Optimal. Leaf size=133 \[ \frac{27 \left (a+b x+c x^2\right )^{7/3}}{455 d^5 \left (b^2-4 a c\right )^3 (b d+2 c d x)^{14/3}}+\frac{9 \left (a+b x+c x^2\right )^{7/3}}{65 d^3 \left (b^2-4 a c\right )^2 (b d+2 c d x)^{20/3}}+\frac{3 \left (a+b x+c x^2\right )^{7/3}}{13 d \left (b^2-4 a c\right ) (b d+2 c d x)^{26/3}} \]
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Rubi [A] time = 0.0658108, antiderivative size = 133, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.071, Rules used = {693, 682} \[ \frac{27 \left (a+b x+c x^2\right )^{7/3}}{455 d^5 \left (b^2-4 a c\right )^3 (b d+2 c d x)^{14/3}}+\frac{9 \left (a+b x+c x^2\right )^{7/3}}{65 d^3 \left (b^2-4 a c\right )^2 (b d+2 c d x)^{20/3}}+\frac{3 \left (a+b x+c x^2\right )^{7/3}}{13 d \left (b^2-4 a c\right ) (b d+2 c d x)^{26/3}} \]
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)^{29/3}} \, dx &=\frac{3 \left (a+b x+c x^2\right )^{7/3}}{13 \left (b^2-4 a c\right ) d (b d+2 c d x)^{26/3}}+\frac{6 \int \frac{\left (a+b x+c x^2\right )^{4/3}}{(b d+2 c d x)^{23/3}} \, dx}{13 \left (b^2-4 a c\right ) d^2}\\ &=\frac{3 \left (a+b x+c x^2\right )^{7/3}}{13 \left (b^2-4 a c\right ) d (b d+2 c d x)^{26/3}}+\frac{9 \left (a+b x+c x^2\right )^{7/3}}{65 \left (b^2-4 a c\right )^2 d^3 (b d+2 c d x)^{20/3}}+\frac{9 \int \frac{\left (a+b x+c x^2\right )^{4/3}}{(b d+2 c d x)^{17/3}} \, dx}{65 \left (b^2-4 a c\right )^2 d^4}\\ &=\frac{3 \left (a+b x+c x^2\right )^{7/3}}{13 \left (b^2-4 a c\right ) d (b d+2 c d x)^{26/3}}+\frac{9 \left (a+b x+c x^2\right )^{7/3}}{65 \left (b^2-4 a c\right )^2 d^3 (b d+2 c d x)^{20/3}}+\frac{27 \left (a+b x+c x^2\right )^{7/3}}{455 \left (b^2-4 a c\right )^3 d^5 (b d+2 c d x)^{14/3}}\\ \end{align*}
Mathematica [A] time = 0.104086, size = 122, normalized size = 0.92 \[ \frac{3 (a+x (b+c x))^{7/3} \left (16 c^2 \left (35 a^2-21 a c x^2+9 c^2 x^4\right )+4 b^2 c \left (75 c x^2-91 a\right )+48 b c^2 x \left (6 c x^2-7 a\right )+156 b^3 c x+65 b^4\right )}{455 d^9 \left (b^2-4 a c\right )^3 (b+2 c x)^8 (d (b+2 c x))^{2/3}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.048, size = 139, normalized size = 1.1 \begin{align*} -{\frac{ \left ( 6\,cx+3\,b \right ) \left ( 144\,{c}^{4}{x}^{4}+288\,b{c}^{3}{x}^{3}-336\,{x}^{2}a{c}^{3}+300\,{x}^{2}{b}^{2}{c}^{2}-336\,ba{c}^{2}x+156\,{b}^{3}cx+560\,{a}^{2}{c}^{2}-364\,ac{b}^{2}+65\,{b}^{4} \right ) }{29120\,{a}^{3}{c}^{3}-21840\,{a}^{2}{b}^{2}{c}^{2}+5460\,a{b}^{4}c-455\,{b}^{6}} \left ( c{x}^{2}+bx+a \right ) ^{{\frac{7}{3}}} \left ( 2\,cdx+bd \right ) ^{-{\frac{29}{3}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 2.55788, size = 1057, normalized size = 7.95 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 4.3618, size = 1536, normalized size = 11.55 \begin{align*} \frac{3 \,{\left (144 \, c^{6} x^{8} + 576 \, b c^{5} x^{7} + 12 \,{\left (85 \, b^{2} c^{4} - 4 \, a c^{5}\right )} x^{6} + 65 \, a^{2} b^{4} - 364 \, a^{3} b^{2} c + 560 \, a^{4} c^{2} + 36 \,{\left (29 \, b^{3} c^{3} - 4 \, a b c^{4}\right )} x^{5} +{\left (677 \, b^{4} c^{2} - 196 \, a b^{2} c^{3} + 32 \, a^{2} c^{4}\right )} x^{4} + 2 \,{\left (143 \, b^{5} c - 76 \, a b^{3} c^{2} + 32 \, a^{2} b c^{3}\right )} x^{3} +{\left (65 \, b^{6} + 78 \, a b^{4} c - 540 \, a^{2} b^{2} c^{2} + 784 \, a^{3} c^{3}\right )} x^{2} + 2 \,{\left (65 \, a b^{5} - 286 \, a^{2} b^{3} c + 392 \, a^{3} b c^{2}\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}}}{455 \,{\left (512 \,{\left (b^{6} c^{9} - 12 \, a b^{4} c^{10} + 48 \, a^{2} b^{2} c^{11} - 64 \, a^{3} c^{12}\right )} d^{10} x^{9} + 2304 \,{\left (b^{7} c^{8} - 12 \, a b^{5} c^{9} + 48 \, a^{2} b^{3} c^{10} - 64 \, a^{3} b c^{11}\right )} d^{10} x^{8} + 4608 \,{\left (b^{8} c^{7} - 12 \, a b^{6} c^{8} + 48 \, a^{2} b^{4} c^{9} - 64 \, a^{3} b^{2} c^{10}\right )} d^{10} x^{7} + 5376 \,{\left (b^{9} c^{6} - 12 \, a b^{7} c^{7} + 48 \, a^{2} b^{5} c^{8} - 64 \, a^{3} b^{3} c^{9}\right )} d^{10} x^{6} + 4032 \,{\left (b^{10} c^{5} - 12 \, a b^{8} c^{6} + 48 \, a^{2} b^{6} c^{7} - 64 \, a^{3} b^{4} c^{8}\right )} d^{10} x^{5} + 2016 \,{\left (b^{11} c^{4} - 12 \, a b^{9} c^{5} + 48 \, a^{2} b^{7} c^{6} - 64 \, a^{3} b^{5} c^{7}\right )} d^{10} x^{4} + 672 \,{\left (b^{12} c^{3} - 12 \, a b^{10} c^{4} + 48 \, a^{2} b^{8} c^{5} - 64 \, a^{3} b^{6} c^{6}\right )} d^{10} x^{3} + 144 \,{\left (b^{13} c^{2} - 12 \, a b^{11} c^{3} + 48 \, a^{2} b^{9} c^{4} - 64 \, a^{3} b^{7} c^{5}\right )} d^{10} x^{2} + 18 \,{\left (b^{14} c - 12 \, a b^{12} c^{2} + 48 \, a^{2} b^{10} c^{3} - 64 \, a^{3} b^{8} c^{4}\right )} d^{10} x +{\left (b^{15} - 12 \, a b^{13} c + 48 \, a^{2} b^{11} c^{2} - 64 \, a^{3} b^{9} c^{3}\right )} d^{10}\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{29}{3}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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