Optimal. Leaf size=84 \[ -\frac {\left (a+\frac {b}{x}\right )^{1+n}}{b c (1+n)}-\frac {d \left (a+\frac {b}{x}\right )^{1+n} \, _2F_1\left (1,1+n;2+n;\frac {c \left (a+\frac {b}{x}\right )}{a c-b d}\right )}{c (a c-b d) (1+n)} \]
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Rubi [A]
time = 0.04, antiderivative size = 84, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {528, 457, 81,
70} \begin {gather*} -\frac {d \left (a+\frac {b}{x}\right )^{n+1} \, _2F_1\left (1,n+1;n+2;\frac {c \left (a+\frac {b}{x}\right )}{a c-b d}\right )}{c (n+1) (a c-b d)}-\frac {\left (a+\frac {b}{x}\right )^{n+1}}{b c (n+1)} \end {gather*}
Antiderivative was successfully verified.
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Rule 70
Rule 81
Rule 457
Rule 528
Rubi steps
\begin {align*} \int \frac {\left (a+\frac {b}{x}\right )^n}{x^2 (c+d x)} \, dx &=\int \frac {\left (a+\frac {b}{x}\right )^n}{\left (d+\frac {c}{x}\right ) x^3} \, dx\\ &=-\text {Subst}\left (\int \frac {x (a+b x)^n}{d+c x} \, dx,x,\frac {1}{x}\right )\\ &=-\frac {\left (a+\frac {b}{x}\right )^{1+n}}{b c (1+n)}+\frac {d \text {Subst}\left (\int \frac {(a+b x)^n}{d+c x} \, dx,x,\frac {1}{x}\right )}{c}\\ &=-\frac {\left (a+\frac {b}{x}\right )^{1+n}}{b c (1+n)}-\frac {d \left (a+\frac {b}{x}\right )^{1+n} \, _2F_1\left (1,1+n;2+n;\frac {c \left (a+\frac {b}{x}\right )}{a c-b d}\right )}{c (a c-b d) (1+n)}\\ \end {align*}
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Mathematica [A]
time = 0.14, size = 77, normalized size = 0.92 \begin {gather*} \frac {\left (a+\frac {b}{x}\right )^n (b+a x) \left (a c-b d+b d \, _2F_1\left (1,1+n;2+n;\frac {c \left (a+\frac {b}{x}\right )}{a c-b d}\right )\right )}{b c (-a c+b d) (1+n) x} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.07, size = 0, normalized size = 0.00 \[\int \frac {\left (a +\frac {b}{x}\right )^{n}}{x^{2} \left (d x +c \right )}\, dx\]
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 + \frac {b}{x}\right )^{n}}{x^{2} \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+\frac {b}{x}\right )}^n}{x^2\,\left (c+d\,x\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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