3.3057 \(\int (a+b x)^m (c+d x)^{-1-m} \, dx\)

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

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.0940878, 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 \[ -\frac{(a+b x)^m (c+d x)^{-m} \left (-\frac{d (a+b x)}{b c-a d}\right )^{-m} \, _2F_1\left (-m,-m;1-m;\frac{b (c+d x)}{b c-a d}\right )}{d m} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 15.346, size = 54, normalized size = 0.72 \[ - \frac{\left (\frac{d \left (a + b x\right )}{a d - b c}\right )^{- m} \left (a + b x\right )^{m} \left (c + d x\right )^{- m}{{}_{2}F_{1}\left (\begin{matrix} - m, - m \\ - m + 1 \end{matrix}\middle |{\frac{b \left (- c - d x\right )}{a d - b c}} \right )}}{d m} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.0516315, size = 74, normalized size = 0.99 \[ -\frac{(a+b x)^m (c+d x)^{-m} \left (\frac{d (a+b x)}{a d-b c}\right )^{-m} \, _2F_1\left (-m,-m;1-m;\frac{b (c+d x)}{b c-a d}\right )}{d m} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Maple [F]  time = 0.102, size = 0, normalized size = 0. \[ \int \left ( bx+a \right ) ^{m} \left ( dx+c \right ) ^{-1-m}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^m*(d*x+c)^(-1-m),x)

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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