Integrand size = 22, antiderivative size = 135 \[ \int \left (d+e x^3\right )^4 \left (a+b x^3+c x^6\right ) \, dx=a d^4 x+\frac {1}{4} d^3 (b d+4 a e) x^4+\frac {1}{7} d^2 \left (c d^2+4 b d e+6 a e^2\right ) x^7+\frac {1}{5} d e \left (2 c d^2+e (3 b d+2 a e)\right ) x^{10}+\frac {1}{13} e^2 \left (6 c d^2+e (4 b d+a e)\right ) x^{13}+\frac {1}{16} e^3 (4 c d+b e) x^{16}+\frac {1}{19} c e^4 x^{19} \]
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Time = 0.08 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {1421} \[ \int \left (d+e x^3\right )^4 \left (a+b x^3+c x^6\right ) \, dx=\frac {1}{13} e^2 x^{13} \left (e (a e+4 b d)+6 c d^2\right )+\frac {1}{7} d^2 x^7 \left (6 a e^2+4 b d e+c d^2\right )+\frac {1}{5} d e x^{10} \left (e (2 a e+3 b d)+2 c d^2\right )+\frac {1}{4} d^3 x^4 (4 a e+b d)+a d^4 x+\frac {1}{16} e^3 x^{16} (b e+4 c d)+\frac {1}{19} c e^4 x^{19} \]
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Rule 1421
Rubi steps \begin{align*} \text {integral}& = \int \left (a d^4+d^3 (b d+4 a e) x^3+d^2 \left (c d^2+4 b d e+6 a e^2\right ) x^6+2 d e \left (2 c d^2+e (3 b d+2 a e)\right ) x^9+e^2 \left (6 c d^2+e (4 b d+a e)\right ) x^{12}+e^3 (4 c d+b e) x^{15}+c e^4 x^{18}\right ) \, dx \\ & = a d^4 x+\frac {1}{4} d^3 (b d+4 a e) x^4+\frac {1}{7} d^2 \left (c d^2+4 b d e+6 a e^2\right ) x^7+\frac {1}{5} d e \left (2 c d^2+e (3 b d+2 a e)\right ) x^{10}+\frac {1}{13} e^2 \left (6 c d^2+e (4 b d+a e)\right ) x^{13}+\frac {1}{16} e^3 (4 c d+b e) x^{16}+\frac {1}{19} c e^4 x^{19} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.00 \[ \int \left (d+e x^3\right )^4 \left (a+b x^3+c x^6\right ) \, dx=a d^4 x+\frac {1}{4} d^3 (b d+4 a e) x^4+\frac {1}{7} d^2 \left (c d^2+4 b d e+6 a e^2\right ) x^7+\frac {1}{5} d e \left (2 c d^2+3 b d e+2 a e^2\right ) x^{10}+\frac {1}{13} e^2 \left (6 c d^2+4 b d e+a e^2\right ) x^{13}+\frac {1}{16} e^3 (4 c d+b e) x^{16}+\frac {1}{19} c e^4 x^{19} \]
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Time = 0.68 (sec) , antiderivative size = 134, normalized size of antiderivative = 0.99
method | result | size |
norman | \(a \,d^{4} x +\left (a \,d^{3} e +\frac {1}{4} d^{4} b \right ) x^{4}+\left (\frac {6}{7} e^{2} d^{2} a +\frac {4}{7} d^{3} e b +\frac {1}{7} d^{4} c \right ) x^{7}+\left (\frac {2}{5} d \,e^{3} a +\frac {3}{5} e^{2} d^{2} b +\frac {2}{5} d^{3} e c \right ) x^{10}+\left (\frac {1}{13} e^{4} a +\frac {4}{13} b d \,e^{3}+\frac {6}{13} e^{2} d^{2} c \right ) x^{13}+\left (\frac {1}{16} b \,e^{4}+\frac {1}{4} d \,e^{3} c \right ) x^{16}+\frac {c \,e^{4} x^{19}}{19}\) | \(134\) |
default | \(\frac {c \,e^{4} x^{19}}{19}+\frac {\left (b \,e^{4}+4 d \,e^{3} c \right ) x^{16}}{16}+\frac {\left (e^{4} a +4 b d \,e^{3}+6 e^{2} d^{2} c \right ) x^{13}}{13}+\frac {\left (4 d \,e^{3} a +6 e^{2} d^{2} b +4 d^{3} e c \right ) x^{10}}{10}+\frac {\left (6 e^{2} d^{2} a +4 d^{3} e b +d^{4} c \right ) x^{7}}{7}+\frac {\left (4 a \,d^{3} e +d^{4} b \right ) x^{4}}{4}+a \,d^{4} x\) | \(136\) |
gosper | \(a \,d^{4} x +x^{4} a \,d^{3} e +\frac {1}{4} b \,x^{4} d^{4}+\frac {6}{7} x^{7} e^{2} d^{2} a +\frac {4}{7} x^{7} d^{3} e b +\frac {1}{7} x^{7} d^{4} c +\frac {2}{5} x^{10} d \,e^{3} a +\frac {3}{5} x^{10} e^{2} d^{2} b +\frac {2}{5} x^{10} d^{3} e c +\frac {1}{13} x^{13} e^{4} a +\frac {4}{13} x^{13} b d \,e^{3}+\frac {6}{13} x^{13} e^{2} d^{2} c +\frac {1}{16} x^{16} b \,e^{4}+\frac {1}{4} x^{16} d \,e^{3} c +\frac {1}{19} c \,e^{4} x^{19}\) | \(148\) |
risch | \(a \,d^{4} x +x^{4} a \,d^{3} e +\frac {1}{4} b \,x^{4} d^{4}+\frac {6}{7} x^{7} e^{2} d^{2} a +\frac {4}{7} x^{7} d^{3} e b +\frac {1}{7} x^{7} d^{4} c +\frac {2}{5} x^{10} d \,e^{3} a +\frac {3}{5} x^{10} e^{2} d^{2} b +\frac {2}{5} x^{10} d^{3} e c +\frac {1}{13} x^{13} e^{4} a +\frac {4}{13} x^{13} b d \,e^{3}+\frac {6}{13} x^{13} e^{2} d^{2} c +\frac {1}{16} x^{16} b \,e^{4}+\frac {1}{4} x^{16} d \,e^{3} c +\frac {1}{19} c \,e^{4} x^{19}\) | \(148\) |
parallelrisch | \(a \,d^{4} x +x^{4} a \,d^{3} e +\frac {1}{4} b \,x^{4} d^{4}+\frac {6}{7} x^{7} e^{2} d^{2} a +\frac {4}{7} x^{7} d^{3} e b +\frac {1}{7} x^{7} d^{4} c +\frac {2}{5} x^{10} d \,e^{3} a +\frac {3}{5} x^{10} e^{2} d^{2} b +\frac {2}{5} x^{10} d^{3} e c +\frac {1}{13} x^{13} e^{4} a +\frac {4}{13} x^{13} b d \,e^{3}+\frac {6}{13} x^{13} e^{2} d^{2} c +\frac {1}{16} x^{16} b \,e^{4}+\frac {1}{4} x^{16} d \,e^{3} c +\frac {1}{19} c \,e^{4} x^{19}\) | \(148\) |
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Time = 0.26 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.00 \[ \int \left (d+e x^3\right )^4 \left (a+b x^3+c x^6\right ) \, dx=\frac {1}{19} \, c e^{4} x^{19} + \frac {1}{16} \, {\left (4 \, c d e^{3} + b e^{4}\right )} x^{16} + \frac {1}{13} \, {\left (6 \, c d^{2} e^{2} + 4 \, b d e^{3} + a e^{4}\right )} x^{13} + \frac {1}{5} \, {\left (2 \, c d^{3} e + 3 \, b d^{2} e^{2} + 2 \, a d e^{3}\right )} x^{10} + \frac {1}{7} \, {\left (c d^{4} + 4 \, b d^{3} e + 6 \, a d^{2} e^{2}\right )} x^{7} + a d^{4} x + \frac {1}{4} \, {\left (b d^{4} + 4 \, a d^{3} e\right )} x^{4} \]
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Time = 0.03 (sec) , antiderivative size = 151, normalized size of antiderivative = 1.12 \[ \int \left (d+e x^3\right )^4 \left (a+b x^3+c x^6\right ) \, dx=a d^{4} x + \frac {c e^{4} x^{19}}{19} + x^{16} \left (\frac {b e^{4}}{16} + \frac {c d e^{3}}{4}\right ) + x^{13} \left (\frac {a e^{4}}{13} + \frac {4 b d e^{3}}{13} + \frac {6 c d^{2} e^{2}}{13}\right ) + x^{10} \cdot \left (\frac {2 a d e^{3}}{5} + \frac {3 b d^{2} e^{2}}{5} + \frac {2 c d^{3} e}{5}\right ) + x^{7} \cdot \left (\frac {6 a d^{2} e^{2}}{7} + \frac {4 b d^{3} e}{7} + \frac {c d^{4}}{7}\right ) + x^{4} \left (a d^{3} e + \frac {b d^{4}}{4}\right ) \]
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Time = 0.20 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.00 \[ \int \left (d+e x^3\right )^4 \left (a+b x^3+c x^6\right ) \, dx=\frac {1}{19} \, c e^{4} x^{19} + \frac {1}{16} \, {\left (4 \, c d e^{3} + b e^{4}\right )} x^{16} + \frac {1}{13} \, {\left (6 \, c d^{2} e^{2} + 4 \, b d e^{3} + a e^{4}\right )} x^{13} + \frac {1}{5} \, {\left (2 \, c d^{3} e + 3 \, b d^{2} e^{2} + 2 \, a d e^{3}\right )} x^{10} + \frac {1}{7} \, {\left (c d^{4} + 4 \, b d^{3} e + 6 \, a d^{2} e^{2}\right )} x^{7} + a d^{4} x + \frac {1}{4} \, {\left (b d^{4} + 4 \, a d^{3} e\right )} x^{4} \]
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Time = 0.29 (sec) , antiderivative size = 147, normalized size of antiderivative = 1.09 \[ \int \left (d+e x^3\right )^4 \left (a+b x^3+c x^6\right ) \, dx=\frac {1}{19} \, c e^{4} x^{19} + \frac {1}{4} \, c d e^{3} x^{16} + \frac {1}{16} \, b e^{4} x^{16} + \frac {6}{13} \, c d^{2} e^{2} x^{13} + \frac {4}{13} \, b d e^{3} x^{13} + \frac {1}{13} \, a e^{4} x^{13} + \frac {2}{5} \, c d^{3} e x^{10} + \frac {3}{5} \, b d^{2} e^{2} x^{10} + \frac {2}{5} \, a d e^{3} x^{10} + \frac {1}{7} \, c d^{4} x^{7} + \frac {4}{7} \, b d^{3} e x^{7} + \frac {6}{7} \, a d^{2} e^{2} x^{7} + \frac {1}{4} \, b d^{4} x^{4} + a d^{3} e x^{4} + a d^{4} x \]
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Time = 0.03 (sec) , antiderivative size = 130, normalized size of antiderivative = 0.96 \[ \int \left (d+e x^3\right )^4 \left (a+b x^3+c x^6\right ) \, dx=x^4\,\left (\frac {b\,d^4}{4}+a\,e\,d^3\right )+x^{16}\,\left (\frac {b\,e^4}{16}+\frac {c\,d\,e^3}{4}\right )+x^7\,\left (\frac {c\,d^4}{7}+\frac {4\,b\,d^3\,e}{7}+\frac {6\,a\,d^2\,e^2}{7}\right )+x^{13}\,\left (\frac {6\,c\,d^2\,e^2}{13}+\frac {4\,b\,d\,e^3}{13}+\frac {a\,e^4}{13}\right )+\frac {c\,e^4\,x^{19}}{19}+a\,d^4\,x+\frac {d\,e\,x^{10}\,\left (2\,c\,d^2+3\,b\,d\,e+2\,a\,e^2\right )}{5} \]
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