3.6.53 \(\int (d+e x)^3 (f+g x)^n (a+2 c d x+c e x^2) \, dx\)
Optimal. Leaf size=275 \[ \frac {(e f-d g)^2 (f+g x)^{n+2} \left (3 a e g^2+c \left (2 d^2 g^2-10 d e f g+5 e^2 f^2\right )\right )}{g^6 (n+2)}-\frac {e (e f-d g) (f+g x)^{n+3} \left (3 a e g^2+c \left (7 d^2 g^2-20 d e f g+10 e^2 f^2\right )\right )}{g^6 (n+3)}+\frac {e^2 (f+g x)^{n+4} \left (a e g^2+c \left (9 d^2 g^2-20 d e f g+10 e^2 f^2\right )\right )}{g^6 (n+4)}-\frac {(e f-d g)^3 (f+g x)^{n+1} \left (a g^2+c f (e f-2 d g)\right )}{g^6 (n+1)}-\frac {5 c e^3 (e f-d g) (f+g x)^{n+5}}{g^6 (n+5)}+\frac {c e^4 (f+g x)^{n+6}}{g^6 (n+6)} \]
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Rubi [A] time = 0.26, 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 =
{947} \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 947
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*}
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Mathematica [A] time = 0.31, size = 249, normalized size = 0.91 \begin {gather*} \frac {(f+g x)^{n+1} \left (\frac {e^2 (f+g x)^3 \left (a e g^2+c \left (9 d^2 g^2-20 d e f g+10 e^2 f^2\right )\right )}{n+4}-\frac {e (f+g x)^2 (e f-d g) \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 )}{n+3}+\frac {(f+g x) (e f-d g)^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 )}{n+2}-\frac {(e f-d g)^3 \left (a g^2+c f (e f-2 d g)\right )}{n+1}-\frac {5 c e^3 (f+g x)^4 (e f-d g)}{n+5}+\frac {c e^4 (f+g x)^5}{n+6}\right )}{g^6} \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)*(-(((e*f - d*g)^3*(a*g^2 + c*f*(e*f - 2*d*g)))/(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) - (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)^2)/(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)^3)/(4
+ n) - (5*c*e^3*(e*f - d*g)*(f + g*x)^4)/(5 + n) + (c*e^4*(f + g*x)^5)/(6 + n)))/g^6
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IntegrateAlgebraic [F] time = 0.16, size = 0, normalized size = 0.00 \begin {gather*} \int (d+e x)^3 (f+g x)^n \left (a+2 c d x+c e x^2\right ) \, dx \end {gather*}
Verification is not applicable to the result.
[In]
IntegrateAlgebraic[(d + e*x)^3*(f + g*x)^n*(a + 2*c*d*x + c*e*x^2),x]
[Out]
Defer[IntegrateAlgebraic][(d + e*x)^3*(f + g*x)^n*(a + 2*c*d*x + c*e*x^2), x]
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fricas [B] time = 0.48, size = 2032, normalized size = 7.39
result too large to display
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 - 120*c*e^4*f^6 + 720*c*d*e^3*f^5*g + 720*a*d^3*f*g^5 - 180*(9*c*d^2*e^2 + a*e^3)*f^4*g^2 + 2
40*(7*c*d^3*e + 3*a*d*e^2)*f^3*g^3 - 360*(2*c*d^4 + 3*a*d^2*e)*f^2*g^4 + (c*e^4*g^6*n^5 + 15*c*e^4*g^6*n^4 + 8
5*c*e^4*g^6*n^3 + 225*c*e^4*g^6*n^2 + 274*c*e^4*g^6*n + 120*c*e^4*g^6)*x^6 + (720*c*d*e^3*g^6 + (c*e^4*f*g^5 +
5*c*d*e^3*g^6)*n^5 + 10*(c*e^4*f*g^5 + 8*c*d*e^3*g^6)*n^4 + 5*(7*c*e^4*f*g^5 + 95*c*d*e^3*g^6)*n^3 + 50*(c*e^
4*f*g^5 + 26*c*d*e^3*g^6)*n^2 + 12*(2*c*e^4*f*g^5 + 135*c*d*e^3*g^6)*n)*x^5 + (20*a*d^3*f*g^5 - (2*c*d^4 + 3*a
*d^2*e)*f^2*g^4)*n^4 + (180*(9*c*d^2*e^2 + a*e^3)*g^6 + (5*c*d*e^3*f*g^5 + (9*c*d^2*e^2 + a*e^3)*g^6)*n^5 - (5
*c*e^4*f^2*g^4 - 60*c*d*e^3*f*g^5 - 17*(9*c*d^2*e^2 + a*e^3)*g^6)*n^4 - (30*c*e^4*f^2*g^4 - 235*c*d*e^3*f*g^5
- 107*(9*c*d^2*e^2 + a*e^3)*g^6)*n^3 - (55*c*e^4*f^2*g^4 - 360*c*d*e^3*f*g^5 - 307*(9*c*d^2*e^2 + a*e^3)*g^6)*
n^2 - 6*(5*c*e^4*f^2*g^4 - 30*c*d*e^3*f*g^5 - 66*(9*c*d^2*e^2 + a*e^3)*g^6)*n)*x^4 + (155*a*d^3*f*g^5 + 2*(7*c
*d^3*e + 3*a*d*e^2)*f^3*g^3 - 18*(2*c*d^4 + 3*a*d^2*e)*f^2*g^4)*n^3 + (240*(7*c*d^3*e + 3*a*d*e^2)*g^6 + ((9*c
*d^2*e^2 + a*e^3)*f*g^5 + (7*c*d^3*e + 3*a*d*e^2)*g^6)*n^5 - 2*(10*c*d*e^3*f^2*g^4 - 7*(9*c*d^2*e^2 + a*e^3)*f
*g^5 - 9*(7*c*d^3*e + 3*a*d*e^2)*g^6)*n^4 + (20*c*e^4*f^3*g^3 - 180*c*d*e^3*f^2*g^4 + 65*(9*c*d^2*e^2 + a*e^3)
*f*g^5 + 121*(7*c*d^3*e + 3*a*d*e^2)*g^6)*n^3 + 4*(15*c*e^4*f^3*g^3 - 100*c*d*e^3*f^2*g^4 + 28*(9*c*d^2*e^2 +
a*e^3)*f*g^5 + 93*(7*c*d^3*e + 3*a*d*e^2)*g^6)*n^2 + 4*(10*c*e^4*f^3*g^3 - 60*c*d*e^3*f^2*g^4 + 15*(9*c*d^2*e^
2 + a*e^3)*f*g^5 + 127*(7*c*d^3*e + 3*a*d*e^2)*g^6)*n)*x^3 + (580*a*d^3*f*g^5 - 6*(9*c*d^2*e^2 + a*e^3)*f^4*g^
2 + 30*(7*c*d^3*e + 3*a*d*e^2)*f^3*g^3 - 119*(2*c*d^4 + 3*a*d^2*e)*f^2*g^4)*n^2 + (360*(2*c*d^4 + 3*a*d^2*e)*g
^6 + ((7*c*d^3*e + 3*a*d*e^2)*f*g^5 + (2*c*d^4 + 3*a*d^2*e)*g^6)*n^5 - (3*(9*c*d^2*e^2 + a*e^3)*f^2*g^4 - 16*(
7*c*d^3*e + 3*a*d*e^2)*f*g^5 - 19*(2*c*d^4 + 3*a*d^2*e)*g^6)*n^4 + (60*c*d*e^3*f^3*g^3 - 36*(9*c*d^2*e^2 + a*e
^3)*f^2*g^4 + 89*(7*c*d^3*e + 3*a*d*e^2)*f*g^5 + 137*(2*c*d^4 + 3*a*d^2*e)*g^6)*n^3 - (60*c*e^4*f^4*g^2 - 420*
c*d*e^3*f^3*g^3 + 123*(9*c*d^2*e^2 + a*e^3)*f^2*g^4 - 194*(7*c*d^3*e + 3*a*d*e^2)*f*g^5 - 461*(2*c*d^4 + 3*a*d
^2*e)*g^6)*n^2 - 6*(10*c*e^4*f^4*g^2 - 60*c*d*e^3*f^3*g^3 + 15*(9*c*d^2*e^2 + a*e^3)*f^2*g^4 - 20*(7*c*d^3*e +
3*a*d*e^2)*f*g^5 - 117*(2*c*d^4 + 3*a*d^2*e)*g^6)*n)*x^2 + 2*(60*c*d*e^3*f^5*g + 522*a*d^3*f*g^5 - 33*(9*c*d^
2*e^2 + a*e^3)*f^4*g^2 + 74*(7*c*d^3*e + 3*a*d*e^2)*f^3*g^3 - 171*(2*c*d^4 + 3*a*d^2*e)*f^2*g^4)*n + (720*a*d^
3*g^6 + (a*d^3*g^6 + (2*c*d^4 + 3*a*d^2*e)*f*g^5)*n^5 + 2*(10*a*d^3*g^6 - (7*c*d^3*e + 3*a*d*e^2)*f^2*g^4 + 9*
(2*c*d^4 + 3*a*d^2*e)*f*g^5)*n^4 + (155*a*d^3*g^6 + 6*(9*c*d^2*e^2 + a*e^3)*f^3*g^3 - 30*(7*c*d^3*e + 3*a*d*e^
2)*f^2*g^4 + 119*(2*c*d^4 + 3*a*d^2*e)*f*g^5)*n^3 - 2*(60*c*d*e^3*f^4*g^2 - 290*a*d^3*g^6 - 33*(9*c*d^2*e^2 +
a*e^3)*f^3*g^3 + 74*(7*c*d^3*e + 3*a*d*e^2)*f^2*g^4 - 171*(2*c*d^4 + 3*a*d^2*e)*f*g^5)*n^2 + 12*(10*c*e^4*f^5*
g - 60*c*d*e^3*f^4*g^2 + 87*a*d^3*g^6 + 15*(9*c*d^2*e^2 + a*e^3)*f^3*g^3 - 20*(7*c*d^3*e + 3*a*d*e^2)*f^2*g^4
+ 30*(2*c*d^4 + 3*a*d^2*e)*f*g^5)*n)*x)*(g*x + f)^n/(g^6*n^6 + 21*g^6*n^5 + 175*g^6*n^4 + 735*g^6*n^3 + 1624*g
^6*n^2 + 1764*g^6*n + 720*g^6)
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giac [B] time = 0.59, size = 3760, normalized size = 13.67
result too large to display
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*f^4*g^2*n^2*e^2 + 6*(g*x + f)^n*a*d*f^3*g^3*n^3*e^2 + 1620*(g*x
+ f)^n*c*d^2*f^3*g^3*n*x*e^2 - 444*(g*x + f)^n*a*d*f^2*g^4*n^2*x*e^2 + 360*(g*x + f)^n*a*d*f*g^5*n*x^2*e^2 +
720*(g*x + f)^n*a*d*g^6*x^3*e^2 + 1036*(g*x + f)^n*c*d^3*f^3*g^3*n*e - 357*(g*x + f)^n*a*d^2*f^2*g^4*n^2*e + 1
080*(g*x + f)^n*a*d^2*f*g^5*n*x*e + 1080*(g*x + f)^n*a*d^2*g^6*x^2*e - 720*(g*x + f)^n*c*d^4*f^2*g^4 + 1044*(g
*x + f)^n*a*d^3*f*g^5*n + 720*(g*x + f)^n*a*d^3*g^6*x - 60*(g*x + f)^n*c*f^4*g^2*n*x^2*e^4 - 720*(g*x + f)^n*c
*d*f^4*g^2*n*x*e^3 + 66*(g*x + f)^n*a*f^3*g^3*n^2*x*e^3 - 90*(g*x + f)^n*a*f^2*g^4*n*x^2*e^3 - 594*(g*x + f)^n
*c*d^2*f^4*g^2*n*e^2 + 90*(g*x + f)^n*a*d*f^3*g^3*n^2*e^2 - 720*(g*x + f)^n*a*d*f^2*g^4*n*x*e^2 + 1680*(g*x +
f)^n*c*d^3*f^3*g^3*e - 1026*(g*x + f)^n*a*d^2*f^2*g^4*n*e + 720*(g*x + f)^n*a*d^3*f*g^5 + 120*(g*x + f)^n*c*f^
5*g*n*x*e^4 + 120*(g*x + f)^n*c*d*f^5*g*n*e^3 - 6*(g*x + f)^n*a*f^4*g^2*n^2*e^3 + 180*(g*x + f)^n*a*f^3*g^3*n*
x*e^3 - 1620*(g*x + f)^n*c*d^2*f^4*g^2*e^2 + 444*(g*x + f)^n*a*d*f^3*g^3*n*e^2 - 1080*(g*x + f)^n*a*d^2*f^2*g^
4*e + 720*(g*x + f)^n*c*d*f^5*g*e^3 - 66*(g*x + f)^n*a*f^4*g^2*n*e^3 + 720*(g*x + f)^n*a*d*f^3*g^3*e^2 - 120*(
g*x + f)^n*c*f^6*e^4 - 180*(g*x + f)^n*a*f^4*g^2*e^3)/(g^6*n^6 + 21*g^6*n^5 + 175*g^6*n^4 + 735*g^6*n^3 + 1624
*g^6*n^2 + 1764*g^6*n + 720*g^6)
________________________________________________________________________________________
maple [B] time = 0.02, size = 2017, normalized size = 7.33
result too large to display
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)
[Out]
(g*x+f)^(n+1)*(c*e^4*g^5*n^5*x^5+5*c*d*e^3*g^5*n^5*x^4+15*c*e^4*g^5*n^4*x^5+9*c*d^2*e^2*g^5*n^5*x^3+80*c*d*e^3
*g^5*n^4*x^4-5*c*e^4*f*g^4*n^4*x^4+85*c*e^4*g^5*n^3*x^5+a*e^3*g^5*n^5*x^3+7*c*d^3*e*g^5*n^5*x^2+153*c*d^2*e^2*
g^5*n^4*x^3-20*c*d*e^3*f*g^4*n^4*x^3+475*c*d*e^3*g^5*n^3*x^4-50*c*e^4*f*g^4*n^3*x^4+225*c*e^4*g^5*n^2*x^5+3*a*
d*e^2*g^5*n^5*x^2+17*a*e^3*g^5*n^4*x^3+2*c*d^4*g^5*n^5*x+126*c*d^3*e*g^5*n^4*x^2-27*c*d^2*e^2*f*g^4*n^4*x^2+96
3*c*d^2*e^2*g^5*n^3*x^3-240*c*d*e^3*f*g^4*n^3*x^3+1300*c*d*e^3*g^5*n^2*x^4+20*c*e^4*f^2*g^3*n^3*x^3-175*c*e^4*
f*g^4*n^2*x^4+274*c*e^4*g^5*n*x^5+3*a*d^2*e*g^5*n^5*x+54*a*d*e^2*g^5*n^4*x^2-3*a*e^3*f*g^4*n^4*x^2+107*a*e^3*g
^5*n^3*x^3+38*c*d^4*g^5*n^4*x-14*c*d^3*e*f*g^4*n^4*x+847*c*d^3*e*g^5*n^3*x^2-378*c*d^2*e^2*f*g^4*n^3*x^2+2763*
c*d^2*e^2*g^5*n^2*x^3+60*c*d*e^3*f^2*g^3*n^3*x^2-940*c*d*e^3*f*g^4*n^2*x^3+1620*c*d*e^3*g^5*n*x^4+120*c*e^4*f^
2*g^3*n^2*x^3-250*c*e^4*f*g^4*n*x^4+120*c*e^4*g^5*x^5+a*d^3*g^5*n^5+57*a*d^2*e*g^5*n^4*x-6*a*d*e^2*f*g^4*n^4*x
+363*a*d*e^2*g^5*n^3*x^2-42*a*e^3*f*g^4*n^3*x^2+307*a*e^3*g^5*n^2*x^3-2*c*d^4*f*g^4*n^4+274*c*d^4*g^5*n^3*x-22
4*c*d^3*e*f*g^4*n^3*x+2604*c*d^3*e*g^5*n^2*x^2+54*c*d^2*e^2*f^2*g^3*n^3*x-1755*c*d^2*e^2*f*g^4*n^2*x^2+3564*c*
d^2*e^2*g^5*n*x^3+540*c*d*e^3*f^2*g^3*n^2*x^2-1440*c*d*e^3*f*g^4*n*x^3+720*c*d*e^3*g^5*x^4-60*c*e^4*f^3*g^2*n^
2*x^2+220*c*e^4*f^2*g^3*n*x^3-120*c*e^4*f*g^4*x^4+20*a*d^3*g^5*n^4-3*a*d^2*e*f*g^4*n^4+411*a*d^2*e*g^5*n^3*x-9
6*a*d*e^2*f*g^4*n^3*x+1116*a*d*e^2*g^5*n^2*x^2+6*a*e^3*f^2*g^3*n^3*x-195*a*e^3*f*g^4*n^2*x^2+396*a*e^3*g^5*n*x
^3-36*c*d^4*f*g^4*n^3+922*c*d^4*g^5*n^2*x+14*c*d^3*e*f^2*g^3*n^3-1246*c*d^3*e*f*g^4*n^2*x+3556*c*d^3*e*g^5*n*x
^2+648*c*d^2*e^2*f^2*g^3*n^2*x-3024*c*d^2*e^2*f*g^4*n*x^2+1620*c*d^2*e^2*g^5*x^3-120*c*d*e^3*f^3*g^2*n^2*x+120
0*c*d*e^3*f^2*g^3*n*x^2-720*c*d*e^3*f*g^4*x^3-180*c*e^4*f^3*g^2*n*x^2+120*c*e^4*f^2*g^3*x^3+155*a*d^3*g^5*n^3-
54*a*d^2*e*f*g^4*n^3+1383*a*d^2*e*g^5*n^2*x+6*a*d*e^2*f^2*g^3*n^3-534*a*d*e^2*f*g^4*n^2*x+1524*a*d*e^2*g^5*n*x
^2+72*a*e^3*f^2*g^3*n^2*x-336*a*e^3*f*g^4*n*x^2+180*a*e^3*g^5*x^3-238*c*d^4*f*g^4*n^2+1404*c*d^4*g^5*n*x+210*c
*d^3*e*f^2*g^3*n^2-2716*c*d^3*e*f*g^4*n*x+1680*c*d^3*e*g^5*x^2-54*c*d^2*e^2*f^3*g^2*n^2+2214*c*d^2*e^2*f^2*g^3
*n*x-1620*c*d^2*e^2*f*g^4*x^2-840*c*d*e^3*f^3*g^2*n*x+720*c*d*e^3*f^2*g^3*x^2+120*c*e^4*f^4*g*n*x-120*c*e^4*f^
3*g^2*x^2+580*a*d^3*g^5*n^2-357*a*d^2*e*f*g^4*n^2+2106*a*d^2*e*g^5*n*x+90*a*d*e^2*f^2*g^3*n^2-1164*a*d*e^2*f*g
^4*n*x+720*a*d*e^2*g^5*x^2-6*a*e^3*f^3*g^2*n^2+246*a*e^3*f^2*g^3*n*x-180*a*e^3*f*g^4*x^2-684*c*d^4*f*g^4*n+720
*c*d^4*g^5*x+1036*c*d^3*e*f^2*g^3*n-1680*c*d^3*e*f*g^4*x-594*c*d^2*e^2*f^3*g^2*n+1620*c*d^2*e^2*f^2*g^3*x+120*
c*d*e^3*f^4*g*n-720*c*d*e^3*f^3*g^2*x+120*c*e^4*f^4*g*x+1044*a*d^3*g^5*n-1026*a*d^2*e*f*g^4*n+1080*a*d^2*e*g^5
*x+444*a*d*e^2*f^2*g^3*n-720*a*d*e^2*f*g^4*x-66*a*e^3*f^3*g^2*n+180*a*e^3*f^2*g^3*x-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)
________________________________________________________________________________________
maxima [B] time = 0.60, 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)
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mupad [B] time = 3.90, size = 1943, normalized size = 7.07
result too large to display
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))
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \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]
Timed out
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