Integrand size = 32, antiderivative size = 326 \[ \int (a+b x)^2 \sqrt [3]{c+d x} \left (A+B x+C x^2+D x^3\right ) \, dx=\frac {3 (b c-a d)^2 \left (c^2 C d-B c d^2+A d^3-c^3 D\right ) (c+d x)^{4/3}}{4 d^6}+\frac {3 (b c-a d) \left (a d \left (2 c C d-B d^2-3 c^2 D\right )-b \left (4 c^2 C d-3 B c d^2+2 A d^3-5 c^3 D\right )\right ) (c+d x)^{7/3}}{7 d^6}+\frac {3 \left (a^2 d^2 (C d-3 c D)-2 a b d \left (3 c C d-B d^2-6 c^2 D\right )+b^2 \left (6 c^2 C d-3 B c d^2+A d^3-10 c^3 D\right )\right ) (c+d x)^{10/3}}{10 d^6}+\frac {3 \left (a^2 d^2 D+2 a b d (C d-4 c D)-b^2 \left (4 c C d-B d^2-10 c^2 D\right )\right ) (c+d x)^{13/3}}{13 d^6}+\frac {3 b (b C d-5 b c D+2 a d D) (c+d x)^{16/3}}{16 d^6}+\frac {3 b^2 D (c+d x)^{19/3}}{19 d^6} \] Output:
3/4*(-a*d+b*c)^2*(A*d^3-B*c*d^2+C*c^2*d-D*c^3)*(d*x+c)^(4/3)/d^6+3/7*(-a*d +b*c)*(a*d*(-B*d^2+2*C*c*d-3*D*c^2)-b*(2*A*d^3-3*B*c*d^2+4*C*c^2*d-5*D*c^3 ))*(d*x+c)^(7/3)/d^6+3/10*(a^2*d^2*(C*d-3*D*c)-2*a*b*d*(-B*d^2+3*C*c*d-6*D *c^2)+b^2*(A*d^3-3*B*c*d^2+6*C*c^2*d-10*D*c^3))*(d*x+c)^(10/3)/d^6+3/13*(a ^2*d^2*D+2*a*b*d*(C*d-4*D*c)-b^2*(-B*d^2+4*C*c*d-10*D*c^2))*(d*x+c)^(13/3) /d^6+3/16*b*(C*b*d+2*D*a*d-5*D*b*c)*(d*x+c)^(16/3)/d^6+3/19*b^2*D*(d*x+c)^ (19/3)/d^6
Time = 0.30 (sec) , antiderivative size = 325, normalized size of antiderivative = 1.00 \[ \int (a+b x)^2 \sqrt [3]{c+d x} \left (A+B x+C x^2+D x^3\right ) \, dx=\frac {3 (c+d x)^{4/3} \left (76 a^2 d^2 \left (-81 c^3 D+9 c^2 d (13 C+12 D x)-3 c d^2 \left (65 B+52 C x+42 D x^2\right )+d^3 \left (455 A+2 x \left (130 B+91 C x+70 D x^2\right )\right )\right )+38 a b d \left (243 c^4 D-324 c^3 d (C+D x)+18 c^2 d^2 (26 B+3 x (8 C+7 D x))+d^4 x \left (1040 A+7 x \left (104 B+80 C x+65 D x^2\right )\right )-12 c d^3 (65 A+x (52 B+7 x (6 C+5 D x)))\right )+b^2 \left (-3645 c^5 D+243 c^4 d (19 C+20 D x)-162 c^3 d^2 (38 B+x (38 C+35 D x))+7 d^5 x^2 (1976 A+5 x (304 B+13 x (19 C+16 D x)))+18 c^2 d^3 (494 A+x (456 B+7 x (57 C+50 D x)))-3 c d^4 x (3952 A+7 x (456 B+5 x (76 C+65 D x)))\right )\right )}{138320 d^6} \] Input:
Integrate[(a + b*x)^2*(c + d*x)^(1/3)*(A + B*x + C*x^2 + D*x^3),x]
Output:
(3*(c + d*x)^(4/3)*(76*a^2*d^2*(-81*c^3*D + 9*c^2*d*(13*C + 12*D*x) - 3*c* d^2*(65*B + 52*C*x + 42*D*x^2) + d^3*(455*A + 2*x*(130*B + 91*C*x + 70*D*x ^2))) + 38*a*b*d*(243*c^4*D - 324*c^3*d*(C + D*x) + 18*c^2*d^2*(26*B + 3*x *(8*C + 7*D*x)) + d^4*x*(1040*A + 7*x*(104*B + 80*C*x + 65*D*x^2)) - 12*c* d^3*(65*A + x*(52*B + 7*x*(6*C + 5*D*x)))) + b^2*(-3645*c^5*D + 243*c^4*d* (19*C + 20*D*x) - 162*c^3*d^2*(38*B + x*(38*C + 35*D*x)) + 7*d^5*x^2*(1976 *A + 5*x*(304*B + 13*x*(19*C + 16*D*x))) + 18*c^2*d^3*(494*A + x*(456*B + 7*x*(57*C + 50*D*x))) - 3*c*d^4*x*(3952*A + 7*x*(456*B + 5*x*(76*C + 65*D* x))))))/(138320*d^6)
Time = 0.59 (sec) , antiderivative size = 326, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {2123, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int (a+b x)^2 \sqrt [3]{c+d x} \left (A+B x+C x^2+D x^3\right ) \, dx\) |
| \(\Big \downarrow \) 2123 |
| \(\displaystyle \int \left (\frac {(c+d x)^{7/3} \left (a^2 d^2 (C d-3 c D)-2 a b d \left (-B d^2-6 c^2 D+3 c C d\right )+b^2 \left (A d^3-3 B c d^2-10 c^3 D+6 c^2 C d\right )\right )}{d^5}+\frac {(c+d x)^{10/3} \left (a^2 d^2 D+2 a b d (C d-4 c D)-\left (b^2 \left (-B d^2-10 c^2 D+4 c C d\right )\right )\right )}{d^5}+\frac {(c+d x)^{4/3} (b c-a d) \left (a d \left (-B d^2-3 c^2 D+2 c C d\right )-b \left (2 A d^3-3 B c d^2-5 c^3 D+4 c^2 C d\right )\right )}{d^5}+\frac {\sqrt [3]{c+d x} (a d-b c)^2 \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{d^5}+\frac {b (c+d x)^{13/3} (2 a d D-5 b c D+b C d)}{d^5}+\frac {b^2 D (c+d x)^{16/3}}{d^5}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {3 (c+d x)^{10/3} \left (a^2 d^2 (C d-3 c D)-2 a b d \left (-B d^2-6 c^2 D+3 c C d\right )+b^2 \left (A d^3-3 B c d^2-10 c^3 D+6 c^2 C d\right )\right )}{10 d^6}+\frac {3 (c+d x)^{13/3} \left (a^2 d^2 D+2 a b d (C d-4 c D)-\left (b^2 \left (-B d^2-10 c^2 D+4 c C d\right )\right )\right )}{13 d^6}+\frac {3 (c+d x)^{7/3} (b c-a d) \left (a d \left (-B d^2-3 c^2 D+2 c C d\right )-b \left (2 A d^3-3 B c d^2-5 c^3 D+4 c^2 C d\right )\right )}{7 d^6}+\frac {3 (c+d x)^{4/3} (b c-a d)^2 \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{4 d^6}+\frac {3 b (c+d x)^{16/3} (2 a d D-5 b c D+b C d)}{16 d^6}+\frac {3 b^2 D (c+d x)^{19/3}}{19 d^6}\) |
Input:
Int[(a + b*x)^2*(c + d*x)^(1/3)*(A + B*x + C*x^2 + D*x^3),x]
Output:
(3*(b*c - a*d)^2*(c^2*C*d - B*c*d^2 + A*d^3 - c^3*D)*(c + d*x)^(4/3))/(4*d ^6) + (3*(b*c - a*d)*(a*d*(2*c*C*d - B*d^2 - 3*c^2*D) - b*(4*c^2*C*d - 3*B *c*d^2 + 2*A*d^3 - 5*c^3*D))*(c + d*x)^(7/3))/(7*d^6) + (3*(a^2*d^2*(C*d - 3*c*D) - 2*a*b*d*(3*c*C*d - B*d^2 - 6*c^2*D) + b^2*(6*c^2*C*d - 3*B*c*d^2 + A*d^3 - 10*c^3*D))*(c + d*x)^(10/3))/(10*d^6) + (3*(a^2*d^2*D + 2*a*b*d *(C*d - 4*c*D) - b^2*(4*c*C*d - B*d^2 - 10*c^2*D))*(c + d*x)^(13/3))/(13*d ^6) + (3*b*(b*C*d - 5*b*c*D + 2*a*d*D)*(c + d*x)^(16/3))/(16*d^6) + (3*b^2 *D*(c + d*x)^(19/3))/(19*d^6)
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])
Time = 1.50 (sec) , antiderivative size = 283, normalized size of antiderivative = 0.87
| method | result | size |
| pseudoelliptic | \(\frac {3 \left (x d +c \right )^{\frac {4}{3}} \left (\left (\frac {2 x^{2} \left (\frac {10}{19} D x^{3}+\frac {5}{8} C \,x^{2}+\frac {10}{13} B x +A \right ) b^{2}}{5}+\frac {8 x \left (\frac {7}{16} D x^{3}+\frac {7}{13} C \,x^{2}+\frac {7}{10} B x +A \right ) a b}{7}+a^{2} \left (\frac {4}{13} D x^{3}+\frac {2}{5} C \,x^{2}+\frac {4}{7} B x +A \right )\right ) d^{5}-\frac {6 \left (\frac {2 x \left (\frac {175}{304} D x^{3}+\frac {35}{52} C \,x^{2}+\frac {21}{26} B x +A \right ) b^{2}}{5}+a \left (\frac {7}{13} D x^{3}+\frac {42}{65} C \,x^{2}+\frac {4}{5} B x +A \right ) b +\frac {\left (\frac {42}{65} D x^{2}+\frac {4}{5} C x +B \right ) a^{2}}{2}\right ) c \,d^{4}}{7}+\frac {9 \left (\left (\frac {175}{247} D x^{3}+\frac {21}{26} C \,x^{2}+\frac {12}{13} B x +A \right ) b^{2}+2 \left (\frac {21}{26} D x^{2}+\frac {12}{13} C x +B \right ) a b +a^{2} \left (\frac {12 D x}{13}+C \right )\right ) c^{2} d^{3}}{35}-\frac {81 \left (\left (\frac {35}{38} D x^{2}+C x +B \right ) b^{2}+2 a \left (D x +C \right ) b +D a^{2}\right ) c^{3} d^{2}}{455}+\frac {243 \left (\left (\frac {20 D x}{19}+C \right ) b +2 D a \right ) c^{4} b d}{1820}-\frac {729 D b^{2} c^{5}}{6916}\right )}{4 d^{6}}\) | \(283\) |
| derivativedivides | \(\frac {\frac {3 b^{2} D \left (x d +c \right )^{\frac {19}{3}}}{19}+\frac {3 \left (2 b \left (a d -b c \right ) D+b^{2} \left (C d -3 D c \right )\right ) \left (x d +c \right )^{\frac {16}{3}}}{16}+\frac {3 \left (\left (a d -b c \right )^{2} D+2 b \left (a d -b c \right ) \left (C d -3 D c \right )+b^{2} \left (B \,d^{2}-2 C c d +3 D c^{2}\right )\right ) \left (x d +c \right )^{\frac {13}{3}}}{13}+\frac {3 \left (\left (a d -b c \right )^{2} \left (C d -3 D c \right )+2 b \left (a d -b c \right ) \left (B \,d^{2}-2 C c d +3 D c^{2}\right )+b^{2} \left (A \,d^{3}-B c \,d^{2}+C \,c^{2} d -D c^{3}\right )\right ) \left (x d +c \right )^{\frac {10}{3}}}{10}+\frac {3 \left (\left (a d -b c \right )^{2} \left (B \,d^{2}-2 C c d +3 D c^{2}\right )+2 b \left (a d -b c \right ) \left (A \,d^{3}-B c \,d^{2}+C \,c^{2} d -D c^{3}\right )\right ) \left (x d +c \right )^{\frac {7}{3}}}{7}+\frac {3 \left (a d -b c \right )^{2} \left (A \,d^{3}-B c \,d^{2}+C \,c^{2} d -D c^{3}\right ) \left (x d +c \right )^{\frac {4}{3}}}{4}}{d^{6}}\) | \(320\) |
| default | \(\frac {\frac {3 b^{2} D \left (x d +c \right )^{\frac {19}{3}}}{19}+\frac {3 \left (2 b \left (a d -b c \right ) D+b^{2} \left (C d -3 D c \right )\right ) \left (x d +c \right )^{\frac {16}{3}}}{16}+\frac {3 \left (\left (a d -b c \right )^{2} D+2 b \left (a d -b c \right ) \left (C d -3 D c \right )+b^{2} \left (B \,d^{2}-2 C c d +3 D c^{2}\right )\right ) \left (x d +c \right )^{\frac {13}{3}}}{13}+\frac {3 \left (\left (a d -b c \right )^{2} \left (C d -3 D c \right )+2 b \left (a d -b c \right ) \left (B \,d^{2}-2 C c d +3 D c^{2}\right )+b^{2} \left (A \,d^{3}-B c \,d^{2}+C \,c^{2} d -D c^{3}\right )\right ) \left (x d +c \right )^{\frac {10}{3}}}{10}+\frac {3 \left (\left (a d -b c \right )^{2} \left (B \,d^{2}-2 C c d +3 D c^{2}\right )+2 b \left (a d -b c \right ) \left (A \,d^{3}-B c \,d^{2}+C \,c^{2} d -D c^{3}\right )\right ) \left (x d +c \right )^{\frac {7}{3}}}{7}+\frac {3 \left (a d -b c \right )^{2} \left (A \,d^{3}-B c \,d^{2}+C \,c^{2} d -D c^{3}\right ) \left (x d +c \right )^{\frac {4}{3}}}{4}}{d^{6}}\) | \(320\) |
| gosper | \(\frac {3 \left (x d +c \right )^{\frac {4}{3}} \left (7280 D x^{5} b^{2} d^{5}+8645 C \,x^{4} b^{2} d^{5}+17290 D x^{4} a b \,d^{5}-6825 D x^{4} b^{2} c \,d^{4}+10640 B \,x^{3} b^{2} d^{5}+21280 C \,x^{3} a b \,d^{5}-7980 C \,x^{3} b^{2} c \,d^{4}+10640 D x^{3} a^{2} d^{5}-15960 D x^{3} a b c \,d^{4}+6300 D x^{3} b^{2} c^{2} d^{3}+13832 A \,x^{2} b^{2} d^{5}+27664 B \,x^{2} a b \,d^{5}-9576 B \,x^{2} b^{2} c \,d^{4}+13832 C \,x^{2} a^{2} d^{5}-19152 C \,x^{2} a b c \,d^{4}+7182 C \,x^{2} b^{2} c^{2} d^{3}-9576 D x^{2} a^{2} c \,d^{4}+14364 D x^{2} a b \,c^{2} d^{3}-5670 D x^{2} b^{2} c^{3} d^{2}+39520 A a b \,d^{5} x -11856 A x \,b^{2} c \,d^{4}+19760 B \,a^{2} d^{5} x -23712 B x a b c \,d^{4}+8208 B \,b^{2} c^{2} d^{3} x -11856 C x \,a^{2} c \,d^{4}+16416 C a b \,c^{2} d^{3} x -6156 C \,b^{2} c^{3} d^{2} x +8208 D a^{2} c^{2} d^{3} x -12312 D a b \,c^{3} d^{2} x +4860 D b^{2} c^{4} d x +34580 A \,a^{2} d^{5}-29640 A a b c \,d^{4}+8892 A \,b^{2} c^{2} d^{3}-14820 B \,a^{2} c \,d^{4}+17784 B a b \,c^{2} d^{3}-6156 B \,b^{2} c^{3} d^{2}+8892 C \,a^{2} c^{2} d^{3}-12312 C a b \,c^{3} d^{2}+4617 C \,b^{2} c^{4} d -6156 D a^{2} c^{3} d^{2}+9234 D a b \,c^{4} d -3645 D b^{2} c^{5}\right )}{138320 d^{6}}\) | \(505\) |
| orering | \(\frac {3 \left (x d +c \right )^{\frac {4}{3}} \left (7280 D x^{5} b^{2} d^{5}+8645 C \,x^{4} b^{2} d^{5}+17290 D x^{4} a b \,d^{5}-6825 D x^{4} b^{2} c \,d^{4}+10640 B \,x^{3} b^{2} d^{5}+21280 C \,x^{3} a b \,d^{5}-7980 C \,x^{3} b^{2} c \,d^{4}+10640 D x^{3} a^{2} d^{5}-15960 D x^{3} a b c \,d^{4}+6300 D x^{3} b^{2} c^{2} d^{3}+13832 A \,x^{2} b^{2} d^{5}+27664 B \,x^{2} a b \,d^{5}-9576 B \,x^{2} b^{2} c \,d^{4}+13832 C \,x^{2} a^{2} d^{5}-19152 C \,x^{2} a b c \,d^{4}+7182 C \,x^{2} b^{2} c^{2} d^{3}-9576 D x^{2} a^{2} c \,d^{4}+14364 D x^{2} a b \,c^{2} d^{3}-5670 D x^{2} b^{2} c^{3} d^{2}+39520 A a b \,d^{5} x -11856 A x \,b^{2} c \,d^{4}+19760 B \,a^{2} d^{5} x -23712 B x a b c \,d^{4}+8208 B \,b^{2} c^{2} d^{3} x -11856 C x \,a^{2} c \,d^{4}+16416 C a b \,c^{2} d^{3} x -6156 C \,b^{2} c^{3} d^{2} x +8208 D a^{2} c^{2} d^{3} x -12312 D a b \,c^{3} d^{2} x +4860 D b^{2} c^{4} d x +34580 A \,a^{2} d^{5}-29640 A a b c \,d^{4}+8892 A \,b^{2} c^{2} d^{3}-14820 B \,a^{2} c \,d^{4}+17784 B a b \,c^{2} d^{3}-6156 B \,b^{2} c^{3} d^{2}+8892 C \,a^{2} c^{2} d^{3}-12312 C a b \,c^{3} d^{2}+4617 C \,b^{2} c^{4} d -6156 D a^{2} c^{3} d^{2}+9234 D a b \,c^{4} d -3645 D b^{2} c^{5}\right )}{138320 d^{6}}\) | \(505\) |
| trager | \(\frac {3 \left (7280 b^{2} D d^{6} x^{6}+8645 C \,b^{2} d^{6} x^{5}+17290 D a b \,d^{6} x^{5}+455 D b^{2} c \,d^{5} x^{5}+10640 B \,b^{2} d^{6} x^{4}+21280 C a b \,d^{6} x^{4}+665 C \,b^{2} c \,d^{5} x^{4}+10640 D a^{2} d^{6} x^{4}+1330 D a b c \,d^{5} x^{4}-525 D b^{2} c^{2} d^{4} x^{4}+13832 A \,b^{2} d^{6} x^{3}+27664 B a b \,d^{6} x^{3}+1064 B \,b^{2} c \,d^{5} x^{3}+13832 C \,a^{2} d^{6} x^{3}+2128 C a b c \,d^{5} x^{3}-798 C \,b^{2} c^{2} d^{4} x^{3}+1064 D a^{2} c \,d^{5} x^{3}-1596 D a b \,c^{2} d^{4} x^{3}+630 D b^{2} c^{3} d^{3} x^{3}+39520 A a b \,d^{6} x^{2}+1976 A \,b^{2} c \,d^{5} x^{2}+19760 B \,a^{2} d^{6} x^{2}+3952 B a b c \,d^{5} x^{2}-1368 B \,b^{2} c^{2} d^{4} x^{2}+1976 C \,a^{2} c \,d^{5} x^{2}-2736 C a b \,c^{2} d^{4} x^{2}+1026 C \,b^{2} c^{3} d^{3} x^{2}-1368 D a^{2} c^{2} d^{4} x^{2}+2052 D a b \,c^{3} d^{3} x^{2}-810 D b^{2} c^{4} d^{2} x^{2}+34580 A \,a^{2} d^{6} x +9880 A a b c \,d^{5} x -2964 A \,b^{2} c^{2} d^{4} x +4940 B \,a^{2} c \,d^{5} x -5928 B a b \,c^{2} d^{4} x +2052 B \,b^{2} c^{3} d^{3} x -2964 C \,a^{2} c^{2} d^{4} x +4104 C a b \,c^{3} d^{3} x -1539 C \,b^{2} c^{4} d^{2} x +2052 D a^{2} c^{3} d^{3} x -3078 D a b \,c^{4} d^{2} x +1215 D b^{2} c^{5} d x +34580 A \,a^{2} c \,d^{5}-29640 A a b \,c^{2} d^{4}+8892 A \,b^{2} c^{3} d^{3}-14820 B \,a^{2} c^{2} d^{4}+17784 B a b \,c^{3} d^{3}-6156 B \,b^{2} c^{4} d^{2}+8892 C \,a^{2} c^{3} d^{3}-12312 C a b \,c^{4} d^{2}+4617 C \,b^{2} c^{5} d -6156 D a^{2} c^{4} d^{2}+9234 D a b \,c^{5} d -3645 D b^{2} c^{6}\right ) \left (x d +c \right )^{\frac {1}{3}}}{138320 d^{6}}\) | \(677\) |
Input:
int((b*x+a)^2*(d*x+c)^(1/3)*(D*x^3+C*x^2+B*x+A),x,method=_RETURNVERBOSE)
Output:
3/4*(d*x+c)^(4/3)*((2/5*x^2*(10/19*D*x^3+5/8*C*x^2+10/13*B*x+A)*b^2+8/7*x* (7/16*D*x^3+7/13*C*x^2+7/10*B*x+A)*a*b+a^2*(4/13*D*x^3+2/5*C*x^2+4/7*B*x+A ))*d^5-6/7*(2/5*x*(175/304*D*x^3+35/52*C*x^2+21/26*B*x+A)*b^2+a*(7/13*D*x^ 3+42/65*C*x^2+4/5*B*x+A)*b+1/2*(42/65*D*x^2+4/5*C*x+B)*a^2)*c*d^4+9/35*((1 75/247*D*x^3+21/26*C*x^2+12/13*B*x+A)*b^2+2*(21/26*D*x^2+12/13*C*x+B)*a*b+ a^2*(12/13*D*x+C))*c^2*d^3-81/455*((35/38*D*x^2+C*x+B)*b^2+2*a*(D*x+C)*b+D *a^2)*c^3*d^2+243/1820*((20/19*D*x+C)*b+2*D*a)*c^4*b*d-729/6916*D*b^2*c^5) /d^6
Time = 0.08 (sec) , antiderivative size = 500, normalized size of antiderivative = 1.53 \[ \int (a+b x)^2 \sqrt [3]{c+d x} \left (A+B x+C x^2+D x^3\right ) \, dx=\frac {3 \, {\left (7280 \, D b^{2} d^{6} x^{6} - 3645 \, D b^{2} c^{6} + 34580 \, A a^{2} c d^{5} + 4617 \, {\left (2 \, D a b + C b^{2}\right )} c^{5} d - 6156 \, {\left (D a^{2} + 2 \, C a b + B b^{2}\right )} c^{4} d^{2} + 8892 \, {\left (C a^{2} + 2 \, B a b + A b^{2}\right )} c^{3} d^{3} - 14820 \, {\left (B a^{2} + 2 \, A a b\right )} c^{2} d^{4} + 455 \, {\left (D b^{2} c d^{5} + 19 \, {\left (2 \, D a b + C b^{2}\right )} d^{6}\right )} x^{5} - 35 \, {\left (15 \, D b^{2} c^{2} d^{4} - 19 \, {\left (2 \, D a b + C b^{2}\right )} c d^{5} - 304 \, {\left (D a^{2} + 2 \, C a b + B b^{2}\right )} d^{6}\right )} x^{4} + 14 \, {\left (45 \, D b^{2} c^{3} d^{3} - 57 \, {\left (2 \, D a b + C b^{2}\right )} c^{2} d^{4} + 76 \, {\left (D a^{2} + 2 \, C a b + B b^{2}\right )} c d^{5} + 988 \, {\left (C a^{2} + 2 \, B a b + A b^{2}\right )} d^{6}\right )} x^{3} - 2 \, {\left (405 \, D b^{2} c^{4} d^{2} - 513 \, {\left (2 \, D a b + C b^{2}\right )} c^{3} d^{3} + 684 \, {\left (D a^{2} + 2 \, C a b + B b^{2}\right )} c^{2} d^{4} - 988 \, {\left (C a^{2} + 2 \, B a b + A b^{2}\right )} c d^{5} - 9880 \, {\left (B a^{2} + 2 \, A a b\right )} d^{6}\right )} x^{2} + {\left (1215 \, D b^{2} c^{5} d + 34580 \, A a^{2} d^{6} - 1539 \, {\left (2 \, D a b + C b^{2}\right )} c^{4} d^{2} + 2052 \, {\left (D a^{2} + 2 \, C a b + B b^{2}\right )} c^{3} d^{3} - 2964 \, {\left (C a^{2} + 2 \, B a b + A b^{2}\right )} c^{2} d^{4} + 4940 \, {\left (B a^{2} + 2 \, A a b\right )} c d^{5}\right )} x\right )} {\left (d x + c\right )}^{\frac {1}{3}}}{138320 \, d^{6}} \] Input:
integrate((b*x+a)^2*(d*x+c)^(1/3)*(D*x^3+C*x^2+B*x+A),x, algorithm="fricas ")
Output:
3/138320*(7280*D*b^2*d^6*x^6 - 3645*D*b^2*c^6 + 34580*A*a^2*c*d^5 + 4617*( 2*D*a*b + C*b^2)*c^5*d - 6156*(D*a^2 + 2*C*a*b + B*b^2)*c^4*d^2 + 8892*(C* a^2 + 2*B*a*b + A*b^2)*c^3*d^3 - 14820*(B*a^2 + 2*A*a*b)*c^2*d^4 + 455*(D* b^2*c*d^5 + 19*(2*D*a*b + C*b^2)*d^6)*x^5 - 35*(15*D*b^2*c^2*d^4 - 19*(2*D *a*b + C*b^2)*c*d^5 - 304*(D*a^2 + 2*C*a*b + B*b^2)*d^6)*x^4 + 14*(45*D*b^ 2*c^3*d^3 - 57*(2*D*a*b + C*b^2)*c^2*d^4 + 76*(D*a^2 + 2*C*a*b + B*b^2)*c* d^5 + 988*(C*a^2 + 2*B*a*b + A*b^2)*d^6)*x^3 - 2*(405*D*b^2*c^4*d^2 - 513* (2*D*a*b + C*b^2)*c^3*d^3 + 684*(D*a^2 + 2*C*a*b + B*b^2)*c^2*d^4 - 988*(C *a^2 + 2*B*a*b + A*b^2)*c*d^5 - 9880*(B*a^2 + 2*A*a*b)*d^6)*x^2 + (1215*D* b^2*c^5*d + 34580*A*a^2*d^6 - 1539*(2*D*a*b + C*b^2)*c^4*d^2 + 2052*(D*a^2 + 2*C*a*b + B*b^2)*c^3*d^3 - 2964*(C*a^2 + 2*B*a*b + A*b^2)*c^2*d^4 + 494 0*(B*a^2 + 2*A*a*b)*c*d^5)*x)*(d*x + c)^(1/3)/d^6
Time = 1.56 (sec) , antiderivative size = 641, normalized size of antiderivative = 1.97 \[ \int (a+b x)^2 \sqrt [3]{c+d x} \left (A+B x+C x^2+D x^3\right ) \, dx=\begin {cases} \frac {3 \left (\frac {D b^{2} \left (c + d x\right )^{\frac {19}{3}}}{19 d^{5}} + \frac {\left (c + d x\right )^{\frac {16}{3}} \left (C b^{2} d + 2 D a b d - 5 D b^{2} c\right )}{16 d^{5}} + \frac {\left (c + d x\right )^{\frac {13}{3}} \left (B b^{2} d^{2} + 2 C a b d^{2} - 4 C b^{2} c d + D a^{2} d^{2} - 8 D a b c d + 10 D b^{2} c^{2}\right )}{13 d^{5}} + \frac {\left (c + d x\right )^{\frac {10}{3}} \left (A b^{2} d^{3} + 2 B a b d^{3} - 3 B b^{2} c d^{2} + C a^{2} d^{3} - 6 C a b c d^{2} + 6 C b^{2} c^{2} d - 3 D a^{2} c d^{2} + 12 D a b c^{2} d - 10 D b^{2} c^{3}\right )}{10 d^{5}} + \frac {\left (c + d x\right )^{\frac {7}{3}} \cdot \left (2 A a b d^{4} - 2 A b^{2} c d^{3} + B a^{2} d^{4} - 4 B a b c d^{3} + 3 B b^{2} c^{2} d^{2} - 2 C a^{2} c d^{3} + 6 C a b c^{2} d^{2} - 4 C b^{2} c^{3} d + 3 D a^{2} c^{2} d^{2} - 8 D a b c^{3} d + 5 D b^{2} c^{4}\right )}{7 d^{5}} + \frac {\left (c + d x\right )^{\frac {4}{3}} \left (A a^{2} d^{5} - 2 A a b c d^{4} + A b^{2} c^{2} d^{3} - B a^{2} c d^{4} + 2 B a b c^{2} d^{3} - B b^{2} c^{3} d^{2} + C a^{2} c^{2} d^{3} - 2 C a b c^{3} d^{2} + C b^{2} c^{4} d - D a^{2} c^{3} d^{2} + 2 D a b c^{4} d - D b^{2} c^{5}\right )}{4 d^{5}}\right )}{d} & \text {for}\: d \neq 0 \\\sqrt [3]{c} \left (A a^{2} x + \frac {D b^{2} x^{6}}{6} + \frac {x^{5} \left (C b^{2} + 2 D a b\right )}{5} + \frac {x^{4} \left (B b^{2} + 2 C a b + D a^{2}\right )}{4} + \frac {x^{3} \left (A b^{2} + 2 B a b + C a^{2}\right )}{3} + \frac {x^{2} \cdot \left (2 A a b + B a^{2}\right )}{2}\right ) & \text {otherwise} \end {cases} \] Input:
integrate((b*x+a)**2*(d*x+c)**(1/3)*(D*x**3+C*x**2+B*x+A),x)
Output:
Piecewise((3*(D*b**2*(c + d*x)**(19/3)/(19*d**5) + (c + d*x)**(16/3)*(C*b* *2*d + 2*D*a*b*d - 5*D*b**2*c)/(16*d**5) + (c + d*x)**(13/3)*(B*b**2*d**2 + 2*C*a*b*d**2 - 4*C*b**2*c*d + D*a**2*d**2 - 8*D*a*b*c*d + 10*D*b**2*c**2 )/(13*d**5) + (c + d*x)**(10/3)*(A*b**2*d**3 + 2*B*a*b*d**3 - 3*B*b**2*c*d **2 + C*a**2*d**3 - 6*C*a*b*c*d**2 + 6*C*b**2*c**2*d - 3*D*a**2*c*d**2 + 1 2*D*a*b*c**2*d - 10*D*b**2*c**3)/(10*d**5) + (c + d*x)**(7/3)*(2*A*a*b*d** 4 - 2*A*b**2*c*d**3 + B*a**2*d**4 - 4*B*a*b*c*d**3 + 3*B*b**2*c**2*d**2 - 2*C*a**2*c*d**3 + 6*C*a*b*c**2*d**2 - 4*C*b**2*c**3*d + 3*D*a**2*c**2*d**2 - 8*D*a*b*c**3*d + 5*D*b**2*c**4)/(7*d**5) + (c + d*x)**(4/3)*(A*a**2*d** 5 - 2*A*a*b*c*d**4 + A*b**2*c**2*d**3 - B*a**2*c*d**4 + 2*B*a*b*c**2*d**3 - B*b**2*c**3*d**2 + C*a**2*c**2*d**3 - 2*C*a*b*c**3*d**2 + C*b**2*c**4*d - D*a**2*c**3*d**2 + 2*D*a*b*c**4*d - D*b**2*c**5)/(4*d**5))/d, Ne(d, 0)), (c**(1/3)*(A*a**2*x + D*b**2*x**6/6 + x**5*(C*b**2 + 2*D*a*b)/5 + x**4*(B *b**2 + 2*C*a*b + D*a**2)/4 + x**3*(A*b**2 + 2*B*a*b + C*a**2)/3 + x**2*(2 *A*a*b + B*a**2)/2), True))
Time = 0.04 (sec) , antiderivative size = 387, normalized size of antiderivative = 1.19 \[ \int (a+b x)^2 \sqrt [3]{c+d x} \left (A+B x+C x^2+D x^3\right ) \, dx=\frac {3 \, {\left (7280 \, {\left (d x + c\right )}^{\frac {19}{3}} D b^{2} - 8645 \, {\left (5 \, D b^{2} c - {\left (2 \, D a b + C b^{2}\right )} d\right )} {\left (d x + c\right )}^{\frac {16}{3}} + 10640 \, {\left (10 \, D b^{2} c^{2} - 4 \, {\left (2 \, D a b + C b^{2}\right )} c d + {\left (D a^{2} + 2 \, C a b + B b^{2}\right )} d^{2}\right )} {\left (d x + c\right )}^{\frac {13}{3}} - 13832 \, {\left (10 \, D b^{2} c^{3} - 6 \, {\left (2 \, D a b + C b^{2}\right )} c^{2} d + 3 \, {\left (D a^{2} + 2 \, C a b + B b^{2}\right )} c d^{2} - {\left (C a^{2} + 2 \, B a b + A b^{2}\right )} d^{3}\right )} {\left (d x + c\right )}^{\frac {10}{3}} + 19760 \, {\left (5 \, D b^{2} c^{4} - 4 \, {\left (2 \, D a b + C b^{2}\right )} c^{3} d + 3 \, {\left (D a^{2} + 2 \, C a b + B b^{2}\right )} c^{2} d^{2} - 2 \, {\left (C a^{2} + 2 \, B a b + A b^{2}\right )} c d^{3} + {\left (B a^{2} + 2 \, A a b\right )} d^{4}\right )} {\left (d x + c\right )}^{\frac {7}{3}} - 34580 \, {\left (D b^{2} c^{5} - A a^{2} d^{5} - {\left (2 \, D a b + C b^{2}\right )} c^{4} d + {\left (D a^{2} + 2 \, C a b + B b^{2}\right )} c^{3} d^{2} - {\left (C a^{2} + 2 \, B a b + A b^{2}\right )} c^{2} d^{3} + {\left (B a^{2} + 2 \, A a b\right )} c d^{4}\right )} {\left (d x + c\right )}^{\frac {4}{3}}\right )}}{138320 \, d^{6}} \] Input:
integrate((b*x+a)^2*(d*x+c)^(1/3)*(D*x^3+C*x^2+B*x+A),x, algorithm="maxima ")
Output:
3/138320*(7280*(d*x + c)^(19/3)*D*b^2 - 8645*(5*D*b^2*c - (2*D*a*b + C*b^2 )*d)*(d*x + c)^(16/3) + 10640*(10*D*b^2*c^2 - 4*(2*D*a*b + C*b^2)*c*d + (D *a^2 + 2*C*a*b + B*b^2)*d^2)*(d*x + c)^(13/3) - 13832*(10*D*b^2*c^3 - 6*(2 *D*a*b + C*b^2)*c^2*d + 3*(D*a^2 + 2*C*a*b + B*b^2)*c*d^2 - (C*a^2 + 2*B*a *b + A*b^2)*d^3)*(d*x + c)^(10/3) + 19760*(5*D*b^2*c^4 - 4*(2*D*a*b + C*b^ 2)*c^3*d + 3*(D*a^2 + 2*C*a*b + B*b^2)*c^2*d^2 - 2*(C*a^2 + 2*B*a*b + A*b^ 2)*c*d^3 + (B*a^2 + 2*A*a*b)*d^4)*(d*x + c)^(7/3) - 34580*(D*b^2*c^5 - A*a ^2*d^5 - (2*D*a*b + C*b^2)*c^4*d + (D*a^2 + 2*C*a*b + B*b^2)*c^3*d^2 - (C* a^2 + 2*B*a*b + A*b^2)*c^2*d^3 + (B*a^2 + 2*A*a*b)*c*d^4)*(d*x + c)^(4/3)) /d^6
Leaf count of result is larger than twice the leaf count of optimal. 1269 vs. \(2 (302) = 604\).
Time = 0.14 (sec) , antiderivative size = 1269, normalized size of antiderivative = 3.89 \[ \int (a+b x)^2 \sqrt [3]{c+d x} \left (A+B x+C x^2+D x^3\right ) \, dx=\text {Too large to display} \] Input:
integrate((b*x+a)^2*(d*x+c)^(1/3)*(D*x^3+C*x^2+B*x+A),x, algorithm="giac")
Output:
3/138320*(138320*(d*x + c)^(1/3)*A*a^2*c + 34580*((d*x + c)^(4/3) - 4*(d*x + c)^(1/3)*c)*A*a^2 + 34580*((d*x + c)^(4/3) - 4*(d*x + c)^(1/3)*c)*B*a^2 *c/d + 69160*((d*x + c)^(4/3) - 4*(d*x + c)^(1/3)*c)*A*a*b*c/d + 9880*(2*( d*x + c)^(7/3) - 7*(d*x + c)^(4/3)*c + 14*(d*x + c)^(1/3)*c^2)*C*a^2*c/d^2 + 19760*(2*(d*x + c)^(7/3) - 7*(d*x + c)^(4/3)*c + 14*(d*x + c)^(1/3)*c^2 )*B*a*b*c/d^2 + 9880*(2*(d*x + c)^(7/3) - 7*(d*x + c)^(4/3)*c + 14*(d*x + c)^(1/3)*c^2)*A*b^2*c/d^2 + 9880*(2*(d*x + c)^(7/3) - 7*(d*x + c)^(4/3)*c + 14*(d*x + c)^(1/3)*c^2)*B*a^2/d + 19760*(2*(d*x + c)^(7/3) - 7*(d*x + c) ^(4/3)*c + 14*(d*x + c)^(1/3)*c^2)*A*a*b/d + 988*(14*(d*x + c)^(10/3) - 60 *(d*x + c)^(7/3)*c + 105*(d*x + c)^(4/3)*c^2 - 140*(d*x + c)^(1/3)*c^3)*D* a^2*c/d^3 + 1976*(14*(d*x + c)^(10/3) - 60*(d*x + c)^(7/3)*c + 105*(d*x + c)^(4/3)*c^2 - 140*(d*x + c)^(1/3)*c^3)*C*a*b*c/d^3 + 988*(14*(d*x + c)^(1 0/3) - 60*(d*x + c)^(7/3)*c + 105*(d*x + c)^(4/3)*c^2 - 140*(d*x + c)^(1/3 )*c^3)*B*b^2*c/d^3 + 988*(14*(d*x + c)^(10/3) - 60*(d*x + c)^(7/3)*c + 105 *(d*x + c)^(4/3)*c^2 - 140*(d*x + c)^(1/3)*c^3)*C*a^2/d^2 + 1976*(14*(d*x + c)^(10/3) - 60*(d*x + c)^(7/3)*c + 105*(d*x + c)^(4/3)*c^2 - 140*(d*x + c)^(1/3)*c^3)*B*a*b/d^2 + 988*(14*(d*x + c)^(10/3) - 60*(d*x + c)^(7/3)*c + 105*(d*x + c)^(4/3)*c^2 - 140*(d*x + c)^(1/3)*c^3)*A*b^2/d^2 + 608*(35*( d*x + c)^(13/3) - 182*(d*x + c)^(10/3)*c + 390*(d*x + c)^(7/3)*c^2 - 455*( d*x + c)^(4/3)*c^3 + 455*(d*x + c)^(1/3)*c^4)*D*a*b*c/d^4 + 304*(35*(d*...
Timed out. \[ \int (a+b x)^2 \sqrt [3]{c+d x} \left (A+B x+C x^2+D x^3\right ) \, dx=\int {\left (a+b\,x\right )}^2\,{\left (c+d\,x\right )}^{1/3}\,\left (A+B\,x+C\,x^2+x^3\,D\right ) \,d x \] Input:
int((a + b*x)^2*(c + d*x)^(1/3)*(A + B*x + C*x^2 + x^3*D),x)
Output:
int((a + b*x)^2*(c + d*x)^(1/3)*(A + B*x + C*x^2 + x^3*D), x)
Time = 0.14 (sec) , antiderivative size = 383, normalized size of antiderivative = 1.17 \[ \int (a+b x)^2 \sqrt [3]{c+d x} \left (A+B x+C x^2+D x^3\right ) \, dx=\frac {3 \left (d x +c \right )^{\frac {1}{3}} \left (3640 b^{2} d^{6} x^{6}+8645 a b \,d^{6} x^{5}+4550 b^{2} c \,d^{5} x^{5}+5320 a^{2} d^{6} x^{4}+11305 a b c \,d^{5} x^{4}+5320 b^{3} d^{5} x^{4}+70 b^{2} c^{2} d^{4} x^{4}+7448 a^{2} c \,d^{5} x^{3}+20748 a \,b^{2} d^{5} x^{3}+266 a b \,c^{2} d^{4} x^{3}+532 b^{3} c \,d^{4} x^{3}-84 b^{2} c^{3} d^{3} x^{3}+29640 a^{2} b \,d^{5} x^{2}+304 a^{2} c^{2} d^{4} x^{2}+2964 a \,b^{2} c \,d^{4} x^{2}-342 a b \,c^{3} d^{3} x^{2}-684 b^{3} c^{2} d^{3} x^{2}+108 b^{2} c^{4} d^{2} x^{2}+17290 a^{3} d^{5} x +7410 a^{2} b c \,d^{4} x -456 a^{2} c^{3} d^{3} x -4446 a \,b^{2} c^{2} d^{3} x +513 a b \,c^{4} d^{2} x +1026 b^{3} c^{3} d^{2} x -162 b^{2} c^{5} d x +17290 a^{3} c \,d^{4}-22230 a^{2} b \,c^{2} d^{3}+1368 a^{2} c^{4} d^{2}+13338 a \,b^{2} c^{3} d^{2}-1539 a b \,c^{5} d -3078 b^{3} c^{4} d +486 b^{2} c^{6}\right )}{69160 d^{5}} \] Input:
int((b*x+a)^2*(d*x+c)^(1/3)*(D*x^3+C*x^2+B*x+A),x)
Output:
(3*(c + d*x)**(1/3)*(17290*a**3*c*d**4 + 17290*a**3*d**5*x - 22230*a**2*b* c**2*d**3 + 7410*a**2*b*c*d**4*x + 29640*a**2*b*d**5*x**2 + 1368*a**2*c**4 *d**2 - 456*a**2*c**3*d**3*x + 304*a**2*c**2*d**4*x**2 + 7448*a**2*c*d**5* x**3 + 5320*a**2*d**6*x**4 + 13338*a*b**2*c**3*d**2 - 4446*a*b**2*c**2*d** 3*x + 2964*a*b**2*c*d**4*x**2 + 20748*a*b**2*d**5*x**3 - 1539*a*b*c**5*d + 513*a*b*c**4*d**2*x - 342*a*b*c**3*d**3*x**2 + 266*a*b*c**2*d**4*x**3 + 1 1305*a*b*c*d**5*x**4 + 8645*a*b*d**6*x**5 - 3078*b**3*c**4*d + 1026*b**3*c **3*d**2*x - 684*b**3*c**2*d**3*x**2 + 532*b**3*c*d**4*x**3 + 5320*b**3*d* *5*x**4 + 486*b**2*c**6 - 162*b**2*c**5*d*x + 108*b**2*c**4*d**2*x**2 - 84 *b**2*c**3*d**3*x**3 + 70*b**2*c**2*d**4*x**4 + 4550*b**2*c*d**5*x**5 + 36 40*b**2*d**6*x**6))/(69160*d**5)