3.3129 \(\int (a+b x)^m (c+d x)^n (e+f x)^{-3-m-n} \, dx\)

Optimal. Leaf size=227 \[ -\frac{(a+b x)^{m+1} (c+d x)^n (e+f x)^{-m-n-1} (a d f (m+1)+b (c f (n+1)-d e (m+n+2))) \left (\frac{(c+d x) (b e-a f)}{(e+f x) (b c-a d)}\right )^{-n} \, _2F_1\left (m+1,-n;m+2;-\frac{(d e-c f) (a+b x)}{(b c-a d) (e+f x)}\right )}{(m+1) (m+n+2) (b e-a f)^2 (d e-c f)}-\frac{f (a+b x)^{m+1} (c+d x)^{n+1} (e+f x)^{-m-n-2}}{(m+n+2) (b e-a f) (d e-c f)} \]

[Out]

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

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Rubi [A]  time = 0.332869, antiderivative size = 226, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.069 \[ -\frac{(a+b x)^{m+1} (c+d x)^n (e+f x)^{-m-n-1} (a d f (m+1)+b c f (n+1)-b d e (m+n+2)) \left (\frac{(c+d x) (b e-a f)}{(e+f x) (b c-a d)}\right )^{-n} \, _2F_1\left (m+1,-n;m+2;-\frac{(d e-c f) (a+b x)}{(b c-a d) (e+f x)}\right )}{(m+1) (m+n+2) (b e-a f)^2 (d e-c f)}-\frac{f (a+b x)^{m+1} (c+d x)^{n+1} (e+f x)^{-m-n-2}}{(m+n+2) (b e-a f) (d e-c f)} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x)^m*(c + d*x)^n*(e + f*x)^(-3 - m - n),x]

[Out]

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

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Rubi in Sympy [A]  time = 58.9355, size = 197, normalized size = 0.87 \[ - \frac{f \left (a + b x\right )^{m + 1} \left (c + d x\right )^{n + 1} \left (e + f x\right )^{- m - n - 2}}{\left (a f - b e\right ) \left (c f - d e\right ) \left (m + n + 2\right )} - \frac{\left (\frac{\left (e + f x\right ) \left (- a d + b c\right )}{\left (a + b x\right ) \left (c f - d e\right )}\right )^{m + n + 2} \left (a + b x\right )^{m + 1} \left (c + d x\right )^{n + 1} \left (e + f x\right )^{- m - n - 2} \left (a d f \left (m + 1\right ) + b c f \left (n + 1\right ) - b d e \left (m + n + 2\right )\right ){{}_{2}F_{1}\left (\begin{matrix} n + 1, m + n + 2 \\ n + 2 \end{matrix}\middle |{\frac{\left (- c - d x\right ) \left (- a f + b e\right )}{\left (a + b x\right ) \left (c f - d e\right )}} \right )}}{\left (n + 1\right ) \left (a d - b c\right ) \left (a f - b e\right ) \left (c f - d e\right ) \left (m + n + 2\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)**m*(d*x+c)**n*(f*x+e)**(-3-m-n),x)

[Out]

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

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Mathematica [B]  time = 18.9999, size = 5212, normalized size = 22.96 \[ \text{Result too large to show} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[(a + b*x)^m*(c + d*x)^n*(e + f*x)^(-3 - m - n),x]

[Out]

Result too large to show

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Maple [F]  time = 0.224, size = 0, normalized size = 0. \[ \int \left ( bx+a \right ) ^{m} \left ( dx+c \right ) ^{n} \left ( fx+e \right ) ^{-3-m-n}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^m*(d*x+c)^n*(f*x+e)^(-3-m-n),x)

[Out]

int((b*x+a)^m*(d*x+c)^n*(f*x+e)^(-3-m-n),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (b x + a\right )}^{m}{\left (d x + c\right )}^{n}{\left (f x + e\right )}^{-m - n - 3}\,{d x} \]

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 - 3),x, algorithm="maxima")

[Out]

integrate((b*x + a)^m*(d*x + c)^n*(f*x + e)^(-m - n - 3), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left ({\left (b x + a\right )}^{m}{\left (d x + c\right )}^{n}{\left (f x + e\right )}^{-m - n - 3}, x\right ) \]

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 - 3),x, algorithm="fricas")

[Out]

integral((b*x + a)^m*(d*x + c)^n*(f*x + e)^(-m - n - 3), 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)**(-3-m-n),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (b x + a\right )}^{m}{\left (d x + c\right )}^{n}{\left (f x + e\right )}^{-m - n - 3}\,{d x} \]

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 - 3),x, algorithm="giac")

[Out]

integrate((b*x + a)^m*(d*x + c)^n*(f*x + e)^(-m - n - 3), x)