Optimal. Leaf size=67 \[ x (b e-c e x)^p \left (b^2+b c x+c^2 x^2\right )^p \left (1-\frac {c^3 x^3}{b^3}\right )^{-p} \, _2F_1\left (\frac {1}{3},-p;\frac {4}{3};\frac {c^3 x^3}{b^3}\right ) \]
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Rubi [A]
time = 0.02, antiderivative size = 67, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.103, Rules used = {727, 252, 251}
\begin {gather*} x \left (b^2+b c x+c^2 x^2\right )^p \left (1-\frac {c^3 x^3}{b^3}\right )^{-p} (b e-c e x)^p \, _2F_1\left (\frac {1}{3},-p;\frac {4}{3};\frac {c^3 x^3}{b^3}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 251
Rule 252
Rule 727
Rubi steps
\begin {align*} \int (b e-c e x)^p \left (b^2+b c x+c^2 x^2\right )^p \, dx &=\left ((b e-c e x)^p \left (b^2+b c x+c^2 x^2\right )^p \left (b^3 e-c^3 e x^3\right )^{-p}\right ) \int \left (b^3 e-c^3 e x^3\right )^p \, dx\\ &=\left ((b e-c e x)^p \left (b^2+b c x+c^2 x^2\right )^p \left (1-\frac {c^3 x^3}{b^3}\right )^{-p}\right ) \int \left (1-\frac {c^3 x^3}{b^3}\right )^p \, dx\\ &=x (b e-c e x)^p \left (b^2+b c x+c^2 x^2\right )^p \left (1-\frac {c^3 x^3}{b^3}\right )^{-p} \, _2F_1\left (\frac {1}{3},-p;\frac {4}{3};\frac {c^3 x^3}{b^3}\right )\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 6 vs. order 5 in
optimal.
time = 0.24, size = 243, normalized size = 3.63 \begin {gather*} \frac {(e (b-c x))^p (-b+c x) \left (\frac {b c-\sqrt {3} \sqrt {-b^2 c^2}+2 c^2 x}{3 b c-\sqrt {3} \sqrt {-b^2 c^2}}\right )^{-p} \left (\frac {b c+\sqrt {3} \sqrt {-b^2 c^2}+2 c^2 x}{3 b c+\sqrt {3} \sqrt {-b^2 c^2}}\right )^{-p} \left (b^2+b c x+c^2 x^2\right )^p F_1\left (1+p;-p,-p;2+p;\frac {2 c (b-c x)}{3 b c+\sqrt {3} \sqrt {-b^2 c^2}},\frac {2 c (b-c x)}{3 b c-\sqrt {3} \sqrt {-b^2 c^2}}\right )}{c (1+p)} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.63, size = 0, normalized size = 0.00 \[\int \left (-c e x +b e \right )^{p} \left (c^{2} x^{2}+b c x +b^{2}\right )^{p}\, dx\]
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 \left (- e \left (- b + c x\right )\right )^{p} \left (b^{2} + b c x + c^{2} x^{2}\right )^{p}\, 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 {\left (b\,e-c\,e\,x\right )}^p\,{\left (b^2+b\,c\,x+c^2\,x^2\right )}^p \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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