Optimal. Leaf size=71 \[ \frac {x \left (1-\frac {c^3 x^3}{b^3}\right )^{2/3} \, _2F_1\left (\frac {1}{3},\frac {2}{3};\frac {4}{3};\frac {c^3 x^3}{b^3}\right )}{(b e-c e x)^{2/3} \left (b^2+b c x+c^2 x^2\right )^{2/3}} \]
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Rubi [A]
time = 0.03, 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 = {727, 252, 251}
\begin {gather*} \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}} \end {gather*}
Antiderivative was successfully verified.
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Rule 251
Rule 252
Rule 727
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] Leaf count is larger than twice the leaf count of optimal. \(258\) vs. \(2(71)=142\).
time = 10.18, size = 258, normalized size = 3.63 \begin {gather*} -\frac {3 \sqrt [3]{e (b-c x)} \left (b-\sqrt {3} \sqrt {-b^2}+2 c x\right ) \left (\frac {3 b^2+\sqrt {3} b \sqrt {-b^2}+3 b c x-\sqrt {3} \sqrt {-b^2} c x}{3 b^2-\sqrt {3} b \sqrt {-b^2}+3 b c x+\sqrt {3} \sqrt {-b^2} c x}\right )^{2/3} \, _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+\sqrt {3} \sqrt {-b^2}-2 c x\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}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.48, 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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \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 {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \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 \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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