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

Optimal. Leaf size=140 \[ \frac{(a+b x)^{m+1} (d g-c h) \, _2F_1\left (1,m+1;m+2;-\frac{h (a+b x)}{b g-a h}\right )}{(m+1) (b g-a h) (f g-e h)}-\frac{(a+b x)^{m+1} (d e-c f) \, _2F_1\left (1,m+1;m+2;-\frac{f (a+b x)}{b e-a f}\right )}{(m+1) (b e-a f) (f g-e h)} \]

[Out]

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

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Rubi [A]  time = 0.247025, antiderivative size = 140, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.074 \[ \frac{(a+b x)^{m+1} (d g-c h) \, _2F_1\left (1,m+1;m+2;-\frac{h (a+b x)}{b g-a h}\right )}{(m+1) (b g-a h) (f g-e h)}-\frac{(a+b x)^{m+1} (d e-c f) \, _2F_1\left (1,m+1;m+2;-\frac{f (a+b x)}{b e-a f}\right )}{(m+1) (b e-a f) (f g-e h)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 28.5081, size = 100, normalized size = 0.71 \[ - \frac{\left (a + b x\right )^{m + 1} \left (c h - d g\right ){{}_{2}F_{1}\left (\begin{matrix} 1, m + 1 \\ m + 2 \end{matrix}\middle |{\frac{h \left (a + b x\right )}{a h - b g}} \right )}}{\left (m + 1\right ) \left (a h - b g\right ) \left (e h - f g\right )} + \frac{\left (a + b x\right )^{m + 1} \left (c f - d e\right ){{}_{2}F_{1}\left (\begin{matrix} 1, m + 1 \\ m + 2 \end{matrix}\middle |{\frac{f \left (a + b x\right )}{a f - b e}} \right )}}{\left (m + 1\right ) \left (a f - b e\right ) \left (e h - f g\right )} \]

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

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Mathematica [C]  time = 3.04029, size = 390, normalized size = 2.79 \[ \frac{1}{2} (a+b x)^m \left (3 a d x^2 \left (\frac{e f F_1\left (2;-m,1;3;-\frac{b x}{a},-\frac{f x}{e}\right )}{(e+f x) (f g-e h) \left (3 a e F_1\left (2;-m,1;3;-\frac{b x}{a},-\frac{f x}{e}\right )+b e m x F_1\left (3;1-m,1;4;-\frac{b x}{a},-\frac{f x}{e}\right )-a f x F_1\left (3;-m,2;4;-\frac{b x}{a},-\frac{f x}{e}\right )\right )}+\frac{g h F_1\left (2;-m,1;3;-\frac{b x}{a},-\frac{h x}{g}\right )}{(g+h x) (e h-f g) \left (3 a g F_1\left (2;-m,1;3;-\frac{b x}{a},-\frac{h x}{g}\right )+b g m x F_1\left (3;1-m,1;4;-\frac{b x}{a},-\frac{h x}{g}\right )-a h x F_1\left (3;-m,2;4;-\frac{b x}{a},-\frac{h x}{g}\right )\right )}\right )+\frac{2 c \left (\frac{f (a+b x)}{b (e+f x)}\right )^{-m} \, _2F_1\left (-m,-m;1-m;\frac{b e-a f}{b e+b f x}\right )-2 c \left (\frac{h (a+b x)}{b (g+h x)}\right )^{-m} \, _2F_1\left (-m,-m;1-m;\frac{b g-a h}{b g+b h x}\right )}{f g m-e h m}\right ) \]

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*(3*a*d*x^2*((e*f*AppellF1[2, -m, 1, 3, -((b*x)/a), -((f*x)/e)])/((f
*g - e*h)*(e + f*x)*(3*a*e*AppellF1[2, -m, 1, 3, -((b*x)/a), -((f*x)/e)] + b*e*m
*x*AppellF1[3, 1 - m, 1, 4, -((b*x)/a), -((f*x)/e)] - a*f*x*AppellF1[3, -m, 2, 4
, -((b*x)/a), -((f*x)/e)])) + (g*h*AppellF1[2, -m, 1, 3, -((b*x)/a), -((h*x)/g)]
)/((-(f*g) + e*h)*(g + h*x)*(3*a*g*AppellF1[2, -m, 1, 3, -((b*x)/a), -((h*x)/g)]
+ b*g*m*x*AppellF1[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)]))) + ((2*c*Hypergeometric2F1[-m, -m, 1 - m, (b
*e - a*f)/(b*e + b*f*x)])/((f*(a + b*x))/(b*(e + f*x)))^m - (2*c*Hypergeometric2
F1[-m, -m, 1 - m, (b*g - a*h)/(b*g + b*h*x)])/((h*(a + b*x))/(b*(g + h*x)))^m)/(
f*g*m - e*h*m)))/2

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^m*(d*x+c)/(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. \[ \int \frac{{\left (d x + c\right )}{\left (b x + a\right )}^{m}}{{\left (f x + e\right )}{\left (h x + g\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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