3.1874 \(\int (a+b x)^n (c+d x)^{-2-n} \, dx\)

Optimal. Leaf size=36 \[ \frac{(a+b x)^{n+1} (c+d x)^{-n-1}}{(n+1) (b c-a d)} \]

[Out]

((a + b*x)^(1 + n)*(c + d*x)^(-1 - n))/((b*c - a*d)*(1 + n))

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Rubi [A]  time = 0.022004, antiderivative size = 36, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.053 \[ \frac{(a+b x)^{n+1} (c+d x)^{-n-1}}{(n+1) (b c-a d)} \]

Antiderivative was successfully verified.

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

[Out]

((a + b*x)^(1 + n)*(c + d*x)^(-1 - n))/((b*c - a*d)*(1 + n))

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Rubi in Sympy [A]  time = 4.28074, size = 29, normalized size = 0.81 \[ - \frac{\left (a + b x\right )^{n + 1} \left (c + d x\right )^{- n - 1}}{\left (n + 1\right ) \left (a d - b c\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(a + b*x)**(n + 1)*(c + d*x)**(-n - 1)/((n + 1)*(a*d - b*c))

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Mathematica [A]  time = 0.0599223, size = 36, normalized size = 1. \[ \frac{(a+b x)^{n+1} (c+d x)^{-n-1}}{(n+1) (b c-a d)} \]

Antiderivative was successfully verified.

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

[Out]

((a + b*x)^(1 + n)*(c + d*x)^(-1 - n))/((b*c - a*d)*(1 + n))

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Maple [A]  time = 0.006, size = 42, normalized size = 1.2 \[ -{\frac{ \left ( bx+a \right ) ^{1+n} \left ( dx+c \right ) ^{-1-n}}{adn-bcn+ad-bc}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^n*(d*x+c)^(-2-n),x)

[Out]

-(b*x+a)^(1+n)*(d*x+c)^(-1-n)/(a*d*n-b*c*n+a*d-b*c)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.232217, size = 78, normalized size = 2.17 \[ \frac{{\left (b d x^{2} + a c +{\left (b c + a d\right )} x\right )}{\left (b x + a\right )}^{n}{\left (d x + c\right )}^{-n - 2}}{b c - a d +{\left (b c - a d\right )} n} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(b*d*x^2 + a*c + (b*c + a*d)*x)*(b*x + a)^n*(d*x + c)^(-n - 2)/(b*c - a*d + (b*c
- a*d)*n)

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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)**n*(d*x+c)**(-2-n),x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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