Optimal. Leaf size=133 \[ \frac {d^2 \left (a+\frac {b}{x}\right )^{n+1}}{c^2 \left (\frac {c}{x}+d\right ) (a c-b d)}-\frac {d \left (a+\frac {b}{x}\right )^{n+1} (2 a c-b d (n+2)) \, _2F_1\left (1,n+1;n+2;\frac {c \left (a+\frac {b}{x}\right )}{a c-b d}\right )}{c^2 (n+1) (a c-b d)^2}-\frac {\left (a+\frac {b}{x}\right )^{n+1}}{b c^2 (n+1)} \]
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Rubi [A] time = 0.13, antiderivative size = 133, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {514, 446, 89, 80, 68} \[ \frac {d^2 \left (a+\frac {b}{x}\right )^{n+1}}{c^2 \left (\frac {c}{x}+d\right ) (a c-b d)}-\frac {d \left (a+\frac {b}{x}\right )^{n+1} (2 a c-b d (n+2)) \, _2F_1\left (1,n+1;n+2;\frac {c \left (a+\frac {b}{x}\right )}{a c-b d}\right )}{c^2 (n+1) (a c-b d)^2}-\frac {\left (a+\frac {b}{x}\right )^{n+1}}{b c^2 (n+1)} \]
Antiderivative was successfully verified.
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Rule 68
Rule 80
Rule 89
Rule 446
Rule 514
Rubi steps
\begin {align*} \int \frac {\left (a+\frac {b}{x}\right )^n}{x^2 (c+d x)^2} \, dx &=\int \frac {\left (a+\frac {b}{x}\right )^n}{\left (d+\frac {c}{x}\right )^2 x^4} \, dx\\ &=-\operatorname {Subst}\left (\int \frac {x^2 (a+b x)^n}{(d+c x)^2} \, dx,x,\frac {1}{x}\right )\\ &=\frac {d^2 \left (a+\frac {b}{x}\right )^{1+n}}{c^2 (a c-b d) \left (d+\frac {c}{x}\right )}-\frac {\operatorname {Subst}\left (\int \frac {(a+b x)^n (-d (a c-b d (1+n))+c (a c-b d) x)}{d+c x} \, dx,x,\frac {1}{x}\right )}{c^2 (a c-b d)}\\ &=-\frac {\left (a+\frac {b}{x}\right )^{1+n}}{b c^2 (1+n)}+\frac {d^2 \left (a+\frac {b}{x}\right )^{1+n}}{c^2 (a c-b d) \left (d+\frac {c}{x}\right )}+\frac {(d (2 a c-b d (2+n))) \operatorname {Subst}\left (\int \frac {(a+b x)^n}{d+c x} \, dx,x,\frac {1}{x}\right )}{c^2 (a c-b d)}\\ &=-\frac {\left (a+\frac {b}{x}\right )^{1+n}}{b c^2 (1+n)}+\frac {d^2 \left (a+\frac {b}{x}\right )^{1+n}}{c^2 (a c-b d) \left (d+\frac {c}{x}\right )}-\frac {d (2 a c-b d (2+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^2 (a c-b d)^2 (1+n)}\\ \end {align*}
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Mathematica [A] time = 0.11, size = 125, normalized size = 0.94 \[ -\frac {(a x+b) \left (a+\frac {b}{x}\right )^n \left (b d (c+d x) (2 a c-b d (n+2)) \, _2F_1\left (1,n+1;n+2;\frac {c \left (a+\frac {b}{x}\right )}{a c-b d}\right )+(a c-b d) (a c (c+d x)-b d (c+d (n+2) x))\right )}{b c^2 (n+1) x (c+d x) (a c-b d)^2} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.87, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\left (\frac {a x + b}{x}\right )^{n}}{d^{2} x^{4} + 2 \, c d x^{3} + c^{2} x^{2}}, x\right ) \]
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 \[ \int \frac {{\left (a + \frac {b}{x}\right )}^{n}}{{\left (d x + c\right )}^{2} x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.51, size = 0, normalized size = 0.00 \[ \int \frac {\left (a +\frac {b}{x}\right )^{n}}{\left (d x +c \right )^{2} x^{2}}\, dx \]
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 \[ \int \frac {{\left (a + \frac {b}{x}\right )}^{n}}{{\left (d x + c\right )}^{2} x^{2}}\,{d x} \]
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 \[ \int \frac {{\left (a+\frac {b}{x}\right )}^n}{x^2\,{\left (c+d\,x\right )}^2} \,d x \]
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 \[ \int \frac {\left (a + \frac {b}{x}\right )^{n}}{x^{2} \left (c + d x\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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