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

Optimal. Leaf size=134 \[ \frac{(a+b x)^{m+1} (d g-c h) (f g-e h) \, _2F_1\left (1,m+1;m+2;-\frac{h (a+b x)}{b g-a h}\right )}{h^2 (m+1) (b g-a h)}-\frac{(a+b x)^{m+1} (a d f h+b (m+2) (-c f h-d e h+d f g)-b d f h (m+1) x)}{b^2 h^2 (m+1) (m+2)} \]

[Out]

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

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Rubi [A]  time = 0.245645, antiderivative size = 134, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.08 \[ \frac{(a+b x)^{m+1} (d g-c h) (f g-e h) \, _2F_1\left (1,m+1;m+2;-\frac{h (a+b x)}{b g-a h}\right )}{h^2 (m+1) (b g-a h)}-\frac{(a+b x)^{m+1} (a d f h+b (m+2) (-c f h-d e h+d f g)-b d f h (m+1) x)}{b^2 h^2 (m+1) (m+2)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 19.181, size = 119, normalized size = 0.89 \[ - \frac{\left (a + b x\right )^{m + 1} \left (c h - d g\right ) \left (e h - f g\right ){{}_{2}F_{1}\left (\begin{matrix} 1, m + 1 \\ m + 2 \end{matrix}\middle |{\frac{h \left (a + b x\right )}{a h - b g}} \right )}}{h^{2} \left (m + 1\right ) \left (a h - b g\right )} - \frac{\left (a + b x\right )^{m + 1} \left (a d f h + b d f g \left (m + 2\right ) - b d f h x \left (m + 1\right ) - b h \left (m + 2\right ) \left (c f + d e\right )\right )}{b^{2} h^{2} \left (m + 1\right ) \left (m + 2\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [C]  time = 1.53985, size = 317, normalized size = 2.37 \[ \frac{1}{6} (a+b x)^m \left (\frac{9 a g x^2 (c f+d e) F_1\left (2;-m,1;3;-\frac{b x}{a},-\frac{h x}{g}\right )}{(g+h x) \left (3 a g F_1\left (2;-m,1;3;-\frac{b x}{a},-\frac{h x}{g}\right )+b g m x F_1\left (3;1-m,1;4;-\frac{b x}{a},-\frac{h x}{g}\right )-a h x F_1\left (3;-m,2;4;-\frac{b x}{a},-\frac{h x}{g}\right )\right )}+\frac{8 a d f g x^3 F_1\left (3;-m,1;4;-\frac{b x}{a},-\frac{h x}{g}\right )}{(g+h x) \left (4 a g F_1\left (3;-m,1;4;-\frac{b x}{a},-\frac{h x}{g}\right )+b g m x F_1\left (4;1-m,1;5;-\frac{b x}{a},-\frac{h x}{g}\right )-a h x F_1\left (4;-m,2;5;-\frac{b x}{a},-\frac{h x}{g}\right )\right )}+\frac{6 c e \left (\frac{h (a+b x)}{b (g+h x)}\right )^{-m} \, _2F_1\left (-m,-m;1-m;\frac{b g-a h}{b g+b h x}\right )}{h m}\right ) \]

Warning: Unable to verify antiderivative.

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

[Out]

((a + b*x)^m*((9*a*(d*e + c*f)*g*x^2*AppellF1[2, -m, 1, 3, -((b*x)/a), -((h*x)/g
)])/((g + h*x)*(3*a*g*AppellF1[2, -m, 1, 3, -((b*x)/a), -((h*x)/g)] + b*g*m*x*Ap
pellF1[3, 1 - m, 1, 4, -((b*x)/a), -((h*x)/g)] - a*h*x*AppellF1[3, -m, 2, 4, -((
b*x)/a), -((h*x)/g)])) + (8*a*d*f*g*x^3*AppellF1[3, -m, 1, 4, -((b*x)/a), -((h*x
)/g)])/((g + h*x)*(4*a*g*AppellF1[3, -m, 1, 4, -((b*x)/a), -((h*x)/g)] + b*g*m*x
*AppellF1[4, 1 - m, 1, 5, -((b*x)/a), -((h*x)/g)] - a*h*x*AppellF1[4, -m, 2, 5,
-((b*x)/a), -((h*x)/g)])) + (6*c*e*Hypergeometric2F1[-m, -m, 1 - m, (b*g - a*h)/
(b*g + b*h*x)])/(h*m*((h*(a + b*x))/(b*(g + h*x)))^m)))/6

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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