Optimal. Leaf size=88 \[ -\frac{\left (b^2-4 a c\right ) (b d+2 c d x)^{7/2}}{56 c^3 d^3}+\frac{\left (b^2-4 a c\right )^2 (b d+2 c d x)^{3/2}}{48 c^3 d}+\frac{(b d+2 c d x)^{11/2}}{176 c^3 d^5} \]
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Rubi [A] time = 0.039476, antiderivative size = 88, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.038, Rules used = {683} \[ -\frac{\left (b^2-4 a c\right ) (b d+2 c d x)^{7/2}}{56 c^3 d^3}+\frac{\left (b^2-4 a c\right )^2 (b d+2 c d x)^{3/2}}{48 c^3 d}+\frac{(b d+2 c d x)^{11/2}}{176 c^3 d^5} \]
Antiderivative was successfully verified.
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Rule 683
Rubi steps
\begin{align*} \int \sqrt{b d+2 c d x} \left (a+b x+c x^2\right )^2 \, dx &=\int \left (\frac{\left (-b^2+4 a c\right )^2 \sqrt{b d+2 c d x}}{16 c^2}+\frac{\left (-b^2+4 a c\right ) (b d+2 c d x)^{5/2}}{8 c^2 d^2}+\frac{(b d+2 c d x)^{9/2}}{16 c^2 d^4}\right ) \, dx\\ &=\frac{\left (b^2-4 a c\right )^2 (b d+2 c d x)^{3/2}}{48 c^3 d}-\frac{\left (b^2-4 a c\right ) (b d+2 c d x)^{7/2}}{56 c^3 d^3}+\frac{(b d+2 c d x)^{11/2}}{176 c^3 d^5}\\ \end{align*}
Mathematica [A] time = 0.0439803, size = 92, normalized size = 1.05 \[ \frac{\left (c^2 \left (77 a^2+66 a c x^2+21 c^2 x^4\right )+b^2 c \left (15 c x^2-22 a\right )+6 b c^2 x \left (11 a+7 c x^2\right )-6 b^3 c x+2 b^4\right ) (d (b+2 c x))^{3/2}}{231 c^3 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.044, size = 96, normalized size = 1.1 \begin{align*}{\frac{ \left ( 2\,cx+b \right ) \left ( 21\,{c}^{4}{x}^{4}+42\,b{x}^{3}{c}^{3}+66\,a{c}^{3}{x}^{2}+15\,{b}^{2}{c}^{2}{x}^{2}+66\,ab{c}^{2}x-6\,{b}^{3}cx+77\,{a}^{2}{c}^{2}-22\,ac{b}^{2}+2\,{b}^{4} \right ) }{231\,{c}^{3}}\sqrt{2\,cdx+bd}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.19286, size = 109, normalized size = 1.24 \begin{align*} -\frac{66 \,{\left (2 \, c d x + b d\right )}^{\frac{7}{2}}{\left (b^{2} - 4 \, a c\right )} d^{2} - 77 \,{\left (b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}\right )}{\left (2 \, c d x + b d\right )}^{\frac{3}{2}} d^{4} - 21 \,{\left (2 \, c d x + b d\right )}^{\frac{11}{2}}}{3696 \, c^{3} d^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.99735, size = 271, normalized size = 3.08 \begin{align*} \frac{{\left (42 \, c^{5} x^{5} + 105 \, b c^{4} x^{4} + 2 \, b^{5} - 22 \, a b^{3} c + 77 \, a^{2} b c^{2} + 12 \,{\left (6 \, b^{2} c^{3} + 11 \, a c^{4}\right )} x^{3} + 3 \,{\left (b^{3} c^{2} + 66 \, a b c^{3}\right )} x^{2} - 2 \,{\left (b^{4} c - 11 \, a b^{2} c^{2} - 77 \, a^{2} c^{3}\right )} x\right )} \sqrt{2 \, c d x + b d}}{231 \, c^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 3.29871, size = 94, normalized size = 1.07 \begin{align*} \frac{\frac{\left (b d + 2 c d x\right )^{\frac{3}{2}} \left (16 a^{2} c^{2} - 8 a b^{2} c + b^{4}\right )}{48 c^{2}} + \frac{\left (4 a c - b^{2}\right ) \left (b d + 2 c d x\right )^{\frac{7}{2}}}{56 c^{2} d^{2}} + \frac{\left (b d + 2 c d x\right )^{\frac{11}{2}}}{176 c^{2} d^{4}}}{c d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.14695, size = 471, normalized size = 5.35 \begin{align*} \frac{18480 \,{\left (2 \, c d x + b d\right )}^{\frac{3}{2}} a^{2} - \frac{3696 \,{\left (5 \,{\left (2 \, c d x + b d\right )}^{\frac{3}{2}} b d - 3 \,{\left (2 \, c d x + b d\right )}^{\frac{5}{2}}\right )} a b}{c d} + \frac{132 \,{\left (35 \,{\left (2 \, c d x + b d\right )}^{\frac{3}{2}} b^{2} d^{2} - 42 \,{\left (2 \, c d x + b d\right )}^{\frac{5}{2}} b d + 15 \,{\left (2 \, c d x + b d\right )}^{\frac{7}{2}}\right )} b^{2}}{c^{2} d^{2}} + \frac{264 \,{\left (35 \,{\left (2 \, c d x + b d\right )}^{\frac{3}{2}} b^{2} d^{2} - 42 \,{\left (2 \, c d x + b d\right )}^{\frac{5}{2}} b d + 15 \,{\left (2 \, c d x + b d\right )}^{\frac{7}{2}}\right )} a}{c d^{2}} - \frac{44 \,{\left (105 \,{\left (2 \, c d x + b d\right )}^{\frac{3}{2}} b^{3} d^{3} - 189 \,{\left (2 \, c d x + b d\right )}^{\frac{5}{2}} b^{2} d^{2} + 135 \,{\left (2 \, c d x + b d\right )}^{\frac{7}{2}} b d - 35 \,{\left (2 \, c d x + b d\right )}^{\frac{9}{2}}\right )} b}{c^{2} d^{3}} + \frac{1155 \,{\left (2 \, c d x + b d\right )}^{\frac{3}{2}} b^{4} d^{4} - 2772 \,{\left (2 \, c d x + b d\right )}^{\frac{5}{2}} b^{3} d^{3} + 2970 \,{\left (2 \, c d x + b d\right )}^{\frac{7}{2}} b^{2} d^{2} - 1540 \,{\left (2 \, c d x + b d\right )}^{\frac{9}{2}} b d + 315 \,{\left (2 \, c d x + b d\right )}^{\frac{11}{2}}}{c^{2} d^{4}}}{55440 \, c d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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