Optimal. Leaf size=131 \[ \frac{b (a+b x)^{m+1} (c+d x)^{-m} (e+f x)^p \left (\frac{b (c+d x)}{b c-a d}\right )^m \left (\frac{b (e+f x)}{b e-a f}\right )^{-p} F_1\left (m+1;m+2,-p;m+2;-\frac{d (a+b x)}{b c-a d},-\frac{f (a+b x)}{b e-a f}\right )}{(m+1) (b c-a d)^2} \]
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Rubi [A] time = 0.368352, antiderivative size = 131, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.115 \[ \frac{b (a+b x)^{m+1} (c+d x)^{-m} (e+f x)^p \left (\frac{b (c+d x)}{b c-a d}\right )^m \left (\frac{b (e+f x)}{b e-a f}\right )^{-p} F_1\left (m+1;m+2,-p;m+2;-\frac{d (a+b x)}{b c-a d},-\frac{f (a+b x)}{b e-a f}\right )}{(m+1) (b c-a d)^2} \]
Antiderivative was successfully verified.
[In] Int[(a + b*x)^m*(c + d*x)^(-2 - m)*(e + f*x)^p,x]
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Rubi in Sympy [A] time = 76.4257, size = 102, normalized size = 0.78 \[ \frac{b \left (\frac{b \left (- c - d x\right )}{a d - b c}\right )^{m} \left (\frac{b \left (- e - f x\right )}{a f - b e}\right )^{- p} \left (a + b x\right )^{m + 1} \left (c + d x\right )^{- m} \left (e + f x\right )^{p} \operatorname{appellf_{1}}{\left (m + 1,- p,m + 2,m + 2,\frac{f \left (a + b x\right )}{a f - b e},\frac{d \left (a + b x\right )}{a d - b c} \right )}}{\left (m + 1\right ) \left (a d - b c\right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b*x+a)**m*(d*x+c)**(-2-m)*(f*x+e)**p,x)
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Mathematica [B] time = 1.51097, size = 300, normalized size = 2.29 \[ \frac{(m+2) (b c-a d) (b e-a f) (a+b x)^{m+1} (c+d x)^{-m-2} (e+f x)^p F_1\left (m+1;m+2,-p;m+2;\frac{d (a+b x)}{a d-b c},\frac{f (a+b x)}{a f-b e}\right )}{b (m+1) \left ((m+2) (b c-a d) (b e-a f) F_1\left (m+1;m+2,-p;m+2;\frac{d (a+b x)}{a d-b c},\frac{f (a+b x)}{a f-b e}\right )-(a+b x) \left (f p (a d-b c) F_1\left (m+2;m+2,1-p;m+3;\frac{d (a+b x)}{a d-b c},\frac{f (a+b x)}{a f-b e}\right )+d (m+2) (b e-a f) F_1\left (m+2;m+3,-p;m+3;\frac{d (a+b x)}{a d-b c},\frac{f (a+b x)}{a f-b e}\right )\right )\right )} \]
Warning: Unable to verify antiderivative.
[In] Integrate[(a + b*x)^m*(c + d*x)^(-2 - m)*(e + f*x)^p,x]
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Maple [F] time = 0.236, size = 0, normalized size = 0. \[ \int \left ( bx+a \right ) ^{m} \left ( dx+c \right ) ^{-2-m} \left ( fx+e \right ) ^{p}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b*x+a)^m*(d*x+c)^(-2-m)*(f*x+e)^p,x)
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int{\left (b x + a\right )}^{m}{\left (d x + c\right )}^{-m - 2}{\left (f x + e\right )}^{p}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x + a)^m*(d*x + c)^(-m - 2)*(f*x + e)^p,x, algorithm="maxima")
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left ({\left (b x + a\right )}^{m}{\left (d x + c\right )}^{-m - 2}{\left (f x + e\right )}^{p}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x + a)^m*(d*x + c)^(-m - 2)*(f*x + e)^p,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)**(-2-m)*(f*x+e)**p,x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int{\left (b x + a\right )}^{m}{\left (d x + c\right )}^{-m - 2}{\left (f x + e\right )}^{p}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x + a)^m*(d*x + c)^(-m - 2)*(f*x + e)^p,x, algorithm="giac")
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