Integrand size = 32, antiderivative size = 455 \[ \int x^3 (c+d x)^2 \left (a+b x^2\right )^{3/2} \left (A+B x+C x^2\right ) \, dx=-\frac {3 a^3 (2 b c (B c+2 A d)-a d (2 c C+B d)) x \sqrt {a+b x^2}}{256 b^3}+\frac {a^2 (2 b c (B c+2 A d)-a d (2 c C+B d)) x^3 \sqrt {a+b x^2}}{128 b^2}+\frac {a (2 b c (B c+2 A d)-a d (2 c C+B d)) x^5 \sqrt {a+b x^2}}{32 b}+\frac {(2 b c (B c+2 A d)-a d (2 c C+B d)) x^5 \left (a+b x^2\right )^{3/2}}{16 b}-\frac {a \left (A b \left (b c^2-a d^2\right )+a \left (a C d^2-b c (c C+2 B d)\right )\right ) \left (a+b x^2\right )^{5/2}}{5 b^4}+\frac {d (2 c C+B d) x^5 \left (a+b x^2\right )^{5/2}}{10 b}+\frac {\left (A b \left (b c^2-2 a d^2\right )+a \left (3 a C d^2-2 b c (c C+2 B d)\right )\right ) \left (a+b x^2\right )^{7/2}}{7 b^4}-\frac {\left (3 a C d^2-b \left (c^2 C+2 B c d+A d^2\right )\right ) \left (a+b x^2\right )^{9/2}}{9 b^4}+\frac {C d^2 \left (a+b x^2\right )^{11/2}}{11 b^4}+\frac {3 a^4 (2 b c (B c+2 A d)-a d (2 c C+B d)) \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{256 b^{7/2}} \] Output:
-3/256*a^3*(2*b*c*(2*A*d+B*c)-a*d*(B*d+2*C*c))*x*(b*x^2+a)^(1/2)/b^3+1/128 *a^2*(2*b*c*(2*A*d+B*c)-a*d*(B*d+2*C*c))*x^3*(b*x^2+a)^(1/2)/b^2+1/32*a*(2 *b*c*(2*A*d+B*c)-a*d*(B*d+2*C*c))*x^5*(b*x^2+a)^(1/2)/b+1/16*(2*b*c*(2*A*d +B*c)-a*d*(B*d+2*C*c))*x^5*(b*x^2+a)^(3/2)/b-1/5*a*(A*b*(-a*d^2+b*c^2)+a*( a*C*d^2-b*c*(2*B*d+C*c)))*(b*x^2+a)^(5/2)/b^4+1/10*d*(B*d+2*C*c)*x^5*(b*x^ 2+a)^(5/2)/b+1/7*(A*b*(-2*a*d^2+b*c^2)+a*(3*a*C*d^2-2*b*c*(2*B*d+C*c)))*(b *x^2+a)^(7/2)/b^4-1/9*(3*a*C*d^2-b*(A*d^2+2*B*c*d+C*c^2))*(b*x^2+a)^(9/2)/ b^4+1/11*C*d^2*(b*x^2+a)^(11/2)/b^4+3/256*a^4*(2*b*c*(2*A*d+B*c)-a*d*(B*d+ 2*C*c))*arctanh(b^(1/2)*x/(b*x^2+a)^(1/2))/b^(7/2)
Time = 2.33 (sec) , antiderivative size = 448, normalized size of antiderivative = 0.98 \[ \int x^3 (c+d x)^2 \left (a+b x^2\right )^{3/2} \left (A+B x+C x^2\right ) \, dx=\frac {\sqrt {a+b x^2} \left (-12288 a^5 C d^2+a^4 b \left (22528 c^2 C+22 c d (2048 B+945 C x)+d^2 \left (22528 A+10395 B x+6144 C x^2\right )\right )+32 b^5 x^6 \left (110 A \left (36 c^2+63 c d x+28 d^2 x^2\right )+7 x \left (11 B \left (45 c^2+80 c d x+36 d^2 x^2\right )+8 C x \left (55 c^2+99 c d x+45 d^2 x^2\right )\right )\right )+12 a^2 b^3 x^2 \left (22 A \left (96 c^2+105 c d x+32 d^2 x^2\right )+x \left (11 B \left (105 c^2+128 c d x+42 d^2 x^2\right )+4 C x \left (176 c^2+231 c d x+80 d^2 x^2\right )\right )\right )-2 a^3 b^2 \left (22 A \left (1152 c^2+945 c d x+256 d^2 x^2\right )+x \left (11 B \left (945 c^2+1024 c d x+315 d^2 x^2\right )+2 C x \left (2816 c^2+3465 c d x+1152 d^2 x^2\right )\right )\right )+16 a b^4 x^4 \left (22 A \left (576 c^2+945 c d x+400 d^2 x^2\right )+x \left (11 B \left (945 c^2+1600 c d x+693 d^2 x^2\right )+2 C x \left (4400 c^2+7623 c d x+3360 d^2 x^2\right )\right )\right )\right )+10395 a^4 \sqrt {b} (-2 b c (B c+2 A d)+a d (2 c C+B d)) \log \left (-\sqrt {b} x+\sqrt {a+b x^2}\right )}{887040 b^4} \] Input:
Integrate[x^3*(c + d*x)^2*(a + b*x^2)^(3/2)*(A + B*x + C*x^2),x]
Output:
(Sqrt[a + b*x^2]*(-12288*a^5*C*d^2 + a^4*b*(22528*c^2*C + 22*c*d*(2048*B + 945*C*x) + d^2*(22528*A + 10395*B*x + 6144*C*x^2)) + 32*b^5*x^6*(110*A*(3 6*c^2 + 63*c*d*x + 28*d^2*x^2) + 7*x*(11*B*(45*c^2 + 80*c*d*x + 36*d^2*x^2 ) + 8*C*x*(55*c^2 + 99*c*d*x + 45*d^2*x^2))) + 12*a^2*b^3*x^2*(22*A*(96*c^ 2 + 105*c*d*x + 32*d^2*x^2) + x*(11*B*(105*c^2 + 128*c*d*x + 42*d^2*x^2) + 4*C*x*(176*c^2 + 231*c*d*x + 80*d^2*x^2))) - 2*a^3*b^2*(22*A*(1152*c^2 + 945*c*d*x + 256*d^2*x^2) + x*(11*B*(945*c^2 + 1024*c*d*x + 315*d^2*x^2) + 2*C*x*(2816*c^2 + 3465*c*d*x + 1152*d^2*x^2))) + 16*a*b^4*x^4*(22*A*(576*c ^2 + 945*c*d*x + 400*d^2*x^2) + x*(11*B*(945*c^2 + 1600*c*d*x + 693*d^2*x^ 2) + 2*C*x*(4400*c^2 + 7623*c*d*x + 3360*d^2*x^2)))) + 10395*a^4*Sqrt[b]*( -2*b*c*(B*c + 2*A*d) + a*d*(2*c*C + B*d))*Log[-(Sqrt[b]*x) + Sqrt[a + b*x^ 2]])/(887040*b^4)
Time = 3.49 (sec) , antiderivative size = 616, normalized size of antiderivative = 1.35, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.438, Rules used = {2185, 25, 2185, 27, 2185, 2185, 25, 27, 687, 676, 211, 211, 224, 219}
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.
| \(\displaystyle \int x^3 \left (a+b x^2\right )^{3/2} (c+d x)^2 \left (A+B x+C x^2\right ) \, dx\) |
| \(\Big \downarrow \) 2185 |
| \(\displaystyle \frac {\int -(c+d x)^2 \left (b x^2+a\right )^{3/2} \left (b d^4 (38 c C-11 B d) x^4+d^3 \left (48 b C c^2-11 A b d^2+6 a C d^2\right ) x^3+2 c C d^2 \left (13 b c^2+9 a d^2\right ) x^2+c^2 C d \left (5 b c^2+18 a d^2\right ) x+6 a c^3 C d^2\right )dx}{11 b d^5}+\frac {C \left (a+b x^2\right )^{5/2} (c+d x)^6}{11 b d^4}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {C \left (a+b x^2\right )^{5/2} (c+d x)^6}{11 b d^4}-\frac {\int (c+d x)^2 \left (b x^2+a\right )^{3/2} \left (b d^4 (38 c C-11 B d) x^4+d^3 \left (48 b C c^2-11 A b d^2+6 a C d^2\right ) x^3+2 c C d^2 \left (13 b c^2+9 a d^2\right ) x^2+c^2 C d \left (5 b c^2+18 a d^2\right ) x+6 a c^3 C d^2\right )dx}{11 b d^5}\) |
| \(\Big \downarrow \) 2185 |
| \(\displaystyle \frac {C \left (a+b x^2\right )^{5/2} (c+d x)^6}{11 b d^4}-\frac {\frac {\int -5 (c+d x)^2 \left (b x^2+a\right )^{3/2} \left (-b \left (12 a C d^2-b \left (94 C c^2-55 B d c+22 A d^2\right )\right ) x^3 d^7+b \left (4 b (25 c C-11 B d) c^2+a d^2 (2 c C-11 B d)\right ) x^2 d^6+a b c^2 (26 c C-11 B d) d^6+b c \left (b (28 c C-11 B d) c^2+2 a d^2 (20 c C-11 B d)\right ) x d^5\right )dx}{10 b d^4}+\frac {1}{10} d \left (a+b x^2\right )^{5/2} (c+d x)^5 (38 c C-11 B d)}{11 b d^5}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {C \left (a+b x^2\right )^{5/2} (c+d x)^6}{11 b d^4}-\frac {\frac {1}{10} d \left (a+b x^2\right )^{5/2} (c+d x)^5 (38 c C-11 B d)-\frac {\int (c+d x)^2 \left (b x^2+a\right )^{3/2} \left (-b \left (12 a C d^2-b \left (94 C c^2-55 B d c+22 A d^2\right )\right ) x^3 d^7+b \left (4 b (25 c C-11 B d) c^2+a d^2 (2 c C-11 B d)\right ) x^2 d^6+a b c^2 (26 c C-11 B d) d^6+b c \left (b (28 c C-11 B d) c^2+2 a d^2 (20 c C-11 B d)\right ) x d^5\right )dx}{2 b d^4}}{11 b d^5}\) |
| \(\Big \downarrow \) 2185 |
| \(\displaystyle \frac {C \left (a+b x^2\right )^{5/2} (c+d x)^6}{11 b d^4}-\frac {\frac {1}{10} d \left (a+b x^2\right )^{5/2} (c+d x)^5 (38 c C-11 B d)-\frac {\frac {\int (c+d x)^2 \left (b x^2+a\right )^{3/2} \left (b^2 \left (3 a d^2 (62 c C-33 B d)-2 b c \left (208 C c^2-187 B d c+154 A d^2\right )\right ) x^2 d^9+a b c \left (48 a C d^2-b \left (142 C c^2-121 B d c+88 A d^2\right )\right ) d^9+2 b \left (24 a^2 C d^4+11 a b \left (2 C c^2+B d c-4 A d^2\right ) d^2-b^2 c^2 \left (109 C c^2-88 B d c+55 A d^2\right )\right ) x d^8\right )dx}{9 b d^3}-\frac {1}{9} d^5 \left (a+b x^2\right )^{5/2} (c+d x)^4 \left (12 a C d^2-b \left (22 A d^2-55 B c d+94 c^2 C\right )\right )}{2 b d^4}}{11 b d^5}\) |
| \(\Big \downarrow \) 2185 |
| \(\displaystyle \frac {C \left (a+b x^2\right )^{5/2} (c+d x)^6}{11 b d^4}-\frac {\frac {1}{10} d \left (a+b x^2\right )^{5/2} (c+d x)^5 (38 c C-11 B d)-\frac {\frac {\frac {\int -b^2 d^{10} (c+d x)^2 \left (a d \left (3 a d^2 (58 c C-99 B d)-2 b c \left (56 C c^2-77 B d c+110 A d^2\right )\right )-\left (384 a^2 C d^4-a b \left (578 C c^2-671 B d c+704 A d^2\right ) d^2+6 b^2 c^2 \left (56 C c^2-77 B d c+110 A d^2\right )\right ) x\right ) \left (b x^2+a\right )^{3/2}dx}{8 b d^2}+\frac {1}{8} b d^8 \left (a+b x^2\right )^{5/2} (c+d x)^3 \left (3 a d^2 (62 c C-33 B d)-2 b c \left (154 A d^2-187 B c d+208 c^2 C\right )\right )}{9 b d^3}-\frac {1}{9} d^5 \left (a+b x^2\right )^{5/2} (c+d x)^4 \left (12 a C d^2-b \left (22 A d^2-55 B c d+94 c^2 C\right )\right )}{2 b d^4}}{11 b d^5}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {C \left (a+b x^2\right )^{5/2} (c+d x)^6}{11 b d^4}-\frac {\frac {1}{10} d \left (a+b x^2\right )^{5/2} (c+d x)^5 (38 c C-11 B d)-\frac {\frac {\frac {1}{8} b d^8 \left (a+b x^2\right )^{5/2} (c+d x)^3 \left (3 a d^2 (62 c C-33 B d)-2 b c \left (154 A d^2-187 B c d+208 c^2 C\right )\right )-\frac {\int b^2 d^{10} (c+d x)^2 \left (a d \left (3 a d^2 (58 c C-99 B d)-2 b c \left (56 C c^2-77 B d c+110 A d^2\right )\right )-\left (384 a^2 C d^4-a b \left (578 C c^2-671 B d c+704 A d^2\right ) d^2+6 b^2 c^2 \left (56 C c^2-77 B d c+110 A d^2\right )\right ) x\right ) \left (b x^2+a\right )^{3/2}dx}{8 b d^2}}{9 b d^3}-\frac {1}{9} d^5 \left (a+b x^2\right )^{5/2} (c+d x)^4 \left (12 a C d^2-b \left (22 A d^2-55 B c d+94 c^2 C\right )\right )}{2 b d^4}}{11 b d^5}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {C \left (a+b x^2\right )^{5/2} (c+d x)^6}{11 b d^4}-\frac {\frac {1}{10} d \left (a+b x^2\right )^{5/2} (c+d x)^5 (38 c C-11 B d)-\frac {\frac {\frac {1}{8} b d^8 \left (a+b x^2\right )^{5/2} (c+d x)^3 \left (3 a d^2 (62 c C-33 B d)-2 b c \left (154 A d^2-187 B c d+208 c^2 C\right )\right )-\frac {1}{8} b d^8 \int (c+d x)^2 \left (a d \left (3 a d^2 (58 c C-99 B d)-2 b c \left (56 C c^2-77 B d c+110 A d^2\right )\right )-\left (384 a^2 C d^4-a b \left (578 C c^2-671 B d c+704 A d^2\right ) d^2+6 b^2 c^2 \left (56 C c^2-77 B d c+110 A d^2\right )\right ) x\right ) \left (b x^2+a\right )^{3/2}dx}{9 b d^3}-\frac {1}{9} d^5 \left (a+b x^2\right )^{5/2} (c+d x)^4 \left (12 a C d^2-b \left (22 A d^2-55 B c d+94 c^2 C\right )\right )}{2 b d^4}}{11 b d^5}\) |
| \(\Big \downarrow \) 687 |
| \(\displaystyle \frac {C \left (a+b x^2\right )^{5/2} (c+d x)^6}{11 b d^4}-\frac {\frac {1}{10} d \left (a+b x^2\right )^{5/2} (c+d x)^5 (38 c C-11 B d)-\frac {\frac {\frac {1}{8} b d^8 \left (a+b x^2\right )^{5/2} (c+d x)^3 \left (3 a d^2 (62 c C-33 B d)-2 b c \left (154 A d^2-187 B c d+208 c^2 C\right )\right )-\frac {1}{8} b d^8 \left (\frac {\int (c+d x) \left (a d \left (768 a^2 C d^4+a b \left (62 C c^2-737 B d c-1408 A d^2\right ) d^2-2 b^2 c^2 \left (56 C c^2-77 B d c+110 A d^2\right )\right )+3 b \left (3 a^2 (50 c C-231 B d) d^4+4 a b c \left (31 C c^2-22 B d c-11 A d^2\right ) d^2-4 b^2 c^3 \left (56 C c^2-77 B d c+110 A d^2\right )\right ) x\right ) \left (b x^2+a\right )^{3/2}dx}{7 b}-\frac {\left (a+b x^2\right )^{5/2} (c+d x)^2 \left (384 a^2 C d^4-a b d^2 \left (704 A d^2-671 B c d+578 c^2 C\right )+6 b^2 c^2 \left (110 A d^2-77 B c d+56 c^2 C\right )\right )}{7 b}\right )}{9 b d^3}-\frac {1}{9} d^5 \left (a+b x^2\right )^{5/2} (c+d x)^4 \left (12 a C d^2-b \left (22 A d^2-55 B c d+94 c^2 C\right )\right )}{2 b d^4}}{11 b d^5}\) |
| \(\Big \downarrow \) 676 |
| \(\displaystyle \frac {C \left (a+b x^2\right )^{5/2} (c+d x)^6}{11 b d^4}-\frac {\frac {1}{10} d \left (a+b x^2\right )^{5/2} (c+d x)^5 (38 c C-11 B d)-\frac {\frac {\frac {1}{8} b d^8 \left (a+b x^2\right )^{5/2} (c+d x)^3 \left (3 a d^2 (62 c C-33 B d)-2 b c \left (154 A d^2-187 B c d+208 c^2 C\right )\right )-\frac {1}{8} b d^8 \left (\frac {-\frac {693}{2} a^2 d^4 (2 b c (2 A d+B c)-a d (B d+2 c C)) \int \left (b x^2+a\right )^{3/2}dx+\frac {1}{2} d x \left (a+b x^2\right )^{5/2} \left (3 a^2 d^4 (50 c C-231 B d)+4 a b c d^2 \left (-11 A d^2-22 B c d+31 c^2 C\right )-4 b^2 c^3 \left (110 A d^2-77 B c d+56 c^2 C\right )\right )+\frac {2 \left (a+b x^2\right )^{5/2} \left (384 a^3 C d^6+64 a^2 b d^4 \left (-11 A d^2-22 B c d+4 c^2 C\right )+a b^2 c^2 d^2 \left (-176 A d^2-55 B c d+130 c^2 C\right )-6 b^3 c^4 \left (110 A d^2-77 B c d+56 c^2 C\right )\right )}{5 b}}{7 b}-\frac {\left (a+b x^2\right )^{5/2} (c+d x)^2 \left (384 a^2 C d^4-a b d^2 \left (704 A d^2-671 B c d+578 c^2 C\right )+6 b^2 c^2 \left (110 A d^2-77 B c d+56 c^2 C\right )\right )}{7 b}\right )}{9 b d^3}-\frac {1}{9} d^5 \left (a+b x^2\right )^{5/2} (c+d x)^4 \left (12 a C d^2-b \left (22 A d^2-55 B c d+94 c^2 C\right )\right )}{2 b d^4}}{11 b d^5}\) |
| \(\Big \downarrow \) 211 |
| \(\displaystyle \frac {C \left (a+b x^2\right )^{5/2} (c+d x)^6}{11 b d^4}-\frac {\frac {1}{10} d \left (a+b x^2\right )^{5/2} (c+d x)^5 (38 c C-11 B d)-\frac {\frac {\frac {1}{8} b d^8 \left (a+b x^2\right )^{5/2} (c+d x)^3 \left (3 a d^2 (62 c C-33 B d)-2 b c \left (154 A d^2-187 B c d+208 c^2 C\right )\right )-\frac {1}{8} b d^8 \left (\frac {-\frac {693}{2} a^2 d^4 (2 b c (2 A d+B c)-a d (B d+2 c C)) \left (\frac {3}{4} a \int \sqrt {b x^2+a}dx+\frac {1}{4} x \left (a+b x^2\right )^{3/2}\right )+\frac {1}{2} d x \left (a+b x^2\right )^{5/2} \left (3 a^2 d^4 (50 c C-231 B d)+4 a b c d^2 \left (-11 A d^2-22 B c d+31 c^2 C\right )-4 b^2 c^3 \left (110 A d^2-77 B c d+56 c^2 C\right )\right )+\frac {2 \left (a+b x^2\right )^{5/2} \left (384 a^3 C d^6+64 a^2 b d^4 \left (-11 A d^2-22 B c d+4 c^2 C\right )+a b^2 c^2 d^2 \left (-176 A d^2-55 B c d+130 c^2 C\right )-6 b^3 c^4 \left (110 A d^2-77 B c d+56 c^2 C\right )\right )}{5 b}}{7 b}-\frac {\left (a+b x^2\right )^{5/2} (c+d x)^2 \left (384 a^2 C d^4-a b d^2 \left (704 A d^2-671 B c d+578 c^2 C\right )+6 b^2 c^2 \left (110 A d^2-77 B c d+56 c^2 C\right )\right )}{7 b}\right )}{9 b d^3}-\frac {1}{9} d^5 \left (a+b x^2\right )^{5/2} (c+d x)^4 \left (12 a C d^2-b \left (22 A d^2-55 B c d+94 c^2 C\right )\right )}{2 b d^4}}{11 b d^5}\) |
| \(\Big \downarrow \) 211 |
| \(\displaystyle \frac {C \left (a+b x^2\right )^{5/2} (c+d x)^6}{11 b d^4}-\frac {\frac {1}{10} d \left (a+b x^2\right )^{5/2} (c+d x)^5 (38 c C-11 B d)-\frac {\frac {\frac {1}{8} b d^8 \left (a+b x^2\right )^{5/2} (c+d x)^3 \left (3 a d^2 (62 c C-33 B d)-2 b c \left (154 A d^2-187 B c d+208 c^2 C\right )\right )-\frac {1}{8} b d^8 \left (\frac {-\frac {693}{2} a^2 d^4 (2 b c (2 A d+B c)-a d (B d+2 c C)) \left (\frac {3}{4} a \left (\frac {1}{2} a \int \frac {1}{\sqrt {b x^2+a}}dx+\frac {1}{2} x \sqrt {a+b x^2}\right )+\frac {1}{4} x \left (a+b x^2\right )^{3/2}\right )+\frac {1}{2} d x \left (a+b x^2\right )^{5/2} \left (3 a^2 d^4 (50 c C-231 B d)+4 a b c d^2 \left (-11 A d^2-22 B c d+31 c^2 C\right )-4 b^2 c^3 \left (110 A d^2-77 B c d+56 c^2 C\right )\right )+\frac {2 \left (a+b x^2\right )^{5/2} \left (384 a^3 C d^6+64 a^2 b d^4 \left (-11 A d^2-22 B c d+4 c^2 C\right )+a b^2 c^2 d^2 \left (-176 A d^2-55 B c d+130 c^2 C\right )-6 b^3 c^4 \left (110 A d^2-77 B c d+56 c^2 C\right )\right )}{5 b}}{7 b}-\frac {\left (a+b x^2\right )^{5/2} (c+d x)^2 \left (384 a^2 C d^4-a b d^2 \left (704 A d^2-671 B c d+578 c^2 C\right )+6 b^2 c^2 \left (110 A d^2-77 B c d+56 c^2 C\right )\right )}{7 b}\right )}{9 b d^3}-\frac {1}{9} d^5 \left (a+b x^2\right )^{5/2} (c+d x)^4 \left (12 a C d^2-b \left (22 A d^2-55 B c d+94 c^2 C\right )\right )}{2 b d^4}}{11 b d^5}\) |
| \(\Big \downarrow \) 224 |
| \(\displaystyle \frac {C \left (a+b x^2\right )^{5/2} (c+d x)^6}{11 b d^4}-\frac {\frac {1}{10} d \left (a+b x^2\right )^{5/2} (c+d x)^5 (38 c C-11 B d)-\frac {\frac {\frac {1}{8} b d^8 \left (a+b x^2\right )^{5/2} (c+d x)^3 \left (3 a d^2 (62 c C-33 B d)-2 b c \left (154 A d^2-187 B c d+208 c^2 C\right )\right )-\frac {1}{8} b d^8 \left (\frac {-\frac {693}{2} a^2 d^4 (2 b c (2 A d+B c)-a d (B d+2 c C)) \left (\frac {3}{4} a \left (\frac {1}{2} a \int \frac {1}{1-\frac {b x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}+\frac {1}{2} x \sqrt {a+b x^2}\right )+\frac {1}{4} x \left (a+b x^2\right )^{3/2}\right )+\frac {1}{2} d x \left (a+b x^2\right )^{5/2} \left (3 a^2 d^4 (50 c C-231 B d)+4 a b c d^2 \left (-11 A d^2-22 B c d+31 c^2 C\right )-4 b^2 c^3 \left (110 A d^2-77 B c d+56 c^2 C\right )\right )+\frac {2 \left (a+b x^2\right )^{5/2} \left (384 a^3 C d^6+64 a^2 b d^4 \left (-11 A d^2-22 B c d+4 c^2 C\right )+a b^2 c^2 d^2 \left (-176 A d^2-55 B c d+130 c^2 C\right )-6 b^3 c^4 \left (110 A d^2-77 B c d+56 c^2 C\right )\right )}{5 b}}{7 b}-\frac {\left (a+b x^2\right )^{5/2} (c+d x)^2 \left (384 a^2 C d^4-a b d^2 \left (704 A d^2-671 B c d+578 c^2 C\right )+6 b^2 c^2 \left (110 A d^2-77 B c d+56 c^2 C\right )\right )}{7 b}\right )}{9 b d^3}-\frac {1}{9} d^5 \left (a+b x^2\right )^{5/2} (c+d x)^4 \left (12 a C d^2-b \left (22 A d^2-55 B c d+94 c^2 C\right )\right )}{2 b d^4}}{11 b d^5}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {C \left (a+b x^2\right )^{5/2} (c+d x)^6}{11 b d^4}-\frac {\frac {1}{10} d \left (a+b x^2\right )^{5/2} (c+d x)^5 (38 c C-11 B d)-\frac {\frac {\frac {1}{8} b d^8 \left (a+b x^2\right )^{5/2} (c+d x)^3 \left (3 a d^2 (62 c C-33 B d)-2 b c \left (154 A d^2-187 B c d+208 c^2 C\right )\right )-\frac {1}{8} b d^8 \left (\frac {-\frac {693}{2} a^2 d^4 \left (\frac {3}{4} a \left (\frac {a \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{2 \sqrt {b}}+\frac {1}{2} x \sqrt {a+b x^2}\right )+\frac {1}{4} x \left (a+b x^2\right )^{3/2}\right ) (2 b c (2 A d+B c)-a d (B d+2 c C))+\frac {1}{2} d x \left (a+b x^2\right )^{5/2} \left (3 a^2 d^4 (50 c C-231 B d)+4 a b c d^2 \left (-11 A d^2-22 B c d+31 c^2 C\right )-4 b^2 c^3 \left (110 A d^2-77 B c d+56 c^2 C\right )\right )+\frac {2 \left (a+b x^2\right )^{5/2} \left (384 a^3 C d^6+64 a^2 b d^4 \left (-11 A d^2-22 B c d+4 c^2 C\right )+a b^2 c^2 d^2 \left (-176 A d^2-55 B c d+130 c^2 C\right )-6 b^3 c^4 \left (110 A d^2-77 B c d+56 c^2 C\right )\right )}{5 b}}{7 b}-\frac {\left (a+b x^2\right )^{5/2} (c+d x)^2 \left (384 a^2 C d^4-a b d^2 \left (704 A d^2-671 B c d+578 c^2 C\right )+6 b^2 c^2 \left (110 A d^2-77 B c d+56 c^2 C\right )\right )}{7 b}\right )}{9 b d^3}-\frac {1}{9} d^5 \left (a+b x^2\right )^{5/2} (c+d x)^4 \left (12 a C d^2-b \left (22 A d^2-55 B c d+94 c^2 C\right )\right )}{2 b d^4}}{11 b d^5}\) |
Input:
Int[x^3*(c + d*x)^2*(a + b*x^2)^(3/2)*(A + B*x + C*x^2),x]
Output:
(C*(c + d*x)^6*(a + b*x^2)^(5/2))/(11*b*d^4) - ((d*(38*c*C - 11*B*d)*(c + d*x)^5*(a + b*x^2)^(5/2))/10 - (-1/9*(d^5*(12*a*C*d^2 - b*(94*c^2*C - 55*B *c*d + 22*A*d^2))*(c + d*x)^4*(a + b*x^2)^(5/2)) + ((b*d^8*(3*a*d^2*(62*c* C - 33*B*d) - 2*b*c*(208*c^2*C - 187*B*c*d + 154*A*d^2))*(c + d*x)^3*(a + b*x^2)^(5/2))/8 - (b*d^8*(-1/7*((384*a^2*C*d^4 + 6*b^2*c^2*(56*c^2*C - 77* B*c*d + 110*A*d^2) - a*b*d^2*(578*c^2*C - 671*B*c*d + 704*A*d^2))*(c + d*x )^2*(a + b*x^2)^(5/2))/b + ((2*(384*a^3*C*d^6 + a*b^2*c^2*d^2*(130*c^2*C - 55*B*c*d - 176*A*d^2) + 64*a^2*b*d^4*(4*c^2*C - 22*B*c*d - 11*A*d^2) - 6* b^3*c^4*(56*c^2*C - 77*B*c*d + 110*A*d^2))*(a + b*x^2)^(5/2))/(5*b) + (d*( 3*a^2*d^4*(50*c*C - 231*B*d) + 4*a*b*c*d^2*(31*c^2*C - 22*B*c*d - 11*A*d^2 ) - 4*b^2*c^3*(56*c^2*C - 77*B*c*d + 110*A*d^2))*x*(a + b*x^2)^(5/2))/2 - (693*a^2*d^4*(2*b*c*(B*c + 2*A*d) - a*d*(2*c*C + B*d))*((x*(a + b*x^2)^(3/ 2))/4 + (3*a*((x*Sqrt[a + b*x^2])/2 + (a*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^ 2]])/(2*Sqrt[b])))/4))/2)/(7*b)))/8)/(9*b*d^3))/(2*b*d^4))/(11*b*d^5)
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[x*((a + b*x^2)^p/(2*p + 1 )), x] + Simp[2*a*(p/(2*p + 1)) Int[(a + b*x^2)^(p - 1), x], x] /; FreeQ[ {a, b}, x] && GtQ[p, 0] && (IntegerQ[4*p] || IntegerQ[6*p])
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a, b}, x] && !GtQ[a, 0]
Int[((d_.) + (e_.)*(x_))*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_), x _Symbol] :> Simp[(e*f + d*g)*((a + c*x^2)^(p + 1)/(2*c*(p + 1))), x] + (Sim p[e*g*x*((a + c*x^2)^(p + 1)/(c*(2*p + 3))), x] - Simp[(a*e*g - c*d*f*(2*p + 3))/(c*(2*p + 3)) Int[(a + c*x^2)^p, x], x]) /; FreeQ[{a, c, d, e, f, g , p}, x] && !LeQ[p, -1]
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p _.), x_Symbol] :> Simp[g*(d + e*x)^m*((a + c*x^2)^(p + 1)/(c*(m + 2*p + 2)) ), x] + Simp[1/(c*(m + 2*p + 2)) Int[(d + e*x)^(m - 1)*(a + c*x^2)^p*Simp [c*d*f*(m + 2*p + 2) - a*e*g*m + c*(e*f*(m + 2*p + 2) + d*g*m)*x, x], x], x ] /; FreeQ[{a, c, d, e, f, g, p}, x] && GtQ[m, 0] && NeQ[m + 2*p + 2, 0] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p]) && !(IGtQ[m, 0] && Eq Q[f, 0])
Int[(Pq_)*((d_) + (e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] : > With[{q = Expon[Pq, x], f = Coeff[Pq, x, Expon[Pq, x]]}, Simp[f*(d + e*x) ^(m + q - 1)*((a + b*x^2)^(p + 1)/(b*e^(q - 1)*(m + q + 2*p + 1))), x] + Si mp[1/(b*e^q*(m + q + 2*p + 1)) Int[(d + e*x)^m*(a + b*x^2)^p*ExpandToSum[ b*e^q*(m + q + 2*p + 1)*Pq - b*f*(m + q + 2*p + 1)*(d + e*x)^q - f*(d + e*x )^(q - 2)*(a*e^2*(m + q - 1) - b*d^2*(m + q + 2*p + 1) - 2*b*d*e*(m + q + p )*x), x], x], x] /; GtQ[q, 1] && NeQ[m + q + 2*p + 1, 0]] /; FreeQ[{a, b, d , e, m, p}, x] && PolyQ[Pq, x] && NeQ[b*d^2 + a*e^2, 0] && !(EqQ[d, 0] && True) && !(IGtQ[m, 0] && RationalQ[a, b, d, e] && (IntegerQ[p] || ILtQ[p + 1/2, 0]))
Time = 0.37 (sec) , antiderivative size = 438, normalized size of antiderivative = 0.96
| method | result | size |
| default | \(c \left (2 A d +B c \right ) \left (\frac {x^{3} \left (b \,x^{2}+a \right )^{\frac {5}{2}}}{8 b}-\frac {3 a \left (\frac {x \left (b \,x^{2}+a \right )^{\frac {5}{2}}}{6 b}-\frac {a \left (\frac {x \left (b \,x^{2}+a \right )^{\frac {3}{2}}}{4}+\frac {3 a \left (\frac {x \sqrt {b \,x^{2}+a}}{2}+\frac {a \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{2 \sqrt {b}}\right )}{4}\right )}{6 b}\right )}{8 b}\right )+d \left (B d +2 C c \right ) \left (\frac {x^{5} \left (b \,x^{2}+a \right )^{\frac {5}{2}}}{10 b}-\frac {a \left (\frac {x^{3} \left (b \,x^{2}+a \right )^{\frac {5}{2}}}{8 b}-\frac {3 a \left (\frac {x \left (b \,x^{2}+a \right )^{\frac {5}{2}}}{6 b}-\frac {a \left (\frac {x \left (b \,x^{2}+a \right )^{\frac {3}{2}}}{4}+\frac {3 a \left (\frac {x \sqrt {b \,x^{2}+a}}{2}+\frac {a \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{2 \sqrt {b}}\right )}{4}\right )}{6 b}\right )}{8 b}\right )}{2 b}\right )+\left (A \,d^{2}+2 B c d +C \,c^{2}\right ) \left (\frac {x^{4} \left (b \,x^{2}+a \right )^{\frac {5}{2}}}{9 b}-\frac {4 a \left (\frac {x^{2} \left (b \,x^{2}+a \right )^{\frac {5}{2}}}{7 b}-\frac {2 a \left (b \,x^{2}+a \right )^{\frac {5}{2}}}{35 b^{2}}\right )}{9 b}\right )+A \,c^{2} \left (\frac {x^{2} \left (b \,x^{2}+a \right )^{\frac {5}{2}}}{7 b}-\frac {2 a \left (b \,x^{2}+a \right )^{\frac {5}{2}}}{35 b^{2}}\right )+C \,d^{2} \left (\frac {x^{6} \left (b \,x^{2}+a \right )^{\frac {5}{2}}}{11 b}-\frac {6 a \left (\frac {x^{4} \left (b \,x^{2}+a \right )^{\frac {5}{2}}}{9 b}-\frac {4 a \left (\frac {x^{2} \left (b \,x^{2}+a \right )^{\frac {5}{2}}}{7 b}-\frac {2 a \left (b \,x^{2}+a \right )^{\frac {5}{2}}}{35 b^{2}}\right )}{9 b}\right )}{11 b}\right )\) | \(438\) |
| risch | \(\frac {\left (80640 C \,b^{5} d^{2} x^{10}+88704 B \,b^{5} d^{2} x^{9}+177408 C \,b^{5} c d \,x^{9}+98560 A \,b^{5} d^{2} x^{8}+197120 B \,b^{5} c d \,x^{8}+107520 C a \,b^{4} d^{2} x^{8}+98560 C \,b^{5} c^{2} x^{8}+221760 A \,b^{5} c d \,x^{7}+121968 B a \,b^{4} d^{2} x^{7}+110880 B \,b^{5} c^{2} x^{7}+243936 C a \,b^{4} c d \,x^{7}+140800 A a \,b^{4} d^{2} x^{6}+126720 A \,b^{5} c^{2} x^{6}+281600 B a \,b^{4} c d \,x^{6}+3840 C \,a^{2} b^{3} d^{2} x^{6}+140800 C a \,b^{4} c^{2} x^{6}+332640 A a \,b^{4} c d \,x^{5}+5544 B \,a^{2} b^{3} d^{2} x^{5}+166320 B a \,b^{4} c^{2} x^{5}+11088 C \,a^{2} b^{3} c d \,x^{5}+8448 A \,a^{2} b^{3} d^{2} x^{4}+202752 A a \,b^{4} c^{2} x^{4}+16896 B \,a^{2} b^{3} c d \,x^{4}-4608 C \,a^{3} b^{2} d^{2} x^{4}+8448 C \,a^{2} b^{3} c^{2} x^{4}+27720 A \,a^{2} b^{3} c d \,x^{3}-6930 B \,a^{3} b^{2} d^{2} x^{3}+13860 B \,a^{2} b^{3} c^{2} x^{3}-13860 C \,a^{3} b^{2} c d \,x^{3}-11264 A \,a^{3} b^{2} d^{2} x^{2}+25344 A \,a^{2} b^{3} c^{2} x^{2}-22528 B \,a^{3} b^{2} c d \,x^{2}+6144 C \,a^{4} b \,d^{2} x^{2}-11264 C \,a^{3} b^{2} c^{2} x^{2}-41580 A \,a^{3} b^{2} c d x +10395 B \,a^{4} b \,d^{2} x -20790 B \,a^{3} b^{2} c^{2} x +20790 C \,a^{4} b c d x +22528 A \,a^{4} b \,d^{2}-50688 A \,a^{3} b^{2} c^{2}+45056 B \,a^{4} b c d -12288 C \,a^{5} d^{2}+22528 C \,a^{4} b \,c^{2}\right ) \sqrt {b \,x^{2}+a}}{887040 b^{4}}+\frac {3 a^{4} \left (4 A b c d -a B \,d^{2}+2 b B \,c^{2}-2 C a c d \right ) \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{256 b^{\frac {7}{2}}}\) | \(616\) |
Input:
int(x^3*(d*x+c)^2*(b*x^2+a)^(3/2)*(C*x^2+B*x+A),x,method=_RETURNVERBOSE)
Output:
c*(2*A*d+B*c)*(1/8*x^3*(b*x^2+a)^(5/2)/b-3/8*a/b*(1/6*x*(b*x^2+a)^(5/2)/b- 1/6*a/b*(1/4*x*(b*x^2+a)^(3/2)+3/4*a*(1/2*x*(b*x^2+a)^(1/2)+1/2*a/b^(1/2)* ln(b^(1/2)*x+(b*x^2+a)^(1/2))))))+d*(B*d+2*C*c)*(1/10*x^5*(b*x^2+a)^(5/2)/ b-1/2*a/b*(1/8*x^3*(b*x^2+a)^(5/2)/b-3/8*a/b*(1/6*x*(b*x^2+a)^(5/2)/b-1/6* a/b*(1/4*x*(b*x^2+a)^(3/2)+3/4*a*(1/2*x*(b*x^2+a)^(1/2)+1/2*a/b^(1/2)*ln(b ^(1/2)*x+(b*x^2+a)^(1/2)))))))+(A*d^2+2*B*c*d+C*c^2)*(1/9*x^4*(b*x^2+a)^(5 /2)/b-4/9*a/b*(1/7*x^2*(b*x^2+a)^(5/2)/b-2/35*a/b^2*(b*x^2+a)^(5/2)))+A*c^ 2*(1/7*x^2*(b*x^2+a)^(5/2)/b-2/35*a/b^2*(b*x^2+a)^(5/2))+C*d^2*(1/11*x^6*( b*x^2+a)^(5/2)/b-6/11*a/b*(1/9*x^4*(b*x^2+a)^(5/2)/b-4/9*a/b*(1/7*x^2*(b*x ^2+a)^(5/2)/b-2/35*a/b^2*(b*x^2+a)^(5/2))))
Time = 0.13 (sec) , antiderivative size = 1152, normalized size of antiderivative = 2.53 \[ \int x^3 (c+d x)^2 \left (a+b x^2\right )^{3/2} \left (A+B x+C x^2\right ) \, dx=\text {Too large to display} \] Input:
integrate(x^3*(d*x+c)^2*(b*x^2+a)^(3/2)*(C*x^2+B*x+A),x, algorithm="fricas ")
Output:
[-1/1774080*(10395*(2*B*a^4*b*c^2 - B*a^5*d^2 - 2*(C*a^5 - 2*A*a^4*b)*c*d) *sqrt(b)*log(-2*b*x^2 + 2*sqrt(b*x^2 + a)*sqrt(b)*x - a) - 2*(80640*C*b^5* d^2*x^10 + 88704*(2*C*b^5*c*d + B*b^5*d^2)*x^9 + 8960*(11*C*b^5*c^2 + 22*B *b^5*c*d + (12*C*a*b^4 + 11*A*b^5)*d^2)*x^8 + 45056*B*a^4*b*c*d + 11088*(1 0*B*b^5*c^2 + 11*B*a*b^4*d^2 + 2*(11*C*a*b^4 + 10*A*b^5)*c*d)*x^7 + 1280*( 220*B*a*b^4*c*d + 11*(10*C*a*b^4 + 9*A*b^5)*c^2 + (3*C*a^2*b^3 + 110*A*a*b ^4)*d^2)*x^6 + 5544*(30*B*a*b^4*c^2 + B*a^2*b^3*d^2 + 2*(C*a^2*b^3 + 30*A* a*b^4)*c*d)*x^5 + 768*(22*B*a^2*b^3*c*d + 11*(C*a^2*b^3 + 24*A*a*b^4)*c^2 - (6*C*a^3*b^2 - 11*A*a^2*b^3)*d^2)*x^4 + 6930*(2*B*a^2*b^3*c^2 - B*a^3*b^ 2*d^2 - 2*(C*a^3*b^2 - 2*A*a^2*b^3)*c*d)*x^3 + 5632*(4*C*a^4*b - 9*A*a^3*b ^2)*c^2 - 2048*(6*C*a^5 - 11*A*a^4*b)*d^2 - 256*(88*B*a^3*b^2*c*d + 11*(4* C*a^3*b^2 - 9*A*a^2*b^3)*c^2 - 4*(6*C*a^4*b - 11*A*a^3*b^2)*d^2)*x^2 - 103 95*(2*B*a^3*b^2*c^2 - B*a^4*b*d^2 - 2*(C*a^4*b - 2*A*a^3*b^2)*c*d)*x)*sqrt (b*x^2 + a))/b^4, -1/887040*(10395*(2*B*a^4*b*c^2 - B*a^5*d^2 - 2*(C*a^5 - 2*A*a^4*b)*c*d)*sqrt(-b)*arctan(sqrt(-b)*x/sqrt(b*x^2 + a)) - (80640*C*b^ 5*d^2*x^10 + 88704*(2*C*b^5*c*d + B*b^5*d^2)*x^9 + 8960*(11*C*b^5*c^2 + 22 *B*b^5*c*d + (12*C*a*b^4 + 11*A*b^5)*d^2)*x^8 + 45056*B*a^4*b*c*d + 11088* (10*B*b^5*c^2 + 11*B*a*b^4*d^2 + 2*(11*C*a*b^4 + 10*A*b^5)*c*d)*x^7 + 1280 *(220*B*a*b^4*c*d + 11*(10*C*a*b^4 + 9*A*b^5)*c^2 + (3*C*a^2*b^3 + 110*A*a *b^4)*d^2)*x^6 + 5544*(30*B*a*b^4*c^2 + B*a^2*b^3*d^2 + 2*(C*a^2*b^3 + ...
Leaf count of result is larger than twice the leaf count of optimal. 1406 vs. \(2 (434) = 868\).
Time = 0.74 (sec) , antiderivative size = 1406, normalized size of antiderivative = 3.09 \[ \int x^3 (c+d x)^2 \left (a+b x^2\right )^{3/2} \left (A+B x+C x^2\right ) \, dx=\text {Too large to display} \] Input:
integrate(x**3*(d*x+c)**2*(b*x**2+a)**(3/2)*(C*x**2+B*x+A),x)
Output:
Piecewise((3*a**2*(2*A*a**2*c*d + B*a**2*c**2 - 5*a*(4*A*a*b*c*d + B*a**2* d**2 + 2*B*a*b*c**2 + 2*C*a**2*c*d - 7*a*(2*A*b**2*c*d + 2*B*a*b*d**2 + B* b**2*c**2 + 4*C*a*b*c*d - 9*a*(B*b**2*d**2 + 2*C*b**2*c*d)/(10*b))/(8*b))/ (6*b))*Piecewise((log(2*sqrt(b)*sqrt(a + b*x**2) + 2*b*x)/sqrt(b), Ne(a, 0 )), (x*log(x)/sqrt(b*x**2), True))/(8*b**2) + sqrt(a + b*x**2)*(C*b*d**2*x **10/11 - 3*a*x*(2*A*a**2*c*d + B*a**2*c**2 - 5*a*(4*A*a*b*c*d + B*a**2*d* *2 + 2*B*a*b*c**2 + 2*C*a**2*c*d - 7*a*(2*A*b**2*c*d + 2*B*a*b*d**2 + B*b* *2*c**2 + 4*C*a*b*c*d - 9*a*(B*b**2*d**2 + 2*C*b**2*c*d)/(10*b))/(8*b))/(6 *b))/(8*b**2) - 2*a*(A*a**2*c**2 - 4*a*(A*a**2*d**2 + 2*A*a*b*c**2 + 2*B*a **2*c*d + C*a**2*c**2 - 6*a*(2*A*a*b*d**2 + A*b**2*c**2 + 4*B*a*b*c*d + C* a**2*d**2 + 2*C*a*b*c**2 - 8*a*(A*b**2*d**2 + 2*B*b**2*c*d + 12*C*a*b*d**2 /11 + C*b**2*c**2)/(9*b))/(7*b))/(5*b))/(3*b**2) + x**9*(B*b**2*d**2 + 2*C *b**2*c*d)/(10*b) + x**8*(A*b**2*d**2 + 2*B*b**2*c*d + 12*C*a*b*d**2/11 + C*b**2*c**2)/(9*b) + x**7*(2*A*b**2*c*d + 2*B*a*b*d**2 + B*b**2*c**2 + 4*C *a*b*c*d - 9*a*(B*b**2*d**2 + 2*C*b**2*c*d)/(10*b))/(8*b) + x**6*(2*A*a*b* d**2 + A*b**2*c**2 + 4*B*a*b*c*d + C*a**2*d**2 + 2*C*a*b*c**2 - 8*a*(A*b** 2*d**2 + 2*B*b**2*c*d + 12*C*a*b*d**2/11 + C*b**2*c**2)/(9*b))/(7*b) + x** 5*(4*A*a*b*c*d + B*a**2*d**2 + 2*B*a*b*c**2 + 2*C*a**2*c*d - 7*a*(2*A*b**2 *c*d + 2*B*a*b*d**2 + B*b**2*c**2 + 4*C*a*b*c*d - 9*a*(B*b**2*d**2 + 2*C*b **2*c*d)/(10*b))/(8*b))/(6*b) + x**4*(A*a**2*d**2 + 2*A*a*b*c**2 + 2*B*...
Time = 0.04 (sec) , antiderivative size = 542, normalized size of antiderivative = 1.19 \[ \int x^3 (c+d x)^2 \left (a+b x^2\right )^{3/2} \left (A+B x+C x^2\right ) \, dx =\text {Too large to display} \] Input:
integrate(x^3*(d*x+c)^2*(b*x^2+a)^(3/2)*(C*x^2+B*x+A),x, algorithm="maxima ")
Output:
1/11*(b*x^2 + a)^(5/2)*C*d^2*x^6/b - 2/33*(b*x^2 + a)^(5/2)*C*a*d^2*x^4/b^ 2 + 1/10*(2*C*c*d + B*d^2)*(b*x^2 + a)^(5/2)*x^5/b + 1/7*(b*x^2 + a)^(5/2) *A*c^2*x^2/b + 8/231*(b*x^2 + a)^(5/2)*C*a^2*d^2*x^2/b^3 + 1/9*(C*c^2 + 2* B*c*d + A*d^2)*(b*x^2 + a)^(5/2)*x^4/b - 1/16*(2*C*c*d + B*d^2)*(b*x^2 + a )^(5/2)*a*x^3/b^2 + 1/8*(B*c^2 + 2*A*c*d)*(b*x^2 + a)^(5/2)*x^3/b - 2/35*( b*x^2 + a)^(5/2)*A*a*c^2/b^2 - 16/1155*(b*x^2 + a)^(5/2)*C*a^3*d^2/b^4 - 4 /63*(C*c^2 + 2*B*c*d + A*d^2)*(b*x^2 + a)^(5/2)*a*x^2/b^2 + 1/32*(2*C*c*d + B*d^2)*(b*x^2 + a)^(5/2)*a^2*x/b^3 - 1/128*(2*C*c*d + B*d^2)*(b*x^2 + a) ^(3/2)*a^3*x/b^3 - 3/256*(2*C*c*d + B*d^2)*sqrt(b*x^2 + a)*a^4*x/b^3 - 1/1 6*(B*c^2 + 2*A*c*d)*(b*x^2 + a)^(5/2)*a*x/b^2 + 1/64*(B*c^2 + 2*A*c*d)*(b* x^2 + a)^(3/2)*a^2*x/b^2 + 3/128*(B*c^2 + 2*A*c*d)*sqrt(b*x^2 + a)*a^3*x/b ^2 - 3/256*(2*C*c*d + B*d^2)*a^5*arcsinh(b*x/sqrt(a*b))/b^(7/2) + 3/128*(B *c^2 + 2*A*c*d)*a^4*arcsinh(b*x/sqrt(a*b))/b^(5/2) + 8/315*(C*c^2 + 2*B*c* d + A*d^2)*(b*x^2 + a)^(5/2)*a^2/b^3
Time = 0.20 (sec) , antiderivative size = 620, normalized size of antiderivative = 1.36 \[ \int x^3 (c+d x)^2 \left (a+b x^2\right )^{3/2} \left (A+B x+C x^2\right ) \, dx=\frac {1}{887040} \, \sqrt {b x^{2} + a} {\left ({\left (2 \, {\left ({\left (4 \, {\left ({\left (2 \, {\left (7 \, {\left (8 \, {\left (9 \, {\left (10 \, C b d^{2} x + \frac {11 \, {\left (2 \, C b^{10} c d + B b^{10} d^{2}\right )}}{b^{9}}\right )} x + \frac {10 \, {\left (11 \, C b^{10} c^{2} + 22 \, B b^{10} c d + 12 \, C a b^{9} d^{2} + 11 \, A b^{10} d^{2}\right )}}{b^{9}}\right )} x + \frac {99 \, {\left (10 \, B b^{10} c^{2} + 22 \, C a b^{9} c d + 20 \, A b^{10} c d + 11 \, B a b^{9} d^{2}\right )}}{b^{9}}\right )} x + \frac {80 \, {\left (110 \, C a b^{9} c^{2} + 99 \, A b^{10} c^{2} + 220 \, B a b^{9} c d + 3 \, C a^{2} b^{8} d^{2} + 110 \, A a b^{9} d^{2}\right )}}{b^{9}}\right )} x + \frac {693 \, {\left (30 \, B a b^{9} c^{2} + 2 \, C a^{2} b^{8} c d + 60 \, A a b^{9} c d + B a^{2} b^{8} d^{2}\right )}}{b^{9}}\right )} x + \frac {96 \, {\left (11 \, C a^{2} b^{8} c^{2} + 264 \, A a b^{9} c^{2} + 22 \, B a^{2} b^{8} c d - 6 \, C a^{3} b^{7} d^{2} + 11 \, A a^{2} b^{8} d^{2}\right )}}{b^{9}}\right )} x + \frac {3465 \, {\left (2 \, B a^{2} b^{8} c^{2} - 2 \, C a^{3} b^{7} c d + 4 \, A a^{2} b^{8} c d - B a^{3} b^{7} d^{2}\right )}}{b^{9}}\right )} x - \frac {128 \, {\left (44 \, C a^{3} b^{7} c^{2} - 99 \, A a^{2} b^{8} c^{2} + 88 \, B a^{3} b^{7} c d - 24 \, C a^{4} b^{6} d^{2} + 44 \, A a^{3} b^{7} d^{2}\right )}}{b^{9}}\right )} x - \frac {10395 \, {\left (2 \, B a^{3} b^{7} c^{2} - 2 \, C a^{4} b^{6} c d + 4 \, A a^{3} b^{7} c d - B a^{4} b^{6} d^{2}\right )}}{b^{9}}\right )} x + \frac {512 \, {\left (44 \, C a^{4} b^{6} c^{2} - 99 \, A a^{3} b^{7} c^{2} + 88 \, B a^{4} b^{6} c d - 24 \, C a^{5} b^{5} d^{2} + 44 \, A a^{4} b^{6} d^{2}\right )}}{b^{9}}\right )} - \frac {3 \, {\left (2 \, B a^{4} b c^{2} - 2 \, C a^{5} c d + 4 \, A a^{4} b c d - B a^{5} d^{2}\right )} \log \left ({\left | -\sqrt {b} x + \sqrt {b x^{2} + a} \right |}\right )}{256 \, b^{\frac {7}{2}}} \] Input:
integrate(x^3*(d*x+c)^2*(b*x^2+a)^(3/2)*(C*x^2+B*x+A),x, algorithm="giac")
Output:
1/887040*sqrt(b*x^2 + a)*((2*((4*((2*(7*(8*(9*(10*C*b*d^2*x + 11*(2*C*b^10 *c*d + B*b^10*d^2)/b^9)*x + 10*(11*C*b^10*c^2 + 22*B*b^10*c*d + 12*C*a*b^9 *d^2 + 11*A*b^10*d^2)/b^9)*x + 99*(10*B*b^10*c^2 + 22*C*a*b^9*c*d + 20*A*b ^10*c*d + 11*B*a*b^9*d^2)/b^9)*x + 80*(110*C*a*b^9*c^2 + 99*A*b^10*c^2 + 2 20*B*a*b^9*c*d + 3*C*a^2*b^8*d^2 + 110*A*a*b^9*d^2)/b^9)*x + 693*(30*B*a*b ^9*c^2 + 2*C*a^2*b^8*c*d + 60*A*a*b^9*c*d + B*a^2*b^8*d^2)/b^9)*x + 96*(11 *C*a^2*b^8*c^2 + 264*A*a*b^9*c^2 + 22*B*a^2*b^8*c*d - 6*C*a^3*b^7*d^2 + 11 *A*a^2*b^8*d^2)/b^9)*x + 3465*(2*B*a^2*b^8*c^2 - 2*C*a^3*b^7*c*d + 4*A*a^2 *b^8*c*d - B*a^3*b^7*d^2)/b^9)*x - 128*(44*C*a^3*b^7*c^2 - 99*A*a^2*b^8*c^ 2 + 88*B*a^3*b^7*c*d - 24*C*a^4*b^6*d^2 + 44*A*a^3*b^7*d^2)/b^9)*x - 10395 *(2*B*a^3*b^7*c^2 - 2*C*a^4*b^6*c*d + 4*A*a^3*b^7*c*d - B*a^4*b^6*d^2)/b^9 )*x + 512*(44*C*a^4*b^6*c^2 - 99*A*a^3*b^7*c^2 + 88*B*a^4*b^6*c*d - 24*C*a ^5*b^5*d^2 + 44*A*a^4*b^6*d^2)/b^9) - 3/256*(2*B*a^4*b*c^2 - 2*C*a^5*c*d + 4*A*a^4*b*c*d - B*a^5*d^2)*log(abs(-sqrt(b)*x + sqrt(b*x^2 + a)))/b^(7/2)
Timed out. \[ \int x^3 (c+d x)^2 \left (a+b x^2\right )^{3/2} \left (A+B x+C x^2\right ) \, dx=\int x^3\,{\left (b\,x^2+a\right )}^{3/2}\,{\left (c+d\,x\right )}^2\,\left (C\,x^2+B\,x+A\right ) \,d x \] Input:
int(x^3*(a + b*x^2)^(3/2)*(c + d*x)^2*(A + B*x + C*x^2),x)
Output:
int(x^3*(a + b*x^2)^(3/2)*(c + d*x)^2*(A + B*x + C*x^2), x)
Time = 24.36 (sec) , antiderivative size = 1005, normalized size of antiderivative = 2.21 \[ \int x^3 (c+d x)^2 \left (a+b x^2\right )^{3/2} \left (A+B x+C x^2\right ) \, dx =\text {Too large to display} \] Input:
int(x^3*(d*x+c)^2*(b*x^2+a)^(3/2)*(C*x^2+B*x+A),x)
Output:
(22528*sqrt(a + b*x**2)*a**5*b*d**2 - 12288*sqrt(a + b*x**2)*a**5*c*d**2 - 50688*sqrt(a + b*x**2)*a**4*b**2*c**2 - 41580*sqrt(a + b*x**2)*a**4*b**2* c*d*x + 45056*sqrt(a + b*x**2)*a**4*b**2*c*d - 11264*sqrt(a + b*x**2)*a**4 *b**2*d**2*x**2 + 10395*sqrt(a + b*x**2)*a**4*b**2*d**2*x + 22528*sqrt(a + b*x**2)*a**4*b*c**3 + 20790*sqrt(a + b*x**2)*a**4*b*c**2*d*x + 6144*sqrt( a + b*x**2)*a**4*b*c*d**2*x**2 + 25344*sqrt(a + b*x**2)*a**3*b**3*c**2*x** 2 - 20790*sqrt(a + b*x**2)*a**3*b**3*c**2*x + 27720*sqrt(a + b*x**2)*a**3* b**3*c*d*x**3 - 22528*sqrt(a + b*x**2)*a**3*b**3*c*d*x**2 + 8448*sqrt(a + b*x**2)*a**3*b**3*d**2*x**4 - 6930*sqrt(a + b*x**2)*a**3*b**3*d**2*x**3 - 11264*sqrt(a + b*x**2)*a**3*b**2*c**3*x**2 - 13860*sqrt(a + b*x**2)*a**3*b **2*c**2*d*x**3 - 4608*sqrt(a + b*x**2)*a**3*b**2*c*d**2*x**4 + 202752*sqr t(a + b*x**2)*a**2*b**4*c**2*x**4 + 13860*sqrt(a + b*x**2)*a**2*b**4*c**2* x**3 + 332640*sqrt(a + b*x**2)*a**2*b**4*c*d*x**5 + 16896*sqrt(a + b*x**2) *a**2*b**4*c*d*x**4 + 140800*sqrt(a + b*x**2)*a**2*b**4*d**2*x**6 + 5544*s qrt(a + b*x**2)*a**2*b**4*d**2*x**5 + 8448*sqrt(a + b*x**2)*a**2*b**3*c**3 *x**4 + 11088*sqrt(a + b*x**2)*a**2*b**3*c**2*d*x**5 + 3840*sqrt(a + b*x** 2)*a**2*b**3*c*d**2*x**6 + 126720*sqrt(a + b*x**2)*a*b**5*c**2*x**6 + 1663 20*sqrt(a + b*x**2)*a*b**5*c**2*x**5 + 221760*sqrt(a + b*x**2)*a*b**5*c*d* x**7 + 281600*sqrt(a + b*x**2)*a*b**5*c*d*x**6 + 98560*sqrt(a + b*x**2)*a* b**5*d**2*x**8 + 121968*sqrt(a + b*x**2)*a*b**5*d**2*x**7 + 140800*sqrt...