Optimal. Leaf size=105 \[ \frac {\left (a+\frac {b}{x}\right )^{n+1} (a c-b d (n+1)) \, _2F_1\left (1,n+1;n+2;\frac {c \left (a+\frac {b}{x}\right )}{a c-b d}\right )}{c (n+1) (a c-b d)^2}-\frac {d \left (a+\frac {b}{x}\right )^{n+1}}{c \left (\frac {c}{x}+d\right ) (a c-b d)} \]
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Rubi [A] time = 0.07, antiderivative size = 105, 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 = {514, 446, 78, 68} \[ \frac {\left (a+\frac {b}{x}\right )^{n+1} (a c-b d (n+1)) \, _2F_1\left (1,n+1;n+2;\frac {c \left (a+\frac {b}{x}\right )}{a c-b d}\right )}{c (n+1) (a c-b d)^2}-\frac {d \left (a+\frac {b}{x}\right )^{n+1}}{c \left (\frac {c}{x}+d\right ) (a c-b d)} \]
Antiderivative was successfully verified.
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Rule 68
Rule 78
Rule 446
Rule 514
Rubi steps
\begin {align*} \int \frac {\left (a+\frac {b}{x}\right )^n}{x (c+d x)^2} \, dx &=\int \frac {\left (a+\frac {b}{x}\right )^n}{\left (d+\frac {c}{x}\right )^2 x^3} \, dx\\ &=-\operatorname {Subst}\left (\int \frac {x (a+b x)^n}{(d+c x)^2} \, dx,x,\frac {1}{x}\right )\\ &=-\frac {d \left (a+\frac {b}{x}\right )^{1+n}}{c (a c-b d) \left (d+\frac {c}{x}\right )}-\frac {(a c-b d (1+n)) \operatorname {Subst}\left (\int \frac {(a+b x)^n}{d+c x} \, dx,x,\frac {1}{x}\right )}{c (a c-b d)}\\ &=-\frac {d \left (a+\frac {b}{x}\right )^{1+n}}{c (a c-b d) \left (d+\frac {c}{x}\right )}+\frac {(a c-b d (1+n)) \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)^2 (1+n)}\\ \end {align*}
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Mathematica [A] time = 0.12, size = 88, normalized size = 0.84 \[ \frac {\left (a+\frac {b}{x}\right )^{n+1} \left (\frac {(a c-b d (n+1)) \, _2F_1\left (1,n+1;n+2;\frac {c \left (a+\frac {b}{x}\right )}{a c-b d}\right )}{n+1}+\frac {d x (b d-a c)}{c+d x}\right )}{c (a c-b d)^2} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.95, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\left (\frac {a x + b}{x}\right )^{n}}{d^{2} x^{3} + 2 \, c d x^{2} + c^{2} x}, x\right ) \]
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 \[ \int \frac {{\left (a + \frac {b}{x}\right )}^{n}}{{\left (d x + c\right )}^{2} x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.56, size = 0, normalized size = 0.00 \[ \int \frac {\left (a +\frac {b}{x}\right )^{n}}{\left (d x +c \right )^{2} x}\, dx \]
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 \[ \int \frac {{\left (a + \frac {b}{x}\right )}^{n}}{{\left (d x + c\right )}^{2} x}\,{d x} \]
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 \[ \int \frac {{\left (a+\frac {b}{x}\right )}^n}{x\,{\left (c+d\,x\right )}^2} \,d x \]
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 \[ \int \frac {\left (a + \frac {b}{x}\right )^{n}}{x \left (c + d x\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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