Optimal. Leaf size=35 \[ -2^{2 p-1} (1-2 x) \, _2F_1\left (\frac{1}{2},-p;\frac{3}{2};\frac{1}{4} (1-2 x)^2\right ) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0133281, antiderivative size = 35, normalized size of antiderivative = 1., 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} \[ -2^{2 p-1} (1-2 x) \, _2F_1\left (\frac{1}{2},-p;\frac{3}{2};\frac{1}{4} (1-2 x)^2\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 619
Rule 245
Rubi steps
\begin{align*} \int \left (3+4 x-4 x^2\right )^p \, dx &=-\left (2^{-3+2 p} \operatorname{Subst}\left (\int \left (1-\frac{x^2}{64}\right )^p \, dx,x,4-8 x\right )\right )\\ &=-2^{-1+2 p} (1-2 x) \, _2F_1\left (\frac{1}{2},-p;\frac{3}{2};\frac{1}{4} (1-2 x)^2\right )\\ \end{align*}
Mathematica [A] time = 0.0077285, size = 35, normalized size = 1. \[ -2^{2 p-3} (4-8 x) \, _2F_1\left (\frac{1}{2},-p;\frac{3}{2};\frac{1}{64} (4-8 x)^2\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.919, size = 0, normalized size = 0. \begin{align*} \int \left ( -4\,{x}^{2}+4\,x+3 \right ) ^{p}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (-4 \, x^{2} + 4 \, x + 3\right )}^{p}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (-4 \, x^{2} + 4 \, x + 3\right )}^{p}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (- 4 x^{2} + 4 x + 3\right )^{p}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (-4 \, x^{2} + 4 \, x + 3\right )}^{p}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]