Optimal. Leaf size=127 \[ \frac {1}{4} b^3 d^2 x^4+\frac {1}{5} b^2 d (3 c d+2 b e) x^5+\frac {1}{6} b \left (3 c^2 d^2+6 b c d e+b^2 e^2\right ) x^6+\frac {1}{7} c \left (c^2 d^2+6 b c d e+3 b^2 e^2\right ) x^7+\frac {1}{8} c^2 e (2 c d+3 b e) x^8+\frac {1}{9} c^3 e^2 x^9 \]
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Rubi [A]
time = 0.08, antiderivative size = 127, 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^2 x^4+\frac {1}{7} c x^7 \left (3 b^2 e^2+6 b c d e+c^2 d^2\right )+\frac {1}{6} b x^6 \left (b^2 e^2+6 b c d e+3 c^2 d^2\right )+\frac {1}{5} b^2 d x^5 (2 b e+3 c d)+\frac {1}{8} c^2 e x^8 (3 b e+2 c d)+\frac {1}{9} c^3 e^2 x^9 \end {gather*}
Antiderivative was successfully verified.
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Rule 712
Rubi steps
\begin {align*} \int (d+e x)^2 \left (b x+c x^2\right )^3 \, dx &=\int \left (b^3 d^2 x^3+b^2 d (3 c d+2 b e) x^4+b \left (3 c^2 d^2+6 b c d e+b^2 e^2\right ) x^5+c \left (c^2 d^2+6 b c d e+3 b^2 e^2\right ) x^6+c^2 e (2 c d+3 b e) x^7+c^3 e^2 x^8\right ) \, dx\\ &=\frac {1}{4} b^3 d^2 x^4+\frac {1}{5} b^2 d (3 c d+2 b e) x^5+\frac {1}{6} b \left (3 c^2 d^2+6 b c d e+b^2 e^2\right ) x^6+\frac {1}{7} c \left (c^2 d^2+6 b c d e+3 b^2 e^2\right ) x^7+\frac {1}{8} c^2 e (2 c d+3 b e) x^8+\frac {1}{9} c^3 e^2 x^9\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 127, normalized size = 1.00 \begin {gather*} \frac {1}{4} b^3 d^2 x^4+\frac {1}{5} b^2 d (3 c d+2 b e) x^5+\frac {1}{6} b \left (3 c^2 d^2+6 b c d e+b^2 e^2\right ) x^6+\frac {1}{7} c \left (c^2 d^2+6 b c d e+3 b^2 e^2\right ) x^7+\frac {1}{8} c^2 e (2 c d+3 b e) x^8+\frac {1}{9} c^3 e^2 x^9 \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.44, size = 128, normalized size = 1.01
method | result | size |
norman | \(\frac {c^{3} e^{2} x^{9}}{9}+\left (\frac {3}{8} e^{2} b \,c^{2}+\frac {1}{4} d e \,c^{3}\right ) x^{8}+\left (\frac {3}{7} b^{2} e^{2} c +\frac {6}{7} d e b \,c^{2}+\frac {1}{7} c^{3} d^{2}\right ) x^{7}+\left (\frac {1}{6} b^{3} e^{2}+b^{2} d c e +\frac {1}{2} d^{2} b \,c^{2}\right ) x^{6}+\left (\frac {2}{5} b^{3} d e +\frac {3}{5} b^{2} c \,d^{2}\right ) x^{5}+\frac {b^{3} d^{2} x^{4}}{4}\) | \(125\) |
default | \(\frac {c^{3} e^{2} x^{9}}{9}+\frac {\left (3 e^{2} b \,c^{2}+2 d e \,c^{3}\right ) x^{8}}{8}+\frac {\left (3 b^{2} e^{2} c +6 d e b \,c^{2}+c^{3} d^{2}\right ) x^{7}}{7}+\frac {\left (b^{3} e^{2}+6 b^{2} d c e +3 d^{2} b \,c^{2}\right ) x^{6}}{6}+\frac {\left (2 b^{3} d e +3 b^{2} c \,d^{2}\right ) x^{5}}{5}+\frac {b^{3} d^{2} x^{4}}{4}\) | \(128\) |
gosper | \(\frac {x^{4} \left (280 c^{3} e^{2} x^{5}+945 x^{4} e^{2} b \,c^{2}+630 x^{4} d e \,c^{3}+1080 x^{3} b^{2} e^{2} c +2160 x^{3} d e b \,c^{2}+360 x^{3} c^{3} d^{2}+420 x^{2} b^{3} e^{2}+2520 x^{2} b^{2} d c e +1260 x^{2} d^{2} b \,c^{2}+1008 x \,b^{3} d e +1512 x \,b^{2} c \,d^{2}+630 d^{2} b^{3}\right )}{2520}\) | \(134\) |
risch | \(\frac {1}{9} c^{3} e^{2} x^{9}+\frac {3}{8} x^{8} e^{2} b \,c^{2}+\frac {1}{4} x^{8} d e \,c^{3}+\frac {3}{7} x^{7} b^{2} e^{2} c +\frac {6}{7} x^{7} d e b \,c^{2}+\frac {1}{7} x^{7} c^{3} d^{2}+\frac {1}{6} x^{6} b^{3} e^{2}+x^{6} b^{2} d c e +\frac {1}{2} x^{6} d^{2} b \,c^{2}+\frac {2}{5} x^{5} b^{3} d e +\frac {3}{5} b^{2} c \,d^{2} x^{5}+\frac {1}{4} b^{3} d^{2} x^{4}\) | \(135\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 127, normalized size = 1.00 \begin {gather*} \frac {1}{9} \, c^{3} x^{9} e^{2} + \frac {1}{4} \, b^{3} d^{2} x^{4} + \frac {1}{8} \, {\left (2 \, c^{3} d e + 3 \, b c^{2} e^{2}\right )} x^{8} + \frac {1}{7} \, {\left (c^{3} d^{2} + 6 \, b c^{2} d e + 3 \, b^{2} c e^{2}\right )} x^{7} + \frac {1}{6} \, {\left (3 \, b c^{2} d^{2} + 6 \, b^{2} c d e + b^{3} e^{2}\right )} x^{6} + \frac {1}{5} \, {\left (3 \, b^{2} c d^{2} + 2 \, b^{3} d e\right )} x^{5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.55, size = 129, normalized size = 1.02 \begin {gather*} \frac {1}{7} \, c^{3} d^{2} x^{7} + \frac {1}{2} \, b c^{2} d^{2} x^{6} + \frac {3}{5} \, b^{2} c d^{2} x^{5} + \frac {1}{4} \, b^{3} d^{2} x^{4} + \frac {1}{504} \, {\left (56 \, c^{3} x^{9} + 189 \, b c^{2} x^{8} + 216 \, b^{2} c x^{7} + 84 \, b^{3} x^{6}\right )} e^{2} + \frac {1}{140} \, {\left (35 \, c^{3} d x^{8} + 120 \, b c^{2} d x^{7} + 140 \, b^{2} c d x^{6} + 56 \, b^{3} d 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 = 138, normalized size = 1.09 \begin {gather*} \frac {b^{3} d^{2} x^{4}}{4} + \frac {c^{3} e^{2} x^{9}}{9} + x^{8} \cdot \left (\frac {3 b c^{2} e^{2}}{8} + \frac {c^{3} d e}{4}\right ) + x^{7} \cdot \left (\frac {3 b^{2} c e^{2}}{7} + \frac {6 b c^{2} d e}{7} + \frac {c^{3} d^{2}}{7}\right ) + x^{6} \left (\frac {b^{3} e^{2}}{6} + b^{2} c d e + \frac {b c^{2} d^{2}}{2}\right ) + x^{5} \cdot \left (\frac {2 b^{3} d e}{5} + \frac {3 b^{2} c d^{2}}{5}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.63, size = 134, normalized size = 1.06 \begin {gather*} \frac {1}{9} \, c^{3} x^{9} e^{2} + \frac {1}{4} \, c^{3} d x^{8} e + \frac {1}{7} \, c^{3} d^{2} x^{7} + \frac {3}{8} \, b c^{2} x^{8} e^{2} + \frac {6}{7} \, b c^{2} d x^{7} e + \frac {1}{2} \, b c^{2} d^{2} x^{6} + \frac {3}{7} \, b^{2} c x^{7} e^{2} + b^{2} c d x^{6} e + \frac {3}{5} \, b^{2} c d^{2} x^{5} + \frac {1}{6} \, b^{3} x^{6} e^{2} + \frac {2}{5} \, b^{3} d x^{5} e + \frac {1}{4} \, b^{3} d^{2} x^{4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.05, size = 118, normalized size = 0.93 \begin {gather*} x^6\,\left (\frac {b^3\,e^2}{6}+b^2\,c\,d\,e+\frac {b\,c^2\,d^2}{2}\right )+x^7\,\left (\frac {3\,b^2\,c\,e^2}{7}+\frac {6\,b\,c^2\,d\,e}{7}+\frac {c^3\,d^2}{7}\right )+\frac {b^3\,d^2\,x^4}{4}+\frac {c^3\,e^2\,x^9}{9}+\frac {b^2\,d\,x^5\,\left (2\,b\,e+3\,c\,d\right )}{5}+\frac {c^2\,e\,x^8\,\left (3\,b\,e+2\,c\,d\right )}{8} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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