Optimal. Leaf size=23 \[ \frac{\left (a+b (c+d x)^4\right )^3}{12 b d} \]
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Rubi [A] time = 0.0931545, antiderivative size = 23, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095, Rules used = {372, 261} \[ \frac{\left (a+b (c+d x)^4\right )^3}{12 b d} \]
Antiderivative was successfully verified.
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Rule 372
Rule 261
Rubi steps
\begin{align*} \int (c+d x)^3 \left (a+b (c+d x)^4\right )^2 \, dx &=\frac{\operatorname{Subst}\left (\int x^3 \left (a+b x^4\right )^2 \, dx,x,c+d x\right )}{d}\\ &=\frac{\left (a+b (c+d x)^4\right )^3}{12 b d}\\ \end{align*}
Mathematica [B] time = 0.0357989, size = 172, normalized size = 7.48 \[ \frac{1}{12} x \left (6 c^2 d x+4 c^3+4 c d^2 x^2+d^3 x^3\right ) \left (3 a^2+3 a b \left (6 c^2 d^2 x^2+4 c^3 d x+2 c^4+4 c d^3 x^3+d^4 x^4\right )+b^2 \left (34 c^6 d^2 x^2+60 c^5 d^3 x^3+71 c^4 d^4 x^4+56 c^3 d^5 x^5+28 c^2 d^6 x^6+12 c^7 d x+3 c^8+8 c d^7 x^7+d^8 x^8\right )\right ) \]
Antiderivative was successfully verified.
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Maple [B] time = 0.002, size = 622, normalized size = 27. \begin{align*}{\frac{{d}^{11}{b}^{2}{x}^{12}}{12}}+c{d}^{10}{b}^{2}{x}^{11}+{\frac{11\,{c}^{2}{d}^{9}{b}^{2}{x}^{10}}{2}}+{\frac{55\,{c}^{3}{b}^{2}{d}^{8}{x}^{9}}{3}}+{\frac{ \left ( 260\,{c}^{4}{b}^{2}{d}^{7}+{d}^{3} \left ( 2\, \left ( b{c}^{4}+a \right ) b{d}^{4}+68\,{c}^{4}{d}^{4}{b}^{2} \right ) \right ){x}^{8}}{8}}+{\frac{ \left ( 196\,{c}^{5}{d}^{6}{b}^{2}+3\,c{d}^{2} \left ( 2\, \left ( b{c}^{4}+a \right ) b{d}^{4}+68\,{c}^{4}{d}^{4}{b}^{2} \right ) +{d}^{3} \left ( 8\, \left ( b{c}^{4}+a \right ) bc{d}^{3}+48\,{c}^{5}{d}^{3}{b}^{2} \right ) \right ){x}^{7}}{7}}+{\frac{ \left ( 56\,{c}^{6}{d}^{5}{b}^{2}+3\,{c}^{2}d \left ( 2\, \left ( b{c}^{4}+a \right ) b{d}^{4}+68\,{c}^{4}{d}^{4}{b}^{2} \right ) +3\,c{d}^{2} \left ( 8\, \left ( b{c}^{4}+a \right ) bc{d}^{3}+48\,{c}^{5}{d}^{3}{b}^{2} \right ) +{d}^{3} \left ( 12\, \left ( b{c}^{4}+a \right ){c}^{2}{d}^{2}b+16\,{c}^{6}{d}^{2}{b}^{2} \right ) \right ){x}^{6}}{6}}+{\frac{ \left ({c}^{3} \left ( 2\, \left ( b{c}^{4}+a \right ) b{d}^{4}+68\,{c}^{4}{d}^{4}{b}^{2} \right ) +3\,{c}^{2}d \left ( 8\, \left ( b{c}^{4}+a \right ) bc{d}^{3}+48\,{c}^{5}{d}^{3}{b}^{2} \right ) +3\,c{d}^{2} \left ( 12\, \left ( b{c}^{4}+a \right ){c}^{2}{d}^{2}b+16\,{c}^{6}{d}^{2}{b}^{2} \right ) +8\,{d}^{4} \left ( b{c}^{4}+a \right ){c}^{3}b \right ){x}^{5}}{5}}+{\frac{ \left ({c}^{3} \left ( 8\, \left ( b{c}^{4}+a \right ) bc{d}^{3}+48\,{c}^{5}{d}^{3}{b}^{2} \right ) +3\,{c}^{2}d \left ( 12\, \left ( b{c}^{4}+a \right ){c}^{2}{d}^{2}b+16\,{c}^{6}{d}^{2}{b}^{2} \right ) +24\,{c}^{4}{d}^{3} \left ( b{c}^{4}+a \right ) b+{d}^{3} \left ( b{c}^{4}+a \right ) ^{2} \right ){x}^{4}}{4}}+{\frac{ \left ({c}^{3} \left ( 12\, \left ( b{c}^{4}+a \right ){c}^{2}{d}^{2}b+16\,{c}^{6}{d}^{2}{b}^{2} \right ) +24\,{c}^{5}{d}^{2} \left ( b{c}^{4}+a \right ) b+3\,c{d}^{2} \left ( b{c}^{4}+a \right ) ^{2} \right ){x}^{3}}{3}}+{\frac{ \left ( 8\,{c}^{6} \left ( b{c}^{4}+a \right ) db+3\,{c}^{2}d \left ( b{c}^{4}+a \right ) ^{2} \right ){x}^{2}}{2}}+{c}^{3} \left ( b{c}^{4}+a \right ) ^{2}x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.979569, size = 28, normalized size = 1.22 \begin{align*} \frac{{\left ({\left (d x + c\right )}^{4} b + a\right )}^{3}}{12 \, b d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.11316, size = 639, normalized size = 27.78 \begin{align*} \frac{1}{12} x^{12} d^{11} b^{2} + x^{11} d^{10} c b^{2} + \frac{11}{2} x^{10} d^{9} c^{2} b^{2} + \frac{55}{3} x^{9} d^{8} c^{3} b^{2} + \frac{165}{4} x^{8} d^{7} c^{4} b^{2} + 66 x^{7} d^{6} c^{5} b^{2} + 77 x^{6} d^{5} c^{6} b^{2} + 66 x^{5} d^{4} c^{7} b^{2} + \frac{165}{4} x^{4} d^{3} c^{8} b^{2} + \frac{1}{4} x^{8} d^{7} b a + \frac{55}{3} x^{3} d^{2} c^{9} b^{2} + 2 x^{7} d^{6} c b a + \frac{11}{2} x^{2} d c^{10} b^{2} + 7 x^{6} d^{5} c^{2} b a + x c^{11} b^{2} + 14 x^{5} d^{4} c^{3} b a + \frac{35}{2} x^{4} d^{3} c^{4} b a + 14 x^{3} d^{2} c^{5} b a + 7 x^{2} d c^{6} b a + 2 x c^{7} b a + \frac{1}{4} x^{4} d^{3} a^{2} + x^{3} d^{2} c a^{2} + \frac{3}{2} x^{2} d c^{2} a^{2} + x c^{3} a^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 0.125388, size = 299, normalized size = 13. \begin{align*} \frac{55 b^{2} c^{3} d^{8} x^{9}}{3} + \frac{11 b^{2} c^{2} d^{9} x^{10}}{2} + b^{2} c d^{10} x^{11} + \frac{b^{2} d^{11} x^{12}}{12} + x^{8} \left (\frac{a b d^{7}}{4} + \frac{165 b^{2} c^{4} d^{7}}{4}\right ) + x^{7} \left (2 a b c d^{6} + 66 b^{2} c^{5} d^{6}\right ) + x^{6} \left (7 a b c^{2} d^{5} + 77 b^{2} c^{6} d^{5}\right ) + x^{5} \left (14 a b c^{3} d^{4} + 66 b^{2} c^{7} d^{4}\right ) + x^{4} \left (\frac{a^{2} d^{3}}{4} + \frac{35 a b c^{4} d^{3}}{2} + \frac{165 b^{2} c^{8} d^{3}}{4}\right ) + x^{3} \left (a^{2} c d^{2} + 14 a b c^{5} d^{2} + \frac{55 b^{2} c^{9} d^{2}}{3}\right ) + x^{2} \left (\frac{3 a^{2} c^{2} d}{2} + 7 a b c^{6} d + \frac{11 b^{2} c^{10} d}{2}\right ) + x \left (a^{2} c^{3} + 2 a b c^{7} + b^{2} c^{11}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.18806, size = 28, normalized size = 1.22 \begin{align*} \frac{{\left ({\left (d x + c\right )}^{4} b + a\right )}^{3}}{12 \, b d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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