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

Optimal. Leaf size=261 \[ \frac{(a+b x)^{m+1} (c+d x)^{-m} \left (-c d (a f h m+2 b (e h+f g))+d f h m x (b c-a d)+b c^2 f h (m+2)+2 b d^2 e g\right )}{2 b d^2 m (b c-a d)}-\frac{(a+b x)^{m+1} (c+d x)^{-m} \left (\frac{b (c+d x)}{b c-a d}\right )^m \, _2F_1\left (m,m+1;m+2;-\frac{d (a+b x)}{b c-a d}\right ) \left (d^2 \left (a^2 (-f) h (1-m) m+2 a b m (e h+f g)+2 b^2 e g\right )-2 b c d (m+1) (a f h m+b e h+b f g)+b^2 c^2 f h (m+1) (m+2)\right )}{2 b^2 d^2 m (m+1) (b c-a d)} \]

[Out]

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

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Rubi [A]  time = 0.567323, antiderivative size = 261, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.103 \[ \frac{(a+b x)^{m+1} (c+d x)^{-m} \left (-c d (a f h m+2 b (e h+f g))+d f h m x (b c-a d)+b c^2 f h (m+2)+2 b d^2 e g\right )}{2 b d^2 m (b c-a d)}-\frac{(a+b x)^{m+1} (c+d x)^{-m} \left (\frac{b (c+d x)}{b c-a d}\right )^m \, _2F_1\left (m,m+1;m+2;-\frac{d (a+b x)}{b c-a d}\right ) \left (d^2 \left (a^2 (-f) h (1-m) m+2 a b m (e h+f g)+2 b^2 e g\right )-2 b c d (m+1) (a f h m+b e h+b f g)+b^2 c^2 f h (m+1) (m+2)\right )}{2 b^2 d^2 m (m+1) (b c-a d)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 61.8651, size = 238, normalized size = 0.91 \[ \frac{\left (a + b x\right )^{m + 1} \left (c + d x\right )^{- m} \left (- b c^{2} f h \left (m + 2\right ) - 2 b d^{2} e g + c d \left (a f h m + 2 b \left (e h + f g\right )\right ) + d f h m x \left (a d - b c\right )\right )}{2 b d^{2} m \left (a d - b c\right )} + \frac{\left (\frac{b \left (- c - d x\right )}{a d - b c}\right )^{m} \left (a + b x\right )^{m + 1} \left (c + d x\right )^{- m} \left (b^{2} c^{2} f h \left (m + 1\right ) \left (m + 2\right ) - 2 b c d \left (m + 1\right ) \left (a f h m + b \left (e h + f g\right )\right ) + d^{2} \left (- a^{2} f h m \left (- m + 1\right ) + 2 a b m \left (e h + f g\right ) + 2 b^{2} e g\right )\right ){{}_{2}F_{1}\left (\begin{matrix} m, m + 1 \\ m + 2 \end{matrix}\middle |{\frac{d \left (a + b x\right )}{a d - b c}} \right )}}{2 b^{2} d^{2} m \left (m + 1\right ) \left (a d - b c\right )} \]

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

[Out]

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

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Mathematica [C]  time = 2.15174, size = 346, normalized size = 1.33 \[ \frac{1}{6} (a+b x)^m (c+d x)^{-m} \left (\frac{9 a c x^2 (e h+f g) F_1\left (2;-m,m+1;3;-\frac{b x}{a},-\frac{d x}{c}\right )}{(c+d x) \left (3 a c F_1\left (2;-m,m+1;3;-\frac{b x}{a},-\frac{d x}{c}\right )+b c m x F_1\left (3;1-m,m+1;4;-\frac{b x}{a},-\frac{d x}{c}\right )-a d (m+1) x F_1\left (3;-m,m+2;4;-\frac{b x}{a},-\frac{d x}{c}\right )\right )}+\frac{8 a c f h x^3 F_1\left (3;-m,m+1;4;-\frac{b x}{a},-\frac{d x}{c}\right )}{(c+d x) \left (4 a c F_1\left (3;-m,m+1;4;-\frac{b x}{a},-\frac{d x}{c}\right )+b c m x F_1\left (4;1-m,m+1;5;-\frac{b x}{a},-\frac{d x}{c}\right )-a d (m+1) x F_1\left (4;-m,m+2;5;-\frac{b x}{a},-\frac{d x}{c}\right )\right )}-\frac{6 e g \left (\frac{d (a+b x)}{a d-b c}\right )^{-m} \, _2F_1\left (-m,-m;1-m;\frac{b (c+d x)}{b c-a d}\right )}{d m}\right ) \]

Warning: Unable to verify antiderivative.

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

[Out]

((a + b*x)^m*((9*a*c*(f*g + e*h)*x^2*AppellF1[2, -m, 1 + m, 3, -((b*x)/a), -((d*
x)/c)])/((c + d*x)*(3*a*c*AppellF1[2, -m, 1 + m, 3, -((b*x)/a), -((d*x)/c)] + b*
c*m*x*AppellF1[3, 1 - m, 1 + m, 4, -((b*x)/a), -((d*x)/c)] - a*d*(1 + m)*x*Appel
lF1[3, -m, 2 + m, 4, -((b*x)/a), -((d*x)/c)])) + (8*a*c*f*h*x^3*AppellF1[3, -m,
1 + m, 4, -((b*x)/a), -((d*x)/c)])/((c + d*x)*(4*a*c*AppellF1[3, -m, 1 + m, 4, -
((b*x)/a), -((d*x)/c)] + b*c*m*x*AppellF1[4, 1 - m, 1 + m, 5, -((b*x)/a), -((d*x
)/c)] - a*d*(1 + m)*x*AppellF1[4, -m, 2 + m, 5, -((b*x)/a), -((d*x)/c)])) - (6*e
*g*Hypergeometric2F1[-m, -m, 1 - m, (b*(c + d*x))/(b*c - a*d)])/(d*m*((d*(a + b*
x))/(-(b*c) + a*d))^m)))/(6*(c + d*x)^m)

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Maple [F]  time = 0.077, size = 0, normalized size = 0. \[ \int \left ( bx+a \right ) ^{m} \left ( dx+c \right ) ^{-1-m} \left ( fx+e \right ) \left ( hx+g \right ) \, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

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 - 1),x, algorithm="maxima")

[Out]

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

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

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 - 1),x, algorithm="fricas")

[Out]

integral((f*h*x^2 + e*g + (f*g + e*h)*x)*(b*x + a)^m*(d*x + c)^(-m - 1), 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)**(-1-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. \[ \int{\left (f x + e\right )}{\left (h x + g\right )}{\left (b x + a\right )}^{m}{\left (d x + c\right )}^{-m - 1}\,{d x} \]

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 - 1),x, algorithm="giac")

[Out]

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