Integrand size = 32, antiderivative size = 450 \[ \int \frac {(c+d x)^2 \left (a+b x^2\right )^{3/2} \left (A+B x+C x^2\right )}{x^{11}} \, dx=-\frac {a A c^2 \sqrt {a+b x^2}}{10 x^{10}}-\frac {\left (10 a c (c C+2 B d)+A \left (11 b c^2+10 a d^2\right )\right ) \sqrt {a+b x^2}}{80 x^8}-\frac {\left (3 A b \left (b c^2+30 a d^2\right )+10 a \left (8 a C d^2+9 b c (c C+2 B d)\right )\right ) \sqrt {a+b x^2}}{480 a x^6}+\frac {b \left (3 A b \left (b c^2-2 a d^2\right )-2 a \left (56 a C d^2+3 b c (c C+2 B d)\right )\right ) \sqrt {a+b x^2}}{384 a^2 x^4}-\frac {b^2 \left (3 A b \left (b c^2-2 a d^2\right )+2 a \left (8 a C d^2-3 b c (c C+2 B d)\right )\right ) \sqrt {a+b x^2}}{256 a^3 x^2}-\frac {c (B c+2 A d) \left (a+b x^2\right )^{5/2}}{9 a x^9}+\frac {(4 b c (B c+2 A d)-9 a d (2 c C+B d)) \left (a+b x^2\right )^{5/2}}{63 a^2 x^7}-\frac {2 b (4 b c (B c+2 A d)-9 a d (2 c C+B d)) \left (a+b x^2\right )^{5/2}}{315 a^3 x^5}+\frac {b^3 \left (3 A b \left (b c^2-2 a d^2\right )+2 a \left (8 a C d^2-3 b c (c C+2 B d)\right )\right ) \text {arctanh}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{256 a^{7/2}} \] Output:
-1/10*a*A*c^2*(b*x^2+a)^(1/2)/x^10-1/80*(10*a*c*(2*B*d+C*c)+A*(10*a*d^2+11 *b*c^2))*(b*x^2+a)^(1/2)/x^8-1/480*(3*A*b*(30*a*d^2+b*c^2)+10*a*(8*a*C*d^2 +9*b*c*(2*B*d+C*c)))*(b*x^2+a)^(1/2)/a/x^6+1/384*b*(3*A*b*(-2*a*d^2+b*c^2) -2*a*(56*a*C*d^2+3*b*c*(2*B*d+C*c)))*(b*x^2+a)^(1/2)/a^2/x^4-1/256*b^2*(3* A*b*(-2*a*d^2+b*c^2)+2*a*(8*a*C*d^2-3*b*c*(2*B*d+C*c)))*(b*x^2+a)^(1/2)/a^ 3/x^2-1/9*c*(2*A*d+B*c)*(b*x^2+a)^(5/2)/a/x^9+1/63*(4*b*c*(2*A*d+B*c)-9*a* d*(B*d+2*C*c))*(b*x^2+a)^(5/2)/a^2/x^7-2/315*b*(4*b*c*(2*A*d+B*c)-9*a*d*(B *d+2*C*c))*(b*x^2+a)^(5/2)/a^3/x^5+1/256*b^3*(3*A*b*(-2*a*d^2+b*c^2)+2*a*( 8*a*C*d^2-3*b*c*(2*B*d+C*c)))*arctanh((b*x^2+a)^(1/2)/a^(1/2))/a^(7/2)
Time = 6.08 (sec) , antiderivative size = 462, normalized size of antiderivative = 1.03 \[ \int \frac {(c+d x)^2 \left (a+b x^2\right )^{3/2} \left (A+B x+C x^2\right )}{x^{11}} \, dx=\frac {-\frac {\sqrt {a} \sqrt {a+b x^2} \left (b^4 c x^8 (945 A c+2048 B c x+4096 A d x)+32 a^4 \left (7 A \left (36 c^2+80 c d x+45 d^2 x^2\right )+5 x \left (3 C x \left (21 c^2+48 c d x+28 d^2 x^2\right )+2 B \left (28 c^2+63 c d x+36 d^2 x^2\right )\right )\right )+12 a^2 b^2 x^4 \left (A \left (42 c^2+128 c d x+105 d^2 x^2\right )+x \left (2 B \left (32 c^2+105 c d x+96 d^2 x^2\right )+3 C x \left (35 c^2+128 c d x+140 d^2 x^2\right )\right )\right )+16 a^3 b x^2 \left (A \left (693 c^2+1600 c d x+945 d^2 x^2\right )+x \left (3 C x \left (315 c^2+768 c d x+490 d^2 x^2\right )+2 B \left (400 c^2+945 c d x+576 d^2 x^2\right )\right )\right )-2 a b^3 x^6 \left (A \left (315 c^2+1024 c d x+945 d^2 x^2\right )+x \left (9 c C x (105 c+512 d x)+2 B \left (256 c^2+945 c d x+1152 d^2 x^2\right )\right )\right )\right )}{x^{10}}-630 b^3 \left (3 A b^2 c^2+16 a^2 C d^2\right ) \text {arctanh}\left (\frac {\sqrt {b} x-\sqrt {a+b x^2}}{\sqrt {a}}\right )-3780 a b^4 \left (c^2 C+2 B c d+A d^2\right ) \text {arctanh}\left (\frac {-\sqrt {b} x+\sqrt {a+b x^2}}{\sqrt {a}}\right )}{80640 a^{7/2}} \] Input:
Integrate[((c + d*x)^2*(a + b*x^2)^(3/2)*(A + B*x + C*x^2))/x^11,x]
Output:
(-((Sqrt[a]*Sqrt[a + b*x^2]*(b^4*c*x^8*(945*A*c + 2048*B*c*x + 4096*A*d*x) + 32*a^4*(7*A*(36*c^2 + 80*c*d*x + 45*d^2*x^2) + 5*x*(3*C*x*(21*c^2 + 48* c*d*x + 28*d^2*x^2) + 2*B*(28*c^2 + 63*c*d*x + 36*d^2*x^2))) + 12*a^2*b^2* x^4*(A*(42*c^2 + 128*c*d*x + 105*d^2*x^2) + x*(2*B*(32*c^2 + 105*c*d*x + 9 6*d^2*x^2) + 3*C*x*(35*c^2 + 128*c*d*x + 140*d^2*x^2))) + 16*a^3*b*x^2*(A* (693*c^2 + 1600*c*d*x + 945*d^2*x^2) + x*(3*C*x*(315*c^2 + 768*c*d*x + 490 *d^2*x^2) + 2*B*(400*c^2 + 945*c*d*x + 576*d^2*x^2))) - 2*a*b^3*x^6*(A*(31 5*c^2 + 1024*c*d*x + 945*d^2*x^2) + x*(9*c*C*x*(105*c + 512*d*x) + 2*B*(25 6*c^2 + 945*c*d*x + 1152*d^2*x^2)))))/x^10) - 630*b^3*(3*A*b^2*c^2 + 16*a^ 2*C*d^2)*ArcTanh[(Sqrt[b]*x - Sqrt[a + b*x^2])/Sqrt[a]] - 3780*a*b^4*(c^2* C + 2*B*c*d + A*d^2)*ArcTanh[(-(Sqrt[b]*x) + Sqrt[a + b*x^2])/Sqrt[a]])/(8 0640*a^(7/2))
Time = 2.12 (sec) , antiderivative size = 392, normalized size of antiderivative = 0.87, number of steps used = 17, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {2338, 27, 2338, 25, 2338, 27, 539, 27, 539, 27, 534, 243, 51, 51, 73, 221}
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.
| \(\displaystyle \int \frac {\left (a+b x^2\right )^{3/2} (c+d x)^2 \left (A+B x+C x^2\right )}{x^{11}} \, dx\) |
| \(\Big \downarrow \) 2338 |
| \(\displaystyle -\frac {\int -\frac {5 \left (b x^2+a\right )^{3/2} \left (2 a C d^2 x^3+2 a d (2 c C+B d) x^2+\left (2 a c (c C+2 B d)-A \left (b c^2-2 a d^2\right )\right ) x+2 a c (B c+2 A d)\right )}{x^{10}}dx}{10 a}-\frac {A c^2 \left (a+b x^2\right )^{5/2}}{10 a x^{10}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {\left (b x^2+a\right )^{3/2} \left (2 a C d^2 x^3+2 a d (2 c C+B d) x^2+\left (2 a c (c C+2 B d)-A \left (b c^2-2 a d^2\right )\right ) x+2 a c (B c+2 A d)\right )}{x^{10}}dx}{2 a}-\frac {A c^2 \left (a+b x^2\right )^{5/2}}{10 a x^{10}}\) |
| \(\Big \downarrow \) 2338 |
| \(\displaystyle \frac {-\frac {\int -\frac {\left (b x^2+a\right )^{3/2} \left (18 a^2 C d^2 x^2-2 a (4 b c (B c+2 A d)-9 a d (2 c C+B d)) x+9 a \left (2 a c (c C+2 B d)-A \left (b c^2-2 a d^2\right )\right )\right )}{x^9}dx}{9 a}-\frac {2 c \left (a+b x^2\right )^{5/2} (2 A d+B c)}{9 x^9}}{2 a}-\frac {A c^2 \left (a+b x^2\right )^{5/2}}{10 a x^{10}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\int \frac {\left (b x^2+a\right )^{3/2} \left (18 a^2 C d^2 x^2-2 a (4 b c (B c+2 A d)-9 a d (2 c C+B d)) x+9 a \left (2 a c (c C+2 B d)-A \left (b c^2-2 a d^2\right )\right )\right )}{x^9}dx}{9 a}-\frac {2 c \left (a+b x^2\right )^{5/2} (2 A d+B c)}{9 x^9}}{2 a}-\frac {A c^2 \left (a+b x^2\right )^{5/2}}{10 a x^{10}}\) |
| \(\Big \downarrow \) 2338 |
| \(\displaystyle \frac {\frac {-\frac {\int \frac {a \left (16 a (4 b c (B c+2 A d)-9 a d (2 c C+B d))-9 \left (3 A b \left (b c^2-2 a d^2\right )+2 a \left (8 a C d^2-3 b c (c C+2 B d)\right )\right ) x\right ) \left (b x^2+a\right )^{3/2}}{x^8}dx}{8 a}-\frac {9 \left (a+b x^2\right )^{5/2} \left (2 a c (2 B d+c C)-A \left (b c^2-2 a d^2\right )\right )}{8 x^8}}{9 a}-\frac {2 c \left (a+b x^2\right )^{5/2} (2 A d+B c)}{9 x^9}}{2 a}-\frac {A c^2 \left (a+b x^2\right )^{5/2}}{10 a x^{10}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {-\frac {1}{8} \int \frac {\left (16 a (4 b c (B c+2 A d)-9 a d (2 c C+B d))-9 \left (3 A b \left (b c^2-2 a d^2\right )+2 a \left (8 a C d^2-3 b c (c C+2 B d)\right )\right ) x\right ) \left (b x^2+a\right )^{3/2}}{x^8}dx-\frac {9 \left (a+b x^2\right )^{5/2} \left (2 a c (2 B d+c C)-A \left (b c^2-2 a d^2\right )\right )}{8 x^8}}{9 a}-\frac {2 c \left (a+b x^2\right )^{5/2} (2 A d+B c)}{9 x^9}}{2 a}-\frac {A c^2 \left (a+b x^2\right )^{5/2}}{10 a x^{10}}\) |
| \(\Big \downarrow \) 539 |
| \(\displaystyle \frac {\frac {\frac {1}{8} \left (\frac {\int \frac {a \left (63 \left (3 A b \left (b c^2-2 a d^2\right )+2 a \left (8 a C d^2-3 b c (c C+2 B d)\right )\right )+32 b (4 b c (B c+2 A d)-9 a d (2 c C+B d)) x\right ) \left (b x^2+a\right )^{3/2}}{x^7}dx}{7 a}+\frac {16 \left (a+b x^2\right )^{5/2} (4 b c (2 A d+B c)-9 a d (B d+2 c C))}{7 x^7}\right )-\frac {9 \left (a+b x^2\right )^{5/2} \left (2 a c (2 B d+c C)-A \left (b c^2-2 a d^2\right )\right )}{8 x^8}}{9 a}-\frac {2 c \left (a+b x^2\right )^{5/2} (2 A d+B c)}{9 x^9}}{2 a}-\frac {A c^2 \left (a+b x^2\right )^{5/2}}{10 a x^{10}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\frac {1}{8} \left (\frac {1}{7} \int \frac {\left (63 \left (3 A b \left (b c^2-2 a d^2\right )+2 a \left (8 a C d^2-3 b c (c C+2 B d)\right )\right )+32 b (4 b c (B c+2 A d)-9 a d (2 c C+B d)) x\right ) \left (b x^2+a\right )^{3/2}}{x^7}dx+\frac {16 \left (a+b x^2\right )^{5/2} (4 b c (2 A d+B c)-9 a d (B d+2 c C))}{7 x^7}\right )-\frac {9 \left (a+b x^2\right )^{5/2} \left (2 a c (2 B d+c C)-A \left (b c^2-2 a d^2\right )\right )}{8 x^8}}{9 a}-\frac {2 c \left (a+b x^2\right )^{5/2} (2 A d+B c)}{9 x^9}}{2 a}-\frac {A c^2 \left (a+b x^2\right )^{5/2}}{10 a x^{10}}\) |
| \(\Big \downarrow \) 539 |
| \(\displaystyle \frac {\frac {\frac {1}{8} \left (\frac {1}{7} \left (-\frac {\int -\frac {3 b \left (64 a (4 b c (B c+2 A d)-9 a d (2 c C+B d))-21 \left (3 A b \left (b c^2-2 a d^2\right )+2 a \left (8 a C d^2-3 b c (c C+2 B d)\right )\right ) x\right ) \left (b x^2+a\right )^{3/2}}{x^6}dx}{6 a}-\frac {21 \left (a+b x^2\right )^{5/2} \left (3 A b \left (b c^2-2 a d^2\right )+2 a \left (8 a C d^2-3 b c (2 B d+c C)\right )\right )}{2 a x^6}\right )+\frac {16 \left (a+b x^2\right )^{5/2} (4 b c (2 A d+B c)-9 a d (B d+2 c C))}{7 x^7}\right )-\frac {9 \left (a+b x^2\right )^{5/2} \left (2 a c (2 B d+c C)-A \left (b c^2-2 a d^2\right )\right )}{8 x^8}}{9 a}-\frac {2 c \left (a+b x^2\right )^{5/2} (2 A d+B c)}{9 x^9}}{2 a}-\frac {A c^2 \left (a+b x^2\right )^{5/2}}{10 a x^{10}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\frac {1}{8} \left (\frac {1}{7} \left (\frac {b \int \frac {\left (64 a (4 b c (B c+2 A d)-9 a d (2 c C+B d))-21 \left (3 A b \left (b c^2-2 a d^2\right )+2 a \left (8 a C d^2-3 b c (c C+2 B d)\right )\right ) x\right ) \left (b x^2+a\right )^{3/2}}{x^6}dx}{2 a}-\frac {21 \left (a+b x^2\right )^{5/2} \left (3 A b \left (b c^2-2 a d^2\right )+2 a \left (8 a C d^2-3 b c (2 B d+c C)\right )\right )}{2 a x^6}\right )+\frac {16 \left (a+b x^2\right )^{5/2} (4 b c (2 A d+B c)-9 a d (B d+2 c C))}{7 x^7}\right )-\frac {9 \left (a+b x^2\right )^{5/2} \left (2 a c (2 B d+c C)-A \left (b c^2-2 a d^2\right )\right )}{8 x^8}}{9 a}-\frac {2 c \left (a+b x^2\right )^{5/2} (2 A d+B c)}{9 x^9}}{2 a}-\frac {A c^2 \left (a+b x^2\right )^{5/2}}{10 a x^{10}}\) |
| \(\Big \downarrow \) 534 |
| \(\displaystyle \frac {\frac {\frac {1}{8} \left (\frac {1}{7} \left (\frac {b \left (-21 \left (3 A b \left (b c^2-2 a d^2\right )+2 a \left (8 a C d^2-3 b c (2 B d+c C)\right )\right ) \int \frac {\left (b x^2+a\right )^{3/2}}{x^5}dx-\frac {64 \left (a+b x^2\right )^{5/2} (4 b c (2 A d+B c)-9 a d (B d+2 c C))}{5 x^5}\right )}{2 a}-\frac {21 \left (a+b x^2\right )^{5/2} \left (3 A b \left (b c^2-2 a d^2\right )+2 a \left (8 a C d^2-3 b c (2 B d+c C)\right )\right )}{2 a x^6}\right )+\frac {16 \left (a+b x^2\right )^{5/2} (4 b c (2 A d+B c)-9 a d (B d+2 c C))}{7 x^7}\right )-\frac {9 \left (a+b x^2\right )^{5/2} \left (2 a c (2 B d+c C)-A \left (b c^2-2 a d^2\right )\right )}{8 x^8}}{9 a}-\frac {2 c \left (a+b x^2\right )^{5/2} (2 A d+B c)}{9 x^9}}{2 a}-\frac {A c^2 \left (a+b x^2\right )^{5/2}}{10 a x^{10}}\) |
| \(\Big \downarrow \) 243 |
| \(\displaystyle \frac {\frac {\frac {1}{8} \left (\frac {1}{7} \left (\frac {b \left (-\frac {21}{2} \left (3 A b \left (b c^2-2 a d^2\right )+2 a \left (8 a C d^2-3 b c (2 B d+c C)\right )\right ) \int \frac {\left (b x^2+a\right )^{3/2}}{x^6}dx^2-\frac {64 \left (a+b x^2\right )^{5/2} (4 b c (2 A d+B c)-9 a d (B d+2 c C))}{5 x^5}\right )}{2 a}-\frac {21 \left (a+b x^2\right )^{5/2} \left (3 A b \left (b c^2-2 a d^2\right )+2 a \left (8 a C d^2-3 b c (2 B d+c C)\right )\right )}{2 a x^6}\right )+\frac {16 \left (a+b x^2\right )^{5/2} (4 b c (2 A d+B c)-9 a d (B d+2 c C))}{7 x^7}\right )-\frac {9 \left (a+b x^2\right )^{5/2} \left (2 a c (2 B d+c C)-A \left (b c^2-2 a d^2\right )\right )}{8 x^8}}{9 a}-\frac {2 c \left (a+b x^2\right )^{5/2} (2 A d+B c)}{9 x^9}}{2 a}-\frac {A c^2 \left (a+b x^2\right )^{5/2}}{10 a x^{10}}\) |
| \(\Big \downarrow \) 51 |
| \(\displaystyle \frac {\frac {\frac {1}{8} \left (\frac {1}{7} \left (\frac {b \left (-\frac {21}{2} \left (3 A b \left (b c^2-2 a d^2\right )+2 a \left (8 a C d^2-3 b c (2 B d+c C)\right )\right ) \left (\frac {3}{4} b \int \frac {\sqrt {b x^2+a}}{x^4}dx^2-\frac {\left (a+b x^2\right )^{3/2}}{2 x^4}\right )-\frac {64 \left (a+b x^2\right )^{5/2} (4 b c (2 A d+B c)-9 a d (B d+2 c C))}{5 x^5}\right )}{2 a}-\frac {21 \left (a+b x^2\right )^{5/2} \left (3 A b \left (b c^2-2 a d^2\right )+2 a \left (8 a C d^2-3 b c (2 B d+c C)\right )\right )}{2 a x^6}\right )+\frac {16 \left (a+b x^2\right )^{5/2} (4 b c (2 A d+B c)-9 a d (B d+2 c C))}{7 x^7}\right )-\frac {9 \left (a+b x^2\right )^{5/2} \left (2 a c (2 B d+c C)-A \left (b c^2-2 a d^2\right )\right )}{8 x^8}}{9 a}-\frac {2 c \left (a+b x^2\right )^{5/2} (2 A d+B c)}{9 x^9}}{2 a}-\frac {A c^2 \left (a+b x^2\right )^{5/2}}{10 a x^{10}}\) |
| \(\Big \downarrow \) 51 |
| \(\displaystyle \frac {\frac {\frac {1}{8} \left (\frac {1}{7} \left (\frac {b \left (-\frac {21}{2} \left (3 A b \left (b c^2-2 a d^2\right )+2 a \left (8 a C d^2-3 b c (2 B d+c C)\right )\right ) \left (\frac {3}{4} b \left (\frac {1}{2} b \int \frac {1}{x^2 \sqrt {b x^2+a}}dx^2-\frac {\sqrt {a+b x^2}}{x^2}\right )-\frac {\left (a+b x^2\right )^{3/2}}{2 x^4}\right )-\frac {64 \left (a+b x^2\right )^{5/2} (4 b c (2 A d+B c)-9 a d (B d+2 c C))}{5 x^5}\right )}{2 a}-\frac {21 \left (a+b x^2\right )^{5/2} \left (3 A b \left (b c^2-2 a d^2\right )+2 a \left (8 a C d^2-3 b c (2 B d+c C)\right )\right )}{2 a x^6}\right )+\frac {16 \left (a+b x^2\right )^{5/2} (4 b c (2 A d+B c)-9 a d (B d+2 c C))}{7 x^7}\right )-\frac {9 \left (a+b x^2\right )^{5/2} \left (2 a c (2 B d+c C)-A \left (b c^2-2 a d^2\right )\right )}{8 x^8}}{9 a}-\frac {2 c \left (a+b x^2\right )^{5/2} (2 A d+B c)}{9 x^9}}{2 a}-\frac {A c^2 \left (a+b x^2\right )^{5/2}}{10 a x^{10}}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {\frac {\frac {1}{8} \left (\frac {1}{7} \left (\frac {b \left (-\frac {21}{2} \left (3 A b \left (b c^2-2 a d^2\right )+2 a \left (8 a C d^2-3 b c (2 B d+c C)\right )\right ) \left (\frac {3}{4} b \left (\int \frac {1}{\frac {x^4}{b}-\frac {a}{b}}d\sqrt {b x^2+a}-\frac {\sqrt {a+b x^2}}{x^2}\right )-\frac {\left (a+b x^2\right )^{3/2}}{2 x^4}\right )-\frac {64 \left (a+b x^2\right )^{5/2} (4 b c (2 A d+B c)-9 a d (B d+2 c C))}{5 x^5}\right )}{2 a}-\frac {21 \left (a+b x^2\right )^{5/2} \left (3 A b \left (b c^2-2 a d^2\right )+2 a \left (8 a C d^2-3 b c (2 B d+c C)\right )\right )}{2 a x^6}\right )+\frac {16 \left (a+b x^2\right )^{5/2} (4 b c (2 A d+B c)-9 a d (B d+2 c C))}{7 x^7}\right )-\frac {9 \left (a+b x^2\right )^{5/2} \left (2 a c (2 B d+c C)-A \left (b c^2-2 a d^2\right )\right )}{8 x^8}}{9 a}-\frac {2 c \left (a+b x^2\right )^{5/2} (2 A d+B c)}{9 x^9}}{2 a}-\frac {A c^2 \left (a+b x^2\right )^{5/2}}{10 a x^{10}}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {\frac {\frac {1}{8} \left (\frac {1}{7} \left (\frac {b \left (-\frac {21}{2} \left (\frac {3}{4} b \left (-\frac {b \text {arctanh}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{\sqrt {a}}-\frac {\sqrt {a+b x^2}}{x^2}\right )-\frac {\left (a+b x^2\right )^{3/2}}{2 x^4}\right ) \left (3 A b \left (b c^2-2 a d^2\right )+2 a \left (8 a C d^2-3 b c (2 B d+c C)\right )\right )-\frac {64 \left (a+b x^2\right )^{5/2} (4 b c (2 A d+B c)-9 a d (B d+2 c C))}{5 x^5}\right )}{2 a}-\frac {21 \left (a+b x^2\right )^{5/2} \left (3 A b \left (b c^2-2 a d^2\right )+2 a \left (8 a C d^2-3 b c (2 B d+c C)\right )\right )}{2 a x^6}\right )+\frac {16 \left (a+b x^2\right )^{5/2} (4 b c (2 A d+B c)-9 a d (B d+2 c C))}{7 x^7}\right )-\frac {9 \left (a+b x^2\right )^{5/2} \left (2 a c (2 B d+c C)-A \left (b c^2-2 a d^2\right )\right )}{8 x^8}}{9 a}-\frac {2 c \left (a+b x^2\right )^{5/2} (2 A d+B c)}{9 x^9}}{2 a}-\frac {A c^2 \left (a+b x^2\right )^{5/2}}{10 a x^{10}}\) |
Input:
Int[((c + d*x)^2*(a + b*x^2)^(3/2)*(A + B*x + C*x^2))/x^11,x]
Output:
-1/10*(A*c^2*(a + b*x^2)^(5/2))/(a*x^10) + ((-2*c*(B*c + 2*A*d)*(a + b*x^2 )^(5/2))/(9*x^9) + ((-9*(2*a*c*(c*C + 2*B*d) - A*(b*c^2 - 2*a*d^2))*(a + b *x^2)^(5/2))/(8*x^8) + ((16*(4*b*c*(B*c + 2*A*d) - 9*a*d*(2*c*C + B*d))*(a + b*x^2)^(5/2))/(7*x^7) + ((-21*(3*A*b*(b*c^2 - 2*a*d^2) + 2*a*(8*a*C*d^2 - 3*b*c*(c*C + 2*B*d)))*(a + b*x^2)^(5/2))/(2*a*x^6) + (b*((-64*(4*b*c*(B *c + 2*A*d) - 9*a*d*(2*c*C + B*d))*(a + b*x^2)^(5/2))/(5*x^5) - (21*(3*A*b *(b*c^2 - 2*a*d^2) + 2*a*(8*a*C*d^2 - 3*b*c*(c*C + 2*B*d)))*(-1/2*(a + b*x ^2)^(3/2)/x^4 + (3*b*(-(Sqrt[a + b*x^2]/x^2) - (b*ArcTanh[Sqrt[a + b*x^2]/ Sqrt[a]])/Sqrt[a]))/4))/2))/(2*a))/7)/8)/(9*a))/(2*a)
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_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^n/(b*(m + 1))), x] - Simp[d*(n/(b*(m + 1))) Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d, n}, x ] && ILtQ[m, -1] && FractionQ[n] && GtQ[n, 0]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ {p = Denominator[m]}, Simp[p/b Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL inearQ[a, b, c, d, m, n, x]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x /Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[1/2 Subst[In t[x^((m - 1)/2)*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, m, p}, x] && I ntegerQ[(m - 1)/2]
Int[(x_)^(m_)*((c_) + (d_.)*(x_))*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-c)*x^(m + 1)*((a + b*x^2)^(p + 1)/(2*a*(p + 1))), x] + Simp[d Int[ x^(m + 1)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, m, p}, x] && ILtQ[m, 0] && GtQ[p, -1] && EqQ[m + 2*p + 3, 0]
Int[(x_)^(m_)*((c_) + (d_.)*(x_))*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[c*x^(m + 1)*((a + b*x^2)^(p + 1)/(a*(m + 1))), x] + Simp[1/(a*(m + 1)) Int[x^(m + 1)*(a + b*x^2)^p*(a*d*(m + 1) - b*c*(m + 2*p + 3)*x), x], x] /; FreeQ[{a, b, c, d, p}, x] && ILtQ[m, -1] && GtQ[p, -1] && IntegerQ[2*p]
Int[(Pq_)*((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{ Q = PolynomialQuotient[Pq, c*x, x], R = PolynomialRemainder[Pq, c*x, x]}, S imp[R*(c*x)^(m + 1)*((a + b*x^2)^(p + 1)/(a*c*(m + 1))), x] + Simp[1/(a*c*( m + 1)) Int[(c*x)^(m + 1)*(a + b*x^2)^p*ExpandToSum[a*c*(m + 1)*Q - b*R*( m + 2*p + 3)*x, x], x], x]] /; FreeQ[{a, b, c, p}, x] && PolyQ[Pq, x] && Lt Q[m, -1] && (IntegerQ[2*p] || NeQ[Expon[Pq, x], 1])
Time = 0.82 (sec) , antiderivative size = 573, normalized size of antiderivative = 1.27
| method | result | size |
| risch | \(-\frac {\sqrt {b \,x^{2}+a}\, \left (4096 A \,b^{4} c d \,x^{9}-4608 B a \,b^{3} d^{2} x^{9}+2048 B \,b^{4} c^{2} x^{9}-9216 C a \,b^{3} c d \,x^{9}-1890 A a \,b^{3} d^{2} x^{8}+945 A \,b^{4} c^{2} x^{8}-3780 B a \,b^{3} c d \,x^{8}+5040 C \,a^{2} b^{2} d^{2} x^{8}-1890 C a \,b^{3} c^{2} x^{8}-2048 A a \,b^{3} c d \,x^{7}+2304 B \,a^{2} b^{2} d^{2} x^{7}-1024 B a \,b^{3} c^{2} x^{7}+4608 C \,a^{2} b^{2} c d \,x^{7}+1260 A \,a^{2} b^{2} d^{2} x^{6}-630 A a \,b^{3} c^{2} x^{6}+2520 B \,a^{2} b^{2} c d \,x^{6}+23520 C \,a^{3} b \,d^{2} x^{6}+1260 C \,a^{2} b^{2} c^{2} x^{6}+1536 A \,a^{2} b^{2} c d \,x^{5}+18432 B \,a^{3} b \,d^{2} x^{5}+768 B \,a^{2} b^{2} c^{2} x^{5}+36864 C \,a^{3} b c d \,x^{5}+15120 A \,a^{3} b \,d^{2} x^{4}+504 A \,a^{2} b^{2} c^{2} x^{4}+30240 B \,a^{3} b c d \,x^{4}+13440 C \,a^{4} d^{2} x^{4}+15120 C \,a^{3} b \,c^{2} x^{4}+25600 A \,a^{3} b c d \,x^{3}+11520 B \,a^{4} d^{2} x^{3}+12800 B \,a^{3} b \,c^{2} x^{3}+23040 C \,a^{4} c d \,x^{3}+10080 A \,a^{4} d^{2} x^{2}+11088 A \,a^{3} b \,c^{2} x^{2}+20160 B \,a^{4} c d \,x^{2}+10080 C \,a^{4} c^{2} x^{2}+17920 A \,a^{4} c d x +8960 B \,a^{4} c^{2} x +8064 A \,a^{4} c^{2}\right )}{80640 x^{10} a^{3}}-\frac {\left (6 A a b \,d^{2}-3 A \,b^{2} c^{2}+12 B a c d b -16 a^{2} C \,d^{2}+6 a b \,c^{2} C \right ) b^{3} \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{2}+a}}{x}\right )}{256 a^{\frac {7}{2}}}\) | \(573\) |
| default | \(\left (A \,d^{2}+2 B c d +C \,c^{2}\right ) \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {5}{2}}}{8 a \,x^{8}}-\frac {3 b \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {5}{2}}}{6 a \,x^{6}}-\frac {b \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {5}{2}}}{4 a \,x^{4}}+\frac {b \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {5}{2}}}{2 a \,x^{2}}+\frac {3 b \left (\frac {\left (b \,x^{2}+a \right )^{\frac {3}{2}}}{3}+a \left (\sqrt {b \,x^{2}+a}-\sqrt {a}\, \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{2}+a}}{x}\right )\right )\right )}{2 a}\right )}{4 a}\right )}{6 a}\right )}{8 a}\right )+A \,c^{2} \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {5}{2}}}{10 a \,x^{10}}-\frac {b \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {5}{2}}}{8 a \,x^{8}}-\frac {3 b \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {5}{2}}}{6 a \,x^{6}}-\frac {b \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {5}{2}}}{4 a \,x^{4}}+\frac {b \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {5}{2}}}{2 a \,x^{2}}+\frac {3 b \left (\frac {\left (b \,x^{2}+a \right )^{\frac {3}{2}}}{3}+a \left (\sqrt {b \,x^{2}+a}-\sqrt {a}\, \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{2}+a}}{x}\right )\right )\right )}{2 a}\right )}{4 a}\right )}{6 a}\right )}{8 a}\right )}{2 a}\right )+C \,d^{2} \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {5}{2}}}{6 a \,x^{6}}-\frac {b \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {5}{2}}}{4 a \,x^{4}}+\frac {b \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {5}{2}}}{2 a \,x^{2}}+\frac {3 b \left (\frac {\left (b \,x^{2}+a \right )^{\frac {3}{2}}}{3}+a \left (\sqrt {b \,x^{2}+a}-\sqrt {a}\, \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{2}+a}}{x}\right )\right )\right )}{2 a}\right )}{4 a}\right )}{6 a}\right )+c \left (2 A d +B c \right ) \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {5}{2}}}{9 a \,x^{9}}-\frac {4 b \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {5}{2}}}{7 a \,x^{7}}+\frac {2 b \left (b \,x^{2}+a \right )^{\frac {5}{2}}}{35 a^{2} x^{5}}\right )}{9 a}\right )+d \left (B d +2 C c \right ) \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {5}{2}}}{7 a \,x^{7}}+\frac {2 b \left (b \,x^{2}+a \right )^{\frac {5}{2}}}{35 a^{2} x^{5}}\right )\) | \(589\) |
Input:
int((d*x+c)^2*(b*x^2+a)^(3/2)*(C*x^2+B*x+A)/x^11,x,method=_RETURNVERBOSE)
Output:
-1/80640*(b*x^2+a)^(1/2)*(4096*A*b^4*c*d*x^9-4608*B*a*b^3*d^2*x^9+2048*B*b ^4*c^2*x^9-9216*C*a*b^3*c*d*x^9-1890*A*a*b^3*d^2*x^8+945*A*b^4*c^2*x^8-378 0*B*a*b^3*c*d*x^8+5040*C*a^2*b^2*d^2*x^8-1890*C*a*b^3*c^2*x^8-2048*A*a*b^3 *c*d*x^7+2304*B*a^2*b^2*d^2*x^7-1024*B*a*b^3*c^2*x^7+4608*C*a^2*b^2*c*d*x^ 7+1260*A*a^2*b^2*d^2*x^6-630*A*a*b^3*c^2*x^6+2520*B*a^2*b^2*c*d*x^6+23520* C*a^3*b*d^2*x^6+1260*C*a^2*b^2*c^2*x^6+1536*A*a^2*b^2*c*d*x^5+18432*B*a^3* b*d^2*x^5+768*B*a^2*b^2*c^2*x^5+36864*C*a^3*b*c*d*x^5+15120*A*a^3*b*d^2*x^ 4+504*A*a^2*b^2*c^2*x^4+30240*B*a^3*b*c*d*x^4+13440*C*a^4*d^2*x^4+15120*C* a^3*b*c^2*x^4+25600*A*a^3*b*c*d*x^3+11520*B*a^4*d^2*x^3+12800*B*a^3*b*c^2* x^3+23040*C*a^4*c*d*x^3+10080*A*a^4*d^2*x^2+11088*A*a^3*b*c^2*x^2+20160*B* a^4*c*d*x^2+10080*C*a^4*c^2*x^2+17920*A*a^4*c*d*x+8960*B*a^4*c^2*x+8064*A* a^4*c^2)/x^10/a^3-1/256*(6*A*a*b*d^2-3*A*b^2*c^2+12*B*a*b*c*d-16*C*a^2*d^2 +6*C*a*b*c^2)*b^3/a^(7/2)*ln((2*a+2*a^(1/2)*(b*x^2+a)^(1/2))/x)
Time = 1.06 (sec) , antiderivative size = 1097, normalized size of antiderivative = 2.44 \[ \int \frac {(c+d x)^2 \left (a+b x^2\right )^{3/2} \left (A+B x+C x^2\right )}{x^{11}} \, dx=\text {Too large to display} \] Input:
integrate((d*x+c)^2*(b*x^2+a)^(3/2)*(C*x^2+B*x+A)/x^11,x, algorithm="frica s")
Output:
[1/161280*(315*(12*B*a*b^4*c*d + 3*(2*C*a*b^4 - A*b^5)*c^2 - 2*(8*C*a^2*b^ 3 - 3*A*a*b^4)*d^2)*sqrt(a)*x^10*log(-(b*x^2 - 2*sqrt(b*x^2 + a)*sqrt(a) + 2*a)/x^2) - 2*(512*(4*B*a*b^4*c^2 - 9*B*a^2*b^3*d^2 - 2*(9*C*a^2*b^3 - 4* A*a*b^4)*c*d)*x^9 - 315*(12*B*a^2*b^3*c*d + 3*(2*C*a^2*b^3 - A*a*b^4)*c^2 - 2*(8*C*a^3*b^2 - 3*A*a^2*b^3)*d^2)*x^8 + 8064*A*a^5*c^2 - 256*(4*B*a^2*b ^3*c^2 - 9*B*a^3*b^2*d^2 - 2*(9*C*a^3*b^2 - 4*A*a^2*b^3)*c*d)*x^7 + 210*(1 2*B*a^3*b^2*c*d + 3*(2*C*a^3*b^2 - A*a^2*b^3)*c^2 + 2*(56*C*a^4*b + 3*A*a^ 3*b^2)*d^2)*x^6 + 768*(B*a^3*b^2*c^2 + 24*B*a^4*b*d^2 + 2*(24*C*a^4*b + A* a^3*b^2)*c*d)*x^5 + 168*(180*B*a^4*b*c*d + 3*(30*C*a^4*b + A*a^3*b^2)*c^2 + 10*(8*C*a^5 + 9*A*a^4*b)*d^2)*x^4 + 1280*(10*B*a^4*b*c^2 + 9*B*a^5*d^2 + 2*(9*C*a^5 + 10*A*a^4*b)*c*d)*x^3 + 1008*(20*B*a^5*c*d + 10*A*a^5*d^2 + ( 10*C*a^5 + 11*A*a^4*b)*c^2)*x^2 + 8960*(B*a^5*c^2 + 2*A*a^5*c*d)*x)*sqrt(b *x^2 + a))/(a^4*x^10), 1/80640*(315*(12*B*a*b^4*c*d + 3*(2*C*a*b^4 - A*b^5 )*c^2 - 2*(8*C*a^2*b^3 - 3*A*a*b^4)*d^2)*sqrt(-a)*x^10*arctan(sqrt(b*x^2 + a)*sqrt(-a)/a) - (512*(4*B*a*b^4*c^2 - 9*B*a^2*b^3*d^2 - 2*(9*C*a^2*b^3 - 4*A*a*b^4)*c*d)*x^9 - 315*(12*B*a^2*b^3*c*d + 3*(2*C*a^2*b^3 - A*a*b^4)*c ^2 - 2*(8*C*a^3*b^2 - 3*A*a^2*b^3)*d^2)*x^8 + 8064*A*a^5*c^2 - 256*(4*B*a^ 2*b^3*c^2 - 9*B*a^3*b^2*d^2 - 2*(9*C*a^3*b^2 - 4*A*a^2*b^3)*c*d)*x^7 + 210 *(12*B*a^3*b^2*c*d + 3*(2*C*a^3*b^2 - A*a^2*b^3)*c^2 + 2*(56*C*a^4*b + 3*A *a^3*b^2)*d^2)*x^6 + 768*(B*a^3*b^2*c^2 + 24*B*a^4*b*d^2 + 2*(24*C*a^4*...
Timed out. \[ \int \frac {(c+d x)^2 \left (a+b x^2\right )^{3/2} \left (A+B x+C x^2\right )}{x^{11}} \, dx=\text {Timed out} \] Input:
integrate((d*x+c)**2*(b*x**2+a)**(3/2)*(C*x**2+B*x+A)/x**11,x)
Output:
Timed out
Time = 0.04 (sec) , antiderivative size = 701, normalized size of antiderivative = 1.56 \[ \int \frac {(c+d x)^2 \left (a+b x^2\right )^{3/2} \left (A+B x+C x^2\right )}{x^{11}} \, dx =\text {Too large to display} \] Input:
integrate((d*x+c)^2*(b*x^2+a)^(3/2)*(C*x^2+B*x+A)/x^11,x, algorithm="maxim a")
Output:
3/256*A*b^5*c^2*arcsinh(a/(sqrt(a*b)*abs(x)))/a^(7/2) + 1/16*C*b^3*d^2*arc sinh(a/(sqrt(a*b)*abs(x)))/a^(3/2) - 1/256*(b*x^2 + a)^(3/2)*A*b^5*c^2/a^5 - 3/256*sqrt(b*x^2 + a)*A*b^5*c^2/a^4 - 1/48*(b*x^2 + a)^(3/2)*C*b^3*d^2/ a^3 - 1/16*sqrt(b*x^2 + a)*C*b^3*d^2/a^2 - 3/128*(C*c^2 + 2*B*c*d + A*d^2) *b^4*arcsinh(a/(sqrt(a*b)*abs(x)))/a^(5/2) + 1/128*(C*c^2 + 2*B*c*d + A*d^ 2)*(b*x^2 + a)^(3/2)*b^4/a^4 + 3/128*(C*c^2 + 2*B*c*d + A*d^2)*sqrt(b*x^2 + a)*b^4/a^3 + 1/256*(b*x^2 + a)^(5/2)*A*b^4*c^2/(a^5*x^2) + 1/48*(b*x^2 + a)^(5/2)*C*b^2*d^2/(a^3*x^2) + 1/128*(b*x^2 + a)^(5/2)*A*b^3*c^2/(a^4*x^4 ) + 1/24*(b*x^2 + a)^(5/2)*C*b*d^2/(a^2*x^4) - 1/128*(C*c^2 + 2*B*c*d + A* d^2)*(b*x^2 + a)^(5/2)*b^3/(a^4*x^2) - 1/32*(b*x^2 + a)^(5/2)*A*b^2*c^2/(a ^3*x^6) - 1/6*(b*x^2 + a)^(5/2)*C*d^2/(a*x^6) - 1/64*(C*c^2 + 2*B*c*d + A* d^2)*(b*x^2 + a)^(5/2)*b^2/(a^3*x^4) + 2/35*(2*C*c*d + B*d^2)*(b*x^2 + a)^ (5/2)*b/(a^2*x^5) - 8/315*(B*c^2 + 2*A*c*d)*(b*x^2 + a)^(5/2)*b^2/(a^3*x^5 ) + 1/16*(b*x^2 + a)^(5/2)*A*b*c^2/(a^2*x^8) + 1/16*(C*c^2 + 2*B*c*d + A*d ^2)*(b*x^2 + a)^(5/2)*b/(a^2*x^6) - 1/7*(2*C*c*d + B*d^2)*(b*x^2 + a)^(5/2 )/(a*x^7) + 4/63*(B*c^2 + 2*A*c*d)*(b*x^2 + a)^(5/2)*b/(a^2*x^7) - 1/10*(b *x^2 + a)^(5/2)*A*c^2/(a*x^10) - 1/8*(C*c^2 + 2*B*c*d + A*d^2)*(b*x^2 + a) ^(5/2)/(a*x^8) - 1/9*(B*c^2 + 2*A*c*d)*(b*x^2 + a)^(5/2)/(a*x^9)
Leaf count of result is larger than twice the leaf count of optimal. 2598 vs. \(2 (410) = 820\).
Time = 0.24 (sec) , antiderivative size = 2598, normalized size of antiderivative = 5.77 \[ \int \frac {(c+d x)^2 \left (a+b x^2\right )^{3/2} \left (A+B x+C x^2\right )}{x^{11}} \, dx=\text {Too large to display} \] Input:
integrate((d*x+c)^2*(b*x^2+a)^(3/2)*(C*x^2+B*x+A)/x^11,x, algorithm="giac" )
Output:
1/128*(6*C*a*b^4*c^2 - 3*A*b^5*c^2 + 12*B*a*b^4*c*d - 16*C*a^2*b^3*d^2 + 6 *A*a*b^4*d^2)*arctan(-(sqrt(b)*x - sqrt(b*x^2 + a))/sqrt(-a))/(sqrt(-a)*a^ 3) - 1/40320*(1890*(sqrt(b)*x - sqrt(b*x^2 + a))^19*C*a*b^4*c^2 - 945*(sqr t(b)*x - sqrt(b*x^2 + a))^19*A*b^5*c^2 + 3780*(sqrt(b)*x - sqrt(b*x^2 + a) )^19*B*a*b^4*c*d - 5040*(sqrt(b)*x - sqrt(b*x^2 + a))^19*C*a^2*b^3*d^2 + 1 890*(sqrt(b)*x - sqrt(b*x^2 + a))^19*A*a*b^4*d^2 - 18270*(sqrt(b)*x - sqrt (b*x^2 + a))^17*C*a^2*b^4*c^2 + 9135*(sqrt(b)*x - sqrt(b*x^2 + a))^17*A*a* b^5*c^2 - 36540*(sqrt(b)*x - sqrt(b*x^2 + a))^17*B*a^2*b^4*c*d - 58800*(sq rt(b)*x - sqrt(b*x^2 + a))^17*C*a^3*b^3*d^2 - 18270*(sqrt(b)*x - sqrt(b*x^ 2 + a))^17*A*a^2*b^4*d^2 - 322560*(sqrt(b)*x - sqrt(b*x^2 + a))^16*C*a^3*b ^(7/2)*c*d - 161280*(sqrt(b)*x - sqrt(b*x^2 + a))^16*B*a^3*b^(7/2)*d^2 - 1 78920*(sqrt(b)*x - sqrt(b*x^2 + a))^15*C*a^3*b^4*c^2 - 39564*(sqrt(b)*x - sqrt(b*x^2 + a))^15*A*a^2*b^5*c^2 - 357840*(sqrt(b)*x - sqrt(b*x^2 + a))^1 5*B*a^3*b^4*c*d + 154560*(sqrt(b)*x - sqrt(b*x^2 + a))^15*C*a^4*b^3*d^2 - 178920*(sqrt(b)*x - sqrt(b*x^2 + a))^15*A*a^3*b^4*d^2 - 430080*(sqrt(b)*x - sqrt(b*x^2 + a))^14*B*a^3*b^(9/2)*c^2 + 645120*(sqrt(b)*x - sqrt(b*x^2 + a))^14*C*a^4*b^(7/2)*c*d - 860160*(sqrt(b)*x - sqrt(b*x^2 + a))^14*A*a^3* b^(9/2)*c*d + 322560*(sqrt(b)*x - sqrt(b*x^2 + a))^14*B*a^4*b^(7/2)*d^2 - 17640*(sqrt(b)*x - sqrt(b*x^2 + a))^13*C*a^4*b^4*c^2 - 636300*(sqrt(b)*x - sqrt(b*x^2 + a))^13*A*a^3*b^5*c^2 - 35280*(sqrt(b)*x - sqrt(b*x^2 + a)...
Timed out. \[ \int \frac {(c+d x)^2 \left (a+b x^2\right )^{3/2} \left (A+B x+C x^2\right )}{x^{11}} \, dx=\int \frac {{\left (b\,x^2+a\right )}^{3/2}\,{\left (c+d\,x\right )}^2\,\left (C\,x^2+B\,x+A\right )}{x^{11}} \,d x \] Input:
int(((a + b*x^2)^(3/2)*(c + d*x)^2*(A + B*x + C*x^2))/x^11,x)
Output:
int(((a + b*x^2)^(3/2)*(c + d*x)^2*(A + B*x + C*x^2))/x^11, x)
Time = 0.70 (sec) , antiderivative size = 1192, normalized size of antiderivative = 2.65 \[ \int \frac {(c+d x)^2 \left (a+b x^2\right )^{3/2} \left (A+B x+C x^2\right )}{x^{11}} \, dx =\text {Too large to display} \] Input:
int((d*x+c)^2*(b*x^2+a)^(3/2)*(C*x^2+B*x+A)/x^11,x)
Output:
( - 8064*sqrt(a + b*x**2)*a**5*c**2 - 17920*sqrt(a + b*x**2)*a**5*c*d*x - 10080*sqrt(a + b*x**2)*a**5*d**2*x**2 - 11088*sqrt(a + b*x**2)*a**4*b*c**2 *x**2 - 8960*sqrt(a + b*x**2)*a**4*b*c**2*x - 25600*sqrt(a + b*x**2)*a**4* b*c*d*x**3 - 20160*sqrt(a + b*x**2)*a**4*b*c*d*x**2 - 15120*sqrt(a + b*x** 2)*a**4*b*d**2*x**4 - 11520*sqrt(a + b*x**2)*a**4*b*d**2*x**3 - 10080*sqrt (a + b*x**2)*a**4*c**3*x**2 - 23040*sqrt(a + b*x**2)*a**4*c**2*d*x**3 - 13 440*sqrt(a + b*x**2)*a**4*c*d**2*x**4 - 504*sqrt(a + b*x**2)*a**3*b**2*c** 2*x**4 - 12800*sqrt(a + b*x**2)*a**3*b**2*c**2*x**3 - 1536*sqrt(a + b*x**2 )*a**3*b**2*c*d*x**5 - 30240*sqrt(a + b*x**2)*a**3*b**2*c*d*x**4 - 1260*sq rt(a + b*x**2)*a**3*b**2*d**2*x**6 - 18432*sqrt(a + b*x**2)*a**3*b**2*d**2 *x**5 - 15120*sqrt(a + b*x**2)*a**3*b*c**3*x**4 - 36864*sqrt(a + b*x**2)*a **3*b*c**2*d*x**5 - 23520*sqrt(a + b*x**2)*a**3*b*c*d**2*x**6 + 630*sqrt(a + b*x**2)*a**2*b**3*c**2*x**6 - 768*sqrt(a + b*x**2)*a**2*b**3*c**2*x**5 + 2048*sqrt(a + b*x**2)*a**2*b**3*c*d*x**7 - 2520*sqrt(a + b*x**2)*a**2*b* *3*c*d*x**6 + 1890*sqrt(a + b*x**2)*a**2*b**3*d**2*x**8 - 2304*sqrt(a + b* x**2)*a**2*b**3*d**2*x**7 - 1260*sqrt(a + b*x**2)*a**2*b**2*c**3*x**6 - 46 08*sqrt(a + b*x**2)*a**2*b**2*c**2*d*x**7 - 5040*sqrt(a + b*x**2)*a**2*b** 2*c*d**2*x**8 - 945*sqrt(a + b*x**2)*a*b**4*c**2*x**8 + 1024*sqrt(a + b*x* *2)*a*b**4*c**2*x**7 - 4096*sqrt(a + b*x**2)*a*b**4*c*d*x**9 + 3780*sqrt(a + b*x**2)*a*b**4*c*d*x**8 + 4608*sqrt(a + b*x**2)*a*b**4*d**2*x**9 + 1...