Optimal. Leaf size=88 \[ -\frac{\left (b^2-4 a c\right ) \sqrt{b d+2 c d x}}{8 c^3 d^3}-\frac{\left (b^2-4 a c\right )^2}{48 c^3 d (b d+2 c d x)^{3/2}}+\frac{(b d+2 c d x)^{5/2}}{80 c^3 d^5} \]
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Rubi [A] time = 0.0403534, 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} \[ -\frac{\left (b^2-4 a c\right ) \sqrt{b d+2 c d x}}{8 c^3 d^3}-\frac{\left (b^2-4 a c\right )^2}{48 c^3 d (b d+2 c d x)^{3/2}}+\frac{(b d+2 c d x)^{5/2}}{80 c^3 d^5} \]
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)^{5/2}} \, dx &=\int \left (\frac{\left (-b^2+4 a c\right )^2}{16 c^2 (b d+2 c d x)^{5/2}}+\frac{-b^2+4 a c}{8 c^2 d^2 \sqrt{b d+2 c d x}}+\frac{(b d+2 c d x)^{3/2}}{16 c^2 d^4}\right ) \, dx\\ &=-\frac{\left (b^2-4 a c\right )^2}{48 c^3 d (b d+2 c d x)^{3/2}}-\frac{\left (b^2-4 a c\right ) \sqrt{b d+2 c d x}}{8 c^3 d^3}+\frac{(b d+2 c d x)^{5/2}}{80 c^3 d^5}\\ \end{align*}
Mathematica [A] time = 0.0398138, size = 91, normalized size = 1.03 \[ \frac{c^2 \left (-5 a^2+30 a c x^2+3 c^2 x^4\right )+b^2 c \left (10 a-3 c x^2\right )+6 b c^2 x \left (5 a+c x^2\right )-6 b^3 c x-2 b^4}{15 c^3 d (d (b+2 c x))^{3/2}} \]
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 ( -3\,{c}^{4}{x}^{4}-6\,b{x}^{3}{c}^{3}-30\,a{c}^{3}{x}^{2}+3\,{b}^{2}{c}^{2}{x}^{2}-30\,ab{c}^{2}x+6\,{b}^{3}cx+5\,{a}^{2}{c}^{2}-10\,ac{b}^{2}+2\,{b}^{4} \right ) }{15\,{c}^{3}} \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 = 1.10376, size = 122, normalized size = 1.39 \begin{align*} -\frac{\frac{5 \,{\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}} c^{2}} + \frac{3 \,{\left (10 \, \sqrt{2 \, c d x + b d}{\left (b^{2} - 4 \, a c\right )} d^{2} -{\left (2 \, c d x + b d\right )}^{\frac{5}{2}}\right )}}{c^{2} d^{4}}}{240 \, c d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.01993, size = 251, normalized size = 2.85 \begin{align*} \frac{{\left (3 \, c^{4} x^{4} + 6 \, b c^{3} x^{3} - 2 \, b^{4} + 10 \, a b^{2} c - 5 \, a^{2} c^{2} - 3 \,{\left (b^{2} c^{2} - 10 \, a c^{3}\right )} x^{2} - 6 \,{\left (b^{3} c - 5 \, a b c^{2}\right )} x\right )} \sqrt{2 \, c d x + b d}}{15 \,{\left (4 \, c^{5} d^{3} x^{2} + 4 \, b c^{4} d^{3} x + b^{2} c^{3} d^{3}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 49.9339, size = 82, normalized size = 0.93 \begin{align*} - \frac{\left (4 a c - b^{2}\right )^{2}}{48 c^{3} d \left (b d + 2 c d x\right )^{\frac{3}{2}}} + \frac{\left (4 a c - b^{2}\right ) \sqrt{b d + 2 c d x}}{8 c^{3} d^{3}} + \frac{\left (b d + 2 c d x\right )^{\frac{5}{2}}}{80 c^{3} d^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.18923, size = 147, normalized size = 1.67 \begin{align*} -\frac{b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}}{48 \,{\left (2 \, c d x + b d\right )}^{\frac{3}{2}} c^{3} d} - \frac{10 \, \sqrt{2 \, c d x + b d} b^{2} c^{12} d^{22} - 40 \, \sqrt{2 \, c d x + b d} a c^{13} d^{22} -{\left (2 \, c d x + b d\right )}^{\frac{5}{2}} c^{12} d^{20}}{80 \, c^{15} d^{25}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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