Optimal. Leaf size=101 \[ -\frac {c \left (a+\frac {b}{x}\right )^{1+m} \, _2F_1\left (1,1+m;2+m;\frac {c \left (a+\frac {b}{x}\right )}{a c-b d}\right )}{d (a c-b d) (1+m)}+\frac {\left (a+\frac {b}{x}\right )^{1+m} \, _2F_1\left (1,1+m;2+m;1+\frac {b}{a x}\right )}{a d (1+m)} \]
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Rubi [A]
time = 0.05, 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 = {445, 457, 88,
67, 70} \begin {gather*} \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)} \end {gather*}
Antiderivative was successfully verified.
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Rule 67
Rule 70
Rule 88
Rule 445
Rule 457
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\\ &=-\text {Subst}\left (\int \frac {(a+b x)^m}{x (d+c x)} \, dx,x,\frac {1}{x}\right )\\ &=-\frac {\text {Subst}\left (\int \frac {(a+b x)^m}{x} \, dx,x,\frac {1}{x}\right )}{d}+\frac {c \text {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 = 0.14, size = 97, normalized size = 0.96 \begin {gather*} \frac {\left (a+\frac {b}{x}\right )^m (b+a x) \left (a c \, _2F_1\left (1,1+m;2+m;\frac {c \left (a+\frac {b}{x}\right )}{a c-b d}\right )+(-a c+b d) \, _2F_1\left (1,1+m;2+m;1+\frac {b}{a x}\right )\right )}{a d (-a c+b d) (1+m) x} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.04, 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 \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 )^{m}}{c + d x}\, 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 )}^m}{c+d\,x} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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