Optimal. Leaf size=217 \[ -\frac{\left (a+\frac{b}{x}\right )^{n+1} \left (d (b d (n+2) (a c+b d (n+3))-a c (a c+b d (3 n+5)))-\frac{c (a c-b d) (a c+b d (n+3))}{x}\right )}{b^2 c^3 (n+1) (n+2) \left (\frac{c}{x}+d\right ) (a c-b d)}+\frac{d^2 \left (a+\frac{b}{x}\right )^{n+1} (3 a c-b d (n+3)) \, _2F_1\left (1,n+1;n+2;\frac{c \left (a+\frac{b}{x}\right )}{a c-b d}\right )}{c^3 (n+1) (a c-b d)^2}-\frac{\left (a+\frac{b}{x}\right )^{n+1}}{b c (n+2) x^2 \left (\frac{c}{x}+d\right )} \]
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Rubi [A] time = 0.263849, antiderivative size = 217, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {514, 446, 100, 146, 68} \[ -\frac{\left (a+\frac{b}{x}\right )^{n+1} \left (d (b d (n+2) (a c+b d (n+3))-a c (a c+b d (3 n+5)))-\frac{c (a c-b d) (a c+b d (n+3))}{x}\right )}{b^2 c^3 (n+1) (n+2) \left (\frac{c}{x}+d\right ) (a c-b d)}+\frac{d^2 \left (a+\frac{b}{x}\right )^{n+1} (3 a c-b d (n+3)) \, _2F_1\left (1,n+1;n+2;\frac{c \left (a+\frac{b}{x}\right )}{a c-b d}\right )}{c^3 (n+1) (a c-b d)^2}-\frac{\left (a+\frac{b}{x}\right )^{n+1}}{b c (n+2) x^2 \left (\frac{c}{x}+d\right )} \]
Antiderivative was successfully verified.
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Rule 514
Rule 446
Rule 100
Rule 146
Rule 68
Rubi steps
\begin{align*} \int \frac{\left (a+\frac{b}{x}\right )^n}{x^3 (c+d x)^2} \, dx &=\int \frac{\left (a+\frac{b}{x}\right )^n}{\left (d+\frac{c}{x}\right )^2 x^5} \, dx\\ &=-\operatorname{Subst}\left (\int \frac{x^3 (a+b x)^n}{(d+c x)^2} \, dx,x,\frac{1}{x}\right )\\ &=-\frac{\left (a+\frac{b}{x}\right )^{1+n}}{b c (2+n) \left (d+\frac{c}{x}\right ) x^2}-\frac{\operatorname{Subst}\left (\int \frac{x (a+b x)^n (-2 a d+(-a c-b d (3+n)) x)}{(d+c x)^2} \, dx,x,\frac{1}{x}\right )}{b c (2+n)}\\ &=-\frac{\left (a+\frac{b}{x}\right )^{1+n} \left (d (b d (2+n) (a c+b d (3+n))-a c (a c+b d (5+3 n)))-\frac{c (a c-b d) (a c+b d (3+n))}{x}\right )}{b^2 c^3 (a c-b d) (1+n) (2+n) \left (d+\frac{c}{x}\right )}-\frac{\left (a+\frac{b}{x}\right )^{1+n}}{b c (2+n) \left (d+\frac{c}{x}\right ) x^2}-\frac{\left (d^2 (3 a c-b d (3+n))\right ) \operatorname{Subst}\left (\int \frac{(a+b x)^n}{d+c x} \, dx,x,\frac{1}{x}\right )}{c^3 (a c-b d)}\\ &=-\frac{\left (a+\frac{b}{x}\right )^{1+n} \left (d (b d (2+n) (a c+b d (3+n))-a c (a c+b d (5+3 n)))-\frac{c (a c-b d) (a c+b d (3+n))}{x}\right )}{b^2 c^3 (a c-b d) (1+n) (2+n) \left (d+\frac{c}{x}\right )}-\frac{\left (a+\frac{b}{x}\right )^{1+n}}{b c (2+n) \left (d+\frac{c}{x}\right ) x^2}+\frac{d^2 (3 a c-b d (3+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 )}{c^3 (a c-b d)^2 (1+n)}\\ \end{align*}
Mathematica [A] time = 0.199432, size = 182, normalized size = 0.84 \[ \frac{\left (a+\frac{b}{x}\right )^{n+1} \left (\frac{-a^2 c^2 (c+d x)-a b c d (c (n+2)+d (2 n+3) x)+b^2 d^2 (n+3) (c+d (n+2) x)}{b c^2 (n+1) (c+d x) (b d-a c)}-\frac{b d^2 (n+2) (b d (n+3)-3 a c) \, _2F_1\left (1,n+1;n+2;\frac{c \left (a+\frac{b}{x}\right )}{a c-b d}\right )}{c^2 (n+1) (a c-b d)^2}-\frac{1}{x (c+d x)}\right )}{b c (n+2)} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.528, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{3} \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}}{{\left (d x + c\right )}^{2} x^{3}}\,{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{\left (\frac{a x + b}{x}\right )^{n}}{d^{2} x^{5} + 2 \, c d x^{4} + c^{2} x^{3}}, 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}}{{\left (d x + c\right )}^{2} x^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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