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

Optimal. Leaf size=157 \[ \frac{(d+e x)^{m+1} \, _2F_1\left (1,m+1;m+2;-\frac{g (d+e x)}{e f-d g}\right ) (c f (2 d g-e f (m+2))-g (a e g m+b (d g-e f (m+1))))}{g^2 (m+1) (e f-d g)^2}+\frac{(d+e x)^{m+1} \left (a+\frac{f (c f-b g)}{g^2}\right )}{(f+g x) (e f-d g)}+\frac{c (d+e x)^{m+1}}{e g^2 (m+1)} \]

[Out]

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

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Rubi [A]  time = 0.533131, antiderivative size = 157, 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 \[ -\frac{(d+e x)^{m+1} \, _2F_1\left (1,m+1;m+2;-\frac{g (d+e x)}{e f-d g}\right ) (g (a e g m+b d g-b e f (m+1))-c f (2 d g-e f (m+2)))}{g^2 (m+1) (e f-d g)^2}+\frac{(d+e x)^{m+1} \left (a+\frac{f (c f-b g)}{g^2}\right )}{(f+g x) (e f-d g)}+\frac{c (d+e x)^{m+1}}{e g^2 (m+1)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 42.9209, size = 126, normalized size = 0.8 \[ \frac{c \left (d + e x\right )^{m + 1}}{e g^{2} \left (m + 1\right )} + \frac{e \left (d + e x\right )^{m + 1} \left (a g^{2} - b f g + c f^{2}\right ){{}_{2}F_{1}\left (\begin{matrix} 2, m + 1 \\ m + 2 \end{matrix}\middle |{\frac{g \left (d + e x\right )}{d g - e f}} \right )}}{g^{2} \left (m + 1\right ) \left (d g - e f\right )^{2}} - \frac{\left (d + e x\right )^{m + 1} \left (b g - 2 c f\right ){{}_{2}F_{1}\left (\begin{matrix} 1, m + 1 \\ m + 2 \end{matrix}\middle |{\frac{g \left (d + e x\right )}{d g - e f}} \right )}}{g^{2} \left (m + 1\right ) \left (d g - e f\right )} \]

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)**2,x)

[Out]

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

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Mathematica [A]  time = 0.210325, size = 0, normalized size = 0. \[ \int \frac{(d+e x)^m \left (a+b x+c x^2\right )}{(f+g x)^2} \, dx \]

Verification is Not applicable to the result.

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

[Out]

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

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Maple [F]  time = 0.083, size = 0, normalized size = 0. \[ \int{\frac{ \left ( ex+d \right ) ^{m} \left ( c{x}^{2}+bx+a \right ) }{ \left ( gx+f \right ) ^{2}}}\, dx \]

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)^2,x)

[Out]

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

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

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)^2,x, algorithm="maxima")

[Out]

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

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

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)^2,x, algorithm="fricas")

[Out]

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

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

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)**2,x)

[Out]

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

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

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)^2,x, algorithm="giac")

[Out]

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