3.929 \(\int \frac{(d+e x)^m \left (a+b x+c x^2\right )^2}{(f+g x)^3} \, dx\)

Optimal. Leaf size=461 \[ \frac{(d+e x)^{m+1} \, _2F_1\left (1,m+1;m+2;-\frac{g (d+e x)}{e f-d g}\right ) \left (-g^2 \left (a^2 e^2 g^2 (1-m) m-2 a b e g m (2 d g-e f (m+1))+b^2 \left (-\left (2 d^2 g^2-4 d e f g (m+1)+e^2 f^2 \left (m^2+3 m+2\right )\right )\right )\right )+2 c g \left (a g \left (2 d^2 g^2-4 d e f g (m+1)+e^2 f^2 \left (m^2+3 m+2\right )\right )-b f \left (6 d^2 g^2-6 d e f g (m+2)+e^2 f^2 \left (m^2+5 m+6\right )\right )\right )+c^2 f^2 \left (12 d^2 g^2-8 d e f g (m+3)+e^2 f^2 \left (m^2+7 m+12\right )\right )\right )}{2 g^4 (m+1) (e f-d g)^3}+\frac{(d+e x)^{m+1} \left (a g^2-b f g+c f^2\right ) (g (a e g (1-m)-b (4 d g-e f (m+3)))+c f (8 d g-e f (m+7)))}{2 g^4 (f+g x) (e f-d g)^2}+\frac{(d+e x)^{m+1} \left (a g^2-b f g+c f^2\right )^2}{2 g^4 (f+g x)^2 (e f-d g)}-\frac{c (d+e x)^{m+1} (-2 b e g+c d g+3 c e f)}{e^2 g^4 (m+1)}+\frac{c^2 (d+e x)^{m+2}}{e^2 g^3 (m+2)} \]

[Out]

-((c*(3*c*e*f + c*d*g - 2*b*e*g)*(d + e*x)^(1 + m))/(e^2*g^4*(1 + m))) + (c^2*(d
+ e*x)^(2 + m))/(e^2*g^3*(2 + m)) + ((c*f^2 - b*f*g + a*g^2)^2*(d + e*x)^(1 + m
))/(2*g^4*(e*f - d*g)*(f + g*x)^2) + ((c*f^2 - b*f*g + a*g^2)*(c*f*(8*d*g - e*f*
(7 + m)) + g*(a*e*g*(1 - m) - b*(4*d*g - e*f*(3 + m))))*(d + e*x)^(1 + m))/(2*g^
4*(e*f - d*g)^2*(f + g*x)) + ((c^2*f^2*(12*d^2*g^2 - 8*d*e*f*g*(3 + m) + e^2*f^2
*(12 + 7*m + m^2)) - g^2*(a^2*e^2*g^2*(1 - m)*m - 2*a*b*e*g*m*(2*d*g - e*f*(1 +
m)) - b^2*(2*d^2*g^2 - 4*d*e*f*g*(1 + m) + e^2*f^2*(2 + 3*m + m^2))) + 2*c*g*(a*
g*(2*d^2*g^2 - 4*d*e*f*g*(1 + m) + e^2*f^2*(2 + 3*m + m^2)) - b*f*(6*d^2*g^2 - 6
*d*e*f*g*(2 + m) + e^2*f^2*(6 + 5*m + m^2))))*(d + e*x)^(1 + m)*Hypergeometric2F
1[1, 1 + m, 2 + m, -((g*(d + e*x))/(e*f - d*g))])/(2*g^4*(e*f - d*g)^3*(1 + m))

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Rubi [A]  time = 4.35087, antiderivative size = 461, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.148 \[ \frac{(d+e x)^{m+1} \, _2F_1\left (1,m+1;m+2;-\frac{g (d+e x)}{e f-d g}\right ) \left (-g^2 \left (a^2 e^2 g^2 (1-m) m-2 a b e g m (2 d g-e f (m+1))+b^2 \left (-\left (2 d^2 g^2-4 d e f g (m+1)+e^2 f^2 \left (m^2+3 m+2\right )\right )\right )\right )+2 c g \left (a g \left (2 d^2 g^2-4 d e f g (m+1)+e^2 f^2 \left (m^2+3 m+2\right )\right )-b f \left (6 d^2 g^2-6 d e f g (m+2)+e^2 f^2 \left (m^2+5 m+6\right )\right )\right )+c^2 f^2 \left (12 d^2 g^2-8 d e f g (m+3)+e^2 f^2 \left (m^2+7 m+12\right )\right )\right )}{2 g^4 (m+1) (e f-d g)^3}-\frac{(d+e x)^{m+1} \left (a g^2-b f g+c f^2\right ) (g (-a e g (1-m)+4 b d g-b e f (m+3))-c f (8 d g-e f (m+7)))}{2 g^4 (f+g x) (e f-d g)^2}+\frac{(d+e x)^{m+1} \left (a g^2-b f g+c f^2\right )^2}{2 g^4 (f+g x)^2 (e f-d g)}-\frac{c (d+e x)^{m+1} (-2 b e g+c d g+3 c e f)}{e^2 g^4 (m+1)}+\frac{c^2 (d+e x)^{m+2}}{e^2 g^3 (m+2)} \]

Antiderivative was successfully verified.

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

[Out]

-((c*(3*c*e*f + c*d*g - 2*b*e*g)*(d + e*x)^(1 + m))/(e^2*g^4*(1 + m))) + (c^2*(d
+ e*x)^(2 + m))/(e^2*g^3*(2 + m)) + ((c*f^2 - b*f*g + a*g^2)^2*(d + e*x)^(1 + m
))/(2*g^4*(e*f - d*g)*(f + g*x)^2) - ((c*f^2 - b*f*g + a*g^2)*(g*(4*b*d*g - a*e*
g*(1 - m) - b*e*f*(3 + m)) - c*f*(8*d*g - e*f*(7 + m)))*(d + e*x)^(1 + m))/(2*g^
4*(e*f - d*g)^2*(f + g*x)) + ((c^2*f^2*(12*d^2*g^2 - 8*d*e*f*g*(3 + m) + e^2*f^2
*(12 + 7*m + m^2)) - g^2*(a^2*e^2*g^2*(1 - m)*m - 2*a*b*e*g*m*(2*d*g - e*f*(1 +
m)) - b^2*(2*d^2*g^2 - 4*d*e*f*g*(1 + m) + e^2*f^2*(2 + 3*m + m^2))) + 2*c*g*(a*
g*(2*d^2*g^2 - 4*d*e*f*g*(1 + m) + e^2*f^2*(2 + 3*m + m^2)) - b*f*(6*d^2*g^2 - 6
*d*e*f*g*(2 + m) + e^2*f^2*(6 + 5*m + m^2))))*(d + e*x)^(1 + m)*Hypergeometric2F
1[1, 1 + m, 2 + m, -((g*(d + e*x))/(e*f - d*g))])/(2*g^4*(e*f - d*g)^3*(1 + m))

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Rubi in Sympy [A]  time = 112.77, size = 265, normalized size = 0.57 \[ \frac{c^{2} \left (d + e x\right )^{m + 2}}{e^{2} g^{3} \left (m + 2\right )} + \frac{c \left (d + e x\right )^{m + 1} \left (2 b e g - c d g - 3 c e f\right )}{e^{2} g^{4} \left (m + 1\right )} - \frac{e^{2} \left (d + e x\right )^{m + 1} \left (a g^{2} - b f g + c f^{2}\right )^{2}{{}_{2}F_{1}\left (\begin{matrix} 3, m + 1 \\ m + 2 \end{matrix}\middle |{\frac{g \left (d + e x\right )}{d g - e f}} \right )}}{g^{4} \left (m + 1\right ) \left (d g - e f\right )^{3}} + \frac{2 e \left (d + e x\right )^{m + 1} \left (b g - 2 c f\right ) \left (a g^{2} - b f g + c f^{2}\right ){{}_{2}F_{1}\left (\begin{matrix} 2, m + 1 \\ m + 2 \end{matrix}\middle |{\frac{g \left (d + e x\right )}{d g - e f}} \right )}}{g^{4} \left (m + 1\right ) \left (d g - e f\right )^{2}} - \frac{\left (d + e x\right )^{m + 1} \left (2 a c g^{2} + b^{2} g^{2} - 6 b c f g + 6 c^{2} f^{2}\right ){{}_{2}F_{1}\left (\begin{matrix} 1, m + 1 \\ m + 2 \end{matrix}\middle |{\frac{g \left (d + e x\right )}{d g - e f}} \right )}}{g^{4} \left (m + 1\right ) \left (d g - e f\right )} \]

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)**3,x)

[Out]

c**2*(d + e*x)**(m + 2)/(e**2*g**3*(m + 2)) + c*(d + e*x)**(m + 1)*(2*b*e*g - c*
d*g - 3*c*e*f)/(e**2*g**4*(m + 1)) - e**2*(d + e*x)**(m + 1)*(a*g**2 - b*f*g + c
*f**2)**2*hyper((3, m + 1), (m + 2,), g*(d + e*x)/(d*g - e*f))/(g**4*(m + 1)*(d*
g - e*f)**3) + 2*e*(d + e*x)**(m + 1)*(b*g - 2*c*f)*(a*g**2 - b*f*g + c*f**2)*hy
per((2, m + 1), (m + 2,), g*(d + e*x)/(d*g - e*f))/(g**4*(m + 1)*(d*g - e*f)**2)
- (d + e*x)**(m + 1)*(2*a*c*g**2 + b**2*g**2 - 6*b*c*f*g + 6*c**2*f**2)*hyper((
1, m + 1), (m + 2,), g*(d + e*x)/(d*g - e*f))/(g**4*(m + 1)*(d*g - e*f))

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Mathematica [A]  time = 0.693177, size = 0, normalized size = 0. \[ \int \frac{(d+e x)^m \left (a+b x+c x^2\right )^2}{(f+g x)^3} \, dx \]

Verification is Not applicable to the result.

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

[Out]

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

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Maple [F]  time = 0.182, size = 0, normalized size = 0. \[ \int{\frac{ \left ( ex+d \right ) ^{m} \left ( c{x}^{2}+bx+a \right ) ^{2}}{ \left ( gx+f \right ) ^{3}}}\, dx \]

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)^3,x)

[Out]

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

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

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)^3,x, algorithm="maxima")

[Out]

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

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

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)^3,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^3*x^3 + 3*f*g^2*x^2 + 3*f^2*g*x + f^3), 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)**3,x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (c x^{2} + b x + a\right )}^{2}{\left (e x + d\right )}^{m}}{{\left (g x + f\right )}^{3}}\,{d x} \]

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)^3,x, algorithm="giac")

[Out]

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