Optimal. Leaf size=128 \[ \frac{(a+b x)^m (c+d x)^{-m} \left (\frac{b (c+d x)}{b c-a d}\right )^m \, _2F_1\left (m,m;m+1;-\frac{d (a+b x)}{b c-a d}\right )}{f m}-\frac{(a+b x)^m (c+d x)^{-m} \, _2F_1\left (1,m;m+1;\frac{(d e-c f) (a+b x)}{(b e-a f) (c+d x)}\right )}{f m} \]
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Rubi [A] time = 0.228846, antiderivative size = 128, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ \frac{(a+b x)^m (c+d x)^{-m} \left (\frac{b (c+d x)}{b c-a d}\right )^m \, _2F_1\left (m,m;m+1;-\frac{d (a+b x)}{b c-a d}\right )}{f m}-\frac{(a+b x)^m (c+d x)^{-m} \, _2F_1\left (1,m;m+1;\frac{(d e-c f) (a+b x)}{(b e-a f) (c+d x)}\right )}{f m} \]
Antiderivative was successfully verified.
[In] Int[(a + b*x)^m/((c + d*x)^m*(e + f*x)),x]
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Rubi in Sympy [A] time = 29.0179, size = 95, normalized size = 0.74 \[ \frac{\left (\frac{b \left (- c - d x\right )}{a d - b c}\right )^{m} \left (a + b x\right )^{m} \left (c + d x\right )^{- m}{{}_{2}F_{1}\left (\begin{matrix} m, m \\ m + 1 \end{matrix}\middle |{\frac{d \left (a + b x\right )}{a d - b c}} \right )}}{f m} - \frac{\left (a + b x\right )^{m} \left (c + d x\right )^{- m}{{}_{2}F_{1}\left (\begin{matrix} m, 1 \\ m + 1 \end{matrix}\middle |{\frac{\left (- a - b x\right ) \left (- c f + d e\right )}{\left (c + d x\right ) \left (a f - b e\right )}} \right )}}{f m} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b*x+a)**m/((d*x+c)**m)/(f*x+e),x)
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Mathematica [C] time = 0.868285, size = 292, normalized size = 2.28 \[ -\frac{(m+2) (b c-a d) (b e-a f)^2 (a+b x)^{m+1} (c+d x)^{-m} F_1\left (m+1;m,1;m+2;\frac{d (a+b x)}{a d-b c},\frac{f (a+b x)}{a f-b e}\right )}{b (m+1) (e+f x) (a f-b e) \left ((m+2) (b c-a d) (b e-a f) F_1\left (m+1;m,1;m+2;\frac{d (a+b x)}{a d-b c},\frac{f (a+b x)}{a f-b e}\right )+(a+b x) \left ((a d f-b c f) F_1\left (m+2;m,2;m+3;\frac{d (a+b x)}{a d-b c},\frac{f (a+b x)}{a f-b e}\right )+d m (a f-b e) F_1\left (m+2;m+1,1;m+3;\frac{d (a+b x)}{a d-b c},\frac{f (a+b x)}{a f-b e}\right )\right )\right )} \]
Warning: Unable to verify antiderivative.
[In] Integrate[(a + b*x)^m/((c + d*x)^m*(e + f*x)),x]
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Maple [F] time = 0.092, size = 0, normalized size = 0. \[ \int{\frac{ \left ( bx+a \right ) ^{m}}{ \left ( dx+c \right ) ^{m} \left ( fx+e \right ) }}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b*x+a)^m/((d*x+c)^m)/(f*x+e),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x + a\right )}^{m}{\left (d x + c\right )}^{-m}}{f x + e}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x + a)^m/((f*x + e)*(d*x + c)^m),x, algorithm="maxima")
[Out]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (b x + a\right )}^{m}}{{\left (f x + e\right )}{\left (d x + c\right )}^{m}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x + a)^m/((f*x + e)*(d*x + c)^m),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)**m/((d*x+c)**m)/(f*x+e),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x + a\right )}^{m}}{{\left (f x + e\right )}{\left (d x + c\right )}^{m}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x + a)^m/((f*x + e)*(d*x + c)^m),x, algorithm="giac")
[Out]