Optimal. Leaf size=131 \[ -\frac {\left (a+\frac {b}{x}\right )^{n+1} (a c-b d n) \, _2F_1\left (1,n+1;n+2;\frac {b}{a x}+1\right )}{a^2 d^2 (n+1)}+\frac {c^2 \left (a+\frac {b}{x}\right )^{n+1} \, _2F_1\left (1,n+1;n+2;\frac {c \left (a+\frac {b}{x}\right )}{a c-b d}\right )}{d^2 (n+1) (a c-b d)}+\frac {x \left (a+\frac {b}{x}\right )^{n+1}}{a d} \]
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Rubi [A] time = 0.10, antiderivative size = 131, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {514, 375, 103, 156, 65, 68} \[ -\frac {\left (a+\frac {b}{x}\right )^{n+1} (a c-b d n) \, _2F_1\left (1,n+1;n+2;\frac {b}{a x}+1\right )}{a^2 d^2 (n+1)}+\frac {c^2 \left (a+\frac {b}{x}\right )^{n+1} \, _2F_1\left (1,n+1;n+2;\frac {c \left (a+\frac {b}{x}\right )}{a c-b d}\right )}{d^2 (n+1) (a c-b d)}+\frac {x \left (a+\frac {b}{x}\right )^{n+1}}{a d} \]
Antiderivative was successfully verified.
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Rule 65
Rule 68
Rule 103
Rule 156
Rule 375
Rule 514
Rubi steps
\begin {align*} \int \frac {\left (a+\frac {b}{x}\right )^n x}{c+d x} \, dx &=\int \frac {\left (a+\frac {b}{x}\right )^n}{d+\frac {c}{x}} \, dx\\ &=-\operatorname {Subst}\left (\int \frac {(a+b x)^n}{x^2 (d+c x)} \, dx,x,\frac {1}{x}\right )\\ &=\frac {\left (a+\frac {b}{x}\right )^{1+n} x}{a d}+\frac {\operatorname {Subst}\left (\int \frac {(a+b x)^n (a c-b d n-b c n x)}{x (d+c x)} \, dx,x,\frac {1}{x}\right )}{a d}\\ &=\frac {\left (a+\frac {b}{x}\right )^{1+n} x}{a d}-\frac {c^2 \operatorname {Subst}\left (\int \frac {(a+b x)^n}{d+c x} \, dx,x,\frac {1}{x}\right )}{d^2}+\frac {(a c-b d n) \operatorname {Subst}\left (\int \frac {(a+b x)^n}{x} \, dx,x,\frac {1}{x}\right )}{a d^2}\\ &=\frac {\left (a+\frac {b}{x}\right )^{1+n} x}{a d}+\frac {c^2 \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 )}{d^2 (a c-b d) (1+n)}-\frac {(a c-b d n) \left (a+\frac {b}{x}\right )^{1+n} \, _2F_1\left (1,1+n;2+n;1+\frac {b}{a x}\right )}{a^2 d^2 (1+n)}\\ \end {align*}
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Mathematica [A] time = 0.08, size = 119, normalized size = 0.91 \[ \frac {(a x+b) \left (a+\frac {b}{x}\right )^n \left (a^2 c^2 \, _2F_1\left (1,n+1;n+2;\frac {c \left (a+\frac {b}{x}\right )}{a c-b d}\right )+(a c-b d) \left ((b d n-a c) \, _2F_1\left (1,n+1;n+2;\frac {b}{a x}+1\right )+a d (n+1) x\right )\right )}{a^2 d^2 (n+1) x (a c-b d)} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.06, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {x \left (\frac {a x + b}{x}\right )^{n}}{d x + c}, x\right ) \]
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 \[ \int \frac {{\left (a + \frac {b}{x}\right )}^{n} x}{d x + c}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.54, size = 0, normalized size = 0.00 \[ \int \frac {x \left (a +\frac {b}{x}\right )^{n}}{d x +c}\, dx \]
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 \[ \int \frac {{\left (a + \frac {b}{x}\right )}^{n} x}{d x + c}\,{d x} \]
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 \[ \int \frac {x\,{\left (a+\frac {b}{x}\right )}^n}{c+d\,x} \,d x \]
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 \[ \int \frac {x \left (a + \frac {b}{x}\right )^{n}}{c + d x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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