3.3038 \(\int (a+b x)^{-n} (c+d x) (e+f x)^{-2+n} \, dx\)

Optimal. Leaf size=118 \[ \frac{d (a+b x)^{-n} (e+f x)^n \left (-\frac{f (a+b x)}{b e-a f}\right )^n \, _2F_1\left (n,n;n+1;\frac{b (e+f x)}{b e-a f}\right )}{f^2 n}-\frac{(a+b x)^{1-n} (d e-c f) (e+f x)^{n-1}}{f (1-n) (b e-a f)} \]

[Out]

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

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Rubi [A]  time = 0.179949, antiderivative size = 118, 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 \[ \frac{d (a+b x)^{-n} (e+f x)^n \left (-\frac{f (a+b x)}{b e-a f}\right )^n \, _2F_1\left (n,n;n+1;\frac{b (e+f x)}{b e-a f}\right )}{f^2 n}-\frac{(a+b x)^{1-n} (d e-c f) (e+f x)^{n-1}}{f (1-n) (b e-a f)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 22.5997, size = 88, normalized size = 0.75 \[ \frac{d \left (\frac{f \left (a + b x\right )}{a f - b e}\right )^{n} \left (a + b x\right )^{- n} \left (e + f x\right )^{n}{{}_{2}F_{1}\left (\begin{matrix} n, n \\ n + 1 \end{matrix}\middle |{\frac{b \left (- e - f x\right )}{a f - b e}} \right )}}{f^{2} n} - \frac{\left (a + b x\right )^{- n + 1} \left (e + f x\right )^{n - 1} \left (c f - d e\right )}{f \left (- n + 1\right ) \left (a f - b e\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.269086, size = 125, normalized size = 1.06 \[ -\frac{(a+b x)^{-n} (e+f x)^{n-1} \left (c f^2 n (a+b x)-d (n-1) (e+f x) (b e-a f) \left (\frac{f (a+b x)}{a f-b e}\right )^n \, _2F_1\left (n,n;n+1;\frac{b (e+f x)}{b e-a f}\right )-d e f n (a+b x)\right )}{f^2 (n-1) n (b e-a f)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [F]  time = 0.076, size = 0, normalized size = 0. \[ \int{\frac{ \left ( dx+c \right ) \left ( fx+e \right ) ^{-2+n}}{ \left ( bx+a \right ) ^{n}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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