Optimal. Leaf size=15 \[ \frac{1}{8} \left (b x+d x^3\right )^8 \]
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Rubi [A] time = 0.0127609, antiderivative size = 15, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.05, Rules used = {1588} \[ \frac{1}{8} \left (b x+d x^3\right )^8 \]
Antiderivative was successfully verified.
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Rule 1588
Rubi steps
\begin{align*} \int \left (b+3 d x^2\right ) \left (b x+d x^3\right )^7 \, dx &=\frac{1}{8} \left (b x+d x^3\right )^8\\ \end{align*}
Mathematica [B] time = 0.0033504, size = 98, normalized size = 6.53 \[ \frac{7}{2} b^2 d^6 x^{20}+7 b^3 d^5 x^{18}+\frac{35}{4} b^4 d^4 x^{16}+7 b^5 d^3 x^{14}+\frac{7}{2} b^6 d^2 x^{12}+b^7 d x^{10}+\frac{b^8 x^8}{8}+b d^7 x^{22}+\frac{d^8 x^{24}}{8} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.002, size = 89, normalized size = 5.9 \begin{align*}{\frac{{d}^{8}{x}^{24}}{8}}+b{d}^{7}{x}^{22}+{\frac{7\,{b}^{2}{d}^{6}{x}^{20}}{2}}+7\,{b}^{3}{d}^{5}{x}^{18}+{\frac{35\,{b}^{4}{d}^{4}{x}^{16}}{4}}+7\,{b}^{5}{d}^{3}{x}^{14}+{\frac{7\,{b}^{6}{d}^{2}{x}^{12}}{2}}+d{b}^{7}{x}^{10}+{\frac{{b}^{8}{x}^{8}}{8}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.01034, size = 18, normalized size = 1.2 \begin{align*} \frac{1}{8} \,{\left (d x^{3} + b x\right )}^{8} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.12616, size = 197, normalized size = 13.13 \begin{align*} \frac{1}{8} x^{24} d^{8} + x^{22} d^{7} b + \frac{7}{2} x^{20} d^{6} b^{2} + 7 x^{18} d^{5} b^{3} + \frac{35}{4} x^{16} d^{4} b^{4} + 7 x^{14} d^{3} b^{5} + \frac{7}{2} x^{12} d^{2} b^{6} + x^{10} d b^{7} + \frac{1}{8} x^{8} b^{8} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 0.089477, size = 97, normalized size = 6.47 \begin{align*} \frac{b^{8} x^{8}}{8} + b^{7} d x^{10} + \frac{7 b^{6} d^{2} x^{12}}{2} + 7 b^{5} d^{3} x^{14} + \frac{35 b^{4} d^{4} x^{16}}{4} + 7 b^{3} d^{5} x^{18} + \frac{7 b^{2} d^{6} x^{20}}{2} + b d^{7} x^{22} + \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 = 1.19029, size = 119, normalized size = 7.93 \begin{align*} \frac{1}{8} \, d^{8} x^{24} + b d^{7} x^{22} + \frac{7}{2} \, b^{2} d^{6} x^{20} + 7 \, b^{3} d^{5} x^{18} + \frac{35}{4} \, b^{4} d^{4} x^{16} + 7 \, b^{5} d^{3} x^{14} + \frac{7}{2} \, b^{6} d^{2} x^{12} + b^{7} d x^{10} + \frac{1}{8} \, b^{8} x^{8} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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