Optimal. Leaf size=112 \[ -\frac {b \left (a+\frac {b}{x}\right )^{m+1} (2 a c-b d (m+1)) \, _2F_1\left (2,m+1;m+2;\frac {c \left (a+\frac {b}{x}\right )}{a c-b d}\right )}{2 c (m+1) (a c-b d)^3}-\frac {d \left (a+\frac {b}{x}\right )^{m+1}}{2 c \left (\frac {c}{x}+d\right )^2 (a c-b d)} \]
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Rubi [A] time = 0.07, antiderivative size = 112, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, Rules used = {434, 446, 78, 68} \[ -\frac {b \left (a+\frac {b}{x}\right )^{m+1} (2 a c-b d (m+1)) \, _2F_1\left (2,m+1;m+2;\frac {c \left (a+\frac {b}{x}\right )}{a c-b d}\right )}{2 c (m+1) (a c-b d)^3}-\frac {d \left (a+\frac {b}{x}\right )^{m+1}}{2 c \left (\frac {c}{x}+d\right )^2 (a c-b d)} \]
Antiderivative was successfully verified.
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Rule 68
Rule 78
Rule 434
Rule 446
Rubi steps
\begin {align*} \int \frac {\left (a+\frac {b}{x}\right )^m}{(c+d x)^3} \, dx &=\int \frac {\left (a+\frac {b}{x}\right )^m}{\left (d+\frac {c}{x}\right )^3 x^3} \, dx\\ &=-\operatorname {Subst}\left (\int \frac {x (a+b x)^m}{(d+c x)^3} \, dx,x,\frac {1}{x}\right )\\ &=-\frac {d \left (a+\frac {b}{x}\right )^{1+m}}{2 c (a c-b d) \left (d+\frac {c}{x}\right )^2}-\frac {(2 a c-b d (1+m)) \operatorname {Subst}\left (\int \frac {(a+b x)^m}{(d+c x)^2} \, dx,x,\frac {1}{x}\right )}{2 c (a c-b d)}\\ &=-\frac {d \left (a+\frac {b}{x}\right )^{1+m}}{2 c (a c-b d) \left (d+\frac {c}{x}\right )^2}-\frac {b (2 a c-b d (1+m)) \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 )}{2 c (a c-b d)^3 (1+m)}\\ \end {align*}
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Mathematica [A] time = 0.08, size = 99, normalized size = 0.88 \[ \frac {\left (a+\frac {b}{x}\right )^{m+1} \left (\frac {b (b d (m+1)-2 a c) \, _2F_1\left (2,m+1;m+2;\frac {b c+a x c}{a c x-b d x}\right )}{(m+1) (a c-b d)^2}-\frac {d x^2}{(c+d x)^2}\right )}{2 c (a c-b d)} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.43, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\left (\frac {a x + b}{x}\right )^{m}}{d^{3} x^{3} + 3 \, c d^{2} x^{2} + 3 \, c^{2} d x + c^{3}}, x\right ) \]
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 \[ \int \frac {{\left (a + \frac {b}{x}\right )}^{m}}{{\left (d x + c\right )}^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.07, size = 0, normalized size = 0.00 \[ \int \frac {\left (a +\frac {b}{x}\right )^{m}}{\left (d x +c \right )^{3}}\, dx \]
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 \[ \int \frac {{\left (a + \frac {b}{x}\right )}^{m}}{{\left (d x + c\right )}^{3}}\,{d x} \]
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 \[ \int \frac {{\left (a+\frac {b}{x}\right )}^m}{{\left (c+d\,x\right )}^3} \,d x \]
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 \[ \int \frac {\left (a + \frac {b}{x}\right )^{m}}{\left (c + d x\right )^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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