3.3031 \(\int (a+b x) (c+d x)^{-1+n} (e+f x)^{-n} \, dx\)

Optimal. Leaf size=152 \[ -\frac{(c+d x)^{n+1} (e+f x)^{-n} \left (\frac{d (e+f x)}{d e-c f}\right )^n (a d f-b (c f (1-n)+d e n)) \, _2F_1\left (n,n+1;n+2;-\frac{f (c+d x)}{d e-c f}\right )}{d^2 n (n+1) (d e-c f)}-\frac{(b c-a d) (c+d x)^n (e+f x)^{1-n}}{d n (d e-c f)} \]

[Out]

-(((b*c - a*d)*(c + d*x)^n*(e + f*x)^(1 - n))/(d*(d*e - c*f)*n)) - ((a*d*f - b*(
c*f*(1 - n) + d*e*n))*(c + d*x)^(1 + n)*((d*(e + f*x))/(d*e - c*f))^n*Hypergeome
tric2F1[n, 1 + n, 2 + n, -((f*(c + d*x))/(d*e - c*f))])/(d^2*(d*e - c*f)*n*(1 +
n)*(e + f*x)^n)

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Rubi [A]  time = 0.232721, antiderivative size = 152, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125 \[ -\frac{(c+d x)^{n+1} (e+f x)^{-n} \left (\frac{d (e+f x)}{d e-c f}\right )^n (a d f-b c f (1-n)-b d e n) \, _2F_1\left (n,n+1;n+2;-\frac{f (c+d x)}{d e-c f}\right )}{d^2 n (n+1) (d e-c f)}-\frac{(b c-a d) (c+d x)^n (e+f x)^{1-n}}{d n (d e-c f)} \]

Antiderivative was successfully verified.

[In]  Int[((a + b*x)*(c + d*x)^(-1 + n))/(e + f*x)^n,x]

[Out]

-(((b*c - a*d)*(c + d*x)^n*(e + f*x)^(1 - n))/(d*(d*e - c*f)*n)) - ((a*d*f - b*c
*f*(1 - n) - b*d*e*n)*(c + d*x)^(1 + n)*((d*(e + f*x))/(d*e - c*f))^n*Hypergeome
tric2F1[n, 1 + n, 2 + n, -((f*(c + d*x))/(d*e - c*f))])/(d^2*(d*e - c*f)*n*(1 +
n)*(e + f*x)^n)

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Rubi in Sympy [A]  time = 28.5923, size = 117, normalized size = 0.77 \[ - \frac{\left (c + d x\right )^{n} \left (e + f x\right )^{- n + 1} \left (a d - b c\right )}{d n \left (c f - d e\right )} - \frac{\left (\frac{d \left (- e - f x\right )}{c f - d e}\right )^{n} \left (c + d x\right )^{n + 1} \left (e + f x\right )^{- n} \left (- a d f + b \left (c f \left (- n + 1\right ) + d e n\right )\right ){{}_{2}F_{1}\left (\begin{matrix} n, n + 1 \\ n + 2 \end{matrix}\middle |{\frac{f \left (c + d x\right )}{c f - d e}} \right )}}{d^{2} n \left (n + 1\right ) \left (c f - d e\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)*(d*x+c)**(-1+n)/((f*x+e)**n),x)

[Out]

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

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Mathematica [A]  time = 0.199788, size = 147, normalized size = 0.97 \[ \frac{(c+d x)^n (e+f x)^{1-n} \left (\frac{f (c+d x)}{c f-d e}\right )^{-n} \left ((a d f-b c f) \, _2F_1\left (1-n,1-n;2-n;\frac{d (e+f x)}{d e-c f}\right )+b (c f-d e) \, _2F_1\left (1-n,-n;2-n;\frac{d (e+f x)}{d e-c f}\right )\right )}{d f (n-1) (d e-c f)} \]

Antiderivative was successfully verified.

[In]  Integrate[((a + b*x)*(c + d*x)^(-1 + n))/(e + f*x)^n,x]

[Out]

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

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Maple [F]  time = 0.08, size = 0, normalized size = 0. \[ \int{\frac{ \left ( bx+a \right ) \left ( dx+c \right ) ^{-1+n}}{ \left ( fx+e \right ) ^{n}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)*(d*x+c)^(-1+n)/((f*x+e)^n),x)

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (b x + a\right )}{\left (d x + c\right )}^{n - 1}{\left (f x + e\right )}^{-n}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)*(d*x + c)^(n - 1)/(f*x + e)^n,x, algorithm="maxima")

[Out]

integrate((b*x + a)*(d*x + c)^(n - 1)*(f*x + e)^(-n), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (b x + a\right )}{\left (d x + c\right )}^{n - 1}}{{\left (f x + e\right )}^{n}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)*(d*x + c)^(n - 1)/(f*x + e)^n,x, algorithm="fricas")

[Out]

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

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x + a\right )}{\left (d x + c\right )}^{n - 1}}{{\left (f x + e\right )}^{n}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)*(d*x + c)^(n - 1)/(f*x + e)^n,x, algorithm="giac")

[Out]

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