Optimal. Leaf size=101 \[ \frac {\left (a+\frac {b}{x}\right )^{m+1} \, _2F_1\left (1,m+1;m+2;\frac {b}{a x}+1\right )}{a d (m+1)}-\frac {c \left (a+\frac {b}{x}\right )^{m+1} \, _2F_1\left (1,m+1;m+2;\frac {c \left (a+\frac {b}{x}\right )}{a c-b d}\right )}{d (m+1) (a c-b d)} \]
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Rubi [A] time = 0.07, antiderivative size = 101, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.294, Rules used = {434, 446, 86, 65, 68} \[ \frac {\left (a+\frac {b}{x}\right )^{m+1} \, _2F_1\left (1,m+1;m+2;\frac {b}{a x}+1\right )}{a d (m+1)}-\frac {c \left (a+\frac {b}{x}\right )^{m+1} \, _2F_1\left (1,m+1;m+2;\frac {c \left (a+\frac {b}{x}\right )}{a c-b d}\right )}{d (m+1) (a c-b d)} \]
Antiderivative was successfully verified.
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Rule 65
Rule 68
Rule 86
Rule 434
Rule 446
Rubi steps
\begin {align*} \int \frac {\left (a+\frac {b}{x}\right )^m}{c+d x} \, dx &=\int \frac {\left (a+\frac {b}{x}\right )^m}{\left (d+\frac {c}{x}\right ) x} \, dx\\ &=-\operatorname {Subst}\left (\int \frac {(a+b x)^m}{x (d+c x)} \, dx,x,\frac {1}{x}\right )\\ &=-\frac {\operatorname {Subst}\left (\int \frac {(a+b x)^m}{x} \, dx,x,\frac {1}{x}\right )}{d}+\frac {c \operatorname {Subst}\left (\int \frac {(a+b x)^m}{d+c x} \, dx,x,\frac {1}{x}\right )}{d}\\ &=-\frac {c \left (a+\frac {b}{x}\right )^{1+m} \, _2F_1\left (1,1+m;2+m;\frac {c \left (a+\frac {b}{x}\right )}{a c-b d}\right )}{d (a c-b d) (1+m)}+\frac {\left (a+\frac {b}{x}\right )^{1+m} \, _2F_1\left (1,1+m;2+m;1+\frac {b}{a x}\right )}{a d (1+m)}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 97, normalized size = 0.96 \[ \frac {(a x+b) \left (a+\frac {b}{x}\right )^m \left (a c \, _2F_1\left (1,m+1;m+2;\frac {c \left (a+\frac {b}{x}\right )}{a c-b d}\right )+(b d-a c) \, _2F_1\left (1,m+1;m+2;\frac {b}{a x}+1\right )\right )}{a d (m+1) x (b d-a c)} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.42, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\left (\frac {a x + b}{x}\right )^{m}}{d x + c}, x\right ) \]
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 \[ \int \frac {{\left (a + \frac {b}{x}\right )}^{m}}{d x + c}\,{d x} \]
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 \[ \int \frac {\left (a +\frac {b}{x}\right )^{m}}{d x +c}\, dx \]
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 \[ \int \frac {{\left (a + \frac {b}{x}\right )}^{m}}{d x + c}\,{d x} \]
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 \[ \int \frac {{\left (a+\frac {b}{x}\right )}^m}{c+d\,x} \,d x \]
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 \[ \int \frac {\left (a + \frac {b}{x}\right )^{m}}{c + d x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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