Optimal. Leaf size=42 \[ \frac {2^n x^{m+1} \, _2F_1\left (\frac {m+1}{2},-n;\frac {m+3}{2};\frac {a^2 x^2}{4}\right )}{m+1} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.01, antiderivative size = 42, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {125, 364} \[ \frac {2^n x^{m+1} \, _2F_1\left (\frac {m+1}{2},-n;\frac {m+3}{2};\frac {a^2 x^2}{4}\right )}{m+1} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 125
Rule 364
Rubi steps
\begin {align*} \int x^m \left (1-\frac {a x}{2}\right )^n (2+a x)^n \, dx &=\int x^m \left (2-\frac {a^2 x^2}{2}\right )^n \, dx\\ &=\frac {2^n x^{1+m} \, _2F_1\left (\frac {1+m}{2},-n;\frac {3+m}{2};\frac {a^2 x^2}{4}\right )}{1+m}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.02, size = 42, normalized size = 1.00 \[ \frac {2^n x^{m+1} \, _2F_1\left (\frac {m+1}{2},-n;\frac {m+3}{2};\frac {a^2 x^2}{4}\right )}{m+1} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.89, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (a x + 2\right )}^{n} {\left (-\frac {1}{2} \, a x + 1\right )}^{n} x^{m}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (a x + 2\right )}^{n} {\left (-\frac {1}{2} \, a x + 1\right )}^{n} x^{m}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 0.16, size = 0, normalized size = 0.00 \[ \int x^{m} \left (-\frac {a x}{2}+1\right )^{n} \left (a x +2\right )^{n}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (a x + 2\right )}^{n} {\left (-\frac {1}{2} \, a x + 1\right )}^{n} x^{m}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int x^m\,{\left (a\,x+2\right )}^n\,{\left (1-\frac {a\,x}{2}\right )}^n \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________