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

Optimal. Leaf size=245 \[ \frac{(d+e x)^{m+1} \, _2F_1\left (1,m+1;m+2;-\frac{g (d+e x)}{e f-d g}\right ) \left (c \left (2 d^2 g^2-4 d e f g (m+1)+e^2 f^2 \left (m^2+3 m+2\right )\right )-e g m (a e g (1-m)-b (2 d g-e f (m+1)))\right )}{2 g^2 (m+1) (e f-d g)^3}+\frac{(d+e x)^{m+1} (g (a e g (1-m)-b (2 d g-e f (m+1)))+c f (4 d g-e f (m+3)))}{2 g^2 (f+g x) (e f-d g)^2}+\frac{(d+e x)^{m+1} \left (a+\frac{f (c f-b g)}{g^2}\right )}{2 (f+g x)^2 (e f-d g)} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.846946, antiderivative size = 243, normalized size of antiderivative = 0.99, number of steps used = 3, number of rules used = 3, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.12 \[ \frac{(d+e x)^{m+1} \, _2F_1\left (1,m+1;m+2;-\frac{g (d+e x)}{e f-d g}\right ) \left (e g m (-a e g (1-m)+2 b d g-b e f (m+1))+c \left (2 d^2 g^2-4 d e f g (m+1)+e^2 f^2 \left (m^2+3 m+2\right )\right )\right )}{2 g^2 (m+1) (e f-d g)^3}-\frac{(d+e x)^{m+1} (g (-a e g (1-m)+2 b d g-b e f (m+1))-c f (4 d g-e f (m+3)))}{2 g^2 (f+g x) (e f-d g)^2}+\frac{(d+e x)^{m+1} \left (a+\frac{f (c f-b g)}{g^2}\right )}{2 (f+g x)^2 (e f-d g)} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 51.2783, size = 155, normalized size = 0.63 \[ - \frac{c \left (d + e x\right )^{m + 1}{{}_{2}F_{1}\left (\begin{matrix} 1, m + 1 \\ m + 2 \end{matrix}\middle |{\frac{g \left (d + e x\right )}{d g - e f}} \right )}}{g^{2} \left (m + 1\right ) \left (d g - e f\right )} - \frac{e^{2} \left (d + e x\right )^{m + 1} \left (a g^{2} - b f g + c f^{2}\right ){{}_{2}F_{1}\left (\begin{matrix} 3, m + 1 \\ m + 2 \end{matrix}\middle |{\frac{g \left (d + e x\right )}{d g - e f}} \right )}}{g^{2} \left (m + 1\right ) \left (d g - e f\right )^{3}} + \frac{e \left (d + e x\right )^{m + 1} \left (b g - 2 c f\right ){{}_{2}F_{1}\left (\begin{matrix} 2, m + 1 \\ m + 2 \end{matrix}\middle |{\frac{g \left (d + e x\right )}{d g - e f}} \right )}}{g^{2} \left (m + 1\right ) \left (d g - e f\right )^{2}} \]

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)**3,x)

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.286995, size = 0, normalized size = 0. \[ \int \frac{(d+e x)^m \left (a+b x+c x^2\right )}{(f+g x)^3} \, dx \]

Verification is Not applicable to the result.

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

[Out]

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

_______________________________________________________________________________________

Maple [F]  time = 0.11, size = 0, normalized size = 0. \[ \int{\frac{ \left ( ex+d \right ) ^{m} \left ( c{x}^{2}+bx+a \right ) }{ \left ( gx+f \right ) ^{3}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

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)^3,x, algorithm="maxima")

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x + a)*(e*x + d)^m/(g*x + f)^3,x, algorithm="fricas")

[Out]

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

_______________________________________________________________________________________

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)/(g*x+f)**3,x)

[Out]

Timed out

_______________________________________________________________________________________

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

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)^3,x, algorithm="giac")

[Out]

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