Optimal. Leaf size=79 \[ \frac{d x^2 \left (a+\frac{b}{x}\right )^{m+1}}{2 a}-\frac{b \left (a+\frac{b}{x}\right )^{m+1} (2 a c-b d (1-m)) \, _2F_1\left (2,m+1;m+2;\frac{b}{a x}+1\right )}{2 a^3 (m+1)} \]
[Out]
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Rubi [A] time = 0.105425, antiderivative size = 79, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267 \[ \frac{d x^2 \left (a+\frac{b}{x}\right )^{m+1}}{2 a}-\frac{b \left (a+\frac{b}{x}\right )^{m+1} (2 a c-b d (1-m)) \, _2F_1\left (2,m+1;m+2;\frac{b}{a x}+1\right )}{2 a^3 (m+1)} \]
Antiderivative was successfully verified.
[In] Int[(a + b/x)^m*(c + d*x),x]
[Out]
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Rubi in Sympy [A] time = 6.47905, size = 58, normalized size = 0.73 \[ \frac{d x^{2} \left (a + \frac{b}{x}\right )^{m + 1}}{2 a} - \frac{b \left (a + \frac{b}{x}\right )^{m + 1} \left (2 a c - b d \left (- m + 1\right )\right ){{}_{2}F_{1}\left (\begin{matrix} 2, m + 1 \\ m + 2 \end{matrix}\middle |{1 + \frac{b}{a x}} \right )}}{2 a^{3} \left (m + 1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((a+b/x)**m*(d*x+c),x)
[Out]
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Mathematica [A] time = 0.048839, size = 88, normalized size = 1.11 \[ -\frac{x \left (a+\frac{b}{x}\right )^m \left (\frac{a x}{b}+1\right )^{-m} \left (c (m-2) \, _2F_1\left (1-m,-m;2-m;-\frac{a x}{b}\right )+d (m-1) x \, _2F_1\left (2-m,-m;3-m;-\frac{a x}{b}\right )\right )}{(m-2) (m-1)} \]
Antiderivative was successfully verified.
[In] Integrate[(a + b/x)^m*(c + d*x),x]
[Out]
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Maple [F] time = 0.028, size = 0, normalized size = 0. \[ \int \left ( a+{\frac{b}{x}} \right ) ^{m} \left ( dx+c \right ) \, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((a+b/x)^m*(d*x+c),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int{\left (d x + c\right )}{\left (a + \frac{b}{x}\right )}^{m}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x + c)*(a + b/x)^m,x, algorithm="maxima")
[Out]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left ({\left (d x + c\right )} \left (\frac{a x + b}{x}\right )^{m}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x + c)*(a + b/x)^m,x, algorithm="fricas")
[Out]
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Sympy [A] time = 10.7185, size = 75, normalized size = 0.95 \[ \frac{b^{m} c x x^{- m} \Gamma \left (- m + 1\right ){{}_{2}F_{1}\left (\begin{matrix} - m, - m + 1 \\ - m + 2 \end{matrix}\middle |{\frac{a x e^{i \pi }}{b}} \right )}}{\Gamma \left (- m + 2\right )} + \frac{b^{m} d x^{2} x^{- m} \Gamma \left (- m + 2\right ){{}_{2}F_{1}\left (\begin{matrix} - m, - m + 2 \\ - m + 3 \end{matrix}\middle |{\frac{a x e^{i \pi }}{b}} \right )}}{\Gamma \left (- m + 3\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((a+b/x)**m*(d*x+c),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int{\left (d x + c\right )}{\left (a + \frac{b}{x}\right )}^{m}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x + c)*(a + b/x)^m,x, algorithm="giac")
[Out]