3.9.5 \(\int (d+e x)^3 (f+g x)^n (a+2 c d x+c e x^2) \, dx\) [805]
Optimal. Leaf size=275 \[ -\frac {(e f-d g)^3 \left (a g^2+c f (e f-2 d g)\right ) (f+g x)^{1+n}}{g^6 (1+n)}+\frac {(e f-d g)^2 \left (3 a e g^2+c \left (5 e^2 f^2-10 d e f g+2 d^2 g^2\right )\right ) (f+g x)^{2+n}}{g^6 (2+n)}-\frac {e (e f-d g) \left (3 a e g^2+c \left (10 e^2 f^2-20 d e f g+7 d^2 g^2\right )\right ) (f+g x)^{3+n}}{g^6 (3+n)}+\frac {e^2 \left (a e g^2+c \left (10 e^2 f^2-20 d e f g+9 d^2 g^2\right )\right ) (f+g x)^{4+n}}{g^6 (4+n)}-\frac {5 c e^3 (e f-d g) (f+g x)^{5+n}}{g^6 (5+n)}+\frac {c e^4 (f+g x)^{6+n}}{g^6 (6+n)} \]
[Out]
-(-d*g+e*f)^3*(a*g^2+c*f*(-2*d*g+e*f))*(g*x+f)^(1+n)/g^6/(1+n)+(-d*g+e*f)^2*(3*a*e*g^2+c*(2*d^2*g^2-10*d*e*f*g
+5*e^2*f^2))*(g*x+f)^(2+n)/g^6/(2+n)-e*(-d*g+e*f)*(3*a*e*g^2+c*(7*d^2*g^2-20*d*e*f*g+10*e^2*f^2))*(g*x+f)^(3+n
)/g^6/(3+n)+e^2*(a*e*g^2+c*(9*d^2*g^2-20*d*e*f*g+10*e^2*f^2))*(g*x+f)^(4+n)/g^6/(4+n)-5*c*e^3*(-d*g+e*f)*(g*x+
f)^(5+n)/g^6/(5+n)+c*e^4*(g*x+f)^(6+n)/g^6/(6+n)
________________________________________________________________________________________
Rubi [A]
time = 0.17, antiderivative size = 275, 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 {(e f-d g)^2 (f+g x)^{n+2} \left (3 a e g^2+c \left (2 d^2 g^2-10 d e f g+5 e^2 f^2\right )\right )}{g^6 (n+2)}-\frac {e (e f-d g) (f+g x)^{n+3} \left (3 a e g^2+c \left (7 d^2 g^2-20 d e f g+10 e^2 f^2\right )\right )}{g^6 (n+3)}+\frac {e^2 (f+g x)^{n+4} \left (a e g^2+c \left (9 d^2 g^2-20 d e f g+10 e^2 f^2\right )\right )}{g^6 (n+4)}-\frac {(e f-d g)^3 (f+g x)^{n+1} \left (a g^2+c f (e f-2 d g)\right )}{g^6 (n+1)}-\frac {5 c e^3 (e f-d g) (f+g x)^{n+5}}{g^6 (n+5)}+\frac {c e^4 (f+g x)^{n+6}}{g^6 (n+6)} \end {gather*}
Antiderivative was successfully verified.
[In]
Int[(d + e*x)^3*(f + g*x)^n*(a + 2*c*d*x + c*e*x^2),x]
[Out]
-(((e*f - d*g)^3*(a*g^2 + c*f*(e*f - 2*d*g))*(f + g*x)^(1 + n))/(g^6*(1 + n))) + ((e*f - d*g)^2*(3*a*e*g^2 + c
*(5*e^2*f^2 - 10*d*e*f*g + 2*d^2*g^2))*(f + g*x)^(2 + n))/(g^6*(2 + n)) - (e*(e*f - d*g)*(3*a*e*g^2 + c*(10*e^
2*f^2 - 20*d*e*f*g + 7*d^2*g^2))*(f + g*x)^(3 + n))/(g^6*(3 + n)) + (e^2*(a*e*g^2 + c*(10*e^2*f^2 - 20*d*e*f*g
+ 9*d^2*g^2))*(f + g*x)^(4 + n))/(g^6*(4 + n)) - (5*c*e^3*(e*f - d*g)*(f + g*x)^(5 + n))/(g^6*(5 + n)) + (c*e
^4*(f + g*x)^(6 + n))/(g^6*(6 + 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)^3 (f+g x)^n \left (a+2 c d x+c e x^2\right ) \, dx &=\int \left (\frac {(e f-d g)^3 \left (-a g^2-c f (e f-2 d g)\right ) (f+g x)^n}{g^5}+\frac {(e f-d g)^2 \left (3 a e g^2+c \left (5 e^2 f^2-10 d e f g+2 d^2 g^2\right )\right ) (f+g x)^{1+n}}{g^5}+\frac {e (e f-d g) \left (-3 a e g^2-c \left (10 e^2 f^2-20 d e f g+7 d^2 g^2\right )\right ) (f+g x)^{2+n}}{g^5}+\frac {e^2 \left (a e g^2+c \left (10 e^2 f^2-20 d e f g+9 d^2 g^2\right )\right ) (f+g x)^{3+n}}{g^5}-\frac {5 c e^3 (e f-d g) (f+g x)^{4+n}}{g^5}+\frac {c e^4 (f+g x)^{5+n}}{g^5}\right ) \, dx\\ &=-\frac {(e f-d g)^3 \left (a g^2+c f (e f-2 d g)\right ) (f+g x)^{1+n}}{g^6 (1+n)}+\frac {(e f-d g)^2 \left (3 a e g^2+c \left (5 e^2 f^2-10 d e f g+2 d^2 g^2\right )\right ) (f+g x)^{2+n}}{g^6 (2+n)}-\frac {e (e f-d g) \left (3 a e g^2+c \left (10 e^2 f^2-20 d e f g+7 d^2 g^2\right )\right ) (f+g x)^{3+n}}{g^6 (3+n)}+\frac {e^2 \left (a e g^2+c \left (10 e^2 f^2-20 d e f g+9 d^2 g^2\right )\right ) (f+g x)^{4+n}}{g^6 (4+n)}-\frac {5 c e^3 (e f-d g) (f+g x)^{5+n}}{g^6 (5+n)}+\frac {c e^4 (f+g x)^{6+n}}{g^6 (6+n)}\\ \end {align*}
________________________________________________________________________________________
Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(577\) vs. \(2(275)=550\).
time = 0.58, size = 577, normalized size = 2.10 \begin {gather*} \frac {(f+g x)^{1+n} \left (a g^2 \left (30+11 n+n^2\right ) \left (d^3 g^3 \left (24+26 n+9 n^2+n^3\right )+3 d^2 e g^2 \left (12+7 n+n^2\right ) (-f+g (1+n) x)+3 d e^2 g (4+n) \left (2 f^2-2 f g (1+n) x+g^2 \left (2+3 n+n^2\right ) x^2\right )+e^3 \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 )\right )+c \left (2 d^4 g^4 \left (360+342 n+119 n^2+18 n^3+n^4\right ) (-f+g (1+n) x)+7 d^3 e g^3 \left (120+74 n+15 n^2+n^3\right ) \left (2 f^2-2 f g (1+n) x+g^2 \left (2+3 n+n^2\right ) x^2\right )+9 d^2 e^2 g^2 \left (30+11 n+n^2\right ) \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 )+5 d e^3 g (6+n) \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 )-e^4 \left (120 f^5-120 f^4 g (1+n) x+60 f^3 g^2 \left (2+3 n+n^2\right ) x^2-20 f^2 g^3 \left (6+11 n+6 n^2+n^3\right ) x^3+5 f g^4 \left (24+50 n+35 n^2+10 n^3+n^4\right ) x^4-g^5 \left (120+274 n+225 n^2+85 n^3+15 n^4+n^5\right ) x^5\right )\right )\right )}{g^6 (1+n) (2+n) (3+n) (4+n) (5+n) (6+n)} \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[(d + e*x)^3*(f + g*x)^n*(a + 2*c*d*x + c*e*x^2),x]
[Out]
((f + g*x)^(1 + n)*(a*g^2*(30 + 11*n + n^2)*(d^3*g^3*(24 + 26*n + 9*n^2 + n^3) + 3*d^2*e*g^2*(12 + 7*n + n^2)*
(-f + g*(1 + n)*x) + 3*d*e^2*g*(4 + n)*(2*f^2 - 2*f*g*(1 + n)*x + g^2*(2 + 3*n + n^2)*x^2) + e^3*(-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)) + c*(2*d^4*g^4*(360 + 342*n
+ 119*n^2 + 18*n^3 + n^4)*(-f + g*(1 + n)*x) + 7*d^3*e*g^3*(120 + 74*n + 15*n^2 + n^3)*(2*f^2 - 2*f*g*(1 + n)*
x + g^2*(2 + 3*n + n^2)*x^2) + 9*d^2*e^2*g^2*(30 + 11*n + n^2)*(-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) + 5*d*e^3*g*(6 + n)*(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) - e^4
*(120*f^5 - 120*f^4*g*(1 + n)*x + 60*f^3*g^2*(2 + 3*n + n^2)*x^2 - 20*f^2*g^3*(6 + 11*n + 6*n^2 + n^3)*x^3 + 5
*f*g^4*(24 + 50*n + 35*n^2 + 10*n^3 + n^4)*x^4 - g^5*(120 + 274*n + 225*n^2 + 85*n^3 + 15*n^4 + n^5)*x^5))))/(
g^6*(1 + n)*(2 + n)*(3 + n)*(4 + n)*(5 + n)*(6 + n))
________________________________________________________________________________________
Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(1785\) vs.
\(2(275)=550\).
time = 0.14, size = 1786, normalized size = 6.49
| | |
| method |
result |
size |
| | |
| norman |
\(\text {Expression too large to display}\) |
\(1786\) |
| gosper |
\(\text {Expression too large to display}\) |
\(2017\) |
| risch |
\(\text {Expression too large to display}\) |
\(2642\) |
| | |
|
|
|
|
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((e*x+d)^3*(g*x+f)^n*(c*e*x^2+2*c*d*x+a),x,method=_RETURNVERBOSE)
[Out]
c*e^4/(6+n)*x^6*exp(n*ln(g*x+f))+f*(a*d^3*g^5*n^5-2*c*d^4*f*g^4*n^4+20*a*d^3*g^5*n^4-3*a*d^2*e*f*g^4*n^4-36*c*
d^4*f*g^4*n^3+14*c*d^3*e*f^2*g^3*n^3+155*a*d^3*g^5*n^3-54*a*d^2*e*f*g^4*n^3+6*a*d*e^2*f^2*g^3*n^3-238*c*d^4*f*
g^4*n^2+210*c*d^3*e*f^2*g^3*n^2-54*c*d^2*e^2*f^3*g^2*n^2+580*a*d^3*g^5*n^2-357*a*d^2*e*f*g^4*n^2+90*a*d*e^2*f^
2*g^3*n^2-6*a*e^3*f^3*g^2*n^2-684*c*d^4*f*g^4*n+1036*c*d^3*e*f^2*g^3*n-594*c*d^2*e^2*f^3*g^2*n+120*c*d*e^3*f^4
*g*n+1044*a*d^3*g^5*n-1026*a*d^2*e*f*g^4*n+444*a*d*e^2*f^2*g^3*n-66*a*e^3*f^3*g^2*n-720*c*d^4*f*g^4+1680*c*d^3
*e*f^2*g^3-1620*c*d^2*e^2*f^3*g^2+720*c*d*e^3*f^4*g-120*c*e^4*f^5+720*a*d^3*g^5-1080*a*d^2*e*f*g^4+720*a*d*e^2
*f^2*g^3-180*a*e^3*f^3*g^2)/g^6/(n^6+21*n^5+175*n^4+735*n^3+1624*n^2+1764*n+720)*exp(n*ln(g*x+f))+(2*c*d^4*g^4
*n^4+7*c*d^3*e*f*g^3*n^4+3*a*d^2*e*g^4*n^4+3*a*d*e^2*f*g^3*n^4+36*c*d^4*g^4*n^3+105*c*d^3*e*f*g^3*n^3-27*c*d^2
*e^2*f^2*g^2*n^3+54*a*d^2*e*g^4*n^3+45*a*d*e^2*f*g^3*n^3-3*a*e^3*f^2*g^2*n^3+238*c*d^4*g^4*n^2+518*c*d^3*e*f*g
^3*n^2-297*c*d^2*e^2*f^2*g^2*n^2+60*c*d*e^3*f^3*g*n^2+357*a*d^2*e*g^4*n^2+222*a*d*e^2*f*g^3*n^2-33*a*e^3*f^2*g
^2*n^2+684*c*d^4*g^4*n+840*c*d^3*e*f*g^3*n-810*c*d^2*e^2*f^2*g^2*n+360*c*d*e^3*f^3*g*n-60*c*e^4*f^4*n+1026*a*d
^2*e*g^4*n+360*a*d*e^2*f*g^3*n-90*a*e^3*f^2*g^2*n+720*c*d^4*g^4+1080*a*d^2*e*g^4)/g^4/(n^5+20*n^4+155*n^3+580*
n^2+1044*n+720)*x^2*exp(n*ln(g*x+f))+(2*c*d^4*f*g^4*n^5+a*d^3*g^5*n^5+3*a*d^2*e*f*g^4*n^5+36*c*d^4*f*g^4*n^4-1
4*c*d^3*e*f^2*g^3*n^4+20*a*d^3*g^5*n^4+54*a*d^2*e*f*g^4*n^4-6*a*d*e^2*f^2*g^3*n^4+238*c*d^4*f*g^4*n^3-210*c*d^
3*e*f^2*g^3*n^3+54*c*d^2*e^2*f^3*g^2*n^3+155*a*d^3*g^5*n^3+357*a*d^2*e*f*g^4*n^3-90*a*d*e^2*f^2*g^3*n^3+6*a*e^
3*f^3*g^2*n^3+684*c*d^4*f*g^4*n^2-1036*c*d^3*e*f^2*g^3*n^2+594*c*d^2*e^2*f^3*g^2*n^2-120*c*d*e^3*f^4*g*n^2+580
*a*d^3*g^5*n^2+1026*a*d^2*e*f*g^4*n^2-444*a*d*e^2*f^2*g^3*n^2+66*a*e^3*f^3*g^2*n^2+720*c*d^4*f*g^4*n-1680*c*d^
3*e*f^2*g^3*n+1620*c*d^2*e^2*f^3*g^2*n-720*c*d*e^3*f^4*g*n+120*c*e^4*f^5*n+1044*a*d^3*g^5*n+1080*a*d^2*e*f*g^4
*n-720*a*d*e^2*f^2*g^3*n+180*a*e^3*f^3*g^2*n+720*a*d^3*g^5)/g^5/(n^6+21*n^5+175*n^4+735*n^3+1624*n^2+1764*n+72
0)*x*exp(n*ln(g*x+f))+(9*c*d^2*g^2*n^2+5*c*d*e*f*g*n^2+a*e*g^2*n^2+99*c*d^2*g^2*n+30*c*d*e*f*g*n-5*c*e^2*f^2*n
+11*a*e*g^2*n+270*c*d^2*g^2+30*a*e*g^2)*e^2/g^2/(n^3+15*n^2+74*n+120)*x^4*exp(n*ln(g*x+f))+(7*c*d^3*g^3*n^3+9*
c*d^2*e*f*g^2*n^3+3*a*d*e*g^3*n^3+a*e^2*f*g^2*n^3+105*c*d^3*g^3*n^2+99*c*d^2*e*f*g^2*n^2-20*c*d*e^2*f^2*g*n^2+
45*a*d*e*g^3*n^2+11*a*e^2*f*g^2*n^2+518*c*d^3*g^3*n+270*c*d^2*e*f*g^2*n-120*c*d*e^2*f^2*g*n+20*c*e^3*f^3*n+222
*a*d*e*g^3*n+30*a*e^2*f*g^2*n+840*c*d^3*g^3+360*a*d*e*g^3)*e/g^3/(n^4+18*n^3+119*n^2+342*n+360)*x^3*exp(n*ln(g
*x+f))+(5*d*g*n+e*f*n+30*d*g)*c/g*e^3/(n^2+11*n+30)*x^5*exp(n*ln(g*x+f))
________________________________________________________________________________________
Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 811 vs.
\(2 (275) = 550\).
time = 0.33, size = 811, normalized size = 2.95 \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^{4}}{{\left (n^{2} + 3 \, n + 2\right )} g^{2}} + \frac {7 \, {\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^{3} e}{{\left (n^{3} + 6 \, n^{2} + 11 \, n + 6\right )} g^{3}} + \frac {3 \, {\left (g^{2} {\left (n + 1\right )} x^{2} + f g n x - f^{2}\right )} {\left (g x + f\right )}^{n} a d^{2} e}{{\left (n^{2} + 3 \, n + 2\right )} g^{2}} + \frac {{\left (g x + f\right )}^{n + 1} a d^{3}}{g {\left (n + 1\right )}} + \frac {9 \, {\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^{2} e^{2}}{{\left (n^{4} + 10 \, n^{3} + 35 \, n^{2} + 50 \, n + 24\right )} g^{4}} + \frac {3 \, {\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 d e^{2}}{{\left (n^{3} + 6 \, n^{2} + 11 \, n + 6\right )} g^{3}} + \frac {5 \, {\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 d e^{3}}{{\left (n^{5} + 15 \, n^{4} + 85 \, n^{3} + 225 \, n^{2} + 274 \, n + 120\right )} g^{5}} + \frac {{\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} a e^{3}}{{\left (n^{4} + 10 \, n^{3} + 35 \, n^{2} + 50 \, n + 24\right )} g^{4}} + \frac {{\left ({\left (n^{5} + 15 \, n^{4} + 85 \, n^{3} + 225 \, n^{2} + 274 \, n + 120\right )} g^{6} x^{6} + {\left (n^{5} + 10 \, n^{4} + 35 \, n^{3} + 50 \, n^{2} + 24 \, n\right )} f g^{5} x^{5} - 5 \, {\left (n^{4} + 6 \, n^{3} + 11 \, n^{2} + 6 \, n\right )} f^{2} g^{4} x^{4} + 20 \, {\left (n^{3} + 3 \, n^{2} + 2 \, n\right )} f^{3} g^{3} x^{3} - 60 \, {\left (n^{2} + n\right )} f^{4} g^{2} x^{2} + 120 \, f^{5} g n x - 120 \, f^{6}\right )} {\left (g x + f\right )}^{n} c e^{4}}{{\left (n^{6} + 21 \, n^{5} + 175 \, n^{4} + 735 \, n^{3} + 1624 \, n^{2} + 1764 \, n + 720\right )} g^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((e*x+d)^3*(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^4/((n^2 + 3*n + 2)*g^2) + 7*((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^3*e/((n^3 + 6*n^2 + 11*n + 6)*g^3) + 3*(g^2*(n + 1)*x^
2 + f*g*n*x - f^2)*(g*x + f)^n*a*d^2*e/((n^2 + 3*n + 2)*g^2) + (g*x + f)^(n + 1)*a*d^3/(g*(n + 1)) + 9*((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^2*e^2/((n^4 + 10*n^3 + 35*n^2 + 50*n + 24)*g^4) + 3*((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*d*e^2/((n^3 + 6*n^2 + 11*n + 6)*g^3) + 5*((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*d*e^3/((n^5 + 15*n^4 + 85*n^3 + 225*n^2 + 274*n + 120)*g^5) +
((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*a*e^3/((n^4 + 10*n^3 + 35*n^2 + 50*n + 24)*g^4) + ((n^5 + 15*n^4 + 85*n^3 + 225*n^2 + 274*n
+ 120)*g^6*x^6 + (n^5 + 10*n^4 + 35*n^3 + 50*n^2 + 24*n)*f*g^5*x^5 - 5*(n^4 + 6*n^3 + 11*n^2 + 6*n)*f^2*g^4*x^
4 + 20*(n^3 + 3*n^2 + 2*n)*f^3*g^3*x^3 - 60*(n^2 + n)*f^4*g^2*x^2 + 120*f^5*g*n*x - 120*f^6)*(g*x + f)^n*c*e^4
/((n^6 + 21*n^5 + 175*n^4 + 735*n^3 + 1624*n^2 + 1764*n + 720)*g^6)
________________________________________________________________________________________
Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 2095 vs.
\(2 (282) = 564\).
time = 2.92, size = 2095, normalized size = 7.62 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((e*x+d)^3*(g*x+f)^n*(c*e*x^2+2*c*d*x+a),x, algorithm="fricas")
[Out]
(a*d^3*f*g^5*n^5 - 720*c*d^4*f^2*g^4 + 720*a*d^3*f*g^5 - 2*(c*d^4*f^2*g^4 - 10*a*d^3*f*g^5)*n^4 - (36*c*d^4*f^
2*g^4 - 155*a*d^3*f*g^5)*n^3 - 2*(119*c*d^4*f^2*g^4 - 290*a*d^3*f*g^5)*n^2 + 2*(c*d^4*g^6*n^5 + 19*c*d^4*g^6*n
^4 + 137*c*d^4*g^6*n^3 + 461*c*d^4*g^6*n^2 + 702*c*d^4*g^6*n + 360*c*d^4*g^6)*x^2 - 36*(19*c*d^4*f^2*g^4 - 29*
a*d^3*f*g^5)*n + (720*a*d^3*g^6 + (2*c*d^4*f*g^5 + a*d^3*g^6)*n^5 + 4*(9*c*d^4*f*g^5 + 5*a*d^3*g^6)*n^4 + (238
*c*d^4*f*g^5 + 155*a*d^3*g^6)*n^3 + 4*(171*c*d^4*f*g^5 + 145*a*d^3*g^6)*n^2 + 36*(20*c*d^4*f*g^5 + 29*a*d^3*g^
6)*n)*x + (120*c*f^5*g*n*x - 120*c*f^6 + (c*g^6*n^5 + 15*c*g^6*n^4 + 85*c*g^6*n^3 + 225*c*g^6*n^2 + 274*c*g^6*
n + 120*c*g^6)*x^6 + (c*f*g^5*n^5 + 10*c*f*g^5*n^4 + 35*c*f*g^5*n^3 + 50*c*f*g^5*n^2 + 24*c*f*g^5*n)*x^5 - 5*(
c*f^2*g^4*n^4 + 6*c*f^2*g^4*n^3 + 11*c*f^2*g^4*n^2 + 6*c*f^2*g^4*n)*x^4 + 20*(c*f^3*g^3*n^3 + 3*c*f^3*g^3*n^2
+ 2*c*f^3*g^3*n)*x^3 - 60*(c*f^4*g^2*n^2 + c*f^4*g^2*n)*x^2)*e^4 - (6*a*f^4*g^2*n^2 - 720*c*d*f^5*g + 180*a*f^
4*g^2 - 5*(c*d*g^6*n^5 + 16*c*d*g^6*n^4 + 95*c*d*g^6*n^3 + 260*c*d*g^6*n^2 + 324*c*d*g^6*n + 144*c*d*g^6)*x^5
- (180*a*g^6 + (5*c*d*f*g^5 + a*g^6)*n^5 + (60*c*d*f*g^5 + 17*a*g^6)*n^4 + (235*c*d*f*g^5 + 107*a*g^6)*n^3 + (
360*c*d*f*g^5 + 307*a*g^6)*n^2 + 36*(5*c*d*f*g^5 + 11*a*g^6)*n)*x^4 - (a*f*g^5*n^5 - 2*(10*c*d*f^2*g^4 - 7*a*f
*g^5)*n^4 - 5*(36*c*d*f^2*g^4 - 13*a*f*g^5)*n^3 - 16*(25*c*d*f^2*g^4 - 7*a*f*g^5)*n^2 - 60*(4*c*d*f^2*g^4 - a*
f*g^5)*n)*x^3 + 3*(a*f^2*g^4*n^4 - 4*(5*c*d*f^3*g^3 - 3*a*f^2*g^4)*n^3 - (140*c*d*f^3*g^3 - 41*a*f^2*g^4)*n^2
- 30*(4*c*d*f^3*g^3 - a*f^2*g^4)*n)*x^2 - 6*(20*c*d*f^5*g - 11*a*f^4*g^2)*n - 6*(a*f^3*g^3*n^3 - (20*c*d*f^4*g
^2 - 11*a*f^3*g^3)*n^2 - 30*(4*c*d*f^4*g^2 - a*f^3*g^3)*n)*x)*e^3 + 3*(2*a*d*f^3*g^3*n^3 - 540*c*d^2*f^4*g^2 +
240*a*d*f^3*g^3 + 3*(c*d^2*g^6*n^5 + 17*c*d^2*g^6*n^4 + 107*c*d^2*g^6*n^3 + 307*c*d^2*g^6*n^2 + 396*c*d^2*g^6
*n + 180*c*d^2*g^6)*x^4 + (240*a*d*g^6 + (3*c*d^2*f*g^5 + a*d*g^6)*n^5 + 6*(7*c*d^2*f*g^5 + 3*a*d*g^6)*n^4 + (
195*c*d^2*f*g^5 + 121*a*d*g^6)*n^3 + 12*(28*c*d^2*f*g^5 + 31*a*d*g^6)*n^2 + 4*(45*c*d^2*f*g^5 + 127*a*d*g^6)*n
)*x^3 - 6*(3*c*d^2*f^4*g^2 - 5*a*d*f^3*g^3)*n^2 + (a*d*f*g^5*n^5 - (9*c*d^2*f^2*g^4 - 16*a*d*f*g^5)*n^4 - (108
*c*d^2*f^2*g^4 - 89*a*d*f*g^5)*n^3 - (369*c*d^2*f^2*g^4 - 194*a*d*f*g^5)*n^2 - 30*(9*c*d^2*f^2*g^4 - 4*a*d*f*g
^5)*n)*x^2 - 2*(99*c*d^2*f^4*g^2 - 74*a*d*f^3*g^3)*n - 2*(a*d*f^2*g^4*n^4 - 3*(3*c*d^2*f^3*g^3 - 5*a*d*f^2*g^4
)*n^3 - (99*c*d^2*f^3*g^3 - 74*a*d*f^2*g^4)*n^2 - 30*(9*c*d^2*f^3*g^3 - 4*a*d*f^2*g^4)*n)*x)*e^2 - (3*a*d^2*f^
2*g^4*n^4 - 1680*c*d^3*f^3*g^3 + 1080*a*d^2*f^2*g^4 - 2*(7*c*d^3*f^3*g^3 - 27*a*d^2*f^2*g^4)*n^3 - 7*(c*d^3*g^
6*n^5 + 18*c*d^3*g^6*n^4 + 121*c*d^3*g^6*n^3 + 372*c*d^3*g^6*n^2 + 508*c*d^3*g^6*n + 240*c*d^3*g^6)*x^3 - 21*(
10*c*d^3*f^3*g^3 - 17*a*d^2*f^2*g^4)*n^2 - (1080*a*d^2*g^6 + (7*c*d^3*f*g^5 + 3*a*d^2*g^6)*n^5 + (112*c*d^3*f*
g^5 + 57*a*d^2*g^6)*n^4 + (623*c*d^3*f*g^5 + 411*a*d^2*g^6)*n^3 + (1358*c*d^3*f*g^5 + 1383*a*d^2*g^6)*n^2 + 6*
(140*c*d^3*f*g^5 + 351*a*d^2*g^6)*n)*x^2 - 2*(518*c*d^3*f^3*g^3 - 513*a*d^2*f^2*g^4)*n - (3*a*d^2*f*g^5*n^5 -
2*(7*c*d^3*f^2*g^4 - 27*a*d^2*f*g^5)*n^4 - 21*(10*c*d^3*f^2*g^4 - 17*a*d^2*f*g^5)*n^3 - 2*(518*c*d^3*f^2*g^4 -
513*a*d^2*f*g^5)*n^2 - 120*(14*c*d^3*f^2*g^4 - 9*a*d^2*f*g^5)*n)*x)*e)*(g*x + f)^n/(g^6*n^6 + 21*g^6*n^5 + 17
5*g^6*n^4 + 735*g^6*n^3 + 1624*g^6*n^2 + 1764*g^6*n + 720*g^6)
________________________________________________________________________________________
Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 24206 vs.
\(2 (260) = 520\).
time = 4.97, size = 24206, normalized size = 88.02 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((e*x+d)**3*(g*x+f)**n*(c*e*x**2+2*c*d*x+a),x)
[Out]
Piecewise((f**n*(a*d**3*x + 3*a*d**2*e*x**2/2 + a*d*e**2*x**3 + a*e**3*x**4/4 + c*d**4*x**2 + 7*c*d**3*e*x**3/
3 + 9*c*d**2*e**2*x**4/4 + c*d*e**3*x**5 + c*e**4*x**6/6), Eq(g, 0)), (-12*a*d**3*g**5/(60*f**5*g**6 + 300*f**
4*g**7*x + 600*f**3*g**8*x**2 + 600*f**2*g**9*x**3 + 300*f*g**10*x**4 + 60*g**11*x**5) - 9*a*d**2*e*f*g**4/(60
*f**5*g**6 + 300*f**4*g**7*x + 600*f**3*g**8*x**2 + 600*f**2*g**9*x**3 + 300*f*g**10*x**4 + 60*g**11*x**5) - 4
5*a*d**2*e*g**5*x/(60*f**5*g**6 + 300*f**4*g**7*x + 600*f**3*g**8*x**2 + 600*f**2*g**9*x**3 + 300*f*g**10*x**4
+ 60*g**11*x**5) - 6*a*d*e**2*f**2*g**3/(60*f**5*g**6 + 300*f**4*g**7*x + 600*f**3*g**8*x**2 + 600*f**2*g**9*
x**3 + 300*f*g**10*x**4 + 60*g**11*x**5) - 30*a*d*e**2*f*g**4*x/(60*f**5*g**6 + 300*f**4*g**7*x + 600*f**3*g**
8*x**2 + 600*f**2*g**9*x**3 + 300*f*g**10*x**4 + 60*g**11*x**5) - 60*a*d*e**2*g**5*x**2/(60*f**5*g**6 + 300*f*
*4*g**7*x + 600*f**3*g**8*x**2 + 600*f**2*g**9*x**3 + 300*f*g**10*x**4 + 60*g**11*x**5) - 3*a*e**3*f**3*g**2/(
60*f**5*g**6 + 300*f**4*g**7*x + 600*f**3*g**8*x**2 + 600*f**2*g**9*x**3 + 300*f*g**10*x**4 + 60*g**11*x**5) -
15*a*e**3*f**2*g**3*x/(60*f**5*g**6 + 300*f**4*g**7*x + 600*f**3*g**8*x**2 + 600*f**2*g**9*x**3 + 300*f*g**10
*x**4 + 60*g**11*x**5) - 30*a*e**3*f*g**4*x**2/(60*f**5*g**6 + 300*f**4*g**7*x + 600*f**3*g**8*x**2 + 600*f**2
*g**9*x**3 + 300*f*g**10*x**4 + 60*g**11*x**5) - 30*a*e**3*g**5*x**3/(60*f**5*g**6 + 300*f**4*g**7*x + 600*f**
3*g**8*x**2 + 600*f**2*g**9*x**3 + 300*f*g**10*x**4 + 60*g**11*x**5) - 6*c*d**4*f*g**4/(60*f**5*g**6 + 300*f**
4*g**7*x + 600*f**3*g**8*x**2 + 600*f**2*g**9*x**3 + 300*f*g**10*x**4 + 60*g**11*x**5) - 30*c*d**4*g**5*x/(60*
f**5*g**6 + 300*f**4*g**7*x + 600*f**3*g**8*x**2 + 600*f**2*g**9*x**3 + 300*f*g**10*x**4 + 60*g**11*x**5) - 14
*c*d**3*e*f**2*g**3/(60*f**5*g**6 + 300*f**4*g**7*x + 600*f**3*g**8*x**2 + 600*f**2*g**9*x**3 + 300*f*g**10*x*
*4 + 60*g**11*x**5) - 70*c*d**3*e*f*g**4*x/(60*f**5*g**6 + 300*f**4*g**7*x + 600*f**3*g**8*x**2 + 600*f**2*g**
9*x**3 + 300*f*g**10*x**4 + 60*g**11*x**5) - 140*c*d**3*e*g**5*x**2/(60*f**5*g**6 + 300*f**4*g**7*x + 600*f**3
*g**8*x**2 + 600*f**2*g**9*x**3 + 300*f*g**10*x**4 + 60*g**11*x**5) - 27*c*d**2*e**2*f**3*g**2/(60*f**5*g**6 +
300*f**4*g**7*x + 600*f**3*g**8*x**2 + 600*f**2*g**9*x**3 + 300*f*g**10*x**4 + 60*g**11*x**5) - 135*c*d**2*e*
*2*f**2*g**3*x/(60*f**5*g**6 + 300*f**4*g**7*x + 600*f**3*g**8*x**2 + 600*f**2*g**9*x**3 + 300*f*g**10*x**4 +
60*g**11*x**5) - 270*c*d**2*e**2*f*g**4*x**2/(60*f**5*g**6 + 300*f**4*g**7*x + 600*f**3*g**8*x**2 + 600*f**2*g
**9*x**3 + 300*f*g**10*x**4 + 60*g**11*x**5) - 270*c*d**2*e**2*g**5*x**3/(60*f**5*g**6 + 300*f**4*g**7*x + 600
*f**3*g**8*x**2 + 600*f**2*g**9*x**3 + 300*f*g**10*x**4 + 60*g**11*x**5) - 60*c*d*e**3*f**4*g/(60*f**5*g**6 +
300*f**4*g**7*x + 600*f**3*g**8*x**2 + 600*f**2*g**9*x**3 + 300*f*g**10*x**4 + 60*g**11*x**5) - 300*c*d*e**3*f
**3*g**2*x/(60*f**5*g**6 + 300*f**4*g**7*x + 600*f**3*g**8*x**2 + 600*f**2*g**9*x**3 + 300*f*g**10*x**4 + 60*g
**11*x**5) - 600*c*d*e**3*f**2*g**3*x**2/(60*f**5*g**6 + 300*f**4*g**7*x + 600*f**3*g**8*x**2 + 600*f**2*g**9*
x**3 + 300*f*g**10*x**4 + 60*g**11*x**5) - 600*c*d*e**3*f*g**4*x**3/(60*f**5*g**6 + 300*f**4*g**7*x + 600*f**3
*g**8*x**2 + 600*f**2*g**9*x**3 + 300*f*g**10*x**4 + 60*g**11*x**5) - 300*c*d*e**3*g**5*x**4/(60*f**5*g**6 + 3
00*f**4*g**7*x + 600*f**3*g**8*x**2 + 600*f**2*g**9*x**3 + 300*f*g**10*x**4 + 60*g**11*x**5) + 60*c*e**4*f**5*
log(f/g + x)/(60*f**5*g**6 + 300*f**4*g**7*x + 600*f**3*g**8*x**2 + 600*f**2*g**9*x**3 + 300*f*g**10*x**4 + 60
*g**11*x**5) + 137*c*e**4*f**5/(60*f**5*g**6 + 300*f**4*g**7*x + 600*f**3*g**8*x**2 + 600*f**2*g**9*x**3 + 300
*f*g**10*x**4 + 60*g**11*x**5) + 300*c*e**4*f**4*g*x*log(f/g + x)/(60*f**5*g**6 + 300*f**4*g**7*x + 600*f**3*g
**8*x**2 + 600*f**2*g**9*x**3 + 300*f*g**10*x**4 + 60*g**11*x**5) + 625*c*e**4*f**4*g*x/(60*f**5*g**6 + 300*f*
*4*g**7*x + 600*f**3*g**8*x**2 + 600*f**2*g**9*x**3 + 300*f*g**10*x**4 + 60*g**11*x**5) + 600*c*e**4*f**3*g**2
*x**2*log(f/g + x)/(60*f**5*g**6 + 300*f**4*g**7*x + 600*f**3*g**8*x**2 + 600*f**2*g**9*x**3 + 300*f*g**10*x**
4 + 60*g**11*x**5) + 1100*c*e**4*f**3*g**2*x**2/(60*f**5*g**6 + 300*f**4*g**7*x + 600*f**3*g**8*x**2 + 600*f**
2*g**9*x**3 + 300*f*g**10*x**4 + 60*g**11*x**5) + 600*c*e**4*f**2*g**3*x**3*log(f/g + x)/(60*f**5*g**6 + 300*f
**4*g**7*x + 600*f**3*g**8*x**2 + 600*f**2*g**9*x**3 + 300*f*g**10*x**4 + 60*g**11*x**5) + 900*c*e**4*f**2*g**
3*x**3/(60*f**5*g**6 + 300*f**4*g**7*x + 600*f**3*g**8*x**2 + 600*f**2*g**9*x**3 + 300*f*g**10*x**4 + 60*g**11
*x**5) + 300*c*e**4*f*g**4*x**4*log(f/g + x)/(60*f**5*g**6 + 300*f**4*g**7*x + 600*f**3*g**8*x**2 + 600*f**2*g
**9*x**3 + 300*f*g**10*x**4 + 60*g**11*x**5) + 300*c*e**4*f*g**4*x**4/(60*f**5*g**6 + 300*f**4*g**7*x + 600*f*
*3*g**8*x**2 + 600*f**2*g**9*x**3 + 300*f*g**10*x**4 + 60*g**11*x**5) + 60*c*e**4*g**5*x**5*log(f/g + x)/(60*f
**5*g**6 + 300*f**4*g**7*x + 600*f**3*g**8*x**2 + 600*f**2*g**9*x**3 + 300*f*g**10*x**4 + 60*g**11*x**5), Eq(n
, -6)), (-3*a*d**3*g**5/(12*f**4*g**6 + 48*f**3...
________________________________________________________________________________________
Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 3760 vs.
\(2 (282) = 564\).
time = 4.41, size = 3760, normalized size = 13.67 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((e*x+d)^3*(g*x+f)^n*(c*e*x^2+2*c*d*x+a),x, algorithm="giac")
[Out]
((g*x + f)^n*c*g^6*n^5*x^6*e^4 + 5*(g*x + f)^n*c*d*g^6*n^5*x^5*e^3 + 9*(g*x + f)^n*c*d^2*g^6*n^5*x^4*e^2 + 7*(
g*x + f)^n*c*d^3*g^6*n^5*x^3*e + 2*(g*x + f)^n*c*d^4*g^6*n^5*x^2 + (g*x + f)^n*c*f*g^5*n^5*x^5*e^4 + 15*(g*x +
f)^n*c*g^6*n^4*x^6*e^4 + 5*(g*x + f)^n*c*d*f*g^5*n^5*x^4*e^3 + 80*(g*x + f)^n*c*d*g^6*n^4*x^5*e^3 + 9*(g*x +
f)^n*c*d^2*f*g^5*n^5*x^3*e^2 + 153*(g*x + f)^n*c*d^2*g^6*n^4*x^4*e^2 + 7*(g*x + f)^n*c*d^3*f*g^5*n^5*x^2*e + 1
26*(g*x + f)^n*c*d^3*g^6*n^4*x^3*e + 2*(g*x + f)^n*c*d^4*f*g^5*n^5*x + 38*(g*x + f)^n*c*d^4*g^6*n^4*x^2 + 10*(
g*x + f)^n*c*f*g^5*n^4*x^5*e^4 + 85*(g*x + f)^n*c*g^6*n^3*x^6*e^4 + 60*(g*x + f)^n*c*d*f*g^5*n^4*x^4*e^3 + (g*
x + f)^n*a*g^6*n^5*x^4*e^3 + 475*(g*x + f)^n*c*d*g^6*n^3*x^5*e^3 + 126*(g*x + f)^n*c*d^2*f*g^5*n^4*x^3*e^2 + 3
*(g*x + f)^n*a*d*g^6*n^5*x^3*e^2 + 963*(g*x + f)^n*c*d^2*g^6*n^3*x^4*e^2 + 112*(g*x + f)^n*c*d^3*f*g^5*n^4*x^2
*e + 3*(g*x + f)^n*a*d^2*g^6*n^5*x^2*e + 847*(g*x + f)^n*c*d^3*g^6*n^3*x^3*e + 36*(g*x + f)^n*c*d^4*f*g^5*n^4*
x + (g*x + f)^n*a*d^3*g^6*n^5*x + 274*(g*x + f)^n*c*d^4*g^6*n^3*x^2 - 5*(g*x + f)^n*c*f^2*g^4*n^4*x^4*e^4 + 35
*(g*x + f)^n*c*f*g^5*n^3*x^5*e^4 + 225*(g*x + f)^n*c*g^6*n^2*x^6*e^4 - 20*(g*x + f)^n*c*d*f^2*g^4*n^4*x^3*e^3
+ (g*x + f)^n*a*f*g^5*n^5*x^3*e^3 + 235*(g*x + f)^n*c*d*f*g^5*n^3*x^4*e^3 + 17*(g*x + f)^n*a*g^6*n^4*x^4*e^3 +
1300*(g*x + f)^n*c*d*g^6*n^2*x^5*e^3 - 27*(g*x + f)^n*c*d^2*f^2*g^4*n^4*x^2*e^2 + 3*(g*x + f)^n*a*d*f*g^5*n^5
*x^2*e^2 + 585*(g*x + f)^n*c*d^2*f*g^5*n^3*x^3*e^2 + 54*(g*x + f)^n*a*d*g^6*n^4*x^3*e^2 + 2763*(g*x + f)^n*c*d
^2*g^6*n^2*x^4*e^2 - 14*(g*x + f)^n*c*d^3*f^2*g^4*n^4*x*e + 3*(g*x + f)^n*a*d^2*f*g^5*n^5*x*e + 623*(g*x + f)^
n*c*d^3*f*g^5*n^3*x^2*e + 57*(g*x + f)^n*a*d^2*g^6*n^4*x^2*e + 2604*(g*x + f)^n*c*d^3*g^6*n^2*x^3*e - 2*(g*x +
f)^n*c*d^4*f^2*g^4*n^4 + (g*x + f)^n*a*d^3*f*g^5*n^5 + 238*(g*x + f)^n*c*d^4*f*g^5*n^3*x + 20*(g*x + f)^n*a*d
^3*g^6*n^4*x + 922*(g*x + f)^n*c*d^4*g^6*n^2*x^2 - 30*(g*x + f)^n*c*f^2*g^4*n^3*x^4*e^4 + 50*(g*x + f)^n*c*f*g
^5*n^2*x^5*e^4 + 274*(g*x + f)^n*c*g^6*n*x^6*e^4 - 180*(g*x + f)^n*c*d*f^2*g^4*n^3*x^3*e^3 + 14*(g*x + f)^n*a*
f*g^5*n^4*x^3*e^3 + 360*(g*x + f)^n*c*d*f*g^5*n^2*x^4*e^3 + 107*(g*x + f)^n*a*g^6*n^3*x^4*e^3 + 1620*(g*x + f)
^n*c*d*g^6*n*x^5*e^3 - 324*(g*x + f)^n*c*d^2*f^2*g^4*n^3*x^2*e^2 + 48*(g*x + f)^n*a*d*f*g^5*n^4*x^2*e^2 + 1008
*(g*x + f)^n*c*d^2*f*g^5*n^2*x^3*e^2 + 363*(g*x + f)^n*a*d*g^6*n^3*x^3*e^2 + 3564*(g*x + f)^n*c*d^2*g^6*n*x^4*
e^2 - 210*(g*x + f)^n*c*d^3*f^2*g^4*n^3*x*e + 54*(g*x + f)^n*a*d^2*f*g^5*n^4*x*e + 1358*(g*x + f)^n*c*d^3*f*g^
5*n^2*x^2*e + 411*(g*x + f)^n*a*d^2*g^6*n^3*x^2*e + 3556*(g*x + f)^n*c*d^3*g^6*n*x^3*e - 36*(g*x + f)^n*c*d^4*
f^2*g^4*n^3 + 20*(g*x + f)^n*a*d^3*f*g^5*n^4 + 684*(g*x + f)^n*c*d^4*f*g^5*n^2*x + 155*(g*x + f)^n*a*d^3*g^6*n
^3*x + 1404*(g*x + f)^n*c*d^4*g^6*n*x^2 + 20*(g*x + f)^n*c*f^3*g^3*n^3*x^3*e^4 - 55*(g*x + f)^n*c*f^2*g^4*n^2*
x^4*e^4 + 24*(g*x + f)^n*c*f*g^5*n*x^5*e^4 + 120*(g*x + f)^n*c*g^6*x^6*e^4 + 60*(g*x + f)^n*c*d*f^3*g^3*n^3*x^
2*e^3 - 3*(g*x + f)^n*a*f^2*g^4*n^4*x^2*e^3 - 400*(g*x + f)^n*c*d*f^2*g^4*n^2*x^3*e^3 + 65*(g*x + f)^n*a*f*g^5
*n^3*x^3*e^3 + 180*(g*x + f)^n*c*d*f*g^5*n*x^4*e^3 + 307*(g*x + f)^n*a*g^6*n^2*x^4*e^3 + 720*(g*x + f)^n*c*d*g
^6*x^5*e^3 + 54*(g*x + f)^n*c*d^2*f^3*g^3*n^3*x*e^2 - 6*(g*x + f)^n*a*d*f^2*g^4*n^4*x*e^2 - 1107*(g*x + f)^n*c
*d^2*f^2*g^4*n^2*x^2*e^2 + 267*(g*x + f)^n*a*d*f*g^5*n^3*x^2*e^2 + 540*(g*x + f)^n*c*d^2*f*g^5*n*x^3*e^2 + 111
6*(g*x + f)^n*a*d*g^6*n^2*x^3*e^2 + 1620*(g*x + f)^n*c*d^2*g^6*x^4*e^2 + 14*(g*x + f)^n*c*d^3*f^3*g^3*n^3*e -
3*(g*x + f)^n*a*d^2*f^2*g^4*n^4*e - 1036*(g*x + f)^n*c*d^3*f^2*g^4*n^2*x*e + 357*(g*x + f)^n*a*d^2*f*g^5*n^3*x
*e + 840*(g*x + f)^n*c*d^3*f*g^5*n*x^2*e + 1383*(g*x + f)^n*a*d^2*g^6*n^2*x^2*e + 1680*(g*x + f)^n*c*d^3*g^6*x
^3*e - 238*(g*x + f)^n*c*d^4*f^2*g^4*n^2 + 155*(g*x + f)^n*a*d^3*f*g^5*n^3 + 720*(g*x + f)^n*c*d^4*f*g^5*n*x +
580*(g*x + f)^n*a*d^3*g^6*n^2*x + 720*(g*x + f)^n*c*d^4*g^6*x^2 + 60*(g*x + f)^n*c*f^3*g^3*n^2*x^3*e^4 - 30*(
g*x + f)^n*c*f^2*g^4*n*x^4*e^4 + 420*(g*x + f)^n*c*d*f^3*g^3*n^2*x^2*e^3 - 36*(g*x + f)^n*a*f^2*g^4*n^3*x^2*e^
3 - 240*(g*x + f)^n*c*d*f^2*g^4*n*x^3*e^3 + 112*(g*x + f)^n*a*f*g^5*n^2*x^3*e^3 + 396*(g*x + f)^n*a*g^6*n*x^4*
e^3 + 594*(g*x + f)^n*c*d^2*f^3*g^3*n^2*x*e^2 - 90*(g*x + f)^n*a*d*f^2*g^4*n^3*x*e^2 - 810*(g*x + f)^n*c*d^2*f
^2*g^4*n*x^2*e^2 + 582*(g*x + f)^n*a*d*f*g^5*n^2*x^2*e^2 + 1524*(g*x + f)^n*a*d*g^6*n*x^3*e^2 + 210*(g*x + f)^
n*c*d^3*f^3*g^3*n^2*e - 54*(g*x + f)^n*a*d^2*f^2*g^4*n^3*e - 1680*(g*x + f)^n*c*d^3*f^2*g^4*n*x*e + 1026*(g*x
+ f)^n*a*d^2*f*g^5*n^2*x*e + 2106*(g*x + f)^n*a*d^2*g^6*n*x^2*e - 684*(g*x + f)^n*c*d^4*f^2*g^4*n + 580*(g*x +
f)^n*a*d^3*f*g^5*n^2 + 1044*(g*x + f)^n*a*d^3*g^6*n*x - 60*(g*x + f)^n*c*f^4*g^2*n^2*x^2*e^4 + 40*(g*x + f)^n
*c*f^3*g^3*n*x^3*e^4 - 120*(g*x + f)^n*c*d*f^4*g^2*n^2*x*e^3 + 6*(g*x + f)^n*a*f^3*g^3*n^3*x*e^3 + 360*(g*x +
f)^n*c*d*f^3*g^3*n*x^2*e^3 - 123*(g*x + f)^n*a*f^2*g^4*n^2*x^2*e^3 + 60*(g*x + f)^n*a*f*g^5*n*x^3*e^3 + 180*(g
*x + f)^n*a*g^6*x^4*e^3 - 54*(g*x + f)^n*c*d^2*...
________________________________________________________________________________________
Mupad [B]
time = 3.90, size = 1943, normalized size = 7.07 \begin {gather*} \frac {x\,{\left (f+g\,x\right )}^n\,\left (2\,c\,d^4\,f\,g^5\,n^5+36\,c\,d^4\,f\,g^5\,n^4+238\,c\,d^4\,f\,g^5\,n^3+684\,c\,d^4\,f\,g^5\,n^2+720\,c\,d^4\,f\,g^5\,n-14\,c\,d^3\,e\,f^2\,g^4\,n^4-210\,c\,d^3\,e\,f^2\,g^4\,n^3-1036\,c\,d^3\,e\,f^2\,g^4\,n^2-1680\,c\,d^3\,e\,f^2\,g^4\,n+a\,d^3\,g^6\,n^5+20\,a\,d^3\,g^6\,n^4+155\,a\,d^3\,g^6\,n^3+580\,a\,d^3\,g^6\,n^2+1044\,a\,d^3\,g^6\,n+720\,a\,d^3\,g^6+54\,c\,d^2\,e^2\,f^3\,g^3\,n^3+594\,c\,d^2\,e^2\,f^3\,g^3\,n^2+1620\,c\,d^2\,e^2\,f^3\,g^3\,n+3\,a\,d^2\,e\,f\,g^5\,n^5+54\,a\,d^2\,e\,f\,g^5\,n^4+357\,a\,d^2\,e\,f\,g^5\,n^3+1026\,a\,d^2\,e\,f\,g^5\,n^2+1080\,a\,d^2\,e\,f\,g^5\,n-120\,c\,d\,e^3\,f^4\,g^2\,n^2-720\,c\,d\,e^3\,f^4\,g^2\,n-6\,a\,d\,e^2\,f^2\,g^4\,n^4-90\,a\,d\,e^2\,f^2\,g^4\,n^3-444\,a\,d\,e^2\,f^2\,g^4\,n^2-720\,a\,d\,e^2\,f^2\,g^4\,n+120\,c\,e^4\,f^5\,g\,n+6\,a\,e^3\,f^3\,g^3\,n^3+66\,a\,e^3\,f^3\,g^3\,n^2+180\,a\,e^3\,f^3\,g^3\,n\right )}{g^6\,\left (n^6+21\,n^5+175\,n^4+735\,n^3+1624\,n^2+1764\,n+720\right )}-\frac {{\left (f+g\,x\right )}^n\,\left (2\,c\,d^4\,f^2\,g^4\,n^4+36\,c\,d^4\,f^2\,g^4\,n^3+238\,c\,d^4\,f^2\,g^4\,n^2+684\,c\,d^4\,f^2\,g^4\,n+720\,c\,d^4\,f^2\,g^4-14\,c\,d^3\,e\,f^3\,g^3\,n^3-210\,c\,d^3\,e\,f^3\,g^3\,n^2-1036\,c\,d^3\,e\,f^3\,g^3\,n-1680\,c\,d^3\,e\,f^3\,g^3-a\,d^3\,f\,g^5\,n^5-20\,a\,d^3\,f\,g^5\,n^4-155\,a\,d^3\,f\,g^5\,n^3-580\,a\,d^3\,f\,g^5\,n^2-1044\,a\,d^3\,f\,g^5\,n-720\,a\,d^3\,f\,g^5+54\,c\,d^2\,e^2\,f^4\,g^2\,n^2+594\,c\,d^2\,e^2\,f^4\,g^2\,n+1620\,c\,d^2\,e^2\,f^4\,g^2+3\,a\,d^2\,e\,f^2\,g^4\,n^4+54\,a\,d^2\,e\,f^2\,g^4\,n^3+357\,a\,d^2\,e\,f^2\,g^4\,n^2+1026\,a\,d^2\,e\,f^2\,g^4\,n+1080\,a\,d^2\,e\,f^2\,g^4-120\,c\,d\,e^3\,f^5\,g\,n-720\,c\,d\,e^3\,f^5\,g-6\,a\,d\,e^2\,f^3\,g^3\,n^3-90\,a\,d\,e^2\,f^3\,g^3\,n^2-444\,a\,d\,e^2\,f^3\,g^3\,n-720\,a\,d\,e^2\,f^3\,g^3+120\,c\,e^4\,f^6+6\,a\,e^3\,f^4\,g^2\,n^2+66\,a\,e^3\,f^4\,g^2\,n+180\,a\,e^3\,f^4\,g^2\right )}{g^6\,\left (n^6+21\,n^5+175\,n^4+735\,n^3+1624\,n^2+1764\,n+720\right )}+\frac {c\,e^4\,x^6\,{\left (f+g\,x\right )}^n\,\left (n^5+15\,n^4+85\,n^3+225\,n^2+274\,n+120\right )}{n^6+21\,n^5+175\,n^4+735\,n^3+1624\,n^2+1764\,n+720}+\frac {x^2\,{\left (f+g\,x\right )}^n\,\left (n+1\right )\,\left (2\,c\,d^4\,g^4\,n^4+36\,c\,d^4\,g^4\,n^3+238\,c\,d^4\,g^4\,n^2+684\,c\,d^4\,g^4\,n+720\,c\,d^4\,g^4+7\,c\,d^3\,e\,f\,g^3\,n^4+105\,c\,d^3\,e\,f\,g^3\,n^3+518\,c\,d^3\,e\,f\,g^3\,n^2+840\,c\,d^3\,e\,f\,g^3\,n-27\,c\,d^2\,e^2\,f^2\,g^2\,n^3-297\,c\,d^2\,e^2\,f^2\,g^2\,n^2-810\,c\,d^2\,e^2\,f^2\,g^2\,n+3\,a\,d^2\,e\,g^4\,n^4+54\,a\,d^2\,e\,g^4\,n^3+357\,a\,d^2\,e\,g^4\,n^2+1026\,a\,d^2\,e\,g^4\,n+1080\,a\,d^2\,e\,g^4+60\,c\,d\,e^3\,f^3\,g\,n^2+360\,c\,d\,e^3\,f^3\,g\,n+3\,a\,d\,e^2\,f\,g^3\,n^4+45\,a\,d\,e^2\,f\,g^3\,n^3+222\,a\,d\,e^2\,f\,g^3\,n^2+360\,a\,d\,e^2\,f\,g^3\,n-60\,c\,e^4\,f^4\,n-3\,a\,e^3\,f^2\,g^2\,n^3-33\,a\,e^3\,f^2\,g^2\,n^2-90\,a\,e^3\,f^2\,g^2\,n\right )}{g^4\,\left (n^6+21\,n^5+175\,n^4+735\,n^3+1624\,n^2+1764\,n+720\right )}+\frac {e\,x^3\,{\left (f+g\,x\right )}^n\,\left (n^2+3\,n+2\right )\,\left (7\,c\,d^3\,g^3\,n^3+105\,c\,d^3\,g^3\,n^2+518\,c\,d^3\,g^3\,n+840\,c\,d^3\,g^3+9\,c\,d^2\,e\,f\,g^2\,n^3+99\,c\,d^2\,e\,f\,g^2\,n^2+270\,c\,d^2\,e\,f\,g^2\,n-20\,c\,d\,e^2\,f^2\,g\,n^2-120\,c\,d\,e^2\,f^2\,g\,n+3\,a\,d\,e\,g^3\,n^3+45\,a\,d\,e\,g^3\,n^2+222\,a\,d\,e\,g^3\,n+360\,a\,d\,e\,g^3+20\,c\,e^3\,f^3\,n+a\,e^2\,f\,g^2\,n^3+11\,a\,e^2\,f\,g^2\,n^2+30\,a\,e^2\,f\,g^2\,n\right )}{g^3\,\left (n^6+21\,n^5+175\,n^4+735\,n^3+1624\,n^2+1764\,n+720\right )}+\frac {e^2\,x^4\,{\left (f+g\,x\right )}^n\,\left (n^3+6\,n^2+11\,n+6\right )\,\left (9\,c\,d^2\,g^2\,n^2+99\,c\,d^2\,g^2\,n+270\,c\,d^2\,g^2+5\,c\,d\,e\,f\,g\,n^2+30\,c\,d\,e\,f\,g\,n-5\,c\,e^2\,f^2\,n+a\,e\,g^2\,n^2+11\,a\,e\,g^2\,n+30\,a\,e\,g^2\right )}{g^2\,\left (n^6+21\,n^5+175\,n^4+735\,n^3+1624\,n^2+1764\,n+720\right )}+\frac {c\,e^3\,x^5\,{\left (f+g\,x\right )}^n\,\left (30\,d\,g+5\,d\,g\,n+e\,f\,n\right )\,\left (n^4+10\,n^3+35\,n^2+50\,n+24\right )}{g\,\left (n^6+21\,n^5+175\,n^4+735\,n^3+1624\,n^2+1764\,n+720\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((f + g*x)^n*(d + e*x)^3*(a + 2*c*d*x + c*e*x^2),x)
[Out]
(x*(f + g*x)^n*(720*a*d^3*g^6 + 580*a*d^3*g^6*n^2 + 155*a*d^3*g^6*n^3 + 20*a*d^3*g^6*n^4 + a*d^3*g^6*n^5 + 104
4*a*d^3*g^6*n + 720*c*d^4*f*g^5*n + 120*c*e^4*f^5*g*n + 180*a*e^3*f^3*g^3*n + 684*c*d^4*f*g^5*n^2 + 238*c*d^4*
f*g^5*n^3 + 36*c*d^4*f*g^5*n^4 + 2*c*d^4*f*g^5*n^5 + 66*a*e^3*f^3*g^3*n^2 + 6*a*e^3*f^3*g^3*n^3 - 444*a*d*e^2*
f^2*g^4*n^2 - 90*a*d*e^2*f^2*g^4*n^3 - 6*a*d*e^2*f^2*g^4*n^4 + 1620*c*d^2*e^2*f^3*g^3*n - 120*c*d*e^3*f^4*g^2*
n^2 - 1036*c*d^3*e*f^2*g^4*n^2 - 210*c*d^3*e*f^2*g^4*n^3 - 14*c*d^3*e*f^2*g^4*n^4 + 1080*a*d^2*e*f*g^5*n + 594
*c*d^2*e^2*f^3*g^3*n^2 + 54*c*d^2*e^2*f^3*g^3*n^3 - 720*a*d*e^2*f^2*g^4*n + 1026*a*d^2*e*f*g^5*n^2 + 357*a*d^2
*e*f*g^5*n^3 + 54*a*d^2*e*f*g^5*n^4 + 3*a*d^2*e*f*g^5*n^5 - 720*c*d*e^3*f^4*g^2*n - 1680*c*d^3*e*f^2*g^4*n))/(
g^6*(1764*n + 1624*n^2 + 735*n^3 + 175*n^4 + 21*n^5 + n^6 + 720)) - ((f + g*x)^n*(120*c*e^4*f^6 + 180*a*e^3*f^
4*g^2 + 720*c*d^4*f^2*g^4 - 720*a*d^3*f*g^5 - 720*c*d*e^3*f^5*g - 1044*a*d^3*f*g^5*n - 720*a*d*e^2*f^3*g^3 + 1
080*a*d^2*e*f^2*g^4 - 1680*c*d^3*e*f^3*g^3 - 580*a*d^3*f*g^5*n^2 - 155*a*d^3*f*g^5*n^3 - 20*a*d^3*f*g^5*n^4 -
a*d^3*f*g^5*n^5 + 66*a*e^3*f^4*g^2*n + 684*c*d^4*f^2*g^4*n + 1620*c*d^2*e^2*f^4*g^2 + 6*a*e^3*f^4*g^2*n^2 + 23
8*c*d^4*f^2*g^4*n^2 + 36*c*d^4*f^2*g^4*n^3 + 2*c*d^4*f^2*g^4*n^4 - 90*a*d*e^2*f^3*g^3*n^2 + 357*a*d^2*e*f^2*g^
4*n^2 - 6*a*d*e^2*f^3*g^3*n^3 + 54*a*d^2*e*f^2*g^4*n^3 + 3*a*d^2*e*f^2*g^4*n^4 + 594*c*d^2*e^2*f^4*g^2*n - 210
*c*d^3*e*f^3*g^3*n^2 - 14*c*d^3*e*f^3*g^3*n^3 - 120*c*d*e^3*f^5*g*n + 54*c*d^2*e^2*f^4*g^2*n^2 - 444*a*d*e^2*f
^3*g^3*n + 1026*a*d^2*e*f^2*g^4*n - 1036*c*d^3*e*f^3*g^3*n))/(g^6*(1764*n + 1624*n^2 + 735*n^3 + 175*n^4 + 21*
n^5 + n^6 + 720)) + (c*e^4*x^6*(f + g*x)^n*(274*n + 225*n^2 + 85*n^3 + 15*n^4 + n^5 + 120))/(1764*n + 1624*n^2
+ 735*n^3 + 175*n^4 + 21*n^5 + n^6 + 720) + (x^2*(f + g*x)^n*(n + 1)*(720*c*d^4*g^4 + 238*c*d^4*g^4*n^2 + 36*
c*d^4*g^4*n^3 + 2*c*d^4*g^4*n^4 + 1080*a*d^2*e*g^4 + 684*c*d^4*g^4*n - 60*c*e^4*f^4*n + 1026*a*d^2*e*g^4*n + 3
57*a*d^2*e*g^4*n^2 + 54*a*d^2*e*g^4*n^3 + 3*a*d^2*e*g^4*n^4 - 90*a*e^3*f^2*g^2*n - 33*a*e^3*f^2*g^2*n^2 - 3*a*
e^3*f^2*g^2*n^3 - 810*c*d^2*e^2*f^2*g^2*n + 360*a*d*e^2*f*g^3*n + 360*c*d*e^3*f^3*g*n + 840*c*d^3*e*f*g^3*n -
297*c*d^2*e^2*f^2*g^2*n^2 - 27*c*d^2*e^2*f^2*g^2*n^3 + 222*a*d*e^2*f*g^3*n^2 + 45*a*d*e^2*f*g^3*n^3 + 3*a*d*e^
2*f*g^3*n^4 + 60*c*d*e^3*f^3*g*n^2 + 518*c*d^3*e*f*g^3*n^2 + 105*c*d^3*e*f*g^3*n^3 + 7*c*d^3*e*f*g^3*n^4))/(g^
4*(1764*n + 1624*n^2 + 735*n^3 + 175*n^4 + 21*n^5 + n^6 + 720)) + (e*x^3*(f + g*x)^n*(3*n + n^2 + 2)*(840*c*d^
3*g^3 + 105*c*d^3*g^3*n^2 + 7*c*d^3*g^3*n^3 + 360*a*d*e*g^3 + 518*c*d^3*g^3*n + 20*c*e^3*f^3*n + 45*a*d*e*g^3*
n^2 + 3*a*d*e*g^3*n^3 + 30*a*e^2*f*g^2*n + 11*a*e^2*f*g^2*n^2 + a*e^2*f*g^2*n^3 + 222*a*d*e*g^3*n - 120*c*d*e^
2*f^2*g*n + 270*c*d^2*e*f*g^2*n - 20*c*d*e^2*f^2*g*n^2 + 99*c*d^2*e*f*g^2*n^2 + 9*c*d^2*e*f*g^2*n^3))/(g^3*(17
64*n + 1624*n^2 + 735*n^3 + 175*n^4 + 21*n^5 + n^6 + 720)) + (e^2*x^4*(f + g*x)^n*(11*n + 6*n^2 + n^3 + 6)*(27
0*c*d^2*g^2 + 30*a*e*g^2 + 9*c*d^2*g^2*n^2 + 11*a*e*g^2*n + a*e*g^2*n^2 + 99*c*d^2*g^2*n - 5*c*e^2*f^2*n + 5*c
*d*e*f*g*n^2 + 30*c*d*e*f*g*n))/(g^2*(1764*n + 1624*n^2 + 735*n^3 + 175*n^4 + 21*n^5 + n^6 + 720)) + (c*e^3*x^
5*(f + g*x)^n*(30*d*g + 5*d*g*n + e*f*n)*(50*n + 35*n^2 + 10*n^3 + n^4 + 24))/(g*(1764*n + 1624*n^2 + 735*n^3
+ 175*n^4 + 21*n^5 + n^6 + 720))
________________________________________________________________________________________