Optimal. Leaf size=73 \[ -\frac{d^5 \left (b^2-4 a c\right ) (b+2 c x)^8}{128 c^3}+\frac{d^5 \left (b^2-4 a c\right )^2 (b+2 c x)^6}{192 c^3}+\frac{d^5 (b+2 c x)^{10}}{320 c^3} \]
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Rubi [A] time = 0.169635, 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} \[ -\frac{d^5 \left (b^2-4 a c\right ) (b+2 c x)^8}{128 c^3}+\frac{d^5 \left (b^2-4 a c\right )^2 (b+2 c x)^6}{192 c^3}+\frac{d^5 (b+2 c x)^{10}}{320 c^3} \]
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 )^2 \, dx &=\int \left (\frac{\left (-b^2+4 a c\right )^2 (b d+2 c d x)^5}{16 c^2}+\frac{\left (-b^2+4 a c\right ) (b d+2 c d x)^7}{8 c^2 d^2}+\frac{(b d+2 c d x)^9}{16 c^2 d^4}\right ) \, dx\\ &=\frac{\left (b^2-4 a c\right )^2 d^5 (b+2 c x)^6}{192 c^3}-\frac{\left (b^2-4 a c\right ) d^5 (b+2 c x)^8}{128 c^3}+\frac{d^5 (b+2 c x)^{10}}{320 c^3}\\ \end{align*}
Mathematica [B] time = 0.0598018, size = 168, normalized size = 2.3 \[ \frac{1}{15} d^5 x (b+c x) \left (5 a^2 \left (28 b^2 c^2 x^2+12 b^3 c x+3 b^4+32 b c^3 x^3+16 c^4 x^4\right )+5 a x \left (56 b^3 c^2 x^2+88 b^2 c^3 x^3+19 b^4 c x+3 b^5+72 b c^4 x^4+24 c^5 x^5\right )+x^2 (b+c x)^2 \left (78 b^2 c^2 x^2+30 b^3 c x+5 b^4+96 b c^3 x^3+48 c^4 x^4\right )\right ) \]
Antiderivative was successfully verified.
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Maple [B] time = 0.059, size = 362, normalized size = 5. \begin{align*}{\frac{16\,{c}^{7}{d}^{5}{x}^{10}}{5}}+16\,b{d}^{5}{c}^{6}{x}^{9}+{\frac{ \left ( 240\,{b}^{2}{d}^{5}{c}^{5}+32\,{c}^{5}{d}^{5} \left ( 2\,ac+{b}^{2} \right ) \right ){x}^{8}}{8}}+{\frac{ \left ( 200\,{b}^{3}{d}^{5}{c}^{4}+80\,b{d}^{5}{c}^{4} \left ( 2\,ac+{b}^{2} \right ) +64\,{c}^{5}{d}^{5}ab \right ){x}^{7}}{7}}+{\frac{ \left ( 90\,{b}^{4}{d}^{5}{c}^{3}+80\,{b}^{2}{d}^{5}{c}^{3} \left ( 2\,ac+{b}^{2} \right ) +160\,{b}^{2}{d}^{5}{c}^{4}a+32\,{c}^{5}{d}^{5}{a}^{2} \right ){x}^{6}}{6}}+{\frac{ \left ( 21\,{b}^{5}{d}^{5}{c}^{2}+40\,{b}^{3}{d}^{5}{c}^{2} \left ( 2\,ac+{b}^{2} \right ) +160\,{b}^{3}{d}^{5}{c}^{3}a+80\,b{d}^{5}{c}^{4}{a}^{2} \right ){x}^{5}}{5}}+{\frac{ \left ( 2\,{b}^{6}{d}^{5}c+10\,{b}^{4}{d}^{5}c \left ( 2\,ac+{b}^{2} \right ) +80\,{b}^{4}{d}^{5}{c}^{2}a+80\,{b}^{2}{d}^{5}{c}^{3}{a}^{2} \right ){x}^{4}}{4}}+{\frac{ \left ({b}^{5}{d}^{5} \left ( 2\,ac+{b}^{2} \right ) +20\,{b}^{5}{d}^{5}ca+40\,{b}^{3}{d}^{5}{c}^{2}{a}^{2} \right ){x}^{3}}{3}}+{\frac{ \left ( 10\,{b}^{4}{d}^{5}c{a}^{2}+2\,{b}^{6}{d}^{5}a \right ){x}^{2}}{2}}+{b}^{5}{d}^{5}{a}^{2}x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.20255, size = 320, normalized size = 4.38 \begin{align*} \frac{16}{5} \, c^{7} d^{5} x^{10} + 16 \, b c^{6} d^{5} x^{9} + 2 \,{\left (17 \, b^{2} c^{5} + 4 \, a c^{6}\right )} d^{5} x^{8} + a^{2} b^{5} d^{5} x + 8 \,{\left (5 \, b^{3} c^{4} + 4 \, a b c^{5}\right )} d^{5} x^{7} + \frac{1}{3} \,{\left (85 \, b^{4} c^{3} + 160 \, a b^{2} c^{4} + 16 \, a^{2} c^{5}\right )} d^{5} x^{6} + \frac{1}{5} \,{\left (61 \, b^{5} c^{2} + 240 \, a b^{3} c^{3} + 80 \, a^{2} b c^{4}\right )} d^{5} x^{5} +{\left (3 \, b^{6} c + 25 \, a b^{4} c^{2} + 20 \, a^{2} b^{2} c^{3}\right )} d^{5} x^{4} + \frac{1}{3} \,{\left (b^{7} + 22 \, a b^{5} c + 40 \, a^{2} b^{3} c^{2}\right )} d^{5} x^{3} +{\left (a b^{6} + 5 \, a^{2} 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 = 2.08502, size = 605, normalized size = 8.29 \begin{align*} \frac{16}{5} x^{10} d^{5} c^{7} + 16 x^{9} d^{5} c^{6} b + 34 x^{8} d^{5} c^{5} b^{2} + 8 x^{8} d^{5} c^{6} a + 40 x^{7} d^{5} c^{4} b^{3} + 32 x^{7} d^{5} c^{5} b a + \frac{85}{3} x^{6} d^{5} c^{3} b^{4} + \frac{160}{3} x^{6} d^{5} c^{4} b^{2} a + \frac{16}{3} x^{6} d^{5} c^{5} a^{2} + \frac{61}{5} x^{5} d^{5} c^{2} b^{5} + 48 x^{5} d^{5} c^{3} b^{3} a + 16 x^{5} d^{5} c^{4} b a^{2} + 3 x^{4} d^{5} c b^{6} + 25 x^{4} d^{5} c^{2} b^{4} a + 20 x^{4} d^{5} c^{3} b^{2} a^{2} + \frac{1}{3} x^{3} d^{5} b^{7} + \frac{22}{3} x^{3} d^{5} c b^{5} a + \frac{40}{3} x^{3} d^{5} c^{2} b^{3} a^{2} + x^{2} d^{5} b^{6} a + 5 x^{2} d^{5} c b^{4} a^{2} + x d^{5} b^{5} a^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 0.136747, size = 291, normalized size = 3.99 \begin{align*} a^{2} b^{5} d^{5} x + 16 b c^{6} d^{5} x^{9} + \frac{16 c^{7} d^{5} x^{10}}{5} + x^{8} \left (8 a c^{6} d^{5} + 34 b^{2} c^{5} d^{5}\right ) + x^{7} \left (32 a b c^{5} d^{5} + 40 b^{3} c^{4} d^{5}\right ) + x^{6} \left (\frac{16 a^{2} c^{5} d^{5}}{3} + \frac{160 a b^{2} c^{4} d^{5}}{3} + \frac{85 b^{4} c^{3} d^{5}}{3}\right ) + x^{5} \left (16 a^{2} b c^{4} d^{5} + 48 a b^{3} c^{3} d^{5} + \frac{61 b^{5} c^{2} d^{5}}{5}\right ) + x^{4} \left (20 a^{2} b^{2} c^{3} d^{5} + 25 a b^{4} c^{2} d^{5} + 3 b^{6} c d^{5}\right ) + x^{3} \left (\frac{40 a^{2} b^{3} c^{2} d^{5}}{3} + \frac{22 a b^{5} c d^{5}}{3} + \frac{b^{7} d^{5}}{3}\right ) + x^{2} \left (5 a^{2} b^{4} c d^{5} + a b^{6} d^{5}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.18512, size = 386, normalized size = 5.29 \begin{align*} \frac{16}{5} \, c^{7} d^{5} x^{10} + 16 \, b c^{6} d^{5} x^{9} + 34 \, b^{2} c^{5} d^{5} x^{8} + 8 \, a c^{6} d^{5} x^{8} + 40 \, b^{3} c^{4} d^{5} x^{7} + 32 \, a b c^{5} d^{5} x^{7} + \frac{85}{3} \, b^{4} c^{3} d^{5} x^{6} + \frac{160}{3} \, a b^{2} c^{4} d^{5} x^{6} + \frac{16}{3} \, a^{2} c^{5} d^{5} x^{6} + \frac{61}{5} \, b^{5} c^{2} d^{5} x^{5} + 48 \, a b^{3} c^{3} d^{5} x^{5} + 16 \, a^{2} b c^{4} d^{5} x^{5} + 3 \, b^{6} c d^{5} x^{4} + 25 \, a b^{4} c^{2} d^{5} x^{4} + 20 \, a^{2} b^{2} c^{3} d^{5} x^{4} + \frac{1}{3} \, b^{7} d^{5} x^{3} + \frac{22}{3} \, a b^{5} c d^{5} x^{3} + \frac{40}{3} \, a^{2} b^{3} c^{2} d^{5} x^{3} + a b^{6} d^{5} x^{2} + 5 \, a^{2} b^{4} c d^{5} x^{2} + a^{2} b^{5} d^{5} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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