Optimal. Leaf size=88 \[ \frac{b^2-4 a c}{24 c^3 d^3 (b d+2 c d x)^{3/2}}-\frac{\left (b^2-4 a c\right )^2}{112 c^3 d (b d+2 c d x)^{7/2}}+\frac{\sqrt{b d+2 c d x}}{16 c^3 d^5} \]
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Rubi [A] time = 0.0374583, 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} \[ \frac{b^2-4 a c}{24 c^3 d^3 (b d+2 c d x)^{3/2}}-\frac{\left (b^2-4 a c\right )^2}{112 c^3 d (b d+2 c d x)^{7/2}}+\frac{\sqrt{b d+2 c d x}}{16 c^3 d^5} \]
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}{(b d+2 c d x)^{9/2}} \, dx &=\int \left (\frac{\left (-b^2+4 a c\right )^2}{16 c^2 (b d+2 c d x)^{9/2}}+\frac{-b^2+4 a c}{8 c^2 d^2 (b d+2 c d x)^{5/2}}+\frac{1}{16 c^2 d^4 \sqrt{b d+2 c d x}}\right ) \, dx\\ &=-\frac{\left (b^2-4 a c\right )^2}{112 c^3 d (b d+2 c d x)^{7/2}}+\frac{b^2-4 a c}{24 c^3 d^3 (b d+2 c d x)^{3/2}}+\frac{\sqrt{b d+2 c d x}}{16 c^3 d^5}\\ \end{align*}
Mathematica [A] time = 0.0476852, size = 63, normalized size = 0.72 \[ \frac{14 \left (b^2-4 a c\right ) (b+2 c x)^2-3 \left (b^2-4 a c\right )^2+21 (b+2 c x)^4}{336 c^3 d (d (b+2 c x))^{7/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.049, 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}+14\,a{c}^{3}{x}^{2}-35\,{b}^{2}{c}^{2}{x}^{2}+14\,ab{c}^{2}x-14\,{b}^{3}cx+3\,{a}^{2}{c}^{2}+2\,ac{b}^{2}-2\,{b}^{4} \right ) }{21\,{c}^{3}} \left ( 2\,cdx+bd \right ) ^{-{\frac{9}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.993463, size = 124, normalized size = 1.41 \begin{align*} \frac{\frac{21 \, \sqrt{2 \, c d x + b d}}{c^{2} d^{4}} + \frac{14 \,{\left (2 \, c d x + b d\right )}^{2}{\left (b^{2} - 4 \, a c\right )} - 3 \,{\left (b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}\right )} d^{2}}{{\left (2 \, c d x + b d\right )}^{\frac{7}{2}} c^{2} d^{2}}}{336 \, c d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.01866, size = 311, normalized size = 3.53 \begin{align*} \frac{{\left (21 \, c^{4} x^{4} + 42 \, b c^{3} x^{3} + 2 \, b^{4} - 2 \, a b^{2} c - 3 \, a^{2} c^{2} + 7 \,{\left (5 \, b^{2} c^{2} - 2 \, a c^{3}\right )} x^{2} + 14 \,{\left (b^{3} c - a b c^{2}\right )} x\right )} \sqrt{2 \, c d x + b d}}{21 \,{\left (16 \, c^{7} d^{5} x^{4} + 32 \, b c^{6} d^{5} x^{3} + 24 \, b^{2} c^{5} d^{5} x^{2} + 8 \, b^{3} c^{4} d^{5} x + b^{4} c^{3} d^{5}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 9.5047, size = 826, normalized size = 9.39 \begin{align*} \begin{cases} - \frac{3 a^{2} c^{2} \sqrt{b d + 2 c d x}}{21 b^{4} c^{3} d^{5} + 168 b^{3} c^{4} d^{5} x + 504 b^{2} c^{5} d^{5} x^{2} + 672 b c^{6} d^{5} x^{3} + 336 c^{7} d^{5} x^{4}} - \frac{2 a b^{2} c \sqrt{b d + 2 c d x}}{21 b^{4} c^{3} d^{5} + 168 b^{3} c^{4} d^{5} x + 504 b^{2} c^{5} d^{5} x^{2} + 672 b c^{6} d^{5} x^{3} + 336 c^{7} d^{5} x^{4}} - \frac{14 a b c^{2} x \sqrt{b d + 2 c d x}}{21 b^{4} c^{3} d^{5} + 168 b^{3} c^{4} d^{5} x + 504 b^{2} c^{5} d^{5} x^{2} + 672 b c^{6} d^{5} x^{3} + 336 c^{7} d^{5} x^{4}} - \frac{14 a c^{3} x^{2} \sqrt{b d + 2 c d x}}{21 b^{4} c^{3} d^{5} + 168 b^{3} c^{4} d^{5} x + 504 b^{2} c^{5} d^{5} x^{2} + 672 b c^{6} d^{5} x^{3} + 336 c^{7} d^{5} x^{4}} + \frac{2 b^{4} \sqrt{b d + 2 c d x}}{21 b^{4} c^{3} d^{5} + 168 b^{3} c^{4} d^{5} x + 504 b^{2} c^{5} d^{5} x^{2} + 672 b c^{6} d^{5} x^{3} + 336 c^{7} d^{5} x^{4}} + \frac{14 b^{3} c x \sqrt{b d + 2 c d x}}{21 b^{4} c^{3} d^{5} + 168 b^{3} c^{4} d^{5} x + 504 b^{2} c^{5} d^{5} x^{2} + 672 b c^{6} d^{5} x^{3} + 336 c^{7} d^{5} x^{4}} + \frac{35 b^{2} c^{2} x^{2} \sqrt{b d + 2 c d x}}{21 b^{4} c^{3} d^{5} + 168 b^{3} c^{4} d^{5} x + 504 b^{2} c^{5} d^{5} x^{2} + 672 b c^{6} d^{5} x^{3} + 336 c^{7} d^{5} x^{4}} + \frac{42 b c^{3} x^{3} \sqrt{b d + 2 c d x}}{21 b^{4} c^{3} d^{5} + 168 b^{3} c^{4} d^{5} x + 504 b^{2} c^{5} d^{5} x^{2} + 672 b c^{6} d^{5} x^{3} + 336 c^{7} d^{5} x^{4}} + \frac{21 c^{4} x^{4} \sqrt{b d + 2 c d x}}{21 b^{4} c^{3} d^{5} + 168 b^{3} c^{4} d^{5} x + 504 b^{2} c^{5} d^{5} x^{2} + 672 b c^{6} d^{5} x^{3} + 336 c^{7} d^{5} x^{4}} & \text{for}\: c \neq 0 \\\frac{a^{2} x + a b x^{2} + \frac{b^{2} x^{3}}{3}}{\left (b d\right )^{\frac{9}{2}}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.16398, size = 135, normalized size = 1.53 \begin{align*} \frac{\sqrt{2 \, c d x + b d}}{16 \, c^{3} d^{5}} - \frac{3 \, b^{4} d^{2} - 24 \, a b^{2} c d^{2} + 48 \, a^{2} c^{2} d^{2} - 14 \,{\left (2 \, c d x + b d\right )}^{2} b^{2} + 56 \,{\left (2 \, c d x + b d\right )}^{2} a c}{336 \,{\left (2 \, c d x + b d\right )}^{\frac{7}{2}} c^{3} d^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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