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

Optimal. Leaf size=298 \[ \frac{(d+e x)^{m+1} \left (2 c e g (a e g-b (d g+2 e f))+b^2 e^2 g^2+c^2 \left (d^2 g^2+2 d e f g+3 e^2 f^2\right )\right )}{e^3 g^4 (m+1)}+\frac{(d+e x)^{m+1} \left (a g^2-b f g+c f^2\right ) \, _2F_1\left (1,m+1;m+2;-\frac{g (d+e x)}{e f-d g}\right ) (c f (4 d g-e f (m+4))-g (a e g m+b (2 d g-e f (m+2))))}{g^4 (m+1) (e f-d g)^2}+\frac{(d+e x)^{m+1} \left (a g^2-b f g+c f^2\right )^2}{g^4 (f+g x) (e f-d g)}-\frac{2 c (d+e x)^{m+2} (-b e g+c d g+c e f)}{e^3 g^3 (m+2)}+\frac{c^2 (d+e x)^{m+3}}{e^3 g^2 (m+3)} \]

[Out]

((b^2*e^2*g^2 + c^2*(3*e^2*f^2 + 2*d*e*f*g + d^2*g^2) + 2*c*e*g*(a*e*g - b*(2*e*
f + d*g)))*(d + e*x)^(1 + m))/(e^3*g^4*(1 + m)) - (2*c*(c*e*f + c*d*g - b*e*g)*(
d + e*x)^(2 + m))/(e^3*g^3*(2 + m)) + (c^2*(d + e*x)^(3 + m))/(e^3*g^2*(3 + m))
+ ((c*f^2 - b*f*g + a*g^2)^2*(d + e*x)^(1 + m))/(g^4*(e*f - d*g)*(f + g*x)) + ((
c*f^2 - b*f*g + a*g^2)*(c*f*(4*d*g - e*f*(4 + m)) - g*(a*e*g*m + b*(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^4*(e*f - d*g)^2*(1 + m))

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Rubi [A]  time = 2.33824, antiderivative size = 298, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111 \[ \frac{(d+e x)^{m+1} \left (2 c e g (a e g-b (d g+2 e f))+b^2 e^2 g^2+c^2 \left (d^2 g^2+2 d e f g+3 e^2 f^2\right )\right )}{e^3 g^4 (m+1)}-\frac{(d+e x)^{m+1} \left (a g^2-b f g+c f^2\right ) \, _2F_1\left (1,m+1;m+2;-\frac{g (d+e x)}{e f-d g}\right ) (g (a e g m+2 b d g-b e f (m+2))-c f (4 d g-e f (m+4)))}{g^4 (m+1) (e f-d g)^2}+\frac{(d+e x)^{m+1} \left (a g^2-b f g+c f^2\right )^2}{g^4 (f+g x) (e f-d g)}-\frac{2 c (d+e x)^{m+2} (-b e g+c d g+c e f)}{e^3 g^3 (m+2)}+\frac{c^2 (d+e x)^{m+3}}{e^3 g^2 (m+3)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 148.022, size = 277, normalized size = 0.93 \[ \frac{c^{2} \left (d + e x\right )^{m + 3}}{e^{3} g^{2} \left (m + 3\right )} + \frac{2 c \left (d + e x\right )^{m + 2} \left (b e g - c d g - c e f\right )}{e^{3} g^{3} \left (m + 2\right )} + \frac{e \left (d + e x\right )^{m + 1} \left (a g^{2} - b f g + c f^{2}\right )^{2}{{}_{2}F_{1}\left (\begin{matrix} 2, m + 1 \\ m + 2 \end{matrix}\middle |{\frac{g \left (d + e x\right )}{d g - e f}} \right )}}{g^{4} \left (m + 1\right ) \left (d g - e f\right )^{2}} - \frac{2 \left (d + e x\right )^{m + 1} \left (b g - 2 c f\right ) \left (a g^{2} - b f g + c f^{2}\right ){{}_{2}F_{1}\left (\begin{matrix} 1, m + 1 \\ m + 2 \end{matrix}\middle |{\frac{g \left (d + e x\right )}{d g - e f}} \right )}}{g^{4} \left (m + 1\right ) \left (d g - e f\right )} + \frac{\left (d + e x\right )^{m + 1} \left (- 2 b c d e g^{2} - 4 b c e^{2} f g + c^{2} d^{2} g^{2} + 2 c^{2} d e f g + 3 c^{2} e^{2} f^{2} + e^{2} g^{2} \left (2 a c + b^{2}\right )\right )}{e^{3} g^{4} \left (m + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

c**2*(d + e*x)**(m + 3)/(e**3*g**2*(m + 3)) + 2*c*(d + e*x)**(m + 2)*(b*e*g - c*
d*g - c*e*f)/(e**3*g**3*(m + 2)) + e*(d + e*x)**(m + 1)*(a*g**2 - b*f*g + c*f**2
)**2*hyper((2, m + 1), (m + 2,), g*(d + e*x)/(d*g - e*f))/(g**4*(m + 1)*(d*g - e
*f)**2) - 2*(d + e*x)**(m + 1)*(b*g - 2*c*f)*(a*g**2 - b*f*g + c*f**2)*hyper((1,
m + 1), (m + 2,), g*(d + e*x)/(d*g - e*f))/(g**4*(m + 1)*(d*g - e*f)) + (d + e*
x)**(m + 1)*(-2*b*c*d*e*g**2 - 4*b*c*e**2*f*g + c**2*d**2*g**2 + 2*c**2*d*e*f*g
+ 3*c**2*e**2*f**2 + e**2*g**2*(2*a*c + b**2))/(e**3*g**4*(m + 1))

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Mathematica [A]  time = 0.40186, size = 0, normalized size = 0. \[ \int \frac{(d+e x)^m \left (a+b x+c x^2\right )^2}{(f+g x)^2} \, dx \]

Verification is Not applicable to the result.

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

[Out]

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

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Maple [F]  time = 0.14, size = 0, normalized size = 0. \[ \int{\frac{ \left ( ex+d \right ) ^{m} \left ( c{x}^{2}+bx+a \right ) ^{2}}{ \left ( gx+f \right ) ^{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (c^{2} x^{4} + 2 \, b c x^{3} + 2 \, a b x +{\left (b^{2} + 2 \, a c\right )} x^{2} + a^{2}\right )}{\left (e x + d\right )}^{m}}{g^{2} x^{2} + 2 \, f g x + f^{2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((c^2*x^4 + 2*b*c*x^3 + 2*a*b*x + (b^2 + 2*a*c)*x^2 + a^2)*(e*x + d)^m/(
g^2*x^2 + 2*f*g*x + f^2), 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((e*x+d)**m*(c*x**2+b*x+a)**2/(g*x+f)**2,x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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