3.1867 \(\int (a+b x)^{-1-n} (c+d x)^n \, dx\)

Optimal. Leaf size=75 \[ -\frac{(a+b x)^{-n} (c+d x)^n \left (\frac{b (c+d x)}{b c-a d}\right )^{-n} \, _2F_1\left (-n,-n;1-n;-\frac{d (a+b x)}{b c-a d}\right )}{b n} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.0671104, antiderivative size = 75, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105 \[ -\frac{(a+b x)^{-n} (c+d x)^n \left (\frac{b (c+d x)}{b c-a d}\right )^{-n} \, _2F_1\left (-n,-n;1-n;-\frac{d (a+b x)}{b c-a d}\right )}{b n} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 14.8189, size = 54, normalized size = 0.72 \[ - \frac{\left (\frac{b \left (- c - d x\right )}{a d - b c}\right )^{- n} \left (a + b x\right )^{- n} \left (c + d x\right )^{n}{{}_{2}F_{1}\left (\begin{matrix} - n, - n \\ - n + 1 \end{matrix}\middle |{\frac{d \left (a + b x\right )}{a d - b c}} \right )}}{b n} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.0729507, size = 81, normalized size = 1.08 \[ -\frac{(a+b x)^{-n} (c+d x)^{n+1} \left (\frac{d (a+b x)}{a d-b c}\right )^n \, _2F_1\left (n+1,n+1;n+2;\frac{b (c+d x)}{b c-a d}\right )}{(n+1) (b c-a d)} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Maple [F]  time = 0.104, size = 0, normalized size = 0. \[ \int \left ( bx+a \right ) ^{-1-n} \left ( dx+c \right ) ^{n}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^(-1-n)*(d*x+c)^n,x)

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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)**(-1-n)*(d*x+c)**n,x)

[Out]

Timed out

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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