Optimal. Leaf size=56 \[ -\frac {b \left (a+\frac {b}{x}\right )^{m+1} \, _2F_1\left (2,m+1;m+2;\frac {c \left (a+\frac {b}{x}\right )}{a c-b d}\right )}{(m+1) (a c-b d)^2} \]
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Rubi [A] time = 0.03, antiderivative size = 56, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {434, 444, 68} \[ -\frac {b \left (a+\frac {b}{x}\right )^{m+1} \, _2F_1\left (2,m+1;m+2;\frac {c \left (a+\frac {b}{x}\right )}{a c-b d}\right )}{(m+1) (a c-b d)^2} \]
Antiderivative was successfully verified.
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Rule 68
Rule 434
Rule 444
Rubi steps
\begin {align*} \int \frac {\left (a+\frac {b}{x}\right )^m}{(c+d x)^2} \, dx &=\int \frac {\left (a+\frac {b}{x}\right )^m}{\left (d+\frac {c}{x}\right )^2 x^2} \, dx\\ &=-\operatorname {Subst}\left (\int \frac {(a+b x)^m}{(d+c x)^2} \, dx,x,\frac {1}{x}\right )\\ &=-\frac {b \left (a+\frac {b}{x}\right )^{1+m} \, _2F_1\left (2,1+m;2+m;\frac {c \left (a+\frac {b}{x}\right )}{a c-b d}\right )}{(a c-b d)^2 (1+m)}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 57, normalized size = 1.02 \[ -\frac {b \left (a+\frac {b}{x}\right )^{m+1} \, _2F_1\left (2,m+1;m+2;-\frac {c \left (a+\frac {b}{x}\right )}{b d-a c}\right )}{(m+1) (b d-a c)^2} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.41, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\left (\frac {a x + b}{x}\right )^{m}}{d^{2} x^{2} + 2 \, c d x + c^{2}}, x\right ) \]
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 \[ \int \frac {{\left (a + \frac {b}{x}\right )}^{m}}{{\left (d x + c\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.06, size = 0, normalized size = 0.00 \[ \int \frac {\left (a +\frac {b}{x}\right )^{m}}{\left (d x +c \right )^{2}}\, dx \]
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 \[ \int \frac {{\left (a + \frac {b}{x}\right )}^{m}}{{\left (d x + c\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {{\left (a+\frac {b}{x}\right )}^m}{{\left (c+d\,x\right )}^2} \,d x \]
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 \[ \int \frac {\left (a + \frac {b}{x}\right )^{m}}{\left (c + d x\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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