Optimal. Leaf size=18 \[ \frac{1}{8} \left (a+c x^2+d x^3\right )^8 \]
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Rubi [A] time = 0.0596462, antiderivative size = 18, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.042, Rules used = {1588} \[ \frac{1}{8} \left (a+c x^2+d x^3\right )^8 \]
Antiderivative was successfully verified.
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Rule 1588
Rubi steps
\begin{align*} \int x (2 c+3 d x) \left (a+c x^2+d x^3\right )^7 \, dx &=\frac{1}{8} \left (a+c x^2+d x^3\right )^8\\ \end{align*}
Mathematica [B] time = 0.0107206, size = 115, normalized size = 6.39 \[ \frac{1}{8} x^2 (c+d x) \left (28 a^2 x^{10} (c+d x)^5+56 a^3 x^8 (c+d x)^4+70 a^4 x^6 (c+d x)^3+56 a^5 x^4 (c+d x)^2+28 a^6 x^2 (c+d x)+8 a^7+8 a x^{12} (c+d x)^6+x^{14} (c+d x)^7\right ) \]
Antiderivative was successfully verified.
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Maple [B] time = 0.002, size = 2205, normalized size = 122.5 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 0.995793, size = 618, normalized size = 34.33 \begin{align*} \frac{1}{8} \, d^{8} x^{24} + c d^{7} x^{23} + \frac{7}{2} \, c^{2} d^{6} x^{22} +{\left (7 \, c^{3} d^{5} + a d^{7}\right )} x^{21} + \frac{7}{4} \,{\left (5 \, c^{4} d^{4} + 4 \, a c d^{6}\right )} x^{20} + 7 \,{\left (c^{5} d^{3} + 3 \, a c^{2} d^{5}\right )} x^{19} + \frac{7}{2} \,{\left (c^{6} d^{2} + 10 \, a c^{3} d^{4} + a^{2} d^{6}\right )} x^{18} +{\left (c^{7} d + 35 \, a c^{4} d^{3} + 21 \, a^{2} c d^{5}\right )} x^{17} + \frac{1}{8} \,{\left (c^{8} + 168 \, a c^{5} d^{2} + 420 \, a^{2} c^{2} d^{4}\right )} x^{16} + 7 \,{\left (a c^{6} d + 10 \, a^{2} c^{3} d^{3} + a^{3} d^{5}\right )} x^{15} + 21 \, a^{5} c^{2} d x^{7} + \frac{1}{2} \,{\left (2 \, a c^{7} + 105 \, a^{2} c^{4} d^{2} + 70 \, a^{3} c d^{4}\right )} x^{14} + 7 \,{\left (3 \, a^{2} c^{5} d + 10 \, a^{3} c^{2} d^{3}\right )} x^{13} + 7 \, a^{6} c d x^{5} + \frac{7}{4} \,{\left (2 \, a^{2} c^{6} + 40 \, a^{3} c^{3} d^{2} + 5 \, a^{4} d^{4}\right )} x^{12} + \frac{7}{2} \, a^{6} c^{2} x^{4} + 35 \,{\left (a^{3} c^{4} d + a^{4} c d^{3}\right )} x^{11} + a^{7} d x^{3} + \frac{7}{2} \,{\left (2 \, a^{3} c^{5} + 15 \, a^{4} c^{2} d^{2}\right )} x^{10} + a^{7} c x^{2} + 7 \,{\left (5 \, a^{4} c^{3} d + a^{5} d^{3}\right )} x^{9} + \frac{7}{4} \,{\left (5 \, a^{4} c^{4} + 12 \, a^{5} c d^{2}\right )} x^{8} + \frac{7}{2} \,{\left (2 \, a^{5} c^{3} + a^{6} d^{2}\right )} x^{6} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.04234, size = 1085, normalized size = 60.28 \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} + x^{21} d^{7} a + \frac{35}{4} x^{20} d^{4} c^{4} + 7 x^{20} d^{6} c a + 7 x^{19} d^{3} c^{5} + 21 x^{19} d^{5} c^{2} a + \frac{7}{2} x^{18} d^{2} c^{6} + 35 x^{18} d^{4} c^{3} a + \frac{7}{2} x^{18} d^{6} a^{2} + x^{17} d c^{7} + 35 x^{17} d^{3} c^{4} a + 21 x^{17} d^{5} c a^{2} + \frac{1}{8} x^{16} c^{8} + 21 x^{16} d^{2} c^{5} a + \frac{105}{2} x^{16} d^{4} c^{2} a^{2} + 7 x^{15} d c^{6} a + 70 x^{15} d^{3} c^{3} a^{2} + 7 x^{15} d^{5} a^{3} + x^{14} c^{7} a + \frac{105}{2} x^{14} d^{2} c^{4} a^{2} + 35 x^{14} d^{4} c a^{3} + 21 x^{13} d c^{5} a^{2} + 70 x^{13} d^{3} c^{2} a^{3} + \frac{7}{2} x^{12} c^{6} a^{2} + 70 x^{12} d^{2} c^{3} a^{3} + \frac{35}{4} x^{12} d^{4} a^{4} + 35 x^{11} d c^{4} a^{3} + 35 x^{11} d^{3} c a^{4} + 7 x^{10} c^{5} a^{3} + \frac{105}{2} x^{10} d^{2} c^{2} a^{4} + 35 x^{9} d c^{3} a^{4} + 7 x^{9} d^{3} a^{5} + \frac{35}{4} x^{8} c^{4} a^{4} + 21 x^{8} d^{2} c a^{5} + 21 x^{7} d c^{2} a^{5} + 7 x^{6} c^{3} a^{5} + \frac{7}{2} x^{6} d^{2} a^{6} + 7 x^{5} d c a^{6} + \frac{7}{2} x^{4} c^{2} a^{6} + x^{3} d a^{7} + x^{2} c a^{7} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 0.151716, size = 484, normalized size = 26.89 \begin{align*} a^{7} c x^{2} + a^{7} d x^{3} + \frac{7 a^{6} c^{2} x^{4}}{2} + 7 a^{6} c d x^{5} + 21 a^{5} c^{2} d x^{7} + \frac{7 c^{2} d^{6} x^{22}}{2} + c d^{7} x^{23} + \frac{d^{8} x^{24}}{8} + x^{21} \left (a d^{7} + 7 c^{3} d^{5}\right ) + x^{20} \left (7 a c d^{6} + \frac{35 c^{4} d^{4}}{4}\right ) + x^{19} \left (21 a c^{2} d^{5} + 7 c^{5} d^{3}\right ) + x^{18} \left (\frac{7 a^{2} d^{6}}{2} + 35 a c^{3} d^{4} + \frac{7 c^{6} d^{2}}{2}\right ) + x^{17} \left (21 a^{2} c d^{5} + 35 a c^{4} d^{3} + c^{7} d\right ) + x^{16} \left (\frac{105 a^{2} c^{2} d^{4}}{2} + 21 a c^{5} d^{2} + \frac{c^{8}}{8}\right ) + x^{15} \left (7 a^{3} d^{5} + 70 a^{2} c^{3} d^{3} + 7 a c^{6} d\right ) + x^{14} \left (35 a^{3} c d^{4} + \frac{105 a^{2} c^{4} d^{2}}{2} + a c^{7}\right ) + x^{13} \left (70 a^{3} c^{2} d^{3} + 21 a^{2} c^{5} d\right ) + x^{12} \left (\frac{35 a^{4} d^{4}}{4} + 70 a^{3} c^{3} d^{2} + \frac{7 a^{2} c^{6}}{2}\right ) + x^{11} \left (35 a^{4} c d^{3} + 35 a^{3} c^{4} d\right ) + x^{10} \left (\frac{105 a^{4} c^{2} d^{2}}{2} + 7 a^{3} c^{5}\right ) + x^{9} \left (7 a^{5} d^{3} + 35 a^{4} c^{3} d\right ) + x^{8} \left (21 a^{5} c d^{2} + \frac{35 a^{4} c^{4}}{4}\right ) + x^{6} \left (\frac{7 a^{6} d^{2}}{2} + 7 a^{5} c^{3}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.19708, size = 659, normalized size = 36.61 \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} + a d^{7} x^{21} + \frac{35}{4} \, c^{4} d^{4} x^{20} + 7 \, a c d^{6} x^{20} + 7 \, c^{5} d^{3} x^{19} + 21 \, a c^{2} d^{5} x^{19} + \frac{7}{2} \, c^{6} d^{2} x^{18} + 35 \, a c^{3} d^{4} x^{18} + \frac{7}{2} \, a^{2} d^{6} x^{18} + c^{7} d x^{17} + 35 \, a c^{4} d^{3} x^{17} + 21 \, a^{2} c d^{5} x^{17} + \frac{1}{8} \, c^{8} x^{16} + 21 \, a c^{5} d^{2} x^{16} + \frac{105}{2} \, a^{2} c^{2} d^{4} x^{16} + 7 \, a c^{6} d x^{15} + 70 \, a^{2} c^{3} d^{3} x^{15} + 7 \, a^{3} d^{5} x^{15} + a c^{7} x^{14} + \frac{105}{2} \, a^{2} c^{4} d^{2} x^{14} + 35 \, a^{3} c d^{4} x^{14} + 21 \, a^{2} c^{5} d x^{13} + 70 \, a^{3} c^{2} d^{3} x^{13} + \frac{7}{2} \, a^{2} c^{6} x^{12} + 70 \, a^{3} c^{3} d^{2} x^{12} + \frac{35}{4} \, a^{4} d^{4} x^{12} + 35 \, a^{3} c^{4} d x^{11} + 35 \, a^{4} c d^{3} x^{11} + 7 \, a^{3} c^{5} x^{10} + \frac{105}{2} \, a^{4} c^{2} d^{2} x^{10} + 35 \, a^{4} c^{3} d x^{9} + 7 \, a^{5} d^{3} x^{9} + \frac{35}{4} \, a^{4} c^{4} x^{8} + 21 \, a^{5} c d^{2} x^{8} + 21 \, a^{5} c^{2} d x^{7} + 7 \, a^{5} c^{3} x^{6} + \frac{7}{2} \, a^{6} d^{2} x^{6} + 7 \, a^{6} c d x^{5} + \frac{7}{2} \, a^{6} c^{2} x^{4} + a^{7} d x^{3} + a^{7} c x^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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