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

Optimal. Leaf size=203 \[ \frac{(a+b x)^{m+1} (c+d x)^{-m-1} \left (-c d (a f h (m+1)+b (e h+f g))+d f h (m+1) x (b c-a d)+b c^2 f h (m+2)+b d^2 e g\right )}{b d^2 (m+1) (b c-a d)}-\frac{(a+b x)^m (c+d x)^{-m} \left (-\frac{d (a+b x)}{b c-a d}\right )^{-m} \, _2F_1\left (-m,-m;1-m;\frac{b (c+d x)}{b c-a d}\right ) (a d f h m+b (d (e h+f g)-c f h (m+2)))}{b d^3 m} \]

[Out]

((a + b*x)^(1 + m)*(c + d*x)^(-1 - m)*(b*d^2*e*g + b*c^2*f*h*(2 + m) - c*d*(b*(f
*g + e*h) + a*f*h*(1 + m)) + d*(b*c - a*d)*f*h*(1 + m)*x))/(b*d^2*(b*c - a*d)*(1
+ m)) - ((a*d*f*h*m + b*(d*(f*g + e*h) - c*f*h*(2 + m)))*(a + b*x)^m*Hypergeome
tric2F1[-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)

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Rubi [A]  time = 0.378012, antiderivative size = 205, normalized size of antiderivative = 1.01, number of steps used = 3, number of rules used = 3, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.103 \[ -\frac{(a+b x)^{m+1} (c+d x)^{-m-1} \left (-d f h (m+1) x (b c-a d)+a c d f h (m+1)-b \left (c^2 f h (m+2)-c d (e h+f g)+d^2 e g\right )\right )}{b d^2 (m+1) (b c-a d)}-\frac{(a+b x)^m (c+d x)^{-m} \left (-\frac{d (a+b x)}{b c-a d}\right )^{-m} \, _2F_1\left (-m,-m;1-m;\frac{b (c+d x)}{b c-a d}\right ) (a d f h m-b c f h (m+2)+b d (e h+f g))}{b d^3 m} \]

Antiderivative was successfully verified.

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

[Out]

-(((a + b*x)^(1 + m)*(c + d*x)^(-1 - m)*(a*c*d*f*h*(1 + m) - b*(d^2*e*g - c*d*(f
*g + e*h) + c^2*f*h*(2 + m)) - d*(b*c - a*d)*f*h*(1 + m)*x))/(b*d^2*(b*c - a*d)*
(1 + m))) - ((b*d*(f*g + e*h) + a*d*f*h*m - b*c*f*h*(2 + m))*(a + b*x)^m*Hyperge
ometric2F1[-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)

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Rubi in Sympy [A]  time = 33.058, size = 175, normalized size = 0.86 \[ \frac{\left (a + b x\right )^{m + 1} \left (c + d x\right )^{- m - 1} \left (- b c^{2} f h \left (m + 2\right ) - b d^{2} e g + c d \left (a f h \left (m + 1\right ) + b \left (e h + f g\right )\right ) + d f h x \left (m + 1\right ) \left (a d - b c\right )\right )}{b d^{2} \left (m + 1\right ) \left (a d - b c\right )} - \frac{\left (\frac{d \left (a + b x\right )}{a d - b c}\right )^{- m} \left (a + b x\right )^{m} \left (c + d x\right )^{- m} \left (- b c f h \left (m + 2\right ) + d \left (a f h m + b \left (e h + f g\right )\right )\right ){{}_{2}F_{1}\left (\begin{matrix} - m, - m \\ - m + 1 \end{matrix}\middle |{\frac{b \left (- c - d x\right )}{a d - b c}} \right )}}{b d^{3} m} \]

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)*(h*x+g),x)

[Out]

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

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Mathematica [C]  time = 2.49988, size = 303, normalized size = 1.49 \[ \frac{1}{6} (a+b x)^m (c+d x)^{-m-2} \left (-\frac{9 a c x^2 (e h+f g) F_1\left (2;-m,m+2;3;-\frac{b x}{a},-\frac{d x}{c}\right )}{-3 a c F_1\left (2;-m,m+2;3;-\frac{b x}{a},-\frac{d x}{c}\right )-b c m x F_1\left (3;1-m,m+2;4;-\frac{b x}{a},-\frac{d x}{c}\right )+a d (m+2) x F_1\left (3;-m,m+3;4;-\frac{b x}{a},-\frac{d x}{c}\right )}-\frac{8 a c f h x^3 F_1\left (3;-m,m+2;4;-\frac{b x}{a},-\frac{d x}{c}\right )}{-4 a c F_1\left (3;-m,m+2;4;-\frac{b x}{a},-\frac{d x}{c}\right )-b c m x F_1\left (4;1-m,m+2;5;-\frac{b x}{a},-\frac{d x}{c}\right )+a d (m+2) x F_1\left (4;-m,m+3;5;-\frac{b x}{a},-\frac{d x}{c}\right )}+\frac{6 e g (a+b x) (c+d x)}{(m+1) (b c-a d)}\right ) \]

Warning: Unable to verify antiderivative.

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

[Out]

((a + b*x)^m*(c + d*x)^(-2 - m)*((6*e*g*(a + b*x)*(c + d*x))/((b*c - a*d)*(1 + m
)) - (9*a*c*(f*g + e*h)*x^2*AppellF1[2, -m, 2 + m, 3, -((b*x)/a), -((d*x)/c)])/(
-3*a*c*AppellF1[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)]) - (8*a*c*f*h*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*AppellF1[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)])))/6

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Maple [F]  time = 0.075, size = 0, normalized size = 0. \[ \int \left ( bx+a \right ) ^{m} \left ( dx+c \right ) ^{-2-m} \left ( fx+e \right ) \left ( hx+g \right ) \, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

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 - 2),x, algorithm="maxima")

[Out]

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

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

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 - 2),x, algorithm="fricas")

[Out]

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

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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)*(h*x+g),x)

[Out]

Timed out

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

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 - 2),x, algorithm="giac")

[Out]

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