Integrand size = 31, antiderivative size = 55 \[ \int (d f+e f x)^3 \left (a+b (d+e x)^2+c (d+e x)^4\right ) \, dx=\frac {a f^3 (d+e x)^4}{4 e}+\frac {b f^3 (d+e x)^6}{6 e}+\frac {c f^3 (d+e x)^8}{8 e} \]
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Time = 0.03 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.065, Rules used = {1156, 14} \[ \int (d f+e f x)^3 \left (a+b (d+e x)^2+c (d+e x)^4\right ) \, dx=\frac {a f^3 (d+e x)^4}{4 e}+\frac {b f^3 (d+e x)^6}{6 e}+\frac {c f^3 (d+e x)^8}{8 e} \]
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Rule 14
Rule 1156
Rubi steps \begin{align*} \text {integral}& = \frac {f^3 \text {Subst}\left (\int x^3 \left (a+b x^2+c x^4\right ) \, dx,x,d+e x\right )}{e} \\ & = \frac {f^3 \text {Subst}\left (\int \left (a x^3+b x^5+c x^7\right ) \, dx,x,d+e x\right )}{e} \\ & = \frac {a f^3 (d+e x)^4}{4 e}+\frac {b f^3 (d+e x)^6}{6 e}+\frac {c f^3 (d+e x)^8}{8 e} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(154\) vs. \(2(55)=110\).
Time = 0.02 (sec) , antiderivative size = 154, normalized size of antiderivative = 2.80 \[ \int (d f+e f x)^3 \left (a+b (d+e x)^2+c (d+e x)^4\right ) \, dx=f^3 \left (d^3 \left (a+b d^2+c d^4\right ) x+\frac {1}{2} d^2 \left (3 a+5 b d^2+7 c d^4\right ) e x^2+\frac {1}{3} d \left (3 a+10 b d^2+21 c d^4\right ) e^2 x^3+\frac {1}{4} \left (a+10 b d^2+35 c d^4\right ) e^3 x^4+d \left (b+7 c d^2\right ) e^4 x^5+\frac {1}{6} \left (b+21 c d^2\right ) e^5 x^6+c d e^6 x^7+\frac {1}{8} c e^7 x^8\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. \(178\) vs. \(2(49)=98\).
Time = 0.63 (sec) , antiderivative size = 179, normalized size of antiderivative = 3.25
method | result | size |
gosper | \(\frac {f^{3} x \left (3 e^{7} c \,x^{7}+24 d \,e^{6} c \,x^{6}+84 x^{5} d^{2} e^{5} c +168 c \,d^{3} e^{4} x^{4}+4 x^{5} b \,e^{5}+210 x^{3} d^{4} c \,e^{3}+24 b d \,e^{4} x^{4}+168 x^{2} c \,d^{5} e^{2}+60 x^{3} b \,d^{2} e^{3}+84 x c \,d^{6} e +80 x^{2} b \,d^{3} e^{2}+24 c \,d^{7}+6 a \,e^{3} x^{3}+60 x b \,d^{4} e +24 x^{2} d \,e^{2} a +24 b \,d^{5}+36 x a \,d^{2} e +24 a \,d^{3}\right )}{24}\) | \(179\) |
norman | \(\left (\frac {7}{2} d^{2} f^{3} e^{5} c +\frac {1}{6} b \,e^{5} f^{3}\right ) x^{6}+\left (7 c \,d^{5} e^{2} f^{3}+\frac {10}{3} b \,d^{3} e^{2} f^{3}+a d \,e^{2} f^{3}\right ) x^{3}+\left (\frac {7}{2} c \,d^{6} e \,f^{3}+\frac {5}{2} b \,d^{4} e \,f^{3}+\frac {3}{2} a \,d^{2} e \,f^{3}\right ) x^{2}+\left (\frac {35}{4} d^{4} f^{3} c \,e^{3}+\frac {5}{2} b \,d^{2} e^{3} f^{3}+\frac {1}{4} a \,e^{3} f^{3}\right ) x^{4}+\left (7 d^{3} f^{3} c \,e^{4}+b d \,e^{4} f^{3}\right ) x^{5}+\left (c \,d^{7} f^{3}+b \,d^{5} f^{3}+a \,d^{3} f^{3}\right ) x +d \,f^{3} e^{6} c \,x^{7}+\frac {e^{7} f^{3} c \,x^{8}}{8}\) | \(216\) |
risch | \(\frac {1}{8} e^{7} f^{3} c \,x^{8}+d \,f^{3} e^{6} c \,x^{7}+\frac {7}{2} f^{3} x^{6} d^{2} e^{5} c +\frac {1}{6} f^{3} x^{6} b \,e^{5}+7 f^{3} c \,d^{3} e^{4} x^{5}+f^{3} b d \,e^{4} x^{5}+\frac {35}{4} f^{3} x^{4} d^{4} c \,e^{3}+\frac {5}{2} f^{3} x^{4} b \,d^{2} e^{3}+\frac {1}{4} f^{3} x^{4} a \,e^{3}+7 f^{3} x^{3} c \,d^{5} e^{2}+\frac {10}{3} f^{3} x^{3} b \,d^{3} e^{2}+f^{3} x^{3} d \,e^{2} a +\frac {7}{2} f^{3} x^{2} c \,d^{6} e +\frac {5}{2} f^{3} x^{2} b \,d^{4} e +\frac {3}{2} f^{3} x^{2} a \,d^{2} e +f^{3} c \,d^{7} x +f^{3} b \,d^{5} x +f^{3} a \,d^{3} x\) | \(230\) |
parallelrisch | \(\frac {1}{8} e^{7} f^{3} c \,x^{8}+d \,f^{3} e^{6} c \,x^{7}+\frac {7}{2} f^{3} x^{6} d^{2} e^{5} c +\frac {1}{6} f^{3} x^{6} b \,e^{5}+7 f^{3} c \,d^{3} e^{4} x^{5}+f^{3} b d \,e^{4} x^{5}+\frac {35}{4} f^{3} x^{4} d^{4} c \,e^{3}+\frac {5}{2} f^{3} x^{4} b \,d^{2} e^{3}+\frac {1}{4} f^{3} x^{4} a \,e^{3}+7 f^{3} x^{3} c \,d^{5} e^{2}+\frac {10}{3} f^{3} x^{3} b \,d^{3} e^{2}+f^{3} x^{3} d \,e^{2} a +\frac {7}{2} f^{3} x^{2} c \,d^{6} e +\frac {5}{2} f^{3} x^{2} b \,d^{4} e +\frac {3}{2} f^{3} x^{2} a \,d^{2} e +f^{3} c \,d^{7} x +f^{3} b \,d^{5} x +f^{3} a \,d^{3} x\) | \(230\) |
default | \(\frac {e^{7} f^{3} c \,x^{8}}{8}+d \,f^{3} e^{6} c \,x^{7}+\frac {\left (15 d^{2} f^{3} e^{5} c +e^{3} f^{3} \left (6 c \,d^{2} e^{2}+b \,e^{2}\right )\right ) x^{6}}{6}+\frac {\left (13 d^{3} f^{3} c \,e^{4}+3 d \,f^{3} e^{2} \left (6 c \,d^{2} e^{2}+b \,e^{2}\right )+e^{3} f^{3} \left (4 d^{3} e c +2 b d e \right )\right ) x^{5}}{5}+\frac {\left (4 d^{4} f^{3} c \,e^{3}+3 d^{2} f^{3} e \left (6 c \,d^{2} e^{2}+b \,e^{2}\right )+3 d \,f^{3} e^{2} \left (4 d^{3} e c +2 b d e \right )+e^{3} f^{3} \left (d^{4} c +b \,d^{2}+a \right )\right ) x^{4}}{4}+\frac {\left (d^{3} f^{3} \left (6 c \,d^{2} e^{2}+b \,e^{2}\right )+3 d^{2} f^{3} e \left (4 d^{3} e c +2 b d e \right )+3 d \,f^{3} e^{2} \left (d^{4} c +b \,d^{2}+a \right )\right ) x^{3}}{3}+\frac {\left (d^{3} f^{3} \left (4 d^{3} e c +2 b d e \right )+3 d^{2} f^{3} e \left (d^{4} c +b \,d^{2}+a \right )\right ) x^{2}}{2}+d^{3} f^{3} \left (d^{4} c +b \,d^{2}+a \right ) x\) | \(349\) |
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Leaf count of result is larger than twice the leaf count of optimal. 166 vs. \(2 (49) = 98\).
Time = 0.27 (sec) , antiderivative size = 166, normalized size of antiderivative = 3.02 \[ \int (d f+e f x)^3 \left (a+b (d+e x)^2+c (d+e x)^4\right ) \, dx=\frac {1}{8} \, c e^{7} f^{3} x^{8} + c d e^{6} f^{3} x^{7} + \frac {1}{6} \, {\left (21 \, c d^{2} + b\right )} e^{5} f^{3} x^{6} + {\left (7 \, c d^{3} + b d\right )} e^{4} f^{3} x^{5} + \frac {1}{4} \, {\left (35 \, c d^{4} + 10 \, b d^{2} + a\right )} e^{3} f^{3} x^{4} + \frac {1}{3} \, {\left (21 \, c d^{5} + 10 \, b d^{3} + 3 \, a d\right )} e^{2} f^{3} x^{3} + \frac {1}{2} \, {\left (7 \, c d^{6} + 5 \, b d^{4} + 3 \, a d^{2}\right )} e f^{3} x^{2} + {\left (c d^{7} + b d^{5} + a d^{3}\right )} f^{3} x \]
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Leaf count of result is larger than twice the leaf count of optimal. 240 vs. \(2 (44) = 88\).
Time = 0.05 (sec) , antiderivative size = 240, normalized size of antiderivative = 4.36 \[ \int (d f+e f x)^3 \left (a+b (d+e x)^2+c (d+e x)^4\right ) \, dx=c d e^{6} f^{3} x^{7} + \frac {c e^{7} f^{3} x^{8}}{8} + x^{6} \left (\frac {b e^{5} f^{3}}{6} + \frac {7 c d^{2} e^{5} f^{3}}{2}\right ) + x^{5} \left (b d e^{4} f^{3} + 7 c d^{3} e^{4} f^{3}\right ) + x^{4} \left (\frac {a e^{3} f^{3}}{4} + \frac {5 b d^{2} e^{3} f^{3}}{2} + \frac {35 c d^{4} e^{3} f^{3}}{4}\right ) + x^{3} \left (a d e^{2} f^{3} + \frac {10 b d^{3} e^{2} f^{3}}{3} + 7 c d^{5} e^{2} f^{3}\right ) + x^{2} \cdot \left (\frac {3 a d^{2} e f^{3}}{2} + \frac {5 b d^{4} e f^{3}}{2} + \frac {7 c d^{6} e f^{3}}{2}\right ) + x \left (a d^{3} f^{3} + b d^{5} f^{3} + c d^{7} f^{3}\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. 166 vs. \(2 (49) = 98\).
Time = 0.21 (sec) , antiderivative size = 166, normalized size of antiderivative = 3.02 \[ \int (d f+e f x)^3 \left (a+b (d+e x)^2+c (d+e x)^4\right ) \, dx=\frac {1}{8} \, c e^{7} f^{3} x^{8} + c d e^{6} f^{3} x^{7} + \frac {1}{6} \, {\left (21 \, c d^{2} + b\right )} e^{5} f^{3} x^{6} + {\left (7 \, c d^{3} + b d\right )} e^{4} f^{3} x^{5} + \frac {1}{4} \, {\left (35 \, c d^{4} + 10 \, b d^{2} + a\right )} e^{3} f^{3} x^{4} + \frac {1}{3} \, {\left (21 \, c d^{5} + 10 \, b d^{3} + 3 \, a d\right )} e^{2} f^{3} x^{3} + \frac {1}{2} \, {\left (7 \, c d^{6} + 5 \, b d^{4} + 3 \, a d^{2}\right )} e f^{3} x^{2} + {\left (c d^{7} + b d^{5} + a d^{3}\right )} f^{3} x \]
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Leaf count of result is larger than twice the leaf count of optimal. 204 vs. \(2 (49) = 98\).
Time = 0.30 (sec) , antiderivative size = 204, normalized size of antiderivative = 3.71 \[ \int (d f+e f x)^3 \left (a+b (d+e x)^2+c (d+e x)^4\right ) \, dx=\frac {1}{2} \, {\left (e f x^{2} + 2 \, d f x\right )} c d^{6} f^{2} + \frac {1}{2} \, {\left (e f x^{2} + 2 \, d f x\right )} b d^{4} f^{2} + \frac {1}{2} \, {\left (e f x^{2} + 2 \, d f x\right )} a d^{2} f^{2} + \frac {18 \, {\left (e f x^{2} + 2 \, d f x\right )}^{2} c d^{4} e f^{2} + 12 \, {\left (e f x^{2} + 2 \, d f x\right )}^{3} c d^{2} e^{2} f + 3 \, {\left (e f x^{2} + 2 \, d f x\right )}^{4} c e^{3} + 12 \, {\left (e f x^{2} + 2 \, d f x\right )}^{2} b d^{2} e f^{2} + 4 \, {\left (e f x^{2} + 2 \, d f x\right )}^{3} b e^{2} f + 6 \, {\left (e f x^{2} + 2 \, d f x\right )}^{2} a e f^{2}}{24 \, f} \]
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Time = 0.05 (sec) , antiderivative size = 164, normalized size of antiderivative = 2.98 \[ \int (d f+e f x)^3 \left (a+b (d+e x)^2+c (d+e x)^4\right ) \, dx=\frac {e^5\,f^3\,x^6\,\left (21\,c\,d^2+b\right )}{6}+\frac {c\,e^7\,f^3\,x^8}{8}+d^3\,f^3\,x\,\left (c\,d^4+b\,d^2+a\right )+\frac {e^3\,f^3\,x^4\,\left (35\,c\,d^4+10\,b\,d^2+a\right )}{4}+\frac {d^2\,e\,f^3\,x^2\,\left (7\,c\,d^4+5\,b\,d^2+3\,a\right )}{2}+\frac {d\,e^2\,f^3\,x^3\,\left (21\,c\,d^4+10\,b\,d^2+3\,a\right )}{3}+d\,e^4\,f^3\,x^5\,\left (7\,c\,d^2+b\right )+c\,d\,e^6\,f^3\,x^7 \]
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