Optimal. Leaf size=75 \[ -\frac{(a+b x)^n (c+d x)^{-n} \left (-\frac{d (a+b x)}{b c-a d}\right )^{-n} \, _2F_1\left (-n,-n;1-n;\frac{b (c+d x)}{b c-a d}\right )}{d n} \]
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Rubi [A] time = 0.0764983, antiderivative size = 75, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105 \[ -\frac{(a+b x)^n (c+d x)^{-n} \left (-\frac{d (a+b x)}{b c-a d}\right )^{-n} \, _2F_1\left (-n,-n;1-n;\frac{b (c+d x)}{b c-a d}\right )}{d n} \]
Antiderivative was successfully verified.
[In] Int[(a + b*x)^n*(c + d*x)^(-1 - n),x]
[Out]
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Rubi in Sympy [A] time = 15.7879, size = 54, normalized size = 0.72 \[ - \frac{\left (\frac{d \left (a + b x\right )}{a d - b c}\right )^{- n} \left (a + b x\right )^{n} \left (c + d x\right )^{- n}{{}_{2}F_{1}\left (\begin{matrix} - n, - n \\ - n + 1 \end{matrix}\middle |{\frac{b \left (- c - d x\right )}{a d - b c}} \right )}}{d n} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b*x+a)**n*(d*x+c)**(-1-n),x)
[Out]
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Mathematica [A] time = 0.0574878, size = 74, normalized size = 0.99 \[ -\frac{(a+b x)^n (c+d x)^{-n} \left (\frac{d (a+b x)}{a d-b c}\right )^{-n} \, _2F_1\left (-n,-n;1-n;\frac{b (c+d x)}{b c-a d}\right )}{d n} \]
Antiderivative was successfully verified.
[In] Integrate[(a + b*x)^n*(c + d*x)^(-1 - n),x]
[Out]
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Maple [F] time = 0.118, size = 0, normalized size = 0. \[ \int \left ( bx+a \right ) ^{n} \left ( dx+c \right ) ^{-1-n}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b*x+a)^n*(d*x+c)^(-1-n),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int{\left (b x + a\right )}^{n}{\left (d x + c\right )}^{-n - 1}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x + a)^n*(d*x + c)^(-n - 1),x, algorithm="maxima")
[Out]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left ({\left (b x + a\right )}^{n}{\left (d x + c\right )}^{-n - 1}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x + a)^n*(d*x + c)^(-n - 1),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*(d*x+c)**(-1-n),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int{\left (b x + a\right )}^{n}{\left (d x + c\right )}^{-n - 1}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x + a)^n*(d*x + c)^(-n - 1),x, algorithm="giac")
[Out]