Optimal. Leaf size=121 \[ -\frac{\left (b^2-4 a c\right )^2}{64 c^4 d^3 (b d+2 c d x)^{3/2}}-\frac{3 \left (b^2-4 a c\right ) \sqrt{b d+2 c d x}}{64 c^4 d^5}+\frac{\left (b^2-4 a c\right )^3}{448 c^4 d (b d+2 c d x)^{7/2}}+\frac{(b d+2 c d x)^{5/2}}{320 c^4 d^7} \]
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Rubi [A] time = 0.0504371, 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} \[ -\frac{\left (b^2-4 a c\right )^2}{64 c^4 d^3 (b d+2 c d x)^{3/2}}-\frac{3 \left (b^2-4 a c\right ) \sqrt{b d+2 c d x}}{64 c^4 d^5}+\frac{\left (b^2-4 a c\right )^3}{448 c^4 d (b d+2 c d x)^{7/2}}+\frac{(b d+2 c d x)^{5/2}}{320 c^4 d^7} \]
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)^{9/2}} \, dx &=\int \left (\frac{\left (-b^2+4 a c\right )^3}{64 c^3 (b d+2 c d x)^{9/2}}+\frac{3 \left (-b^2+4 a c\right )^2}{64 c^3 d^2 (b d+2 c d x)^{5/2}}+\frac{3 \left (-b^2+4 a c\right )}{64 c^3 d^4 \sqrt{b d+2 c d x}}+\frac{(b d+2 c d x)^{3/2}}{64 c^3 d^6}\right ) \, dx\\ &=\frac{\left (b^2-4 a c\right )^3}{448 c^4 d (b d+2 c d x)^{7/2}}-\frac{\left (b^2-4 a c\right )^2}{64 c^4 d^3 (b d+2 c d x)^{3/2}}-\frac{3 \left (b^2-4 a c\right ) \sqrt{b d+2 c d x}}{64 c^4 d^5}+\frac{(b d+2 c d x)^{5/2}}{320 c^4 d^7}\\ \end{align*}
Mathematica [A] time = 0.0701657, size = 83, normalized size = 0.69 \[ \frac{-105 \left (b^2-4 a c\right ) (b+2 c x)^4-35 \left (b^2-4 a c\right )^2 (b+2 c x)^2+5 \left (b^2-4 a c\right )^3+7 (b+2 c x)^6}{2240 c^4 d (d (b+2 c x))^{7/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.045, size = 163, normalized size = 1.4 \begin{align*} -{\frac{ \left ( 2\,cx+b \right ) \left ( -7\,{c}^{6}{x}^{6}-21\,b{c}^{5}{x}^{5}-105\,a{c}^{5}{x}^{4}-210\,ab{c}^{4}{x}^{3}+35\,{b}^{3}{c}^{3}{x}^{3}+35\,{a}^{2}{c}^{4}{x}^{2}-175\,a{b}^{2}{c}^{3}{x}^{2}+35\,{b}^{4}{c}^{2}{x}^{2}+35\,{a}^{2}b{c}^{3}x-70\,a{b}^{3}{c}^{2}x+14\,{b}^{5}cx+5\,{a}^{3}{c}^{3}+5\,{a}^{2}{b}^{2}{c}^{2}-10\,a{b}^{4}c+2\,{b}^{6} \right ) }{35\,{c}^{4}} \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 = 1.14639, size = 192, normalized size = 1.59 \begin{align*} -\frac{\frac{5 \,{\left (7 \,{\left (b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}\right )}{\left (2 \, c d x + b d\right )}^{2} -{\left (b^{6} - 12 \, a b^{4} c + 48 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}\right )} d^{2}\right )}}{{\left (2 \, c d x + b d\right )}^{\frac{7}{2}} c^{3} d^{2}} + \frac{7 \,{\left (15 \, \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^{3} d^{6}}}{2240 \, c d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.07601, size = 440, normalized size = 3.64 \begin{align*} \frac{{\left (7 \, c^{6} x^{6} + 21 \, b c^{5} x^{5} + 105 \, a c^{5} x^{4} - 2 \, b^{6} + 10 \, a b^{4} c - 5 \, a^{2} b^{2} c^{2} - 5 \, a^{3} c^{3} - 35 \,{\left (b^{3} c^{3} - 6 \, a b c^{4}\right )} x^{3} - 35 \,{\left (b^{4} c^{2} - 5 \, a b^{2} c^{3} + a^{2} c^{4}\right )} x^{2} - 7 \,{\left (2 \, b^{5} c - 10 \, a b^{3} c^{2} + 5 \, a^{2} b c^{3}\right )} x\right )} \sqrt{2 \, c d x + b d}}{35 \,{\left (16 \, c^{8} d^{5} x^{4} + 32 \, b c^{7} d^{5} x^{3} + 24 \, b^{2} c^{6} d^{5} x^{2} + 8 \, b^{3} c^{5} d^{5} x + b^{4} c^{4} d^{5}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 15.2915, size = 1394, normalized size = 11.52 \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 = 1.20658, size = 251, normalized size = 2.07 \begin{align*} \frac{b^{6} d^{2} - 12 \, a b^{4} c d^{2} + 48 \, a^{2} b^{2} c^{2} d^{2} - 64 \, a^{3} c^{3} d^{2} - 7 \,{\left (2 \, c d x + b d\right )}^{2} b^{4} + 56 \,{\left (2 \, c d x + b d\right )}^{2} a b^{2} c - 112 \,{\left (2 \, c d x + b d\right )}^{2} a^{2} c^{2}}{448 \,{\left (2 \, c d x + b d\right )}^{\frac{7}{2}} c^{4} d^{3}} - \frac{15 \, \sqrt{2 \, c d x + b d} b^{2} c^{16} d^{30} - 60 \, \sqrt{2 \, c d x + b d} a c^{17} d^{30} -{\left (2 \, c d x + b d\right )}^{\frac{5}{2}} c^{16} d^{28}}{320 \, c^{20} d^{35}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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