Optimal. Leaf size=88 \[ \frac{b^2-4 a c}{40 c^3 d^3 (b d+2 c d x)^{5/2}}-\frac{\left (b^2-4 a c\right )^2}{144 c^3 d (b d+2 c d x)^{9/2}}-\frac{1}{16 c^3 d^5 \sqrt{b d+2 c d x}} \]
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Rubi [A] time = 0.0372918, 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} \[ \frac{b^2-4 a c}{40 c^3 d^3 (b d+2 c d x)^{5/2}}-\frac{\left (b^2-4 a c\right )^2}{144 c^3 d (b d+2 c d x)^{9/2}}-\frac{1}{16 c^3 d^5 \sqrt{b d+2 c d x}} \]
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)^{11/2}} \, dx &=\int \left (\frac{\left (-b^2+4 a c\right )^2}{16 c^2 (b d+2 c d x)^{11/2}}+\frac{-b^2+4 a c}{8 c^2 d^2 (b d+2 c d x)^{7/2}}+\frac{1}{16 c^2 d^4 (b d+2 c d x)^{3/2}}\right ) \, dx\\ &=-\frac{\left (b^2-4 a c\right )^2}{144 c^3 d (b d+2 c d x)^{9/2}}+\frac{b^2-4 a c}{40 c^3 d^3 (b d+2 c d x)^{5/2}}-\frac{1}{16 c^3 d^5 \sqrt{b d+2 c d x}}\\ \end{align*}
Mathematica [A] time = 0.0507299, size = 63, normalized size = 0.72 \[ \frac{18 \left (b^2-4 a c\right ) (b+2 c x)^2-5 \left (b^2-4 a c\right )^2-45 (b+2 c x)^4}{720 c^3 d (d (b+2 c x))^{9/2}} \]
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 ( 45\,{c}^{4}{x}^{4}+90\,b{x}^{3}{c}^{3}+18\,a{c}^{3}{x}^{2}+63\,{b}^{2}{c}^{2}{x}^{2}+18\,ab{c}^{2}x+18\,{b}^{3}cx+5\,{a}^{2}{c}^{2}+2\,ac{b}^{2}+2\,{b}^{4} \right ) }{45\,{c}^{3}} \left ( 2\,cdx+bd \right ) ^{-{\frac{11}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.07437, size = 109, normalized size = 1.24 \begin{align*} \frac{18 \,{\left (2 \, c d x + b d\right )}^{2}{\left (b^{2} - 4 \, a c\right )} d^{2} - 5 \,{\left (b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}\right )} d^{4} - 45 \,{\left (2 \, c d x + b d\right )}^{4}}{720 \,{\left (2 \, c d x + b d\right )}^{\frac{9}{2}} c^{3} d^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.97052, size = 342, normalized size = 3.89 \begin{align*} -\frac{{\left (45 \, c^{4} x^{4} + 90 \, b c^{3} x^{3} + 2 \, b^{4} + 2 \, a b^{2} c + 5 \, a^{2} c^{2} + 9 \,{\left (7 \, b^{2} c^{2} + 2 \, a c^{3}\right )} x^{2} + 18 \,{\left (b^{3} c + a b c^{2}\right )} x\right )} \sqrt{2 \, c d x + b d}}{45 \,{\left (32 \, c^{8} d^{6} x^{5} + 80 \, b c^{7} d^{6} x^{4} + 80 \, b^{2} c^{6} d^{6} x^{3} + 40 \, b^{3} c^{5} d^{6} x^{2} + 10 \, b^{4} c^{4} d^{6} x + b^{5} c^{3} d^{6}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 20.5602, size = 966, normalized size = 10.98 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.16077, size = 134, normalized size = 1.52 \begin{align*} -\frac{5 \, b^{4} d^{4} - 40 \, a b^{2} c d^{4} + 80 \, a^{2} c^{2} d^{4} - 18 \,{\left (2 \, c d x + b d\right )}^{2} b^{2} d^{2} + 72 \,{\left (2 \, c d x + b d\right )}^{2} a c d^{2} + 45 \,{\left (2 \, c d x + b d\right )}^{4}}{720 \,{\left (2 \, c d x + b d\right )}^{\frac{9}{2}} c^{3} d^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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