Optimal. Leaf size=44 \[ \frac{3 \left (a+b x+c x^2\right )^{7/3}}{7 d \left (b^2-4 a c\right ) (b d+2 c d x)^{14/3}} \]
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Rubi [A] time = 0.0177368, antiderivative size = 44, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.036, Rules used = {682} \[ \frac{3 \left (a+b x+c x^2\right )^{7/3}}{7 d \left (b^2-4 a c\right ) (b d+2 c d x)^{14/3}} \]
Antiderivative was successfully verified.
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Rule 682
Rubi steps
\begin{align*} \int \frac{\left (a+b x+c x^2\right )^{4/3}}{(b d+2 c d x)^{17/3}} \, dx &=\frac{3 \left (a+b x+c x^2\right )^{7/3}}{7 \left (b^2-4 a c\right ) d (b d+2 c d x)^{14/3}}\\ \end{align*}
Mathematica [A] time = 0.0500632, size = 50, normalized size = 1.14 \[ \frac{3 (a+x (b+c x))^{7/3} \sqrt [3]{d (b+2 c x)}}{7 d^6 \left (b^2-4 a c\right ) (b+2 c x)^5} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.046, size = 44, normalized size = 1. \begin{align*} -{\frac{6\,cx+3\,b}{28\,ac-7\,{b}^{2}} \left ( c{x}^{2}+bx+a \right ) ^{{\frac{7}{3}}} \left ( 2\,cdx+bd \right ) ^{-{\frac{17}{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{17}{3}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.95271, size = 416, normalized size = 9.45 \begin{align*} \frac{3 \,{\left (c^{2} x^{4} + 2 \, b c x^{3} + 2 \, a b x +{\left (b^{2} + 2 \, a c\right )} x^{2} + a^{2}\right )}{\left (2 \, c d x + b d\right )}^{\frac{1}{3}}{\left (c x^{2} + b x + a\right )}^{\frac{1}{3}}}{7 \,{\left (32 \,{\left (b^{2} c^{5} - 4 \, a c^{6}\right )} d^{6} x^{5} + 80 \,{\left (b^{3} c^{4} - 4 \, a b c^{5}\right )} d^{6} x^{4} + 80 \,{\left (b^{4} c^{3} - 4 \, a b^{2} c^{4}\right )} d^{6} x^{3} + 40 \,{\left (b^{5} c^{2} - 4 \, a b^{3} c^{3}\right )} d^{6} x^{2} + 10 \,{\left (b^{6} c - 4 \, a b^{4} c^{2}\right )} d^{6} x +{\left (b^{7} - 4 \, a b^{5} c\right )} d^{6}\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{17}{3}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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