Optimal. Leaf size=52 \[ -\frac {(a c-b c x)^{1+n} \, _2F_1\left (1,1+n;2+n;\frac {a-b x}{2 a}\right )}{2 a b c (1+n)} \]
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Rubi [A]
time = 0.02, antiderivative size = 52, normalized size of antiderivative = 1.00, number of steps
used = 1, number of rules used = 1, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {70}
\begin {gather*} -\frac {(a c-b c x)^{n+1} \, _2F_1\left (1,n+1;n+2;\frac {a-b x}{2 a}\right )}{2 a b c (n+1)} \end {gather*}
Antiderivative was successfully verified.
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Rule 70
Rubi steps
\begin {align*} \int \frac {(a c-b c x)^n}{a+b x} \, dx &=-\frac {(a c-b c x)^{1+n} \, _2F_1\left (1,1+n;2+n;\frac {a-b x}{2 a}\right )}{2 a b c (1+n)}\\ \end {align*}
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Mathematica [A]
time = 0.04, size = 52, normalized size = 1.00 \begin {gather*} -\frac {(a-b x) (c (a-b x))^n \, _2F_1\left (1,1+n;2+n;\frac {a-b x}{2 a}\right )}{2 a b (1+n)} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.06, size = 0, normalized size = 0.00 \[\int \frac {\left (-b c x +a c \right )^{n}}{b x +a}\, dx\]
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 [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (- c \left (- a + b x\right )\right )^{n}}{a + b x}\, dx \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 {{\left (a\,c-b\,c\,x\right )}^n}{a+b\,x} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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