3.3064 \(\int (a+b x)^m (c+d x)^{-2-m} (e+f x)^2 \, dx\)

Optimal. Leaf size=204 \[ -\frac{f (a+b x)^m (c+d x)^{-m} \left (-\frac{d (a+b x)}{b c-a d}\right )^{-m} (a d f m+b (2 d e-c f (m+2))) \, _2F_1\left (-m,-m;1-m;\frac{b (c+d x)}{b c-a d}\right )}{b d^3 m}+\frac{(a+b x)^{m+1} (d e-c f) (c+d x)^{-m-1} (a d f (m+1)+b (d e-c f (m+2)))}{b d^2 (m+1) (b c-a d)}+\frac{f (e+f x) (a+b x)^{m+1} (c+d x)^{-m-1}}{b d} \]

[Out]

((d*e - c*f)*(a*d*f*(1 + m) + b*(d*e - c*f*(2 + m)))*(a + b*x)^(1 + m)*(c + d*x)
^(-1 - m))/(b*d^2*(b*c - a*d)*(1 + m)) + (f*(a + b*x)^(1 + m)*(c + d*x)^(-1 - m)
*(e + f*x))/(b*d) - (f*(a*d*f*m + b*(2*d*e - c*f*(2 + m)))*(a + b*x)^m*Hypergeom
etric2F1[-m, -m, 1 - m, (b*(c + d*x))/(b*c - a*d)])/(b*d^3*m*(-((d*(a + b*x))/(b
*c - a*d)))^m*(c + d*x)^m)

_______________________________________________________________________________________

Rubi [A]  time = 0.498389, antiderivative size = 202, normalized size of antiderivative = 0.99, number of steps used = 4, number of rules used = 4, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ -\frac{f (a+b x)^m (c+d x)^{-m} \left (-\frac{d (a+b x)}{b c-a d}\right )^{-m} (a d f m-b c f (m+2)+2 b d e) \, _2F_1\left (-m,-m;1-m;\frac{b (c+d x)}{b c-a d}\right )}{b d^3 m}+\frac{(a+b x)^{m+1} (d e-c f) (c+d x)^{-m-1} (a d f (m+1)-b c f (m+2)+b d e)}{b d^2 (m+1) (b c-a d)}+\frac{f (e+f x) (a+b x)^{m+1} (c+d x)^{-m-1}}{b d} \]

Antiderivative was successfully verified.

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

[Out]

((d*e - c*f)*(b*d*e + a*d*f*(1 + m) - b*c*f*(2 + m))*(a + b*x)^(1 + m)*(c + d*x)
^(-1 - m))/(b*d^2*(b*c - a*d)*(1 + m)) + (f*(a + b*x)^(1 + m)*(c + d*x)^(-1 - m)
*(e + f*x))/(b*d) - (f*(2*b*d*e + a*d*f*m - b*c*f*(2 + m))*(a + b*x)^m*Hypergeom
etric2F1[-m, -m, 1 - m, (b*(c + d*x))/(b*c - a*d)])/(b*d^3*m*(-((d*(a + b*x))/(b
*c - a*d)))^m*(c + d*x)^m)

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 66.8433, size = 178, normalized size = 0.87 \[ \frac{f \left (a + b x\right )^{m + 1} \left (c + d x\right )^{- m - 1} \left (e + f x\right )}{b d} + \frac{\left (a + b x\right )^{m + 1} \left (c + d x\right )^{- m - 1} \left (c f - d e\right ) \left (a d f m + a d f - b c f m - 2 b c f + b d e\right )}{b d^{2} \left (m + 1\right ) \left (a d - b c\right )} - \frac{f \left (\frac{d \left (a + b x\right )}{a d - b c}\right )^{- m} \left (a + b x\right )^{m} \left (c + d x\right )^{- m} \left (2 b d e + f \left (a d m - b c \left (m + 2\right )\right )\right ){{}_{2}F_{1}\left (\begin{matrix} - m, - m \\ - m + 1 \end{matrix}\middle |{\frac{b \left (- c - d x\right )}{a d - b c}} \right )}}{b d^{3} m} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

f*(a + b*x)**(m + 1)*(c + d*x)**(-m - 1)*(e + f*x)/(b*d) + (a + b*x)**(m + 1)*(c
+ d*x)**(-m - 1)*(c*f - d*e)*(a*d*f*m + a*d*f - b*c*f*m - 2*b*c*f + b*d*e)/(b*d
**2*(m + 1)*(a*d - b*c)) - f*(d*(a + b*x)/(a*d - b*c))**(-m)*(a + b*x)**m*(c + d
*x)**(-m)*(2*b*d*e + f*(a*d*m - b*c*(m + 2)))*hyper((-m, -m), (-m + 1,), b*(-c -
d*x)/(a*d - b*c))/(b*d**3*m)

_______________________________________________________________________________________

Mathematica [C]  time = 1.6592, size = 300, normalized size = 1.47 \[ \frac{1}{3} (a+b x)^m (c+d x)^{-m-2} \left (-\frac{9 a c e f x^2 F_1\left (2;-m,m+2;3;-\frac{b x}{a},-\frac{d x}{c}\right )}{-3 a c F_1\left (2;-m,m+2;3;-\frac{b x}{a},-\frac{d x}{c}\right )-b c m x F_1\left (3;1-m,m+2;4;-\frac{b x}{a},-\frac{d x}{c}\right )+a d (m+2) x F_1\left (3;-m,m+3;4;-\frac{b x}{a},-\frac{d x}{c}\right )}-\frac{4 a c f^2 x^3 F_1\left (3;-m,m+2;4;-\frac{b x}{a},-\frac{d x}{c}\right )}{-4 a c F_1\left (3;-m,m+2;4;-\frac{b x}{a},-\frac{d x}{c}\right )-b c m x F_1\left (4;1-m,m+2;5;-\frac{b x}{a},-\frac{d x}{c}\right )+a d (m+2) x F_1\left (4;-m,m+3;5;-\frac{b x}{a},-\frac{d x}{c}\right )}+\frac{3 e^2 (a+b x) (c+d x)}{(m+1) (b c-a d)}\right ) \]

Warning: Unable to verify antiderivative.

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

[Out]

((a + b*x)^m*(c + d*x)^(-2 - m)*((3*e^2*(a + b*x)*(c + d*x))/((b*c - a*d)*(1 + m
)) - (9*a*c*e*f*x^2*AppellF1[2, -m, 2 + m, 3, -((b*x)/a), -((d*x)/c)])/(-3*a*c*A
ppellF1[2, -m, 2 + m, 3, -((b*x)/a), -((d*x)/c)] - b*c*m*x*AppellF1[3, 1 - m, 2
+ m, 4, -((b*x)/a), -((d*x)/c)] + a*d*(2 + m)*x*AppellF1[3, -m, 3 + m, 4, -((b*x
)/a), -((d*x)/c)]) - (4*a*c*f^2*x^3*AppellF1[3, -m, 2 + m, 4, -((b*x)/a), -((d*x
)/c)])/(-4*a*c*AppellF1[3, -m, 2 + m, 4, -((b*x)/a), -((d*x)/c)] - b*c*m*x*Appel
lF1[4, 1 - m, 2 + m, 5, -((b*x)/a), -((d*x)/c)] + a*d*(2 + m)*x*AppellF1[4, -m,
3 + m, 5, -((b*x)/a), -((d*x)/c)])))/3

_______________________________________________________________________________________

Maple [F]  time = 0.083, size = 0, normalized size = 0. \[ \int \left ( bx+a \right ) ^{m} \left ( dx+c \right ) ^{-2-m} \left ( fx+e \right ) ^{2}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^m*(d*x+c)^(-2-m)*(f*x+e)^2,x)

[Out]

int((b*x+a)^m*(d*x+c)^(-2-m)*(f*x+e)^2,x)

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((f*x + e)^2*(b*x + a)^m*(d*x + c)^(-m - 2), x)

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((f^2*x^2 + 2*e*f*x + e^2)*(b*x + a)^m*(d*x + c)^(-m - 2), 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)**(-2-m)*(f*x+e)**2,x)

[Out]

Timed out

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((f*x + e)^2*(b*x + a)^m*(d*x + c)^(-m - 2), x)