Optimal. Leaf size=88 \[ \frac{b^2-4 a c}{8 c^3 d^3 \sqrt{b d+2 c d x}}-\frac{\left (b^2-4 a c\right )^2}{80 c^3 d (b d+2 c d x)^{5/2}}+\frac{(b d+2 c d x)^{3/2}}{48 c^3 d^5} \]
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Rubi [A] time = 0.0383003, 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} \[ \frac{b^2-4 a c}{8 c^3 d^3 \sqrt{b d+2 c d x}}-\frac{\left (b^2-4 a c\right )^2}{80 c^3 d (b d+2 c d x)^{5/2}}+\frac{(b d+2 c d x)^{3/2}}{48 c^3 d^5} \]
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)^{7/2}} \, dx &=\int \left (\frac{\left (-b^2+4 a c\right )^2}{16 c^2 (b d+2 c d x)^{7/2}}+\frac{-b^2+4 a c}{8 c^2 d^2 (b d+2 c d x)^{3/2}}+\frac{\sqrt{b d+2 c d x}}{16 c^2 d^4}\right ) \, dx\\ &=-\frac{\left (b^2-4 a c\right )^2}{80 c^3 d (b d+2 c d x)^{5/2}}+\frac{b^2-4 a c}{8 c^3 d^3 \sqrt{b d+2 c d x}}+\frac{(b d+2 c d x)^{3/2}}{48 c^3 d^5}\\ \end{align*}
Mathematica [A] time = 0.0431497, size = 92, normalized size = 1.05 \[ \frac{c^2 \left (-3 a^2-30 a c x^2+5 c^2 x^4\right )+3 b^2 c \left (5 c x^2-2 a\right )+10 b c^2 x \left (c x^2-3 a\right )+10 b^3 c x+2 b^4}{15 c^3 d (d (b+2 c x))^{5/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.046, 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}+30\,a{c}^{3}{x}^{2}-15\,{b}^{2}{c}^{2}{x}^{2}+30\,ab{c}^{2}x-10\,{b}^{3}cx+3\,{a}^{2}{c}^{2}+6\,ac{b}^{2}-2\,{b}^{4} \right ) }{15\,{c}^{3}} \left ( 2\,cdx+bd \right ) ^{-{\frac{7}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.00077, size = 126, normalized size = 1.43 \begin{align*} \frac{\frac{5 \,{\left (2 \, c d x + b d\right )}^{\frac{3}{2}}}{c^{2} d^{4}} + \frac{3 \,{\left (10 \,{\left (2 \, c d x + b d\right )}^{2}{\left (b^{2} - 4 \, a c\right )} -{\left (b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}\right )} d^{2}\right )}}{{\left (2 \, c d x + b d\right )}^{\frac{5}{2}} c^{2} d^{2}}}{240 \, c d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.01388, size = 281, normalized size = 3.19 \begin{align*} \frac{{\left (5 \, c^{4} x^{4} + 10 \, b c^{3} x^{3} + 2 \, b^{4} - 6 \, a b^{2} c - 3 \, a^{2} c^{2} + 15 \,{\left (b^{2} c^{2} - 2 \, a c^{3}\right )} x^{2} + 10 \,{\left (b^{3} c - 3 \, a b c^{2}\right )} x\right )} \sqrt{2 \, c d x + b d}}{15 \,{\left (8 \, c^{6} d^{4} x^{3} + 12 \, b c^{5} d^{4} x^{2} + 6 \, b^{2} c^{4} d^{4} x + b^{3} c^{3} d^{4}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 5.93285, size = 688, normalized size = 7.82 \begin{align*} \begin{cases} - \frac{3 a^{2} c^{2} \sqrt{b d + 2 c d x}}{15 b^{3} c^{3} d^{4} + 90 b^{2} c^{4} d^{4} x + 180 b c^{5} d^{4} x^{2} + 120 c^{6} d^{4} x^{3}} - \frac{6 a b^{2} c \sqrt{b d + 2 c d x}}{15 b^{3} c^{3} d^{4} + 90 b^{2} c^{4} d^{4} x + 180 b c^{5} d^{4} x^{2} + 120 c^{6} d^{4} x^{3}} - \frac{30 a b c^{2} x \sqrt{b d + 2 c d x}}{15 b^{3} c^{3} d^{4} + 90 b^{2} c^{4} d^{4} x + 180 b c^{5} d^{4} x^{2} + 120 c^{6} d^{4} x^{3}} - \frac{30 a c^{3} x^{2} \sqrt{b d + 2 c d x}}{15 b^{3} c^{3} d^{4} + 90 b^{2} c^{4} d^{4} x + 180 b c^{5} d^{4} x^{2} + 120 c^{6} d^{4} x^{3}} + \frac{2 b^{4} \sqrt{b d + 2 c d x}}{15 b^{3} c^{3} d^{4} + 90 b^{2} c^{4} d^{4} x + 180 b c^{5} d^{4} x^{2} + 120 c^{6} d^{4} x^{3}} + \frac{10 b^{3} c x \sqrt{b d + 2 c d x}}{15 b^{3} c^{3} d^{4} + 90 b^{2} c^{4} d^{4} x + 180 b c^{5} d^{4} x^{2} + 120 c^{6} d^{4} x^{3}} + \frac{15 b^{2} c^{2} x^{2} \sqrt{b d + 2 c d x}}{15 b^{3} c^{3} d^{4} + 90 b^{2} c^{4} d^{4} x + 180 b c^{5} d^{4} x^{2} + 120 c^{6} d^{4} x^{3}} + \frac{10 b c^{3} x^{3} \sqrt{b d + 2 c d x}}{15 b^{3} c^{3} d^{4} + 90 b^{2} c^{4} d^{4} x + 180 b c^{5} d^{4} x^{2} + 120 c^{6} d^{4} x^{3}} + \frac{5 c^{4} x^{4} \sqrt{b d + 2 c d x}}{15 b^{3} c^{3} d^{4} + 90 b^{2} c^{4} d^{4} x + 180 b c^{5} d^{4} x^{2} + 120 c^{6} d^{4} x^{3}} & \text{for}\: c \neq 0 \\\frac{a^{2} x + a b x^{2} + \frac{b^{2} x^{3}}{3}}{\left (b d\right )^{\frac{7}{2}}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.18467, size = 134, normalized size = 1.52 \begin{align*} \frac{{\left (2 \, c d x + b d\right )}^{\frac{3}{2}}}{48 \, c^{3} d^{5}} - \frac{b^{4} d^{2} - 8 \, a b^{2} c d^{2} + 16 \, a^{2} c^{2} d^{2} - 10 \,{\left (2 \, c d x + b d\right )}^{2} b^{2} + 40 \,{\left (2 \, c d x + b d\right )}^{2} a c}{80 \,{\left (2 \, c d x + b d\right )}^{\frac{5}{2}} c^{3} d^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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