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 = 0.0334323, antiderivative size = 67, normalized size of antiderivative = 1., 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 713
Rule 246
Rule 245
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*}
Mathematica [C] time = 0.306608, 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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Maple [F] time = 3.016, size = 0, normalized size = 0. \begin{align*} \int \left ( -cex+be \right ) ^{p} \left ({c}^{2}{x}^{2}+bcx+{b}^{2} \right ) ^{p}\, 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{\left (c^{2} x^{2} + b c x + b^{2}\right )}^{p}{\left (-c e x + b e\right )}^{p}\,{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 ({\left (c^{2} x^{2} + b c x + b^{2}\right )}^{p}{\left (-c e x + b e\right )}^{p}, 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 \left (- e \left (- b + c x\right )\right )^{p} \left (b^{2} + b c x + c^{2} x^{2}\right )^{p}\, 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{\left (c^{2} x^{2} + b c x + b^{2}\right )}^{p}{\left (-c e x + b e\right )}^{p}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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