Optimal. Leaf size=35 \[ \frac {2^n (b x)^{m+1} \, _2F_1\left (m+1,-n;m+2;-\frac {d x}{2}\right )}{b (m+1)} \]
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Rubi [A] time = 0.01, antiderivative size = 35, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {64} \[ \frac {2^n (b x)^{m+1} \, _2F_1\left (m+1,-n;m+2;-\frac {d x}{2}\right )}{b (m+1)} \]
Antiderivative was successfully verified.
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Rule 64
Rubi steps
\begin {align*} \int (b x)^m (2+d x)^n \, dx &=\frac {2^n (b x)^{1+m} \, _2F_1\left (1+m,-n;2+m;-\frac {d x}{2}\right )}{b (1+m)}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 31, normalized size = 0.89 \[ \frac {2^n x (b x)^m \, _2F_1\left (m+1,-n;m+2;-\frac {d x}{2}\right )}{m+1} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.46, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\left (b x\right )^{m} {\left (d x + 2\right )}^{n}, x\right ) \]
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 \[ \int \left (b x\right )^{m} {\left (d x + 2\right )}^{n}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.09, size = 32, normalized size = 0.91 \[ \frac {x 2^{n} \left (b x \right )^{m} \hypergeom \left (\left [-n , m +1\right ], \left [m +2\right ], -\frac {d x}{2}\right )}{m +1} \]
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 \[ \int \left (b x\right )^{m} {\left (d x + 2\right )}^{n}\,{d x} \]
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 \[ \int {\left (b\,x\right )}^m\,{\left (d\,x+2\right )}^n \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 2.53, size = 37, normalized size = 1.06 \[ \frac {2^{n} b^{m} x x^{m} \Gamma \left (m + 1\right ) {{}_{2}F_{1}\left (\begin {matrix} - n, m + 1 \\ m + 2 \end {matrix}\middle | {\frac {d x e^{i \pi }}{2}} \right )}}{\Gamma \left (m + 2\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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