Optimal. Leaf size=138 \[ -\frac{b \left (a+\frac{b}{x}\right )^{m+1} \left (6 a^2 c^2-6 a b c d (1-m)+b^2 d^2 \left (m^2-3 m+2\right )\right ) \, _2F_1\left (2,m+1;m+2;\frac{b}{a x}+1\right )}{6 a^4 (m+1)}+\frac{d x^2 \left (a+\frac{b}{x}\right )^{m+1} (6 a c-b d (2-m))}{6 a^2}+\frac{d^2 x^3 \left (a+\frac{b}{x}\right )^{m+1}}{3 a} \]
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Rubi [A] time = 0.116717, antiderivative size = 138, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.294, Rules used = {434, 446, 89, 78, 65} \[ -\frac{b \left (a+\frac{b}{x}\right )^{m+1} \left (6 a^2 c^2-6 a b c d (1-m)+b^2 d^2 \left (m^2-3 m+2\right )\right ) \, _2F_1\left (2,m+1;m+2;\frac{b}{a x}+1\right )}{6 a^4 (m+1)}+\frac{d x^2 \left (a+\frac{b}{x}\right )^{m+1} (6 a c-b d (2-m))}{6 a^2}+\frac{d^2 x^3 \left (a+\frac{b}{x}\right )^{m+1}}{3 a} \]
Antiderivative was successfully verified.
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Rule 434
Rule 446
Rule 89
Rule 78
Rule 65
Rubi steps
\begin{align*} \int \left (a+\frac{b}{x}\right )^m (c+d x)^2 \, dx &=\int \left (a+\frac{b}{x}\right )^m \left (d+\frac{c}{x}\right )^2 x^2 \, dx\\ &=-\operatorname{Subst}\left (\int \frac{(a+b x)^m (d+c x)^2}{x^4} \, dx,x,\frac{1}{x}\right )\\ &=\frac{d^2 \left (a+\frac{b}{x}\right )^{1+m} x^3}{3 a}-\frac{\operatorname{Subst}\left (\int \frac{(a+b x)^m \left (d (6 a c-b d (2-m))+3 a c^2 x\right )}{x^3} \, dx,x,\frac{1}{x}\right )}{3 a}\\ &=\frac{d (6 a c-b d (2-m)) \left (a+\frac{b}{x}\right )^{1+m} x^2}{6 a^2}+\frac{d^2 \left (a+\frac{b}{x}\right )^{1+m} x^3}{3 a}-\frac{1}{6} \left (6 c^2-\frac{b d (6 a c-b d (2-m)) (1-m)}{a^2}\right ) \operatorname{Subst}\left (\int \frac{(a+b x)^m}{x^2} \, dx,x,\frac{1}{x}\right )\\ &=\frac{d (6 a c-b d (2-m)) \left (a+\frac{b}{x}\right )^{1+m} x^2}{6 a^2}+\frac{d^2 \left (a+\frac{b}{x}\right )^{1+m} x^3}{3 a}-\frac{b \left (6 a^2 c^2-6 a b c d (1-m)+b^2 d^2 \left (2-3 m+m^2\right )\right ) \left (a+\frac{b}{x}\right )^{1+m} \, _2F_1\left (2,1+m;2+m;1+\frac{b}{a x}\right )}{6 a^4 (1+m)}\\ \end{align*}
Mathematica [A] time = 0.0750939, size = 112, normalized size = 0.81 \[ \frac{(a x+b) \left (a+\frac{b}{x}\right )^m \left (a^2 d (m+1) x^2 (2 a (3 c+d x)+b d (m-2))-b \left (6 a^2 c^2+6 a b c d (m-1)+b^2 d^2 \left (m^2-3 m+2\right )\right ) \, _2F_1\left (2,m+1;m+2;\frac{b}{a x}+1\right )\right )}{6 a^4 (m+1) x} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.029, size = 0, normalized size = 0. \begin{align*} \int \left ( a+{\frac{b}{x}} \right ) ^{m} \left ( dx+c \right ) ^{2}\, 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{\left (d x + c\right )}^{2}{\left (a + \frac{b}{x}\right )}^{m}\,{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 ({\left (d^{2} x^{2} + 2 \, c d x + c^{2}\right )} \left (\frac{a x + b}{x}\right )^{m}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 5.36515, size = 121, normalized size = 0.88 \begin{align*} \frac{b^{m} c^{2} x x^{- m} \Gamma \left (1 - m\right ){{}_{2}F_{1}\left (\begin{matrix} - m, 1 - m \\ 2 - m \end{matrix}\middle |{\frac{a x e^{i \pi }}{b}} \right )}}{\Gamma \left (2 - m\right )} + \frac{2 b^{m} c d x^{2} x^{- m} \Gamma \left (2 - m\right ){{}_{2}F_{1}\left (\begin{matrix} - m, 2 - m \\ 3 - m \end{matrix}\middle |{\frac{a x e^{i \pi }}{b}} \right )}}{\Gamma \left (3 - m\right )} + \frac{b^{m} d^{2} x^{3} x^{- m} \Gamma \left (3 - m\right ){{}_{2}F_{1}\left (\begin{matrix} - m, 3 - m \\ 4 - m \end{matrix}\middle |{\frac{a x e^{i \pi }}{b}} \right )}}{\Gamma \left (4 - m\right )} \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{\left (d x + c\right )}^{2}{\left (a + \frac{b}{x}\right )}^{m}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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