Optimal. Leaf size=138 \[ \frac {d x^2 \left (a+\frac {b}{x}\right )^{m+1} (6 a c-b d (2-m))}{6 a^2}-\frac {b \left (a+\frac {b}{x}\right )^{m+1} \left (6 a^2 c^2-6 a b c d (1-m)+b^2 d^2 \left (m^2-3 m+2\right )\right ) \, _2F_1\left (2,m+1;m+2;\frac {b}{a x}+1\right )}{6 a^4 (m+1)}+\frac {d^2 x^3 \left (a+\frac {b}{x}\right )^{m+1}}{3 a} \]
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Rubi [A] time = 0.12, antiderivative size = 138, normalized size of antiderivative = 1.00, 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 65
Rule 78
Rule 89
Rule 434
Rule 446
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*}
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Mathematica [A] time = 0.08, 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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fricas [F] time = 0.42, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (d^{2} x^{2} + 2 \, c d x + c^{2}\right )} \left (\frac {a x + b}{x}\right )^{m}, x\right ) \]
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 \[ \int {\left (d x + c\right )}^{2} {\left (a + \frac {b}{x}\right )}^{m}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.05, size = 0, normalized size = 0.00 \[ \int \left (d x +c \right )^{2} \left (a +\frac {b}{x}\right )^{m}\, dx \]
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 \[ \int {\left (d x + c\right )}^{2} {\left (a + \frac {b}{x}\right )}^{m}\,{d x} \]
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 \[ \int {\left (a+\frac {b}{x}\right )}^m\,{\left (c+d\,x\right )}^2 \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 6.14, size = 121, normalized size = 0.88 \[ \frac {b^{m} c^{2} x x^{- m} \Gamma \left (1 - m\right ) {{}_{2}F_{1}\left (\begin {matrix} - m, 1 - m \\ 2 - m \end {matrix}\middle | {\frac {a x e^{i \pi }}{b}} \right )}}{\Gamma \left (2 - m\right )} + \frac {2 b^{m} c d x^{2} x^{- m} \Gamma \left (2 - m\right ) {{}_{2}F_{1}\left (\begin {matrix} - m, 2 - m \\ 3 - m \end {matrix}\middle | {\frac {a x e^{i \pi }}{b}} \right )}}{\Gamma \left (3 - m\right )} + \frac {b^{m} d^{2} x^{3} x^{- m} \Gamma \left (3 - m\right ) {{}_{2}F_{1}\left (\begin {matrix} - m, 3 - m \\ 4 - m \end {matrix}\middle | {\frac {a x e^{i \pi }}{b}} \right )}}{\Gamma \left (4 - m\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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