Optimal. Leaf size=67 \[ x \left (b^2+b c x+c^2 x^2\right )^p \left (1-\frac {c^3 x^3}{b^3}\right )^{-p} (b e-c e x)^p \, _2F_1\left (\frac {1}{3},-p;\frac {4}{3};\frac {c^3 x^3}{b^3}\right ) \]
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Rubi [A] time = 0.03, 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 = {713, 246, 245} \[ x \left (b^2+b c x+c^2 x^2\right )^p \left (1-\frac {c^3 x^3}{b^3}\right )^{-p} (b e-c e x)^p \, _2F_1\left (\frac {1}{3},-p;\frac {4}{3};\frac {c^3 x^3}{b^3}\right ) \]
Antiderivative was successfully verified.
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Rule 245
Rule 246
Rule 713
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] time = 0.30, size = 243, normalized size = 3.63 \[ \frac {(c x-b) \left (\frac {-\sqrt {3} \sqrt {-b^2 c^2}+b c+2 c^2 x}{3 b c-\sqrt {3} \sqrt {-b^2 c^2}}\right )^{-p} \left (\frac {\sqrt {3} \sqrt {-b^2 c^2}+b c+2 c^2 x}{\sqrt {3} \sqrt {-b^2 c^2}+3 b c}\right )^{-p} \left (b^2+b c x+c^2 x^2\right )^p F_1\left (p+1;-p,-p;p+2;\frac {2 c (b-c x)}{3 b c+\sqrt {3} \sqrt {-b^2 c^2}},\frac {2 c (b-c x)}{3 b c-\sqrt {3} \sqrt {-b^2 c^2}}\right ) (e (b-c x))^p}{c (p+1)} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.62, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (c^{2} x^{2} + b c x + b^{2}\right )}^{p} {\left (-c e x + b e\right )}^{p}, x\right ) \]
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 \[ \int {\left (c^{2} x^{2} + b c x + b^{2}\right )}^{p} {\left (-c e x + b e\right )}^{p}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 2.12, 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 \[ \int {\left (c^{2} x^{2} + b c x + b^{2}\right )}^{p} {\left (-c e x + b e\right )}^{p}\,{d x} \]
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 \[ \int {\left (b\,e-c\,e\,x\right )}^p\,{\left (b^2+b\,c\,x+c^2\,x^2\right )}^p \,d x \]
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 \[ \int \left (- e \left (- b + c x\right )\right )^{p} \left (b^{2} + b c x + c^{2} x^{2}\right )^{p}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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