Optimal. Leaf size=138 \[ \frac{(a+b x)^{m+1} (e+f x)^n (c+d x)^{-m-n} \left (\frac{b (e+f x)}{b e-a f}\right )^{-n} \left (\frac{b (c+d x)}{b c-a d}\right )^{m+n} F_1\left (m+1;m+n,1-n;m+2;-\frac{d (a+b x)}{b c-a d},-\frac{f (a+b x)}{b e-a f}\right )}{(m+1) (b e-a f)} \]
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Rubi [A] time = 0.333286, antiderivative size = 138, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1 \[ \frac{(a+b x)^{m+1} (e+f x)^n (c+d x)^{-m-n} \left (\frac{b (e+f x)}{b e-a f}\right )^{-n} \left (\frac{b (c+d x)}{b c-a d}\right )^{m+n} F_1\left (m+1;m+n,1-n;m+2;-\frac{d (a+b x)}{b c-a d},-\frac{f (a+b x)}{b e-a f}\right )}{(m+1) (b e-a f)} \]
Antiderivative was successfully verified.
[In] Int[(a + b*x)^m*(c + d*x)^(-m - n)*(e + f*x)^(-1 + n),x]
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Rubi in Sympy [A] time = 77.4004, size = 105, normalized size = 0.76 \[ - \frac{\left (\frac{b \left (- c - d x\right )}{a d - b c}\right )^{m + n} \left (\frac{b \left (- e - f x\right )}{a f - b e}\right )^{- n} \left (a + b x\right )^{m + 1} \left (c + d x\right )^{- m - n} \left (e + f x\right )^{n} \operatorname{appellf_{1}}{\left (m + 1,m + n,- n + 1,m + 2,\frac{d \left (a + b x\right )}{a d - b c},\frac{f \left (a + b x\right )}{a f - b e} \right )}}{\left (m + 1\right ) \left (a f - b e\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b*x+a)**m*(d*x+c)**(-m-n)*(f*x+e)**(-1+n),x)
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Mathematica [B] time = 1.55274, size = 315, normalized size = 2.28 \[ -\frac{(m+2) (b c-a d) (b e-a f) (a+b x)^{m+1} (e+f x)^{n-1} (c+d x)^{-m-n} F_1\left (m+1;m+n,1-n;m+2;\frac{d (a+b x)}{a d-b c},\frac{f (a+b x)}{a f-b e}\right )}{b (m+1) \left ((a+b x) \left (d (m+n) (b e-a f) F_1\left (m+2;m+n+1,1-n;m+3;\frac{d (a+b x)}{a d-b c},\frac{f (a+b x)}{a f-b e}\right )-f (n-1) (b c-a d) F_1\left (m+2;m+n,2-n;m+3;\frac{d (a+b x)}{a d-b c},\frac{f (a+b x)}{a f-b e}\right )\right )-(m+2) (b c-a d) (b e-a f) F_1\left (m+1;m+n,1-n;m+2;\frac{d (a+b x)}{a d-b c},\frac{f (a+b x)}{a f-b e}\right )\right )} \]
Warning: Unable to verify antiderivative.
[In] Integrate[(a + b*x)^m*(c + d*x)^(-m - n)*(e + f*x)^(-1 + n),x]
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Maple [F] time = 0.237, size = 0, normalized size = 0. \[ \int \left ( bx+a \right ) ^{m} \left ( dx+c \right ) ^{-m-n} \left ( fx+e \right ) ^{-1+n}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b*x+a)^m*(d*x+c)^(-m-n)*(f*x+e)^(-1+n),x)
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int{\left (b x + a\right )}^{m}{\left (d x + c\right )}^{-m - n}{\left (f x + e\right )}^{n - 1}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x + a)^m*(d*x + c)^(-m - n)*(f*x + e)^(n - 1),x, algorithm="maxima")
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left ({\left (b x + a\right )}^{m}{\left (d x + c\right )}^{-m - n}{\left (f x + e\right )}^{n - 1}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x + a)^m*(d*x + c)^(-m - n)*(f*x + e)^(n - 1),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)**(-m-n)*(f*x+e)**(-1+n),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int{\left (b x + a\right )}^{m}{\left (d x + c\right )}^{-m - n}{\left (f x + e\right )}^{n - 1}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x + a)^m*(d*x + c)^(-m - n)*(f*x + e)^(n - 1),x, algorithm="giac")
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