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

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

[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*Hypergeome
tric2F1[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)

_______________________________________________________________________________________

Rubi [A]  time = 2.33733, 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 \[ \frac{(a+b x)^{m+1} (c+d x)^{n+1} \left (a^2 d^2 D \left (m^2+m (3 n+8)+3 \left (n^2+5 n+6\right )\right )+a b d \left (c D (m+2) (m+3 n+6)-C d \left (m^2+m (3 n+8)+2 \left (n^2+6 n+8\right )\right )\right )+b^2 \left (B d^2 \left (m^2+m (2 n+7)+n^2+7 n+12\right )+c^2 D \left (m^2+5 m+6\right )-c C d (m+2) (m+n+4)\right )\right )}{b^3 d^3 (m+n+2) (m+n+3) (m+n+4)}+\frac{(a+b x)^{m+1} (c+d x)^n \left (\frac{b (c+d x)}{b c-a d}\right )^{-n} \, _2F_1\left (m+1,-n;m+2;-\frac{d (a+b x)}{b c-a d}\right ) \left (a^3 d^2 D (n+1) (m+2 n+6)-\frac{(a d (n+1)+b c (m+1)) \left (a^2 d^2 D \left (m^2+m (3 n+8)+3 \left (n^2+5 n+6\right )\right )+a b d \left (c D (m+2) (m+3 n+6)-C d \left (m^2+m (3 n+8)+2 \left (n^2+6 n+8\right )\right )\right )+b^2 \left (B d^2 \left (m^2+m (2 n+7)+n^2+7 n+12\right )+c^2 D \left (m^2+5 m+6\right )-c C d (m+2) (m+n+4)\right )\right )}{d (m+n+2)}-a^2 b d (C d (n+1) (m+n+4)-c D (m+2) (m+3 n+6))+a b^2 c (m+2) (c D (m+3)-C d (m+n+4))+A b^3 d^2 \left (m^2+m (2 n+7)+n^2+7 n+12\right )\right )}{b^4 d^2 (m+1) (m+n+3) (m+n+4)}-\frac{(a+b x)^{m+2} (c+d x)^{n+1} (a d D (2 m+3 n+9)+b c D (m+3)-b C d (m+n+4))}{b^3 d^2 (m+n+3) (m+n+4)}+\frac{D (a+b x)^{m+3} (c+d x)^{n+1}}{b^3 d (m+n+4)} \]

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*Hypergeo
metric2F1[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)

_______________________________________________________________________________________

Rubi in Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

_______________________________________________________________________________________

Mathematica [C]  time = 3.40405, size = 446, normalized size = 0.73 \[ \frac{1}{12} (a+b x)^m (c+d x)^n \left (\frac{12 A (c+d x) \left (\frac{d (a+b x)}{a d-b c}\right )^{-m} \, _2F_1\left (-m,n+1;n+2;\frac{b (c+d x)}{b c-a d}\right )}{d (n+1)}+\frac{18 a B c x^2 F_1\left (2;-m,-n;3;-\frac{b x}{a},-\frac{d x}{c}\right )}{3 a c F_1\left (2;-m,-n;3;-\frac{b x}{a},-\frac{d x}{c}\right )+b c m x F_1\left (3;1-m,-n;4;-\frac{b x}{a},-\frac{d x}{c}\right )+a d n x F_1\left (3;-m,1-n;4;-\frac{b x}{a},-\frac{d x}{c}\right )}+\frac{16 a c C x^3 F_1\left (3;-m,-n;4;-\frac{b x}{a},-\frac{d x}{c}\right )}{4 a c F_1\left (3;-m,-n;4;-\frac{b x}{a},-\frac{d x}{c}\right )+b c m x F_1\left (4;1-m,-n;5;-\frac{b x}{a},-\frac{d x}{c}\right )+a d n x F_1\left (4;-m,1-n;5;-\frac{b x}{a},-\frac{d x}{c}\right )}+\frac{15 a c D x^4 F_1\left (4;-m,-n;5;-\frac{b x}{a},-\frac{d x}{c}\right )}{5 a c F_1\left (4;-m,-n;5;-\frac{b x}{a},-\frac{d x}{c}\right )+b c m x F_1\left (5;1-m,-n;6;-\frac{b x}{a},-\frac{d x}{c}\right )+a d n x F_1\left (5;-m,1-n;6;-\frac{b x}{a},-\frac{d x}{c}\right )}\right ) \]

Warning: Unable to verify antiderivative.

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

[Out]

((a + b*x)^m*(c + d*x)^n*((18*a*B*c*x^2*AppellF1[2, -m, -n, 3, -((b*x)/a), -((d*
x)/c)])/(3*a*c*AppellF1[2, -m, -n, 3, -((b*x)/a), -((d*x)/c)] + b*c*m*x*AppellF1
[3, 1 - m, -n, 4, -((b*x)/a), -((d*x)/c)] + a*d*n*x*AppellF1[3, -m, 1 - n, 4, -(
(b*x)/a), -((d*x)/c)]) + (16*a*c*C*x^3*AppellF1[3, -m, -n, 4, -((b*x)/a), -((d*x
)/c)])/(4*a*c*AppellF1[3, -m, -n, 4, -((b*x)/a), -((d*x)/c)] + b*c*m*x*AppellF1[
4, 1 - m, -n, 5, -((b*x)/a), -((d*x)/c)] + a*d*n*x*AppellF1[4, -m, 1 - n, 5, -((
b*x)/a), -((d*x)/c)]) + (15*a*c*D*x^4*AppellF1[4, -m, -n, 5, -((b*x)/a), -((d*x)
/c)])/(5*a*c*AppellF1[4, -m, -n, 5, -((b*x)/a), -((d*x)/c)] + b*c*m*x*AppellF1[5
, 1 - m, -n, 6, -((b*x)/a), -((d*x)/c)] + a*d*n*x*AppellF1[5, -m, 1 - n, 6, -((b
*x)/a), -((d*x)/c)]) + (12*A*(c + d*x)*Hypergeometric2F1[-m, 1 + n, 2 + n, (b*(c
+ d*x))/(b*c - a*d)])/(d*(1 + n)*((d*(a + b*x))/(-(b*c) + a*d))^m)))/12

_______________________________________________________________________________________

Maple [F]  time = 0.051, size = 0, normalized size = 0. \[ \int \left ( bx+a \right ) ^{m} \left ( dx+c \right ) ^{n} \left ( D{x}^{3}+C{x}^{2}+Bx+A \right ) \, dx \]

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)

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left ({\left (D x^{3} + C x^{2} + B x + A\right )}{\left (b x + a\right )}^{m}{\left (d x + c\right )}^{n}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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