Optimal. Leaf size=99 \[ \frac{3 \left (b^2-4 a c\right ) \sqrt [3]{a+b x+c x^2} \, _2F_1\left (-\frac{4}{3},-\frac{1}{6};\frac{5}{6};\frac{(b+2 c x)^2}{b^2-4 a c}\right )}{8 c^2 d \sqrt [3]{1-\frac{(b+2 c x)^2}{b^2-4 a c}} \sqrt [3]{d (b+2 c x)}} \]
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Rubi [A] time = 0.124247, antiderivative size = 99, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.107, Rules used = {694, 365, 364} \[ \frac{3 \left (b^2-4 a c\right ) \sqrt [3]{a+b x+c x^2} \, _2F_1\left (-\frac{4}{3},-\frac{1}{6};\frac{5}{6};\frac{(b+2 c x)^2}{b^2-4 a c}\right )}{8 c^2 d \sqrt [3]{1-\frac{(b+2 c x)^2}{b^2-4 a c}} \sqrt [3]{d (b+2 c x)}} \]
Antiderivative was successfully verified.
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Rule 694
Rule 365
Rule 364
Rubi steps
\begin{align*} \int \frac{\left (a+b x+c x^2\right )^{4/3}}{(b d+2 c d x)^{4/3}} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\left (a-\frac{b^2}{4 c}+\frac{x^2}{4 c d^2}\right )^{4/3}}{x^{4/3}} \, dx,x,b d+2 c d x\right )}{2 c d}\\ &=\frac{\left (\left (a-\frac{b^2}{4 c}\right ) \sqrt [3]{a+b x+c x^2}\right ) \operatorname{Subst}\left (\int \frac{\left (1+\frac{x^2}{4 \left (a-\frac{b^2}{4 c}\right ) c d^2}\right )^{4/3}}{x^{4/3}} \, dx,x,b d+2 c d x\right )}{\sqrt [3]{2} c d \sqrt [3]{4+\frac{(b d+2 c d x)^2}{\left (a-\frac{b^2}{4 c}\right ) c d^2}}}\\ &=\frac{3 \left (b^2-4 a c\right ) \sqrt [3]{a+b x+c x^2} \, _2F_1\left (-\frac{4}{3},-\frac{1}{6};\frac{5}{6};\frac{(b+2 c x)^2}{b^2-4 a c}\right )}{8 c^2 d \sqrt [3]{d (b+2 c x)} \sqrt [3]{1-\frac{(b+2 c x)^2}{b^2-4 a c}}}\\ \end{align*}
Mathematica [A] time = 0.0669767, size = 104, normalized size = 1.05 \[ \frac{3 \left (b^2-4 a c\right ) \sqrt [3]{a+x (b+c x)} \, _2F_1\left (-\frac{4}{3},-\frac{1}{6};\frac{5}{6};\frac{(b+2 c x)^2}{b^2-4 a c}\right )}{8\ 2^{2/3} c^2 d \sqrt [3]{\frac{c (a+x (b+c x))}{4 a c-b^2}} \sqrt [3]{d (b+2 c x)}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.413, size = 0, normalized size = 0. \begin{align*} \int{ \left ( c{x}^{2}+bx+a \right ) ^{{\frac{4}{3}}} \left ( 2\,cdx+bd \right ) ^{-{\frac{4}{3}}}}\, dx \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{4}{3}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (2 \, c d x + b d\right )}^{\frac{2}{3}}{\left (c x^{2} + b x + a\right )}^{\frac{4}{3}}}{4 \, c^{2} d^{2} x^{2} + 4 \, b c d^{2} x + b^{2} d^{2}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (a + b x + c x^{2}\right )^{\frac{4}{3}}}{\left (d \left (b + 2 c x\right )\right )^{\frac{4}{3}}}\, dx \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{4}{3}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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