3.126 \(\int (a+b x)^m (c+d x)^{-m} (e+f x) (g+h x) \, dx\)

Optimal. Leaf size=235 \[ \frac{(a+b x)^{m+1} (c+d x)^{-m} \left (\frac{b (c+d x)}{b c-a d}\right )^m \, _2F_1\left (m,m+1;m+2;-\frac{d (a+b x)}{b c-a d}\right ) \left (a^2 d^2 f h \left (m^2-3 m+2\right )-a b d (1-m) (3 d (e h+f g)-2 c f h (m+1))+b^2 \left (c^2 f h \left (m^2+3 m+2\right )-3 c d (m+1) (e h+f g)+6 d^2 e g\right )\right )}{6 b^3 d^2 (m+1)}+\frac{(a+b x)^{m+1} (c+d x)^{1-m} (-a d f h (2-m)-b c f h (m+2)+3 b d (e h+f g)+2 b d f h x)}{6 b^2 d^2} \]

[Out]

((a + b*x)^(1 + m)*(c + d*x)^(1 - m)*(3*b*d*(f*g + e*h) - a*d*f*h*(2 - m) - b*c*
f*h*(2 + m) + 2*b*d*f*h*x))/(6*b^2*d^2) + ((a^2*d^2*f*h*(2 - 3*m + m^2) - a*b*d*
(1 - m)*(3*d*(f*g + e*h) - 2*c*f*h*(1 + m)) + b^2*(6*d^2*e*g - 3*c*d*(f*g + e*h)
*(1 + m) + c^2*f*h*(2 + 3*m + m^2)))*(a + b*x)^(1 + m)*((b*(c + d*x))/(b*c - a*d
))^m*Hypergeometric2F1[m, 1 + m, 2 + m, -((d*(a + b*x))/(b*c - a*d))])/(6*b^3*d^
2*(1 + m)*(c + d*x)^m)

_______________________________________________________________________________________

Rubi [A]  time = 0.394816, antiderivative size = 235, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111 \[ \frac{(a+b x)^{m+1} (c+d x)^{-m} \left (\frac{b (c+d x)}{b c-a d}\right )^m \, _2F_1\left (m,m+1;m+2;-\frac{d (a+b x)}{b c-a d}\right ) \left (a^2 d^2 f h \left (m^2-3 m+2\right )-a b d (1-m) (3 d (e h+f g)-2 c f h (m+1))+b^2 \left (c^2 f h \left (m^2+3 m+2\right )-3 c d (m+1) (e h+f g)+6 d^2 e g\right )\right )}{6 b^3 d^2 (m+1)}+\frac{(a+b x)^{m+1} (c+d x)^{1-m} (-a d f h (2-m)-b c f h (m+2)+3 b d (e h+f g)+2 b d f h x)}{6 b^2 d^2} \]

Antiderivative was successfully verified.

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

[Out]

((a + b*x)^(1 + m)*(c + d*x)^(1 - m)*(3*b*d*(f*g + e*h) - a*d*f*h*(2 - m) - b*c*
f*h*(2 + m) + 2*b*d*f*h*x))/(6*b^2*d^2) + ((a^2*d^2*f*h*(2 - 3*m + m^2) - a*b*d*
(1 - m)*(3*d*(f*g + e*h) - 2*c*f*h*(1 + m)) + b^2*(6*d^2*e*g - 3*c*d*(f*g + e*h)
*(1 + m) + c^2*f*h*(2 + 3*m + m^2)))*(a + b*x)^(1 + m)*((b*(c + d*x))/(b*c - a*d
))^m*Hypergeometric2F1[m, 1 + m, 2 + m, -((d*(a + b*x))/(b*c - a*d))])/(6*b^3*d^
2*(1 + m)*(c + d*x)^m)

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 39.5191, size = 216, normalized size = 0.92 \[ \frac{\left (a + b x\right )^{m + 1} \left (c + d x\right )^{- m + 1} \left (- a d f h \left (- m + 2\right ) - b c f h \left (m + 2\right ) + 2 b d f h x + 3 b d \left (e h + f g\right )\right )}{6 b^{2} d^{2}} + \frac{\left (\frac{b \left (- c - d x\right )}{a d - b c}\right )^{m} \left (a + b x\right )^{m + 1} \left (c + d x\right )^{- m} \left (a^{2} d^{2} f h \left (- m + 1\right ) \left (- m + 2\right ) - a b d \left (- m + 1\right ) \left (- 2 c f h \left (m + 1\right ) + 3 d \left (e h + f g\right )\right ) + b^{2} \left (c^{2} f h \left (m + 1\right ) \left (m + 2\right ) - 3 c d \left (m + 1\right ) \left (e h + f g\right ) + 6 d^{2} e g\right )\right ){{}_{2}F_{1}\left (\begin{matrix} m, m + 1 \\ m + 2 \end{matrix}\middle |{\frac{d \left (a + b x\right )}{a d - b c}} \right )}}{6 b^{3} d^{2} \left (m + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(a + b*x)**(m + 1)*(c + d*x)**(-m + 1)*(-a*d*f*h*(-m + 2) - b*c*f*h*(m + 2) + 2*
b*d*f*h*x + 3*b*d*(e*h + f*g))/(6*b**2*d**2) + (b*(-c - d*x)/(a*d - b*c))**m*(a
+ b*x)**(m + 1)*(c + d*x)**(-m)*(a**2*d**2*f*h*(-m + 1)*(-m + 2) - a*b*d*(-m + 1
)*(-2*c*f*h*(m + 1) + 3*d*(e*h + f*g)) + b**2*(c**2*f*h*(m + 1)*(m + 2) - 3*c*d*
(m + 1)*(e*h + f*g) + 6*d**2*e*g))*hyper((m, m + 1), (m + 2,), d*(a + b*x)/(a*d
- b*c))/(6*b**3*d**2*(m + 1))

_______________________________________________________________________________________

Mathematica [C]  time = 1.31731, size = 324, normalized size = 1.38 \[ (a+b x)^m (c+d x)^{-m} \left (\frac{3 a c x^2 (e h+f g) F_1\left (2;-m,m;3;-\frac{b x}{a},-\frac{d x}{c}\right )}{6 a c F_1\left (2;-m,m;3;-\frac{b x}{a},-\frac{d x}{c}\right )+2 m x \left (b c F_1\left (3;1-m,m;4;-\frac{b x}{a},-\frac{d x}{c}\right )-a d F_1\left (3;-m,m+1;4;-\frac{b x}{a},-\frac{d x}{c}\right )\right )}+\frac{4 a c f h x^3 F_1\left (3;-m,m;4;-\frac{b x}{a},-\frac{d x}{c}\right )}{12 a c F_1\left (3;-m,m;4;-\frac{b x}{a},-\frac{d x}{c}\right )+3 b c m x F_1\left (4;1-m,m;5;-\frac{b x}{a},-\frac{d x}{c}\right )-3 a d m x F_1\left (4;-m,m+1;5;-\frac{b x}{a},-\frac{d x}{c}\right )}-\frac{e g (c+d x) \left (\frac{d (a+b x)}{a d-b c}\right )^{-m} \, _2F_1\left (1-m,-m;2-m;\frac{b (c+d x)}{b c-a d}\right )}{d (m-1)}\right ) \]

Warning: Unable to verify antiderivative.

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

[Out]

((a + b*x)^m*((3*a*c*(f*g + e*h)*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*AppellF1[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*c*f*h*x^3*AppellF1[3, -m, m, 4, -((b*x)/a), -((d*x)/c)])
/(12*a*c*AppellF1[3, -m, m, 4, -((b*x)/a), -((d*x)/c)] + 3*b*c*m*x*AppellF1[4, 1
- m, m, 5, -((b*x)/a), -((d*x)/c)] - 3*a*d*m*x*AppellF1[4, -m, 1 + m, 5, -((b*x
)/a), -((d*x)/c)]) - (e*g*(c + d*x)*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)))/(c + d*x)^m

_______________________________________________________________________________________

Maple [F]  time = 0.087, size = 0, normalized size = 0. \[ \int{\frac{ \left ( bx+a \right ) ^{m} \left ( fx+e \right ) \left ( hx+g \right ) }{ \left ( dx+c \right ) ^{m}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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