Optimal. Leaf size=89 \[ \frac{9 \left (a+b x+c x^2\right )^{7/3}}{70 d^3 \left (b^2-4 a c\right )^2 (b d+2 c d x)^{14/3}}+\frac{3 \left (a+b x+c x^2\right )^{7/3}}{10 d \left (b^2-4 a c\right ) (b d+2 c d x)^{20/3}} \]
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Rubi [A] time = 0.0395993, antiderivative size = 89, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.071, Rules used = {693, 682} \[ \frac{9 \left (a+b x+c x^2\right )^{7/3}}{70 d^3 \left (b^2-4 a c\right )^2 (b d+2 c d x)^{14/3}}+\frac{3 \left (a+b x+c x^2\right )^{7/3}}{10 d \left (b^2-4 a c\right ) (b d+2 c d x)^{20/3}} \]
Antiderivative was successfully verified.
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Rule 693
Rule 682
Rubi steps
\begin{align*} \int \frac{\left (a+b x+c x^2\right )^{4/3}}{(b d+2 c d x)^{23/3}} \, dx &=\frac{3 \left (a+b x+c x^2\right )^{7/3}}{10 \left (b^2-4 a c\right ) d (b d+2 c d x)^{20/3}}+\frac{3 \int \frac{\left (a+b x+c x^2\right )^{4/3}}{(b d+2 c d x)^{17/3}} \, dx}{10 \left (b^2-4 a c\right ) d^2}\\ &=\frac{3 \left (a+b x+c x^2\right )^{7/3}}{10 \left (b^2-4 a c\right ) d (b d+2 c d x)^{20/3}}+\frac{9 \left (a+b x+c x^2\right )^{7/3}}{70 \left (b^2-4 a c\right )^2 d^3 (b d+2 c d x)^{14/3}}\\ \end{align*}
Mathematica [A] time = 0.0776626, size = 74, normalized size = 0.83 \[ \frac{3 (a+x (b+c x))^{7/3} \left (2 c \left (3 c x^2-7 a\right )+5 b^2+6 b c x\right ) \sqrt [3]{d (b+2 c x)}}{35 d^8 \left (b^2-4 a c\right )^2 (b+2 c x)^7} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.044, size = 76, normalized size = 0.9 \begin{align*} -{\frac{ \left ( 6\,cx+3\,b \right ) \left ( -6\,{c}^{2}{x}^{2}-6\,bcx+14\,ac-5\,{b}^{2} \right ) }{560\,{a}^{2}{c}^{2}-280\,ac{b}^{2}+35\,{b}^{4}} \left ( c{x}^{2}+bx+a \right ) ^{{\frac{7}{3}}} \left ( 2\,cdx+bd \right ) ^{-{\frac{23}{3}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (c x^{2} + b x + a\right )}^{\frac{4}{3}}}{{\left (2 \, c d x + b d\right )}^{\frac{23}{3}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 3.58114, size = 868, normalized size = 9.75 \begin{align*} \frac{3 \,{\left (6 \, c^{4} x^{6} + 18 \, b c^{3} x^{5} +{\left (23 \, b^{2} c^{2} - 2 \, a c^{3}\right )} x^{4} + 5 \, a^{2} b^{2} - 14 \, a^{3} c + 4 \,{\left (4 \, b^{3} c - a b c^{2}\right )} x^{3} +{\left (5 \, b^{4} + 8 \, a b^{2} c - 22 \, a^{2} c^{2}\right )} x^{2} + 2 \,{\left (5 \, a b^{3} - 11 \, a^{2} b c\right )} x\right )}{\left (2 \, c d x + b d\right )}^{\frac{1}{3}}{\left (c x^{2} + b x + a\right )}^{\frac{1}{3}}}{35 \,{\left (128 \,{\left (b^{4} c^{7} - 8 \, a b^{2} c^{8} + 16 \, a^{2} c^{9}\right )} d^{8} x^{7} + 448 \,{\left (b^{5} c^{6} - 8 \, a b^{3} c^{7} + 16 \, a^{2} b c^{8}\right )} d^{8} x^{6} + 672 \,{\left (b^{6} c^{5} - 8 \, a b^{4} c^{6} + 16 \, a^{2} b^{2} c^{7}\right )} d^{8} x^{5} + 560 \,{\left (b^{7} c^{4} - 8 \, a b^{5} c^{5} + 16 \, a^{2} b^{3} c^{6}\right )} d^{8} x^{4} + 280 \,{\left (b^{8} c^{3} - 8 \, a b^{6} c^{4} + 16 \, a^{2} b^{4} c^{5}\right )} d^{8} x^{3} + 84 \,{\left (b^{9} c^{2} - 8 \, a b^{7} c^{3} + 16 \, a^{2} b^{5} c^{4}\right )} d^{8} x^{2} + 14 \,{\left (b^{10} c - 8 \, a b^{8} c^{2} + 16 \, a^{2} b^{6} c^{3}\right )} d^{8} x +{\left (b^{11} - 8 \, a b^{9} c + 16 \, a^{2} b^{7} c^{2}\right )} d^{8}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (c x^{2} + b x + a\right )}^{\frac{4}{3}}}{{\left (2 \, c d x + b d\right )}^{\frac{23}{3}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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