3.922 \(\int \frac{(d+e x)^m \left (a+b x+c x^2\right )}{f+g x} \, dx\)

Optimal. Leaf size=129 \[ \frac{(d+e x)^{m+1} \left (a g^2-b f g+c f^2\right ) \, _2F_1\left (1,m+1;m+2;-\frac{g (d+e x)}{e f-d g}\right )}{g^2 (m+1) (e f-d g)}-\frac{(d+e x)^{m+1} (-b e g+c d g+c e f)}{e^2 g^2 (m+1)}+\frac{c (d+e x)^{m+2}}{e^2 g (m+2)} \]

[Out]

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

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Rubi [A]  time = 0.399742, antiderivative size = 129, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.12 \[ \frac{(d+e x)^{m+1} \left (a g^2-b f g+c f^2\right ) \, _2F_1\left (1,m+1;m+2;-\frac{g (d+e x)}{e f-d g}\right )}{g^2 (m+1) (e f-d g)}-\frac{(d+e x)^{m+1} (-b e g+c d g+c e f)}{e^2 g^2 (m+1)}+\frac{c (d+e x)^{m+2}}{e^2 g (m+2)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 41.7642, size = 107, normalized size = 0.83 \[ \frac{c \left (d + e x\right )^{m + 2}}{e^{2} g \left (m + 2\right )} - \frac{\left (d + e x\right )^{m + 1} \left (a g^{2} - b f g + c f^{2}\right ){{}_{2}F_{1}\left (\begin{matrix} 1, m + 1 \\ m + 2 \end{matrix}\middle |{\frac{g \left (d + e x\right )}{d g - e f}} \right )}}{g^{2} \left (m + 1\right ) \left (d g - e f\right )} + \frac{\left (d + e x\right )^{m + 1} \left (b e g - c d g - c e f\right )}{e^{2} g^{2} \left (m + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x+d)**m*(c*x**2+b*x+a)/(g*x+f),x)

[Out]

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

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Mathematica [A]  time = 1.15942, size = 166, normalized size = 1.29 \[ \frac{(d+e x)^m \left (\frac{\left (g (a g-b f)+c f^2\right ) \left (\frac{g (d+e x)}{e (f+g x)}\right )^{-m} \, _2F_1\left (-m,-m;1-m;\frac{e f-d g}{e f+e g x}\right )}{m}+\frac{g \left (b e g (m+2) (d+e x)+c \left (d^2 g \left (\left (\frac{e x}{d}+1\right )^{-m}-1\right )+d e (g m x-f (m+2))+e^2 x (g (m+1) x-f (m+2))\right )\right )}{e^2 (m+1) (m+2)}\right )}{g^3} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [F]  time = 0.062, size = 0, normalized size = 0. \[ \int{\frac{ \left ( ex+d \right ) ^{m} \left ( c{x}^{2}+bx+a \right ) }{gx+f}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x + a)*(e*x + d)^m/(g*x + f),x, algorithm="maxima")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x + a)*(e*x + d)^m/(g*x + f),x, algorithm="fricas")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x+d)**m*(c*x**2+b*x+a)/(g*x+f),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x + a)*(e*x + d)^m/(g*x + f),x, algorithm="giac")

[Out]

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