Integrand size = 32, antiderivative size = 324 \[ \int \frac {(a+b x)^2 \left (A+B x+C x^2+D x^3\right )}{(c+d x)^{2/3}} \, dx=\frac {3 (b c-a d)^2 \left (c^2 C d-B c d^2+A d^3-c^3 D\right ) \sqrt [3]{c+d x}}{d^6}+\frac {3 (b c-a d) \left (a d \left (2 c C d-B d^2-3 c^2 D\right )-b \left (4 c^2 C d-3 B c d^2+2 A d^3-5 c^3 D\right )\right ) (c+d x)^{4/3}}{4 d^6}+\frac {3 \left (a^2 d^2 (C d-3 c D)-2 a b d \left (3 c C d-B d^2-6 c^2 D\right )+b^2 \left (6 c^2 C d-3 B c d^2+A d^3-10 c^3 D\right )\right ) (c+d x)^{7/3}}{7 d^6}+\frac {3 \left (a^2 d^2 D+2 a b d (C d-4 c D)-b^2 \left (4 c C d-B d^2-10 c^2 D\right )\right ) (c+d x)^{10/3}}{10 d^6}+\frac {3 b (b C d-5 b c D+2 a d D) (c+d x)^{13/3}}{13 d^6}+\frac {3 b^2 D (c+d x)^{16/3}}{16 d^6} \] Output:
3*(-a*d+b*c)^2*(A*d^3-B*c*d^2+C*c^2*d-D*c^3)*(d*x+c)^(1/3)/d^6+3/4*(-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)^(4/3)/d^6+3/7*(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)^(7/3)/d^6+3/10*(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)^(10/3)/d^6 +3/13*b*(C*b*d+2*D*a*d-5*D*b*c)*(d*x+c)^(13/3)/d^6+3/16*b^2*D*(d*x+c)^(16/ 3)/d^6
Time = 0.29 (sec) , antiderivative size = 324, normalized size of antiderivative = 1.00 \[ \int \frac {(a+b x)^2 \left (A+B x+C x^2+D x^3\right )}{(c+d x)^{2/3}} \, dx=\frac {3 \sqrt [3]{c+d x} \left (52 a^2 d^2 \left (-81 c^3 D+9 c^2 d (10 C+3 D x)-3 c d^2 (35 B+2 x (5 C+3 D x))+d^3 \left (140 A+x \left (35 B+20 C x+14 D x^2\right )\right )\right )+8 a b d \left (972 c^4 D-81 c^3 d (13 C+4 D x)+9 c^2 d^2 (130 B+3 x (13 C+8 D x))-3 c d^3 \left (455 A+2 x \left (65 B+39 C x+28 D x^2\right )\right )+d^4 x \left (455 A+2 x \left (130 B+91 C x+70 D x^2\right )\right )\right )+b^2 \left (-3645 c^5 D+243 c^4 d (16 C+5 D x)-162 c^3 d^2 (26 B+x (8 C+5 D x))+d^5 x^2 \left (1040 A+7 x \left (104 B+80 C x+65 D x^2\right )\right )-3 c d^4 x (520 A+x (312 B+7 x (32 C+25 D x)))+18 c^2 d^3 (260 A+x (78 B+x (48 C+35 D x)))\right )\right )}{7280 d^6} \] Input:
Integrate[((a + b*x)^2*(A + B*x + C*x^2 + D*x^3))/(c + d*x)^(2/3),x]
Output:
(3*(c + d*x)^(1/3)*(52*a^2*d^2*(-81*c^3*D + 9*c^2*d*(10*C + 3*D*x) - 3*c*d ^2*(35*B + 2*x*(5*C + 3*D*x)) + d^3*(140*A + x*(35*B + 20*C*x + 14*D*x^2)) ) + 8*a*b*d*(972*c^4*D - 81*c^3*d*(13*C + 4*D*x) + 9*c^2*d^2*(130*B + 3*x* (13*C + 8*D*x)) - 3*c*d^3*(455*A + 2*x*(65*B + 39*C*x + 28*D*x^2)) + d^4*x *(455*A + 2*x*(130*B + 91*C*x + 70*D*x^2))) + b^2*(-3645*c^5*D + 243*c^4*d *(16*C + 5*D*x) - 162*c^3*d^2*(26*B + x*(8*C + 5*D*x)) + d^5*x^2*(1040*A + 7*x*(104*B + 80*C*x + 65*D*x^2)) - 3*c*d^4*x*(520*A + x*(312*B + 7*x*(32* C + 25*D*x))) + 18*c^2*d^3*(260*A + x*(78*B + x*(48*C + 35*D*x))))))/(7280 *d^6)
Time = 0.61 (sec) , antiderivative size = 324, 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.
| \(\displaystyle \int \frac {(a+b x)^2 \left (A+B x+C x^2+D x^3\right )}{(c+d x)^{2/3}} \, dx\) |
| \(\Big \downarrow \) 2123 |
| \(\displaystyle \int \left (\frac {(c+d x)^{4/3} \left (a^2 d^2 (C d-3 c D)-2 a b d \left (-B d^2-6 c^2 D+3 c C d\right )+b^2 \left (A d^3-3 B c d^2-10 c^3 D+6 c^2 C d\right )\right )}{d^5}+\frac {(c+d x)^{7/3} \left (a^2 d^2 D+2 a b d (C d-4 c D)-\left (b^2 \left (-B d^2-10 c^2 D+4 c C d\right )\right )\right )}{d^5}+\frac {\sqrt [3]{c+d x} (b c-a d) \left (a d \left (-B d^2-3 c^2 D+2 c C d\right )-b \left (2 A d^3-3 B c d^2-5 c^3 D+4 c^2 C d\right )\right )}{d^5}+\frac {(a d-b c)^2 \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{d^5 (c+d x)^{2/3}}+\frac {b (c+d x)^{10/3} (2 a d D-5 b c D+b C d)}{d^5}+\frac {b^2 D (c+d x)^{13/3}}{d^5}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {3 (c+d x)^{7/3} \left (a^2 d^2 (C d-3 c D)-2 a b d \left (-B d^2-6 c^2 D+3 c C d\right )+b^2 \left (A d^3-3 B c d^2-10 c^3 D+6 c^2 C d\right )\right )}{7 d^6}+\frac {3 (c+d x)^{10/3} \left (a^2 d^2 D+2 a b d (C d-4 c D)-\left (b^2 \left (-B d^2-10 c^2 D+4 c C d\right )\right )\right )}{10 d^6}+\frac {3 (c+d x)^{4/3} (b c-a d) \left (a d \left (-B d^2-3 c^2 D+2 c C d\right )-b \left (2 A d^3-3 B c d^2-5 c^3 D+4 c^2 C d\right )\right )}{4 d^6}+\frac {3 \sqrt [3]{c+d x} (b c-a d)^2 \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{d^6}+\frac {3 b (c+d x)^{13/3} (2 a d D-5 b c D+b C d)}{13 d^6}+\frac {3 b^2 D (c+d x)^{16/3}}{16 d^6}\) |
Input:
Int[((a + b*x)^2*(A + B*x + C*x^2 + D*x^3))/(c + d*x)^(2/3),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)^(1/3))/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)^(4/3))/(4*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)^(7/3))/(7*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)^(10/3))/(10*d^6) + (3*b*(b*C*d - 5*b*c*D + 2*a*d*D)*(c + d*x)^(13/3))/(13*d^6) + (3*b^2*D*(c + d*x)^(16/3))/(16*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 = 1.48 (sec) , antiderivative size = 285, normalized size of antiderivative = 0.88
| method | result | size |
| pseudoelliptic | \(\frac {3 \left (\left (\frac {x^{2} \left (\frac {7}{16} D x^{3}+\frac {7}{13} C \,x^{2}+\frac {7}{10} B x +A \right ) b^{2}}{7}+\frac {x \left (\frac {4}{13} D x^{3}+\frac {2}{5} C \,x^{2}+\frac {4}{7} B x +A \right ) a b}{2}+a^{2} \left (\frac {1}{10} D x^{3}+\frac {1}{7} C \,x^{2}+\frac {1}{4} B x +A \right )\right ) d^{5}-\frac {3 \left (\frac {\left (\frac {35}{104} D x^{3}+\frac {28}{65} C \,x^{2}+\frac {3}{5} B x +A \right ) x \,b^{2}}{7}+a \left (\frac {8}{65} D x^{3}+\frac {6}{35} C \,x^{2}+\frac {2}{7} B x +A \right ) b +\frac {\left (\frac {6}{35} D x^{2}+\frac {2}{7} C x +B \right ) a^{2}}{2}\right ) c \,d^{4}}{2}+\frac {9 \left (\left (\frac {7}{52} D x^{3}+\frac {12}{65} C \,x^{2}+\frac {3}{10} B x +A \right ) b^{2}+2 \left (\frac {12}{65} D x^{2}+\frac {3}{10} C x +B \right ) a b +a^{2} \left (\frac {3 D x}{10}+C \right )\right ) c^{2} d^{3}}{14}-\frac {81 \left (\left (\frac {5}{26} D x^{2}+\frac {4}{13} C x +B \right ) b^{2}+2 \left (\frac {4 D x}{13}+C \right ) a b +D a^{2}\right ) c^{3} d^{2}}{140}+\frac {243 \left (\left (\frac {5 D x}{16}+C \right ) b +2 D a \right ) c^{4} b d}{455}-\frac {729 D b^{2} c^{5}}{1456}\right ) \left (x d +c \right )^{\frac {1}{3}}}{d^{6}}\) | \(285\) |
| derivativedivides | \(\frac {\frac {3 b^{2} D \left (x d +c \right )^{\frac {16}{3}}}{16}+\frac {3 \left (2 b \left (a d -b c \right ) D+b^{2} \left (C d -3 D c \right )\right ) \left (x d +c \right )^{\frac {13}{3}}}{13}+\frac {3 \left (\left (a d -b c \right )^{2} D+2 b \left (a d -b c \right ) \left (C d -3 D c \right )+b^{2} \left (B \,d^{2}-2 C c d +3 D c^{2}\right )\right ) \left (x d +c \right )^{\frac {10}{3}}}{10}+\frac {3 \left (\left (a d -b c \right )^{2} \left (C d -3 D c \right )+2 b \left (a d -b c \right ) \left (B \,d^{2}-2 C c d +3 D c^{2}\right )+b^{2} \left (A \,d^{3}-B c \,d^{2}+C \,c^{2} d -D c^{3}\right )\right ) \left (x d +c \right )^{\frac {7}{3}}}{7}+\frac {3 \left (\left (a d -b c \right )^{2} \left (B \,d^{2}-2 C c d +3 D c^{2}\right )+2 b \left (a d -b c \right ) \left (A \,d^{3}-B c \,d^{2}+C \,c^{2} d -D c^{3}\right )\right ) \left (x d +c \right )^{\frac {4}{3}}}{4}+3 \left (a d -b c \right )^{2} \left (A \,d^{3}-B c \,d^{2}+C \,c^{2} d -D c^{3}\right ) \left (x d +c \right )^{\frac {1}{3}}}{d^{6}}\) | \(319\) |
| default | \(\frac {\frac {3 b^{2} D \left (x d +c \right )^{\frac {16}{3}}}{16}+\frac {3 \left (2 b \left (a d -b c \right ) D+b^{2} \left (C d -3 D c \right )\right ) \left (x d +c \right )^{\frac {13}{3}}}{13}+\frac {3 \left (\left (a d -b c \right )^{2} D+2 b \left (a d -b c \right ) \left (C d -3 D c \right )+b^{2} \left (B \,d^{2}-2 C c d +3 D c^{2}\right )\right ) \left (x d +c \right )^{\frac {10}{3}}}{10}+\frac {3 \left (\left (a d -b c \right )^{2} \left (C d -3 D c \right )+2 b \left (a d -b c \right ) \left (B \,d^{2}-2 C c d +3 D c^{2}\right )+b^{2} \left (A \,d^{3}-B c \,d^{2}+C \,c^{2} d -D c^{3}\right )\right ) \left (x d +c \right )^{\frac {7}{3}}}{7}+\frac {3 \left (\left (a d -b c \right )^{2} \left (B \,d^{2}-2 C c d +3 D c^{2}\right )+2 b \left (a d -b c \right ) \left (A \,d^{3}-B c \,d^{2}+C \,c^{2} d -D c^{3}\right )\right ) \left (x d +c \right )^{\frac {4}{3}}}{4}+3 \left (a d -b c \right )^{2} \left (A \,d^{3}-B c \,d^{2}+C \,c^{2} d -D c^{3}\right ) \left (x d +c \right )^{\frac {1}{3}}}{d^{6}}\) | \(319\) |
| gosper | \(\frac {3 \left (x d +c \right )^{\frac {1}{3}} \left (455 D x^{5} b^{2} d^{5}+560 C \,x^{4} b^{2} d^{5}+1120 D x^{4} a b \,d^{5}-525 D x^{4} b^{2} c \,d^{4}+728 B \,x^{3} b^{2} d^{5}+1456 C \,x^{3} a b \,d^{5}-672 C \,x^{3} b^{2} c \,d^{4}+728 D x^{3} a^{2} d^{5}-1344 D x^{3} a b c \,d^{4}+630 D x^{3} b^{2} c^{2} d^{3}+1040 A \,x^{2} b^{2} d^{5}+2080 B \,x^{2} a b \,d^{5}-936 B \,x^{2} b^{2} c \,d^{4}+1040 C \,x^{2} a^{2} d^{5}-1872 C \,x^{2} a b c \,d^{4}+864 C \,x^{2} b^{2} c^{2} d^{3}-936 D x^{2} a^{2} c \,d^{4}+1728 D x^{2} a b \,c^{2} d^{3}-810 D x^{2} b^{2} c^{3} d^{2}+3640 A a b \,d^{5} x -1560 A x \,b^{2} c \,d^{4}+1820 B \,a^{2} d^{5} x -3120 B x a b c \,d^{4}+1404 B \,b^{2} c^{2} d^{3} x -1560 C x \,a^{2} c \,d^{4}+2808 C a b \,c^{2} d^{3} x -1296 C \,b^{2} c^{3} d^{2} x +1404 D a^{2} c^{2} d^{3} x -2592 D a b \,c^{3} d^{2} x +1215 D b^{2} c^{4} d x +7280 A \,a^{2} d^{5}-10920 A a b c \,d^{4}+4680 A \,b^{2} c^{2} d^{3}-5460 B \,a^{2} c \,d^{4}+9360 B a b \,c^{2} d^{3}-4212 B \,b^{2} c^{3} d^{2}+4680 C \,a^{2} c^{2} d^{3}-8424 C a b \,c^{3} d^{2}+3888 C \,b^{2} c^{4} d -4212 D a^{2} c^{3} d^{2}+7776 D a b \,c^{4} d -3645 D b^{2} c^{5}\right )}{7280 d^{6}}\) | \(505\) |
| trager | \(\frac {3 \left (x d +c \right )^{\frac {1}{3}} \left (455 D x^{5} b^{2} d^{5}+560 C \,x^{4} b^{2} d^{5}+1120 D x^{4} a b \,d^{5}-525 D x^{4} b^{2} c \,d^{4}+728 B \,x^{3} b^{2} d^{5}+1456 C \,x^{3} a b \,d^{5}-672 C \,x^{3} b^{2} c \,d^{4}+728 D x^{3} a^{2} d^{5}-1344 D x^{3} a b c \,d^{4}+630 D x^{3} b^{2} c^{2} d^{3}+1040 A \,x^{2} b^{2} d^{5}+2080 B \,x^{2} a b \,d^{5}-936 B \,x^{2} b^{2} c \,d^{4}+1040 C \,x^{2} a^{2} d^{5}-1872 C \,x^{2} a b c \,d^{4}+864 C \,x^{2} b^{2} c^{2} d^{3}-936 D x^{2} a^{2} c \,d^{4}+1728 D x^{2} a b \,c^{2} d^{3}-810 D x^{2} b^{2} c^{3} d^{2}+3640 A a b \,d^{5} x -1560 A x \,b^{2} c \,d^{4}+1820 B \,a^{2} d^{5} x -3120 B x a b c \,d^{4}+1404 B \,b^{2} c^{2} d^{3} x -1560 C x \,a^{2} c \,d^{4}+2808 C a b \,c^{2} d^{3} x -1296 C \,b^{2} c^{3} d^{2} x +1404 D a^{2} c^{2} d^{3} x -2592 D a b \,c^{3} d^{2} x +1215 D b^{2} c^{4} d x +7280 A \,a^{2} d^{5}-10920 A a b c \,d^{4}+4680 A \,b^{2} c^{2} d^{3}-5460 B \,a^{2} c \,d^{4}+9360 B a b \,c^{2} d^{3}-4212 B \,b^{2} c^{3} d^{2}+4680 C \,a^{2} c^{2} d^{3}-8424 C a b \,c^{3} d^{2}+3888 C \,b^{2} c^{4} d -4212 D a^{2} c^{3} d^{2}+7776 D a b \,c^{4} d -3645 D b^{2} c^{5}\right )}{7280 d^{6}}\) | \(505\) |
| orering | \(\frac {3 \left (x d +c \right )^{\frac {1}{3}} \left (455 D x^{5} b^{2} d^{5}+560 C \,x^{4} b^{2} d^{5}+1120 D x^{4} a b \,d^{5}-525 D x^{4} b^{2} c \,d^{4}+728 B \,x^{3} b^{2} d^{5}+1456 C \,x^{3} a b \,d^{5}-672 C \,x^{3} b^{2} c \,d^{4}+728 D x^{3} a^{2} d^{5}-1344 D x^{3} a b c \,d^{4}+630 D x^{3} b^{2} c^{2} d^{3}+1040 A \,x^{2} b^{2} d^{5}+2080 B \,x^{2} a b \,d^{5}-936 B \,x^{2} b^{2} c \,d^{4}+1040 C \,x^{2} a^{2} d^{5}-1872 C \,x^{2} a b c \,d^{4}+864 C \,x^{2} b^{2} c^{2} d^{3}-936 D x^{2} a^{2} c \,d^{4}+1728 D x^{2} a b \,c^{2} d^{3}-810 D x^{2} b^{2} c^{3} d^{2}+3640 A a b \,d^{5} x -1560 A x \,b^{2} c \,d^{4}+1820 B \,a^{2} d^{5} x -3120 B x a b c \,d^{4}+1404 B \,b^{2} c^{2} d^{3} x -1560 C x \,a^{2} c \,d^{4}+2808 C a b \,c^{2} d^{3} x -1296 C \,b^{2} c^{3} d^{2} x +1404 D a^{2} c^{2} d^{3} x -2592 D a b \,c^{3} d^{2} x +1215 D b^{2} c^{4} d x +7280 A \,a^{2} d^{5}-10920 A a b c \,d^{4}+4680 A \,b^{2} c^{2} d^{3}-5460 B \,a^{2} c \,d^{4}+9360 B a b \,c^{2} d^{3}-4212 B \,b^{2} c^{3} d^{2}+4680 C \,a^{2} c^{2} d^{3}-8424 C a b \,c^{3} d^{2}+3888 C \,b^{2} c^{4} d -4212 D a^{2} c^{3} d^{2}+7776 D a b \,c^{4} d -3645 D b^{2} c^{5}\right )}{7280 d^{6}}\) | \(505\) |
Input:
int((b*x+a)^2*(D*x^3+C*x^2+B*x+A)/(d*x+c)^(2/3),x,method=_RETURNVERBOSE)
Output:
3*((1/7*x^2*(7/16*D*x^3+7/13*C*x^2+7/10*B*x+A)*b^2+1/2*x*(4/13*D*x^3+2/5*C *x^2+4/7*B*x+A)*a*b+a^2*(1/10*D*x^3+1/7*C*x^2+1/4*B*x+A))*d^5-3/2*(1/7*(35 /104*D*x^3+28/65*C*x^2+3/5*B*x+A)*x*b^2+a*(8/65*D*x^3+6/35*C*x^2+2/7*B*x+A )*b+1/2*(6/35*D*x^2+2/7*C*x+B)*a^2)*c*d^4+9/14*((7/52*D*x^3+12/65*C*x^2+3/ 10*B*x+A)*b^2+2*(12/65*D*x^2+3/10*C*x+B)*a*b+a^2*(3/10*D*x+C))*c^2*d^3-81/ 140*((5/26*D*x^2+4/13*C*x+B)*b^2+2*(4/13*D*x+C)*a*b+D*a^2)*c^3*d^2+243/455 *((5/16*D*x+C)*b+2*D*a)*c^4*b*d-729/1456*D*b^2*c^5)*(d*x+c)^(1/3)/d^6
Time = 0.07 (sec) , antiderivative size = 387, normalized size of antiderivative = 1.19 \[ \int \frac {(a+b x)^2 \left (A+B x+C x^2+D x^3\right )}{(c+d x)^{2/3}} \, dx=\frac {3 \, {\left (455 \, D b^{2} d^{5} x^{5} - 3645 \, D b^{2} c^{5} + 7280 \, A a^{2} d^{5} + 3888 \, {\left (2 \, D a b + C b^{2}\right )} c^{4} d - 4212 \, {\left (D a^{2} + 2 \, C a b + B b^{2}\right )} c^{3} d^{2} + 4680 \, {\left (C a^{2} + 2 \, B a b + A b^{2}\right )} c^{2} d^{3} - 5460 \, {\left (B a^{2} + 2 \, A a b\right )} c d^{4} - 35 \, {\left (15 \, D b^{2} c d^{4} - 16 \, {\left (2 \, D a b + C b^{2}\right )} d^{5}\right )} x^{4} + 14 \, {\left (45 \, D b^{2} c^{2} d^{3} - 48 \, {\left (2 \, D a b + C b^{2}\right )} c d^{4} + 52 \, {\left (D a^{2} + 2 \, C a b + B b^{2}\right )} d^{5}\right )} x^{3} - 2 \, {\left (405 \, D b^{2} c^{3} d^{2} - 432 \, {\left (2 \, D a b + C b^{2}\right )} c^{2} d^{3} + 468 \, {\left (D a^{2} + 2 \, C a b + B b^{2}\right )} c d^{4} - 520 \, {\left (C a^{2} + 2 \, B a b + A b^{2}\right )} d^{5}\right )} x^{2} + {\left (1215 \, D b^{2} c^{4} d - 1296 \, {\left (2 \, D a b + C b^{2}\right )} c^{3} d^{2} + 1404 \, {\left (D a^{2} + 2 \, C a b + B b^{2}\right )} c^{2} d^{3} - 1560 \, {\left (C a^{2} + 2 \, B a b + A b^{2}\right )} c d^{4} + 1820 \, {\left (B a^{2} + 2 \, A a b\right )} d^{5}\right )} x\right )} {\left (d x + c\right )}^{\frac {1}{3}}}{7280 \, d^{6}} \] Input:
integrate((b*x+a)^2*(D*x^3+C*x^2+B*x+A)/(d*x+c)^(2/3),x, algorithm="fricas ")
Output:
3/7280*(455*D*b^2*d^5*x^5 - 3645*D*b^2*c^5 + 7280*A*a^2*d^5 + 3888*(2*D*a* b + C*b^2)*c^4*d - 4212*(D*a^2 + 2*C*a*b + B*b^2)*c^3*d^2 + 4680*(C*a^2 + 2*B*a*b + A*b^2)*c^2*d^3 - 5460*(B*a^2 + 2*A*a*b)*c*d^4 - 35*(15*D*b^2*c*d ^4 - 16*(2*D*a*b + C*b^2)*d^5)*x^4 + 14*(45*D*b^2*c^2*d^3 - 48*(2*D*a*b + C*b^2)*c*d^4 + 52*(D*a^2 + 2*C*a*b + B*b^2)*d^5)*x^3 - 2*(405*D*b^2*c^3*d^ 2 - 432*(2*D*a*b + C*b^2)*c^2*d^3 + 468*(D*a^2 + 2*C*a*b + B*b^2)*c*d^4 - 520*(C*a^2 + 2*B*a*b + A*b^2)*d^5)*x^2 + (1215*D*b^2*c^4*d - 1296*(2*D*a*b + C*b^2)*c^3*d^2 + 1404*(D*a^2 + 2*C*a*b + B*b^2)*c^2*d^3 - 1560*(C*a^2 + 2*B*a*b + A*b^2)*c*d^4 + 1820*(B*a^2 + 2*A*a*b)*d^5)*x)*(d*x + c)^(1/3)/d ^6
Time = 1.42 (sec) , antiderivative size = 639, normalized size of antiderivative = 1.97 \[ \int \frac {(a+b x)^2 \left (A+B x+C x^2+D x^3\right )}{(c+d x)^{2/3}} \, dx=\begin {cases} \frac {3 \left (\frac {D b^{2} \left (c + d x\right )^{\frac {16}{3}}}{16 d^{5}} + \frac {\left (c + d x\right )^{\frac {13}{3}} \left (C b^{2} d + 2 D a b d - 5 D b^{2} c\right )}{13 d^{5}} + \frac {\left (c + d x\right )^{\frac {10}{3}} \left (B b^{2} d^{2} + 2 C a b d^{2} - 4 C b^{2} c d + D a^{2} d^{2} - 8 D a b c d + 10 D b^{2} c^{2}\right )}{10 d^{5}} + \frac {\left (c + d x\right )^{\frac {7}{3}} \left (A b^{2} d^{3} + 2 B a b d^{3} - 3 B b^{2} c d^{2} + C a^{2} d^{3} - 6 C a b c d^{2} + 6 C b^{2} c^{2} d - 3 D a^{2} c d^{2} + 12 D a b c^{2} d - 10 D b^{2} c^{3}\right )}{7 d^{5}} + \frac {\left (c + d x\right )^{\frac {4}{3}} \cdot \left (2 A a b d^{4} - 2 A b^{2} c d^{3} + B a^{2} d^{4} - 4 B a b c d^{3} + 3 B b^{2} c^{2} d^{2} - 2 C a^{2} c d^{3} + 6 C a b c^{2} d^{2} - 4 C b^{2} c^{3} d + 3 D a^{2} c^{2} d^{2} - 8 D a b c^{3} d + 5 D b^{2} c^{4}\right )}{4 d^{5}} + \frac {\sqrt [3]{c + d x} \left (A a^{2} d^{5} - 2 A a b c d^{4} + A b^{2} c^{2} d^{3} - B a^{2} c d^{4} + 2 B a b c^{2} d^{3} - B b^{2} c^{3} d^{2} + C a^{2} c^{2} d^{3} - 2 C a b c^{3} d^{2} + C b^{2} c^{4} d - D a^{2} c^{3} d^{2} + 2 D a b c^{4} d - D b^{2} c^{5}\right )}{d^{5}}\right )}{d} & \text {for}\: d \neq 0 \\\frac {A a^{2} x + \frac {D b^{2} x^{6}}{6} + \frac {x^{5} \left (C b^{2} + 2 D a b\right )}{5} + \frac {x^{4} \left (B b^{2} + 2 C a b + D a^{2}\right )}{4} + \frac {x^{3} \left (A b^{2} + 2 B a b + C a^{2}\right )}{3} + \frac {x^{2} \cdot \left (2 A a b + B a^{2}\right )}{2}}{c^{\frac {2}{3}}} & \text {otherwise} \end {cases} \] Input:
integrate((b*x+a)**2*(D*x**3+C*x**2+B*x+A)/(d*x+c)**(2/3),x)
Output:
Piecewise((3*(D*b**2*(c + d*x)**(16/3)/(16*d**5) + (c + d*x)**(13/3)*(C*b* *2*d + 2*D*a*b*d - 5*D*b**2*c)/(13*d**5) + (c + d*x)**(10/3)*(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 )/(10*d**5) + (c + d*x)**(7/3)*(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 + 12 *D*a*b*c**2*d - 10*D*b**2*c**3)/(7*d**5) + (c + d*x)**(4/3)*(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)/(4*d**5) + (c + d*x)**(1/3)*(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)/d**5)/d, Ne(d, 0)), ((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)/c**(2/3), True))
Time = 0.04 (sec) , antiderivative size = 387, normalized size of antiderivative = 1.19 \[ \int \frac {(a+b x)^2 \left (A+B x+C x^2+D x^3\right )}{(c+d x)^{2/3}} \, dx=\frac {3 \, {\left (455 \, {\left (d x + c\right )}^{\frac {16}{3}} D b^{2} - 560 \, {\left (5 \, D b^{2} c - {\left (2 \, D a b + C b^{2}\right )} d\right )} {\left (d x + c\right )}^{\frac {13}{3}} + 728 \, {\left (10 \, D b^{2} c^{2} - 4 \, {\left (2 \, D a b + C b^{2}\right )} c d + {\left (D a^{2} + 2 \, C a b + B b^{2}\right )} d^{2}\right )} {\left (d x + c\right )}^{\frac {10}{3}} - 1040 \, {\left (10 \, D b^{2} c^{3} - 6 \, {\left (2 \, D a b + C b^{2}\right )} c^{2} d + 3 \, {\left (D a^{2} + 2 \, C a b + B b^{2}\right )} c d^{2} - {\left (C a^{2} + 2 \, B a b + A b^{2}\right )} d^{3}\right )} {\left (d x + c\right )}^{\frac {7}{3}} + 1820 \, {\left (5 \, D b^{2} c^{4} - 4 \, {\left (2 \, D a b + C b^{2}\right )} c^{3} d + 3 \, {\left (D a^{2} + 2 \, C a b + B b^{2}\right )} c^{2} d^{2} - 2 \, {\left (C a^{2} + 2 \, B a b + A b^{2}\right )} c d^{3} + {\left (B a^{2} + 2 \, A a b\right )} d^{4}\right )} {\left (d x + c\right )}^{\frac {4}{3}} - 7280 \, {\left (D b^{2} c^{5} - A a^{2} d^{5} - {\left (2 \, D a b + C b^{2}\right )} c^{4} d + {\left (D a^{2} + 2 \, C a b + B b^{2}\right )} c^{3} d^{2} - {\left (C a^{2} + 2 \, B a b + A b^{2}\right )} c^{2} d^{3} + {\left (B a^{2} + 2 \, A a b\right )} c d^{4}\right )} {\left (d x + c\right )}^{\frac {1}{3}}\right )}}{7280 \, d^{6}} \] Input:
integrate((b*x+a)^2*(D*x^3+C*x^2+B*x+A)/(d*x+c)^(2/3),x, algorithm="maxima ")
Output:
3/7280*(455*(d*x + c)^(16/3)*D*b^2 - 560*(5*D*b^2*c - (2*D*a*b + C*b^2)*d) *(d*x + c)^(13/3) + 728*(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)^(10/3) - 1040*(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)^(7/3) + 1820*(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)^(4/3) - 7280*(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)^(1/3))/d^6
Time = 0.13 (sec) , antiderivative size = 558, normalized size of antiderivative = 1.72 \[ \int \frac {(a+b x)^2 \left (A+B x+C x^2+D x^3\right )}{(c+d x)^{2/3}} \, dx =\text {Too large to display} \] Input:
integrate((b*x+a)^2*(D*x^3+C*x^2+B*x+A)/(d*x+c)^(2/3),x, algorithm="giac")
Output:
3/7280*(7280*(d*x + c)^(1/3)*A*a^2 + 1820*((d*x + c)^(4/3) - 4*(d*x + c)^( 1/3)*c)*B*a^2/d + 3640*((d*x + c)^(4/3) - 4*(d*x + c)^(1/3)*c)*A*a*b/d + 5 20*(2*(d*x + c)^(7/3) - 7*(d*x + c)^(4/3)*c + 14*(d*x + c)^(1/3)*c^2)*C*a^ 2/d^2 + 1040*(2*(d*x + c)^(7/3) - 7*(d*x + c)^(4/3)*c + 14*(d*x + c)^(1/3) *c^2)*B*a*b/d^2 + 520*(2*(d*x + c)^(7/3) - 7*(d*x + c)^(4/3)*c + 14*(d*x + c)^(1/3)*c^2)*A*b^2/d^2 + 52*(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/d^3 + 104*(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/d^3 + 52*(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*b^2/d^3 + 32*(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/d^4 + 16*(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)*C*b^2/d^4 + 5*(91*(d*x + c)^(16/3) - 560*(d*x + c)^(13/3)*c + 1456*(d*x + c)^(10/3)*c^2 - 2080*(d*x + c)^(7/ 3)*c^3 + 1820*(d*x + c)^(4/3)*c^4 - 1456*(d*x + c)^(1/3)*c^5)*D*b^2/d^5)/d
Timed out. \[ \int \frac {(a+b x)^2 \left (A+B x+C x^2+D x^3\right )}{(c+d x)^{2/3}} \, dx=\int \frac {{\left (a+b\,x\right )}^2\,\left (A+B\,x+C\,x^2+x^3\,D\right )}{{\left (c+d\,x\right )}^{2/3}} \,d x \] Input:
int(((a + b*x)^2*(A + B*x + C*x^2 + x^3*D))/(c + d*x)^(2/3),x)
Output:
int(((a + b*x)^2*(A + B*x + C*x^2 + x^3*D))/(c + d*x)^(2/3), x)
Time = 0.14 (sec) , antiderivative size = 288, normalized size of antiderivative = 0.89 \[ \int \frac {(a+b x)^2 \left (A+B x+C x^2+D x^3\right )}{(c+d x)^{2/3}} \, dx=\frac {3 \left (d x +c \right )^{\frac {1}{3}} \left (455 b^{2} d^{5} x^{5}+1120 a b \,d^{5} x^{4}+35 b^{2} c \,d^{4} x^{4}+728 a^{2} d^{5} x^{3}+112 a b c \,d^{4} x^{3}+728 b^{3} d^{4} x^{3}-42 b^{2} c^{2} d^{3} x^{3}+104 a^{2} c \,d^{4} x^{2}+3120 a \,b^{2} d^{4} x^{2}-144 a b \,c^{2} d^{3} x^{2}-936 b^{3} c \,d^{3} x^{2}+54 b^{2} c^{3} d^{2} x^{2}+5460 a^{2} b \,d^{4} x -156 a^{2} c^{2} d^{3} x -4680 a \,b^{2} c \,d^{3} x +216 a b \,c^{3} d^{2} x +1404 b^{3} c^{2} d^{2} x -81 b^{2} c^{4} d x +7280 a^{3} d^{4}-16380 a^{2} b c \,d^{3}+468 a^{2} c^{3} d^{2}+14040 a \,b^{2} c^{2} d^{2}-648 a b \,c^{4} d -4212 b^{3} c^{3} d +243 b^{2} c^{5}\right )}{7280 d^{5}} \] Input:
int((b*x+a)^2*(D*x^3+C*x^2+B*x+A)/(d*x+c)^(2/3),x)
Output:
(3*(c + d*x)**(1/3)*(7280*a**3*d**4 - 16380*a**2*b*c*d**3 + 5460*a**2*b*d* *4*x + 468*a**2*c**3*d**2 - 156*a**2*c**2*d**3*x + 104*a**2*c*d**4*x**2 + 728*a**2*d**5*x**3 + 14040*a*b**2*c**2*d**2 - 4680*a*b**2*c*d**3*x + 3120* a*b**2*d**4*x**2 - 648*a*b*c**4*d + 216*a*b*c**3*d**2*x - 144*a*b*c**2*d** 3*x**2 + 112*a*b*c*d**4*x**3 + 1120*a*b*d**5*x**4 - 4212*b**3*c**3*d + 140 4*b**3*c**2*d**2*x - 936*b**3*c*d**3*x**2 + 728*b**3*d**4*x**3 + 243*b**2* c**5 - 81*b**2*c**4*d*x + 54*b**2*c**3*d**2*x**2 - 42*b**2*c**2*d**3*x**3 + 35*b**2*c*d**4*x**4 + 455*b**2*d**5*x**5))/(7280*d**5)