3.3097 \(\int (a+b x)^m (c+d x)^{1-m} (e+f x)^2 \, dx\)

Optimal. Leaf size=260 \[ \frac{(b c-a d) (a+b x)^{m+1} (c+d x)^{-m} \left (\frac{b (c+d x)}{b c-a d}\right )^m \left (a^2 d^2 f^2 \left (m^2-5 m+6\right )-2 a b d f (2-m) (4 d e-c f (m+1))+b^2 \left (c^2 f^2 \left (m^2+3 m+2\right )-8 c d e f (m+1)+12 d^2 e^2\right )\right ) \, _2F_1\left (m-1,m+1;m+2;-\frac{d (a+b x)}{b c-a d}\right )}{12 b^4 d^2 (m+1)}-\frac{f (a+b x)^{m+1} (c+d x)^{2-m} (a d f (3-m)-b (5 d e-c f (m+2)))}{12 b^2 d^2}+\frac{f (e+f x) (a+b x)^{m+1} (c+d x)^{2-m}}{4 b d} \]

[Out]

-(f*(a*d*f*(3 - m) - b*(5*d*e - c*f*(2 + m)))*(a + b*x)^(1 + m)*(c + d*x)^(2 - m
))/(12*b^2*d^2) + (f*(a + b*x)^(1 + m)*(c + d*x)^(2 - m)*(e + f*x))/(4*b*d) + ((
b*c - a*d)*(a^2*d^2*f^2*(6 - 5*m + m^2) - 2*a*b*d*f*(2 - m)*(4*d*e - c*f*(1 + m)
) + b^2*(12*d^2*e^2 - 8*c*d*e*f*(1 + m) + c^2*f^2*(2 + 3*m + m^2)))*(a + b*x)^(1
+ m)*((b*(c + d*x))/(b*c - a*d))^m*Hypergeometric2F1[-1 + m, 1 + m, 2 + m, -((d
*(a + b*x))/(b*c - a*d))])/(12*b^4*d^2*(1 + m)*(c + d*x)^m)

_______________________________________________________________________________________

Rubi [A]  time = 0.638674, antiderivative size = 259, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ \frac{(b c-a d) (a+b x)^{m+1} (c+d x)^{-m} \left (\frac{b (c+d x)}{b c-a d}\right )^m \left (a^2 d^2 f^2 \left (m^2-5 m+6\right )-2 a b d f (2-m) (4 d e-c f (m+1))+b^2 \left (c^2 f^2 \left (m^2+3 m+2\right )-8 c d e f (m+1)+12 d^2 e^2\right )\right ) \, _2F_1\left (m-1,m+1;m+2;-\frac{d (a+b x)}{b c-a d}\right )}{12 b^4 d^2 (m+1)}+\frac{f (a+b x)^{m+1} (c+d x)^{2-m} (-a d f (3-m)-b c f (m+2)+5 b d e)}{12 b^2 d^2}+\frac{f (e+f x) (a+b x)^{m+1} (c+d x)^{2-m}}{4 b d} \]

Antiderivative was successfully verified.

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

[Out]

(f*(5*b*d*e - a*d*f*(3 - m) - b*c*f*(2 + m))*(a + b*x)^(1 + m)*(c + d*x)^(2 - m)
)/(12*b^2*d^2) + (f*(a + b*x)^(1 + m)*(c + d*x)^(2 - m)*(e + f*x))/(4*b*d) + ((b
*c - a*d)*(a^2*d^2*f^2*(6 - 5*m + m^2) - 2*a*b*d*f*(2 - m)*(4*d*e - c*f*(1 + m))
+ b^2*(12*d^2*e^2 - 8*c*d*e*f*(1 + m) + c^2*f^2*(2 + 3*m + m^2)))*(a + b*x)^(1
+ m)*((b*(c + d*x))/(b*c - a*d))^m*Hypergeometric2F1[-1 + m, 1 + m, 2 + m, -((d*
(a + b*x))/(b*c - a*d))])/(12*b^4*d^2*(1 + m)*(c + d*x)^m)

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 52.5514, size = 221, normalized size = 0.85 \[ \frac{f \left (a + b x\right )^{m + 1} \left (c + d x\right )^{- m + 2} \left (e + f x\right )}{4 b d} - \frac{f \left (a + b x\right )^{m + 1} \left (c + d x\right )^{- m + 2} \left (- 5 b d e + f \left (a d \left (- m + 3\right ) + b c \left (m + 2\right )\right )\right )}{12 b^{2} d^{2}} + \frac{\left (\frac{b \left (- c - d x\right )}{a d - b c}\right )^{m} \left (a + b x\right )^{m + 1} \left (c + d x\right )^{- m} \left (a d - b c\right ) \left (3 b d \left (- 4 b d e^{2} + f \left (a c f + e \left (a d \left (- m + 2\right ) + b c \left (m + 1\right )\right )\right )\right ) - f \left (a d \left (- m + 2\right ) + b c \left (m + 1\right )\right ) \left (- 5 b d e + f \left (a d \left (- m + 3\right ) + b c \left (m + 2\right )\right )\right )\right ){{}_{2}F_{1}\left (\begin{matrix} m - 1, m + 1 \\ m + 2 \end{matrix}\middle |{\frac{d \left (a + b x\right )}{a d - b c}} \right )}}{12 b^{4} d^{2} \left (m + 1\right )} \]

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)**2,x)

[Out]

f*(a + b*x)**(m + 1)*(c + d*x)**(-m + 2)*(e + f*x)/(4*b*d) - f*(a + b*x)**(m + 1
)*(c + d*x)**(-m + 2)*(-5*b*d*e + f*(a*d*(-m + 3) + b*c*(m + 2)))/(12*b**2*d**2)
+ (b*(-c - d*x)/(a*d - b*c))**m*(a + b*x)**(m + 1)*(c + d*x)**(-m)*(a*d - b*c)*
(3*b*d*(-4*b*d*e**2 + f*(a*c*f + e*(a*d*(-m + 2) + b*c*(m + 1)))) - f*(a*d*(-m +
2) + b*c*(m + 1))*(-5*b*d*e + f*(a*d*(-m + 3) + b*c*(m + 2))))*hyper((m - 1, m
+ 1), (m + 2,), d*(a + b*x)/(a*d - b*c))/(12*b**4*d**2*(m + 1))

_______________________________________________________________________________________

Mathematica [C]  time = 5.86664, size = 510, normalized size = 1.96 \[ c (a+b x)^m (c+d x)^{-m} \left (\frac{4 a f x^3 (c f+2 d e) F_1\left (3;-m,m;4;-\frac{b x}{a},-\frac{d x}{c}\right )}{3 \left (4 a c F_1\left (3;-m,m;4;-\frac{b x}{a},-\frac{d x}{c}\right )+b c m x F_1\left (4;1-m,m;5;-\frac{b x}{a},-\frac{d x}{c}\right )-a d m x F_1\left (4;-m,m+1;5;-\frac{b x}{a},-\frac{d x}{c}\right )\right )}+\frac{3 a e x^2 (2 c f+d e) F_1\left (2;-m,m;3;-\frac{b x}{a},-\frac{d x}{c}\right )}{6 a c F_1\left (2;-m,m;3;-\frac{b x}{a},-\frac{d x}{c}\right )+2 m x \left (b c F_1\left (3;1-m,m;4;-\frac{b x}{a},-\frac{d x}{c}\right )-a d F_1\left (3;-m,m+1;4;-\frac{b x}{a},-\frac{d x}{c}\right )\right )}+\frac{5 a d f^2 x^4 F_1\left (4;-m,m;5;-\frac{b x}{a},-\frac{d x}{c}\right )}{20 a c F_1\left (4;-m,m;5;-\frac{b x}{a},-\frac{d x}{c}\right )+4 b c m x F_1\left (5;1-m,m;6;-\frac{b x}{a},-\frac{d x}{c}\right )-4 a d m x F_1\left (5;-m,m+1;6;-\frac{b x}{a},-\frac{d x}{c}\right )}-\frac{e^2 x \left (\frac{d (a+b x)}{a d-b c}\right )^{-m} \, _2F_1\left (1-m,-m;2-m;\frac{b (c+d x)}{b c-a d}\right )}{m-1}-\frac{c e^2 \left (\frac{d (a+b x)}{a d-b c}\right )^{-m} \, _2F_1\left (1-m,-m;2-m;\frac{b (c+d x)}{b c-a d}\right )}{d (m-1)}\right ) \]

Warning: Unable to verify antiderivative.

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

[Out]

(c*(a + b*x)^m*((3*a*e*(d*e + 2*c*f)*x^2*AppellF1[2, -m, m, 3, -((b*x)/a), -((d*
x)/c)])/(6*a*c*AppellF1[2, -m, m, 3, -((b*x)/a), -((d*x)/c)] + 2*m*x*(b*c*Appell
F1[3, 1 - m, m, 4, -((b*x)/a), -((d*x)/c)] - a*d*AppellF1[3, -m, 1 + m, 4, -((b*
x)/a), -((d*x)/c)])) + (4*a*f*(2*d*e + c*f)*x^3*AppellF1[3, -m, m, 4, -((b*x)/a)
, -((d*x)/c)])/(3*(4*a*c*AppellF1[3, -m, m, 4, -((b*x)/a), -((d*x)/c)] + b*c*m*x
*AppellF1[4, 1 - m, m, 5, -((b*x)/a), -((d*x)/c)] - a*d*m*x*AppellF1[4, -m, 1 +
m, 5, -((b*x)/a), -((d*x)/c)])) + (5*a*d*f^2*x^4*AppellF1[4, -m, m, 5, -((b*x)/a
), -((d*x)/c)])/(20*a*c*AppellF1[4, -m, m, 5, -((b*x)/a), -((d*x)/c)] + 4*b*c*m*
x*AppellF1[5, 1 - m, m, 6, -((b*x)/a), -((d*x)/c)] - 4*a*d*m*x*AppellF1[5, -m, 1
+ m, 6, -((b*x)/a), -((d*x)/c)]) - (c*e^2*Hypergeometric2F1[1 - m, -m, 2 - m, (
b*(c + d*x))/(b*c - a*d)])/(d*(-1 + m)*((d*(a + b*x))/(-(b*c) + a*d))^m) - (e^2*
x*Hypergeometric2F1[1 - m, -m, 2 - m, (b*(c + d*x))/(b*c - a*d)])/((-1 + m)*((d*
(a + b*x))/(-(b*c) + a*d))^m)))/(c + d*x)^m

_______________________________________________________________________________________

Maple [F]  time = 0.097, size = 0, normalized size = 0. \[ \int \left ( bx+a \right ) ^{m} \left ( dx+c \right ) ^{1-m} \left ( fx+e \right ) ^{2}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

int((b*x+a)^m*(d*x+c)^(1-m)*(f*x+e)^2,x)

_______________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (f x + e\right )}^{2}{\left (b x + a\right )}^{m}{\left (d x + c\right )}^{-m + 1}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((f*x + e)^2*(b*x + a)^m*(d*x + c)^(-m + 1),x, algorithm="maxima")

[Out]

integrate((f*x + e)^2*(b*x + a)^m*(d*x + c)^(-m + 1), x)

_______________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left ({\left (f^{2} x^{2} + 2 \, e f x + e^{2}\right )}{\left (b x + a\right )}^{m}{\left (d x + c\right )}^{-m + 1}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((f*x + e)^2*(b*x + a)^m*(d*x + c)^(-m + 1),x, algorithm="fricas")

[Out]

integral((f^2*x^2 + 2*e*f*x + e^2)*(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)*(f*x+e)**2,x)

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (f x + e\right )}^{2}{\left (b x + a\right )}^{m}{\left (d x + c\right )}^{-m + 1}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((f*x + e)^2*(b*x + a)^m*(d*x + c)^(-m + 1),x, algorithm="giac")

[Out]

integrate((f*x + e)^2*(b*x + a)^m*(d*x + c)^(-m + 1), x)