Optimal. Leaf size=45 \[ \frac{d^4 (b+2 c x)^7}{56 c^2}-\frac{d^4 \left (b^2-4 a c\right ) (b+2 c x)^5}{40 c^2} \]
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Rubi [A] time = 0.0751143, antiderivative size = 45, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.045, Rules used = {683} \[ \frac{d^4 (b+2 c x)^7}{56 c^2}-\frac{d^4 \left (b^2-4 a c\right ) (b+2 c x)^5}{40 c^2} \]
Antiderivative was successfully verified.
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Rule 683
Rubi steps
\begin{align*} \int (b d+2 c d x)^4 \left (a+b x+c x^2\right ) \, dx &=\int \left (\frac{\left (-b^2+4 a c\right ) (b d+2 c d x)^4}{4 c}+\frac{(b d+2 c d x)^6}{4 c d^2}\right ) \, dx\\ &=-\frac{\left (b^2-4 a c\right ) d^4 (b+2 c x)^5}{40 c^2}+\frac{d^4 (b+2 c x)^7}{56 c^2}\\ \end{align*}
Mathematica [B] time = 0.0171741, size = 102, normalized size = 2.27 \[ d^4 \left (\frac{8}{5} c^3 x^5 \left (2 a c+7 b^2\right )+8 b c^2 x^4 \left (a c+b^2\right )+b^2 c x^3 \left (8 a c+3 b^2\right )+\frac{1}{2} b^3 x^2 \left (8 a c+b^2\right )+a b^4 x+8 b c^4 x^6+\frac{16 c^5 x^7}{7}\right ) \]
Antiderivative was successfully verified.
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Maple [B] time = 0.04, size = 137, normalized size = 3. \begin{align*}{\frac{16\,{c}^{5}{d}^{4}{x}^{7}}{7}}+8\,b{d}^{4}{c}^{4}{x}^{6}+{\frac{ \left ( 16\,{c}^{4}{d}^{4}a+56\,{b}^{2}{d}^{4}{c}^{3} \right ){x}^{5}}{5}}+{\frac{ \left ( 32\,b{d}^{4}{c}^{3}a+32\,{b}^{3}{d}^{4}{c}^{2} \right ){x}^{4}}{4}}+{\frac{ \left ( 24\,{b}^{2}{d}^{4}{c}^{2}a+9\,{b}^{4}{d}^{4}c \right ){x}^{3}}{3}}+{\frac{ \left ( 8\,{b}^{3}{d}^{4}ca+{b}^{5}{d}^{4} \right ){x}^{2}}{2}}+{b}^{4}{d}^{4}ax \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.22842, size = 162, normalized size = 3.6 \begin{align*} \frac{16}{7} \, c^{5} d^{4} x^{7} + 8 \, b c^{4} d^{4} x^{6} + a b^{4} d^{4} x + \frac{8}{5} \,{\left (7 \, b^{2} c^{3} + 2 \, a c^{4}\right )} d^{4} x^{5} + 8 \,{\left (b^{3} c^{2} + a b c^{3}\right )} d^{4} x^{4} +{\left (3 \, b^{4} c + 8 \, a b^{2} c^{2}\right )} d^{4} x^{3} + \frac{1}{2} \,{\left (b^{5} + 8 \, a b^{3} c\right )} d^{4} x^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.02305, size = 286, normalized size = 6.36 \begin{align*} \frac{16}{7} x^{7} d^{4} c^{5} + 8 x^{6} d^{4} c^{4} b + \frac{56}{5} x^{5} d^{4} c^{3} b^{2} + \frac{16}{5} x^{5} d^{4} c^{4} a + 8 x^{4} d^{4} c^{2} b^{3} + 8 x^{4} d^{4} c^{3} b a + 3 x^{3} d^{4} c b^{4} + 8 x^{3} d^{4} c^{2} b^{2} a + \frac{1}{2} x^{2} d^{4} b^{5} + 4 x^{2} d^{4} c b^{3} a + x d^{4} b^{4} a \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 0.123086, size = 143, normalized size = 3.18 \begin{align*} a b^{4} d^{4} x + 8 b c^{4} d^{4} x^{6} + \frac{16 c^{5} d^{4} x^{7}}{7} + x^{5} \left (\frac{16 a c^{4} d^{4}}{5} + \frac{56 b^{2} c^{3} d^{4}}{5}\right ) + x^{4} \left (8 a b c^{3} d^{4} + 8 b^{3} c^{2} d^{4}\right ) + x^{3} \left (8 a b^{2} c^{2} d^{4} + 3 b^{4} c d^{4}\right ) + x^{2} \left (4 a b^{3} c d^{4} + \frac{b^{5} d^{4}}{2}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.21602, size = 185, normalized size = 4.11 \begin{align*} \frac{16}{7} \, c^{5} d^{4} x^{7} + 8 \, b c^{4} d^{4} x^{6} + \frac{56}{5} \, b^{2} c^{3} d^{4} x^{5} + \frac{16}{5} \, a c^{4} d^{4} x^{5} + 8 \, b^{3} c^{2} d^{4} x^{4} + 8 \, a b c^{3} d^{4} x^{4} + 3 \, b^{4} c d^{4} x^{3} + 8 \, a b^{2} c^{2} d^{4} x^{3} + \frac{1}{2} \, b^{5} d^{4} x^{2} + 4 \, a b^{3} c d^{4} x^{2} + a b^{4} d^{4} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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