Optimal. Leaf size=75 \[ \frac{(a+b x)^{m+1} (c+d x)^{-m-1} \, _2F_1\left (1,m+1;m+2;\frac{(d e-c f) (a+b x)}{(b e-a f) (c+d x)}\right )}{(m+1) (b e-a f)} \]
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Rubi [A] time = 0.0878635, antiderivative size = 75, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.038 \[ \frac{(a+b x)^{m+1} (c+d x)^{-m-1} \, _2F_1\left (1,m+1;m+2;\frac{(d e-c f) (a+b x)}{(b e-a f) (c+d x)}\right )}{(m+1) (b e-a f)} \]
Antiderivative was successfully verified.
[In] Int[((a + b*x)^m*(c + d*x)^(-1 - m))/(e + f*x),x]
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Rubi in Sympy [A] time = 9.09276, size = 51, normalized size = 0.68 \[ \frac{\left (a + b x\right )^{m} \left (c + d x\right )^{- m}{{}_{2}F_{1}\left (\begin{matrix} - m, 1 \\ - m + 1 \end{matrix}\middle |{\frac{\left (- c - d x\right ) \left (- a f + b e\right )}{\left (a + b x\right ) \left (c f - d e\right )}} \right )}}{m \left (c f - d e\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b*x+a)**m*(d*x+c)**(-1-m)/(f*x+e),x)
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Mathematica [C] time = 0.708028, size = 362, normalized size = 4.83 \[ \frac{(a+b x)^m (c+d x)^{-m} \left (\frac{f (m+2) (a+b x) (b c-a d) (b e-a f)^2 F_1\left (m+1;m,1;m+2;\frac{d (a+b x)}{a d-b c},\frac{f (a+b x)}{a f-b e}\right )}{b (m+1) (e+f x) (a f-b e) \left ((m+2) (b c-a d) (b e-a f) F_1\left (m+1;m,1;m+2;\frac{d (a+b x)}{a d-b c},\frac{f (a+b x)}{a f-b e}\right )+(a+b x) \left ((a d f-b c f) F_1\left (m+2;m,2;m+3;\frac{d (a+b x)}{a d-b c},\frac{f (a+b x)}{a f-b e}\right )+d m (a f-b e) F_1\left (m+2;m+1,1;m+3;\frac{d (a+b x)}{a d-b c},\frac{f (a+b x)}{a f-b e}\right )\right )\right )}-\frac{\left (\frac{d (a+b x)}{a d-b c}\right )^{-m} \, _2F_1\left (-m,-m;1-m;\frac{b (c+d x)}{b c-a d}\right )}{m}\right )}{d e-c f} \]
Warning: Unable to verify antiderivative.
[In] Integrate[((a + b*x)^m*(c + d*x)^(-1 - m))/(e + f*x),x]
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Maple [F] time = 0.091, size = 0, normalized size = 0. \[ \int{\frac{ \left ( bx+a \right ) ^{m} \left ( dx+c \right ) ^{-1-m}}{fx+e}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b*x+a)^m*(d*x+c)^(-1-m)/(f*x+e),x)
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x + a\right )}^{m}{\left (d x + c\right )}^{-m - 1}}{f x + e}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x + a)^m*(d*x + c)^(-m - 1)/(f*x + e),x, algorithm="maxima")
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (b x + a\right )}^{m}{\left (d x + c\right )}^{-m - 1}}{f x + e}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x + a)^m*(d*x + c)^(-m - 1)/(f*x + e),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)**m*(d*x+c)**(-1-m)/(f*x+e),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x + a\right )}^{m}{\left (d x + c\right )}^{-m - 1}}{f x + e}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x + a)^m*(d*x + c)^(-m - 1)/(f*x + e),x, algorithm="giac")
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