3.953 \(\int (d+e x)^m (f+g x)^n \left (a+b x+c x^2\right ) \, dx\)

Optimal. Leaf size=265 \[ \frac{(d+e x)^{m+1} (f+g x)^n \left (\frac{e (f+g x)}{e f-d g}\right )^{-n} \, _2F_1\left (m+1,-n;m+2;-\frac{g (d+e x)}{e f-d g}\right ) \left (g (m+n+2) \left (a e^2 g (m+n+3)-c d (d g (n+1)+e f (m+2))\right )-(d g (n+1)+e f (m+1)) (b e g (m+n+3)-c (d g (m+2 n+4)+e f (m+2)))\right )}{e^3 g^2 (m+1) (m+n+2) (m+n+3)}+\frac{(d+e x)^{m+1} (f+g x)^{n+1} (b e g (m+n+3)-c (d g (m+2 n+4)+e f (m+2)))}{e^2 g^2 (m+n+2) (m+n+3)}+\frac{c (d+e x)^{m+2} (f+g x)^{n+1}}{e^2 g (m+n+3)} \]

[Out]

((b*e*g*(3 + m + n) - c*(e*f*(2 + m) + d*g*(4 + m + 2*n)))*(d + e*x)^(1 + m)*(f
+ g*x)^(1 + n))/(e^2*g^2*(2 + m + n)*(3 + m + n)) + (c*(d + e*x)^(2 + m)*(f + g*
x)^(1 + n))/(e^2*g*(3 + m + n)) + ((g*(2 + m + n)*(a*e^2*g*(3 + m + n) - c*d*(e*
f*(2 + m) + d*g*(1 + n))) - (e*f*(1 + m) + d*g*(1 + n))*(b*e*g*(3 + m + n) - c*(
e*f*(2 + m) + d*g*(4 + m + 2*n))))*(d + e*x)^(1 + m)*(f + g*x)^n*Hypergeometric2
F1[1 + m, -n, 2 + m, -((g*(d + e*x))/(e*f - d*g))])/(e^3*g^2*(1 + m)*(2 + m + n)
*(3 + m + n)*((e*(f + g*x))/(e*f - d*g))^n)

_______________________________________________________________________________________

Rubi [A]  time = 0.853422, antiderivative size = 263, normalized size of antiderivative = 0.99, number of steps used = 4, number of rules used = 4, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.16 \[ \frac{(d+e x)^{m+1} (f+g x)^n \left (\frac{e (f+g x)}{e f-d g}\right )^{-n} \, _2F_1\left (m+1,-n;m+2;-\frac{g (d+e x)}{e f-d g}\right ) \left (g (m+n+2) \left (a e^2 g (m+n+3)-c d (d g (n+1)+e f (m+2))\right )+(d g (n+1)+e f (m+1)) (-b e g (m+n+3)+c d g (m+2 n+4)+c e f (m+2))\right )}{e^3 g^2 (m+1) (m+n+2) (m+n+3)}-\frac{(d+e x)^{m+1} (f+g x)^{n+1} (-b e g (m+n+3)+c d g (m+2 n+4)+c e f (m+2))}{e^2 g^2 (m+n+2) (m+n+3)}+\frac{c (d+e x)^{m+2} (f+g x)^{n+1}}{e^2 g (m+n+3)} \]

Antiderivative was successfully verified.

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

[Out]

-(((c*e*f*(2 + m) - b*e*g*(3 + m + n) + c*d*g*(4 + m + 2*n))*(d + e*x)^(1 + m)*(
f + g*x)^(1 + n))/(e^2*g^2*(2 + m + n)*(3 + m + n))) + (c*(d + e*x)^(2 + m)*(f +
g*x)^(1 + n))/(e^2*g*(3 + m + n)) + (((e*f*(1 + m) + d*g*(1 + n))*(c*e*f*(2 + m
) - b*e*g*(3 + m + n) + c*d*g*(4 + m + 2*n)) + g*(2 + m + n)*(a*e^2*g*(3 + m + n
) - c*d*(e*f*(2 + m) + d*g*(1 + n))))*(d + e*x)^(1 + m)*(f + g*x)^n*Hypergeometr
ic2F1[1 + m, -n, 2 + m, -((g*(d + e*x))/(e*f - d*g))])/(e^3*g^2*(1 + m)*(2 + m +
n)*(3 + m + n)*((e*(f + g*x))/(e*f - d*g))^n)

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 97.6807, size = 230, normalized size = 0.87 \[ \frac{c \left (\frac{e \left (- f - g x\right )}{d g - e f}\right )^{- n} \left (d + e x\right )^{m + 1} \left (f + g x\right )^{n} \left (d g - e f\right )^{2}{{}_{2}F_{1}\left (\begin{matrix} - n - 2, m + 1 \\ m + 2 \end{matrix}\middle |{\frac{g \left (d + e x\right )}{d g - e f}} \right )}}{e^{3} g^{2} \left (m + 1\right )} + \frac{\left (\frac{e \left (- f - g x\right )}{d g - e f}\right )^{- n} \left (d + e x\right )^{m + 1} \left (f + g x\right )^{n} \left (a g^{2} - b f g + c f^{2}\right ){{}_{2}F_{1}\left (\begin{matrix} - n, m + 1 \\ m + 2 \end{matrix}\middle |{\frac{g \left (d + e x\right )}{d g - e f}} \right )}}{e g^{2} \left (m + 1\right )} - \frac{\left (\frac{e \left (- f - g x\right )}{d g - e f}\right )^{- n} \left (d + e x\right )^{m + 1} \left (f + g x\right )^{n} \left (b g - 2 c f\right ) \left (d g - e f\right ){{}_{2}F_{1}\left (\begin{matrix} - n - 1, m + 1 \\ m + 2 \end{matrix}\middle |{\frac{g \left (d + e x\right )}{d g - e f}} \right )}}{e^{2} g^{2} \left (m + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Mathematica [C]  time = 0.926809, size = 327, normalized size = 1.23 \[ \frac{1}{3} (d+e x)^m (f+g x)^n \left (\frac{3 a (f+g x) \left (\frac{g (d+e x)}{d g-e f}\right )^{-m} \, _2F_1\left (-m,n+1;n+2;\frac{e (f+g x)}{e f-d g}\right )}{g (n+1)}+\frac{9 b d f x^2 F_1\left (2;-m,-n;3;-\frac{e x}{d},-\frac{g x}{f}\right )}{6 d f F_1\left (2;-m,-n;3;-\frac{e x}{d},-\frac{g x}{f}\right )+2 e f m x F_1\left (3;1-m,-n;4;-\frac{e x}{d},-\frac{g x}{f}\right )+2 d g n x F_1\left (3;-m,1-n;4;-\frac{e x}{d},-\frac{g x}{f}\right )}+\frac{4 c d f x^3 F_1\left (3;-m,-n;4;-\frac{e x}{d},-\frac{g x}{f}\right )}{4 d f F_1\left (3;-m,-n;4;-\frac{e x}{d},-\frac{g x}{f}\right )+e f m x F_1\left (4;1-m,-n;5;-\frac{e x}{d},-\frac{g x}{f}\right )+d g n x F_1\left (4;-m,1-n;5;-\frac{e x}{d},-\frac{g x}{f}\right )}\right ) \]

Warning: Unable to verify antiderivative.

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

[Out]

((d + e*x)^m*(f + g*x)^n*((9*b*d*f*x^2*AppellF1[2, -m, -n, 3, -((e*x)/d), -((g*x
)/f)])/(6*d*f*AppellF1[2, -m, -n, 3, -((e*x)/d), -((g*x)/f)] + 2*e*f*m*x*AppellF
1[3, 1 - m, -n, 4, -((e*x)/d), -((g*x)/f)] + 2*d*g*n*x*AppellF1[3, -m, 1 - n, 4,
-((e*x)/d), -((g*x)/f)]) + (4*c*d*f*x^3*AppellF1[3, -m, -n, 4, -((e*x)/d), -((g
*x)/f)])/(4*d*f*AppellF1[3, -m, -n, 4, -((e*x)/d), -((g*x)/f)] + e*f*m*x*AppellF
1[4, 1 - m, -n, 5, -((e*x)/d), -((g*x)/f)] + d*g*n*x*AppellF1[4, -m, 1 - n, 5, -
((e*x)/d), -((g*x)/f)]) + (3*a*(f + g*x)*Hypergeometric2F1[-m, 1 + n, 2 + n, (e*
(f + g*x))/(e*f - d*g)])/(g*(1 + n)*((g*(d + e*x))/(-(e*f) + d*g))^m)))/3

_______________________________________________________________________________________

Maple [F]  time = 0.089, size = 0, normalized size = 0. \[ \int \left ( ex+d \right ) ^{m} \left ( gx+f \right ) ^{n} \left ( c{x}^{2}+bx+a \right ) \, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

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)^n,x, algorithm="maxima")

[Out]

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

_______________________________________________________________________________________

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

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)^n,x, algorithm="fricas")

[Out]

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

[Out]

Timed out

_______________________________________________________________________________________

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

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)^n,x, algorithm="giac")

[Out]

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