Optimal. Leaf size=31 \[ -5^p (1-x) \, _2F_1\left (\frac {1}{2},-p;\frac {3}{2};\frac {2}{5} (1-x)^2\right ) \]
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Rubi [A] time = 0.01, antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {619, 245} \[ -5^p (1-x) \, _2F_1\left (\frac {1}{2},-p;\frac {3}{2};\frac {2}{5} (1-x)^2\right ) \]
Antiderivative was successfully verified.
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Rule 245
Rule 619
Rubi steps
\begin {align*} \int \left (3+4 x-2 x^2\right )^p \, dx &=-\left (\frac {1}{4} 5^p \operatorname {Subst}\left (\int \left (1-\frac {x^2}{40}\right )^p \, dx,x,4-4 x\right )\right )\\ &=-5^p (1-x) \, _2F_1\left (\frac {1}{2},-p;\frac {3}{2};\frac {2}{5} (1-x)^2\right )\\ \end {align*}
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Mathematica [A] time = 0.01, size = 26, normalized size = 0.84 \[ 5^p (x-1) \, _2F_1\left (\frac {1}{2},-p;\frac {3}{2};\frac {2}{5} (x-1)^2\right ) \]
Antiderivative was successfully verified.
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fricas [F] time = 1.01, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (-2 \, x^{2} + 4 \, x + 3\right )}^{p}, x\right ) \]
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 \[ \int {\left (-2 \, x^{2} + 4 \, x + 3\right )}^{p}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 1.12, size = 0, normalized size = 0.00 \[ \int \left (-2 x^{2}+4 x +3\right )^{p}\, dx \]
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 \[ \int {\left (-2 \, x^{2} + 4 \, x + 3\right )}^{p}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.03 \[ \int {\left (-2\,x^2+4\,x+3\right )}^p \,d x \]
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 \[ \int \left (- 2 x^{2} + 4 x + 3\right )^{p}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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