3.3058 \(\int \frac{(a+b x)^m (c+d x)^{-1-m}}{e+f x} \, dx\)

Optimal. Leaf size=75 \[ \frac{(a+b x)^{m+1} (c+d x)^{-m-1} \, _2F_1\left (1,m+1;m+2;\frac{(d e-c f) (a+b x)}{(b e-a f) (c+d x)}\right )}{(m+1) (b e-a f)} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.0878635, antiderivative size = 75, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.038 \[ \frac{(a+b x)^{m+1} (c+d x)^{-m-1} \, _2F_1\left (1,m+1;m+2;\frac{(d e-c f) (a+b x)}{(b e-a f) (c+d x)}\right )}{(m+1) (b e-a f)} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 9.09276, size = 51, normalized size = 0.68 \[ \frac{\left (a + b x\right )^{m} \left (c + d x\right )^{- m}{{}_{2}F_{1}\left (\begin{matrix} - m, 1 \\ - m + 1 \end{matrix}\middle |{\frac{\left (- c - d x\right ) \left (- a f + b e\right )}{\left (a + b x\right ) \left (c f - d e\right )}} \right )}}{m \left (c f - d e\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Mathematica [C]  time = 0.708028, size = 362, normalized size = 4.83 \[ \frac{(a+b x)^m (c+d x)^{-m} \left (\frac{f (m+2) (a+b x) (b c-a d) (b e-a f)^2 F_1\left (m+1;m,1;m+2;\frac{d (a+b x)}{a d-b c},\frac{f (a+b x)}{a f-b e}\right )}{b (m+1) (e+f x) (a f-b e) \left ((m+2) (b c-a d) (b e-a f) F_1\left (m+1;m,1;m+2;\frac{d (a+b x)}{a d-b c},\frac{f (a+b x)}{a f-b e}\right )+(a+b x) \left ((a d f-b c f) F_1\left (m+2;m,2;m+3;\frac{d (a+b x)}{a d-b c},\frac{f (a+b x)}{a f-b e}\right )+d m (a f-b e) F_1\left (m+2;m+1,1;m+3;\frac{d (a+b x)}{a d-b c},\frac{f (a+b x)}{a f-b e}\right )\right )\right )}-\frac{\left (\frac{d (a+b x)}{a d-b c}\right )^{-m} \, _2F_1\left (-m,-m;1-m;\frac{b (c+d x)}{b c-a d}\right )}{m}\right )}{d e-c f} \]

Warning: Unable to verify antiderivative.

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

[Out]

((a + b*x)^m*(((b*c - a*d)*f*(b*e - a*f)^2*(2 + m)*(a + b*x)*AppellF1[1 + m, m,
1, 2 + m, (d*(a + b*x))/(-(b*c) + a*d), (f*(a + b*x))/(-(b*e) + a*f)])/(b*(-(b*e
) + a*f)*(1 + m)*(e + f*x)*((b*c - a*d)*(b*e - a*f)*(2 + m)*AppellF1[1 + m, m, 1
, 2 + m, (d*(a + b*x))/(-(b*c) + a*d), (f*(a + b*x))/(-(b*e) + a*f)] + (a + b*x)
*((-(b*c*f) + a*d*f)*AppellF1[2 + m, m, 2, 3 + m, (d*(a + b*x))/(-(b*c) + a*d),
(f*(a + b*x))/(-(b*e) + a*f)] + d*(-(b*e) + a*f)*m*AppellF1[2 + m, 1 + m, 1, 3 +
m, (d*(a + b*x))/(-(b*c) + a*d), (f*(a + b*x))/(-(b*e) + a*f)]))) - Hypergeomet
ric2F1[-m, -m, 1 - m, (b*(c + d*x))/(b*c - a*d)]/(m*((d*(a + b*x))/(-(b*c) + a*d
))^m)))/((d*e - c*f)*(c + d*x)^m)

_______________________________________________________________________________________

Maple [F]  time = 0.091, size = 0, normalized size = 0. \[ \int{\frac{ \left ( bx+a \right ) ^{m} \left ( dx+c \right ) ^{-1-m}}{fx+e}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^m*(d*x+c)^(-1-m)/(f*x+e),x)

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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