Optimal. Leaf size=108 \[ \frac{(a+b x)^n (c+d x)^{-n} \left (\frac{b (c+d x)}{b c-a d}\right )^n \, _2F_1\left (n,n;n+1;-\frac{d (a+b x)}{b c-a d}\right )}{n}-\frac{(a+b x)^n (c+d x)^{-n} \, _2F_1\left (1,n;n+1;\frac{c (a+b x)}{a (c+d x)}\right )}{n} \]
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Rubi [A] time = 0.149724, antiderivative size = 108, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ \frac{(a+b x)^n (c+d x)^{-n} \left (\frac{b (c+d x)}{b c-a d}\right )^n \, _2F_1\left (n,n;n+1;-\frac{d (a+b x)}{b c-a d}\right )}{n}-\frac{(a+b x)^n (c+d x)^{-n} \, _2F_1\left (1,n;n+1;\frac{c (a+b x)}{a (c+d x)}\right )}{n} \]
Antiderivative was successfully verified.
[In] Int[(a + b*x)^n/(x*(c + d*x)^n),x]
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Rubi in Sympy [A] time = 23.1398, size = 90, normalized size = 0.83 \[ - \frac{a \left (a + b x\right )^{n - 1} \left (c + d x\right )^{- n + 1}{{}_{2}F_{1}\left (\begin{matrix} - n + 1, 1 \\ - n + 2 \end{matrix}\middle |{\frac{a \left (c + d x\right )}{c \left (a + b x\right )}} \right )}}{c \left (- n + 1\right )} + \frac{\left (\frac{b \left (- c - d x\right )}{a d - b c}\right )^{n} \left (a + b x\right )^{n} \left (c + d x\right )^{- n}{{}_{2}F_{1}\left (\begin{matrix} n, n \\ n + 1 \end{matrix}\middle |{\frac{d \left (a + b x\right )}{a d - b c}} \right )}}{n} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b*x+a)**n/x/((d*x+c)**n),x)
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Mathematica [C] time = 0.406255, size = 216, normalized size = 2. \[ \frac{a (n+2) (a d-b c) (a+b x)^{n+1} (c+d x)^{-n} F_1\left (n+1;n,1;n+2;\frac{d (a+b x)}{a d-b c},\frac{b x}{a}+1\right )}{b (n+1) x \left (a (n+2) (a d-b c) F_1\left (n+1;n,1;n+2;\frac{d (a+b x)}{a d-b c},\frac{b x}{a}+1\right )+(a+b x) \left ((a d-b c) F_1\left (n+2;n,2;n+3;\frac{d (a+b x)}{a d-b c},\frac{b x}{a}+1\right )+a d n F_1\left (n+2;n+1,1;n+3;\frac{d (a+b x)}{a d-b c},\frac{b x}{a}+1\right )\right )\right )} \]
Warning: Unable to verify antiderivative.
[In] Integrate[(a + b*x)^n/(x*(c + d*x)^n),x]
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Maple [F] time = 0.079, size = 0, normalized size = 0. \[ \int{\frac{ \left ( bx+a \right ) ^{n}}{x \left ( dx+c \right ) ^{n}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b*x+a)^n/x/((d*x+c)^n),x)
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x + a\right )}^{n}{\left (d x + c\right )}^{-n}}{x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x + a)^n/((d*x + c)^n*x),x, algorithm="maxima")
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (b x + a\right )}^{n}}{{\left (d x + c\right )}^{n} x}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x + a)^n/((d*x + c)^n*x),x, algorithm="fricas")
[Out]
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x+a)**n/x/((d*x+c)**n),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x + a\right )}^{n}}{{\left (d x + c\right )}^{n} x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x + a)^n/((d*x + c)^n*x),x, algorithm="giac")
[Out]