Optimal. Leaf size=101 \[ -\frac{3 d^5 \left (b^2-4 a c\right ) (b+2 c x)^{10}}{1280 c^4}+\frac{3 d^5 \left (b^2-4 a c\right )^2 (b+2 c x)^8}{1024 c^4}-\frac{d^5 \left (b^2-4 a c\right )^3 (b+2 c x)^6}{768 c^4}+\frac{d^5 (b+2 c x)^{12}}{1536 c^4} \]
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Rubi [A] time = 0.244876, 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} \[ -\frac{3 d^5 \left (b^2-4 a c\right ) (b+2 c x)^{10}}{1280 c^4}+\frac{3 d^5 \left (b^2-4 a c\right )^2 (b+2 c x)^8}{1024 c^4}-\frac{d^5 \left (b^2-4 a c\right )^3 (b+2 c x)^6}{768 c^4}+\frac{d^5 (b+2 c x)^{12}}{1536 c^4} \]
Antiderivative was successfully verified.
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Rule 683
Rubi steps
\begin{align*} \int (b d+2 c d x)^5 \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)^5}{64 c^3}+\frac{3 \left (-b^2+4 a c\right )^2 (b d+2 c d x)^7}{64 c^3 d^2}+\frac{3 \left (-b^2+4 a c\right ) (b d+2 c d x)^9}{64 c^3 d^4}+\frac{(b d+2 c d x)^{11}}{64 c^3 d^6}\right ) \, dx\\ &=-\frac{\left (b^2-4 a c\right )^3 d^5 (b+2 c x)^6}{768 c^4}+\frac{3 \left (b^2-4 a c\right )^2 d^5 (b+2 c x)^8}{1024 c^4}-\frac{3 \left (b^2-4 a c\right ) d^5 (b+2 c x)^{10}}{1280 c^4}+\frac{d^5 (b+2 c x)^{12}}{1536 c^4}\\ \end{align*}
Mathematica [B] time = 0.084249, size = 224, normalized size = 2.22 \[ \frac{1}{60} d^5 x (b+c x) \left (20 a^3 \left (28 b^2 c^2 x^2+12 b^3 c x+3 b^4+32 b c^3 x^3+16 c^4 x^4\right )+30 a^2 x \left (56 b^3 c^2 x^2+88 b^2 c^3 x^3+19 b^4 c x+3 b^5+72 b c^4 x^4+24 c^5 x^5\right )+12 a x^2 (b+c x)^2 \left (78 b^2 c^2 x^2+30 b^3 c x+5 b^4+96 b c^3 x^3+48 c^4 x^4\right )+x^3 (b+c x)^3 \left (256 b^2 c^2 x^2+96 b^3 c x+15 b^4+320 b c^3 x^3+160 c^4 x^4\right )\right ) \]
Antiderivative was successfully verified.
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Maple [B] time = 0.041, size = 810, normalized size = 8. \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 = 1.1377, size = 462, normalized size = 4.57 \begin{align*} \frac{8}{3} \, c^{8} d^{5} x^{12} + 16 \, b c^{7} d^{5} x^{11} + \frac{16}{5} \,{\left (13 \, b^{2} c^{6} + 3 \, a c^{7}\right )} d^{5} x^{10} + \frac{8}{3} \,{\left (23 \, b^{3} c^{5} + 18 \, a b c^{6}\right )} d^{5} x^{9} + a^{3} b^{5} d^{5} x + \frac{3}{4} \,{\left (75 \, b^{4} c^{4} + 136 \, a b^{2} c^{5} + 16 \, a^{2} c^{6}\right )} d^{5} x^{8} + 3 \,{\left (11 \, b^{5} c^{3} + 40 \, a b^{3} c^{4} + 16 \, a^{2} b c^{5}\right )} d^{5} x^{7} + \frac{1}{6} \,{\left (73 \, b^{6} c^{2} + 510 \, a b^{4} c^{3} + 480 \, a^{2} b^{2} c^{4} + 32 \, a^{3} c^{5}\right )} d^{5} x^{6} + \frac{1}{5} \,{\left (13 \, b^{7} c + 183 \, a b^{5} c^{2} + 360 \, a^{2} b^{3} c^{3} + 80 \, a^{3} b c^{4}\right )} d^{5} x^{5} + \frac{1}{4} \,{\left (b^{8} + 36 \, a b^{6} c + 150 \, a^{2} b^{4} c^{2} + 80 \, a^{3} b^{2} c^{3}\right )} d^{5} x^{4} + \frac{1}{3} \,{\left (3 \, a b^{7} + 33 \, a^{2} b^{5} c + 40 \, a^{3} b^{3} c^{2}\right )} d^{5} x^{3} + \frac{1}{2} \,{\left (3 \, a^{2} b^{6} + 10 \, a^{3} b^{4} c\right )} d^{5} x^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.72415, size = 907, normalized size = 8.98 \begin{align*} \frac{8}{3} x^{12} d^{5} c^{8} + 16 x^{11} d^{5} c^{7} b + \frac{208}{5} x^{10} d^{5} c^{6} b^{2} + \frac{48}{5} x^{10} d^{5} c^{7} a + \frac{184}{3} x^{9} d^{5} c^{5} b^{3} + 48 x^{9} d^{5} c^{6} b a + \frac{225}{4} x^{8} d^{5} c^{4} b^{4} + 102 x^{8} d^{5} c^{5} b^{2} a + 12 x^{8} d^{5} c^{6} a^{2} + 33 x^{7} d^{5} c^{3} b^{5} + 120 x^{7} d^{5} c^{4} b^{3} a + 48 x^{7} d^{5} c^{5} b a^{2} + \frac{73}{6} x^{6} d^{5} c^{2} b^{6} + 85 x^{6} d^{5} c^{3} b^{4} a + 80 x^{6} d^{5} c^{4} b^{2} a^{2} + \frac{16}{3} x^{6} d^{5} c^{5} a^{3} + \frac{13}{5} x^{5} d^{5} c b^{7} + \frac{183}{5} x^{5} d^{5} c^{2} b^{5} a + 72 x^{5} d^{5} c^{3} b^{3} a^{2} + 16 x^{5} d^{5} c^{4} b a^{3} + \frac{1}{4} x^{4} d^{5} b^{8} + 9 x^{4} d^{5} c b^{6} a + \frac{75}{2} x^{4} d^{5} c^{2} b^{4} a^{2} + 20 x^{4} d^{5} c^{3} b^{2} a^{3} + x^{3} d^{5} b^{7} a + 11 x^{3} d^{5} c b^{5} a^{2} + \frac{40}{3} x^{3} d^{5} c^{2} b^{3} a^{3} + \frac{3}{2} x^{2} d^{5} b^{6} a^{2} + 5 x^{2} d^{5} c b^{4} a^{3} + x d^{5} b^{5} a^{3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 0.127715, size = 428, normalized size = 4.24 \begin{align*} a^{3} b^{5} d^{5} x + 16 b c^{7} d^{5} x^{11} + \frac{8 c^{8} d^{5} x^{12}}{3} + x^{10} \left (\frac{48 a c^{7} d^{5}}{5} + \frac{208 b^{2} c^{6} d^{5}}{5}\right ) + x^{9} \left (48 a b c^{6} d^{5} + \frac{184 b^{3} c^{5} d^{5}}{3}\right ) + x^{8} \left (12 a^{2} c^{6} d^{5} + 102 a b^{2} c^{5} d^{5} + \frac{225 b^{4} c^{4} d^{5}}{4}\right ) + x^{7} \left (48 a^{2} b c^{5} d^{5} + 120 a b^{3} c^{4} d^{5} + 33 b^{5} c^{3} d^{5}\right ) + x^{6} \left (\frac{16 a^{3} c^{5} d^{5}}{3} + 80 a^{2} b^{2} c^{4} d^{5} + 85 a b^{4} c^{3} d^{5} + \frac{73 b^{6} c^{2} d^{5}}{6}\right ) + x^{5} \left (16 a^{3} b c^{4} d^{5} + 72 a^{2} b^{3} c^{3} d^{5} + \frac{183 a b^{5} c^{2} d^{5}}{5} + \frac{13 b^{7} c d^{5}}{5}\right ) + x^{4} \left (20 a^{3} b^{2} c^{3} d^{5} + \frac{75 a^{2} b^{4} c^{2} d^{5}}{2} + 9 a b^{6} c d^{5} + \frac{b^{8} d^{5}}{4}\right ) + x^{3} \left (\frac{40 a^{3} b^{3} c^{2} d^{5}}{3} + 11 a^{2} b^{5} c d^{5} + a b^{7} d^{5}\right ) + x^{2} \left (5 a^{3} b^{4} c d^{5} + \frac{3 a^{2} b^{6} d^{5}}{2}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.23557, size = 572, normalized size = 5.66 \begin{align*} \frac{8}{3} \, c^{8} d^{5} x^{12} + 16 \, b c^{7} d^{5} x^{11} + \frac{208}{5} \, b^{2} c^{6} d^{5} x^{10} + \frac{48}{5} \, a c^{7} d^{5} x^{10} + \frac{184}{3} \, b^{3} c^{5} d^{5} x^{9} + 48 \, a b c^{6} d^{5} x^{9} + \frac{225}{4} \, b^{4} c^{4} d^{5} x^{8} + 102 \, a b^{2} c^{5} d^{5} x^{8} + 12 \, a^{2} c^{6} d^{5} x^{8} + 33 \, b^{5} c^{3} d^{5} x^{7} + 120 \, a b^{3} c^{4} d^{5} x^{7} + 48 \, a^{2} b c^{5} d^{5} x^{7} + \frac{73}{6} \, b^{6} c^{2} d^{5} x^{6} + 85 \, a b^{4} c^{3} d^{5} x^{6} + 80 \, a^{2} b^{2} c^{4} d^{5} x^{6} + \frac{16}{3} \, a^{3} c^{5} d^{5} x^{6} + \frac{13}{5} \, b^{7} c d^{5} x^{5} + \frac{183}{5} \, a b^{5} c^{2} d^{5} x^{5} + 72 \, a^{2} b^{3} c^{3} d^{5} x^{5} + 16 \, a^{3} b c^{4} d^{5} x^{5} + \frac{1}{4} \, b^{8} d^{5} x^{4} + 9 \, a b^{6} c d^{5} x^{4} + \frac{75}{2} \, a^{2} b^{4} c^{2} d^{5} x^{4} + 20 \, a^{3} b^{2} c^{3} d^{5} x^{4} + a b^{7} d^{5} x^{3} + 11 \, a^{2} b^{5} c d^{5} x^{3} + \frac{40}{3} \, a^{3} b^{3} c^{2} d^{5} x^{3} + \frac{3}{2} \, a^{2} b^{6} d^{5} x^{2} + 5 \, a^{3} b^{4} c d^{5} x^{2} + a^{3} b^{5} d^{5} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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