3.3128 \(\int (a+b x)^m (c+d x)^n (e+f x)^{-2-m-n} \, dx\)

Optimal. Leaf size=123 \[ \frac{(a+b x)^{m+1} (c+d x)^n (e+f x)^{-m-n-1} \left (\frac{(c+d x) (b e-a f)}{(e+f x) (b c-a d)}\right )^{-n} \, _2F_1\left (m+1,-n;m+2;-\frac{(d e-c f) (a+b x)}{(b c-a d) (e+f x)}\right )}{(m+1) (b e-a f)} \]

[Out]

((a + b*x)^(1 + m)*(c + d*x)^n*(e + f*x)^(-1 - m - n)*Hypergeometric2F1[1 + m, -
n, 2 + m, -(((d*e - c*f)*(a + b*x))/((b*c - a*d)*(e + f*x)))])/((b*e - a*f)*(1 +
m)*(((b*e - a*f)*(c + d*x))/((b*c - a*d)*(e + f*x)))^n)

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Rubi [A]  time = 0.105173, antiderivative size = 123, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.034 \[ \frac{(a+b x)^{m+1} (c+d x)^n (e+f x)^{-m-n-1} \left (\frac{(c+d x) (b e-a f)}{(e+f x) (b c-a d)}\right )^{-n} \, _2F_1\left (m+1,-n;m+2;-\frac{(d e-c f) (a+b x)}{(b c-a d) (e+f x)}\right )}{(m+1) (b e-a f)} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x)^m*(c + d*x)^n*(e + f*x)^(-2 - m - n),x]

[Out]

((a + b*x)^(1 + m)*(c + d*x)^n*(e + f*x)^(-1 - m - n)*Hypergeometric2F1[1 + m, -
n, 2 + m, -(((d*e - c*f)*(a + b*x))/((b*c - a*d)*(e + f*x)))])/((b*e - a*f)*(1 +
m)*(((b*e - a*f)*(c + d*x))/((b*c - a*d)*(e + f*x)))^n)

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Rubi in Sympy [A]  time = 10.5188, size = 95, normalized size = 0.77 \[ - \frac{\left (\frac{\left (c + d x\right ) \left (a f - b e\right )}{\left (e + f x\right ) \left (a d - b c\right )}\right )^{- n} \left (a + b x\right )^{m + 1} \left (c + d x\right )^{n} \left (e + f x\right )^{- m - n - 1}{{}_{2}F_{1}\left (\begin{matrix} m + 1, - n \\ m + 2 \end{matrix}\middle |{\frac{\left (- a - b x\right ) \left (c f - d e\right )}{\left (e + f x\right ) \left (a d - b c\right )}} \right )}}{\left (m + 1\right ) \left (a f - b e\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)**m*(d*x+c)**n*(f*x+e)**(-2-m-n),x)

[Out]

-((c + d*x)*(a*f - b*e)/((e + f*x)*(a*d - b*c)))**(-n)*(a + b*x)**(m + 1)*(c + d
*x)**n*(e + f*x)**(-m - n - 1)*hyper((m + 1, -n), (m + 2,), (-a - b*x)*(c*f - d*
e)/((e + f*x)*(a*d - b*c)))/((m + 1)*(a*f - b*e))

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Mathematica [A]  time = 0.968004, size = 122, normalized size = 0.99 \[ \frac{(a+b x)^{m+1} (c+d x)^n (e+f x)^{-m-n-1} \left (\frac{(c+d x) (b e-a f)}{(e+f x) (b c-a d)}\right )^{-n} \, _2F_1\left (m+1,-n;m+2;\frac{(c f-d e) (a+b x)}{(b c-a d) (e+f x)}\right )}{(m+1) (b e-a f)} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x)^m*(c + d*x)^n*(e + f*x)^(-2 - m - n),x]

[Out]

((a + b*x)^(1 + m)*(c + d*x)^n*(e + f*x)^(-1 - m - n)*Hypergeometric2F1[1 + m, -
n, 2 + m, ((-(d*e) + c*f)*(a + b*x))/((b*c - a*d)*(e + f*x))])/((b*e - a*f)*(1 +
m)*(((b*e - a*f)*(c + d*x))/((b*c - a*d)*(e + f*x)))^n)

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Maple [F]  time = 0.227, size = 0, normalized size = 0. \[ \int \left ( bx+a \right ) ^{m} \left ( dx+c \right ) ^{n} \left ( fx+e \right ) ^{-2-m-n}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^m*(d*x+c)^n*(f*x+e)^(-2-m-n),x)

[Out]

int((b*x+a)^m*(d*x+c)^n*(f*x+e)^(-2-m-n),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (b x + a\right )}^{m}{\left (d x + c\right )}^{n}{\left (f x + e\right )}^{-m - n - 2}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^m*(d*x + c)^n*(f*x + e)^(-m - n - 2),x, algorithm="maxima")

[Out]

integrate((b*x + a)^m*(d*x + c)^n*(f*x + e)^(-m - n - 2), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left ({\left (b x + a\right )}^{m}{\left (d x + c\right )}^{n}{\left (f x + e\right )}^{-m - n - 2}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^m*(d*x + c)^n*(f*x + e)^(-m - n - 2),x, algorithm="fricas")

[Out]

integral((b*x + a)^m*(d*x + c)^n*(f*x + e)^(-m - n - 2), x)

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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)**n*(f*x+e)**(-2-m-n),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (b x + a\right )}^{m}{\left (d x + c\right )}^{n}{\left (f x + e\right )}^{-m - n - 2}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^m*(d*x + c)^n*(f*x + e)^(-m - n - 2),x, algorithm="giac")

[Out]

integrate((b*x + a)^m*(d*x + c)^n*(f*x + e)^(-m - n - 2), x)