Optimal. Leaf size=54 \[ \frac{\left (a+\frac{b}{x}\right )^{n+1} \, _2F_1\left (1,n+1;n+2;\frac{c \left (a+\frac{b}{x}\right )}{a c-b d}\right )}{(n+1) (a c-b d)} \]
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Rubi [A] time = 0.106344, antiderivative size = 54, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15 \[ \frac{\left (a+\frac{b}{x}\right )^{n+1} \, _2F_1\left (1,n+1;n+2;\frac{c \left (a+\frac{b}{x}\right )}{a c-b d}\right )}{(n+1) (a c-b d)} \]
Antiderivative was successfully verified.
[In] Int[(a + b/x)^n/(x*(c + d*x)),x]
[Out]
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Rubi in Sympy [A] time = 14.1783, size = 36, normalized size = 0.67 \[ \frac{\left (a + \frac{b}{x}\right )^{n + 1}{{}_{2}F_{1}\left (\begin{matrix} 1, n + 1 \\ n + 2 \end{matrix}\middle |{\frac{c \left (a + \frac{b}{x}\right )}{a c - b d}} \right )}}{\left (n + 1\right ) \left (a c - b d\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((a+b/x)**n/x/(d*x+c),x)
[Out]
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Mathematica [A] time = 0.064411, size = 0, normalized size = 0. \[ \int \frac{\left (a+\frac{b}{x}\right )^n}{x (c+d x)} \, dx \]
Verification is Not applicable to the result.
[In] Integrate[(a + b/x)^n/(x*(c + d*x)),x]
[Out]
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Maple [F] time = 0.056, size = 0, normalized size = 0. \[ \int{\frac{1}{x \left ( dx+c \right ) } \left ( a+{\frac{b}{x}} \right ) ^{n}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((a+b/x)^n/x/(d*x+c),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (a + \frac{b}{x}\right )}^{n}}{{\left (d x + c\right )} x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((a + b/x)^n/((d*x + c)*x),x, algorithm="maxima")
[Out]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{\left (\frac{a x + b}{x}\right )^{n}}{d x^{2} + c x}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((a + b/x)^n/((d*x + c)*x),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (a + \frac{b}{x}\right )^{n}}{x \left (c + d x\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((a+b/x)**n/x/(d*x+c),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (a + \frac{b}{x}\right )}^{n}}{{\left (d x + c\right )} x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((a + b/x)^n/((d*x + c)*x),x, algorithm="giac")
[Out]