Optimal. Leaf size=124 \[ \frac{(a+b x)^{m+1} (e+f x)^{n-1} (c+d x)^{-m-n} \left (\frac{(c+d x) (b e-a f)}{(e+f x) (b c-a d)}\right )^{m+n} \, _2F_1\left (m+1,m+n;m+2;-\frac{(d e-c f) (a+b x)}{(b c-a d) (e+f x)}\right )}{(m+1) (b e-a f)} \]
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Rubi [A] time = 0.113879, antiderivative size = 124, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.033 \[ \frac{(a+b x)^{m+1} (e+f x)^{n-1} (c+d x)^{-m-n} \left (\frac{(c+d x) (b e-a f)}{(e+f x) (b c-a d)}\right )^{m+n} \, _2F_1\left (m+1,m+n;m+2;-\frac{(d e-c f) (a+b x)}{(b c-a d) (e+f x)}\right )}{(m+1) (b e-a f)} \]
Antiderivative was successfully verified.
[In] Int[(a + b*x)^m*(c + d*x)^(-m - n)*(e + f*x)^(-2 + n),x]
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Rubi in Sympy [A] time = 11.0015, size = 97, normalized size = 0.78 \[ - \frac{\left (\frac{\left (c + d x\right ) \left (a f - b e\right )}{\left (e + f x\right ) \left (a d - b c\right )}\right )^{m + n} \left (a + b x\right )^{m + 1} \left (c + d x\right )^{- m - n} \left (e + f x\right )^{n - 1}{{}_{2}F_{1}\left (\begin{matrix} m + 1, m + n \\ m + 2 \end{matrix}\middle |{\frac{\left (- a - b x\right ) \left (c f - d e\right )}{\left (e + f x\right ) \left (a d - b c\right )}} \right )}}{\left (m + 1\right ) \left (a f - b e\right )} \]
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)**(-2+n),x)
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Mathematica [A] time = 1.07769, size = 123, normalized size = 0.99 \[ \frac{(a+b x)^{m+1} (e+f x)^{n-1} (c+d x)^{-m-n} \left (\frac{(c+d x) (b e-a f)}{(e+f x) (b c-a d)}\right )^{m+n} \, _2F_1\left (m+1,m+n;m+2;\frac{(c f-d e) (a+b x)}{(b c-a d) (e+f x)}\right )}{(m+1) (b e-a f)} \]
Antiderivative was successfully verified.
[In] Integrate[(a + b*x)^m*(c + d*x)^(-m - n)*(e + f*x)^(-2 + n),x]
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Maple [F] time = 0.249, size = 0, normalized size = 0. \[ \int \left ( bx+a \right ) ^{m} \left ( dx+c \right ) ^{-m-n} \left ( fx+e \right ) ^{-2+n}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b*x+a)^m*(d*x+c)^(-m-n)*(f*x+e)^(-2+n),x)
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int{\left (b x + a\right )}^{m}{\left (d x + c\right )}^{-m - n}{\left (f x + e\right )}^{n - 2}\,{d x} \]
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 - 2),x, algorithm="maxima")
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left ({\left (b x + a\right )}^{m}{\left (d x + c\right )}^{-m - n}{\left (f x + e\right )}^{n - 2}, x\right ) \]
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 - 2),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)**(-2+n),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int{\left (b x + a\right )}^{m}{\left (d x + c\right )}^{-m - n}{\left (f x + e\right )}^{n - 2}\,{d x} \]
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 - 2),x, algorithm="giac")
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