Optimal. Leaf size=20 \[ \frac {1}{8} \left (b x+c x^2+d x^3\right )^8 \]
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Rubi [A] time = 0.05, antiderivative size = 20, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.034, Rules used = {1588} \[ \frac {1}{8} \left (b x+c x^2+d x^3\right )^8 \]
Antiderivative was successfully verified.
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Rule 1588
Rubi steps
\begin {align*} \int \left (b+2 c x+3 d x^2\right ) \left (b x+c x^2+d x^3\right )^7 \, dx &=\frac {1}{8} \left (b x+c x^2+d x^3\right )^8\\ \end {align*}
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Mathematica [A] time = 0.04, size = 18, normalized size = 0.90 \[ \frac {1}{8} x^8 (b+x (c+d x))^8 \]
Antiderivative was successfully verified.
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fricas [B] time = 0.50, size = 496, normalized size = 24.80 \[ \frac {1}{8} x^{24} d^{8} + x^{23} d^{7} c + \frac {7}{2} x^{22} d^{6} c^{2} + x^{22} d^{7} b + 7 x^{21} d^{5} c^{3} + 7 x^{21} d^{6} c b + \frac {35}{4} x^{20} d^{4} c^{4} + 21 x^{20} d^{5} c^{2} b + \frac {7}{2} x^{20} d^{6} b^{2} + 7 x^{19} d^{3} c^{5} + 35 x^{19} d^{4} c^{3} b + 21 x^{19} d^{5} c b^{2} + \frac {7}{2} x^{18} d^{2} c^{6} + 35 x^{18} d^{3} c^{4} b + \frac {105}{2} x^{18} d^{4} c^{2} b^{2} + 7 x^{18} d^{5} b^{3} + x^{17} d c^{7} + 21 x^{17} d^{2} c^{5} b + 70 x^{17} d^{3} c^{3} b^{2} + 35 x^{17} d^{4} c b^{3} + \frac {1}{8} x^{16} c^{8} + 7 x^{16} d c^{6} b + \frac {105}{2} x^{16} d^{2} c^{4} b^{2} + 70 x^{16} d^{3} c^{2} b^{3} + \frac {35}{4} x^{16} d^{4} b^{4} + x^{15} c^{7} b + 21 x^{15} d c^{5} b^{2} + 70 x^{15} d^{2} c^{3} b^{3} + 35 x^{15} d^{3} c b^{4} + \frac {7}{2} x^{14} c^{6} b^{2} + 35 x^{14} d c^{4} b^{3} + \frac {105}{2} x^{14} d^{2} c^{2} b^{4} + 7 x^{14} d^{3} b^{5} + 7 x^{13} c^{5} b^{3} + 35 x^{13} d c^{3} b^{4} + 21 x^{13} d^{2} c b^{5} + \frac {35}{4} x^{12} c^{4} b^{4} + 21 x^{12} d c^{2} b^{5} + \frac {7}{2} x^{12} d^{2} b^{6} + 7 x^{11} c^{3} b^{5} + 7 x^{11} d c b^{6} + \frac {7}{2} x^{10} c^{2} b^{6} + x^{10} d b^{7} + x^{9} c b^{7} + \frac {1}{8} x^{8} b^{8} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.31, size = 18, normalized size = 0.90 \[ \frac {1}{8} \, {\left (d x^{3} + c x^{2} + b x\right )}^{8} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.00, size = 5596, normalized size = 279.80 \[ \text {output too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.63, size = 18, normalized size = 0.90 \[ \frac {1}{8} \, {\left (d x^{3} + c x^{2} + b x\right )}^{8} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.32, size = 418, normalized size = 20.90 \[ x^{14}\,\left (7\,b^5\,d^3+\frac {105\,b^4\,c^2\,d^2}{2}+35\,b^3\,c^4\,d+\frac {7\,b^2\,c^6}{2}\right )+x^{18}\,\left (7\,b^3\,d^5+\frac {105\,b^2\,c^2\,d^4}{2}+35\,b\,c^4\,d^3+\frac {7\,c^6\,d^2}{2}\right )+x^{12}\,\left (\frac {7\,b^6\,d^2}{2}+21\,b^5\,c^2\,d+\frac {35\,b^4\,c^4}{4}\right )+x^{20}\,\left (\frac {7\,b^2\,d^6}{2}+21\,b\,c^2\,d^5+\frac {35\,c^4\,d^4}{4}\right )+x^{16}\,\left (\frac {35\,b^4\,d^4}{4}+70\,b^3\,c^2\,d^3+\frac {105\,b^2\,c^4\,d^2}{2}+7\,b\,c^6\,d+\frac {c^8}{8}\right )+\frac {b^8\,x^8}{8}+\frac {d^8\,x^{24}}{8}+x^{10}\,\left (d\,b^7+\frac {7\,b^6\,c^2}{2}\right )+b^7\,c\,x^9+c\,d^7\,x^{23}+\frac {d^6\,x^{22}\,\left (7\,c^2+2\,b\,d\right )}{2}+7\,b^3\,c\,x^{13}\,\left (3\,b^2\,d^2+5\,b\,c^2\,d+c^4\right )+7\,c\,d^3\,x^{19}\,\left (3\,b^2\,d^2+5\,b\,c^2\,d+c^4\right )+b\,c\,x^{15}\,\left (35\,b^3\,d^3+70\,b^2\,c^2\,d^2+21\,b\,c^4\,d+c^6\right )+c\,d\,x^{17}\,\left (35\,b^3\,d^3+70\,b^2\,c^2\,d^2+21\,b\,c^4\,d+c^6\right )+7\,b^5\,c\,x^{11}\,\left (c^2+b\,d\right )+7\,c\,d^5\,x^{21}\,\left (c^2+b\,d\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.18, size = 469, normalized size = 23.45 \[ \frac {b^{8} x^{8}}{8} + b^{7} c x^{9} + c d^{7} x^{23} + \frac {d^{8} x^{24}}{8} + x^{22} \left (b d^{7} + \frac {7 c^{2} d^{6}}{2}\right ) + x^{21} \left (7 b c d^{6} + 7 c^{3} d^{5}\right ) + x^{20} \left (\frac {7 b^{2} d^{6}}{2} + 21 b c^{2} d^{5} + \frac {35 c^{4} d^{4}}{4}\right ) + x^{19} \left (21 b^{2} c d^{5} + 35 b c^{3} d^{4} + 7 c^{5} d^{3}\right ) + x^{18} \left (7 b^{3} d^{5} + \frac {105 b^{2} c^{2} d^{4}}{2} + 35 b c^{4} d^{3} + \frac {7 c^{6} d^{2}}{2}\right ) + x^{17} \left (35 b^{3} c d^{4} + 70 b^{2} c^{3} d^{3} + 21 b c^{5} d^{2} + c^{7} d\right ) + x^{16} \left (\frac {35 b^{4} d^{4}}{4} + 70 b^{3} c^{2} d^{3} + \frac {105 b^{2} c^{4} d^{2}}{2} + 7 b c^{6} d + \frac {c^{8}}{8}\right ) + x^{15} \left (35 b^{4} c d^{3} + 70 b^{3} c^{3} d^{2} + 21 b^{2} c^{5} d + b c^{7}\right ) + x^{14} \left (7 b^{5} d^{3} + \frac {105 b^{4} c^{2} d^{2}}{2} + 35 b^{3} c^{4} d + \frac {7 b^{2} c^{6}}{2}\right ) + x^{13} \left (21 b^{5} c d^{2} + 35 b^{4} c^{3} d + 7 b^{3} c^{5}\right ) + x^{12} \left (\frac {7 b^{6} d^{2}}{2} + 21 b^{5} c^{2} d + \frac {35 b^{4} c^{4}}{4}\right ) + x^{11} \left (7 b^{6} c d + 7 b^{5} c^{3}\right ) + x^{10} \left (b^{7} d + \frac {7 b^{6} c^{2}}{2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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