Optimal. Leaf size=121 \[ -\frac{3 \left (b^2-4 a c\right ) (b d+2 c d x)^{5/2}}{320 c^4 d^5}+\frac{3 \left (b^2-4 a c\right )^2 \sqrt{b d+2 c d x}}{64 c^4 d^3}+\frac{\left (b^2-4 a c\right )^3}{192 c^4 d (b d+2 c d x)^{3/2}}+\frac{(b d+2 c d x)^{9/2}}{576 c^4 d^7} \]
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Rubi [A] time = 0.049903, antiderivative size = 121, 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} \[ -\frac{3 \left (b^2-4 a c\right ) (b d+2 c d x)^{5/2}}{320 c^4 d^5}+\frac{3 \left (b^2-4 a c\right )^2 \sqrt{b d+2 c d x}}{64 c^4 d^3}+\frac{\left (b^2-4 a c\right )^3}{192 c^4 d (b d+2 c d x)^{3/2}}+\frac{(b d+2 c d x)^{9/2}}{576 c^4 d^7} \]
Antiderivative was successfully verified.
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Rule 683
Rubi steps
\begin{align*} \int \frac{\left (a+b x+c x^2\right )^3}{(b d+2 c d x)^{5/2}} \, dx &=\int \left (\frac{\left (-b^2+4 a c\right )^3}{64 c^3 (b d+2 c d x)^{5/2}}+\frac{3 \left (-b^2+4 a c\right )^2}{64 c^3 d^2 \sqrt{b d+2 c d x}}+\frac{3 \left (-b^2+4 a c\right ) (b d+2 c d x)^{3/2}}{64 c^3 d^4}+\frac{(b d+2 c d x)^{7/2}}{64 c^3 d^6}\right ) \, dx\\ &=\frac{\left (b^2-4 a c\right )^3}{192 c^4 d (b d+2 c d x)^{3/2}}+\frac{3 \left (b^2-4 a c\right )^2 \sqrt{b d+2 c d x}}{64 c^4 d^3}-\frac{3 \left (b^2-4 a c\right ) (b d+2 c d x)^{5/2}}{320 c^4 d^5}+\frac{(b d+2 c d x)^{9/2}}{576 c^4 d^7}\\ \end{align*}
Mathematica [A] time = 0.0692922, size = 83, normalized size = 0.69 \[ \frac{-27 \left (b^2-4 a c\right ) (b+2 c x)^4+135 \left (b^2-4 a c\right )^2 (b+2 c x)^2+15 \left (b^2-4 a c\right )^3+5 (b+2 c x)^6}{2880 c^4 d (d (b+2 c x))^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.043, size = 173, normalized size = 1.4 \begin{align*} -{\frac{ \left ( 2\,cx+b \right ) \left ( -5\,{c}^{6}{x}^{6}-15\,b{c}^{5}{x}^{5}-27\,a{c}^{5}{x}^{4}-12\,{b}^{2}{c}^{4}{x}^{4}-54\,ab{c}^{4}{x}^{3}+{b}^{3}{c}^{3}{x}^{3}-135\,{a}^{2}{c}^{4}{x}^{2}+27\,a{b}^{2}{c}^{3}{x}^{2}-3\,{b}^{4}{c}^{2}{x}^{2}-135\,{a}^{2}b{c}^{3}x+54\,a{b}^{3}{c}^{2}x-6\,{b}^{5}cx+15\,{a}^{3}{c}^{3}-45\,{a}^{2}{b}^{2}{c}^{2}+18\,a{b}^{4}c-2\,{b}^{6} \right ) }{45\,{c}^{4}} \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 = 1.04897, size = 184, normalized size = 1.52 \begin{align*} \frac{\frac{15 \,{\left (b^{6} - 12 \, a b^{4} c + 48 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}\right )}}{{\left (2 \, c d x + b d\right )}^{\frac{3}{2}} c^{3}} - \frac{27 \,{\left (2 \, c d x + b d\right )}^{\frac{5}{2}}{\left (b^{2} - 4 \, a c\right )} d^{2} - 135 \,{\left (b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}\right )} \sqrt{2 \, c d x + b d} d^{4} - 5 \,{\left (2 \, c d x + b d\right )}^{\frac{9}{2}}}{c^{3} d^{6}}}{2880 \, c d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.10276, size = 405, normalized size = 3.35 \begin{align*} \frac{{\left (5 \, c^{6} x^{6} + 15 \, b c^{5} x^{5} + 2 \, b^{6} - 18 \, a b^{4} c + 45 \, a^{2} b^{2} c^{2} - 15 \, a^{3} c^{3} + 3 \,{\left (4 \, b^{2} c^{4} + 9 \, a c^{5}\right )} x^{4} -{\left (b^{3} c^{3} - 54 \, a b c^{4}\right )} x^{3} + 3 \,{\left (b^{4} c^{2} - 9 \, a b^{2} c^{3} + 45 \, a^{2} c^{4}\right )} x^{2} + 3 \,{\left (2 \, b^{5} c - 18 \, a b^{3} c^{2} + 45 \, a^{2} b c^{3}\right )} x\right )} \sqrt{2 \, c d x + b d}}{45 \,{\left (4 \, c^{6} d^{3} x^{2} + 4 \, b c^{5} d^{3} x + b^{2} c^{4} d^{3}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 95.0958, size = 128, normalized size = 1.06 \begin{align*} - \frac{\left (4 a c - b^{2}\right )^{3}}{192 c^{4} d \left (b d + 2 c d x\right )^{\frac{3}{2}}} + \frac{\sqrt{b d + 2 c d x} \left (48 a^{2} c^{2} - 24 a b^{2} c + 3 b^{4}\right )}{64 c^{4} d^{3}} + \frac{\left (12 a c - 3 b^{2}\right ) \left (b d + 2 c d x\right )^{\frac{5}{2}}}{320 c^{4} d^{5}} + \frac{\left (b d + 2 c d x\right )^{\frac{9}{2}}}{576 c^{4} d^{7}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.18633, size = 252, normalized size = 2.08 \begin{align*} \frac{b^{6} - 12 \, a b^{4} c + 48 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}}{192 \,{\left (2 \, c d x + b d\right )}^{\frac{3}{2}} c^{4} d} + \frac{135 \, \sqrt{2 \, c d x + b d} b^{4} c^{32} d^{60} - 1080 \, \sqrt{2 \, c d x + b d} a b^{2} c^{33} d^{60} + 2160 \, \sqrt{2 \, c d x + b d} a^{2} c^{34} d^{60} - 27 \,{\left (2 \, c d x + b d\right )}^{\frac{5}{2}} b^{2} c^{32} d^{58} + 108 \,{\left (2 \, c d x + b d\right )}^{\frac{5}{2}} a c^{33} d^{58} + 5 \,{\left (2 \, c d x + b d\right )}^{\frac{9}{2}} c^{32} d^{56}}{2880 \, c^{36} d^{63}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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