Optimal. Leaf size=72 \[ \frac{(a+b x)^{n+3} (c+d x)^{-n} \left (\frac{b (c+d x)}{b c-a d}\right )^n \, _2F_1\left (n,n+3;n+4;-\frac{d (a+b x)}{b c-a d}\right )}{b (n+3)} \]
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Rubi [A] time = 0.0646881, antiderivative size = 72, 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 \[ \frac{(a+b x)^{n+3} (c+d x)^{-n} \left (\frac{b (c+d x)}{b c-a d}\right )^n \, _2F_1\left (n,n+3;n+4;-\frac{d (a+b x)}{b c-a d}\right )}{b (n+3)} \]
Antiderivative was successfully verified.
[In] Int[(a + b*x)^(2 + n)/(c + d*x)^n,x]
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Rubi in Sympy [A] time = 18.0169, size = 68, normalized size = 0.94 \[ \frac{\left (\frac{d \left (a + b x\right )}{a d - b c}\right )^{- n} \left (a + b x\right )^{n} \left (c + d x\right )^{- n + 1} \left (a d - b c\right )^{2}{{}_{2}F_{1}\left (\begin{matrix} - n - 2, - n + 1 \\ - n + 2 \end{matrix}\middle |{\frac{b \left (- c - d x\right )}{a d - b c}} \right )}}{d^{3} \left (- n + 1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b*x+a)**(2+n)/((d*x+c)**n),x)
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Mathematica [C] time = 0.997663, size = 317, normalized size = 4.4 \[ a (a+b x)^n (c+d x)^{-n} \left (\frac{4 b^2 c x^3 F_1\left (3;-n,n;4;-\frac{b x}{a},-\frac{d x}{c}\right )}{12 a c F_1\left (3;-n,n;4;-\frac{b x}{a},-\frac{d x}{c}\right )+3 b c n x F_1\left (4;1-n,n;5;-\frac{b x}{a},-\frac{d x}{c}\right )-3 a d n x F_1\left (4;-n,n+1;5;-\frac{b x}{a},-\frac{d x}{c}\right )}+\frac{3 a b c x^2 F_1\left (2;-n,n;3;-\frac{b x}{a},-\frac{d x}{c}\right )}{3 a c F_1\left (2;-n,n;3;-\frac{b x}{a},-\frac{d x}{c}\right )+n x \left (b c F_1\left (3;1-n,n;4;-\frac{b x}{a},-\frac{d x}{c}\right )-a d F_1\left (3;-n,n+1;4;-\frac{b x}{a},-\frac{d x}{c}\right )\right )}-\frac{a (c+d x) \left (\frac{d (a+b x)}{a d-b c}\right )^{-n} \, _2F_1\left (1-n,-n;2-n;\frac{b (c+d x)}{b c-a d}\right )}{d (n-1)}\right ) \]
Warning: Unable to verify antiderivative.
[In] Integrate[(a + b*x)^(2 + n)/(c + d*x)^n,x]
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Maple [F] time = 0.114, size = 0, normalized size = 0. \[ \int{\frac{ \left ( bx+a \right ) ^{2+n}}{ \left ( dx+c \right ) ^{n}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b*x+a)^(2+n)/((d*x+c)^n),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int{\left (b x + a\right )}^{n + 2}{\left (d x + c\right )}^{-n}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x + a)^(n + 2)/(d*x + c)^n,x, algorithm="maxima")
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (b x + a\right )}^{n + 2}}{{\left (d x + c\right )}^{n}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x + a)^(n + 2)/(d*x + c)^n,x, algorithm="fricas")
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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)**(2+n)/((d*x+c)**n),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x + a\right )}^{n + 2}}{{\left (d x + c\right )}^{n}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x + a)^(n + 2)/(d*x + c)^n,x, algorithm="giac")
[Out]