Optimal. Leaf size=73 \[ -\frac{d^2 \left (b^2-4 a c\right ) (b+2 c x)^5}{80 c^3}+\frac{d^2 \left (b^2-4 a c\right )^2 (b+2 c x)^3}{96 c^3}+\frac{d^2 (b+2 c x)^7}{224 c^3} \]
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Rubi [A] time = 0.0843479, antiderivative size = 73, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.042, Rules used = {683} \[ -\frac{d^2 \left (b^2-4 a c\right ) (b+2 c x)^5}{80 c^3}+\frac{d^2 \left (b^2-4 a c\right )^2 (b+2 c x)^3}{96 c^3}+\frac{d^2 (b+2 c x)^7}{224 c^3} \]
Antiderivative was successfully verified.
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Rule 683
Rubi steps
\begin{align*} \int (b d+2 c d x)^2 \left (a+b x+c x^2\right )^2 \, dx &=\int \left (\frac{\left (-b^2+4 a c\right )^2 (b d+2 c d x)^2}{16 c^2}+\frac{\left (-b^2+4 a c\right ) (b d+2 c d x)^4}{8 c^2 d^2}+\frac{(b d+2 c d x)^6}{16 c^2 d^4}\right ) \, dx\\ &=\frac{\left (b^2-4 a c\right )^2 d^2 (b+2 c x)^3}{96 c^3}-\frac{\left (b^2-4 a c\right ) d^2 (b+2 c x)^5}{80 c^3}+\frac{d^2 (b+2 c x)^7}{224 c^3}\\ \end{align*}
Mathematica [A] time = 0.0183955, size = 111, normalized size = 1.52 \[ d^2 \left (\frac{1}{3} x^3 \left (4 a^2 c^2+10 a b^2 c+b^4\right )+a^2 b^2 x+\frac{1}{5} c^2 x^5 \left (8 a c+13 b^2\right )+\frac{1}{2} b c x^4 \left (8 a c+3 b^2\right )+a b x^2 \left (2 a c+b^2\right )+2 b c^3 x^6+\frac{4 c^4 x^7}{7}\right ) \]
Antiderivative was successfully verified.
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Maple [B] time = 0.039, size = 176, normalized size = 2.4 \begin{align*}{\frac{4\,{c}^{4}{d}^{2}{x}^{7}}{7}}+2\,b{d}^{2}{c}^{3}{x}^{6}+{\frac{ \left ( 9\,{b}^{2}{d}^{2}{c}^{2}+4\,{c}^{2}{d}^{2} \left ( 2\,ac+{b}^{2} \right ) \right ){x}^{5}}{5}}+{\frac{ \left ( 2\,{b}^{3}{d}^{2}c+4\,b{d}^{2}c \left ( 2\,ac+{b}^{2} \right ) +8\,{c}^{2}{d}^{2}ab \right ){x}^{4}}{4}}+{\frac{ \left ({b}^{2}{d}^{2} \left ( 2\,ac+{b}^{2} \right ) +8\,{b}^{2}{d}^{2}ca+4\,{c}^{2}{d}^{2}{a}^{2} \right ){x}^{3}}{3}}+{\frac{ \left ( 4\,cb{a}^{2}{d}^{2}+2\,{b}^{3}{d}^{2}a \right ){x}^{2}}{2}}+{b}^{2}{d}^{2}{a}^{2}x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.14253, size = 171, normalized size = 2.34 \begin{align*} \frac{4}{7} \, c^{4} d^{2} x^{7} + 2 \, b c^{3} d^{2} x^{6} + \frac{1}{5} \,{\left (13 \, b^{2} c^{2} + 8 \, a c^{3}\right )} d^{2} x^{5} + a^{2} b^{2} d^{2} x + \frac{1}{2} \,{\left (3 \, b^{3} c + 8 \, a b c^{2}\right )} d^{2} x^{4} + \frac{1}{3} \,{\left (b^{4} + 10 \, a b^{2} c + 4 \, a^{2} c^{2}\right )} d^{2} x^{3} +{\left (a b^{3} + 2 \, a^{2} b c\right )} d^{2} x^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.67429, size = 315, normalized size = 4.32 \begin{align*} \frac{4}{7} x^{7} d^{2} c^{4} + 2 x^{6} d^{2} c^{3} b + \frac{13}{5} x^{5} d^{2} c^{2} b^{2} + \frac{8}{5} x^{5} d^{2} c^{3} a + \frac{3}{2} x^{4} d^{2} c b^{3} + 4 x^{4} d^{2} c^{2} b a + \frac{1}{3} x^{3} d^{2} b^{4} + \frac{10}{3} x^{3} d^{2} c b^{2} a + \frac{4}{3} x^{3} d^{2} c^{2} a^{2} + x^{2} d^{2} b^{3} a + 2 x^{2} d^{2} c b a^{2} + x d^{2} b^{2} a^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 0.099276, size = 156, normalized size = 2.14 \begin{align*} a^{2} b^{2} d^{2} x + 2 b c^{3} d^{2} x^{6} + \frac{4 c^{4} d^{2} x^{7}}{7} + x^{5} \left (\frac{8 a c^{3} d^{2}}{5} + \frac{13 b^{2} c^{2} d^{2}}{5}\right ) + x^{4} \left (4 a b c^{2} d^{2} + \frac{3 b^{3} c d^{2}}{2}\right ) + x^{3} \left (\frac{4 a^{2} c^{2} d^{2}}{3} + \frac{10 a b^{2} c d^{2}}{3} + \frac{b^{4} d^{2}}{3}\right ) + x^{2} \left (2 a^{2} b c d^{2} + a b^{3} d^{2}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.18166, size = 200, normalized size = 2.74 \begin{align*} \frac{4}{7} \, c^{4} d^{2} x^{7} + 2 \, b c^{3} d^{2} x^{6} + \frac{13}{5} \, b^{2} c^{2} d^{2} x^{5} + \frac{8}{5} \, a c^{3} d^{2} x^{5} + \frac{3}{2} \, b^{3} c d^{2} x^{4} + 4 \, a b c^{2} d^{2} x^{4} + \frac{1}{3} \, b^{4} d^{2} x^{3} + \frac{10}{3} \, a b^{2} c d^{2} x^{3} + \frac{4}{3} \, a^{2} c^{2} d^{2} x^{3} + a b^{3} d^{2} x^{2} + 2 \, a^{2} b c d^{2} x^{2} + a^{2} b^{2} d^{2} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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