Optimal. Leaf size=202 \[ -\frac{\left (a+\frac{b}{x}\right )^{n+1} (2 a c-b d n) \, _2F_1\left (1,n+1;n+2;\frac{b}{a x}+1\right )}{a^2 d^3 (n+1)}+\frac{c^2 \left (a+\frac{b}{x}\right )^{n+1} (2 a c-b d (2-n)) \, _2F_1\left (1,n+1;n+2;\frac{c \left (a+\frac{b}{x}\right )}{a c-b d}\right )}{d^3 (n+1) (a c-b d)^2}+\frac{c (2 a c-b d) \left (a+\frac{b}{x}\right )^{n+1}}{a d^2 \left (\frac{c}{x}+d\right ) (a c-b d)}+\frac{x \left (a+\frac{b}{x}\right )^{n+1}}{a d \left (\frac{c}{x}+d\right )} \]
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Rubi [A] time = 0.238604, antiderivative size = 202, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.35, Rules used = {514, 375, 103, 151, 156, 65, 68} \[ -\frac{\left (a+\frac{b}{x}\right )^{n+1} (2 a c-b d n) \, _2F_1\left (1,n+1;n+2;\frac{b}{a x}+1\right )}{a^2 d^3 (n+1)}+\frac{c^2 \left (a+\frac{b}{x}\right )^{n+1} (2 a c-b d (2-n)) \, _2F_1\left (1,n+1;n+2;\frac{c \left (a+\frac{b}{x}\right )}{a c-b d}\right )}{d^3 (n+1) (a c-b d)^2}+\frac{c (2 a c-b d) \left (a+\frac{b}{x}\right )^{n+1}}{a d^2 \left (\frac{c}{x}+d\right ) (a c-b d)}+\frac{x \left (a+\frac{b}{x}\right )^{n+1}}{a d \left (\frac{c}{x}+d\right )} \]
Antiderivative was successfully verified.
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Rule 514
Rule 375
Rule 103
Rule 151
Rule 156
Rule 65
Rule 68
Rubi steps
\begin{align*} \int \frac{\left (a+\frac{b}{x}\right )^n x^2}{(c+d x)^2} \, dx &=\int \frac{\left (a+\frac{b}{x}\right )^n}{\left (d+\frac{c}{x}\right )^2} \, dx\\ &=-\operatorname{Subst}\left (\int \frac{(a+b x)^n}{x^2 (d+c x)^2} \, dx,x,\frac{1}{x}\right )\\ &=\frac{\left (a+\frac{b}{x}\right )^{1+n} x}{a d \left (d+\frac{c}{x}\right )}+\frac{\operatorname{Subst}\left (\int \frac{(a+b x)^n (2 a c-b d n+b c (1-n) x)}{x (d+c x)^2} \, dx,x,\frac{1}{x}\right )}{a d}\\ &=\frac{c (2 a c-b d) \left (a+\frac{b}{x}\right )^{1+n}}{a d^2 (a c-b d) \left (d+\frac{c}{x}\right )}+\frac{\left (a+\frac{b}{x}\right )^{1+n} x}{a d \left (d+\frac{c}{x}\right )}+\frac{\operatorname{Subst}\left (\int \frac{(a+b x)^n ((a c-b d) (2 a c-b d n)-b c (2 a c-b d) n x)}{x (d+c x)} \, dx,x,\frac{1}{x}\right )}{a d^2 (a c-b d)}\\ &=\frac{c (2 a c-b d) \left (a+\frac{b}{x}\right )^{1+n}}{a d^2 (a c-b d) \left (d+\frac{c}{x}\right )}+\frac{\left (a+\frac{b}{x}\right )^{1+n} x}{a d \left (d+\frac{c}{x}\right )}-\frac{\left (c^2 (2 a c-b d (2-n))\right ) \operatorname{Subst}\left (\int \frac{(a+b x)^n}{d+c x} \, dx,x,\frac{1}{x}\right )}{d^3 (a c-b d)}+\frac{(2 a c-b d n) \operatorname{Subst}\left (\int \frac{(a+b x)^n}{x} \, dx,x,\frac{1}{x}\right )}{a d^3}\\ &=\frac{c (2 a c-b d) \left (a+\frac{b}{x}\right )^{1+n}}{a d^2 (a c-b d) \left (d+\frac{c}{x}\right )}+\frac{\left (a+\frac{b}{x}\right )^{1+n} x}{a d \left (d+\frac{c}{x}\right )}+\frac{c^2 (2 a c-b d (2-n)) \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)^2 (1+n)}-\frac{(2 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^3 (1+n)}\\ \end{align*}
Mathematica [A] time = 0.167212, size = 179, normalized size = 0.89 \[ \frac{\left (a+\frac{b}{x}\right )^{n+1} \left ((c+d x) \left (a^2 c^2 (2 a c+b d (n-2)) \, _2F_1\left (1,n+1;n+2;\frac{c \left (a+\frac{b}{x}\right )}{a c-b d}\right )-(a c-b d)^2 (2 a c-b d n) \, _2F_1\left (1,n+1;n+2;\frac{b}{a x}+1\right )\right )+a d^2 (n+1) x^2 (a c-b d)^2+a c d (n+1) x (a c-b d) (2 a c-b d)\right )}{a^2 d^3 (n+1) (c+d x) (a c-b d)^2} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.53, size = 0, normalized size = 0. \begin{align*} \int{\frac{{x}^{2}}{ \left ( dx+c \right ) ^{2}} \left ( a+{\frac{b}{x}} \right ) ^{n}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (a + \frac{b}{x}\right )}^{n} x^{2}}{{\left (d x + c\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{x^{2} \left (\frac{a x + b}{x}\right )^{n}}{d^{2} x^{2} + 2 \, c d x + c^{2}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (a + \frac{b}{x}\right )}^{n} x^{2}}{{\left (d x + c\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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