Optimal. Leaf size=95 \[ -\frac {d (a+b x)^{1+n} \, _2F_1\left (1,1+n;2+n;-\frac {d (a+b x)}{b c-a d}\right )}{c (b c-a d) (1+n)}-\frac {(a+b x)^{1+n} \, _2F_1\left (1,1+n;2+n;1+\frac {b x}{a}\right )}{a c (1+n)} \]
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Rubi [A]
time = 0.02, antiderivative size = 95, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {88, 67, 70}
\begin {gather*} -\frac {d (a+b x)^{n+1} \, _2F_1\left (1,n+1;n+2;-\frac {d (a+b x)}{b c-a d}\right )}{c (n+1) (b c-a d)}-\frac {(a+b x)^{n+1} \, _2F_1\left (1,n+1;n+2;\frac {b x}{a}+1\right )}{a c (n+1)} \end {gather*}
Antiderivative was successfully verified.
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Rule 67
Rule 70
Rule 88
Rubi steps
\begin {align*} \int \frac {(a+b x)^n}{x (c+d x)} \, dx &=\frac {\int \frac {(a+b x)^n}{x} \, dx}{c}-\frac {d \int \frac {(a+b x)^n}{c+d x} \, dx}{c}\\ &=-\frac {d (a+b x)^{1+n} \, _2F_1\left (1,1+n;2+n;-\frac {d (a+b x)}{b c-a d}\right )}{c (b c-a d) (1+n)}-\frac {(a+b x)^{1+n} \, _2F_1\left (1,1+n;2+n;1+\frac {b x}{a}\right )}{a c (1+n)}\\ \end {align*}
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Mathematica [A]
time = 0.07, size = 85, normalized size = 0.89 \begin {gather*} \frac {(a+b x)^{1+n} \left (a d \, _2F_1\left (1,1+n;2+n;\frac {d (a+b x)}{-b c+a d}\right )+(b c-a d) \, _2F_1\left (1,1+n;2+n;1+\frac {b x}{a}\right )\right )}{a c (-b c+a d) (1+n)} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int \frac {\left (b x +a \right )^{n}}{x \left (d x +c \right )}\, dx\]
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 (a + b x\right )^{n}}{x \left (c + d x\right )}\, 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 = -0.01 \begin {gather*} \int \frac {{\left (a+b\,x\right )}^n}{x\,\left (c+d\,x\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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