Optimal. Leaf size=51 \[ \frac{a^2 (c+d x)^4}{4 d}+\frac{a b (c+d x)^6}{3 d}+\frac{b^2 (c+d x)^8}{8 d} \]
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Rubi [A] time = 0.0673907, antiderivative size = 51, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {372, 266, 43} \[ \frac{a^2 (c+d x)^4}{4 d}+\frac{a b (c+d x)^6}{3 d}+\frac{b^2 (c+d x)^8}{8 d} \]
Antiderivative was successfully verified.
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Rule 372
Rule 266
Rule 43
Rubi steps
\begin{align*} \int (c+d x)^3 \left (a+b (c+d x)^2\right )^2 \, dx &=\frac{\operatorname{Subst}\left (\int x^3 \left (a+b x^2\right )^2 \, dx,x,c+d x\right )}{d}\\ &=\frac{\operatorname{Subst}\left (\int x (a+b x)^2 \, dx,x,(c+d x)^2\right )}{2 d}\\ &=\frac{\operatorname{Subst}\left (\int \left (a^2 x+2 a b x^2+b^2 x^3\right ) \, dx,x,(c+d x)^2\right )}{2 d}\\ &=\frac{a^2 (c+d x)^4}{4 d}+\frac{a b (c+d x)^6}{3 d}+\frac{b^2 (c+d x)^8}{8 d}\\ \end{align*}
Mathematica [B] time = 0.0241508, size = 172, normalized size = 3.37 \[ \frac{1}{4} d^3 x^4 \left (a^2+20 a b c^2+35 b^2 c^4\right )+\frac{1}{3} c d^2 x^3 \left (3 a^2+20 a b c^2+21 b^2 c^4\right )+\frac{1}{2} c^2 d x^2 \left (3 a^2+10 a b c^2+7 b^2 c^4\right )+\frac{1}{6} b d^5 x^6 \left (2 a+21 b c^2\right )+b c d^4 x^5 \left (2 a+7 b c^2\right )+c^3 x \left (a+b c^2\right )^2+b^2 c d^6 x^7+\frac{1}{8} b^2 d^7 x^8 \]
Antiderivative was successfully verified.
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Maple [B] time = 0.002, size = 324, normalized size = 6.4 \begin{align*}{\frac{{d}^{7}{b}^{2}{x}^{8}}{8}}+c{d}^{6}{b}^{2}{x}^{7}+{\frac{ \left ( 15\,{c}^{2}{d}^{5}{b}^{2}+{d}^{3} \left ( 2\, \left ( b{c}^{2}+a \right ) b{d}^{2}+4\,{b}^{2}{c}^{2}{d}^{2} \right ) \right ){x}^{6}}{6}}+{\frac{ \left ( 13\,{c}^{3}{b}^{2}{d}^{4}+3\,c{d}^{2} \left ( 2\, \left ( b{c}^{2}+a \right ) b{d}^{2}+4\,{b}^{2}{c}^{2}{d}^{2} \right ) +4\,{d}^{4} \left ( b{c}^{2}+a \right ) bc \right ){x}^{5}}{5}}+{\frac{ \left ( 4\,{c}^{4}{b}^{2}{d}^{3}+3\,{c}^{2}d \left ( 2\, \left ( b{c}^{2}+a \right ) b{d}^{2}+4\,{b}^{2}{c}^{2}{d}^{2} \right ) +12\,{c}^{2}{d}^{3} \left ( b{c}^{2}+a \right ) b+{d}^{3} \left ( b{c}^{2}+a \right ) ^{2} \right ){x}^{4}}{4}}+{\frac{ \left ({c}^{3} \left ( 2\, \left ( b{c}^{2}+a \right ) b{d}^{2}+4\,{b}^{2}{c}^{2}{d}^{2} \right ) +12\,{c}^{3}{d}^{2} \left ( b{c}^{2}+a \right ) b+3\,c{d}^{2} \left ( b{c}^{2}+a \right ) ^{2} \right ){x}^{3}}{3}}+{\frac{ \left ( 4\,{c}^{4} \left ( b{c}^{2}+a \right ) bd+3\,{c}^{2}d \left ( b{c}^{2}+a \right ) ^{2} \right ){x}^{2}}{2}}+{c}^{3} \left ( b{c}^{2}+a \right ) ^{2}x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.19215, size = 238, normalized size = 4.67 \begin{align*} \frac{1}{8} \, b^{2} d^{7} x^{8} + b^{2} c d^{6} x^{7} + \frac{1}{6} \,{\left (21 \, b^{2} c^{2} + 2 \, a b\right )} d^{5} x^{6} +{\left (7 \, b^{2} c^{3} + 2 \, a b c\right )} d^{4} x^{5} + \frac{1}{4} \,{\left (35 \, b^{2} c^{4} + 20 \, a b c^{2} + a^{2}\right )} d^{3} x^{4} + \frac{1}{3} \,{\left (21 \, b^{2} c^{5} + 20 \, a b c^{3} + 3 \, a^{2} c\right )} d^{2} x^{3} + \frac{1}{2} \,{\left (7 \, b^{2} c^{6} + 10 \, a b c^{4} + 3 \, a^{2} c^{2}\right )} d x^{2} +{\left (b^{2} c^{7} + 2 \, a b c^{5} + 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 = 1.23625, size = 441, normalized size = 8.65 \begin{align*} \frac{1}{8} x^{8} d^{7} b^{2} + x^{7} d^{6} c b^{2} + \frac{7}{2} x^{6} d^{5} c^{2} b^{2} + 7 x^{5} d^{4} c^{3} b^{2} + \frac{35}{4} x^{4} d^{3} c^{4} b^{2} + \frac{1}{3} x^{6} d^{5} b a + 7 x^{3} d^{2} c^{5} b^{2} + 2 x^{5} d^{4} c b a + \frac{7}{2} x^{2} d c^{6} b^{2} + 5 x^{4} d^{3} c^{2} b a + x c^{7} b^{2} + \frac{20}{3} x^{3} d^{2} c^{3} b a + 5 x^{2} d c^{4} b a + \frac{1}{4} x^{4} d^{3} a^{2} + 2 x c^{5} b a + 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 = 0.105779, size = 209, normalized size = 4.1 \begin{align*} b^{2} c d^{6} x^{7} + \frac{b^{2} d^{7} x^{8}}{8} + x^{6} \left (\frac{a b d^{5}}{3} + \frac{7 b^{2} c^{2} d^{5}}{2}\right ) + x^{5} \left (2 a b c d^{4} + 7 b^{2} c^{3} d^{4}\right ) + x^{4} \left (\frac{a^{2} d^{3}}{4} + 5 a b c^{2} d^{3} + \frac{35 b^{2} c^{4} d^{3}}{4}\right ) + x^{3} \left (a^{2} c d^{2} + \frac{20 a b c^{3} d^{2}}{3} + 7 b^{2} c^{5} d^{2}\right ) + x^{2} \left (\frac{3 a^{2} c^{2} d}{2} + 5 a b c^{4} d + \frac{7 b^{2} c^{6} d}{2}\right ) + x \left (a^{2} c^{3} + 2 a b c^{5} + b^{2} c^{7}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.0928, size = 279, normalized size = 5.47 \begin{align*} \frac{1}{8} \, b^{2} d^{7} x^{8} + b^{2} c d^{6} x^{7} + \frac{7}{2} \, b^{2} c^{2} d^{5} x^{6} + 7 \, b^{2} c^{3} d^{4} x^{5} + \frac{35}{4} \, b^{2} c^{4} d^{3} x^{4} + \frac{1}{3} \, a b d^{5} x^{6} + 7 \, b^{2} c^{5} d^{2} x^{3} + 2 \, a b c d^{4} x^{5} + \frac{7}{2} \, b^{2} c^{6} d x^{2} + 5 \, a b c^{2} d^{3} x^{4} + b^{2} c^{7} x + \frac{20}{3} \, a b c^{3} d^{2} x^{3} + 5 \, a b c^{4} d x^{2} + \frac{1}{4} \, a^{2} d^{3} x^{4} + 2 \, a b c^{5} x + 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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