Optimal. Leaf size=73 \[ -\frac{d^4 \left (b^2-4 a c\right ) (b+2 c x)^7}{112 c^3}+\frac{d^4 \left (b^2-4 a c\right )^2 (b+2 c x)^5}{160 c^3}+\frac{d^4 (b+2 c x)^9}{288 c^3} \]
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Rubi [A] time = 0.124237, 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} \[ -\frac{d^4 \left (b^2-4 a c\right ) (b+2 c x)^7}{112 c^3}+\frac{d^4 \left (b^2-4 a c\right )^2 (b+2 c x)^5}{160 c^3}+\frac{d^4 (b+2 c x)^9}{288 c^3} \]
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 )^2 \, dx &=\int \left (\frac{\left (-b^2+4 a c\right )^2 (b d+2 c d x)^4}{16 c^2}+\frac{\left (-b^2+4 a c\right ) (b d+2 c d x)^6}{8 c^2 d^2}+\frac{(b d+2 c d x)^8}{16 c^2 d^4}\right ) \, dx\\ &=\frac{\left (b^2-4 a c\right )^2 d^4 (b+2 c x)^5}{160 c^3}-\frac{\left (b^2-4 a c\right ) d^4 (b+2 c x)^7}{112 c^3}+\frac{d^4 (b+2 c x)^9}{288 c^3}\\ \end{align*}
Mathematica [B] time = 0.026301, size = 179, normalized size = 2.45 \[ d^4 \left (\frac{1}{5} c^2 x^5 \left (16 a^2 c^2+112 a b^2 c+41 b^4\right )+\frac{1}{2} b c x^4 \left (16 a^2 c^2+32 a b^2 c+5 b^4\right )+\frac{1}{3} b^2 x^3 \left (24 a^2 c^2+18 a b^2 c+b^4\right )+a^2 b^4 x+\frac{8}{7} c^4 x^7 \left (4 a c+13 b^2\right )+\frac{4}{3} b c^3 x^6 \left (12 a c+11 b^2\right )+a b^3 x^2 \left (4 a c+b^2\right )+8 b c^5 x^8+\frac{16 c^6 x^9}{9}\right ) \]
Antiderivative was successfully verified.
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Maple [B] time = 0.039, size = 300, normalized size = 4.1 \begin{align*}{\frac{16\,{c}^{6}{d}^{4}{x}^{9}}{9}}+8\,b{d}^{4}{c}^{5}{x}^{8}+{\frac{ \left ( 88\,{b}^{2}{d}^{4}{c}^{4}+16\,{c}^{4}{d}^{4} \left ( 2\,ac+{b}^{2} \right ) \right ){x}^{7}}{7}}+{\frac{ \left ( 56\,{b}^{3}{d}^{4}{c}^{3}+32\,b{d}^{4}{c}^{3} \left ( 2\,ac+{b}^{2} \right ) +32\,{c}^{4}{d}^{4}ab \right ){x}^{6}}{6}}+{\frac{ \left ( 17\,{b}^{4}{d}^{4}{c}^{2}+24\,{b}^{2}{d}^{4}{c}^{2} \left ( 2\,ac+{b}^{2} \right ) +64\,{b}^{2}{d}^{4}{c}^{3}a+16\,{c}^{4}{d}^{4}{a}^{2} \right ){x}^{5}}{5}}+{\frac{ \left ( 2\,{b}^{5}{d}^{4}c+8\,{b}^{3}{d}^{4}c \left ( 2\,ac+{b}^{2} \right ) +48\,{b}^{3}{d}^{4}{c}^{2}a+32\,b{d}^{4}{c}^{3}{a}^{2} \right ){x}^{4}}{4}}+{\frac{ \left ({b}^{4}{d}^{4} \left ( 2\,ac+{b}^{2} \right ) +16\,{b}^{4}{d}^{4}ca+24\,{b}^{2}{d}^{4}{c}^{2}{a}^{2} \right ){x}^{3}}{3}}+{\frac{ \left ( 8\,{b}^{3}{d}^{4}c{a}^{2}+2\,{b}^{5}{d}^{4}a \right ){x}^{2}}{2}}+{b}^{4}{d}^{4}{a}^{2}x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.20309, size = 271, normalized size = 3.71 \begin{align*} \frac{16}{9} \, c^{6} d^{4} x^{9} + 8 \, b c^{5} d^{4} x^{8} + \frac{8}{7} \,{\left (13 \, b^{2} c^{4} + 4 \, a c^{5}\right )} d^{4} x^{7} + a^{2} b^{4} d^{4} x + \frac{4}{3} \,{\left (11 \, b^{3} c^{3} + 12 \, a b c^{4}\right )} d^{4} x^{6} + \frac{1}{5} \,{\left (41 \, b^{4} c^{2} + 112 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}\right )} d^{4} x^{5} + \frac{1}{2} \,{\left (5 \, b^{5} c + 32 \, a b^{3} c^{2} + 16 \, a^{2} b c^{3}\right )} d^{4} x^{4} + \frac{1}{3} \,{\left (b^{6} + 18 \, a b^{4} c + 24 \, a^{2} b^{2} c^{2}\right )} d^{4} x^{3} +{\left (a b^{5} + 4 \, a^{2} 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 = 1.93283, size = 510, normalized size = 6.99 \begin{align*} \frac{16}{9} x^{9} d^{4} c^{6} + 8 x^{8} d^{4} c^{5} b + \frac{104}{7} x^{7} d^{4} c^{4} b^{2} + \frac{32}{7} x^{7} d^{4} c^{5} a + \frac{44}{3} x^{6} d^{4} c^{3} b^{3} + 16 x^{6} d^{4} c^{4} b a + \frac{41}{5} x^{5} d^{4} c^{2} b^{4} + \frac{112}{5} x^{5} d^{4} c^{3} b^{2} a + \frac{16}{5} x^{5} d^{4} c^{4} a^{2} + \frac{5}{2} x^{4} d^{4} c b^{5} + 16 x^{4} d^{4} c^{2} b^{3} a + 8 x^{4} d^{4} c^{3} b a^{2} + \frac{1}{3} x^{3} d^{4} b^{6} + 6 x^{3} d^{4} c b^{4} a + 8 x^{3} d^{4} c^{2} b^{2} a^{2} + x^{2} d^{4} b^{5} a + 4 x^{2} d^{4} c b^{3} a^{2} + x d^{4} b^{4} a^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 0.105083, size = 248, normalized size = 3.4 \begin{align*} a^{2} b^{4} d^{4} x + 8 b c^{5} d^{4} x^{8} + \frac{16 c^{6} d^{4} x^{9}}{9} + x^{7} \left (\frac{32 a c^{5} d^{4}}{7} + \frac{104 b^{2} c^{4} d^{4}}{7}\right ) + x^{6} \left (16 a b c^{4} d^{4} + \frac{44 b^{3} c^{3} d^{4}}{3}\right ) + x^{5} \left (\frac{16 a^{2} c^{4} d^{4}}{5} + \frac{112 a b^{2} c^{3} d^{4}}{5} + \frac{41 b^{4} c^{2} d^{4}}{5}\right ) + x^{4} \left (8 a^{2} b c^{3} d^{4} + 16 a b^{3} c^{2} d^{4} + \frac{5 b^{5} c d^{4}}{2}\right ) + x^{3} \left (8 a^{2} b^{2} c^{2} d^{4} + 6 a b^{4} c d^{4} + \frac{b^{6} d^{4}}{3}\right ) + x^{2} \left (4 a^{2} b^{3} c d^{4} + a b^{5} d^{4}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.27151, size = 324, normalized size = 4.44 \begin{align*} \frac{16}{9} \, c^{6} d^{4} x^{9} + 8 \, b c^{5} d^{4} x^{8} + \frac{104}{7} \, b^{2} c^{4} d^{4} x^{7} + \frac{32}{7} \, a c^{5} d^{4} x^{7} + \frac{44}{3} \, b^{3} c^{3} d^{4} x^{6} + 16 \, a b c^{4} d^{4} x^{6} + \frac{41}{5} \, b^{4} c^{2} d^{4} x^{5} + \frac{112}{5} \, a b^{2} c^{3} d^{4} x^{5} + \frac{16}{5} \, a^{2} c^{4} d^{4} x^{5} + \frac{5}{2} \, b^{5} c d^{4} x^{4} + 16 \, a b^{3} c^{2} d^{4} x^{4} + 8 \, a^{2} b c^{3} d^{4} x^{4} + \frac{1}{3} \, b^{6} d^{4} x^{3} + 6 \, a b^{4} c d^{4} x^{3} + 8 \, a^{2} b^{2} c^{2} d^{4} x^{3} + a b^{5} d^{4} x^{2} + 4 \, a^{2} b^{3} c d^{4} x^{2} + a^{2} b^{4} d^{4} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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