Optimal. Leaf size=71 \[ \frac{x \left (1-\frac{c^3 x^3}{b^3}\right )^{2/3} \, _2F_1\left (\frac{1}{3},\frac{2}{3};\frac{4}{3};\frac{c^3 x^3}{b^3}\right )}{\left (b^2+b c x+c^2 x^2\right )^{2/3} (b e-c e x)^{2/3}} \]
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Rubi [A] time = 0.0391936, antiderivative size = 71, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {713, 246, 245} \[ \frac{x \left (1-\frac{c^3 x^3}{b^3}\right )^{2/3} \, _2F_1\left (\frac{1}{3},\frac{2}{3};\frac{4}{3};\frac{c^3 x^3}{b^3}\right )}{\left (b^2+b c x+c^2 x^2\right )^{2/3} (b e-c e x)^{2/3}} \]
Antiderivative was successfully verified.
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Rule 713
Rule 246
Rule 245
Rubi steps
\begin{align*} \int \frac{1}{(b e-c e x)^{2/3} \left (b^2+b c x+c^2 x^2\right )^{2/3}} \, dx &=\frac{\left (b^3 e-c^3 e x^3\right )^{2/3} \int \frac{1}{\left (b^3 e-c^3 e x^3\right )^{2/3}} \, dx}{(b e-c e x)^{2/3} \left (b^2+b c x+c^2 x^2\right )^{2/3}}\\ &=\frac{\left (1-\frac{c^3 x^3}{b^3}\right )^{2/3} \int \frac{1}{\left (1-\frac{c^3 x^3}{b^3}\right )^{2/3}} \, dx}{(b e-c e x)^{2/3} \left (b^2+b c x+c^2 x^2\right )^{2/3}}\\ &=\frac{x \left (1-\frac{c^3 x^3}{b^3}\right )^{2/3} \, _2F_1\left (\frac{1}{3},\frac{2}{3};\frac{4}{3};\frac{c^3 x^3}{b^3}\right )}{(b e-c e x)^{2/3} \left (b^2+b c x+c^2 x^2\right )^{2/3}}\\ \end{align*}
Mathematica [B] time = 0.217441, size = 258, normalized size = 3.63 \[ -\frac{3 \left (-\sqrt{3} \sqrt{-b^2}+b+2 c x\right ) \left (\frac{-\sqrt{3} \sqrt{-b^2} c x+3 b^2+\sqrt{3} \sqrt{-b^2} b+3 b c x}{\sqrt{3} \sqrt{-b^2} c x+3 b^2-\sqrt{3} \sqrt{-b^2} b+3 b c x}\right )^{2/3} \sqrt [3]{e (b-c x)} \, _2F_1\left (\frac{1}{3},\frac{2}{3};\frac{4}{3};\frac{4 \sqrt{3} \sqrt{-b^2} (b-c x)}{\left (3 b+\sqrt{3} \sqrt{-b^2}\right ) \left (-b-2 c x+\sqrt{3} \sqrt{-b^2}\right )}\right )}{\left (3 b-\sqrt{3} \sqrt{-b^2}\right ) c e \left (b^2+b c x+c^2 x^2\right )^{2/3}} \]
Antiderivative was successfully verified.
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Maple [F] time = 2.985, size = 0, normalized size = 0. \begin{align*} \int{ \left ( -cex+be \right ) ^{-{\frac{2}{3}}} \left ({c}^{2}{x}^{2}+bcx+{b}^{2} \right ) ^{-{\frac{2}{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{1}{{\left (c^{2} x^{2} + b c x + b^{2}\right )}^{\frac{2}{3}}{\left (-c e x + b e\right )}^{\frac{2}{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 (c^{2} x^{2} + b c x + b^{2}\right )}^{\frac{1}{3}}{\left (-c e x + b e\right )}^{\frac{1}{3}}}{c^{3} e x^{3} - b^{3} e}, 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{1}{\left (- e \left (- b + c x\right )\right )^{\frac{2}{3}} \left (b^{2} + b c x + c^{2} x^{2}\right )^{\frac{2}{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{1}{{\left (c^{2} x^{2} + b c x + b^{2}\right )}^{\frac{2}{3}}{\left (-c e x + b e\right )}^{\frac{2}{3}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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