3.9.6 \(\int (d+e x)^2 (f+g x)^n (a+2 c d x+c e x^2) \, dx\) [806]

Optimal. Leaf size=208 \[ \frac {(e f-d g)^2 \left (a g^2+c f (e f-2 d g)\right ) (f+g x)^{1+n}}{g^5 (1+n)}-\frac {2 (e f-d g) \left (a e g^2+c \left (2 e^2 f^2-4 d e f g+d^2 g^2\right )\right ) (f+g x)^{2+n}}{g^5 (2+n)}+\frac {e \left (a e g^2+c \left (6 e^2 f^2-12 d e f g+5 d^2 g^2\right )\right ) (f+g x)^{3+n}}{g^5 (3+n)}-\frac {4 c e^2 (e f-d g) (f+g x)^{4+n}}{g^5 (4+n)}+\frac {c e^3 (f+g x)^{5+n}}{g^5 (5+n)} \]

[Out]

(-d*g+e*f)^2*(a*g^2+c*f*(-2*d*g+e*f))*(g*x+f)^(1+n)/g^5/(1+n)-2*(-d*g+e*f)*(a*e*g^2+c*(d^2*g^2-4*d*e*f*g+2*e^2
*f^2))*(g*x+f)^(2+n)/g^5/(2+n)+e*(a*e*g^2+c*(5*d^2*g^2-12*d*e*f*g+6*e^2*f^2))*(g*x+f)^(3+n)/g^5/(3+n)-4*c*e^2*
(-d*g+e*f)*(g*x+f)^(4+n)/g^5/(4+n)+c*e^3*(g*x+f)^(5+n)/g^5/(5+n)

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Rubi [A]
time = 0.11, antiderivative size = 208, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.036, Rules used = {961} \begin {gather*} -\frac {2 (e f-d g) (f+g x)^{n+2} \left (a e g^2+c \left (d^2 g^2-4 d e f g+2 e^2 f^2\right )\right )}{g^5 (n+2)}+\frac {e (f+g x)^{n+3} \left (a e g^2+c \left (5 d^2 g^2-12 d e f g+6 e^2 f^2\right )\right )}{g^5 (n+3)}+\frac {(e f-d g)^2 (f+g x)^{n+1} \left (a g^2+c f (e f-2 d g)\right )}{g^5 (n+1)}-\frac {4 c e^2 (e f-d g) (f+g x)^{n+4}}{g^5 (n+4)}+\frac {c e^3 (f+g x)^{n+5}}{g^5 (n+5)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((e*f - d*g)^2*(a*g^2 + c*f*(e*f - 2*d*g))*(f + g*x)^(1 + n))/(g^5*(1 + n)) - (2*(e*f - d*g)*(a*e*g^2 + c*(2*e
^2*f^2 - 4*d*e*f*g + d^2*g^2))*(f + g*x)^(2 + n))/(g^5*(2 + n)) + (e*(a*e*g^2 + c*(6*e^2*f^2 - 12*d*e*f*g + 5*
d^2*g^2))*(f + g*x)^(3 + n))/(g^5*(3 + n)) - (4*c*e^2*(e*f - d*g)*(f + g*x)^(4 + n))/(g^5*(4 + n)) + (c*e^3*(f
+ g*x)^(5 + n))/(g^5*(5 + n))

Rule 961

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :
> Int[ExpandIntegrand[(d + e*x)^m*(f + g*x)^n*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] &
& NeQ[e*f - d*g, 0] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IGtQ[p, 0] && (IGtQ[m, 0] || (E
qQ[m, -2] && EqQ[p, 1] && EqQ[2*c*d - b*e, 0]))

Rubi steps

\begin {align*} \int (d+e x)^2 (f+g x)^n \left (a+2 c d x+c e x^2\right ) \, dx &=\int \left (\frac {(e f-d g)^2 \left (a g^2+c f (e f-2 d g)\right ) (f+g x)^n}{g^4}+\frac {2 (e f-d g) \left (-a e g^2-c \left (2 e^2 f^2-4 d e f g+d^2 g^2\right )\right ) (f+g x)^{1+n}}{g^4}+\frac {e \left (a e g^2+c \left (6 e^2 f^2-12 d e f g+5 d^2 g^2\right )\right ) (f+g x)^{2+n}}{g^4}-\frac {4 c e^2 (e f-d g) (f+g x)^{3+n}}{g^4}+\frac {c e^3 (f+g x)^{4+n}}{g^4}\right ) \, dx\\ &=\frac {(e f-d g)^2 \left (a g^2+c f (e f-2 d g)\right ) (f+g x)^{1+n}}{g^5 (1+n)}-\frac {2 (e f-d g) \left (a e g^2+c \left (2 e^2 f^2-4 d e f g+d^2 g^2\right )\right ) (f+g x)^{2+n}}{g^5 (2+n)}+\frac {e \left (a e g^2+c \left (6 e^2 f^2-12 d e f g+5 d^2 g^2\right )\right ) (f+g x)^{3+n}}{g^5 (3+n)}-\frac {4 c e^2 (e f-d g) (f+g x)^{4+n}}{g^5 (4+n)}+\frac {c e^3 (f+g x)^{5+n}}{g^5 (5+n)}\\ \end {align*}

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Mathematica [A]
time = 0.31, size = 348, normalized size = 1.67 \begin {gather*} \frac {(f+g x)^{1+n} \left (a g^2 \left (20+9 n+n^2\right ) \left (d^2 g^2 \left (6+5 n+n^2\right )+2 d e g (3+n) (-f+g (1+n) x)+e^2 \left (2 f^2-2 f g (1+n) x+g^2 \left (2+3 n+n^2\right ) x^2\right )\right )+c \left (2 d^3 g^3 \left (60+47 n+12 n^2+n^3\right ) (-f+g (1+n) x)+5 d^2 e g^2 \left (20+9 n+n^2\right ) \left (2 f^2-2 f g (1+n) x+g^2 \left (2+3 n+n^2\right ) x^2\right )+4 d e^2 g (5+n) \left (-6 f^3+6 f^2 g (1+n) x-3 f g^2 \left (2+3 n+n^2\right ) x^2+g^3 \left (6+11 n+6 n^2+n^3\right ) x^3\right )+e^3 \left (24 f^4-24 f^3 g (1+n) x+12 f^2 g^2 \left (2+3 n+n^2\right ) x^2-4 f g^3 \left (6+11 n+6 n^2+n^3\right ) x^3+g^4 \left (24+50 n+35 n^2+10 n^3+n^4\right ) x^4\right )\right )\right )}{g^5 (1+n) (2+n) (3+n) (4+n) (5+n)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((f + g*x)^(1 + n)*(a*g^2*(20 + 9*n + n^2)*(d^2*g^2*(6 + 5*n + n^2) + 2*d*e*g*(3 + n)*(-f + g*(1 + n)*x) + e^2
*(2*f^2 - 2*f*g*(1 + n)*x + g^2*(2 + 3*n + n^2)*x^2)) + c*(2*d^3*g^3*(60 + 47*n + 12*n^2 + n^3)*(-f + g*(1 + n
)*x) + 5*d^2*e*g^2*(20 + 9*n + n^2)*(2*f^2 - 2*f*g*(1 + n)*x + g^2*(2 + 3*n + n^2)*x^2) + 4*d*e^2*g*(5 + n)*(-
6*f^3 + 6*f^2*g*(1 + n)*x - 3*f*g^2*(2 + 3*n + n^2)*x^2 + g^3*(6 + 11*n + 6*n^2 + n^3)*x^3) + e^3*(24*f^4 - 24
*f^3*g*(1 + n)*x + 12*f^2*g^2*(2 + 3*n + n^2)*x^2 - 4*f*g^3*(6 + 11*n + 6*n^2 + n^3)*x^3 + g^4*(24 + 50*n + 35
*n^2 + 10*n^3 + n^4)*x^4))))/(g^5*(1 + n)*(2 + n)*(3 + n)*(4 + n)*(5 + n))

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(1028\) vs. \(2(208)=416\).
time = 0.11, size = 1029, normalized size = 4.95 Too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^2*(g*x+f)^n*(c*e*x^2+2*c*d*x+a),x,method=_RETURNVERBOSE)

[Out]

c*e^3/(5+n)*x^5*exp(n*ln(g*x+f))+f*(a*d^2*g^4*n^4-2*c*d^3*f*g^3*n^3+14*a*d^2*g^4*n^3-2*a*d*e*f*g^3*n^3-24*c*d^
3*f*g^3*n^2+10*c*d^2*e*f^2*g^2*n^2+71*a*d^2*g^4*n^2-24*a*d*e*f*g^3*n^2+2*a*e^2*f^2*g^2*n^2-94*c*d^3*f*g^3*n+90
*c*d^2*e*f^2*g^2*n-24*c*d*e^2*f^3*g*n+154*a*d^2*g^4*n-94*a*d*e*f*g^3*n+18*a*e^2*f^2*g^2*n-120*c*d^3*f*g^3+200*
c*d^2*e*f^2*g^2-120*c*d*e^2*f^3*g+24*c*e^3*f^4+120*a*d^2*g^4-120*a*d*e*f*g^3+40*a*e^2*f^2*g^2)/g^5/(n^5+15*n^4
+85*n^3+225*n^2+274*n+120)*exp(n*ln(g*x+f))+(2*c*d^3*g^3*n^3+5*c*d^2*e*f*g^2*n^3+2*a*d*e*g^3*n^3+a*e^2*f*g^2*n
^3+24*c*d^3*g^3*n^2+45*c*d^2*e*f*g^2*n^2-12*c*d*e^2*f^2*g*n^2+24*a*d*e*g^3*n^2+9*a*e^2*f*g^2*n^2+94*c*d^3*g^3*
n+100*c*d^2*e*f*g^2*n-60*c*d*e^2*f^2*g*n+12*c*e^3*f^3*n+94*a*d*e*g^3*n+20*a*e^2*f*g^2*n+120*c*d^3*g^3+120*a*d*
e*g^3)/g^3/(n^4+14*n^3+71*n^2+154*n+120)*x^2*exp(n*ln(g*x+f))+(2*c*d^3*f*g^3*n^4+a*d^2*g^4*n^4+2*a*d*e*f*g^3*n
^4+24*c*d^3*f*g^3*n^3-10*c*d^2*e*f^2*g^2*n^3+14*a*d^2*g^4*n^3+24*a*d*e*f*g^3*n^3-2*a*e^2*f^2*g^2*n^3+94*c*d^3*
f*g^3*n^2-90*c*d^2*e*f^2*g^2*n^2+24*c*d*e^2*f^3*g*n^2+71*a*d^2*g^4*n^2+94*a*d*e*f*g^3*n^2-18*a*e^2*f^2*g^2*n^2
+120*c*d^3*f*g^3*n-200*c*d^2*e*f^2*g^2*n+120*c*d*e^2*f^3*g*n-24*c*e^3*f^4*n+154*a*d^2*g^4*n+120*a*d*e*f*g^3*n-
40*a*e^2*f^2*g^2*n+120*a*d^2*g^4)/g^4/(n^5+15*n^4+85*n^3+225*n^2+274*n+120)*x*exp(n*ln(g*x+f))+(5*c*d^2*g^2*n^
2+4*c*d*e*f*g*n^2+a*e*g^2*n^2+45*c*d^2*g^2*n+20*c*d*e*f*g*n-4*c*e^2*f^2*n+9*a*e*g^2*n+100*c*d^2*g^2+20*a*e*g^2
)*e/g^2/(n^3+12*n^2+47*n+60)*x^3*exp(n*ln(g*x+f))+(4*d*g*n+e*f*n+20*d*g)*c*e^2/g/(n^2+9*n+20)*x^4*exp(n*ln(g*x
+f))

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 512 vs. \(2 (208) = 416\).
time = 0.31, size = 512, normalized size = 2.46 \begin {gather*} \frac {2 \, {\left (g^{2} {\left (n + 1\right )} x^{2} + f g n x - f^{2}\right )} {\left (g x + f\right )}^{n} c d^{3}}{{\left (n^{2} + 3 \, n + 2\right )} g^{2}} + \frac {5 \, {\left ({\left (n^{2} + 3 \, n + 2\right )} g^{3} x^{3} + {\left (n^{2} + n\right )} f g^{2} x^{2} - 2 \, f^{2} g n x + 2 \, f^{3}\right )} {\left (g x + f\right )}^{n} c d^{2} e}{{\left (n^{3} + 6 \, n^{2} + 11 \, n + 6\right )} g^{3}} + \frac {2 \, {\left (g^{2} {\left (n + 1\right )} x^{2} + f g n x - f^{2}\right )} {\left (g x + f\right )}^{n} a d e}{{\left (n^{2} + 3 \, n + 2\right )} g^{2}} + \frac {{\left (g x + f\right )}^{n + 1} a d^{2}}{g {\left (n + 1\right )}} + \frac {4 \, {\left ({\left (n^{3} + 6 \, n^{2} + 11 \, n + 6\right )} g^{4} x^{4} + {\left (n^{3} + 3 \, n^{2} + 2 \, n\right )} f g^{3} x^{3} - 3 \, {\left (n^{2} + n\right )} f^{2} g^{2} x^{2} + 6 \, f^{3} g n x - 6 \, f^{4}\right )} {\left (g x + f\right )}^{n} c d e^{2}}{{\left (n^{4} + 10 \, n^{3} + 35 \, n^{2} + 50 \, n + 24\right )} g^{4}} + \frac {{\left ({\left (n^{2} + 3 \, n + 2\right )} g^{3} x^{3} + {\left (n^{2} + n\right )} f g^{2} x^{2} - 2 \, f^{2} g n x + 2 \, f^{3}\right )} {\left (g x + f\right )}^{n} a e^{2}}{{\left (n^{3} + 6 \, n^{2} + 11 \, n + 6\right )} g^{3}} + \frac {{\left ({\left (n^{4} + 10 \, n^{3} + 35 \, n^{2} + 50 \, n + 24\right )} g^{5} x^{5} + {\left (n^{4} + 6 \, n^{3} + 11 \, n^{2} + 6 \, n\right )} f g^{4} x^{4} - 4 \, {\left (n^{3} + 3 \, n^{2} + 2 \, n\right )} f^{2} g^{3} x^{3} + 12 \, {\left (n^{2} + n\right )} f^{3} g^{2} x^{2} - 24 \, f^{4} g n x + 24 \, f^{5}\right )} {\left (g x + f\right )}^{n} c e^{3}}{{\left (n^{5} + 15 \, n^{4} + 85 \, n^{3} + 225 \, n^{2} + 274 \, n + 120\right )} g^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^2*(g*x+f)^n*(c*e*x^2+2*c*d*x+a),x, algorithm="maxima")

[Out]

2*(g^2*(n + 1)*x^2 + f*g*n*x - f^2)*(g*x + f)^n*c*d^3/((n^2 + 3*n + 2)*g^2) + 5*((n^2 + 3*n + 2)*g^3*x^3 + (n^
2 + n)*f*g^2*x^2 - 2*f^2*g*n*x + 2*f^3)*(g*x + f)^n*c*d^2*e/((n^3 + 6*n^2 + 11*n + 6)*g^3) + 2*(g^2*(n + 1)*x^
2 + f*g*n*x - f^2)*(g*x + f)^n*a*d*e/((n^2 + 3*n + 2)*g^2) + (g*x + f)^(n + 1)*a*d^2/(g*(n + 1)) + 4*((n^3 + 6
*n^2 + 11*n + 6)*g^4*x^4 + (n^3 + 3*n^2 + 2*n)*f*g^3*x^3 - 3*(n^2 + n)*f^2*g^2*x^2 + 6*f^3*g*n*x - 6*f^4)*(g*x
+ f)^n*c*d*e^2/((n^4 + 10*n^3 + 35*n^2 + 50*n + 24)*g^4) + ((n^2 + 3*n + 2)*g^3*x^3 + (n^2 + n)*f*g^2*x^2 - 2
*f^2*g*n*x + 2*f^3)*(g*x + f)^n*a*e^2/((n^3 + 6*n^2 + 11*n + 6)*g^3) + ((n^4 + 10*n^3 + 35*n^2 + 50*n + 24)*g^
5*x^5 + (n^4 + 6*n^3 + 11*n^2 + 6*n)*f*g^4*x^4 - 4*(n^3 + 3*n^2 + 2*n)*f^2*g^3*x^3 + 12*(n^2 + n)*f^3*g^2*x^2
- 24*f^4*g*n*x + 24*f^5)*(g*x + f)^n*c*e^3/((n^5 + 15*n^4 + 85*n^3 + 225*n^2 + 274*n + 120)*g^5)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1188 vs. \(2 (216) = 432\).
time = 1.89, size = 1188, normalized size = 5.71 \begin {gather*} \frac {{\left (a d^{2} f g^{4} n^{4} - 120 \, c d^{3} f^{2} g^{3} + 120 \, a d^{2} f g^{4} - 2 \, {\left (c d^{3} f^{2} g^{3} - 7 \, a d^{2} f g^{4}\right )} n^{3} - {\left (24 \, c d^{3} f^{2} g^{3} - 71 \, a d^{2} f g^{4}\right )} n^{2} + 2 \, {\left (c d^{3} g^{5} n^{4} + 13 \, c d^{3} g^{5} n^{3} + 59 \, c d^{3} g^{5} n^{2} + 107 \, c d^{3} g^{5} n + 60 \, c d^{3} g^{5}\right )} x^{2} - 2 \, {\left (47 \, c d^{3} f^{2} g^{3} - 77 \, a d^{2} f g^{4}\right )} n + {\left (120 \, a d^{2} g^{5} + {\left (2 \, c d^{3} f g^{4} + a d^{2} g^{5}\right )} n^{4} + 2 \, {\left (12 \, c d^{3} f g^{4} + 7 \, a d^{2} g^{5}\right )} n^{3} + {\left (94 \, c d^{3} f g^{4} + 71 \, a d^{2} g^{5}\right )} n^{2} + 2 \, {\left (60 \, c d^{3} f g^{4} + 77 \, a d^{2} g^{5}\right )} n\right )} x - {\left (24 \, c f^{4} g n x - 24 \, c f^{5} - {\left (c g^{5} n^{4} + 10 \, c g^{5} n^{3} + 35 \, c g^{5} n^{2} + 50 \, c g^{5} n + 24 \, c g^{5}\right )} x^{5} - {\left (c f g^{4} n^{4} + 6 \, c f g^{4} n^{3} + 11 \, c f g^{4} n^{2} + 6 \, c f g^{4} n\right )} x^{4} + 4 \, {\left (c f^{2} g^{3} n^{3} + 3 \, c f^{2} g^{3} n^{2} + 2 \, c f^{2} g^{3} n\right )} x^{3} - 12 \, {\left (c f^{3} g^{2} n^{2} + c f^{3} g^{2} n\right )} x^{2}\right )} e^{3} + {\left (2 \, a f^{3} g^{2} n^{2} - 120 \, c d f^{4} g + 40 \, a f^{3} g^{2} + 4 \, {\left (c d g^{5} n^{4} + 11 \, c d g^{5} n^{3} + 41 \, c d g^{5} n^{2} + 61 \, c d g^{5} n + 30 \, c d g^{5}\right )} x^{4} + {\left (40 \, a g^{5} + {\left (4 \, c d f g^{4} + a g^{5}\right )} n^{4} + 4 \, {\left (8 \, c d f g^{4} + 3 \, a g^{5}\right )} n^{3} + {\left (68 \, c d f g^{4} + 49 \, a g^{5}\right )} n^{2} + 2 \, {\left (20 \, c d f g^{4} + 39 \, a g^{5}\right )} n\right )} x^{3} + {\left (a f g^{4} n^{4} - 2 \, {\left (6 \, c d f^{2} g^{3} - 5 \, a f g^{4}\right )} n^{3} - {\left (72 \, c d f^{2} g^{3} - 29 \, a f g^{4}\right )} n^{2} - 20 \, {\left (3 \, c d f^{2} g^{3} - a f g^{4}\right )} n\right )} x^{2} - 6 \, {\left (4 \, c d f^{4} g - 3 \, a f^{3} g^{2}\right )} n - 2 \, {\left (a f^{2} g^{3} n^{3} - 3 \, {\left (4 \, c d f^{3} g^{2} - 3 \, a f^{2} g^{3}\right )} n^{2} - 20 \, {\left (3 \, c d f^{3} g^{2} - a f^{2} g^{3}\right )} n\right )} x\right )} e^{2} - {\left (2 \, a d f^{2} g^{3} n^{3} - 200 \, c d^{2} f^{3} g^{2} + 120 \, a d f^{2} g^{3} - 5 \, {\left (c d^{2} g^{5} n^{4} + 12 \, c d^{2} g^{5} n^{3} + 49 \, c d^{2} g^{5} n^{2} + 78 \, c d^{2} g^{5} n + 40 \, c d^{2} g^{5}\right )} x^{3} - 2 \, {\left (5 \, c d^{2} f^{3} g^{2} - 12 \, a d f^{2} g^{3}\right )} n^{2} - {\left (120 \, a d g^{5} + {\left (5 \, c d^{2} f g^{4} + 2 \, a d g^{5}\right )} n^{4} + 2 \, {\left (25 \, c d^{2} f g^{4} + 13 \, a d g^{5}\right )} n^{3} + {\left (145 \, c d^{2} f g^{4} + 118 \, a d g^{5}\right )} n^{2} + 2 \, {\left (50 \, c d^{2} f g^{4} + 107 \, a d g^{5}\right )} n\right )} x^{2} - 2 \, {\left (45 \, c d^{2} f^{3} g^{2} - 47 \, a d f^{2} g^{3}\right )} n - 2 \, {\left (a d f g^{4} n^{4} - {\left (5 \, c d^{2} f^{2} g^{3} - 12 \, a d f g^{4}\right )} n^{3} - {\left (45 \, c d^{2} f^{2} g^{3} - 47 \, a d f g^{4}\right )} n^{2} - 20 \, {\left (5 \, c d^{2} f^{2} g^{3} - 3 \, a d f g^{4}\right )} n\right )} x\right )} e\right )} {\left (g x + f\right )}^{n}}{g^{5} n^{5} + 15 \, g^{5} n^{4} + 85 \, g^{5} n^{3} + 225 \, g^{5} n^{2} + 274 \, g^{5} n + 120 \, g^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^2*(g*x+f)^n*(c*e*x^2+2*c*d*x+a),x, algorithm="fricas")

[Out]

(a*d^2*f*g^4*n^4 - 120*c*d^3*f^2*g^3 + 120*a*d^2*f*g^4 - 2*(c*d^3*f^2*g^3 - 7*a*d^2*f*g^4)*n^3 - (24*c*d^3*f^2
*g^3 - 71*a*d^2*f*g^4)*n^2 + 2*(c*d^3*g^5*n^4 + 13*c*d^3*g^5*n^3 + 59*c*d^3*g^5*n^2 + 107*c*d^3*g^5*n + 60*c*d
^3*g^5)*x^2 - 2*(47*c*d^3*f^2*g^3 - 77*a*d^2*f*g^4)*n + (120*a*d^2*g^5 + (2*c*d^3*f*g^4 + a*d^2*g^5)*n^4 + 2*(
12*c*d^3*f*g^4 + 7*a*d^2*g^5)*n^3 + (94*c*d^3*f*g^4 + 71*a*d^2*g^5)*n^2 + 2*(60*c*d^3*f*g^4 + 77*a*d^2*g^5)*n)
*x - (24*c*f^4*g*n*x - 24*c*f^5 - (c*g^5*n^4 + 10*c*g^5*n^3 + 35*c*g^5*n^2 + 50*c*g^5*n + 24*c*g^5)*x^5 - (c*f
*g^4*n^4 + 6*c*f*g^4*n^3 + 11*c*f*g^4*n^2 + 6*c*f*g^4*n)*x^4 + 4*(c*f^2*g^3*n^3 + 3*c*f^2*g^3*n^2 + 2*c*f^2*g^
3*n)*x^3 - 12*(c*f^3*g^2*n^2 + c*f^3*g^2*n)*x^2)*e^3 + (2*a*f^3*g^2*n^2 - 120*c*d*f^4*g + 40*a*f^3*g^2 + 4*(c*
d*g^5*n^4 + 11*c*d*g^5*n^3 + 41*c*d*g^5*n^2 + 61*c*d*g^5*n + 30*c*d*g^5)*x^4 + (40*a*g^5 + (4*c*d*f*g^4 + a*g^
5)*n^4 + 4*(8*c*d*f*g^4 + 3*a*g^5)*n^3 + (68*c*d*f*g^4 + 49*a*g^5)*n^2 + 2*(20*c*d*f*g^4 + 39*a*g^5)*n)*x^3 +
(a*f*g^4*n^4 - 2*(6*c*d*f^2*g^3 - 5*a*f*g^4)*n^3 - (72*c*d*f^2*g^3 - 29*a*f*g^4)*n^2 - 20*(3*c*d*f^2*g^3 - a*f
*g^4)*n)*x^2 - 6*(4*c*d*f^4*g - 3*a*f^3*g^2)*n - 2*(a*f^2*g^3*n^3 - 3*(4*c*d*f^3*g^2 - 3*a*f^2*g^3)*n^2 - 20*(
3*c*d*f^3*g^2 - a*f^2*g^3)*n)*x)*e^2 - (2*a*d*f^2*g^3*n^3 - 200*c*d^2*f^3*g^2 + 120*a*d*f^2*g^3 - 5*(c*d^2*g^5
*n^4 + 12*c*d^2*g^5*n^3 + 49*c*d^2*g^5*n^2 + 78*c*d^2*g^5*n + 40*c*d^2*g^5)*x^3 - 2*(5*c*d^2*f^3*g^2 - 12*a*d*
f^2*g^3)*n^2 - (120*a*d*g^5 + (5*c*d^2*f*g^4 + 2*a*d*g^5)*n^4 + 2*(25*c*d^2*f*g^4 + 13*a*d*g^5)*n^3 + (145*c*d
^2*f*g^4 + 118*a*d*g^5)*n^2 + 2*(50*c*d^2*f*g^4 + 107*a*d*g^5)*n)*x^2 - 2*(45*c*d^2*f^3*g^2 - 47*a*d*f^2*g^3)*
n - 2*(a*d*f*g^4*n^4 - (5*c*d^2*f^2*g^3 - 12*a*d*f*g^4)*n^3 - (45*c*d^2*f^2*g^3 - 47*a*d*f*g^4)*n^2 - 20*(5*c*
d^2*f^2*g^3 - 3*a*d*f*g^4)*n)*x)*e)*(g*x + f)^n/(g^5*n^5 + 15*g^5*n^4 + 85*g^5*n^3 + 225*g^5*n^2 + 274*g^5*n +
120*g^5)

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 11946 vs. \(2 (197) = 394\).
time = 2.50, size = 11946, normalized size = 57.43 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((f**n*(a*d**2*x + a*d*e*x**2 + a*e**2*x**3/3 + c*d**3*x**2 + 5*c*d**2*e*x**3/3 + c*d*e**2*x**4 + c*e
**3*x**5/5), Eq(g, 0)), (-3*a*d**2*g**4/(12*f**4*g**5 + 48*f**3*g**6*x + 72*f**2*g**7*x**2 + 48*f*g**8*x**3 +
12*g**9*x**4) - 2*a*d*e*f*g**3/(12*f**4*g**5 + 48*f**3*g**6*x + 72*f**2*g**7*x**2 + 48*f*g**8*x**3 + 12*g**9*x
**4) - 8*a*d*e*g**4*x/(12*f**4*g**5 + 48*f**3*g**6*x + 72*f**2*g**7*x**2 + 48*f*g**8*x**3 + 12*g**9*x**4) - a*
e**2*f**2*g**2/(12*f**4*g**5 + 48*f**3*g**6*x + 72*f**2*g**7*x**2 + 48*f*g**8*x**3 + 12*g**9*x**4) - 4*a*e**2*
f*g**3*x/(12*f**4*g**5 + 48*f**3*g**6*x + 72*f**2*g**7*x**2 + 48*f*g**8*x**3 + 12*g**9*x**4) - 6*a*e**2*g**4*x
**2/(12*f**4*g**5 + 48*f**3*g**6*x + 72*f**2*g**7*x**2 + 48*f*g**8*x**3 + 12*g**9*x**4) - 2*c*d**3*f*g**3/(12*
f**4*g**5 + 48*f**3*g**6*x + 72*f**2*g**7*x**2 + 48*f*g**8*x**3 + 12*g**9*x**4) - 8*c*d**3*g**4*x/(12*f**4*g**
5 + 48*f**3*g**6*x + 72*f**2*g**7*x**2 + 48*f*g**8*x**3 + 12*g**9*x**4) - 5*c*d**2*e*f**2*g**2/(12*f**4*g**5 +
48*f**3*g**6*x + 72*f**2*g**7*x**2 + 48*f*g**8*x**3 + 12*g**9*x**4) - 20*c*d**2*e*f*g**3*x/(12*f**4*g**5 + 48
*f**3*g**6*x + 72*f**2*g**7*x**2 + 48*f*g**8*x**3 + 12*g**9*x**4) - 30*c*d**2*e*g**4*x**2/(12*f**4*g**5 + 48*f
**3*g**6*x + 72*f**2*g**7*x**2 + 48*f*g**8*x**3 + 12*g**9*x**4) - 12*c*d*e**2*f**3*g/(12*f**4*g**5 + 48*f**3*g
**6*x + 72*f**2*g**7*x**2 + 48*f*g**8*x**3 + 12*g**9*x**4) - 48*c*d*e**2*f**2*g**2*x/(12*f**4*g**5 + 48*f**3*g
**6*x + 72*f**2*g**7*x**2 + 48*f*g**8*x**3 + 12*g**9*x**4) - 72*c*d*e**2*f*g**3*x**2/(12*f**4*g**5 + 48*f**3*g
**6*x + 72*f**2*g**7*x**2 + 48*f*g**8*x**3 + 12*g**9*x**4) - 48*c*d*e**2*g**4*x**3/(12*f**4*g**5 + 48*f**3*g**
6*x + 72*f**2*g**7*x**2 + 48*f*g**8*x**3 + 12*g**9*x**4) + 12*c*e**3*f**4*log(f/g + x)/(12*f**4*g**5 + 48*f**3
*g**6*x + 72*f**2*g**7*x**2 + 48*f*g**8*x**3 + 12*g**9*x**4) + 25*c*e**3*f**4/(12*f**4*g**5 + 48*f**3*g**6*x +
72*f**2*g**7*x**2 + 48*f*g**8*x**3 + 12*g**9*x**4) + 48*c*e**3*f**3*g*x*log(f/g + x)/(12*f**4*g**5 + 48*f**3*
g**6*x + 72*f**2*g**7*x**2 + 48*f*g**8*x**3 + 12*g**9*x**4) + 88*c*e**3*f**3*g*x/(12*f**4*g**5 + 48*f**3*g**6*
x + 72*f**2*g**7*x**2 + 48*f*g**8*x**3 + 12*g**9*x**4) + 72*c*e**3*f**2*g**2*x**2*log(f/g + x)/(12*f**4*g**5 +
48*f**3*g**6*x + 72*f**2*g**7*x**2 + 48*f*g**8*x**3 + 12*g**9*x**4) + 108*c*e**3*f**2*g**2*x**2/(12*f**4*g**5
+ 48*f**3*g**6*x + 72*f**2*g**7*x**2 + 48*f*g**8*x**3 + 12*g**9*x**4) + 48*c*e**3*f*g**3*x**3*log(f/g + x)/(1
2*f**4*g**5 + 48*f**3*g**6*x + 72*f**2*g**7*x**2 + 48*f*g**8*x**3 + 12*g**9*x**4) + 48*c*e**3*f*g**3*x**3/(12*
f**4*g**5 + 48*f**3*g**6*x + 72*f**2*g**7*x**2 + 48*f*g**8*x**3 + 12*g**9*x**4) + 12*c*e**3*g**4*x**4*log(f/g
+ x)/(12*f**4*g**5 + 48*f**3*g**6*x + 72*f**2*g**7*x**2 + 48*f*g**8*x**3 + 12*g**9*x**4), Eq(n, -5)), (-a*d**2
*g**4/(3*f**3*g**5 + 9*f**2*g**6*x + 9*f*g**7*x**2 + 3*g**8*x**3) - a*d*e*f*g**3/(3*f**3*g**5 + 9*f**2*g**6*x
+ 9*f*g**7*x**2 + 3*g**8*x**3) - 3*a*d*e*g**4*x/(3*f**3*g**5 + 9*f**2*g**6*x + 9*f*g**7*x**2 + 3*g**8*x**3) -
a*e**2*f**2*g**2/(3*f**3*g**5 + 9*f**2*g**6*x + 9*f*g**7*x**2 + 3*g**8*x**3) - 3*a*e**2*f*g**3*x/(3*f**3*g**5
+ 9*f**2*g**6*x + 9*f*g**7*x**2 + 3*g**8*x**3) - 3*a*e**2*g**4*x**2/(3*f**3*g**5 + 9*f**2*g**6*x + 9*f*g**7*x*
*2 + 3*g**8*x**3) - c*d**3*f*g**3/(3*f**3*g**5 + 9*f**2*g**6*x + 9*f*g**7*x**2 + 3*g**8*x**3) - 3*c*d**3*g**4*
x/(3*f**3*g**5 + 9*f**2*g**6*x + 9*f*g**7*x**2 + 3*g**8*x**3) - 5*c*d**2*e*f**2*g**2/(3*f**3*g**5 + 9*f**2*g**
6*x + 9*f*g**7*x**2 + 3*g**8*x**3) - 15*c*d**2*e*f*g**3*x/(3*f**3*g**5 + 9*f**2*g**6*x + 9*f*g**7*x**2 + 3*g**
8*x**3) - 15*c*d**2*e*g**4*x**2/(3*f**3*g**5 + 9*f**2*g**6*x + 9*f*g**7*x**2 + 3*g**8*x**3) + 12*c*d*e**2*f**3
*g*log(f/g + x)/(3*f**3*g**5 + 9*f**2*g**6*x + 9*f*g**7*x**2 + 3*g**8*x**3) + 22*c*d*e**2*f**3*g/(3*f**3*g**5
+ 9*f**2*g**6*x + 9*f*g**7*x**2 + 3*g**8*x**3) + 36*c*d*e**2*f**2*g**2*x*log(f/g + x)/(3*f**3*g**5 + 9*f**2*g*
*6*x + 9*f*g**7*x**2 + 3*g**8*x**3) + 54*c*d*e**2*f**2*g**2*x/(3*f**3*g**5 + 9*f**2*g**6*x + 9*f*g**7*x**2 + 3
*g**8*x**3) + 36*c*d*e**2*f*g**3*x**2*log(f/g + x)/(3*f**3*g**5 + 9*f**2*g**6*x + 9*f*g**7*x**2 + 3*g**8*x**3)
+ 36*c*d*e**2*f*g**3*x**2/(3*f**3*g**5 + 9*f**2*g**6*x + 9*f*g**7*x**2 + 3*g**8*x**3) + 12*c*d*e**2*g**4*x**3
*log(f/g + x)/(3*f**3*g**5 + 9*f**2*g**6*x + 9*f*g**7*x**2 + 3*g**8*x**3) - 12*c*e**3*f**4*log(f/g + x)/(3*f**
3*g**5 + 9*f**2*g**6*x + 9*f*g**7*x**2 + 3*g**8*x**3) - 22*c*e**3*f**4/(3*f**3*g**5 + 9*f**2*g**6*x + 9*f*g**7
*x**2 + 3*g**8*x**3) - 36*c*e**3*f**3*g*x*log(f/g + x)/(3*f**3*g**5 + 9*f**2*g**6*x + 9*f*g**7*x**2 + 3*g**8*x
**3) - 54*c*e**3*f**3*g*x/(3*f**3*g**5 + 9*f**2*g**6*x + 9*f*g**7*x**2 + 3*g**8*x**3) - 36*c*e**3*f**2*g**2*x*
*2*log(f/g + x)/(3*f**3*g**5 + 9*f**2*g**6*x + 9*f*g**7*x**2 + 3*g**8*x**3) - 36*c*e**3*f**2*g**2*x**2/(3*f**3
*g**5 + 9*f**2*g**6*x + 9*f*g**7*x**2 + 3*g**8*x**3) - 12*c*e**3*f*g**3*x**3*log(f/g + x)/(3*f**3*g**5 + 9*f**
2*g**6*x + 9*f*g**7*x**2 + 3*g**8*x**3) + 3*c*e**3*g**4*x**4/(3*f**3*g**5 + 9*f**2*g**6*x + 9*f*g**7*x**2 + 3*
g**8*x**3), Eq(n, -4)), (-a*d**2*g**4/(2*f**2*g...

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 2114 vs. \(2 (216) = 432\).
time = 3.91, size = 2114, normalized size = 10.16 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^2*(g*x+f)^n*(c*e*x^2+2*c*d*x+a),x, algorithm="giac")

[Out]

((g*x + f)^n*c*g^5*n^4*x^5*e^3 + 4*(g*x + f)^n*c*d*g^5*n^4*x^4*e^2 + 5*(g*x + f)^n*c*d^2*g^5*n^4*x^3*e + 2*(g*
x + f)^n*c*d^3*g^5*n^4*x^2 + (g*x + f)^n*c*f*g^4*n^4*x^4*e^3 + 10*(g*x + f)^n*c*g^5*n^3*x^5*e^3 + 4*(g*x + f)^
n*c*d*f*g^4*n^4*x^3*e^2 + 44*(g*x + f)^n*c*d*g^5*n^3*x^4*e^2 + 5*(g*x + f)^n*c*d^2*f*g^4*n^4*x^2*e + 60*(g*x +
f)^n*c*d^2*g^5*n^3*x^3*e + 2*(g*x + f)^n*c*d^3*f*g^4*n^4*x + 26*(g*x + f)^n*c*d^3*g^5*n^3*x^2 + 6*(g*x + f)^n
*c*f*g^4*n^3*x^4*e^3 + 35*(g*x + f)^n*c*g^5*n^2*x^5*e^3 + 32*(g*x + f)^n*c*d*f*g^4*n^3*x^3*e^2 + (g*x + f)^n*a
*g^5*n^4*x^3*e^2 + 164*(g*x + f)^n*c*d*g^5*n^2*x^4*e^2 + 50*(g*x + f)^n*c*d^2*f*g^4*n^3*x^2*e + 2*(g*x + f)^n*
a*d*g^5*n^4*x^2*e + 245*(g*x + f)^n*c*d^2*g^5*n^2*x^3*e + 24*(g*x + f)^n*c*d^3*f*g^4*n^3*x + (g*x + f)^n*a*d^2
*g^5*n^4*x + 118*(g*x + f)^n*c*d^3*g^5*n^2*x^2 - 4*(g*x + f)^n*c*f^2*g^3*n^3*x^3*e^3 + 11*(g*x + f)^n*c*f*g^4*
n^2*x^4*e^3 + 50*(g*x + f)^n*c*g^5*n*x^5*e^3 - 12*(g*x + f)^n*c*d*f^2*g^3*n^3*x^2*e^2 + (g*x + f)^n*a*f*g^4*n^
4*x^2*e^2 + 68*(g*x + f)^n*c*d*f*g^4*n^2*x^3*e^2 + 12*(g*x + f)^n*a*g^5*n^3*x^3*e^2 + 244*(g*x + f)^n*c*d*g^5*
n*x^4*e^2 - 10*(g*x + f)^n*c*d^2*f^2*g^3*n^3*x*e + 2*(g*x + f)^n*a*d*f*g^4*n^4*x*e + 145*(g*x + f)^n*c*d^2*f*g
^4*n^2*x^2*e + 26*(g*x + f)^n*a*d*g^5*n^3*x^2*e + 390*(g*x + f)^n*c*d^2*g^5*n*x^3*e - 2*(g*x + f)^n*c*d^3*f^2*
g^3*n^3 + (g*x + f)^n*a*d^2*f*g^4*n^4 + 94*(g*x + f)^n*c*d^3*f*g^4*n^2*x + 14*(g*x + f)^n*a*d^2*g^5*n^3*x + 21
4*(g*x + f)^n*c*d^3*g^5*n*x^2 - 12*(g*x + f)^n*c*f^2*g^3*n^2*x^3*e^3 + 6*(g*x + f)^n*c*f*g^4*n*x^4*e^3 + 24*(g
*x + f)^n*c*g^5*x^5*e^3 - 72*(g*x + f)^n*c*d*f^2*g^3*n^2*x^2*e^2 + 10*(g*x + f)^n*a*f*g^4*n^3*x^2*e^2 + 40*(g*
x + f)^n*c*d*f*g^4*n*x^3*e^2 + 49*(g*x + f)^n*a*g^5*n^2*x^3*e^2 + 120*(g*x + f)^n*c*d*g^5*x^4*e^2 - 90*(g*x +
f)^n*c*d^2*f^2*g^3*n^2*x*e + 24*(g*x + f)^n*a*d*f*g^4*n^3*x*e + 100*(g*x + f)^n*c*d^2*f*g^4*n*x^2*e + 118*(g*x
+ f)^n*a*d*g^5*n^2*x^2*e + 200*(g*x + f)^n*c*d^2*g^5*x^3*e - 24*(g*x + f)^n*c*d^3*f^2*g^3*n^2 + 14*(g*x + f)^
n*a*d^2*f*g^4*n^3 + 120*(g*x + f)^n*c*d^3*f*g^4*n*x + 71*(g*x + f)^n*a*d^2*g^5*n^2*x + 120*(g*x + f)^n*c*d^3*g
^5*x^2 + 12*(g*x + f)^n*c*f^3*g^2*n^2*x^2*e^3 - 8*(g*x + f)^n*c*f^2*g^3*n*x^3*e^3 + 24*(g*x + f)^n*c*d*f^3*g^2
*n^2*x*e^2 - 2*(g*x + f)^n*a*f^2*g^3*n^3*x*e^2 - 60*(g*x + f)^n*c*d*f^2*g^3*n*x^2*e^2 + 29*(g*x + f)^n*a*f*g^4
*n^2*x^2*e^2 + 78*(g*x + f)^n*a*g^5*n*x^3*e^2 + 10*(g*x + f)^n*c*d^2*f^3*g^2*n^2*e - 2*(g*x + f)^n*a*d*f^2*g^3
*n^3*e - 200*(g*x + f)^n*c*d^2*f^2*g^3*n*x*e + 94*(g*x + f)^n*a*d*f*g^4*n^2*x*e + 214*(g*x + f)^n*a*d*g^5*n*x^
2*e - 94*(g*x + f)^n*c*d^3*f^2*g^3*n + 71*(g*x + f)^n*a*d^2*f*g^4*n^2 + 154*(g*x + f)^n*a*d^2*g^5*n*x + 12*(g*
x + f)^n*c*f^3*g^2*n*x^2*e^3 + 120*(g*x + f)^n*c*d*f^3*g^2*n*x*e^2 - 18*(g*x + f)^n*a*f^2*g^3*n^2*x*e^2 + 20*(
g*x + f)^n*a*f*g^4*n*x^2*e^2 + 40*(g*x + f)^n*a*g^5*x^3*e^2 + 90*(g*x + f)^n*c*d^2*f^3*g^2*n*e - 24*(g*x + f)^
n*a*d*f^2*g^3*n^2*e + 120*(g*x + f)^n*a*d*f*g^4*n*x*e + 120*(g*x + f)^n*a*d*g^5*x^2*e - 120*(g*x + f)^n*c*d^3*
f^2*g^3 + 154*(g*x + f)^n*a*d^2*f*g^4*n + 120*(g*x + f)^n*a*d^2*g^5*x - 24*(g*x + f)^n*c*f^4*g*n*x*e^3 - 24*(g
*x + f)^n*c*d*f^4*g*n*e^2 + 2*(g*x + f)^n*a*f^3*g^2*n^2*e^2 - 40*(g*x + f)^n*a*f^2*g^3*n*x*e^2 + 200*(g*x + f)
^n*c*d^2*f^3*g^2*e - 94*(g*x + f)^n*a*d*f^2*g^3*n*e + 120*(g*x + f)^n*a*d^2*f*g^4 - 120*(g*x + f)^n*c*d*f^4*g*
e^2 + 18*(g*x + f)^n*a*f^3*g^2*n*e^2 - 120*(g*x + f)^n*a*d*f^2*g^3*e + 24*(g*x + f)^n*c*f^5*e^3 + 40*(g*x + f)
^n*a*f^3*g^2*e^2)/(g^5*n^5 + 15*g^5*n^4 + 85*g^5*n^3 + 225*g^5*n^2 + 274*g^5*n + 120*g^5)

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Mupad [B]
time = 3.52, size = 1133, normalized size = 5.45 \begin {gather*} \frac {{\left (f+g\,x\right )}^n\,\left (-2\,c\,d^3\,f^2\,g^3\,n^3-24\,c\,d^3\,f^2\,g^3\,n^2-94\,c\,d^3\,f^2\,g^3\,n-120\,c\,d^3\,f^2\,g^3+10\,c\,d^2\,e\,f^3\,g^2\,n^2+90\,c\,d^2\,e\,f^3\,g^2\,n+200\,c\,d^2\,e\,f^3\,g^2+a\,d^2\,f\,g^4\,n^4+14\,a\,d^2\,f\,g^4\,n^3+71\,a\,d^2\,f\,g^4\,n^2+154\,a\,d^2\,f\,g^4\,n+120\,a\,d^2\,f\,g^4-24\,c\,d\,e^2\,f^4\,g\,n-120\,c\,d\,e^2\,f^4\,g-2\,a\,d\,e\,f^2\,g^3\,n^3-24\,a\,d\,e\,f^2\,g^3\,n^2-94\,a\,d\,e\,f^2\,g^3\,n-120\,a\,d\,e\,f^2\,g^3+24\,c\,e^3\,f^5+2\,a\,e^2\,f^3\,g^2\,n^2+18\,a\,e^2\,f^3\,g^2\,n+40\,a\,e^2\,f^3\,g^2\right )}{g^5\,\left (n^5+15\,n^4+85\,n^3+225\,n^2+274\,n+120\right )}+\frac {x\,{\left (f+g\,x\right )}^n\,\left (2\,c\,d^3\,f\,g^4\,n^4+24\,c\,d^3\,f\,g^4\,n^3+94\,c\,d^3\,f\,g^4\,n^2+120\,c\,d^3\,f\,g^4\,n-10\,c\,d^2\,e\,f^2\,g^3\,n^3-90\,c\,d^2\,e\,f^2\,g^3\,n^2-200\,c\,d^2\,e\,f^2\,g^3\,n+a\,d^2\,g^5\,n^4+14\,a\,d^2\,g^5\,n^3+71\,a\,d^2\,g^5\,n^2+154\,a\,d^2\,g^5\,n+120\,a\,d^2\,g^5+24\,c\,d\,e^2\,f^3\,g^2\,n^2+120\,c\,d\,e^2\,f^3\,g^2\,n+2\,a\,d\,e\,f\,g^4\,n^4+24\,a\,d\,e\,f\,g^4\,n^3+94\,a\,d\,e\,f\,g^4\,n^2+120\,a\,d\,e\,f\,g^4\,n-24\,c\,e^3\,f^4\,g\,n-2\,a\,e^2\,f^2\,g^3\,n^3-18\,a\,e^2\,f^2\,g^3\,n^2-40\,a\,e^2\,f^2\,g^3\,n\right )}{g^5\,\left (n^5+15\,n^4+85\,n^3+225\,n^2+274\,n+120\right )}+\frac {c\,e^3\,x^5\,{\left (f+g\,x\right )}^n\,\left (n^4+10\,n^3+35\,n^2+50\,n+24\right )}{n^5+15\,n^4+85\,n^3+225\,n^2+274\,n+120}+\frac {x^2\,{\left (f+g\,x\right )}^n\,\left (n+1\right )\,\left (2\,c\,d^3\,g^3\,n^3+24\,c\,d^3\,g^3\,n^2+94\,c\,d^3\,g^3\,n+120\,c\,d^3\,g^3+5\,c\,d^2\,e\,f\,g^2\,n^3+45\,c\,d^2\,e\,f\,g^2\,n^2+100\,c\,d^2\,e\,f\,g^2\,n-12\,c\,d\,e^2\,f^2\,g\,n^2-60\,c\,d\,e^2\,f^2\,g\,n+2\,a\,d\,e\,g^3\,n^3+24\,a\,d\,e\,g^3\,n^2+94\,a\,d\,e\,g^3\,n+120\,a\,d\,e\,g^3+12\,c\,e^3\,f^3\,n+a\,e^2\,f\,g^2\,n^3+9\,a\,e^2\,f\,g^2\,n^2+20\,a\,e^2\,f\,g^2\,n\right )}{g^3\,\left (n^5+15\,n^4+85\,n^3+225\,n^2+274\,n+120\right )}+\frac {e\,x^3\,{\left (f+g\,x\right )}^n\,\left (n^2+3\,n+2\right )\,\left (5\,c\,d^2\,g^2\,n^2+45\,c\,d^2\,g^2\,n+100\,c\,d^2\,g^2+4\,c\,d\,e\,f\,g\,n^2+20\,c\,d\,e\,f\,g\,n-4\,c\,e^2\,f^2\,n+a\,e\,g^2\,n^2+9\,a\,e\,g^2\,n+20\,a\,e\,g^2\right )}{g^2\,\left (n^5+15\,n^4+85\,n^3+225\,n^2+274\,n+120\right )}+\frac {c\,e^2\,x^4\,{\left (f+g\,x\right )}^n\,\left (20\,d\,g+4\,d\,g\,n+e\,f\,n\right )\,\left (n^3+6\,n^2+11\,n+6\right )}{g\,\left (n^5+15\,n^4+85\,n^3+225\,n^2+274\,n+120\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((f + g*x)^n*(24*c*e^3*f^5 + 40*a*e^2*f^3*g^2 - 120*c*d^3*f^2*g^3 + 120*a*d^2*f*g^4 - 120*a*d*e*f^2*g^3 - 120*
c*d*e^2*f^4*g + 154*a*d^2*f*g^4*n + 200*c*d^2*e*f^3*g^2 + 71*a*d^2*f*g^4*n^2 + 14*a*d^2*f*g^4*n^3 + a*d^2*f*g^
4*n^4 + 18*a*e^2*f^3*g^2*n - 94*c*d^3*f^2*g^3*n + 2*a*e^2*f^3*g^2*n^2 - 24*c*d^3*f^2*g^3*n^2 - 2*c*d^3*f^2*g^3
*n^3 + 10*c*d^2*e*f^3*g^2*n^2 - 94*a*d*e*f^2*g^3*n - 24*c*d*e^2*f^4*g*n - 24*a*d*e*f^2*g^3*n^2 - 2*a*d*e*f^2*g
^3*n^3 + 90*c*d^2*e*f^3*g^2*n))/(g^5*(274*n + 225*n^2 + 85*n^3 + 15*n^4 + n^5 + 120)) + (x*(f + g*x)^n*(120*a*
d^2*g^5 + 71*a*d^2*g^5*n^2 + 14*a*d^2*g^5*n^3 + a*d^2*g^5*n^4 + 154*a*d^2*g^5*n + 120*c*d^3*f*g^4*n - 24*c*e^3
*f^4*g*n - 40*a*e^2*f^2*g^3*n + 94*c*d^3*f*g^4*n^2 + 24*c*d^3*f*g^4*n^3 + 2*c*d^3*f*g^4*n^4 - 18*a*e^2*f^2*g^3
*n^2 - 2*a*e^2*f^2*g^3*n^3 + 120*a*d*e*f*g^4*n + 24*c*d*e^2*f^3*g^2*n^2 - 90*c*d^2*e*f^2*g^3*n^2 - 10*c*d^2*e*
f^2*g^3*n^3 + 94*a*d*e*f*g^4*n^2 + 24*a*d*e*f*g^4*n^3 + 2*a*d*e*f*g^4*n^4 + 120*c*d*e^2*f^3*g^2*n - 200*c*d^2*
e*f^2*g^3*n))/(g^5*(274*n + 225*n^2 + 85*n^3 + 15*n^4 + n^5 + 120)) + (c*e^3*x^5*(f + g*x)^n*(50*n + 35*n^2 +
10*n^3 + n^4 + 24))/(274*n + 225*n^2 + 85*n^3 + 15*n^4 + n^5 + 120) + (x^2*(f + g*x)^n*(n + 1)*(120*c*d^3*g^3
+ 24*c*d^3*g^3*n^2 + 2*c*d^3*g^3*n^3 + 120*a*d*e*g^3 + 94*c*d^3*g^3*n + 12*c*e^3*f^3*n + 24*a*d*e*g^3*n^2 + 2*
a*d*e*g^3*n^3 + 20*a*e^2*f*g^2*n + 9*a*e^2*f*g^2*n^2 + a*e^2*f*g^2*n^3 + 94*a*d*e*g^3*n - 60*c*d*e^2*f^2*g*n +
100*c*d^2*e*f*g^2*n - 12*c*d*e^2*f^2*g*n^2 + 45*c*d^2*e*f*g^2*n^2 + 5*c*d^2*e*f*g^2*n^3))/(g^3*(274*n + 225*n
^2 + 85*n^3 + 15*n^4 + n^5 + 120)) + (e*x^3*(f + g*x)^n*(3*n + n^2 + 2)*(100*c*d^2*g^2 + 20*a*e*g^2 + 5*c*d^2*
g^2*n^2 + 9*a*e*g^2*n + a*e*g^2*n^2 + 45*c*d^2*g^2*n - 4*c*e^2*f^2*n + 4*c*d*e*f*g*n^2 + 20*c*d*e*f*g*n))/(g^2
*(274*n + 225*n^2 + 85*n^3 + 15*n^4 + n^5 + 120)) + (c*e^2*x^4*(f + g*x)^n*(20*d*g + 4*d*g*n + e*f*n)*(11*n +
6*n^2 + n^3 + 6))/(g*(274*n + 225*n^2 + 85*n^3 + 15*n^4 + n^5 + 120))

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