Optimal. Leaf size=88 \[ -\frac{\left (b^2-4 a c\right ) (b d+2 c d x)^{5/2}}{40 c^3 d^3}+\frac{\left (b^2-4 a c\right )^2 \sqrt{b d+2 c d x}}{16 c^3 d}+\frac{(b d+2 c d x)^{9/2}}{144 c^3 d^5} \]
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Rubi [A] time = 0.0381218, 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} \[ -\frac{\left (b^2-4 a c\right ) (b d+2 c d x)^{5/2}}{40 c^3 d^3}+\frac{\left (b^2-4 a c\right )^2 \sqrt{b d+2 c d x}}{16 c^3 d}+\frac{(b d+2 c d x)^{9/2}}{144 c^3 d^5} \]
Antiderivative was successfully verified.
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Rule 683
Rubi steps
\begin{align*} \int \frac{\left (a+b x+c x^2\right )^2}{\sqrt{b d+2 c d x}} \, dx &=\int \left (\frac{\left (-b^2+4 a c\right )^2}{16 c^2 \sqrt{b d+2 c d x}}+\frac{\left (-b^2+4 a c\right ) (b d+2 c d x)^{3/2}}{8 c^2 d^2}+\frac{(b d+2 c d x)^{7/2}}{16 c^2 d^4}\right ) \, dx\\ &=\frac{\left (b^2-4 a c\right )^2 \sqrt{b d+2 c d x}}{16 c^3 d}-\frac{\left (b^2-4 a c\right ) (b d+2 c d x)^{5/2}}{40 c^3 d^3}+\frac{(b d+2 c d x)^{9/2}}{144 c^3 d^5}\\ \end{align*}
Mathematica [A] time = 0.0426301, size = 92, normalized size = 1.05 \[ \frac{\left (c^2 \left (45 a^2+18 a c x^2+5 c^2 x^4\right )+3 b^2 c \left (c x^2-6 a\right )+2 b c^2 x \left (9 a+5 c x^2\right )-2 b^3 c x+2 b^4\right ) \sqrt{d (b+2 c x)}}{45 c^3 d} \]
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 ( 5\,{c}^{4}{x}^{4}+10\,b{x}^{3}{c}^{3}+18\,a{c}^{3}{x}^{2}+3\,{b}^{2}{c}^{2}{x}^{2}+18\,ab{c}^{2}x-2\,{b}^{3}cx+45\,{a}^{2}{c}^{2}-18\,ac{b}^{2}+2\,{b}^{4} \right ) }{45\,{c}^{3}}{\frac{1}{\sqrt{2\,cdx+bd}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.02153, size = 474, normalized size = 5.39 \begin{align*} \frac{5040 \, \sqrt{2 \, c d x + b d} a^{2} - 168 \, a{\left (\frac{10 \,{\left (3 \, \sqrt{2 \, c d x + b d} b d -{\left (2 \, c d x + b d\right )}^{\frac{3}{2}}\right )} b}{c d} - \frac{15 \, \sqrt{2 \, c d x + b d} b^{2} d^{2} - 10 \,{\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}}}{c d^{2}}\right )} + \frac{84 \,{\left (15 \, \sqrt{2 \, c d x + b d} b^{2} d^{2} - 10 \,{\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^{2}}{c^{2} d^{2}} - \frac{36 \,{\left (35 \, \sqrt{2 \, c d x + b d} b^{3} d^{3} - 35 \,{\left (2 \, c d x + b d\right )}^{\frac{3}{2}} b^{2} d^{2} + 21 \,{\left (2 \, c d x + b d\right )}^{\frac{5}{2}} b d - 5 \,{\left (2 \, c d x + b d\right )}^{\frac{7}{2}}\right )} b}{c^{2} d^{3}} + \frac{315 \, \sqrt{2 \, c d x + b d} b^{4} d^{4} - 420 \,{\left (2 \, c d x + b d\right )}^{\frac{3}{2}} b^{3} d^{3} + 378 \,{\left (2 \, c d x + b d\right )}^{\frac{5}{2}} b^{2} d^{2} - 180 \,{\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}}}{c^{2} d^{4}}}{5040 \, c d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.02639, size = 201, normalized size = 2.28 \begin{align*} \frac{{\left (5 \, c^{4} x^{4} + 10 \, b c^{3} x^{3} + 2 \, b^{4} - 18 \, a b^{2} c + 45 \, a^{2} c^{2} + 3 \,{\left (b^{2} c^{2} + 6 \, a c^{3}\right )} x^{2} - 2 \,{\left (b^{3} c - 9 \, a b c^{2}\right )} x\right )} \sqrt{2 \, c d x + b d}}{45 \, c^{3} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 63.1151, size = 668, normalized size = 7.59 \begin{align*} \begin{cases} - \frac{\frac{a^{2} b}{\sqrt{b d + 2 c d x}} + \frac{a^{2} \left (- \frac{b d}{\sqrt{b d + 2 c d x}} - \sqrt{b d + 2 c d x}\right )}{d} + \frac{a b^{2} \left (- \frac{b d}{\sqrt{b d + 2 c d x}} - \sqrt{b d + 2 c d x}\right )}{c d} + \frac{3 a b \left (\frac{b^{2} d^{2}}{\sqrt{b d + 2 c d x}} + 2 b d \sqrt{b d + 2 c d x} - \frac{\left (b d + 2 c d x\right )^{\frac{3}{2}}}{3}\right )}{2 c d^{2}} + \frac{a \left (- \frac{b^{3} d^{3}}{\sqrt{b d + 2 c d x}} - 3 b^{2} d^{2} \sqrt{b d + 2 c d x} + b d \left (b d + 2 c d x\right )^{\frac{3}{2}} - \frac{\left (b d + 2 c d x\right )^{\frac{5}{2}}}{5}\right )}{2 c d^{3}} + \frac{b^{3} \left (\frac{b^{2} d^{2}}{\sqrt{b d + 2 c d x}} + 2 b d \sqrt{b d + 2 c d x} - \frac{\left (b d + 2 c d x\right )^{\frac{3}{2}}}{3}\right )}{4 c^{2} d^{2}} + \frac{b^{2} \left (- \frac{b^{3} d^{3}}{\sqrt{b d + 2 c d x}} - 3 b^{2} d^{2} \sqrt{b d + 2 c d x} + b d \left (b d + 2 c d x\right )^{\frac{3}{2}} - \frac{\left (b d + 2 c d x\right )^{\frac{5}{2}}}{5}\right )}{2 c^{2} d^{3}} + \frac{5 b \left (\frac{b^{4} d^{4}}{\sqrt{b d + 2 c d x}} + 4 b^{3} d^{3} \sqrt{b d + 2 c d x} - 2 b^{2} d^{2} \left (b d + 2 c d x\right )^{\frac{3}{2}} + \frac{4 b d \left (b d + 2 c d x\right )^{\frac{5}{2}}}{5} - \frac{\left (b d + 2 c d x\right )^{\frac{7}{2}}}{7}\right )}{16 c^{2} d^{4}} + \frac{- \frac{b^{5} d^{5}}{\sqrt{b d + 2 c d x}} - 5 b^{4} d^{4} \sqrt{b d + 2 c d x} + \frac{10 b^{3} d^{3} \left (b d + 2 c d x\right )^{\frac{3}{2}}}{3} - 2 b^{2} d^{2} \left (b d + 2 c d x\right )^{\frac{5}{2}} + \frac{5 b d \left (b d + 2 c d x\right )^{\frac{7}{2}}}{7} - \frac{\left (b d + 2 c d x\right )^{\frac{9}{2}}}{9}}{16 c^{2} d^{5}}}{c} & \text{for}\: c \neq 0 \\\frac{\begin{cases} a^{2} x & \text{for}\: b = 0 \\\frac{\left (a + b x\right )^{3}}{3 b} & \text{otherwise} \end{cases}}{\sqrt{b d}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.13943, size = 471, normalized size = 5.35 \begin{align*} \frac{5040 \, \sqrt{2 \, c d x + b d} a^{2} - \frac{1680 \,{\left (3 \, \sqrt{2 \, c d x + b d} b d -{\left (2 \, c d x + b d\right )}^{\frac{3}{2}}\right )} a b}{c d} + \frac{84 \,{\left (15 \, \sqrt{2 \, c d x + b d} b^{2} d^{2} - 10 \,{\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^{2}}{c^{2} d^{2}} + \frac{168 \,{\left (15 \, \sqrt{2 \, c d x + b d} b^{2} d^{2} - 10 \,{\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}{c d^{2}} - \frac{36 \,{\left (35 \, \sqrt{2 \, c d x + b d} b^{3} d^{3} - 35 \,{\left (2 \, c d x + b d\right )}^{\frac{3}{2}} b^{2} d^{2} + 21 \,{\left (2 \, c d x + b d\right )}^{\frac{5}{2}} b d - 5 \,{\left (2 \, c d x + b d\right )}^{\frac{7}{2}}\right )} b}{c^{2} d^{3}} + \frac{315 \, \sqrt{2 \, c d x + b d} b^{4} d^{4} - 420 \,{\left (2 \, c d x + b d\right )}^{\frac{3}{2}} b^{3} d^{3} + 378 \,{\left (2 \, c d x + b d\right )}^{\frac{5}{2}} b^{2} d^{2} - 180 \,{\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}}}{c^{2} d^{4}}}{5040 \, c d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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