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

Optimal. Leaf size=224 \[ \frac{d^2 (a+b x)^{m+1} \, _2F_1\left (1,m+1;m+2;-\frac{d (a+b x)}{b c-a d}\right )}{(m+1) (b c-a d) (d e-c f) (d g-c h)}-\frac{f^2 (a+b x)^{m+1} \, _2F_1\left (1,m+1;m+2;-\frac{f (a+b x)}{b e-a f}\right )}{(m+1) (b e-a f) (d e-c f) (f g-e h)}+\frac{h^2 (a+b x)^{m+1} \, _2F_1\left (1,m+1;m+2;-\frac{h (a+b x)}{b g-a h}\right )}{(m+1) (b g-a h) (d g-c h) (f g-e h)} \]

[Out]

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

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Rubi [A]  time = 0.58024, antiderivative size = 224, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 2, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.069 \[ \frac{d^2 (a+b x)^{m+1} \, _2F_1\left (1,m+1;m+2;-\frac{d (a+b x)}{b c-a d}\right )}{(m+1) (b c-a d) (d e-c f) (d g-c h)}-\frac{f^2 (a+b x)^{m+1} \, _2F_1\left (1,m+1;m+2;-\frac{f (a+b x)}{b e-a f}\right )}{(m+1) (b e-a f) (d e-c f) (f g-e h)}+\frac{h^2 (a+b x)^{m+1} \, _2F_1\left (1,m+1;m+2;-\frac{h (a+b x)}{b g-a h}\right )}{(m+1) (b g-a h) (d g-c h) (f g-e h)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 81.2976, size = 162, normalized size = 0.72 \[ - \frac{d^{2} \left (a + b x\right )^{m + 1}{{}_{2}F_{1}\left (\begin{matrix} 1, m + 1 \\ m + 2 \end{matrix}\middle |{\frac{d \left (a + b x\right )}{a d - b c}} \right )}}{\left (m + 1\right ) \left (a d - b c\right ) \left (c f - d e\right ) \left (c h - d g\right )} + \frac{f^{2} \left (a + b x\right )^{m + 1}{{}_{2}F_{1}\left (\begin{matrix} 1, m + 1 \\ m + 2 \end{matrix}\middle |{\frac{f \left (a + b x\right )}{a f - b e}} \right )}}{\left (m + 1\right ) \left (a f - b e\right ) \left (c f - d e\right ) \left (e h - f g\right )} - \frac{h^{2} \left (a + b x\right )^{m + 1}{{}_{2}F_{1}\left (\begin{matrix} 1, m + 1 \\ m + 2 \end{matrix}\middle |{\frac{h \left (a + b x\right )}{a h - b g}} \right )}}{\left (m + 1\right ) \left (a h - b g\right ) \left (c h - d g\right ) \left (e h - f g\right )} \]

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]

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

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Mathematica [A]  time = 0.740344, size = 229, normalized size = 1.02 \[ \frac{(a+b x)^m \left (d (f g-e h) \left (\frac{d (a+b x)}{b (c+d x)}\right )^{-m} \, _2F_1\left (-m,-m;1-m;\frac{b c-a d}{b c+b d x}\right )-f (d g-c h) \left (\frac{f (a+b x)}{b (e+f x)}\right )^{-m} \, _2F_1\left (-m,-m;1-m;\frac{b e-a f}{b e+b f x}\right )+h (d e-c f) \left (\frac{h (a+b x)}{b (g+h x)}\right )^{-m} \, _2F_1\left (-m,-m;1-m;\frac{b g-a h}{b g+b h x}\right )\right )}{m (d e-c f) (d g-c h) (f g-e h)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [F]  time = 0.131, size = 0, normalized size = 0. \[ \int{\frac{ \left ( bx+a \right ) ^{m}}{ \left ( hx+g \right ) \left ( fx+e \right ) \left ( dx+c \right ) }}\, dx \]

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

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, algorithm="maxima")

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (b x + a\right )}^{m}}{d f h x^{3} + c e g +{\left (d f g +{\left (d e + c f\right )} h\right )} x^{2} +{\left (c e h +{\left (d e + c f\right )} g\right )} x}, x\right ) \]

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, algorithm="fricas")

[Out]

integral((b*x + a)^m/(d*f*h*x^3 + c*e*g + (d*f*g + (d*e + c*f)*h)*x^2 + (c*e*h +
(d*e + c*f)*g)*x), 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)/(f*x+e)/(h*x+g),x)

[Out]

Timed out

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

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, algorithm="giac")

[Out]

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