3.25 \(\int (a+b x)^3 (c+d x)^n (A+B x+C x^2+D x^3) \, dx\)
Optimal. Leaf size=455 \[ -\frac {(b c-a d) (c+d x)^{n+3} \left (a^2 d^2 (C d-3 c D)-a b d \left (-3 B d^2-15 c^2 D+8 c C d\right )+b^2 \left (3 A d^3-6 B c d^2-15 c^3 D+10 c^2 C d\right )\right )}{d^7 (n+3)}+\frac {b (c+d x)^{n+5} \left (3 a^2 d^2 D+3 a b d (C d-5 c D)-\left (b^2 \left (-B d^2-15 c^2 D+5 c C d\right )\right )\right )}{d^7 (n+5)}+\frac {(c+d x)^{n+4} \left (a^3 d^3 D+3 a^2 b d^2 (C d-4 c D)-3 a b^2 d \left (-B d^2-10 c^2 D+4 c C d\right )+b^3 \left (A d^3-4 B c d^2-20 c^3 D+10 c^2 C d\right )\right )}{d^7 (n+4)}-\frac {(b c-a d)^3 (c+d x)^{n+1} \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{d^7 (n+1)}-\frac {(b c-a d)^2 (c+d x)^{n+2} \left (a d \left (-B d^2-3 c^2 D+2 c C d\right )-b \left (3 A d^3-4 B c d^2-6 c^3 D+5 c^2 C d\right )\right )}{d^7 (n+2)}+\frac {b^2 (c+d x)^{n+6} (3 a d D-6 b c D+b C d)}{d^7 (n+6)}+\frac {b^3 D (c+d x)^{n+7}}{d^7 (n+7)} \]
[Out]
-(-a*d+b*c)^3*(A*d^3-B*c*d^2+C*c^2*d-D*c^3)*(d*x+c)^(1+n)/d^7/(1+n)-(-a*d+b*c)^2*(a*d*(-B*d^2+2*C*c*d-3*D*c^2)
-b*(3*A*d^3-4*B*c*d^2+5*C*c^2*d-6*D*c^3))*(d*x+c)^(2+n)/d^7/(2+n)-(-a*d+b*c)*(a^2*d^2*(C*d-3*D*c)-a*b*d*(-3*B*
d^2+8*C*c*d-15*D*c^2)+b^2*(3*A*d^3-6*B*c*d^2+10*C*c^2*d-15*D*c^3))*(d*x+c)^(3+n)/d^7/(3+n)+(a^3*d^3*D+3*a^2*b*
d^2*(C*d-4*D*c)-3*a*b^2*d*(-B*d^2+4*C*c*d-10*D*c^2)+b^3*(A*d^3-4*B*c*d^2+10*C*c^2*d-20*D*c^3))*(d*x+c)^(4+n)/d
^7/(4+n)+b*(3*a^2*d^2*D+3*a*b*d*(C*d-5*D*c)-b^2*(-B*d^2+5*C*c*d-15*D*c^2))*(d*x+c)^(5+n)/d^7/(5+n)+b^2*(C*b*d+
3*D*a*d-6*D*b*c)*(d*x+c)^(6+n)/d^7/(6+n)+b^3*D*(d*x+c)^(7+n)/d^7/(7+n)
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Rubi [A] time = 0.34, antiderivative size = 455, normalized size of antiderivative = 1.00,
number of steps used = 2, number of rules used = 1, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.033, Rules used =
{1620} \[ -\frac {(b c-a d) (c+d x)^{n+3} \left (a^2 d^2 (C d-3 c D)-a b d \left (-3 B d^2-15 c^2 D+8 c C d\right )+b^2 \left (3 A d^3-6 B c d^2+10 c^2 C d-15 c^3 D\right )\right )}{d^7 (n+3)}+\frac {(c+d x)^{n+4} \left (3 a^2 b d^2 (C d-4 c D)+a^3 d^3 D-3 a b^2 d \left (-B d^2-10 c^2 D+4 c C d\right )+b^3 \left (A d^3-4 B c d^2+10 c^2 C d-20 c^3 D\right )\right )}{d^7 (n+4)}+\frac {b (c+d x)^{n+5} \left (3 a^2 d^2 D+3 a b d (C d-5 c D)+b^2 \left (-\left (-B d^2-15 c^2 D+5 c C d\right )\right )\right )}{d^7 (n+5)}-\frac {(b c-a d)^3 (c+d x)^{n+1} \left (A d^3-B c d^2+c^2 C d+c^3 (-D)\right )}{d^7 (n+1)}-\frac {(b c-a d)^2 (c+d x)^{n+2} \left (a d \left (-B d^2-3 c^2 D+2 c C d\right )-b \left (3 A d^3-4 B c d^2+5 c^2 C d-6 c^3 D\right )\right )}{d^7 (n+2)}+\frac {b^2 (c+d x)^{n+6} (3 a d D-6 b c D+b C d)}{d^7 (n+6)}+\frac {b^3 D (c+d x)^{n+7}}{d^7 (n+7)} \]
Antiderivative was successfully verified.
[In]
Int[(a + b*x)^3*(c + d*x)^n*(A + B*x + C*x^2 + D*x^3),x]
[Out]
-(((b*c - a*d)^3*(c^2*C*d - B*c*d^2 + A*d^3 - c^3*D)*(c + d*x)^(1 + n))/(d^7*(1 + n))) - ((b*c - a*d)^2*(a*d*(
2*c*C*d - B*d^2 - 3*c^2*D) - b*(5*c^2*C*d - 4*B*c*d^2 + 3*A*d^3 - 6*c^3*D))*(c + d*x)^(2 + n))/(d^7*(2 + n)) -
((b*c - a*d)*(a^2*d^2*(C*d - 3*c*D) - a*b*d*(8*c*C*d - 3*B*d^2 - 15*c^2*D) + b^2*(10*c^2*C*d - 6*B*c*d^2 + 3*
A*d^3 - 15*c^3*D))*(c + d*x)^(3 + n))/(d^7*(3 + n)) + ((a^3*d^3*D + 3*a^2*b*d^2*(C*d - 4*c*D) - 3*a*b^2*d*(4*c
*C*d - B*d^2 - 10*c^2*D) + b^3*(10*c^2*C*d - 4*B*c*d^2 + A*d^3 - 20*c^3*D))*(c + d*x)^(4 + n))/(d^7*(4 + n)) +
(b*(3*a^2*d^2*D + 3*a*b*d*(C*d - 5*c*D) - b^2*(5*c*C*d - B*d^2 - 15*c^2*D))*(c + d*x)^(5 + n))/(d^7*(5 + n))
+ (b^2*(b*C*d - 6*b*c*D + 3*a*d*D)*(c + d*x)^(6 + n))/(d^7*(6 + n)) + (b^3*D*(c + d*x)^(7 + n))/(d^7*(7 + n))
Rule 1620
Int[(Px_)*((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[Px*(a + b*x)
^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && PolyQ[Px, x] && (IntegersQ[m, n] || IGtQ[m, -2]) &&
GtQ[Expon[Px, x], 2]
Rubi steps
\begin {align*} \int (a+b x)^3 (c+d x)^n \left (A+B x+C x^2+D x^3\right ) \, dx &=\int \left (\frac {(-b c+a d)^3 \left (c^2 C d-B c d^2+A d^3-c^3 D\right ) (c+d x)^n}{d^6}+\frac {(b c-a d)^2 \left (-a d \left (2 c C d-B d^2-3 c^2 D\right )+b \left (5 c^2 C d-4 B c d^2+3 A d^3-6 c^3 D\right )\right ) (c+d x)^{1+n}}{d^6}+\frac {(b c-a d) \left (-a^2 d^2 (C d-3 c D)+a b d \left (8 c C d-3 B d^2-15 c^2 D\right )-b^2 \left (10 c^2 C d-6 B c d^2+3 A d^3-15 c^3 D\right )\right ) (c+d x)^{2+n}}{d^6}+\frac {\left (a^3 d^3 D+3 a^2 b d^2 (C d-4 c D)-3 a b^2 d \left (4 c C d-B d^2-10 c^2 D\right )+b^3 \left (10 c^2 C d-4 B c d^2+A d^3-20 c^3 D\right )\right ) (c+d x)^{3+n}}{d^6}+\frac {b \left (3 a^2 d^2 D+3 a b d (C d-5 c D)-b^2 \left (5 c C d-B d^2-15 c^2 D\right )\right ) (c+d x)^{4+n}}{d^6}+\frac {b^2 (b C d-6 b c D+3 a d D) (c+d x)^{5+n}}{d^6}+\frac {b^3 D (c+d x)^{6+n}}{d^6}\right ) \, dx\\ &=-\frac {(b c-a d)^3 \left (c^2 C d-B c d^2+A d^3-c^3 D\right ) (c+d x)^{1+n}}{d^7 (1+n)}-\frac {(b c-a d)^2 \left (a d \left (2 c C d-B d^2-3 c^2 D\right )-b \left (5 c^2 C d-4 B c d^2+3 A d^3-6 c^3 D\right )\right ) (c+d x)^{2+n}}{d^7 (2+n)}-\frac {(b c-a d) \left (a^2 d^2 (C d-3 c D)-a b d \left (8 c C d-3 B d^2-15 c^2 D\right )+b^2 \left (10 c^2 C d-6 B c d^2+3 A d^3-15 c^3 D\right )\right ) (c+d x)^{3+n}}{d^7 (3+n)}+\frac {\left (a^3 d^3 D+3 a^2 b d^2 (C d-4 c D)-3 a b^2 d \left (4 c C d-B d^2-10 c^2 D\right )+b^3 \left (10 c^2 C d-4 B c d^2+A d^3-20 c^3 D\right )\right ) (c+d x)^{4+n}}{d^7 (4+n)}+\frac {b \left (3 a^2 d^2 D+3 a b d (C d-5 c D)-b^2 \left (5 c C d-B d^2-15 c^2 D\right )\right ) (c+d x)^{5+n}}{d^7 (5+n)}+\frac {b^2 (b C d-6 b c D+3 a d D) (c+d x)^{6+n}}{d^7 (6+n)}+\frac {b^3 D (c+d x)^{7+n}}{d^7 (7+n)}\\ \end {align*}
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Mathematica [A] time = 0.68, size = 418, normalized size = 0.92 \[ \frac {(c+d x)^{n+1} \left (\frac {(c+d x)^2 (b c-a d) \left (a^2 d^2 (3 c D-C d)+a b d \left (-3 B d^2-15 c^2 D+8 c C d\right )+b^2 \left (-3 A d^3+6 B c d^2+15 c^3 D-10 c^2 C d\right )\right )}{n+3}+\frac {b (c+d x)^4 \left (3 a^2 d^2 D+3 a b d (C d-5 c D)+b^2 \left (B d^2+15 c^2 D-5 c C d\right )\right )}{n+5}+\frac {(c+d x)^3 \left (a^3 d^3 D+3 a^2 b d^2 (C d-4 c D)+3 a b^2 d \left (B d^2+10 c^2 D-4 c C d\right )+b^3 \left (A d^3-4 B c d^2-20 c^3 D+10 c^2 C d\right )\right )}{n+4}-\frac {(c+d x) (b c-a d)^2 \left (b \left (-3 A d^3+4 B c d^2+6 c^3 D-5 c^2 C d\right )-a d \left (B d^2+3 c^2 D-2 c C d\right )\right )}{n+2}+\frac {(b c-a d)^3 \left (-A d^3+B c d^2+c^3 D-c^2 C d\right )}{n+1}+\frac {b^2 (c+d x)^5 (3 a d D-6 b c D+b C d)}{n+6}+\frac {b^3 D (c+d x)^6}{n+7}\right )}{d^7} \]
Antiderivative was successfully verified.
[In]
Integrate[(a + b*x)^3*(c + d*x)^n*(A + B*x + C*x^2 + D*x^3),x]
[Out]
((c + d*x)^(1 + n)*(((b*c - a*d)^3*(-(c^2*C*d) + B*c*d^2 - A*d^3 + c^3*D))/(1 + n) - ((b*c - a*d)^2*(-(a*d*(-2
*c*C*d + B*d^2 + 3*c^2*D)) + b*(-5*c^2*C*d + 4*B*c*d^2 - 3*A*d^3 + 6*c^3*D))*(c + d*x))/(2 + n) + ((b*c - a*d)
*(a^2*d^2*(-(C*d) + 3*c*D) + a*b*d*(8*c*C*d - 3*B*d^2 - 15*c^2*D) + b^2*(-10*c^2*C*d + 6*B*c*d^2 - 3*A*d^3 + 1
5*c^3*D))*(c + d*x)^2)/(3 + n) + ((a^3*d^3*D + 3*a^2*b*d^2*(C*d - 4*c*D) + 3*a*b^2*d*(-4*c*C*d + B*d^2 + 10*c^
2*D) + b^3*(10*c^2*C*d - 4*B*c*d^2 + A*d^3 - 20*c^3*D))*(c + d*x)^3)/(4 + n) + (b*(3*a^2*d^2*D + 3*a*b*d*(C*d
- 5*c*D) + b^2*(-5*c*C*d + B*d^2 + 15*c^2*D))*(c + d*x)^4)/(5 + n) + (b^2*(b*C*d - 6*b*c*D + 3*a*d*D)*(c + d*x
)^5)/(6 + n) + (b^3*D*(c + d*x)^6)/(7 + n)))/d^7
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fricas [B] time = 1.07, size = 4115, normalized size = 9.04 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((b*x+a)^3*(d*x+c)^n*(D*x^3+C*x^2+B*x+A),x, algorithm="fricas")
[Out]
(A*a^3*c*d^6*n^6 + 720*D*b^3*c^7 + 5040*A*a^3*c*d^6 + 1680*(C*a^3 + 3*B*a^2*b + 3*A*a*b^2)*c^3*d^4 - 2520*(B*a
^3 + 3*A*a^2*b)*c^2*d^5 + (D*b^3*d^7*n^6 + 21*D*b^3*d^7*n^5 + 175*D*b^3*d^7*n^4 + 735*D*b^3*d^7*n^3 + 1624*D*b
^3*d^7*n^2 + 1764*D*b^3*d^7*n + 720*D*b^3*d^7)*x^7 + (840*(3*D*a*b^2 + C*b^3)*d^7 + (D*b^3*c*d^6 + (3*D*a*b^2
+ C*b^3)*d^7)*n^6 + (15*D*b^3*c*d^6 + 22*(3*D*a*b^2 + C*b^3)*d^7)*n^5 + 5*(17*D*b^3*c*d^6 + 38*(3*D*a*b^2 + C*
b^3)*d^7)*n^4 + 5*(45*D*b^3*c*d^6 + 164*(3*D*a*b^2 + C*b^3)*d^7)*n^3 + (274*D*b^3*c*d^6 + 1849*(3*D*a*b^2 + C*
b^3)*d^7)*n^2 + 2*(60*D*b^3*c*d^6 + 1019*(3*D*a*b^2 + C*b^3)*d^7)*n)*x^6 + (27*A*a^3*c*d^6 - (B*a^3 + 3*A*a^2*
b)*c^2*d^5)*n^5 + (1008*(3*D*a^2*b + 3*C*a*b^2 + B*b^3)*d^7 + ((3*D*a^2*b + 3*C*a*b^2 + B*b^3)*d^7 + (3*D*a*b^
2*c + C*b^3*c)*d^6)*n^6 - (6*D*b^3*c^2*d^5 - 23*(3*D*a^2*b + 3*C*a*b^2 + B*b^3)*d^7 - 17*(3*D*a*b^2*c + C*b^3*
c)*d^6)*n^5 - 3*(20*D*b^3*c^2*d^5 - 69*(3*D*a^2*b + 3*C*a*b^2 + B*b^3)*d^7 - 35*(3*D*a*b^2*c + C*b^3*c)*d^6)*n
^4 - 5*(42*D*b^3*c^2*d^5 - 185*(3*D*a^2*b + 3*C*a*b^2 + B*b^3)*d^7 - 59*(3*D*a*b^2*c + C*b^3*c)*d^6)*n^3 - 2*(
150*D*b^3*c^2*d^5 - 1072*(3*D*a^2*b + 3*C*a*b^2 + B*b^3)*d^7 - 187*(3*D*a*b^2*c + C*b^3*c)*d^6)*n^2 - 12*(12*D
*b^3*c^2*d^5 - 201*(3*D*a^2*b + 3*C*a*b^2 + B*b^3)*d^7 - 14*(3*D*a*b^2*c + C*b^3*c)*d^6)*n)*x^5 + (295*A*a^3*c
*d^6 + 2*(C*a^3 + 3*B*a^2*b + 3*A*a*b^2)*c^3*d^4 - 25*(B*a^3 + 3*A*a^2*b)*c^2*d^5)*n^4 + (1260*(D*a^3 + 3*C*a^
2*b + 3*B*a*b^2 + A*b^3)*d^7 + ((D*a^3 + 3*C*a^2*b + 3*B*a*b^2 + A*b^3)*d^7 + (3*D*a^2*b*c + (3*C*a*b^2 + B*b^
3)*c)*d^6)*n^6 + (24*(D*a^3 + 3*C*a^2*b + 3*B*a*b^2 + A*b^3)*d^7 + 19*(3*D*a^2*b*c + (3*C*a*b^2 + B*b^3)*c)*d^
6 - 5*(3*D*a*b^2*c^2 + C*b^3*c^2)*d^5)*n^5 + (30*D*b^3*c^3*d^4 + 226*(D*a^3 + 3*C*a^2*b + 3*B*a*b^2 + A*b^3)*d
^7 + 131*(3*D*a^2*b*c + (3*C*a*b^2 + B*b^3)*c)*d^6 - 65*(3*D*a*b^2*c^2 + C*b^3*c^2)*d^5)*n^4 + (180*D*b^3*c^3*
d^4 + 1056*(D*a^3 + 3*C*a^2*b + 3*B*a*b^2 + A*b^3)*d^7 + 401*(3*D*a^2*b*c + (3*C*a*b^2 + B*b^3)*c)*d^6 - 265*(
3*D*a*b^2*c^2 + C*b^3*c^2)*d^5)*n^3 + 5*(66*D*b^3*c^3*d^4 + 509*(D*a^3 + 3*C*a^2*b + 3*B*a*b^2 + A*b^3)*d^7 +
108*(3*D*a^2*b*c + (3*C*a*b^2 + B*b^3)*c)*d^6 - 83*(3*D*a*b^2*c^2 + C*b^3*c^2)*d^5)*n^2 + 6*(30*D*b^3*c^3*d^4
+ 492*(D*a^3 + 3*C*a^2*b + 3*B*a*b^2 + A*b^3)*d^7 + 42*(3*D*a^2*b*c + (3*C*a*b^2 + B*b^3)*c)*d^6 - 35*(3*D*a*b
^2*c^2 + C*b^3*c^2)*d^5)*n)*x^4 - 1260*(D*a^3*c^4 + (3*C*a^2*b + 3*B*a*b^2 + A*b^3)*c^4)*d^3 + (1665*A*a^3*c*d
^6 + 44*(C*a^3 + 3*B*a^2*b + 3*A*a*b^2)*c^3*d^4 - 245*(B*a^3 + 3*A*a^2*b)*c^2*d^5 - 6*(D*a^3*c^4 + (3*C*a^2*b
+ 3*B*a*b^2 + A*b^3)*c^4)*d^3)*n^3 + (1680*(C*a^3 + 3*B*a^2*b + 3*A*a*b^2)*d^7 + ((C*a^3 + 3*B*a^2*b + 3*A*a*b
^2)*d^7 + (D*a^3*c + (3*C*a^2*b + 3*B*a*b^2 + A*b^3)*c)*d^6)*n^6 + (25*(C*a^3 + 3*B*a^2*b + 3*A*a*b^2)*d^7 + 2
1*(D*a^3*c + (3*C*a^2*b + 3*B*a*b^2 + A*b^3)*c)*d^6 - 4*(3*D*a^2*b*c^2 + (3*C*a*b^2 + B*b^3)*c^2)*d^5)*n^5 + (
247*(C*a^3 + 3*B*a^2*b + 3*A*a*b^2)*d^7 + 163*(D*a^3*c + (3*C*a^2*b + 3*B*a*b^2 + A*b^3)*c)*d^6 - 64*(3*D*a^2*
b*c^2 + (3*C*a*b^2 + B*b^3)*c^2)*d^5 + 20*(3*D*a*b^2*c^3 + C*b^3*c^3)*d^4)*n^4 - (120*D*b^3*c^4*d^3 - 1219*(C*
a^3 + 3*B*a^2*b + 3*A*a*b^2)*d^7 - 567*(D*a^3*c + (3*C*a^2*b + 3*B*a*b^2 + A*b^3)*c)*d^6 + 332*(3*D*a^2*b*c^2
+ (3*C*a*b^2 + B*b^3)*c^2)*d^5 - 200*(3*D*a*b^2*c^3 + C*b^3*c^3)*d^4)*n^3 - 4*(90*D*b^3*c^4*d^3 - 778*(C*a^3 +
3*B*a^2*b + 3*A*a*b^2)*d^7 - 211*(D*a^3*c + (3*C*a^2*b + 3*B*a*b^2 + A*b^3)*c)*d^6 + 152*(3*D*a^2*b*c^2 + (3*
C*a*b^2 + B*b^3)*c^2)*d^5 - 115*(3*D*a*b^2*c^3 + C*b^3*c^3)*d^4)*n^2 - 4*(60*D*b^3*c^4*d^3 - 949*(C*a^3 + 3*B*
a^2*b + 3*A*a*b^2)*d^7 - 105*(D*a^3*c + (3*C*a^2*b + 3*B*a*b^2 + A*b^3)*c)*d^6 + 84*(3*D*a^2*b*c^2 + (3*C*a*b^
2 + B*b^3)*c^2)*d^5 - 70*(3*D*a*b^2*c^3 + C*b^3*c^3)*d^4)*n)*x^3 + 1008*(3*D*a^2*b*c^5 + (3*C*a*b^2 + B*b^3)*c
^5)*d^2 + (5104*A*a^3*c*d^6 + 358*(C*a^3 + 3*B*a^2*b + 3*A*a*b^2)*c^3*d^4 - 1175*(B*a^3 + 3*A*a^2*b)*c^2*d^5 -
108*(D*a^3*c^4 + (3*C*a^2*b + 3*B*a*b^2 + A*b^3)*c^4)*d^3 + 24*(3*D*a^2*b*c^5 + (3*C*a*b^2 + B*b^3)*c^5)*d^2)
*n^2 + (2520*(B*a^3 + 3*A*a^2*b)*d^7 + ((C*a^3 + 3*B*a^2*b + 3*A*a*b^2)*c*d^6 + (B*a^3 + 3*A*a^2*b)*d^7)*n^6 +
(23*(C*a^3 + 3*B*a^2*b + 3*A*a*b^2)*c*d^6 + 26*(B*a^3 + 3*A*a^2*b)*d^7 - 3*(D*a^3*c^2 + (3*C*a^2*b + 3*B*a*b^
2 + A*b^3)*c^2)*d^5)*n^5 + 3*(67*(C*a^3 + 3*B*a^2*b + 3*A*a*b^2)*c*d^6 + 90*(B*a^3 + 3*A*a^2*b)*d^7 - 19*(D*a^
3*c^2 + (3*C*a^2*b + 3*B*a*b^2 + A*b^3)*c^2)*d^5 + 4*(3*D*a^2*b*c^3 + (3*C*a*b^2 + B*b^3)*c^3)*d^4)*n^4 + (817
*(C*a^3 + 3*B*a^2*b + 3*A*a*b^2)*c*d^6 + 1420*(B*a^3 + 3*A*a^2*b)*d^7 - 375*(D*a^3*c^2 + (3*C*a^2*b + 3*B*a*b^
2 + A*b^3)*c^2)*d^5 + 168*(3*D*a^2*b*c^3 + (3*C*a*b^2 + B*b^3)*c^3)*d^4 - 60*(3*D*a*b^2*c^4 + C*b^3*c^4)*d^3)*
n^3 + (360*D*b^3*c^5*d^2 + 1478*(C*a^3 + 3*B*a^2*b + 3*A*a*b^2)*c*d^6 + 3929*(B*a^3 + 3*A*a^2*b)*d^7 - 951*(D*
a^3*c^2 + (3*C*a^2*b + 3*B*a*b^2 + A*b^3)*c^2)*d^5 + 660*(3*D*a^2*b*c^3 + (3*C*a*b^2 + B*b^3)*c^3)*d^4 - 480*(
3*D*a*b^2*c^4 + C*b^3*c^4)*d^3)*n^2 + 6*(60*D*b^3*c^5*d^2 + 140*(C*a^3 + 3*B*a^2*b + 3*A*a*b^2)*c*d^6 + 879*(B
*a^3 + 3*A*a^2*b)*d^7 - 105*(D*a^3*c^2 + (3*C*a^2*b + 3*B*a*b^2 + A*b^3)*c^2)*d^5 + 84*(3*D*a^2*b*c^3 + (3*C*a
*b^2 + B*b^3)*c^3)*d^4 - 70*(3*D*a*b^2*c^4 + C*b^3*c^4)*d^3)*n)*x^2 - 840*(3*D*a*b^2*c^6 + C*b^3*c^6)*d + 2*(4
014*A*a^3*c*d^6 + 638*(C*a^3 + 3*B*a^2*b + 3*A*a*b^2)*c^3*d^4 - 1377*(B*a^3 + 3*A*a^2*b)*c^2*d^5 - 321*(D*a^3*
c^4 + (3*C*a^2*b + 3*B*a*b^2 + A*b^3)*c^4)*d^3 + 156*(3*D*a^2*b*c^5 + (3*C*a*b^2 + B*b^3)*c^5)*d^2 - 60*(3*D*a
*b^2*c^6 + C*b^3*c^6)*d)*n + (5040*A*a^3*d^7 + (A*a^3*d^7 + (B*a^3 + 3*A*a^2*b)*c*d^6)*n^6 + (27*A*a^3*d^7 - 2
*(C*a^3 + 3*B*a^2*b + 3*A*a*b^2)*c^2*d^5 + 25*(B*a^3 + 3*A*a^2*b)*c*d^6)*n^5 + (295*A*a^3*d^7 - 44*(C*a^3 + 3*
B*a^2*b + 3*A*a*b^2)*c^2*d^5 + 245*(B*a^3 + 3*A*a^2*b)*c*d^6 + 6*(D*a^3*c^3 + (3*C*a^2*b + 3*B*a*b^2 + A*b^3)*
c^3)*d^4)*n^4 + (1665*A*a^3*d^7 - 358*(C*a^3 + 3*B*a^2*b + 3*A*a*b^2)*c^2*d^5 + 1175*(B*a^3 + 3*A*a^2*b)*c*d^6
+ 108*(D*a^3*c^3 + (3*C*a^2*b + 3*B*a*b^2 + A*b^3)*c^3)*d^4 - 24*(3*D*a^2*b*c^4 + (3*C*a*b^2 + B*b^3)*c^4)*d^
3)*n^3 + 2*(2552*A*a^3*d^7 - 638*(C*a^3 + 3*B*a^2*b + 3*A*a*b^2)*c^2*d^5 + 1377*(B*a^3 + 3*A*a^2*b)*c*d^6 + 32
1*(D*a^3*c^3 + (3*C*a^2*b + 3*B*a*b^2 + A*b^3)*c^3)*d^4 - 156*(3*D*a^2*b*c^4 + (3*C*a*b^2 + B*b^3)*c^4)*d^3 +
60*(3*D*a*b^2*c^5 + C*b^3*c^5)*d^2)*n^2 - 12*(60*D*b^3*c^6*d - 669*A*a^3*d^7 + 140*(C*a^3 + 3*B*a^2*b + 3*A*a*
b^2)*c^2*d^5 - 210*(B*a^3 + 3*A*a^2*b)*c*d^6 - 105*(D*a^3*c^3 + (3*C*a^2*b + 3*B*a*b^2 + A*b^3)*c^3)*d^4 + 84*
(3*D*a^2*b*c^4 + (3*C*a*b^2 + B*b^3)*c^4)*d^3 - 70*(3*D*a*b^2*c^5 + C*b^3*c^5)*d^2)*n)*x)*(d*x + c)^n/(d^7*n^7
+ 28*d^7*n^6 + 322*d^7*n^5 + 1960*d^7*n^4 + 6769*d^7*n^3 + 13132*d^7*n^2 + 13068*d^7*n + 5040*d^7)
________________________________________________________________________________________
giac [B] time = 1.86, size = 9032, normalized size = 19.85 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((b*x+a)^3*(d*x+c)^n*(D*x^3+C*x^2+B*x+A),x, algorithm="giac")
[Out]
((d*x + c)^n*D*b^3*d^7*n^6*x^7 + (d*x + c)^n*D*b^3*c*d^6*n^6*x^6 + 3*(d*x + c)^n*D*a*b^2*d^7*n^6*x^6 + (d*x +
c)^n*C*b^3*d^7*n^6*x^6 + 21*(d*x + c)^n*D*b^3*d^7*n^5*x^7 + 3*(d*x + c)^n*D*a*b^2*c*d^6*n^6*x^5 + (d*x + c)^n*
C*b^3*c*d^6*n^6*x^5 + 3*(d*x + c)^n*D*a^2*b*d^7*n^6*x^5 + 3*(d*x + c)^n*C*a*b^2*d^7*n^6*x^5 + (d*x + c)^n*B*b^
3*d^7*n^6*x^5 + 15*(d*x + c)^n*D*b^3*c*d^6*n^5*x^6 + 66*(d*x + c)^n*D*a*b^2*d^7*n^5*x^6 + 22*(d*x + c)^n*C*b^3
*d^7*n^5*x^6 + 175*(d*x + c)^n*D*b^3*d^7*n^4*x^7 + 3*(d*x + c)^n*D*a^2*b*c*d^6*n^6*x^4 + 3*(d*x + c)^n*C*a*b^2
*c*d^6*n^6*x^4 + (d*x + c)^n*B*b^3*c*d^6*n^6*x^4 + (d*x + c)^n*D*a^3*d^7*n^6*x^4 + 3*(d*x + c)^n*C*a^2*b*d^7*n
^6*x^4 + 3*(d*x + c)^n*B*a*b^2*d^7*n^6*x^4 + (d*x + c)^n*A*b^3*d^7*n^6*x^4 - 6*(d*x + c)^n*D*b^3*c^2*d^5*n^5*x
^5 + 51*(d*x + c)^n*D*a*b^2*c*d^6*n^5*x^5 + 17*(d*x + c)^n*C*b^3*c*d^6*n^5*x^5 + 69*(d*x + c)^n*D*a^2*b*d^7*n^
5*x^5 + 69*(d*x + c)^n*C*a*b^2*d^7*n^5*x^5 + 23*(d*x + c)^n*B*b^3*d^7*n^5*x^5 + 85*(d*x + c)^n*D*b^3*c*d^6*n^4
*x^6 + 570*(d*x + c)^n*D*a*b^2*d^7*n^4*x^6 + 190*(d*x + c)^n*C*b^3*d^7*n^4*x^6 + 735*(d*x + c)^n*D*b^3*d^7*n^3
*x^7 + (d*x + c)^n*D*a^3*c*d^6*n^6*x^3 + 3*(d*x + c)^n*C*a^2*b*c*d^6*n^6*x^3 + 3*(d*x + c)^n*B*a*b^2*c*d^6*n^6
*x^3 + (d*x + c)^n*A*b^3*c*d^6*n^6*x^3 + (d*x + c)^n*C*a^3*d^7*n^6*x^3 + 3*(d*x + c)^n*B*a^2*b*d^7*n^6*x^3 + 3
*(d*x + c)^n*A*a*b^2*d^7*n^6*x^3 - 15*(d*x + c)^n*D*a*b^2*c^2*d^5*n^5*x^4 - 5*(d*x + c)^n*C*b^3*c^2*d^5*n^5*x^
4 + 57*(d*x + c)^n*D*a^2*b*c*d^6*n^5*x^4 + 57*(d*x + c)^n*C*a*b^2*c*d^6*n^5*x^4 + 19*(d*x + c)^n*B*b^3*c*d^6*n
^5*x^4 + 24*(d*x + c)^n*D*a^3*d^7*n^5*x^4 + 72*(d*x + c)^n*C*a^2*b*d^7*n^5*x^4 + 72*(d*x + c)^n*B*a*b^2*d^7*n^
5*x^4 + 24*(d*x + c)^n*A*b^3*d^7*n^5*x^4 - 60*(d*x + c)^n*D*b^3*c^2*d^5*n^4*x^5 + 315*(d*x + c)^n*D*a*b^2*c*d^
6*n^4*x^5 + 105*(d*x + c)^n*C*b^3*c*d^6*n^4*x^5 + 621*(d*x + c)^n*D*a^2*b*d^7*n^4*x^5 + 621*(d*x + c)^n*C*a*b^
2*d^7*n^4*x^5 + 207*(d*x + c)^n*B*b^3*d^7*n^4*x^5 + 225*(d*x + c)^n*D*b^3*c*d^6*n^3*x^6 + 2460*(d*x + c)^n*D*a
*b^2*d^7*n^3*x^6 + 820*(d*x + c)^n*C*b^3*d^7*n^3*x^6 + 1624*(d*x + c)^n*D*b^3*d^7*n^2*x^7 + (d*x + c)^n*C*a^3*
c*d^6*n^6*x^2 + 3*(d*x + c)^n*B*a^2*b*c*d^6*n^6*x^2 + 3*(d*x + c)^n*A*a*b^2*c*d^6*n^6*x^2 + (d*x + c)^n*B*a^3*
d^7*n^6*x^2 + 3*(d*x + c)^n*A*a^2*b*d^7*n^6*x^2 - 12*(d*x + c)^n*D*a^2*b*c^2*d^5*n^5*x^3 - 12*(d*x + c)^n*C*a*
b^2*c^2*d^5*n^5*x^3 - 4*(d*x + c)^n*B*b^3*c^2*d^5*n^5*x^3 + 21*(d*x + c)^n*D*a^3*c*d^6*n^5*x^3 + 63*(d*x + c)^
n*C*a^2*b*c*d^6*n^5*x^3 + 63*(d*x + c)^n*B*a*b^2*c*d^6*n^5*x^3 + 21*(d*x + c)^n*A*b^3*c*d^6*n^5*x^3 + 25*(d*x
+ c)^n*C*a^3*d^7*n^5*x^3 + 75*(d*x + c)^n*B*a^2*b*d^7*n^5*x^3 + 75*(d*x + c)^n*A*a*b^2*d^7*n^5*x^3 + 30*(d*x +
c)^n*D*b^3*c^3*d^4*n^4*x^4 - 195*(d*x + c)^n*D*a*b^2*c^2*d^5*n^4*x^4 - 65*(d*x + c)^n*C*b^3*c^2*d^5*n^4*x^4 +
393*(d*x + c)^n*D*a^2*b*c*d^6*n^4*x^4 + 393*(d*x + c)^n*C*a*b^2*c*d^6*n^4*x^4 + 131*(d*x + c)^n*B*b^3*c*d^6*n
^4*x^4 + 226*(d*x + c)^n*D*a^3*d^7*n^4*x^4 + 678*(d*x + c)^n*C*a^2*b*d^7*n^4*x^4 + 678*(d*x + c)^n*B*a*b^2*d^7
*n^4*x^4 + 226*(d*x + c)^n*A*b^3*d^7*n^4*x^4 - 210*(d*x + c)^n*D*b^3*c^2*d^5*n^3*x^5 + 885*(d*x + c)^n*D*a*b^2
*c*d^6*n^3*x^5 + 295*(d*x + c)^n*C*b^3*c*d^6*n^3*x^5 + 2775*(d*x + c)^n*D*a^2*b*d^7*n^3*x^5 + 2775*(d*x + c)^n
*C*a*b^2*d^7*n^3*x^5 + 925*(d*x + c)^n*B*b^3*d^7*n^3*x^5 + 274*(d*x + c)^n*D*b^3*c*d^6*n^2*x^6 + 5547*(d*x + c
)^n*D*a*b^2*d^7*n^2*x^6 + 1849*(d*x + c)^n*C*b^3*d^7*n^2*x^6 + 1764*(d*x + c)^n*D*b^3*d^7*n*x^7 + (d*x + c)^n*
B*a^3*c*d^6*n^6*x + 3*(d*x + c)^n*A*a^2*b*c*d^6*n^6*x + (d*x + c)^n*A*a^3*d^7*n^6*x - 3*(d*x + c)^n*D*a^3*c^2*
d^5*n^5*x^2 - 9*(d*x + c)^n*C*a^2*b*c^2*d^5*n^5*x^2 - 9*(d*x + c)^n*B*a*b^2*c^2*d^5*n^5*x^2 - 3*(d*x + c)^n*A*
b^3*c^2*d^5*n^5*x^2 + 23*(d*x + c)^n*C*a^3*c*d^6*n^5*x^2 + 69*(d*x + c)^n*B*a^2*b*c*d^6*n^5*x^2 + 69*(d*x + c)
^n*A*a*b^2*c*d^6*n^5*x^2 + 26*(d*x + c)^n*B*a^3*d^7*n^5*x^2 + 78*(d*x + c)^n*A*a^2*b*d^7*n^5*x^2 + 60*(d*x + c
)^n*D*a*b^2*c^3*d^4*n^4*x^3 + 20*(d*x + c)^n*C*b^3*c^3*d^4*n^4*x^3 - 192*(d*x + c)^n*D*a^2*b*c^2*d^5*n^4*x^3 -
192*(d*x + c)^n*C*a*b^2*c^2*d^5*n^4*x^3 - 64*(d*x + c)^n*B*b^3*c^2*d^5*n^4*x^3 + 163*(d*x + c)^n*D*a^3*c*d^6*
n^4*x^3 + 489*(d*x + c)^n*C*a^2*b*c*d^6*n^4*x^3 + 489*(d*x + c)^n*B*a*b^2*c*d^6*n^4*x^3 + 163*(d*x + c)^n*A*b^
3*c*d^6*n^4*x^3 + 247*(d*x + c)^n*C*a^3*d^7*n^4*x^3 + 741*(d*x + c)^n*B*a^2*b*d^7*n^4*x^3 + 741*(d*x + c)^n*A*
a*b^2*d^7*n^4*x^3 + 180*(d*x + c)^n*D*b^3*c^3*d^4*n^3*x^4 - 795*(d*x + c)^n*D*a*b^2*c^2*d^5*n^3*x^4 - 265*(d*x
+ c)^n*C*b^3*c^2*d^5*n^3*x^4 + 1203*(d*x + c)^n*D*a^2*b*c*d^6*n^3*x^4 + 1203*(d*x + c)^n*C*a*b^2*c*d^6*n^3*x^
4 + 401*(d*x + c)^n*B*b^3*c*d^6*n^3*x^4 + 1056*(d*x + c)^n*D*a^3*d^7*n^3*x^4 + 3168*(d*x + c)^n*C*a^2*b*d^7*n^
3*x^4 + 3168*(d*x + c)^n*B*a*b^2*d^7*n^3*x^4 + 1056*(d*x + c)^n*A*b^3*d^7*n^3*x^4 - 300*(d*x + c)^n*D*b^3*c^2*
d^5*n^2*x^5 + 1122*(d*x + c)^n*D*a*b^2*c*d^6*n^2*x^5 + 374*(d*x + c)^n*C*b^3*c*d^6*n^2*x^5 + 6432*(d*x + c)^n*
D*a^2*b*d^7*n^2*x^5 + 6432*(d*x + c)^n*C*a*b^2*d^7*n^2*x^5 + 2144*(d*x + c)^n*B*b^3*d^7*n^2*x^5 + 120*(d*x + c
)^n*D*b^3*c*d^6*n*x^6 + 6114*(d*x + c)^n*D*a*b^2*d^7*n*x^6 + 2038*(d*x + c)^n*C*b^3*d^7*n*x^6 + 720*(d*x + c)^
n*D*b^3*d^7*x^7 + (d*x + c)^n*A*a^3*c*d^6*n^6 - 2*(d*x + c)^n*C*a^3*c^2*d^5*n^5*x - 6*(d*x + c)^n*B*a^2*b*c^2*
d^5*n^5*x - 6*(d*x + c)^n*A*a*b^2*c^2*d^5*n^5*x + 25*(d*x + c)^n*B*a^3*c*d^6*n^5*x + 75*(d*x + c)^n*A*a^2*b*c*
d^6*n^5*x + 27*(d*x + c)^n*A*a^3*d^7*n^5*x + 36*(d*x + c)^n*D*a^2*b*c^3*d^4*n^4*x^2 + 36*(d*x + c)^n*C*a*b^2*c
^3*d^4*n^4*x^2 + 12*(d*x + c)^n*B*b^3*c^3*d^4*n^4*x^2 - 57*(d*x + c)^n*D*a^3*c^2*d^5*n^4*x^2 - 171*(d*x + c)^n
*C*a^2*b*c^2*d^5*n^4*x^2 - 171*(d*x + c)^n*B*a*b^2*c^2*d^5*n^4*x^2 - 57*(d*x + c)^n*A*b^3*c^2*d^5*n^4*x^2 + 20
1*(d*x + c)^n*C*a^3*c*d^6*n^4*x^2 + 603*(d*x + c)^n*B*a^2*b*c*d^6*n^4*x^2 + 603*(d*x + c)^n*A*a*b^2*c*d^6*n^4*
x^2 + 270*(d*x + c)^n*B*a^3*d^7*n^4*x^2 + 810*(d*x + c)^n*A*a^2*b*d^7*n^4*x^2 - 120*(d*x + c)^n*D*b^3*c^4*d^3*
n^3*x^3 + 600*(d*x + c)^n*D*a*b^2*c^3*d^4*n^3*x^3 + 200*(d*x + c)^n*C*b^3*c^3*d^4*n^3*x^3 - 996*(d*x + c)^n*D*
a^2*b*c^2*d^5*n^3*x^3 - 996*(d*x + c)^n*C*a*b^2*c^2*d^5*n^3*x^3 - 332*(d*x + c)^n*B*b^3*c^2*d^5*n^3*x^3 + 567*
(d*x + c)^n*D*a^3*c*d^6*n^3*x^3 + 1701*(d*x + c)^n*C*a^2*b*c*d^6*n^3*x^3 + 1701*(d*x + c)^n*B*a*b^2*c*d^6*n^3*
x^3 + 567*(d*x + c)^n*A*b^3*c*d^6*n^3*x^3 + 1219*(d*x + c)^n*C*a^3*d^7*n^3*x^3 + 3657*(d*x + c)^n*B*a^2*b*d^7*
n^3*x^3 + 3657*(d*x + c)^n*A*a*b^2*d^7*n^3*x^3 + 330*(d*x + c)^n*D*b^3*c^3*d^4*n^2*x^4 - 1245*(d*x + c)^n*D*a*
b^2*c^2*d^5*n^2*x^4 - 415*(d*x + c)^n*C*b^3*c^2*d^5*n^2*x^4 + 1620*(d*x + c)^n*D*a^2*b*c*d^6*n^2*x^4 + 1620*(d
*x + c)^n*C*a*b^2*c*d^6*n^2*x^4 + 540*(d*x + c)^n*B*b^3*c*d^6*n^2*x^4 + 2545*(d*x + c)^n*D*a^3*d^7*n^2*x^4 + 7
635*(d*x + c)^n*C*a^2*b*d^7*n^2*x^4 + 7635*(d*x + c)^n*B*a*b^2*d^7*n^2*x^4 + 2545*(d*x + c)^n*A*b^3*d^7*n^2*x^
4 - 144*(d*x + c)^n*D*b^3*c^2*d^5*n*x^5 + 504*(d*x + c)^n*D*a*b^2*c*d^6*n*x^5 + 168*(d*x + c)^n*C*b^3*c*d^6*n*
x^5 + 7236*(d*x + c)^n*D*a^2*b*d^7*n*x^5 + 7236*(d*x + c)^n*C*a*b^2*d^7*n*x^5 + 2412*(d*x + c)^n*B*b^3*d^7*n*x
^5 + 2520*(d*x + c)^n*D*a*b^2*d^7*x^6 + 840*(d*x + c)^n*C*b^3*d^7*x^6 - (d*x + c)^n*B*a^3*c^2*d^5*n^5 - 3*(d*x
+ c)^n*A*a^2*b*c^2*d^5*n^5 + 27*(d*x + c)^n*A*a^3*c*d^6*n^5 + 6*(d*x + c)^n*D*a^3*c^3*d^4*n^4*x + 18*(d*x + c
)^n*C*a^2*b*c^3*d^4*n^4*x + 18*(d*x + c)^n*B*a*b^2*c^3*d^4*n^4*x + 6*(d*x + c)^n*A*b^3*c^3*d^4*n^4*x - 44*(d*x
+ c)^n*C*a^3*c^2*d^5*n^4*x - 132*(d*x + c)^n*B*a^2*b*c^2*d^5*n^4*x - 132*(d*x + c)^n*A*a*b^2*c^2*d^5*n^4*x +
245*(d*x + c)^n*B*a^3*c*d^6*n^4*x + 735*(d*x + c)^n*A*a^2*b*c*d^6*n^4*x + 295*(d*x + c)^n*A*a^3*d^7*n^4*x - 18
0*(d*x + c)^n*D*a*b^2*c^4*d^3*n^3*x^2 - 60*(d*x + c)^n*C*b^3*c^4*d^3*n^3*x^2 + 504*(d*x + c)^n*D*a^2*b*c^3*d^4
*n^3*x^2 + 504*(d*x + c)^n*C*a*b^2*c^3*d^4*n^3*x^2 + 168*(d*x + c)^n*B*b^3*c^3*d^4*n^3*x^2 - 375*(d*x + c)^n*D
*a^3*c^2*d^5*n^3*x^2 - 1125*(d*x + c)^n*C*a^2*b*c^2*d^5*n^3*x^2 - 1125*(d*x + c)^n*B*a*b^2*c^2*d^5*n^3*x^2 - 3
75*(d*x + c)^n*A*b^3*c^2*d^5*n^3*x^2 + 817*(d*x + c)^n*C*a^3*c*d^6*n^3*x^2 + 2451*(d*x + c)^n*B*a^2*b*c*d^6*n^
3*x^2 + 2451*(d*x + c)^n*A*a*b^2*c*d^6*n^3*x^2 + 1420*(d*x + c)^n*B*a^3*d^7*n^3*x^2 + 4260*(d*x + c)^n*A*a^2*b
*d^7*n^3*x^2 - 360*(d*x + c)^n*D*b^3*c^4*d^3*n^2*x^3 + 1380*(d*x + c)^n*D*a*b^2*c^3*d^4*n^2*x^3 + 460*(d*x + c
)^n*C*b^3*c^3*d^4*n^2*x^3 - 1824*(d*x + c)^n*D*a^2*b*c^2*d^5*n^2*x^3 - 1824*(d*x + c)^n*C*a*b^2*c^2*d^5*n^2*x^
3 - 608*(d*x + c)^n*B*b^3*c^2*d^5*n^2*x^3 + 844*(d*x + c)^n*D*a^3*c*d^6*n^2*x^3 + 2532*(d*x + c)^n*C*a^2*b*c*d
^6*n^2*x^3 + 2532*(d*x + c)^n*B*a*b^2*c*d^6*n^2*x^3 + 844*(d*x + c)^n*A*b^3*c*d^6*n^2*x^3 + 3112*(d*x + c)^n*C
*a^3*d^7*n^2*x^3 + 9336*(d*x + c)^n*B*a^2*b*d^7*n^2*x^3 + 9336*(d*x + c)^n*A*a*b^2*d^7*n^2*x^3 + 180*(d*x + c)
^n*D*b^3*c^3*d^4*n*x^4 - 630*(d*x + c)^n*D*a*b^2*c^2*d^5*n*x^4 - 210*(d*x + c)^n*C*b^3*c^2*d^5*n*x^4 + 756*(d*
x + c)^n*D*a^2*b*c*d^6*n*x^4 + 756*(d*x + c)^n*C*a*b^2*c*d^6*n*x^4 + 252*(d*x + c)^n*B*b^3*c*d^6*n*x^4 + 2952*
(d*x + c)^n*D*a^3*d^7*n*x^4 + 8856*(d*x + c)^n*C*a^2*b*d^7*n*x^4 + 8856*(d*x + c)^n*B*a*b^2*d^7*n*x^4 + 2952*(
d*x + c)^n*A*b^3*d^7*n*x^4 + 3024*(d*x + c)^n*D*a^2*b*d^7*x^5 + 3024*(d*x + c)^n*C*a*b^2*d^7*x^5 + 1008*(d*x +
c)^n*B*b^3*d^7*x^5 + 2*(d*x + c)^n*C*a^3*c^3*d^4*n^4 + 6*(d*x + c)^n*B*a^2*b*c^3*d^4*n^4 + 6*(d*x + c)^n*A*a*
b^2*c^3*d^4*n^4 - 25*(d*x + c)^n*B*a^3*c^2*d^5*n^4 - 75*(d*x + c)^n*A*a^2*b*c^2*d^5*n^4 + 295*(d*x + c)^n*A*a^
3*c*d^6*n^4 - 72*(d*x + c)^n*D*a^2*b*c^4*d^3*n^3*x - 72*(d*x + c)^n*C*a*b^2*c^4*d^3*n^3*x - 24*(d*x + c)^n*B*b
^3*c^4*d^3*n^3*x + 108*(d*x + c)^n*D*a^3*c^3*d^4*n^3*x + 324*(d*x + c)^n*C*a^2*b*c^3*d^4*n^3*x + 324*(d*x + c)
^n*B*a*b^2*c^3*d^4*n^3*x + 108*(d*x + c)^n*A*b^3*c^3*d^4*n^3*x - 358*(d*x + c)^n*C*a^3*c^2*d^5*n^3*x - 1074*(d
*x + c)^n*B*a^2*b*c^2*d^5*n^3*x - 1074*(d*x + c)^n*A*a*b^2*c^2*d^5*n^3*x + 1175*(d*x + c)^n*B*a^3*c*d^6*n^3*x
+ 3525*(d*x + c)^n*A*a^2*b*c*d^6*n^3*x + 1665*(d*x + c)^n*A*a^3*d^7*n^3*x + 360*(d*x + c)^n*D*b^3*c^5*d^2*n^2*
x^2 - 1440*(d*x + c)^n*D*a*b^2*c^4*d^3*n^2*x^2 - 480*(d*x + c)^n*C*b^3*c^4*d^3*n^2*x^2 + 1980*(d*x + c)^n*D*a^
2*b*c^3*d^4*n^2*x^2 + 1980*(d*x + c)^n*C*a*b^2*c^3*d^4*n^2*x^2 + 660*(d*x + c)^n*B*b^3*c^3*d^4*n^2*x^2 - 951*(
d*x + c)^n*D*a^3*c^2*d^5*n^2*x^2 - 2853*(d*x + c)^n*C*a^2*b*c^2*d^5*n^2*x^2 - 2853*(d*x + c)^n*B*a*b^2*c^2*d^5
*n^2*x^2 - 951*(d*x + c)^n*A*b^3*c^2*d^5*n^2*x^2 + 1478*(d*x + c)^n*C*a^3*c*d^6*n^2*x^2 + 4434*(d*x + c)^n*B*a
^2*b*c*d^6*n^2*x^2 + 4434*(d*x + c)^n*A*a*b^2*c*d^6*n^2*x^2 + 3929*(d*x + c)^n*B*a^3*d^7*n^2*x^2 + 11787*(d*x
+ c)^n*A*a^2*b*d^7*n^2*x^2 - 240*(d*x + c)^n*D*b^3*c^4*d^3*n*x^3 + 840*(d*x + c)^n*D*a*b^2*c^3*d^4*n*x^3 + 280
*(d*x + c)^n*C*b^3*c^3*d^4*n*x^3 - 1008*(d*x + c)^n*D*a^2*b*c^2*d^5*n*x^3 - 1008*(d*x + c)^n*C*a*b^2*c^2*d^5*n
*x^3 - 336*(d*x + c)^n*B*b^3*c^2*d^5*n*x^3 + 420*(d*x + c)^n*D*a^3*c*d^6*n*x^3 + 1260*(d*x + c)^n*C*a^2*b*c*d^
6*n*x^3 + 1260*(d*x + c)^n*B*a*b^2*c*d^6*n*x^3 + 420*(d*x + c)^n*A*b^3*c*d^6*n*x^3 + 3796*(d*x + c)^n*C*a^3*d^
7*n*x^3 + 11388*(d*x + c)^n*B*a^2*b*d^7*n*x^3 + 11388*(d*x + c)^n*A*a*b^2*d^7*n*x^3 + 1260*(d*x + c)^n*D*a^3*d
^7*x^4 + 3780*(d*x + c)^n*C*a^2*b*d^7*x^4 + 3780*(d*x + c)^n*B*a*b^2*d^7*x^4 + 1260*(d*x + c)^n*A*b^3*d^7*x^4
- 6*(d*x + c)^n*D*a^3*c^4*d^3*n^3 - 18*(d*x + c)^n*C*a^2*b*c^4*d^3*n^3 - 18*(d*x + c)^n*B*a*b^2*c^4*d^3*n^3 -
6*(d*x + c)^n*A*b^3*c^4*d^3*n^3 + 44*(d*x + c)^n*C*a^3*c^3*d^4*n^3 + 132*(d*x + c)^n*B*a^2*b*c^3*d^4*n^3 + 132
*(d*x + c)^n*A*a*b^2*c^3*d^4*n^3 - 245*(d*x + c)^n*B*a^3*c^2*d^5*n^3 - 735*(d*x + c)^n*A*a^2*b*c^2*d^5*n^3 + 1
665*(d*x + c)^n*A*a^3*c*d^6*n^3 + 360*(d*x + c)^n*D*a*b^2*c^5*d^2*n^2*x + 120*(d*x + c)^n*C*b^3*c^5*d^2*n^2*x
- 936*(d*x + c)^n*D*a^2*b*c^4*d^3*n^2*x - 936*(d*x + c)^n*C*a*b^2*c^4*d^3*n^2*x - 312*(d*x + c)^n*B*b^3*c^4*d^
3*n^2*x + 642*(d*x + c)^n*D*a^3*c^3*d^4*n^2*x + 1926*(d*x + c)^n*C*a^2*b*c^3*d^4*n^2*x + 1926*(d*x + c)^n*B*a*
b^2*c^3*d^4*n^2*x + 642*(d*x + c)^n*A*b^3*c^3*d^4*n^2*x - 1276*(d*x + c)^n*C*a^3*c^2*d^5*n^2*x - 3828*(d*x + c
)^n*B*a^2*b*c^2*d^5*n^2*x - 3828*(d*x + c)^n*A*a*b^2*c^2*d^5*n^2*x + 2754*(d*x + c)^n*B*a^3*c*d^6*n^2*x + 8262
*(d*x + c)^n*A*a^2*b*c*d^6*n^2*x + 5104*(d*x + c)^n*A*a^3*d^7*n^2*x + 360*(d*x + c)^n*D*b^3*c^5*d^2*n*x^2 - 12
60*(d*x + c)^n*D*a*b^2*c^4*d^3*n*x^2 - 420*(d*x + c)^n*C*b^3*c^4*d^3*n*x^2 + 1512*(d*x + c)^n*D*a^2*b*c^3*d^4*
n*x^2 + 1512*(d*x + c)^n*C*a*b^2*c^3*d^4*n*x^2 + 504*(d*x + c)^n*B*b^3*c^3*d^4*n*x^2 - 630*(d*x + c)^n*D*a^3*c
^2*d^5*n*x^2 - 1890*(d*x + c)^n*C*a^2*b*c^2*d^5*n*x^2 - 1890*(d*x + c)^n*B*a*b^2*c^2*d^5*n*x^2 - 630*(d*x + c)
^n*A*b^3*c^2*d^5*n*x^2 + 840*(d*x + c)^n*C*a^3*c*d^6*n*x^2 + 2520*(d*x + c)^n*B*a^2*b*c*d^6*n*x^2 + 2520*(d*x
+ c)^n*A*a*b^2*c*d^6*n*x^2 + 5274*(d*x + c)^n*B*a^3*d^7*n*x^2 + 15822*(d*x + c)^n*A*a^2*b*d^7*n*x^2 + 1680*(d*
x + c)^n*C*a^3*d^7*x^3 + 5040*(d*x + c)^n*B*a^2*b*d^7*x^3 + 5040*(d*x + c)^n*A*a*b^2*d^7*x^3 + 72*(d*x + c)^n*
D*a^2*b*c^5*d^2*n^2 + 72*(d*x + c)^n*C*a*b^2*c^5*d^2*n^2 + 24*(d*x + c)^n*B*b^3*c^5*d^2*n^2 - 108*(d*x + c)^n*
D*a^3*c^4*d^3*n^2 - 324*(d*x + c)^n*C*a^2*b*c^4*d^3*n^2 - 324*(d*x + c)^n*B*a*b^2*c^4*d^3*n^2 - 108*(d*x + c)^
n*A*b^3*c^4*d^3*n^2 + 358*(d*x + c)^n*C*a^3*c^3*d^4*n^2 + 1074*(d*x + c)^n*B*a^2*b*c^3*d^4*n^2 + 1074*(d*x + c
)^n*A*a*b^2*c^3*d^4*n^2 - 1175*(d*x + c)^n*B*a^3*c^2*d^5*n^2 - 3525*(d*x + c)^n*A*a^2*b*c^2*d^5*n^2 + 5104*(d*
x + c)^n*A*a^3*c*d^6*n^2 - 720*(d*x + c)^n*D*b^3*c^6*d*n*x + 2520*(d*x + c)^n*D*a*b^2*c^5*d^2*n*x + 840*(d*x +
c)^n*C*b^3*c^5*d^2*n*x - 3024*(d*x + c)^n*D*a^2*b*c^4*d^3*n*x - 3024*(d*x + c)^n*C*a*b^2*c^4*d^3*n*x - 1008*(
d*x + c)^n*B*b^3*c^4*d^3*n*x + 1260*(d*x + c)^n*D*a^3*c^3*d^4*n*x + 3780*(d*x + c)^n*C*a^2*b*c^3*d^4*n*x + 378
0*(d*x + c)^n*B*a*b^2*c^3*d^4*n*x + 1260*(d*x + c)^n*A*b^3*c^3*d^4*n*x - 1680*(d*x + c)^n*C*a^3*c^2*d^5*n*x -
5040*(d*x + c)^n*B*a^2*b*c^2*d^5*n*x - 5040*(d*x + c)^n*A*a*b^2*c^2*d^5*n*x + 2520*(d*x + c)^n*B*a^3*c*d^6*n*x
+ 7560*(d*x + c)^n*A*a^2*b*c*d^6*n*x + 8028*(d*x + c)^n*A*a^3*d^7*n*x + 2520*(d*x + c)^n*B*a^3*d^7*x^2 + 7560
*(d*x + c)^n*A*a^2*b*d^7*x^2 - 360*(d*x + c)^n*D*a*b^2*c^6*d*n - 120*(d*x + c)^n*C*b^3*c^6*d*n + 936*(d*x + c)
^n*D*a^2*b*c^5*d^2*n + 936*(d*x + c)^n*C*a*b^2*c^5*d^2*n + 312*(d*x + c)^n*B*b^3*c^5*d^2*n - 642*(d*x + c)^n*D
*a^3*c^4*d^3*n - 1926*(d*x + c)^n*C*a^2*b*c^4*d^3*n - 1926*(d*x + c)^n*B*a*b^2*c^4*d^3*n - 642*(d*x + c)^n*A*b
^3*c^4*d^3*n + 1276*(d*x + c)^n*C*a^3*c^3*d^4*n + 3828*(d*x + c)^n*B*a^2*b*c^3*d^4*n + 3828*(d*x + c)^n*A*a*b^
2*c^3*d^4*n - 2754*(d*x + c)^n*B*a^3*c^2*d^5*n - 8262*(d*x + c)^n*A*a^2*b*c^2*d^5*n + 8028*(d*x + c)^n*A*a^3*c
*d^6*n + 5040*(d*x + c)^n*A*a^3*d^7*x + 720*(d*x + c)^n*D*b^3*c^7 - 2520*(d*x + c)^n*D*a*b^2*c^6*d - 840*(d*x
+ c)^n*C*b^3*c^6*d + 3024*(d*x + c)^n*D*a^2*b*c^5*d^2 + 3024*(d*x + c)^n*C*a*b^2*c^5*d^2 + 1008*(d*x + c)^n*B*
b^3*c^5*d^2 - 1260*(d*x + c)^n*D*a^3*c^4*d^3 - 3780*(d*x + c)^n*C*a^2*b*c^4*d^3 - 3780*(d*x + c)^n*B*a*b^2*c^4
*d^3 - 1260*(d*x + c)^n*A*b^3*c^4*d^3 + 1680*(d*x + c)^n*C*a^3*c^3*d^4 + 5040*(d*x + c)^n*B*a^2*b*c^3*d^4 + 50
40*(d*x + c)^n*A*a*b^2*c^3*d^4 - 2520*(d*x + c)^n*B*a^3*c^2*d^5 - 7560*(d*x + c)^n*A*a^2*b*c^2*d^5 + 5040*(d*x
+ c)^n*A*a^3*c*d^6)/(d^7*n^7 + 28*d^7*n^6 + 322*d^7*n^5 + 1960*d^7*n^4 + 6769*d^7*n^3 + 13132*d^7*n^2 + 13068
*d^7*n + 5040*d^7)
________________________________________________________________________________________
maple [B] time = 0.04, size = 5003, normalized size = 11.00 \[ \text {output too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((b*x+a)^3*(d*x+c)^n*(D*x^3+C*x^2+B*x+A),x)
[Out]
result too large to display
________________________________________________________________________________________
maxima [B] time = 0.76, size = 1802, normalized size = 3.96 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((b*x+a)^3*(d*x+c)^n*(D*x^3+C*x^2+B*x+A),x, algorithm="maxima")
[Out]
(d^2*(n + 1)*x^2 + c*d*n*x - c^2)*(d*x + c)^n*B*a^3/((n^2 + 3*n + 2)*d^2) + 3*(d^2*(n + 1)*x^2 + c*d*n*x - c^2
)*(d*x + c)^n*A*a^2*b/((n^2 + 3*n + 2)*d^2) + (d*x + c)^(n + 1)*A*a^3/(d*(n + 1)) + ((n^2 + 3*n + 2)*d^3*x^3 +
(n^2 + n)*c*d^2*x^2 - 2*c^2*d*n*x + 2*c^3)*(d*x + c)^n*C*a^3/((n^3 + 6*n^2 + 11*n + 6)*d^3) + 3*((n^2 + 3*n +
2)*d^3*x^3 + (n^2 + n)*c*d^2*x^2 - 2*c^2*d*n*x + 2*c^3)*(d*x + c)^n*B*a^2*b/((n^3 + 6*n^2 + 11*n + 6)*d^3) +
3*((n^2 + 3*n + 2)*d^3*x^3 + (n^2 + n)*c*d^2*x^2 - 2*c^2*d*n*x + 2*c^3)*(d*x + c)^n*A*a*b^2/((n^3 + 6*n^2 + 11
*n + 6)*d^3) + ((n^3 + 6*n^2 + 11*n + 6)*d^4*x^4 + (n^3 + 3*n^2 + 2*n)*c*d^3*x^3 - 3*(n^2 + n)*c^2*d^2*x^2 + 6
*c^3*d*n*x - 6*c^4)*(d*x + c)^n*D*a^3/((n^4 + 10*n^3 + 35*n^2 + 50*n + 24)*d^4) + 3*((n^3 + 6*n^2 + 11*n + 6)*
d^4*x^4 + (n^3 + 3*n^2 + 2*n)*c*d^3*x^3 - 3*(n^2 + n)*c^2*d^2*x^2 + 6*c^3*d*n*x - 6*c^4)*(d*x + c)^n*C*a^2*b/(
(n^4 + 10*n^3 + 35*n^2 + 50*n + 24)*d^4) + 3*((n^3 + 6*n^2 + 11*n + 6)*d^4*x^4 + (n^3 + 3*n^2 + 2*n)*c*d^3*x^3
- 3*(n^2 + n)*c^2*d^2*x^2 + 6*c^3*d*n*x - 6*c^4)*(d*x + c)^n*B*a*b^2/((n^4 + 10*n^3 + 35*n^2 + 50*n + 24)*d^4
) + ((n^3 + 6*n^2 + 11*n + 6)*d^4*x^4 + (n^3 + 3*n^2 + 2*n)*c*d^3*x^3 - 3*(n^2 + n)*c^2*d^2*x^2 + 6*c^3*d*n*x
- 6*c^4)*(d*x + c)^n*A*b^3/((n^4 + 10*n^3 + 35*n^2 + 50*n + 24)*d^4) + 3*((n^4 + 10*n^3 + 35*n^2 + 50*n + 24)*
d^5*x^5 + (n^4 + 6*n^3 + 11*n^2 + 6*n)*c*d^4*x^4 - 4*(n^3 + 3*n^2 + 2*n)*c^2*d^3*x^3 + 12*(n^2 + n)*c^3*d^2*x^
2 - 24*c^4*d*n*x + 24*c^5)*(d*x + c)^n*D*a^2*b/((n^5 + 15*n^4 + 85*n^3 + 225*n^2 + 274*n + 120)*d^5) + 3*((n^4
+ 10*n^3 + 35*n^2 + 50*n + 24)*d^5*x^5 + (n^4 + 6*n^3 + 11*n^2 + 6*n)*c*d^4*x^4 - 4*(n^3 + 3*n^2 + 2*n)*c^2*d
^3*x^3 + 12*(n^2 + n)*c^3*d^2*x^2 - 24*c^4*d*n*x + 24*c^5)*(d*x + c)^n*C*a*b^2/((n^5 + 15*n^4 + 85*n^3 + 225*n
^2 + 274*n + 120)*d^5) + ((n^4 + 10*n^3 + 35*n^2 + 50*n + 24)*d^5*x^5 + (n^4 + 6*n^3 + 11*n^2 + 6*n)*c*d^4*x^4
- 4*(n^3 + 3*n^2 + 2*n)*c^2*d^3*x^3 + 12*(n^2 + n)*c^3*d^2*x^2 - 24*c^4*d*n*x + 24*c^5)*(d*x + c)^n*B*b^3/((n
^5 + 15*n^4 + 85*n^3 + 225*n^2 + 274*n + 120)*d^5) + 3*((n^5 + 15*n^4 + 85*n^3 + 225*n^2 + 274*n + 120)*d^6*x^
6 + (n^5 + 10*n^4 + 35*n^3 + 50*n^2 + 24*n)*c*d^5*x^5 - 5*(n^4 + 6*n^3 + 11*n^2 + 6*n)*c^2*d^4*x^4 + 20*(n^3 +
3*n^2 + 2*n)*c^3*d^3*x^3 - 60*(n^2 + n)*c^4*d^2*x^2 + 120*c^5*d*n*x - 120*c^6)*(d*x + c)^n*D*a*b^2/((n^6 + 21
*n^5 + 175*n^4 + 735*n^3 + 1624*n^2 + 1764*n + 720)*d^6) + ((n^5 + 15*n^4 + 85*n^3 + 225*n^2 + 274*n + 120)*d^
6*x^6 + (n^5 + 10*n^4 + 35*n^3 + 50*n^2 + 24*n)*c*d^5*x^5 - 5*(n^4 + 6*n^3 + 11*n^2 + 6*n)*c^2*d^4*x^4 + 20*(n
^3 + 3*n^2 + 2*n)*c^3*d^3*x^3 - 60*(n^2 + n)*c^4*d^2*x^2 + 120*c^5*d*n*x - 120*c^6)*(d*x + c)^n*C*b^3/((n^6 +
21*n^5 + 175*n^4 + 735*n^3 + 1624*n^2 + 1764*n + 720)*d^6) + ((n^6 + 21*n^5 + 175*n^4 + 735*n^3 + 1624*n^2 + 1
764*n + 720)*d^7*x^7 + (n^6 + 15*n^5 + 85*n^4 + 225*n^3 + 274*n^2 + 120*n)*c*d^6*x^6 - 6*(n^5 + 10*n^4 + 35*n^
3 + 50*n^2 + 24*n)*c^2*d^5*x^5 + 30*(n^4 + 6*n^3 + 11*n^2 + 6*n)*c^3*d^4*x^4 - 120*(n^3 + 3*n^2 + 2*n)*c^4*d^3
*x^3 + 360*(n^2 + n)*c^5*d^2*x^2 - 720*c^6*d*n*x + 720*c^7)*(d*x + c)^n*D*b^3/((n^7 + 28*n^6 + 322*n^5 + 1960*
n^4 + 6769*n^3 + 13132*n^2 + 13068*n + 5040)*d^7)
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int {\left (a+b\,x\right )}^3\,{\left (c+d\,x\right )}^n\,\left (A+B\,x+C\,x^2+x^3\,D\right ) \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((a + b*x)^3*(c + d*x)^n*(A + B*x + C*x^2 + x^3*D),x)
[Out]
int((a + b*x)^3*(c + d*x)^n*(A + B*x + C*x^2 + x^3*D), x)
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((b*x+a)**3*(d*x+c)**n*(D*x**3+C*x**2+B*x+A),x)
[Out]
Timed out
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