Optimal. Leaf size=55 \[ \frac{(b d+2 c d x)^{11/2}}{44 c^2 d^3}-\frac{\left (b^2-4 a c\right ) (b d+2 c d x)^{7/2}}{28 c^2 d} \]
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Rubi [A] time = 0.0238854, antiderivative size = 55, 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} \[ \frac{(b d+2 c d x)^{11/2}}{44 c^2 d^3}-\frac{\left (b^2-4 a c\right ) (b d+2 c d x)^{7/2}}{28 c^2 d} \]
Antiderivative was successfully verified.
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Rule 683
Rubi steps
\begin{align*} \int (b d+2 c d x)^{5/2} \left (a+b x+c x^2\right ) \, dx &=\int \left (\frac{\left (-b^2+4 a c\right ) (b d+2 c d x)^{5/2}}{4 c}+\frac{(b d+2 c d x)^{9/2}}{4 c d^2}\right ) \, dx\\ &=-\frac{\left (b^2-4 a c\right ) (b d+2 c d x)^{7/2}}{28 c^2 d}+\frac{(b d+2 c d x)^{11/2}}{44 c^2 d^3}\\ \end{align*}
Mathematica [A] time = 0.0436917, size = 45, normalized size = 0.82 \[ \frac{\left (c \left (11 a+7 c x^2\right )-b^2+7 b c x\right ) (d (b+2 c x))^{7/2}}{77 c^2 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.041, size = 46, normalized size = 0.8 \begin{align*}{\frac{ \left ( 2\,cx+b \right ) \left ( 7\,{c}^{2}{x}^{2}+7\,bcx+11\,ac-{b}^{2} \right ) }{77\,{c}^{2}} \left ( 2\,cdx+bd \right ) ^{{\frac{5}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.098, size = 62, normalized size = 1.13 \begin{align*} -\frac{11 \,{\left (2 \, c d x + b d\right )}^{\frac{7}{2}}{\left (b^{2} - 4 \, a c\right )} d^{2} - 7 \,{\left (2 \, c d x + b d\right )}^{\frac{11}{2}}}{308 \, c^{2} d^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.98296, size = 265, normalized size = 4.82 \begin{align*} \frac{{\left (56 \, c^{5} d^{2} x^{5} + 140 \, b c^{4} d^{2} x^{4} + 2 \,{\left (59 \, b^{2} c^{3} + 44 \, a c^{4}\right )} d^{2} x^{3} +{\left (37 \, b^{3} c^{2} + 132 \, a b c^{3}\right )} d^{2} x^{2} +{\left (b^{4} c + 66 \, a b^{2} c^{2}\right )} d^{2} x -{\left (b^{5} - 11 \, a b^{3} c\right )} d^{2}\right )} \sqrt{2 \, c d x + b d}}{77 \, c^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 5.2453, size = 289, normalized size = 5.25 \begin{align*} \begin{cases} \frac{a b^{3} d^{2} \sqrt{b d + 2 c d x}}{7 c} + \frac{6 a b^{2} d^{2} x \sqrt{b d + 2 c d x}}{7} + \frac{12 a b c d^{2} x^{2} \sqrt{b d + 2 c d x}}{7} + \frac{8 a c^{2} d^{2} x^{3} \sqrt{b d + 2 c d x}}{7} - \frac{b^{5} d^{2} \sqrt{b d + 2 c d x}}{77 c^{2}} + \frac{b^{4} d^{2} x \sqrt{b d + 2 c d x}}{77 c} + \frac{37 b^{3} d^{2} x^{2} \sqrt{b d + 2 c d x}}{77} + \frac{118 b^{2} c d^{2} x^{3} \sqrt{b d + 2 c d x}}{77} + \frac{20 b c^{2} d^{2} x^{4} \sqrt{b d + 2 c d x}}{11} + \frac{8 c^{3} d^{2} x^{5} \sqrt{b d + 2 c d x}}{11} & \text{for}\: c \neq 0 \\\left (b d\right )^{\frac{5}{2}} \left (a x + \frac{b x^{2}}{2}\right ) & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.14354, size = 508, normalized size = 9.24 \begin{align*} \frac{4620 \,{\left (2 \, c d x + b d\right )}^{\frac{3}{2}} a b^{2} d - 1848 \,{\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 - \frac{462 \,{\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 )} b^{3}}{c} + \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 )} a}{d} + \frac{165 \,{\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 d} - \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 d^{2}} + \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 d^{3}}}{13860 \, c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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