3.34 \(\int (a+b x)^m (c+d x)^n (A+B x+C x^2+D x^3) \, dx\)

Optimal. Leaf size=610 \[ \frac{(a+b x)^{m+1} (c+d x)^n \left (\frac{b (c+d x)}{b c-a d}\right )^{-n} \, _2F_1\left (m+1,-n;m+2;-\frac{d (a+b x)}{b c-a d}\right ) \left (d (m+n+2) \left (-a^2 b d (C d (n+1) (m+n+4)-c D (m+2) (m+3 n+6))+a^3 d^2 D (n+1) (m+2 n+6)+a b^2 c (m+2) (c D (m+3)-C d (m+n+4))+A b^3 d^2 \left (m^2+m (2 n+7)+n^2+7 n+12\right )\right )-(a d (n+1)+b c (m+1)) \left (a^2 d^2 D \left (m^2+m (3 n+8)+3 \left (n^2+5 n+6\right )\right )+a b d \left (c D (m+2) (m+3 n+6)-C d \left (m^2+m (3 n+8)+2 \left (n^2+6 n+8\right )\right )\right )+b^2 \left (B d^2 \left (m^2+m (2 n+7)+n^2+7 n+12\right )+c^2 D \left (m^2+5 m+6\right )-c C d (m+2) (m+n+4)\right )\right )\right )}{b^4 d^3 (m+1) (m+n+2) (m+n+3) (m+n+4)}+\frac{(a+b x)^{m+1} (c+d x)^{n+1} \left (a^2 d^2 D \left (m^2+m (3 n+8)+3 \left (n^2+5 n+6\right )\right )+a b d \left (c D (m+2) (m+3 n+6)-C d \left (m^2+m (3 n+8)+2 \left (n^2+6 n+8\right )\right )\right )+b^2 \left (B d^2 \left (m^2+m (2 n+7)+n^2+7 n+12\right )+c^2 D \left (m^2+5 m+6\right )-c C d (m+2) (m+n+4)\right )\right )}{b^3 d^3 (m+n+2) (m+n+3) (m+n+4)}-\frac{(a+b x)^{m+2} (c+d x)^{n+1} (a d D (2 m+3 n+9)+b (c D (m+3)-C d (m+n+4)))}{b^3 d^2 (m+n+3) (m+n+4)}+\frac{D (a+b x)^{m+3} (c+d x)^{n+1}}{b^3 d (m+n+4)} \]

[Out]

((a^2*d^2*D*(m^2 + m*(8 + 3*n) + 3*(6 + 5*n + n^2)) + b^2*(c^2*D*(6 + 5*m + m^2) - c*C*d*(2 + m)*(4 + m + n) +
B*d^2*(12 + m^2 + 7*n + n^2 + m*(7 + 2*n))) + a*b*d*(c*D*(2 + m)*(6 + m + 3*n) - C*d*(m^2 + m*(8 + 3*n) + 2*(
8 + 6*n + n^2))))*(a + b*x)^(1 + m)*(c + d*x)^(1 + n))/(b^3*d^3*(2 + m + n)*(3 + m + n)*(4 + m + n)) - ((a*d*D
*(9 + 2*m + 3*n) + b*(c*D*(3 + m) - C*d*(4 + m + n)))*(a + b*x)^(2 + m)*(c + d*x)^(1 + n))/(b^3*d^2*(3 + m + n
)*(4 + m + n)) + (D*(a + b*x)^(3 + m)*(c + d*x)^(1 + n))/(b^3*d*(4 + m + n)) + ((d*(2 + m + n)*(a^3*d^2*D*(1 +
n)*(6 + m + 2*n) + a*b^2*c*(2 + m)*(c*D*(3 + m) - C*d*(4 + m + n)) + A*b^3*d^2*(12 + m^2 + 7*n + n^2 + m*(7 +
2*n)) - a^2*b*d*(C*d*(1 + n)*(4 + m + n) - c*D*(2 + m)*(6 + m + 3*n))) - (b*c*(1 + m) + a*d*(1 + n))*(a^2*d^2
*D*(m^2 + m*(8 + 3*n) + 3*(6 + 5*n + n^2)) + b^2*(c^2*D*(6 + 5*m + m^2) - c*C*d*(2 + m)*(4 + m + n) + B*d^2*(1
2 + m^2 + 7*n + n^2 + m*(7 + 2*n))) + a*b*d*(c*D*(2 + m)*(6 + m + 3*n) - C*d*(m^2 + m*(8 + 3*n) + 2*(8 + 6*n +
n^2)))))*(a + b*x)^(1 + m)*(c + d*x)^n*Hypergeometric2F1[1 + m, -n, 2 + m, -((d*(a + b*x))/(b*c - a*d))])/(b^
4*d^3*(1 + m)*(2 + m + n)*(3 + m + n)*(4 + m + n)*((b*(c + d*x))/(b*c - a*d))^n)

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Rubi [A]  time = 1.06912, antiderivative size = 605, normalized size of antiderivative = 0.99, number of steps used = 5, number of rules used = 5, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {1623, 951, 80, 70, 69} \[ \frac{(a+b x)^{m+1} (c+d x)^n \left (\frac{b (c+d x)}{b c-a d}\right )^{-n} \, _2F_1\left (m+1,-n;m+2;-\frac{d (a+b x)}{b c-a d}\right ) \left (-\frac{(a d (n+1)+b c (m+1)) \left (a^2 d^2 D \left (m^2+m (3 n+8)+3 \left (n^2+5 n+6\right )\right )+a b d \left (c D (m+2) (m+3 n+6)-C d \left (m^2+m (3 n+8)+2 \left (n^2+6 n+8\right )\right )\right )+b^2 \left (B d^2 \left (m^2+m (2 n+7)+n^2+7 n+12\right )+c^2 D \left (m^2+5 m+6\right )-c C d (m+2) (m+n+4)\right )\right )}{d (m+n+2)}-a^2 b d (C d (n+1) (m+n+4)-c D (m+2) (m+3 n+6))+a^3 d^2 D (n+1) (m+2 n+6)+a b^2 c (m+2) (c D (m+3)-C d (m+n+4))+A b^3 d^2 \left (m^2+m (2 n+7)+n^2+7 n+12\right )\right )}{b^4 d^2 (m+1) (m+n+3) (m+n+4)}+\frac{(a+b x)^{m+1} (c+d x)^{n+1} \left (a^2 d^2 D \left (m^2+m (3 n+8)+3 \left (n^2+5 n+6\right )\right )+a b d \left (c D (m+2) (m+3 n+6)-C d \left (m^2+m (3 n+8)+2 \left (n^2+6 n+8\right )\right )\right )+b^2 \left (B d^2 \left (m^2+m (2 n+7)+n^2+7 n+12\right )+c^2 D \left (m^2+5 m+6\right )-c C d (m+2) (m+n+4)\right )\right )}{b^3 d^3 (m+n+2) (m+n+3) (m+n+4)}-\frac{(a+b x)^{m+2} (c+d x)^{n+1} (a d D (2 m+3 n+9)+b c D (m+3)-b C d (m+n+4))}{b^3 d^2 (m+n+3) (m+n+4)}+\frac{D (a+b x)^{m+3} (c+d x)^{n+1}}{b^3 d (m+n+4)} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^m*(c + d*x)^n*(A + B*x + C*x^2 + D*x^3),x]

[Out]

((a^2*d^2*D*(m^2 + m*(8 + 3*n) + 3*(6 + 5*n + n^2)) + b^2*(c^2*D*(6 + 5*m + m^2) - c*C*d*(2 + m)*(4 + m + n) +
B*d^2*(12 + m^2 + 7*n + n^2 + m*(7 + 2*n))) + a*b*d*(c*D*(2 + m)*(6 + m + 3*n) - C*d*(m^2 + m*(8 + 3*n) + 2*(
8 + 6*n + n^2))))*(a + b*x)^(1 + m)*(c + d*x)^(1 + n))/(b^3*d^3*(2 + m + n)*(3 + m + n)*(4 + m + n)) - ((b*c*D
*(3 + m) - b*C*d*(4 + m + n) + a*d*D*(9 + 2*m + 3*n))*(a + b*x)^(2 + m)*(c + d*x)^(1 + n))/(b^3*d^2*(3 + m + n
)*(4 + m + n)) + (D*(a + b*x)^(3 + m)*(c + d*x)^(1 + n))/(b^3*d*(4 + m + n)) + ((a^3*d^2*D*(1 + n)*(6 + m + 2*
n) + a*b^2*c*(2 + m)*(c*D*(3 + m) - C*d*(4 + m + n)) + A*b^3*d^2*(12 + m^2 + 7*n + n^2 + m*(7 + 2*n)) - a^2*b*
d*(C*d*(1 + n)*(4 + m + n) - c*D*(2 + m)*(6 + m + 3*n)) - ((b*c*(1 + m) + a*d*(1 + n))*(a^2*d^2*D*(m^2 + m*(8
+ 3*n) + 3*(6 + 5*n + n^2)) + b^2*(c^2*D*(6 + 5*m + m^2) - c*C*d*(2 + m)*(4 + m + n) + B*d^2*(12 + m^2 + 7*n +
n^2 + m*(7 + 2*n))) + a*b*d*(c*D*(2 + m)*(6 + m + 3*n) - C*d*(m^2 + m*(8 + 3*n) + 2*(8 + 6*n + n^2)))))/(d*(2
+ m + n)))*(a + b*x)^(1 + m)*(c + d*x)^n*Hypergeometric2F1[1 + m, -n, 2 + m, -((d*(a + b*x))/(b*c - a*d))])/(
b^4*d^2*(1 + m)*(3 + m + n)*(4 + m + n)*((b*(c + d*x))/(b*c - a*d))^n)

Rule 1623

Int[(Px_)*((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> With[{q = Expon[Px, x], k = Coef
f[Px, x, Expon[Px, x]]}, Simp[(k*(a + b*x)^(m + q)*(c + d*x)^(n + 1))/(d*b^q*(m + n + q + 1)), x] + Dist[1/(d*
b^q*(m + n + q + 1)), Int[(a + b*x)^m*(c + d*x)^n*ExpandToSum[d*b^q*(m + n + q + 1)*Px - d*k*(m + n + q + 1)*(
a + b*x)^q - k*(b*c - a*d)*(m + q)*(a + b*x)^(q - 1), x], x], x] /; NeQ[m + n + q + 1, 0]] /; FreeQ[{a, b, c,
d, m, n}, x] && PolyQ[Px, x] && GtQ[Expon[Px, x], 2]

Rule 951

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :
> Simp[(c^p*(d + e*x)^(m + 2*p)*(f + g*x)^(n + 1))/(g*e^(2*p)*(m + n + 2*p + 1)), x] + Dist[1/(g*e^(2*p)*(m +
n + 2*p + 1)), Int[(d + e*x)^m*(f + g*x)^n*ExpandToSum[g*(m + n + 2*p + 1)*(e^(2*p)*(a + b*x + c*x^2)^p - c^p*
(d + e*x)^(2*p)) - c^p*(e*f - d*g)*(m + 2*p)*(d + e*x)^(2*p - 1), x], 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] && NeQ[m + n + 2*
p + 1, 0] && (IntegerQ[n] ||  !IntegerQ[m])

Rule 80

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*(c + d*x)
^(n + 1)*(e + f*x)^(p + 1))/(d*f*(n + p + 2)), x] + Dist[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(
d*f*(n + p + 2)), Int[(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 2,
0]

Rule 70

Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Dist[(c + d*x)^FracPart[n]/((b/(b*c - a*d)
)^IntPart[n]*((b*(c + d*x))/(b*c - a*d))^FracPart[n]), Int[(a + b*x)^m*Simp[(b*c)/(b*c - a*d) + (b*d*x)/(b*c -
a*d), x]^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] &&  !IntegerQ[m] &&  !IntegerQ[n] &&
(RationalQ[m] ||  !SimplerQ[n + 1, m + 1])

Rule 69

Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*Hypergeometric2F1[
-n, m + 1, m + 2, -((d*(a + b*x))/(b*c - a*d))])/(b*(m + 1)*(b/(b*c - a*d))^n), x] /; FreeQ[{a, b, c, d, m, n}
, x] && NeQ[b*c - a*d, 0] &&  !IntegerQ[m] &&  !IntegerQ[n] && GtQ[b/(b*c - a*d), 0] && (RationalQ[m] ||  !(Ra
tionalQ[n] && GtQ[-(d/(b*c - a*d)), 0]))

Rubi steps

\begin{align*} \int (a+b x)^m (c+d x)^n \left (A+B x+C x^2+D x^3\right ) \, dx &=\frac{D (a+b x)^{3+m} (c+d x)^{1+n}}{b^3 d (4+m+n)}+\frac{\int (a+b x)^m (c+d x)^n \left (A b^3 d (4+m+n)-a^2 D (b c (3+m)+a d (1+n))-b \left (2 a b c D (3+m)-b^2 B d (4+m+n)+a^2 d D (6+m+3 n)\right ) x-b^2 (b c D (3+m)-b C d (4+m+n)+a d D (9+2 m+3 n)) x^2\right ) \, dx}{b^3 d (4+m+n)}\\ &=-\frac{(b c D (3+m)-b C d (4+m+n)+a d D (9+2 m+3 n)) (a+b x)^{2+m} (c+d x)^{1+n}}{b^3 d^2 (3+m+n) (4+m+n)}+\frac{D (a+b x)^{3+m} (c+d x)^{1+n}}{b^3 d (4+m+n)}+\frac{\int (a+b x)^m (c+d x)^n \left (b^2 \left (a^3 d^2 D (1+n) (6+m+2 n)+a b^2 c (2+m) (c D (3+m)-C d (4+m+n))+A b^3 d^2 \left (12+m^2+7 n+n^2+m (7+2 n)\right )-a^2 b d (C d (1+n) (4+m+n)-c D (2+m) (6+m+3 n))\right )+b^3 \left (a^2 d^2 D \left (m^2+m (8+3 n)+3 \left (6+5 n+n^2\right )\right )+b^2 \left (c^2 D \left (6+5 m+m^2\right )-c C d (2+m) (4+m+n)+B d^2 \left (12+m^2+7 n+n^2+m (7+2 n)\right )\right )+a b d \left (c D (2+m) (6+m+3 n)-C d \left (m^2+m (8+3 n)+2 \left (8+6 n+n^2\right )\right )\right )\right ) x\right ) \, dx}{b^5 d^2 (3+m+n) (4+m+n)}\\ &=\frac{\left (a^2 d^2 D \left (m^2+m (8+3 n)+3 \left (6+5 n+n^2\right )\right )+b^2 \left (c^2 D \left (6+5 m+m^2\right )-c C d (2+m) (4+m+n)+B d^2 \left (12+m^2+7 n+n^2+m (7+2 n)\right )\right )+a b d \left (c D (2+m) (6+m+3 n)-C d \left (m^2+m (8+3 n)+2 \left (8+6 n+n^2\right )\right )\right )\right ) (a+b x)^{1+m} (c+d x)^{1+n}}{b^3 d^3 (2+m+n) (3+m+n) (4+m+n)}-\frac{(b c D (3+m)-b C d (4+m+n)+a d D (9+2 m+3 n)) (a+b x)^{2+m} (c+d x)^{1+n}}{b^3 d^2 (3+m+n) (4+m+n)}+\frac{D (a+b x)^{3+m} (c+d x)^{1+n}}{b^3 d (4+m+n)}+\frac{\left (a^3 d^2 D (1+n) (6+m+2 n)+a b^2 c (2+m) (c D (3+m)-C d (4+m+n))+A b^3 d^2 \left (12+m^2+7 n+n^2+m (7+2 n)\right )-a^2 b d (C d (1+n) (4+m+n)-c D (2+m) (6+m+3 n))-\frac{(b c (1+m)+a d (1+n)) \left (a^2 d^2 D \left (m^2+m (8+3 n)+3 \left (6+5 n+n^2\right )\right )+b^2 \left (c^2 D \left (6+5 m+m^2\right )-c C d (2+m) (4+m+n)+B d^2 \left (12+m^2+7 n+n^2+m (7+2 n)\right )\right )+a b d \left (c D (2+m) (6+m+3 n)-C d \left (m^2+m (8+3 n)+2 \left (8+6 n+n^2\right )\right )\right )\right )}{d (2+m+n)}\right ) \int (a+b x)^m (c+d x)^n \, dx}{b^3 d^2 (3+m+n) (4+m+n)}\\ &=\frac{\left (a^2 d^2 D \left (m^2+m (8+3 n)+3 \left (6+5 n+n^2\right )\right )+b^2 \left (c^2 D \left (6+5 m+m^2\right )-c C d (2+m) (4+m+n)+B d^2 \left (12+m^2+7 n+n^2+m (7+2 n)\right )\right )+a b d \left (c D (2+m) (6+m+3 n)-C d \left (m^2+m (8+3 n)+2 \left (8+6 n+n^2\right )\right )\right )\right ) (a+b x)^{1+m} (c+d x)^{1+n}}{b^3 d^3 (2+m+n) (3+m+n) (4+m+n)}-\frac{(b c D (3+m)-b C d (4+m+n)+a d D (9+2 m+3 n)) (a+b x)^{2+m} (c+d x)^{1+n}}{b^3 d^2 (3+m+n) (4+m+n)}+\frac{D (a+b x)^{3+m} (c+d x)^{1+n}}{b^3 d (4+m+n)}+\frac{\left (\left (a^3 d^2 D (1+n) (6+m+2 n)+a b^2 c (2+m) (c D (3+m)-C d (4+m+n))+A b^3 d^2 \left (12+m^2+7 n+n^2+m (7+2 n)\right )-a^2 b d (C d (1+n) (4+m+n)-c D (2+m) (6+m+3 n))-\frac{(b c (1+m)+a d (1+n)) \left (a^2 d^2 D \left (m^2+m (8+3 n)+3 \left (6+5 n+n^2\right )\right )+b^2 \left (c^2 D \left (6+5 m+m^2\right )-c C d (2+m) (4+m+n)+B d^2 \left (12+m^2+7 n+n^2+m (7+2 n)\right )\right )+a b d \left (c D (2+m) (6+m+3 n)-C d \left (m^2+m (8+3 n)+2 \left (8+6 n+n^2\right )\right )\right )\right )}{d (2+m+n)}\right ) (c+d x)^n \left (\frac{b (c+d x)}{b c-a d}\right )^{-n}\right ) \int (a+b x)^m \left (\frac{b c}{b c-a d}+\frac{b d x}{b c-a d}\right )^n \, dx}{b^3 d^2 (3+m+n) (4+m+n)}\\ &=\frac{\left (a^2 d^2 D \left (m^2+m (8+3 n)+3 \left (6+5 n+n^2\right )\right )+b^2 \left (c^2 D \left (6+5 m+m^2\right )-c C d (2+m) (4+m+n)+B d^2 \left (12+m^2+7 n+n^2+m (7+2 n)\right )\right )+a b d \left (c D (2+m) (6+m+3 n)-C d \left (m^2+m (8+3 n)+2 \left (8+6 n+n^2\right )\right )\right )\right ) (a+b x)^{1+m} (c+d x)^{1+n}}{b^3 d^3 (2+m+n) (3+m+n) (4+m+n)}-\frac{(b c D (3+m)-b C d (4+m+n)+a d D (9+2 m+3 n)) (a+b x)^{2+m} (c+d x)^{1+n}}{b^3 d^2 (3+m+n) (4+m+n)}+\frac{D (a+b x)^{3+m} (c+d x)^{1+n}}{b^3 d (4+m+n)}+\frac{\left (a^3 d^2 D (1+n) (6+m+2 n)+a b^2 c (2+m) (c D (3+m)-C d (4+m+n))+A b^3 d^2 \left (12+m^2+7 n+n^2+m (7+2 n)\right )-a^2 b d (C d (1+n) (4+m+n)-c D (2+m) (6+m+3 n))-\frac{(b c (1+m)+a d (1+n)) \left (a^2 d^2 D \left (m^2+m (8+3 n)+3 \left (6+5 n+n^2\right )\right )+b^2 \left (c^2 D \left (6+5 m+m^2\right )-c C d (2+m) (4+m+n)+B d^2 \left (12+m^2+7 n+n^2+m (7+2 n)\right )\right )+a b d \left (c D (2+m) (6+m+3 n)-C d \left (m^2+m (8+3 n)+2 \left (8+6 n+n^2\right )\right )\right )\right )}{d (2+m+n)}\right ) (a+b x)^{1+m} (c+d x)^n \left (\frac{b (c+d x)}{b c-a d}\right )^{-n} \, _2F_1\left (1+m,-n;2+m;-\frac{d (a+b x)}{b c-a d}\right )}{b^4 d^2 (1+m) (3+m+n) (4+m+n)}\\ \end{align*}

Mathematica [A]  time = 0.246051, size = 254, normalized size = 0.42 \[ \frac{(a+b x)^{m+1} (c+d x)^n \left (\frac{b (c+d x)}{b c-a d}\right )^{-n} \left (b^3 \left (A d^3-B c d^2+c^2 C d+c^3 (-D)\right ) \, _2F_1\left (m+1,-n;m+2;\frac{d (a+b x)}{a d-b c}\right )+b^2 (b c-a d) \left (B d^2+3 c^2 D-2 c C d\right ) \, _2F_1\left (m+1,-n-1;m+2;\frac{d (a+b x)}{a d-b c}\right )+b (b c-a d)^2 (C d-3 c D) \, _2F_1\left (m+1,-n-2;m+2;\frac{d (a+b x)}{a d-b c}\right )+D (b c-a d)^3 \, _2F_1\left (m+1,-n-3;m+2;\frac{d (a+b x)}{a d-b c}\right )\right )}{b^4 d^3 (m+1)} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^m*(c + d*x)^n*(A + B*x + C*x^2 + D*x^3),x]

[Out]

((a + b*x)^(1 + m)*(c + d*x)^n*((b*c - a*d)^3*D*Hypergeometric2F1[1 + m, -3 - n, 2 + m, (d*(a + b*x))/(-(b*c)
+ a*d)] + b*(b*c - a*d)^2*(C*d - 3*c*D)*Hypergeometric2F1[1 + m, -2 - n, 2 + m, (d*(a + b*x))/(-(b*c) + a*d)]
+ b^2*(b*c - a*d)*(-2*c*C*d + B*d^2 + 3*c^2*D)*Hypergeometric2F1[1 + m, -1 - n, 2 + m, (d*(a + b*x))/(-(b*c) +
a*d)] + b^3*(c^2*C*d - B*c*d^2 + A*d^3 - c^3*D)*Hypergeometric2F1[1 + m, -n, 2 + m, (d*(a + b*x))/(-(b*c) + a
*d)]))/(b^4*d^3*(1 + m)*((b*(c + d*x))/(b*c - a*d))^n)

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Maple [F]  time = 0.039, size = 0, normalized size = 0. \begin{align*} \int \left ( bx+a \right ) ^{m} \left ( dx+c \right ) ^{n} \left ( D{x}^{3}+C{x}^{2}+Bx+A \right ) \, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^m*(d*x+c)^n*(D*x^3+C*x^2+B*x+A),x)

[Out]

int((b*x+a)^m*(d*x+c)^n*(D*x^3+C*x^2+B*x+A),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (D x^{3} + C x^{2} + B x + A\right )}{\left (b x + a\right )}^{m}{\left (d x + c\right )}^{n}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^m*(d*x+c)^n*(D*x^3+C*x^2+B*x+A),x, algorithm="maxima")

[Out]

integrate((D*x^3 + C*x^2 + B*x + A)*(b*x + a)^m*(d*x + c)^n, x)

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Fricas [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^m*(d*x+c)^n*(D*x^3+C*x^2+B*x+A),x, algorithm="fricas")

[Out]

Exception raised: UnboundLocalError

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**m*(d*x+c)**n*(D*x**3+C*x**2+B*x+A),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (D x^{3} + C x^{2} + B x + A\right )}{\left (b x + a\right )}^{m}{\left (d x + c\right )}^{n}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^m*(d*x+c)^n*(D*x^3+C*x^2+B*x+A),x, algorithm="giac")

[Out]

integrate((D*x^3 + C*x^2 + B*x + A)*(b*x + a)^m*(d*x + c)^n, x)