Optimal. Leaf size=14 \[ \frac{1}{8} x^{16} (c+d x)^8 \]
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Rubi [A] time = 0.0219512, antiderivative size = 14, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.087, Rules used = {1584, 74} \[ \frac{1}{8} x^{16} (c+d x)^8 \]
Antiderivative was successfully verified.
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Rule 1584
Rule 74
Rubi steps
\begin{align*} \int x (2 c+3 d x) \left (c x^2+d x^3\right )^7 \, dx &=\int x^{15} (c+d x)^7 (2 c+3 d x) \, dx\\ &=\frac{1}{8} x^{16} (c+d x)^8\\ \end{align*}
Mathematica [B] time = 0.0027242, size = 98, normalized size = 7. \[ \frac{7}{2} c^2 d^6 x^{22}+7 c^3 d^5 x^{21}+\frac{35}{4} c^4 d^4 x^{20}+7 c^5 d^3 x^{19}+\frac{7}{2} c^6 d^2 x^{18}+c^7 d x^{17}+\frac{c^8 x^{16}}{8}+c d^7 x^{23}+\frac{d^8 x^{24}}{8} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.001, size = 89, normalized size = 6.4 \begin{align*}{\frac{{d}^{8}{x}^{24}}{8}}+c{d}^{7}{x}^{23}+{\frac{7\,{c}^{2}{d}^{6}{x}^{22}}{2}}+7\,{c}^{3}{d}^{5}{x}^{21}+{\frac{35\,{c}^{4}{d}^{4}{x}^{20}}{4}}+7\,{c}^{5}{d}^{3}{x}^{19}+{\frac{7\,{c}^{6}{d}^{2}{x}^{18}}{2}}+{c}^{7}d{x}^{17}+{\frac{{c}^{8}{x}^{16}}{8}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.15751, size = 119, normalized size = 8.5 \begin{align*} \frac{1}{8} \, d^{8} x^{24} + c d^{7} x^{23} + \frac{7}{2} \, c^{2} d^{6} x^{22} + 7 \, c^{3} d^{5} x^{21} + \frac{35}{4} \, c^{4} d^{4} x^{20} + 7 \, c^{5} d^{3} x^{19} + \frac{7}{2} \, c^{6} d^{2} x^{18} + c^{7} d x^{17} + \frac{1}{8} \, c^{8} x^{16} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.07967, size = 198, normalized size = 14.14 \begin{align*} \frac{1}{8} x^{24} d^{8} + x^{23} d^{7} c + \frac{7}{2} x^{22} d^{6} c^{2} + 7 x^{21} d^{5} c^{3} + \frac{35}{4} x^{20} d^{4} c^{4} + 7 x^{19} d^{3} c^{5} + \frac{7}{2} x^{18} d^{2} c^{6} + x^{17} d c^{7} + \frac{1}{8} x^{16} c^{8} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 0.089397, size = 97, normalized size = 6.93 \begin{align*} \frac{c^{8} x^{16}}{8} + c^{7} d x^{17} + \frac{7 c^{6} d^{2} x^{18}}{2} + 7 c^{5} d^{3} x^{19} + \frac{35 c^{4} d^{4} x^{20}}{4} + 7 c^{3} d^{5} x^{21} + \frac{7 c^{2} d^{6} x^{22}}{2} + c d^{7} x^{23} + \frac{d^{8} x^{24}}{8} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.18284, size = 119, normalized size = 8.5 \begin{align*} \frac{1}{8} \, d^{8} x^{24} + c d^{7} x^{23} + \frac{7}{2} \, c^{2} d^{6} x^{22} + 7 \, c^{3} d^{5} x^{21} + \frac{35}{4} \, c^{4} d^{4} x^{20} + 7 \, c^{5} d^{3} x^{19} + \frac{7}{2} \, c^{6} d^{2} x^{18} + c^{7} d x^{17} + \frac{1}{8} \, c^{8} x^{16} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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