Optimal. Leaf size=101 \[ -\frac{d^4 \left (b^2-4 a c\right ) (b+2 c x)^9}{384 c^4}+\frac{3 d^4 \left (b^2-4 a c\right )^2 (b+2 c x)^7}{896 c^4}-\frac{d^4 \left (b^2-4 a c\right )^3 (b+2 c x)^5}{640 c^4}+\frac{d^4 (b+2 c x)^{11}}{1408 c^4} \]
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Rubi [A] time = 0.191634, antiderivative size = 101, 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} \[ -\frac{d^4 \left (b^2-4 a c\right ) (b+2 c x)^9}{384 c^4}+\frac{3 d^4 \left (b^2-4 a c\right )^2 (b+2 c x)^7}{896 c^4}-\frac{d^4 \left (b^2-4 a c\right )^3 (b+2 c x)^5}{640 c^4}+\frac{d^4 (b+2 c x)^{11}}{1408 c^4} \]
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 )^3 \, dx &=\int \left (\frac{\left (-b^2+4 a c\right )^3 (b d+2 c d x)^4}{64 c^3}+\frac{3 \left (-b^2+4 a c\right )^2 (b d+2 c d x)^6}{64 c^3 d^2}+\frac{3 \left (-b^2+4 a c\right ) (b d+2 c d x)^8}{64 c^3 d^4}+\frac{(b d+2 c d x)^{10}}{64 c^3 d^6}\right ) \, dx\\ &=-\frac{\left (b^2-4 a c\right )^3 d^4 (b+2 c x)^5}{640 c^4}+\frac{3 \left (b^2-4 a c\right )^2 d^4 (b+2 c x)^7}{896 c^4}-\frac{\left (b^2-4 a c\right ) d^4 (b+2 c x)^9}{384 c^4}+\frac{d^4 (b+2 c x)^{11}}{1408 c^4}\\ \end{align*}
Mathematica [B] time = 0.042036, size = 259, normalized size = 2.56 \[ d^4 \left (\frac{3}{7} c^3 x^7 \left (16 a^2 c^2+104 a b^2 c+43 b^4\right )+\frac{1}{2} b c^2 x^6 \left (48 a^2 c^2+88 a b^2 c+17 b^4\right )+\frac{1}{5} c x^5 \left (168 a^2 b^2 c^2+16 a^3 c^3+123 a b^4 c+11 b^6\right )+\frac{1}{4} b x^4 \left (96 a^2 b^2 c^2+32 a^3 c^3+30 a b^4 c+b^6\right )+a b^2 x^3 \left (8 a^2 c^2+9 a b^2 c+b^4\right )+\frac{1}{2} a^2 b^3 x^2 \left (8 a c+3 b^2\right )+a^3 b^4 x+\frac{8}{3} c^5 x^9 \left (2 a c+7 b^2\right )+24 b c^4 x^8 \left (a c+b^2\right )+8 b c^6 x^{10}+\frac{16 c^7 x^{11}}{11}\right ) \]
Antiderivative was successfully verified.
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Maple [B] time = 0.038, size = 672, normalized size = 6.7 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.16763, size = 392, normalized size = 3.88 \begin{align*} \frac{16}{11} \, c^{7} d^{4} x^{11} + 8 \, b c^{6} d^{4} x^{10} + \frac{8}{3} \,{\left (7 \, b^{2} c^{5} + 2 \, a c^{6}\right )} d^{4} x^{9} + 24 \,{\left (b^{3} c^{4} + a b c^{5}\right )} d^{4} x^{8} + a^{3} b^{4} d^{4} x + \frac{3}{7} \,{\left (43 \, b^{4} c^{3} + 104 \, a b^{2} c^{4} + 16 \, a^{2} c^{5}\right )} d^{4} x^{7} + \frac{1}{2} \,{\left (17 \, b^{5} c^{2} + 88 \, a b^{3} c^{3} + 48 \, a^{2} b c^{4}\right )} d^{4} x^{6} + \frac{1}{5} \,{\left (11 \, b^{6} c + 123 \, a b^{4} c^{2} + 168 \, a^{2} b^{2} c^{3} + 16 \, a^{3} c^{4}\right )} d^{4} x^{5} + \frac{1}{4} \,{\left (b^{7} + 30 \, a b^{5} c + 96 \, a^{2} b^{3} c^{2} + 32 \, a^{3} b c^{3}\right )} d^{4} x^{4} +{\left (a b^{6} + 9 \, a^{2} b^{4} c + 8 \, a^{3} b^{2} c^{2}\right )} d^{4} x^{3} + \frac{1}{2} \,{\left (3 \, a^{2} b^{5} + 8 \, a^{3} 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 = 1.70301, size = 776, normalized size = 7.68 \begin{align*} \frac{16}{11} x^{11} d^{4} c^{7} + 8 x^{10} d^{4} c^{6} b + \frac{56}{3} x^{9} d^{4} c^{5} b^{2} + \frac{16}{3} x^{9} d^{4} c^{6} a + 24 x^{8} d^{4} c^{4} b^{3} + 24 x^{8} d^{4} c^{5} b a + \frac{129}{7} x^{7} d^{4} c^{3} b^{4} + \frac{312}{7} x^{7} d^{4} c^{4} b^{2} a + \frac{48}{7} x^{7} d^{4} c^{5} a^{2} + \frac{17}{2} x^{6} d^{4} c^{2} b^{5} + 44 x^{6} d^{4} c^{3} b^{3} a + 24 x^{6} d^{4} c^{4} b a^{2} + \frac{11}{5} x^{5} d^{4} c b^{6} + \frac{123}{5} x^{5} d^{4} c^{2} b^{4} a + \frac{168}{5} x^{5} d^{4} c^{3} b^{2} a^{2} + \frac{16}{5} x^{5} d^{4} c^{4} a^{3} + \frac{1}{4} x^{4} d^{4} b^{7} + \frac{15}{2} x^{4} d^{4} c b^{5} a + 24 x^{4} d^{4} c^{2} b^{3} a^{2} + 8 x^{4} d^{4} c^{3} b a^{3} + x^{3} d^{4} b^{6} a + 9 x^{3} d^{4} c b^{4} a^{2} + 8 x^{3} d^{4} c^{2} b^{2} a^{3} + \frac{3}{2} x^{2} d^{4} b^{5} a^{2} + 4 x^{2} d^{4} c b^{3} a^{3} + x d^{4} b^{4} a^{3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 0.126409, size = 371, normalized size = 3.67 \begin{align*} a^{3} b^{4} d^{4} x + 8 b c^{6} d^{4} x^{10} + \frac{16 c^{7} d^{4} x^{11}}{11} + x^{9} \left (\frac{16 a c^{6} d^{4}}{3} + \frac{56 b^{2} c^{5} d^{4}}{3}\right ) + x^{8} \left (24 a b c^{5} d^{4} + 24 b^{3} c^{4} d^{4}\right ) + x^{7} \left (\frac{48 a^{2} c^{5} d^{4}}{7} + \frac{312 a b^{2} c^{4} d^{4}}{7} + \frac{129 b^{4} c^{3} d^{4}}{7}\right ) + x^{6} \left (24 a^{2} b c^{4} d^{4} + 44 a b^{3} c^{3} d^{4} + \frac{17 b^{5} c^{2} d^{4}}{2}\right ) + x^{5} \left (\frac{16 a^{3} c^{4} d^{4}}{5} + \frac{168 a^{2} b^{2} c^{3} d^{4}}{5} + \frac{123 a b^{4} c^{2} d^{4}}{5} + \frac{11 b^{6} c d^{4}}{5}\right ) + x^{4} \left (8 a^{3} b c^{3} d^{4} + 24 a^{2} b^{3} c^{2} d^{4} + \frac{15 a b^{5} c d^{4}}{2} + \frac{b^{7} d^{4}}{4}\right ) + x^{3} \left (8 a^{3} b^{2} c^{2} d^{4} + 9 a^{2} b^{4} c d^{4} + a b^{6} d^{4}\right ) + x^{2} \left (4 a^{3} b^{3} c d^{4} + \frac{3 a^{2} 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 = 1.28277, size = 487, normalized size = 4.82 \begin{align*} \frac{16}{11} \, c^{7} d^{4} x^{11} + 8 \, b c^{6} d^{4} x^{10} + \frac{56}{3} \, b^{2} c^{5} d^{4} x^{9} + \frac{16}{3} \, a c^{6} d^{4} x^{9} + 24 \, b^{3} c^{4} d^{4} x^{8} + 24 \, a b c^{5} d^{4} x^{8} + \frac{129}{7} \, b^{4} c^{3} d^{4} x^{7} + \frac{312}{7} \, a b^{2} c^{4} d^{4} x^{7} + \frac{48}{7} \, a^{2} c^{5} d^{4} x^{7} + \frac{17}{2} \, b^{5} c^{2} d^{4} x^{6} + 44 \, a b^{3} c^{3} d^{4} x^{6} + 24 \, a^{2} b c^{4} d^{4} x^{6} + \frac{11}{5} \, b^{6} c d^{4} x^{5} + \frac{123}{5} \, a b^{4} c^{2} d^{4} x^{5} + \frac{168}{5} \, a^{2} b^{2} c^{3} d^{4} x^{5} + \frac{16}{5} \, a^{3} c^{4} d^{4} x^{5} + \frac{1}{4} \, b^{7} d^{4} x^{4} + \frac{15}{2} \, a b^{5} c d^{4} x^{4} + 24 \, a^{2} b^{3} c^{2} d^{4} x^{4} + 8 \, a^{3} b c^{3} d^{4} x^{4} + a b^{6} d^{4} x^{3} + 9 \, a^{2} b^{4} c d^{4} x^{3} + 8 \, a^{3} b^{2} c^{2} d^{4} x^{3} + \frac{3}{2} \, a^{2} b^{5} d^{4} x^{2} + 4 \, a^{3} b^{3} c d^{4} x^{2} + a^{3} b^{4} d^{4} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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