Optimal. Leaf size=162 \[ \frac {1}{4} b^3 d^3 x^4+\frac {3}{5} b^2 d^2 (c d+b e) x^5+\frac {1}{2} b d \left (c^2 d^2+3 b c d e+b^2 e^2\right ) x^6+\frac {1}{7} (c d+b e) \left (c^2 d^2+8 b c d e+b^2 e^2\right ) x^7+\frac {3}{8} c e \left (c^2 d^2+3 b c d e+b^2 e^2\right ) x^8+\frac {1}{3} c^2 e^2 (c d+b e) x^9+\frac {1}{10} c^3 e^3 x^{10} \]
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Rubi [A]
time = 0.11, antiderivative size = 162, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 1, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {712}
\begin {gather*} \frac {1}{4} b^3 d^3 x^4+\frac {3}{8} c e x^8 \left (b^2 e^2+3 b c d e+c^2 d^2\right )+\frac {1}{7} x^7 (b e+c d) \left (b^2 e^2+8 b c d e+c^2 d^2\right )+\frac {1}{2} b d x^6 \left (b^2 e^2+3 b c d e+c^2 d^2\right )+\frac {3}{5} b^2 d^2 x^5 (b e+c d)+\frac {1}{3} c^2 e^2 x^9 (b e+c d)+\frac {1}{10} c^3 e^3 x^{10} \end {gather*}
Antiderivative was successfully verified.
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Rule 712
Rubi steps
\begin {align*} \int (d+e x)^3 \left (b x+c x^2\right )^3 \, dx &=\int \left (b^3 d^3 x^3+3 b^2 d^2 (c d+b e) x^4+3 b d \left (c^2 d^2+3 b c d e+b^2 e^2\right ) x^5+(c d+b e) \left (c^2 d^2+8 b c d e+b^2 e^2\right ) x^6+3 c e \left (c^2 d^2+3 b c d e+b^2 e^2\right ) x^7+3 c^2 e^2 (c d+b e) x^8+c^3 e^3 x^9\right ) \, dx\\ &=\frac {1}{4} b^3 d^3 x^4+\frac {3}{5} b^2 d^2 (c d+b e) x^5+\frac {1}{2} b d \left (c^2 d^2+3 b c d e+b^2 e^2\right ) x^6+\frac {1}{7} (c d+b e) \left (c^2 d^2+8 b c d e+b^2 e^2\right ) x^7+\frac {3}{8} c e \left (c^2 d^2+3 b c d e+b^2 e^2\right ) x^8+\frac {1}{3} c^2 e^2 (c d+b e) x^9+\frac {1}{10} c^3 e^3 x^{10}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 169, normalized size = 1.04 \begin {gather*} \frac {1}{4} b^3 d^3 x^4+\frac {3}{5} b^2 d^2 (c d+b e) x^5+\frac {1}{2} b d \left (c^2 d^2+3 b c d e+b^2 e^2\right ) x^6+\frac {1}{7} \left (c^3 d^3+9 b c^2 d^2 e+9 b^2 c d e^2+b^3 e^3\right ) x^7+\frac {3}{8} c e \left (c^2 d^2+3 b c d e+b^2 e^2\right ) x^8+\frac {1}{3} c^2 e^2 (c d+b e) x^9+\frac {1}{10} c^3 e^3 x^{10} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.43, size = 180, normalized size = 1.11
method | result | size |
norman | \(\frac {c^{3} e^{3} x^{10}}{10}+\left (\frac {1}{3} e^{3} b \,c^{2}+\frac {1}{3} d \,e^{2} c^{3}\right ) x^{9}+\left (\frac {3}{8} e^{3} b^{2} c +\frac {9}{8} d \,e^{2} b \,c^{2}+\frac {3}{8} d^{2} e \,c^{3}\right ) x^{8}+\left (\frac {1}{7} b^{3} e^{3}+\frac {9}{7} b^{2} d \,e^{2} c +\frac {9}{7} b \,c^{2} d^{2} e +\frac {1}{7} c^{3} d^{3}\right ) x^{7}+\left (\frac {1}{2} b^{3} d \,e^{2}+\frac {3}{2} d^{2} e \,b^{2} c +\frac {1}{2} d^{3} b \,c^{2}\right ) x^{6}+\left (\frac {3}{5} d^{2} e \,b^{3}+\frac {3}{5} b^{2} c \,d^{3}\right ) x^{5}+\frac {b^{3} d^{3} x^{4}}{4}\) | \(177\) |
default | \(\frac {c^{3} e^{3} x^{10}}{10}+\frac {\left (3 e^{3} b \,c^{2}+3 d \,e^{2} c^{3}\right ) x^{9}}{9}+\frac {\left (3 e^{3} b^{2} c +9 d \,e^{2} b \,c^{2}+3 d^{2} e \,c^{3}\right ) x^{8}}{8}+\frac {\left (b^{3} e^{3}+9 b^{2} d \,e^{2} c +9 b \,c^{2} d^{2} e +c^{3} d^{3}\right ) x^{7}}{7}+\frac {\left (3 b^{3} d \,e^{2}+9 d^{2} e \,b^{2} c +3 d^{3} b \,c^{2}\right ) x^{6}}{6}+\frac {\left (3 d^{2} e \,b^{3}+3 b^{2} c \,d^{3}\right ) x^{5}}{5}+\frac {b^{3} d^{3} x^{4}}{4}\) | \(180\) |
gosper | \(\frac {x^{4} \left (84 c^{3} e^{3} x^{6}+280 x^{5} e^{3} b \,c^{2}+280 x^{5} d \,e^{2} c^{3}+315 x^{4} e^{3} b^{2} c +945 x^{4} d \,e^{2} b \,c^{2}+315 x^{4} d^{2} e \,c^{3}+120 x^{3} b^{3} e^{3}+1080 x^{3} b^{2} d \,e^{2} c +1080 x^{3} b \,c^{2} d^{2} e +120 x^{3} c^{3} d^{3}+420 x^{2} b^{3} d \,e^{2}+1260 x^{2} d^{2} e \,b^{2} c +420 x^{2} d^{3} b \,c^{2}+504 x \,d^{2} e \,b^{3}+504 x \,b^{2} c \,d^{3}+210 b^{3} d^{3}\right )}{840}\) | \(192\) |
risch | \(\frac {1}{10} c^{3} e^{3} x^{10}+\frac {1}{3} x^{9} e^{3} b \,c^{2}+\frac {1}{3} c^{3} d \,e^{2} x^{9}+\frac {3}{8} x^{8} e^{3} b^{2} c +\frac {9}{8} x^{8} d \,e^{2} b \,c^{2}+\frac {3}{8} x^{8} d^{2} e \,c^{3}+\frac {1}{7} x^{7} b^{3} e^{3}+\frac {9}{7} x^{7} b^{2} d \,e^{2} c +\frac {9}{7} x^{7} b \,c^{2} d^{2} e +\frac {1}{7} x^{7} c^{3} d^{3}+\frac {1}{2} x^{6} b^{3} d \,e^{2}+\frac {3}{2} x^{6} d^{2} e \,b^{2} c +\frac {1}{2} x^{6} d^{3} b \,c^{2}+\frac {3}{5} x^{5} d^{2} e \,b^{3}+\frac {3}{5} x^{5} b^{2} c \,d^{3}+\frac {1}{4} b^{3} d^{3} x^{4}\) | \(194\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 167, normalized size = 1.03 \begin {gather*} \frac {1}{10} \, c^{3} x^{10} e^{3} + \frac {1}{4} \, b^{3} d^{3} x^{4} + \frac {1}{3} \, {\left (c^{3} d e^{2} + b c^{2} e^{3}\right )} x^{9} + \frac {3}{8} \, {\left (c^{3} d^{2} e + 3 \, b c^{2} d e^{2} + b^{2} c e^{3}\right )} x^{8} + \frac {1}{7} \, {\left (c^{3} d^{3} + 9 \, b c^{2} d^{2} e + 9 \, b^{2} c d e^{2} + b^{3} e^{3}\right )} x^{7} + \frac {1}{2} \, {\left (b c^{2} d^{3} + 3 \, b^{2} c d^{2} e + b^{3} d e^{2}\right )} x^{6} + \frac {3}{5} \, {\left (b^{2} c d^{3} + b^{3} d^{2} e\right )} x^{5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.59, size = 180, normalized size = 1.11 \begin {gather*} \frac {1}{7} \, c^{3} d^{3} x^{7} + \frac {1}{2} \, b c^{2} d^{3} x^{6} + \frac {3}{5} \, b^{2} c d^{3} x^{5} + \frac {1}{4} \, b^{3} d^{3} x^{4} + \frac {1}{840} \, {\left (84 \, c^{3} x^{10} + 280 \, b c^{2} x^{9} + 315 \, b^{2} c x^{8} + 120 \, b^{3} x^{7}\right )} e^{3} + \frac {1}{168} \, {\left (56 \, c^{3} d x^{9} + 189 \, b c^{2} d x^{8} + 216 \, b^{2} c d x^{7} + 84 \, b^{3} d x^{6}\right )} e^{2} + \frac {3}{280} \, {\left (35 \, c^{3} d^{2} x^{8} + 120 \, b c^{2} d^{2} x^{7} + 140 \, b^{2} c d^{2} x^{6} + 56 \, b^{3} d^{2} x^{5}\right )} e \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.02, size = 199, normalized size = 1.23 \begin {gather*} \frac {b^{3} d^{3} x^{4}}{4} + \frac {c^{3} e^{3} x^{10}}{10} + x^{9} \left (\frac {b c^{2} e^{3}}{3} + \frac {c^{3} d e^{2}}{3}\right ) + x^{8} \cdot \left (\frac {3 b^{2} c e^{3}}{8} + \frac {9 b c^{2} d e^{2}}{8} + \frac {3 c^{3} d^{2} e}{8}\right ) + x^{7} \left (\frac {b^{3} e^{3}}{7} + \frac {9 b^{2} c d e^{2}}{7} + \frac {9 b c^{2} d^{2} e}{7} + \frac {c^{3} d^{3}}{7}\right ) + x^{6} \left (\frac {b^{3} d e^{2}}{2} + \frac {3 b^{2} c d^{2} e}{2} + \frac {b c^{2} d^{3}}{2}\right ) + x^{5} \cdot \left (\frac {3 b^{3} d^{2} e}{5} + \frac {3 b^{2} c d^{3}}{5}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.54, size = 189, normalized size = 1.17 \begin {gather*} \frac {1}{10} \, c^{3} x^{10} e^{3} + \frac {1}{3} \, c^{3} d x^{9} e^{2} + \frac {3}{8} \, c^{3} d^{2} x^{8} e + \frac {1}{7} \, c^{3} d^{3} x^{7} + \frac {1}{3} \, b c^{2} x^{9} e^{3} + \frac {9}{8} \, b c^{2} d x^{8} e^{2} + \frac {9}{7} \, b c^{2} d^{2} x^{7} e + \frac {1}{2} \, b c^{2} d^{3} x^{6} + \frac {3}{8} \, b^{2} c x^{8} e^{3} + \frac {9}{7} \, b^{2} c d x^{7} e^{2} + \frac {3}{2} \, b^{2} c d^{2} x^{6} e + \frac {3}{5} \, b^{2} c d^{3} x^{5} + \frac {1}{7} \, b^{3} x^{7} e^{3} + \frac {1}{2} \, b^{3} d x^{6} e^{2} + \frac {3}{5} \, b^{3} d^{2} x^{5} e + \frac {1}{4} \, b^{3} d^{3} x^{4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.18, size = 156, normalized size = 0.96 \begin {gather*} x^7\,\left (\frac {b^3\,e^3}{7}+\frac {9\,b^2\,c\,d\,e^2}{7}+\frac {9\,b\,c^2\,d^2\,e}{7}+\frac {c^3\,d^3}{7}\right )+\frac {b^3\,d^3\,x^4}{4}+\frac {c^3\,e^3\,x^{10}}{10}+\frac {b\,d\,x^6\,\left (b^2\,e^2+3\,b\,c\,d\,e+c^2\,d^2\right )}{2}+\frac {3\,c\,e\,x^8\,\left (b^2\,e^2+3\,b\,c\,d\,e+c^2\,d^2\right )}{8}+\frac {3\,b^2\,d^2\,x^5\,\left (b\,e+c\,d\right )}{5}+\frac {c^2\,e^2\,x^9\,\left (b\,e+c\,d\right )}{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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