3.1878 \(\int (a+b x)^{-2+n} (c+d x)^{1-n} \, dx\)

Optimal. Leaf size=83 \[ -\frac{(b c-a d) (a+b x)^{n-1} (c+d x)^{-n} \left (\frac{b (c+d x)}{b c-a d}\right )^n \, _2F_1\left (n-1,n-1;n;-\frac{d (a+b x)}{b c-a d}\right )}{b^2 (1-n)} \]

[Out]

-(((b*c - a*d)*(a + b*x)^(-1 + n)*((b*(c + d*x))/(b*c - a*d))^n*Hypergeometric2F
1[-1 + n, -1 + n, n, -((d*(a + b*x))/(b*c - a*d))])/(b^2*(1 - n)*(c + d*x)^n))

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Rubi [A]  time = 0.0944577, antiderivative size = 83, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095 \[ -\frac{(b c-a d) (a+b x)^{n-1} (c+d x)^{-n} \left (\frac{b (c+d x)}{b c-a d}\right )^n \, _2F_1\left (n-1,n-1;n;-\frac{d (a+b x)}{b c-a d}\right )}{b^2 (1-n)} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x)^(-2 + n)*(c + d*x)^(1 - n),x]

[Out]

-(((b*c - a*d)*(a + b*x)^(-1 + n)*((b*(c + d*x))/(b*c - a*d))^n*Hypergeometric2F
1[-1 + n, -1 + n, n, -((d*(a + b*x))/(b*c - a*d))])/(b^2*(1 - n)*(c + d*x)^n))

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Rubi in Sympy [A]  time = 17.3383, size = 63, normalized size = 0.76 \[ \frac{\left (\frac{b \left (- c - d x\right )}{a d - b c}\right )^{n} \left (a + b x\right )^{n - 1} \left (c + d x\right )^{- n} \left (a d - b c\right ){{}_{2}F_{1}\left (\begin{matrix} n - 1, n - 1 \\ n \end{matrix}\middle |{\frac{d \left (a + b x\right )}{a d - b c}} \right )}}{b^{2} \left (- n + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)**(-2+n)*(d*x+c)**(1-n),x)

[Out]

(b*(-c - d*x)/(a*d - b*c))**n*(a + b*x)**(n - 1)*(c + d*x)**(-n)*(a*d - b*c)*hyp
er((n - 1, n - 1), (n,), d*(a + b*x)/(a*d - b*c))/(b**2*(-n + 1))

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Mathematica [A]  time = 0.267464, size = 101, normalized size = 1.22 \[ \frac{(a+b x)^n (c+d x)^{1-n} \left (\frac{d \left (\frac{d (a+b x)}{a d-b c}\right )^{-n} \, _2F_1\left (1-n,1-n;2-n;\frac{b (c+d x)}{b c-a d}\right )}{b c-a d}+\frac{1}{a+b x}\right )}{b (n-1)} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x)^(-2 + n)*(c + d*x)^(1 - n),x]

[Out]

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

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Maple [F]  time = 0.105, size = 0, normalized size = 0. \[ \int \left ( bx+a \right ) ^{-2+n} \left ( dx+c \right ) ^{1-n}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^(-2+n)*(d*x+c)^(1-n),x)

[Out]

int((b*x+a)^(-2+n)*(d*x+c)^(1-n),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (b x + a\right )}^{n - 2}{\left (d x + c\right )}^{-n + 1}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^(n - 2)*(d*x + c)^(-n + 1),x, algorithm="maxima")

[Out]

integrate((b*x + a)^(n - 2)*(d*x + c)^(-n + 1), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left ({\left (b x + a\right )}^{n - 2}{\left (d x + c\right )}^{-n + 1}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^(n - 2)*(d*x + c)^(-n + 1),x, algorithm="fricas")

[Out]

integral((b*x + a)^(n - 2)*(d*x + c)^(-n + 1), 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)**(-2+n)*(d*x+c)**(1-n),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (b x + a\right )}^{n - 2}{\left (d x + c\right )}^{-n + 1}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^(n - 2)*(d*x + c)^(-n + 1),x, algorithm="giac")

[Out]

integrate((b*x + a)^(n - 2)*(d*x + c)^(-n + 1), x)