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 = 0.04, antiderivative size = 71, normalized size of antiderivative = 1.00, 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 245
Rule 246
Rule 713
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*}
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Mathematica [B] time = 0.21, 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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fricas [F] time = 0.82, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {{\left (c^{2} x^{2} + b c x + b^{2}\right )}^{\frac {1}{3}} {\left (-c e x + b e\right )}^{\frac {1}{3}}}{c^{3} e x^{3} - b^{3} e}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (c^{2} x^{2} + b c x + b^{2}\right )}^{\frac {2}{3}} {\left (-c e x + b e\right )}^{\frac {2}{3}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 3.10, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (-c e x +b e \right )^{\frac {2}{3}} \left (c^{2} x^{2}+b c x +b^{2}\right )^{\frac {2}{3}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (c^{2} x^{2} + b c x + b^{2}\right )}^{\frac {2}{3}} {\left (-c e x + b e\right )}^{\frac {2}{3}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {1}{{\left (b\,e-c\,e\,x\right )}^{2/3}\,{\left (b^2+b\,c\,x+c^2\,x^2\right )}^{2/3}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (- e \left (- b + c x\right )\right )^{\frac {2}{3}} \left (b^{2} + b c x + c^{2} x^{2}\right )^{\frac {2}{3}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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