Optimal. Leaf size=45 \[ \frac {x^{2+n} (a+b x)^{-n} \left (1+\frac {b x}{a}\right )^n \, _2F_1\left (n,2+n;3+n;-\frac {b x}{a}\right )}{2+n} \]
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Rubi [A]
time = 0.01, antiderivative size = 45, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {68, 66}
\begin {gather*} \frac {x^{n+2} (a+b x)^{-n} \left (\frac {b x}{a}+1\right )^n \, _2F_1\left (n,n+2;n+3;-\frac {b x}{a}\right )}{n+2} \end {gather*}
Antiderivative was successfully verified.
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Rule 66
Rule 68
Rubi steps
\begin {align*} \int x^{1+n} (a+b x)^{-n} \, dx &=\left ((a+b x)^{-n} \left (1+\frac {b x}{a}\right )^n\right ) \int x^{1+n} \left (1+\frac {b x}{a}\right )^{-n} \, dx\\ &=\frac {x^{2+n} (a+b x)^{-n} \left (1+\frac {b x}{a}\right )^n \, _2F_1\left (n,2+n;3+n;-\frac {b x}{a}\right )}{2+n}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 45, normalized size = 1.00 \begin {gather*} \frac {x^{2+n} (a+b x)^{-n} \left (1+\frac {b x}{a}\right )^n \, _2F_1\left (n,2+n;3+n;-\frac {b x}{a}\right )}{2+n} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.05, size = 0, normalized size = 0.00 \[\int x^{1+n} \left (b x +a \right )^{-n}\, dx\]
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] Result contains complex when optimal does not.
time = 108.86, size = 34, normalized size = 0.76 \begin {gather*} \frac {a^{- n} x^{2} x^{n} \Gamma \left (n + 2\right ) {{}_{2}F_{1}\left (\begin {matrix} n, n + 2 \\ n + 3 \end {matrix}\middle | {\frac {b x e^{i \pi }}{a}} \right )}}{\Gamma \left (n + 3\right )} \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {x^{n+1}}{{\left (a+b\,x\right )}^n} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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