Optimal. Leaf size=45 \[ \frac {x^{1+n} (a+b x)^{-n} \left (1+\frac {b x}{a}\right )^n \, _2F_1\left (n,1+n;2+n;-\frac {b x}{a}\right )}{1+n} \]
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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 = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {68, 66}
\begin {gather*} \frac {x^{n+1} (a+b x)^{-n} \left (\frac {b x}{a}+1\right )^n \, _2F_1\left (n,n+1;n+2;-\frac {b x}{a}\right )}{n+1} \end {gather*}
Antiderivative was successfully verified.
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Rule 66
Rule 68
Rubi steps
\begin {align*} \int x^n (a+b x)^{-n} \, dx &=\left ((a+b x)^{-n} \left (1+\frac {b x}{a}\right )^n\right ) \int x^n \left (1+\frac {b x}{a}\right )^{-n} \, dx\\ &=\frac {x^{1+n} (a+b x)^{-n} \left (1+\frac {b x}{a}\right )^n \, _2F_1\left (n,1+n;2+n;-\frac {b x}{a}\right )}{1+n}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 45, normalized size = 1.00 \begin {gather*} \frac {x^{1+n} (a+b x)^{-n} \left (1+\frac {b x}{a}\right )^n \, _2F_1\left (n,1+n;2+n;-\frac {b x}{a}\right )}{1+n} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.04, size = 0, normalized size = 0.00 \[\int x^{n} \left (b x +a \right )^{-n}\, dx\]
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 = 14.33, size = 32, normalized size = 0.71 \begin {gather*} \frac {a^{- n} x x^{n} \Gamma \left (n + 1\right ) {{}_{2}F_{1}\left (\begin {matrix} n, n + 1 \\ n + 2 \end {matrix}\middle | {\frac {b x e^{i \pi }}{a}} \right )}}{\Gamma \left (n + 2\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}{{\left (a+b\,x\right )}^n} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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