Optimal. Leaf size=47 \[ \frac {x^{m+1} (a+b x)^n \left (\frac {b x}{a}+1\right )^{-n} \, _2F_1\left (m+1,-n;m+2;-\frac {b x}{a}\right )}{m+1} \]
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Rubi [A] time = 0.01, antiderivative size = 47, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {66, 64} \[ \frac {x^{m+1} (a+b x)^n \left (\frac {b x}{a}+1\right )^{-n} \, _2F_1\left (m+1,-n;m+2;-\frac {b x}{a}\right )}{m+1} \]
Antiderivative was successfully verified.
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Rule 64
Rule 66
Rubi steps
\begin {align*} \int x^m (a+b x)^n \, dx &=\left ((a+b x)^n \left (1+\frac {b x}{a}\right )^{-n}\right ) \int x^m \left (1+\frac {b x}{a}\right )^n \, dx\\ &=\frac {x^{1+m} (a+b x)^n \left (1+\frac {b x}{a}\right )^{-n} \, _2F_1\left (1+m,-n;2+m;-\frac {b x}{a}\right )}{1+m}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 47, normalized size = 1.00 \[ \frac {x^{m+1} (a+b x)^n \left (\frac {b x}{a}+1\right )^{-n} \, _2F_1\left (m+1,-n;m+2;-\frac {b x}{a}\right )}{m+1} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.52, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (b x + a\right )}^{n} x^{m}, x\right ) \]
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 \[ \int {\left (b x + a\right )}^{n} x^{m}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.07, size = 0, normalized size = 0.00 \[ \int x^{m} \left (b x +a \right )^{n}\, dx \]
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 \[ \int {\left (b x + a\right )}^{n} x^{m}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int x^m\,{\left (a+b\,x\right )}^n \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 3.57, size = 34, normalized size = 0.72 \[ \frac {a^{n} x x^{m} \Gamma \left (m + 1\right ) {{}_{2}F_{1}\left (\begin {matrix} - n, m + 1 \\ m + 2 \end {matrix}\middle | {\frac {b x e^{i \pi }}{a}} \right )}}{\Gamma \left (m + 2\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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