Optimal. Leaf size=51 \[ \frac{a^2 (c+d x)^4}{4 d}+\frac{2 a b (c+d x)^7}{7 d}+\frac{b^2 (c+d x)^{10}}{10 d} \]
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Rubi [A] time = 0.0632952, antiderivative size = 51, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095, Rules used = {372, 270} \[ \frac{a^2 (c+d x)^4}{4 d}+\frac{2 a b (c+d x)^7}{7 d}+\frac{b^2 (c+d x)^{10}}{10 d} \]
Antiderivative was successfully verified.
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Rule 372
Rule 270
Rubi steps
\begin{align*} \int (c+d x)^3 \left (a+b (c+d x)^3\right )^2 \, dx &=\frac{\operatorname{Subst}\left (\int x^3 \left (a+b x^3\right )^2 \, dx,x,c+d x\right )}{d}\\ &=\frac{\operatorname{Subst}\left (\int \left (a^2 x^3+2 a b x^6+b^2 x^9\right ) \, dx,x,c+d x\right )}{d}\\ &=\frac{a^2 (c+d x)^4}{4 d}+\frac{2 a b (c+d x)^7}{7 d}+\frac{b^2 (c+d x)^{10}}{10 d}\\ \end{align*}
Mathematica [B] time = 0.0254531, size = 203, normalized size = 3.98 \[ \frac{1}{4} d^3 x^4 \left (a^2+40 a b c^3+84 b^2 c^6\right )+c d^2 x^3 \left (a^2+10 a b c^3+12 b^2 c^6\right )+\frac{3}{2} c^2 d x^2 \left (a^2+4 a b c^3+3 b^2 c^6\right )+\frac{2}{7} b d^6 x^7 \left (a+42 b c^3\right )+b c d^5 x^6 \left (2 a+21 b c^3\right )+\frac{6}{5} b c^2 d^4 x^5 \left (5 a+21 b c^3\right )+c^3 x \left (a+b c^3\right )^2+\frac{9}{2} b^2 c^2 d^7 x^8+b^2 c d^8 x^9+\frac{1}{10} b^2 d^9 x^{10} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.002, size = 470, normalized size = 9.2 \begin{align*}{\frac{{d}^{9}{b}^{2}{x}^{10}}{10}}+c{d}^{8}{b}^{2}{x}^{9}+{\frac{9\,{c}^{2}{d}^{7}{b}^{2}{x}^{8}}{2}}+{\frac{ \left ( 64\,{c}^{3}{b}^{2}{d}^{6}+{d}^{3} \left ( 2\, \left ( b{c}^{3}+a \right ) b{d}^{3}+18\,{b}^{2}{c}^{3}{d}^{3} \right ) \right ){x}^{7}}{7}}+{\frac{ \left ( 51\,{c}^{4}{b}^{2}{d}^{5}+3\,c{d}^{2} \left ( 2\, \left ( b{c}^{3}+a \right ) b{d}^{3}+18\,{b}^{2}{c}^{3}{d}^{3} \right ) +{d}^{3} \left ( 6\, \left ( b{c}^{3}+a \right ) bc{d}^{2}+9\,{b}^{2}{c}^{4}{d}^{2} \right ) \right ){x}^{6}}{6}}+{\frac{ \left ( 15\,{c}^{5}{b}^{2}{d}^{4}+3\,{c}^{2}d \left ( 2\, \left ( b{c}^{3}+a \right ) b{d}^{3}+18\,{b}^{2}{c}^{3}{d}^{3} \right ) +3\,c{d}^{2} \left ( 6\, \left ( b{c}^{3}+a \right ) bc{d}^{2}+9\,{b}^{2}{c}^{4}{d}^{2} \right ) +6\,{d}^{4} \left ( b{c}^{3}+a \right ) b{c}^{2} \right ){x}^{5}}{5}}+{\frac{ \left ({c}^{3} \left ( 2\, \left ( b{c}^{3}+a \right ) b{d}^{3}+18\,{b}^{2}{c}^{3}{d}^{3} \right ) +3\,{c}^{2}d \left ( 6\, \left ( b{c}^{3}+a \right ) bc{d}^{2}+9\,{b}^{2}{c}^{4}{d}^{2} \right ) +18\,{c}^{3}{d}^{3} \left ( b{c}^{3}+a \right ) b+{d}^{3} \left ( b{c}^{3}+a \right ) ^{2} \right ){x}^{4}}{4}}+{\frac{ \left ({c}^{3} \left ( 6\, \left ( b{c}^{3}+a \right ) bc{d}^{2}+9\,{b}^{2}{c}^{4}{d}^{2} \right ) +18\,{c}^{4}{d}^{2} \left ( b{c}^{3}+a \right ) b+3\,c{d}^{2} \left ( b{c}^{3}+a \right ) ^{2} \right ){x}^{3}}{3}}+{\frac{ \left ( 6\,{c}^{5} \left ( b{c}^{3}+a \right ) bd+3\,{c}^{2}d \left ( b{c}^{3}+a \right ) ^{2} \right ){x}^{2}}{2}}+{c}^{3} \left ( b{c}^{3}+a \right ) ^{2}x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.07773, size = 284, normalized size = 5.57 \begin{align*} \frac{1}{10} \, b^{2} d^{9} x^{10} + b^{2} c d^{8} x^{9} + \frac{9}{2} \, b^{2} c^{2} d^{7} x^{8} + \frac{2}{7} \,{\left (42 \, b^{2} c^{3} + a b\right )} d^{6} x^{7} +{\left (21 \, b^{2} c^{4} + 2 \, a b c\right )} d^{5} x^{6} + \frac{6}{5} \,{\left (21 \, b^{2} c^{5} + 5 \, a b c^{2}\right )} d^{4} x^{5} + \frac{1}{4} \,{\left (84 \, b^{2} c^{6} + 40 \, a b c^{3} + a^{2}\right )} d^{3} x^{4} +{\left (12 \, b^{2} c^{7} + 10 \, a b c^{4} + a^{2} c\right )} d^{2} x^{3} + \frac{3}{2} \,{\left (3 \, b^{2} c^{8} + 4 \, a b c^{5} + a^{2} c^{2}\right )} d x^{2} +{\left (b^{2} c^{9} + 2 \, a b c^{6} + a^{2} c^{3}\right )} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.27641, size = 531, normalized size = 10.41 \begin{align*} \frac{1}{10} x^{10} d^{9} b^{2} + x^{9} d^{8} c b^{2} + \frac{9}{2} x^{8} d^{7} c^{2} b^{2} + 12 x^{7} d^{6} c^{3} b^{2} + 21 x^{6} d^{5} c^{4} b^{2} + \frac{126}{5} x^{5} d^{4} c^{5} b^{2} + 21 x^{4} d^{3} c^{6} b^{2} + \frac{2}{7} x^{7} d^{6} b a + 12 x^{3} d^{2} c^{7} b^{2} + 2 x^{6} d^{5} c b a + \frac{9}{2} x^{2} d c^{8} b^{2} + 6 x^{5} d^{4} c^{2} b a + x c^{9} b^{2} + 10 x^{4} d^{3} c^{3} b a + 10 x^{3} d^{2} c^{4} b a + 6 x^{2} d c^{5} b a + 2 x c^{6} 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 = 0.116357, size = 252, normalized size = 4.94 \begin{align*} \frac{9 b^{2} c^{2} d^{7} x^{8}}{2} + b^{2} c d^{8} x^{9} + \frac{b^{2} d^{9} x^{10}}{10} + x^{7} \left (\frac{2 a b d^{6}}{7} + 12 b^{2} c^{3} d^{6}\right ) + x^{6} \left (2 a b c d^{5} + 21 b^{2} c^{4} d^{5}\right ) + x^{5} \left (6 a b c^{2} d^{4} + \frac{126 b^{2} c^{5} d^{4}}{5}\right ) + x^{4} \left (\frac{a^{2} d^{3}}{4} + 10 a b c^{3} d^{3} + 21 b^{2} c^{6} d^{3}\right ) + x^{3} \left (a^{2} c d^{2} + 10 a b c^{4} d^{2} + 12 b^{2} c^{7} d^{2}\right ) + x^{2} \left (\frac{3 a^{2} c^{2} d}{2} + 6 a b c^{5} d + \frac{9 b^{2} c^{8} d}{2}\right ) + x \left (a^{2} c^{3} + 2 a b c^{6} + b^{2} c^{9}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.0942, size = 335, normalized size = 6.57 \begin{align*} \frac{1}{10} \, b^{2} d^{9} x^{10} + b^{2} c d^{8} x^{9} + \frac{9}{2} \, b^{2} c^{2} d^{7} x^{8} + 12 \, b^{2} c^{3} d^{6} x^{7} + 21 \, b^{2} c^{4} d^{5} x^{6} + \frac{126}{5} \, b^{2} c^{5} d^{4} x^{5} + 21 \, b^{2} c^{6} d^{3} x^{4} + \frac{2}{7} \, a b d^{6} x^{7} + 12 \, b^{2} c^{7} d^{2} x^{3} + 2 \, a b c d^{5} x^{6} + \frac{9}{2} \, b^{2} c^{8} d x^{2} + 6 \, a b c^{2} d^{4} x^{5} + b^{2} c^{9} x + 10 \, a b c^{3} d^{3} x^{4} + 10 \, a b c^{4} d^{2} x^{3} + 6 \, a b c^{5} d x^{2} + 2 \, a b c^{6} x + \frac{1}{4} \, a^{2} d^{3} x^{4} + a^{2} c d^{2} x^{3} + \frac{3}{2} \, a^{2} c^{2} d x^{2} + a^{2} c^{3} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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