Integrand size = 24, antiderivative size = 30 \[ \int \left (a+c x^2\right ) \left (1+\left (a x+\frac {c x^3}{3}\right )^5\right ) \, dx=a x+\frac {c x^3}{3}+\frac {1}{6} \left (a x+\frac {c x^3}{3}\right )^6 \]
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Time = 0.01 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.042, Rules used = {1605} \[ \int \left (a+c x^2\right ) \left (1+\left (a x+\frac {c x^3}{3}\right )^5\right ) \, dx=\frac {1}{6} \left (a x+\frac {c x^3}{3}\right )^6+a x+\frac {c x^3}{3} \]
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Rule 1605
Rubi steps \begin{align*} \text {integral}& = \text {Subst}\left (\int \left (1+x^5\right ) \, dx,x,a x+\frac {c x^3}{3}\right ) \\ & = a x+\frac {c x^3}{3}+\frac {1}{6} \left (a x+\frac {c x^3}{3}\right )^6 \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(93\) vs. \(2(30)=60\).
Time = 0.01 (sec) , antiderivative size = 93, normalized size of antiderivative = 3.10 \[ \int \left (a+c x^2\right ) \left (1+\left (a x+\frac {c x^3}{3}\right )^5\right ) \, dx=a x+\frac {c x^3}{3}+\frac {a^6 x^6}{6}+\frac {1}{3} a^5 c x^8+\frac {5}{18} a^4 c^2 x^{10}+\frac {10}{81} a^3 c^3 x^{12}+\frac {5}{162} a^2 c^4 x^{14}+\frac {1}{243} a c^5 x^{16}+\frac {c^6 x^{18}}{4374} \]
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Time = 0.80 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.83
method | result | size |
default | \(a x +\frac {c \,x^{3}}{3}+\frac {\left (a x +\frac {1}{3} c \,x^{3}\right )^{6}}{6}\) | \(25\) |
norman | \(a x +\frac {1}{6} a^{6} x^{6}+\frac {1}{3} c \,x^{3}+\frac {1}{4374} c^{6} x^{18}+\frac {1}{243} a \,c^{5} x^{16}+\frac {5}{162} a^{2} c^{4} x^{14}+\frac {5}{18} a^{4} c^{2} x^{10}+\frac {1}{3} c \,a^{5} x^{8}+\frac {10}{81} c^{3} a^{3} x^{12}\) | \(78\) |
risch | \(a x +\frac {1}{6} a^{6} x^{6}+\frac {1}{3} c \,x^{3}+\frac {1}{4374} c^{6} x^{18}+\frac {1}{243} a \,c^{5} x^{16}+\frac {5}{162} a^{2} c^{4} x^{14}+\frac {5}{18} a^{4} c^{2} x^{10}+\frac {1}{3} c \,a^{5} x^{8}+\frac {10}{81} c^{3} a^{3} x^{12}\) | \(78\) |
parallelrisch | \(a x +\frac {1}{6} a^{6} x^{6}+\frac {1}{3} c \,x^{3}+\frac {1}{4374} c^{6} x^{18}+\frac {1}{243} a \,c^{5} x^{16}+\frac {5}{162} a^{2} c^{4} x^{14}+\frac {5}{18} a^{4} c^{2} x^{10}+\frac {1}{3} c \,a^{5} x^{8}+\frac {10}{81} c^{3} a^{3} x^{12}\) | \(78\) |
gosper | \(\frac {x \left (c^{6} x^{17}+18 a \,c^{5} x^{15}+135 a^{2} c^{4} x^{13}+540 c^{3} a^{3} x^{11}+1215 a^{4} c^{2} x^{9}+1458 c \,a^{5} x^{7}+729 a^{6} x^{5}+1458 c \,x^{2}+4374 a \right )}{4374}\) | \(80\) |
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Leaf count of result is larger than twice the leaf count of optimal. 77 vs. \(2 (24) = 48\).
Time = 0.25 (sec) , antiderivative size = 77, normalized size of antiderivative = 2.57 \[ \int \left (a+c x^2\right ) \left (1+\left (a x+\frac {c x^3}{3}\right )^5\right ) \, dx=\frac {1}{4374} \, c^{6} x^{18} + \frac {1}{243} \, a c^{5} x^{16} + \frac {5}{162} \, a^{2} c^{4} x^{14} + \frac {10}{81} \, a^{3} c^{3} x^{12} + \frac {5}{18} \, a^{4} c^{2} x^{10} + \frac {1}{3} \, a^{5} c x^{8} + \frac {1}{6} \, a^{6} x^{6} + \frac {1}{3} \, c x^{3} + a x \]
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Leaf count of result is larger than twice the leaf count of optimal. 87 vs. \(2 (22) = 44\).
Time = 0.04 (sec) , antiderivative size = 87, normalized size of antiderivative = 2.90 \[ \int \left (a+c x^2\right ) \left (1+\left (a x+\frac {c x^3}{3}\right )^5\right ) \, dx=\frac {a^{6} x^{6}}{6} + \frac {a^{5} c x^{8}}{3} + \frac {5 a^{4} c^{2} x^{10}}{18} + \frac {10 a^{3} c^{3} x^{12}}{81} + \frac {5 a^{2} c^{4} x^{14}}{162} + \frac {a c^{5} x^{16}}{243} + a x + \frac {c^{6} x^{18}}{4374} + \frac {c x^{3}}{3} \]
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Leaf count of result is larger than twice the leaf count of optimal. 77 vs. \(2 (24) = 48\).
Time = 0.19 (sec) , antiderivative size = 77, normalized size of antiderivative = 2.57 \[ \int \left (a+c x^2\right ) \left (1+\left (a x+\frac {c x^3}{3}\right )^5\right ) \, dx=\frac {1}{4374} \, c^{6} x^{18} + \frac {1}{243} \, a c^{5} x^{16} + \frac {5}{162} \, a^{2} c^{4} x^{14} + \frac {10}{81} \, a^{3} c^{3} x^{12} + \frac {5}{18} \, a^{4} c^{2} x^{10} + \frac {1}{3} \, a^{5} c x^{8} + \frac {1}{6} \, a^{6} x^{6} + \frac {1}{3} \, c x^{3} + a x \]
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none
Time = 0.28 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.80 \[ \int \left (a+c x^2\right ) \left (1+\left (a x+\frac {c x^3}{3}\right )^5\right ) \, dx=\frac {1}{4374} \, {\left (c x^{3} + 3 \, a x\right )}^{6} + \frac {1}{3} \, c x^{3} + a x \]
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Time = 9.21 (sec) , antiderivative size = 77, normalized size of antiderivative = 2.57 \[ \int \left (a+c x^2\right ) \left (1+\left (a x+\frac {c x^3}{3}\right )^5\right ) \, dx=\frac {a^6\,x^6}{6}+\frac {a^5\,c\,x^8}{3}+\frac {5\,a^4\,c^2\,x^{10}}{18}+\frac {10\,a^3\,c^3\,x^{12}}{81}+\frac {5\,a^2\,c^4\,x^{14}}{162}+\frac {a\,c^5\,x^{16}}{243}+a\,x+\frac {c^6\,x^{18}}{4374}+\frac {c\,x^3}{3} \]
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