaf count of result is larger than twice the leaf count of optimal. 80 vs. \(2 (28) = 56\).
Time = 0.20 (sec) , antiderivative size = 80, normalized size of antiderivative = 2.35 \[ \int \left (b x+c x^2\right ) \left (1+\left (\frac {b x^2}{2}+\frac {c x^3}{3}\right )^5\right ) \, dx=\frac {1}{4374} \, c^{6} x^{18} + \frac {1}{486} \, b c^{5} x^{17} + \frac {5}{648} \, b^{2} c^{4} x^{16} + \frac {5}{324} \, b^{3} c^{3} x^{15} + \frac {5}{288} \, b^{4} c^{2} x^{14} + \frac {1}{96} \, b^{5} c x^{13} + \frac {1}{384} \, b^{6} x^{12} + \frac {1}{3} \, c x^{3} + \frac {1}{2} \, b x^{2} \]
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none
Time = 0.28 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.88 \[ \int \left (b x+c x^2\right ) \left (1+\left (\frac {b x^2}{2}+\frac {c x^3}{3}\right )^5\right ) \, dx=\frac {1}{279936} \, {\left (2 \, c x^{3} + 3 \, b x^{2}\right )}^{6} + \frac {1}{3} \, c x^{3} + \frac {1}{2} \, b x^{2} \]
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Time = 0.04 (sec) , antiderivative size = 80, normalized size of antiderivative = 2.35 \[ \int \left (b x+c x^2\right ) \left (1+\left (\frac {b x^2}{2}+\frac {c x^3}{3}\right )^5\right ) \, dx=\frac {b^6\,x^{12}}{384}+\frac {b^5\,c\,x^{13}}{96}+\frac {5\,b^4\,c^2\,x^{14}}{288}+\frac {5\,b^3\,c^3\,x^{15}}{324}+\frac {5\,b^2\,c^4\,x^{16}}{648}+\frac {b\,c^5\,x^{17}}{486}+\frac {b\,x^2}{2}+\frac {c^6\,x^{18}}{4374}+\frac {c\,x^3}{3} \]
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