Optimal. Leaf size=195 \[ -\frac {x \left (a+\frac {b}{x}\right )^{n+1} (2 a c+b d (1-n))}{2 a^2 d^2}+\frac {\left (a+\frac {b}{x}\right )^{n+1} \left (2 a^2 c^2-2 a b c d n-b^2 d^2 (1-n) n\right ) \, _2F_1\left (1,n+1;n+2;\frac {b}{a x}+1\right )}{2 a^3 d^3 (n+1)}-\frac {c^3 \left (a+\frac {b}{x}\right )^{n+1} \, _2F_1\left (1,n+1;n+2;\frac {c \left (a+\frac {b}{x}\right )}{a c-b d}\right )}{d^3 (n+1) (a c-b d)}+\frac {x^2 \left (a+\frac {b}{x}\right )^{n+1}}{2 a d} \]
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Rubi [A] time = 0.22, antiderivative size = 195, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.350, Rules used = {514, 446, 103, 151, 156, 65, 68} \[ \frac {\left (a+\frac {b}{x}\right )^{n+1} \left (2 a^2 c^2-2 a b c d n-b^2 d^2 (1-n) n\right ) \, _2F_1\left (1,n+1;n+2;\frac {b}{a x}+1\right )}{2 a^3 d^3 (n+1)}-\frac {x \left (a+\frac {b}{x}\right )^{n+1} (2 a c+b d (1-n))}{2 a^2 d^2}-\frac {c^3 \left (a+\frac {b}{x}\right )^{n+1} \, _2F_1\left (1,n+1;n+2;\frac {c \left (a+\frac {b}{x}\right )}{a c-b d}\right )}{d^3 (n+1) (a c-b d)}+\frac {x^2 \left (a+\frac {b}{x}\right )^{n+1}}{2 a d} \]
Antiderivative was successfully verified.
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Rule 65
Rule 68
Rule 103
Rule 151
Rule 156
Rule 446
Rule 514
Rubi steps
\begin {align*} \int \frac {\left (a+\frac {b}{x}\right )^n x^2}{c+d x} \, dx &=\int \frac {\left (a+\frac {b}{x}\right )^n x}{d+\frac {c}{x}} \, dx\\ &=-\operatorname {Subst}\left (\int \frac {(a+b x)^n}{x^3 (d+c x)} \, dx,x,\frac {1}{x}\right )\\ &=\frac {\left (a+\frac {b}{x}\right )^{1+n} x^2}{2 a d}+\frac {\operatorname {Subst}\left (\int \frac {(a+b x)^n (2 a c+b d (1-n)+b c (1-n) x)}{x^2 (d+c x)} \, dx,x,\frac {1}{x}\right )}{2 a d}\\ &=-\frac {(2 a c+b d (1-n)) \left (a+\frac {b}{x}\right )^{1+n} x}{2 a^2 d^2}+\frac {\left (a+\frac {b}{x}\right )^{1+n} x^2}{2 a d}-\frac {\operatorname {Subst}\left (\int \frac {(a+b x)^n \left (2 a^2 c^2-2 a b c d n-b^2 d^2 (1-n) n-b c (2 a c+b d (1-n)) n x\right )}{x (d+c x)} \, dx,x,\frac {1}{x}\right )}{2 a^2 d^2}\\ &=-\frac {(2 a c+b d (1-n)) \left (a+\frac {b}{x}\right )^{1+n} x}{2 a^2 d^2}+\frac {\left (a+\frac {b}{x}\right )^{1+n} x^2}{2 a d}+\frac {c^3 \operatorname {Subst}\left (\int \frac {(a+b x)^n}{d+c x} \, dx,x,\frac {1}{x}\right )}{d^3}-\frac {\left (2 a^2 c^2-2 a b c d n-b^2 d^2 (1-n) n\right ) \operatorname {Subst}\left (\int \frac {(a+b x)^n}{x} \, dx,x,\frac {1}{x}\right )}{2 a^2 d^3}\\ &=-\frac {(2 a c+b d (1-n)) \left (a+\frac {b}{x}\right )^{1+n} x}{2 a^2 d^2}+\frac {\left (a+\frac {b}{x}\right )^{1+n} x^2}{2 a d}-\frac {c^3 \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^3 (a c-b d) (1+n)}+\frac {\left (2 a^2 c^2-2 a b c d n-b^2 d^2 (1-n) n\right ) \left (a+\frac {b}{x}\right )^{1+n} \, _2F_1\left (1,1+n;2+n;1+\frac {b}{a x}\right )}{2 a^3 d^3 (1+n)}\\ \end {align*}
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Mathematica [A] time = 0.17, size = 157, normalized size = 0.81 \[ \frac {(a x+b) \left (a+\frac {b}{x}\right )^n \left ((a c-b d) \left (\left (2 a^2 c^2-2 a b c d n+b^2 d^2 (n-1) n\right ) \, _2F_1\left (1,n+1;n+2;\frac {b}{a x}+1\right )+a d (n+1) x (a (d x-2 c)+b d (n-1))\right )-2 a^3 c^3 \, _2F_1\left (1,n+1;n+2;\frac {c \left (a+\frac {b}{x}\right )}{a c-b d}\right )\right )}{2 a^3 d^3 (n+1) x (a c-b d)} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.00, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {x^{2} \left (\frac {a x + b}{x}\right )^{n}}{d x + c}, x\right ) \]
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 \[ \int \frac {{\left (a + \frac {b}{x}\right )}^{n} x^{2}}{d x + c}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.51, size = 0, normalized size = 0.00 \[ \int \frac {x^{2} \left (a +\frac {b}{x}\right )^{n}}{d x +c}\, dx \]
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 \[ \int \frac {{\left (a + \frac {b}{x}\right )}^{n} x^{2}}{d x + c}\,{d x} \]
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 \[ \int \frac {x^2\,{\left (a+\frac {b}{x}\right )}^n}{c+d\,x} \,d x \]
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 \[ \int \frac {x^{2} \left (a + \frac {b}{x}\right )^{n}}{c + d x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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