Integrand size = 24, antiderivative size = 344 \[ \int \frac {(A+B x) \left (a+c x^2\right )^3}{(d+e x)^{3/2}} \, dx=\frac {2 (B d-A e) \left (c d^2+a e^2\right )^3}{e^8 \sqrt {d+e x}}+\frac {2 \left (c d^2+a e^2\right )^2 \left (7 B c d^2-6 A c d e+a B e^2\right ) \sqrt {d+e x}}{e^8}-\frac {2 c \left (c d^2+a e^2\right ) \left (7 B c d^3-5 A c d^2 e+3 a B d e^2-a A e^3\right ) (d+e x)^{3/2}}{e^8}-\frac {2 c \left (4 A c d e \left (5 c d^2+3 a e^2\right )-B \left (35 c^2 d^4+30 a c d^2 e^2+3 a^2 e^4\right )\right ) (d+e x)^{5/2}}{5 e^8}-\frac {2 c^2 \left (35 B c d^3-15 A c d^2 e+15 a B d e^2-3 a A e^3\right ) (d+e x)^{7/2}}{7 e^8}+\frac {2 c^2 \left (7 B c d^2-2 A c d e+a B e^2\right ) (d+e x)^{9/2}}{3 e^8}-\frac {2 c^3 (7 B d-A e) (d+e x)^{11/2}}{11 e^8}+\frac {2 B c^3 (d+e x)^{13/2}}{13 e^8} \] Output:
2*(-A*e+B*d)*(a*e^2+c*d^2)^3/e^8/(e*x+d)^(1/2)+2*(a*e^2+c*d^2)^2*(-6*A*c*d *e+B*a*e^2+7*B*c*d^2)*(e*x+d)^(1/2)/e^8-2*c*(a*e^2+c*d^2)*(-A*a*e^3-5*A*c* d^2*e+3*B*a*d*e^2+7*B*c*d^3)*(e*x+d)^(3/2)/e^8-2/5*c*(4*A*c*d*e*(3*a*e^2+5 *c*d^2)-B*(3*a^2*e^4+30*a*c*d^2*e^2+35*c^2*d^4))*(e*x+d)^(5/2)/e^8-2/7*c^2 *(-3*A*a*e^3-15*A*c*d^2*e+15*B*a*d*e^2+35*B*c*d^3)*(e*x+d)^(7/2)/e^8+2/3*c ^2*(-2*A*c*d*e+B*a*e^2+7*B*c*d^2)*(e*x+d)^(9/2)/e^8-2/11*c^3*(-A*e+7*B*d)* (e*x+d)^(11/2)/e^8+2/13*B*c^3*(e*x+d)^(13/2)/e^8
Time = 0.65 (sec) , antiderivative size = 373, normalized size of antiderivative = 1.08 \[ \int \frac {(A+B x) \left (a+c x^2\right )^3}{(d+e x)^{3/2}} \, dx=\frac {-26 A e \left (1155 a^3 e^6+1155 a^2 c e^4 \left (8 d^2+4 d e x-e^2 x^2\right )+99 a c^2 e^2 \left (128 d^4+64 d^3 e x-16 d^2 e^2 x^2+8 d e^3 x^3-5 e^4 x^4\right )+5 c^3 \left (1024 d^6+512 d^5 e x-128 d^4 e^2 x^2+64 d^3 e^3 x^3-40 d^2 e^4 x^4+28 d e^5 x^5-21 e^6 x^6\right )\right )+2 B \left (15015 a^3 e^6 (2 d+e x)+9009 a^2 c e^4 \left (16 d^3+8 d^2 e x-2 d e^2 x^2+e^3 x^3\right )+715 a c^2 e^2 \left (256 d^5+128 d^4 e x-32 d^3 e^2 x^2+16 d^2 e^3 x^3-10 d e^4 x^4+7 e^5 x^5\right )+35 c^3 \left (2048 d^7+1024 d^6 e x-256 d^5 e^2 x^2+128 d^4 e^3 x^3-80 d^3 e^4 x^4+56 d^2 e^5 x^5-42 d e^6 x^6+33 e^7 x^7\right )\right )}{15015 e^8 \sqrt {d+e x}} \] Input:
Integrate[((A + B*x)*(a + c*x^2)^3)/(d + e*x)^(3/2),x]
Output:
(-26*A*e*(1155*a^3*e^6 + 1155*a^2*c*e^4*(8*d^2 + 4*d*e*x - e^2*x^2) + 99*a *c^2*e^2*(128*d^4 + 64*d^3*e*x - 16*d^2*e^2*x^2 + 8*d*e^3*x^3 - 5*e^4*x^4) + 5*c^3*(1024*d^6 + 512*d^5*e*x - 128*d^4*e^2*x^2 + 64*d^3*e^3*x^3 - 40*d ^2*e^4*x^4 + 28*d*e^5*x^5 - 21*e^6*x^6)) + 2*B*(15015*a^3*e^6*(2*d + e*x) + 9009*a^2*c*e^4*(16*d^3 + 8*d^2*e*x - 2*d*e^2*x^2 + e^3*x^3) + 715*a*c^2* e^2*(256*d^5 + 128*d^4*e*x - 32*d^3*e^2*x^2 + 16*d^2*e^3*x^3 - 10*d*e^4*x^ 4 + 7*e^5*x^5) + 35*c^3*(2048*d^7 + 1024*d^6*e*x - 256*d^5*e^2*x^2 + 128*d ^4*e^3*x^3 - 80*d^3*e^4*x^4 + 56*d^2*e^5*x^5 - 42*d*e^6*x^6 + 33*e^7*x^7)) )/(15015*e^8*Sqrt[d + e*x])
Time = 0.48 (sec) , antiderivative size = 344, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {652, 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.
| \(\displaystyle \int \frac {\left (a+c x^2\right )^3 (A+B x)}{(d+e x)^{3/2}} \, dx\) |
| \(\Big \downarrow \) 652 |
| \(\displaystyle \int \left (-\frac {c (d+e x)^{3/2} \left (-3 a^2 B e^4+12 a A c d e^3-30 a B c d^2 e^2+20 A c^2 d^3 e-35 B c^2 d^4\right )}{e^7}-\frac {3 c^2 (d+e x)^{7/2} \left (-a B e^2+2 A c d e-7 B c d^2\right )}{e^7}+\frac {c^2 (d+e x)^{5/2} \left (3 a A e^3-15 a B d e^2+15 A c d^2 e-35 B c d^3\right )}{e^7}+\frac {\left (a e^2+c d^2\right )^2 \left (a B e^2-6 A c d e+7 B c d^2\right )}{e^7 \sqrt {d+e x}}+\frac {\left (a e^2+c d^2\right )^3 (A e-B d)}{e^7 (d+e x)^{3/2}}+\frac {3 c \sqrt {d+e x} \left (a e^2+c d^2\right ) \left (a A e^3-3 a B d e^2+5 A c d^2 e-7 B c d^3\right )}{e^7}+\frac {c^3 (d+e x)^{9/2} (A e-7 B d)}{e^7}+\frac {B c^3 (d+e x)^{11/2}}{e^7}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {2 c (d+e x)^{5/2} \left (4 A c d e \left (3 a e^2+5 c d^2\right )-B \left (3 a^2 e^4+30 a c d^2 e^2+35 c^2 d^4\right )\right )}{5 e^8}+\frac {2 c^2 (d+e x)^{9/2} \left (a B e^2-2 A c d e+7 B c d^2\right )}{3 e^8}-\frac {2 c^2 (d+e x)^{7/2} \left (-3 a A e^3+15 a B d e^2-15 A c d^2 e+35 B c d^3\right )}{7 e^8}+\frac {2 \sqrt {d+e x} \left (a e^2+c d^2\right )^2 \left (a B e^2-6 A c d e+7 B c d^2\right )}{e^8}+\frac {2 \left (a e^2+c d^2\right )^3 (B d-A e)}{e^8 \sqrt {d+e x}}-\frac {2 c (d+e x)^{3/2} \left (a e^2+c d^2\right ) \left (-a A e^3+3 a B d e^2-5 A c d^2 e+7 B c d^3\right )}{e^8}-\frac {2 c^3 (d+e x)^{11/2} (7 B d-A e)}{11 e^8}+\frac {2 B c^3 (d+e x)^{13/2}}{13 e^8}\) |
Input:
Int[((A + B*x)*(a + c*x^2)^3)/(d + e*x)^(3/2),x]
Output:
(2*(B*d - A*e)*(c*d^2 + a*e^2)^3)/(e^8*Sqrt[d + e*x]) + (2*(c*d^2 + a*e^2) ^2*(7*B*c*d^2 - 6*A*c*d*e + a*B*e^2)*Sqrt[d + e*x])/e^8 - (2*c*(c*d^2 + a* e^2)*(7*B*c*d^3 - 5*A*c*d^2*e + 3*a*B*d*e^2 - a*A*e^3)*(d + e*x)^(3/2))/e^ 8 - (2*c*(4*A*c*d*e*(5*c*d^2 + 3*a*e^2) - B*(35*c^2*d^4 + 30*a*c*d^2*e^2 + 3*a^2*e^4))*(d + e*x)^(5/2))/(5*e^8) - (2*c^2*(35*B*c*d^3 - 15*A*c*d^2*e + 15*a*B*d*e^2 - 3*a*A*e^3)*(d + e*x)^(7/2))/(7*e^8) + (2*c^2*(7*B*c*d^2 - 2*A*c*d*e + a*B*e^2)*(d + e*x)^(9/2))/(3*e^8) - (2*c^3*(7*B*d - A*e)*(d + e*x)^(11/2))/(11*e^8) + (2*B*c^3*(d + e*x)^(13/2))/(13*e^8)
Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.)*((a_) + (c_.)*(x_ )^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*(f + g*x)^n*(a + c *x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g, m, n}, x] && IGtQ[p, 0]
Time = 1.09 (sec) , antiderivative size = 298, normalized size of antiderivative = 0.87
| method | result | size |
| pseudoelliptic | \(-\frac {2 \left (\left (\left (-\frac {1}{13} B \,x^{7}-\frac {1}{11} A \,x^{6}\right ) c^{3}-\frac {3 \left (\frac {7 B x}{9}+A \right ) x^{4} a \,c^{2}}{7}-\left (\frac {3 B x}{5}+A \right ) x^{2} a^{2} c +a^{3} \left (-B x +A \right )\right ) e^{7}+4 d \left (\frac {x^{5} \left (\frac {21 B x}{26}+A \right ) c^{3}}{33}+\frac {6 x^{3} \left (\frac {25 B x}{36}+A \right ) a \,c^{2}}{35}+a^{2} x \left (\frac {3 B x}{10}+A \right ) c -\frac {B \,a^{3}}{2}\right ) e^{6}+8 c \left (-\frac {5 \left (\frac {49 B x}{65}+A \right ) x^{4} c^{2}}{231}-\frac {6 \left (\frac {5 B x}{9}+A \right ) x^{2} a c}{35}+a^{2} \left (-\frac {3 B x}{5}+A \right )\right ) d^{2} e^{5}+\frac {192 c \,d^{3} \left (\frac {5 x^{3} \left (\frac {35 B x}{52}+A \right ) c^{2}}{99}+a x \left (\frac {5 B x}{18}+A \right ) c -\frac {7 a^{2} B}{4}\right ) e^{4}}{35}+\frac {384 c^{2} d^{4} \left (-\frac {5 x^{2} \left (\frac {7 B x}{13}+A \right ) c}{99}+a \left (-\frac {5 B x}{9}+A \right )\right ) e^{3}}{35}+\frac {512 \left (x \left (\frac {7 B x}{26}+A \right ) c -\frac {11 B a}{2}\right ) c^{2} d^{5} e^{2}}{231}+\frac {1024 c^{3} d^{6} \left (-\frac {7 B x}{13}+A \right ) e}{231}-\frac {2048 B \,c^{3} d^{7}}{429}\right )}{\sqrt {e x +d}\, e^{8}}\) | \(298\) |
| risch | \(-\frac {2 \left (-1155 e^{6} B \,c^{3} x^{6}-1365 e^{6} A \,c^{3} x^{5}+2625 d \,e^{5} B \,c^{3} x^{5}+3185 d \,e^{5} A \,c^{3} x^{4}-5005 e^{6} B a \,c^{2} x^{4}-4585 c^{3} d^{2} e^{4} B \,x^{4}-6435 e^{6} A a \,c^{2} x^{3}-5785 c^{3} d^{2} e^{4} A \,x^{3}+12155 d \,e^{5} B \,x^{3} a \,c^{2}+7385 c^{3} d^{3} e^{3} B \,x^{3}+16731 d \,e^{5} a \,c^{2} x^{2} A +9945 c^{3} d^{3} e^{3} A \,x^{2}-9009 e^{6} B \,a^{2} c \,x^{2}-23595 c^{2} d^{2} e^{4} B a \,x^{2}-11865 c^{3} d^{4} e^{2} B \,x^{2}-15015 e^{6} c x \,a^{2} A -37323 c^{2} d^{2} e^{4} a x A -18265 c^{3} d^{4} e^{2} A x +27027 d \,e^{5} a^{2} x B c +46475 c^{2} d^{3} e^{3} B a x +20825 c^{3} d^{5} e B x +75075 A \,a^{2} c d \,e^{5}+119691 A a \,c^{2} d^{3} e^{3}+51545 A \,c^{3} d^{5} e -15015 B \,a^{3} e^{6}-99099 B \,a^{2} c \,d^{2} e^{4}-137995 B a \,c^{2} d^{4} e^{2}-56665 B \,c^{3} d^{6}\right ) \sqrt {e x +d}}{15015 e^{8}}-\frac {2 \left (A \,a^{3} e^{7}+3 A \,a^{2} c \,d^{2} e^{5}+3 A a \,c^{2} d^{4} e^{3}+A \,c^{3} d^{6} e -B \,a^{3} d \,e^{6}-3 B \,a^{2} c \,d^{3} e^{4}-3 B a \,c^{2} d^{5} e^{2}-B \,c^{3} d^{7}\right )}{e^{8} \sqrt {e x +d}}\) | \(477\) |
| gosper | \(-\frac {2 \left (-1155 B \,x^{7} c^{3} e^{7}-1365 A \,x^{6} c^{3} e^{7}+1470 B \,x^{6} c^{3} d \,e^{6}+1820 A \,x^{5} c^{3} d \,e^{6}-5005 B \,x^{5} a \,c^{2} e^{7}-1960 B \,x^{5} c^{3} d^{2} e^{5}-6435 A \,x^{4} a \,c^{2} e^{7}-2600 A \,x^{4} c^{3} d^{2} e^{5}+7150 B \,x^{4} a \,c^{2} d \,e^{6}+2800 B \,x^{4} c^{3} d^{3} e^{4}+10296 A \,x^{3} a \,c^{2} d \,e^{6}+4160 A \,x^{3} c^{3} d^{3} e^{4}-9009 B \,x^{3} a^{2} c \,e^{7}-11440 B \,x^{3} a \,c^{2} d^{2} e^{5}-4480 B \,x^{3} c^{3} d^{4} e^{3}-15015 A \,x^{2} a^{2} c \,e^{7}-20592 A \,x^{2} a \,c^{2} d^{2} e^{5}-8320 A \,x^{2} c^{3} d^{4} e^{3}+18018 B \,x^{2} a^{2} c d \,e^{6}+22880 B \,x^{2} a \,c^{2} d^{3} e^{4}+8960 B \,x^{2} c^{3} d^{5} e^{2}+60060 A x \,a^{2} c d \,e^{6}+82368 A x a \,c^{2} d^{3} e^{4}+33280 A x \,c^{3} d^{5} e^{2}-15015 B x \,a^{3} e^{7}-72072 B x \,a^{2} c \,d^{2} e^{5}-91520 B x a \,c^{2} d^{4} e^{3}-35840 B x \,c^{3} d^{6} e +15015 A \,a^{3} e^{7}+120120 A \,a^{2} c \,d^{2} e^{5}+164736 A a \,c^{2} d^{4} e^{3}+66560 A \,c^{3} d^{6} e -30030 B \,a^{3} d \,e^{6}-144144 B \,a^{2} c \,d^{3} e^{4}-183040 B a \,c^{2} d^{5} e^{2}-71680 B \,c^{3} d^{7}\right )}{15015 \sqrt {e x +d}\, e^{8}}\) | \(489\) |
| trager | \(-\frac {2 \left (-1155 B \,x^{7} c^{3} e^{7}-1365 A \,x^{6} c^{3} e^{7}+1470 B \,x^{6} c^{3} d \,e^{6}+1820 A \,x^{5} c^{3} d \,e^{6}-5005 B \,x^{5} a \,c^{2} e^{7}-1960 B \,x^{5} c^{3} d^{2} e^{5}-6435 A \,x^{4} a \,c^{2} e^{7}-2600 A \,x^{4} c^{3} d^{2} e^{5}+7150 B \,x^{4} a \,c^{2} d \,e^{6}+2800 B \,x^{4} c^{3} d^{3} e^{4}+10296 A \,x^{3} a \,c^{2} d \,e^{6}+4160 A \,x^{3} c^{3} d^{3} e^{4}-9009 B \,x^{3} a^{2} c \,e^{7}-11440 B \,x^{3} a \,c^{2} d^{2} e^{5}-4480 B \,x^{3} c^{3} d^{4} e^{3}-15015 A \,x^{2} a^{2} c \,e^{7}-20592 A \,x^{2} a \,c^{2} d^{2} e^{5}-8320 A \,x^{2} c^{3} d^{4} e^{3}+18018 B \,x^{2} a^{2} c d \,e^{6}+22880 B \,x^{2} a \,c^{2} d^{3} e^{4}+8960 B \,x^{2} c^{3} d^{5} e^{2}+60060 A x \,a^{2} c d \,e^{6}+82368 A x a \,c^{2} d^{3} e^{4}+33280 A x \,c^{3} d^{5} e^{2}-15015 B x \,a^{3} e^{7}-72072 B x \,a^{2} c \,d^{2} e^{5}-91520 B x a \,c^{2} d^{4} e^{3}-35840 B x \,c^{3} d^{6} e +15015 A \,a^{3} e^{7}+120120 A \,a^{2} c \,d^{2} e^{5}+164736 A a \,c^{2} d^{4} e^{3}+66560 A \,c^{3} d^{6} e -30030 B \,a^{3} d \,e^{6}-144144 B \,a^{2} c \,d^{3} e^{4}-183040 B a \,c^{2} d^{5} e^{2}-71680 B \,c^{3} d^{7}\right )}{15015 \sqrt {e x +d}\, e^{8}}\) | \(489\) |
| orering | \(-\frac {2 \left (-1155 B \,x^{7} c^{3} e^{7}-1365 A \,x^{6} c^{3} e^{7}+1470 B \,x^{6} c^{3} d \,e^{6}+1820 A \,x^{5} c^{3} d \,e^{6}-5005 B \,x^{5} a \,c^{2} e^{7}-1960 B \,x^{5} c^{3} d^{2} e^{5}-6435 A \,x^{4} a \,c^{2} e^{7}-2600 A \,x^{4} c^{3} d^{2} e^{5}+7150 B \,x^{4} a \,c^{2} d \,e^{6}+2800 B \,x^{4} c^{3} d^{3} e^{4}+10296 A \,x^{3} a \,c^{2} d \,e^{6}+4160 A \,x^{3} c^{3} d^{3} e^{4}-9009 B \,x^{3} a^{2} c \,e^{7}-11440 B \,x^{3} a \,c^{2} d^{2} e^{5}-4480 B \,x^{3} c^{3} d^{4} e^{3}-15015 A \,x^{2} a^{2} c \,e^{7}-20592 A \,x^{2} a \,c^{2} d^{2} e^{5}-8320 A \,x^{2} c^{3} d^{4} e^{3}+18018 B \,x^{2} a^{2} c d \,e^{6}+22880 B \,x^{2} a \,c^{2} d^{3} e^{4}+8960 B \,x^{2} c^{3} d^{5} e^{2}+60060 A x \,a^{2} c d \,e^{6}+82368 A x a \,c^{2} d^{3} e^{4}+33280 A x \,c^{3} d^{5} e^{2}-15015 B x \,a^{3} e^{7}-72072 B x \,a^{2} c \,d^{2} e^{5}-91520 B x a \,c^{2} d^{4} e^{3}-35840 B x \,c^{3} d^{6} e +15015 A \,a^{3} e^{7}+120120 A \,a^{2} c \,d^{2} e^{5}+164736 A a \,c^{2} d^{4} e^{3}+66560 A \,c^{3} d^{6} e -30030 B \,a^{3} d \,e^{6}-144144 B \,a^{2} c \,d^{3} e^{4}-183040 B a \,c^{2} d^{5} e^{2}-71680 B \,c^{3} d^{7}\right )}{15015 \sqrt {e x +d}\, e^{8}}\) | \(489\) |
| derivativedivides | \(\frac {12 B a \,c^{2} d^{2} e^{2} \left (e x +d \right )^{\frac {5}{2}}-14 B \,c^{3} d^{5} \left (e x +d \right )^{\frac {3}{2}}-6 B \,a^{2} c d \,e^{4} \left (e x +d \right )^{\frac {3}{2}}-20 B a \,c^{2} d^{3} e^{2} \left (e x +d \right )^{\frac {3}{2}}+12 A a \,c^{2} d^{2} e^{3} \left (e x +d \right )^{\frac {3}{2}}+18 B \,a^{2} c \,d^{2} e^{4} \sqrt {e x +d}-12 A \,a^{2} c d \,e^{5} \sqrt {e x +d}-\frac {24 A a \,c^{2} d \,e^{3} \left (e x +d \right )^{\frac {5}{2}}}{5}+\frac {30 A \,c^{3} d^{2} e \left (e x +d \right )^{\frac {7}{2}}}{7}-\frac {14 B \,c^{3} d \left (e x +d \right )^{\frac {11}{2}}}{11}+14 B \,c^{3} d^{4} \left (e x +d \right )^{\frac {5}{2}}+14 B \,c^{3} d^{6} \sqrt {e x +d}+\frac {2 B a \,c^{2} e^{2} \left (e x +d \right )^{\frac {9}{2}}}{3}+\frac {6 A a \,c^{2} e^{3} \left (e x +d \right )^{\frac {7}{2}}}{7}+\frac {14 B \,c^{3} d^{2} \left (e x +d \right )^{\frac {9}{2}}}{3}-10 B \,c^{3} d^{3} \left (e x +d \right )^{\frac {7}{2}}+2 B \,a^{3} e^{6} \sqrt {e x +d}+\frac {2 A \,c^{3} e \left (e x +d \right )^{\frac {11}{2}}}{11}-\frac {4 A \,c^{3} d e \left (e x +d \right )^{\frac {9}{2}}}{3}+10 A \,c^{3} d^{4} e \left (e x +d \right )^{\frac {3}{2}}+2 A \,a^{2} c \,e^{5} \left (e x +d \right )^{\frac {3}{2}}-8 A \,c^{3} d^{3} e \left (e x +d \right )^{\frac {5}{2}}+\frac {6 B \,a^{2} c \,e^{4} \left (e x +d \right )^{\frac {5}{2}}}{5}-12 A \,c^{3} d^{5} e \sqrt {e x +d}-24 A a \,c^{2} d^{3} e^{3} \sqrt {e x +d}-\frac {30 B a \,c^{2} d \,e^{2} \left (e x +d \right )^{\frac {7}{2}}}{7}+30 B a \,c^{2} d^{4} e^{2} \sqrt {e x +d}-\frac {2 \left (A \,a^{3} e^{7}+3 A \,a^{2} c \,d^{2} e^{5}+3 A a \,c^{2} d^{4} e^{3}+A \,c^{3} d^{6} e -B \,a^{3} d \,e^{6}-3 B \,a^{2} c \,d^{3} e^{4}-3 B a \,c^{2} d^{5} e^{2}-B \,c^{3} d^{7}\right )}{\sqrt {e x +d}}+\frac {2 B \,c^{3} \left (e x +d \right )^{\frac {13}{2}}}{13}}{e^{8}}\) | \(583\) |
| default | \(\frac {12 B a \,c^{2} d^{2} e^{2} \left (e x +d \right )^{\frac {5}{2}}-14 B \,c^{3} d^{5} \left (e x +d \right )^{\frac {3}{2}}-6 B \,a^{2} c d \,e^{4} \left (e x +d \right )^{\frac {3}{2}}-20 B a \,c^{2} d^{3} e^{2} \left (e x +d \right )^{\frac {3}{2}}+12 A a \,c^{2} d^{2} e^{3} \left (e x +d \right )^{\frac {3}{2}}+18 B \,a^{2} c \,d^{2} e^{4} \sqrt {e x +d}-12 A \,a^{2} c d \,e^{5} \sqrt {e x +d}-\frac {24 A a \,c^{2} d \,e^{3} \left (e x +d \right )^{\frac {5}{2}}}{5}+\frac {30 A \,c^{3} d^{2} e \left (e x +d \right )^{\frac {7}{2}}}{7}-\frac {14 B \,c^{3} d \left (e x +d \right )^{\frac {11}{2}}}{11}+14 B \,c^{3} d^{4} \left (e x +d \right )^{\frac {5}{2}}+14 B \,c^{3} d^{6} \sqrt {e x +d}+\frac {2 B a \,c^{2} e^{2} \left (e x +d \right )^{\frac {9}{2}}}{3}+\frac {6 A a \,c^{2} e^{3} \left (e x +d \right )^{\frac {7}{2}}}{7}+\frac {14 B \,c^{3} d^{2} \left (e x +d \right )^{\frac {9}{2}}}{3}-10 B \,c^{3} d^{3} \left (e x +d \right )^{\frac {7}{2}}+2 B \,a^{3} e^{6} \sqrt {e x +d}+\frac {2 A \,c^{3} e \left (e x +d \right )^{\frac {11}{2}}}{11}-\frac {4 A \,c^{3} d e \left (e x +d \right )^{\frac {9}{2}}}{3}+10 A \,c^{3} d^{4} e \left (e x +d \right )^{\frac {3}{2}}+2 A \,a^{2} c \,e^{5} \left (e x +d \right )^{\frac {3}{2}}-8 A \,c^{3} d^{3} e \left (e x +d \right )^{\frac {5}{2}}+\frac {6 B \,a^{2} c \,e^{4} \left (e x +d \right )^{\frac {5}{2}}}{5}-12 A \,c^{3} d^{5} e \sqrt {e x +d}-24 A a \,c^{2} d^{3} e^{3} \sqrt {e x +d}-\frac {30 B a \,c^{2} d \,e^{2} \left (e x +d \right )^{\frac {7}{2}}}{7}+30 B a \,c^{2} d^{4} e^{2} \sqrt {e x +d}-\frac {2 \left (A \,a^{3} e^{7}+3 A \,a^{2} c \,d^{2} e^{5}+3 A a \,c^{2} d^{4} e^{3}+A \,c^{3} d^{6} e -B \,a^{3} d \,e^{6}-3 B \,a^{2} c \,d^{3} e^{4}-3 B a \,c^{2} d^{5} e^{2}-B \,c^{3} d^{7}\right )}{\sqrt {e x +d}}+\frac {2 B \,c^{3} \left (e x +d \right )^{\frac {13}{2}}}{13}}{e^{8}}\) | \(583\) |
Input:
int((B*x+A)*(c*x^2+a)^3/(e*x+d)^(3/2),x,method=_RETURNVERBOSE)
Output:
-2/(e*x+d)^(1/2)*(((-1/13*B*x^7-1/11*A*x^6)*c^3-3/7*(7/9*B*x+A)*x^4*a*c^2- (3/5*B*x+A)*x^2*a^2*c+a^3*(-B*x+A))*e^7+4*d*(1/33*x^5*(21/26*B*x+A)*c^3+6/ 35*x^3*(25/36*B*x+A)*a*c^2+a^2*x*(3/10*B*x+A)*c-1/2*B*a^3)*e^6+8*c*(-5/231 *(49/65*B*x+A)*x^4*c^2-6/35*(5/9*B*x+A)*x^2*a*c+a^2*(-3/5*B*x+A))*d^2*e^5+ 192/35*c*d^3*(5/99*x^3*(35/52*B*x+A)*c^2+a*x*(5/18*B*x+A)*c-7/4*a^2*B)*e^4 +384/35*c^2*d^4*(-5/99*x^2*(7/13*B*x+A)*c+a*(-5/9*B*x+A))*e^3+512/231*(x*( 7/26*B*x+A)*c-11/2*B*a)*c^2*d^5*e^2+1024/231*c^3*d^6*(-7/13*B*x+A)*e-2048/ 429*B*c^3*d^7)/e^8
Time = 0.08 (sec) , antiderivative size = 463, normalized size of antiderivative = 1.35 \[ \int \frac {(A+B x) \left (a+c x^2\right )^3}{(d+e x)^{3/2}} \, dx=\frac {2 \, {\left (1155 \, B c^{3} e^{7} x^{7} + 71680 \, B c^{3} d^{7} - 66560 \, A c^{3} d^{6} e + 183040 \, B a c^{2} d^{5} e^{2} - 164736 \, A a c^{2} d^{4} e^{3} + 144144 \, B a^{2} c d^{3} e^{4} - 120120 \, A a^{2} c d^{2} e^{5} + 30030 \, B a^{3} d e^{6} - 15015 \, A a^{3} e^{7} - 105 \, {\left (14 \, B c^{3} d e^{6} - 13 \, A c^{3} e^{7}\right )} x^{6} + 35 \, {\left (56 \, B c^{3} d^{2} e^{5} - 52 \, A c^{3} d e^{6} + 143 \, B a c^{2} e^{7}\right )} x^{5} - 5 \, {\left (560 \, B c^{3} d^{3} e^{4} - 520 \, A c^{3} d^{2} e^{5} + 1430 \, B a c^{2} d e^{6} - 1287 \, A a c^{2} e^{7}\right )} x^{4} + {\left (4480 \, B c^{3} d^{4} e^{3} - 4160 \, A c^{3} d^{3} e^{4} + 11440 \, B a c^{2} d^{2} e^{5} - 10296 \, A a c^{2} d e^{6} + 9009 \, B a^{2} c e^{7}\right )} x^{3} - {\left (8960 \, B c^{3} d^{5} e^{2} - 8320 \, A c^{3} d^{4} e^{3} + 22880 \, B a c^{2} d^{3} e^{4} - 20592 \, A a c^{2} d^{2} e^{5} + 18018 \, B a^{2} c d e^{6} - 15015 \, A a^{2} c e^{7}\right )} x^{2} + {\left (35840 \, B c^{3} d^{6} e - 33280 \, A c^{3} d^{5} e^{2} + 91520 \, B a c^{2} d^{4} e^{3} - 82368 \, A a c^{2} d^{3} e^{4} + 72072 \, B a^{2} c d^{2} e^{5} - 60060 \, A a^{2} c d e^{6} + 15015 \, B a^{3} e^{7}\right )} x\right )} \sqrt {e x + d}}{15015 \, {\left (e^{9} x + d e^{8}\right )}} \] Input:
integrate((B*x+A)*(c*x^2+a)^3/(e*x+d)^(3/2),x, algorithm="fricas")
Output:
2/15015*(1155*B*c^3*e^7*x^7 + 71680*B*c^3*d^7 - 66560*A*c^3*d^6*e + 183040 *B*a*c^2*d^5*e^2 - 164736*A*a*c^2*d^4*e^3 + 144144*B*a^2*c*d^3*e^4 - 12012 0*A*a^2*c*d^2*e^5 + 30030*B*a^3*d*e^6 - 15015*A*a^3*e^7 - 105*(14*B*c^3*d* e^6 - 13*A*c^3*e^7)*x^6 + 35*(56*B*c^3*d^2*e^5 - 52*A*c^3*d*e^6 + 143*B*a* c^2*e^7)*x^5 - 5*(560*B*c^3*d^3*e^4 - 520*A*c^3*d^2*e^5 + 1430*B*a*c^2*d*e ^6 - 1287*A*a*c^2*e^7)*x^4 + (4480*B*c^3*d^4*e^3 - 4160*A*c^3*d^3*e^4 + 11 440*B*a*c^2*d^2*e^5 - 10296*A*a*c^2*d*e^6 + 9009*B*a^2*c*e^7)*x^3 - (8960* B*c^3*d^5*e^2 - 8320*A*c^3*d^4*e^3 + 22880*B*a*c^2*d^3*e^4 - 20592*A*a*c^2 *d^2*e^5 + 18018*B*a^2*c*d*e^6 - 15015*A*a^2*c*e^7)*x^2 + (35840*B*c^3*d^6 *e - 33280*A*c^3*d^5*e^2 + 91520*B*a*c^2*d^4*e^3 - 82368*A*a*c^2*d^3*e^4 + 72072*B*a^2*c*d^2*e^5 - 60060*A*a^2*c*d*e^6 + 15015*B*a^3*e^7)*x)*sqrt(e* x + d)/(e^9*x + d*e^8)
Time = 17.63 (sec) , antiderivative size = 561, normalized size of antiderivative = 1.63 \[ \int \frac {(A+B x) \left (a+c x^2\right )^3}{(d+e x)^{3/2}} \, dx=\begin {cases} \frac {2 \left (\frac {B c^{3} \left (d + e x\right )^{\frac {13}{2}}}{13 e^{7}} + \frac {\left (d + e x\right )^{\frac {11}{2}} \left (A c^{3} e - 7 B c^{3} d\right )}{11 e^{7}} + \frac {\left (d + e x\right )^{\frac {9}{2}} \left (- 6 A c^{3} d e + 3 B a c^{2} e^{2} + 21 B c^{3} d^{2}\right )}{9 e^{7}} + \frac {\left (d + e x\right )^{\frac {7}{2}} \cdot \left (3 A a c^{2} e^{3} + 15 A c^{3} d^{2} e - 15 B a c^{2} d e^{2} - 35 B c^{3} d^{3}\right )}{7 e^{7}} + \frac {\left (d + e x\right )^{\frac {5}{2}} \left (- 12 A a c^{2} d e^{3} - 20 A c^{3} d^{3} e + 3 B a^{2} c e^{4} + 30 B a c^{2} d^{2} e^{2} + 35 B c^{3} d^{4}\right )}{5 e^{7}} + \frac {\left (d + e x\right )^{\frac {3}{2}} \cdot \left (3 A a^{2} c e^{5} + 18 A a c^{2} d^{2} e^{3} + 15 A c^{3} d^{4} e - 9 B a^{2} c d e^{4} - 30 B a c^{2} d^{3} e^{2} - 21 B c^{3} d^{5}\right )}{3 e^{7}} + \frac {\sqrt {d + e x} \left (- 6 A a^{2} c d e^{5} - 12 A a c^{2} d^{3} e^{3} - 6 A c^{3} d^{5} e + B a^{3} e^{6} + 9 B a^{2} c d^{2} e^{4} + 15 B a c^{2} d^{4} e^{2} + 7 B c^{3} d^{6}\right )}{e^{7}} + \frac {\left (- A e + B d\right ) \left (a e^{2} + c d^{2}\right )^{3}}{e^{7} \sqrt {d + e x}}\right )}{e} & \text {for}\: e \neq 0 \\\frac {A a^{3} x + A a^{2} c x^{3} + \frac {3 A a c^{2} x^{5}}{5} + \frac {A c^{3} x^{7}}{7} + \frac {B a^{3} x^{2}}{2} + \frac {3 B a^{2} c x^{4}}{4} + \frac {B a c^{2} x^{6}}{2} + \frac {B c^{3} x^{8}}{8}}{d^{\frac {3}{2}}} & \text {otherwise} \end {cases} \] Input:
integrate((B*x+A)*(c*x**2+a)**3/(e*x+d)**(3/2),x)
Output:
Piecewise((2*(B*c**3*(d + e*x)**(13/2)/(13*e**7) + (d + e*x)**(11/2)*(A*c* *3*e - 7*B*c**3*d)/(11*e**7) + (d + e*x)**(9/2)*(-6*A*c**3*d*e + 3*B*a*c** 2*e**2 + 21*B*c**3*d**2)/(9*e**7) + (d + e*x)**(7/2)*(3*A*a*c**2*e**3 + 15 *A*c**3*d**2*e - 15*B*a*c**2*d*e**2 - 35*B*c**3*d**3)/(7*e**7) + (d + e*x) **(5/2)*(-12*A*a*c**2*d*e**3 - 20*A*c**3*d**3*e + 3*B*a**2*c*e**4 + 30*B*a *c**2*d**2*e**2 + 35*B*c**3*d**4)/(5*e**7) + (d + e*x)**(3/2)*(3*A*a**2*c* e**5 + 18*A*a*c**2*d**2*e**3 + 15*A*c**3*d**4*e - 9*B*a**2*c*d*e**4 - 30*B *a*c**2*d**3*e**2 - 21*B*c**3*d**5)/(3*e**7) + sqrt(d + e*x)*(-6*A*a**2*c* d*e**5 - 12*A*a*c**2*d**3*e**3 - 6*A*c**3*d**5*e + B*a**3*e**6 + 9*B*a**2* c*d**2*e**4 + 15*B*a*c**2*d**4*e**2 + 7*B*c**3*d**6)/e**7 + (-A*e + B*d)*( a*e**2 + c*d**2)**3/(e**7*sqrt(d + e*x)))/e, Ne(e, 0)), ((A*a**3*x + A*a** 2*c*x**3 + 3*A*a*c**2*x**5/5 + A*c**3*x**7/7 + B*a**3*x**2/2 + 3*B*a**2*c* x**4/4 + B*a*c**2*x**6/2 + B*c**3*x**8/8)/d**(3/2), True))
Time = 0.05 (sec) , antiderivative size = 461, normalized size of antiderivative = 1.34 \[ \int \frac {(A+B x) \left (a+c x^2\right )^3}{(d+e x)^{3/2}} \, dx=\frac {2 \, {\left (\frac {1155 \, {\left (e x + d\right )}^{\frac {13}{2}} B c^{3} - 1365 \, {\left (7 \, B c^{3} d - A c^{3} e\right )} {\left (e x + d\right )}^{\frac {11}{2}} + 5005 \, {\left (7 \, B c^{3} d^{2} - 2 \, A c^{3} d e + B a c^{2} e^{2}\right )} {\left (e x + d\right )}^{\frac {9}{2}} - 2145 \, {\left (35 \, B c^{3} d^{3} - 15 \, A c^{3} d^{2} e + 15 \, B a c^{2} d e^{2} - 3 \, A a c^{2} e^{3}\right )} {\left (e x + d\right )}^{\frac {7}{2}} + 3003 \, {\left (35 \, B c^{3} d^{4} - 20 \, A c^{3} d^{3} e + 30 \, B a c^{2} d^{2} e^{2} - 12 \, A a c^{2} d e^{3} + 3 \, B a^{2} c e^{4}\right )} {\left (e x + d\right )}^{\frac {5}{2}} - 15015 \, {\left (7 \, B c^{3} d^{5} - 5 \, A c^{3} d^{4} e + 10 \, B a c^{2} d^{3} e^{2} - 6 \, A a c^{2} d^{2} e^{3} + 3 \, B a^{2} c d e^{4} - A a^{2} c e^{5}\right )} {\left (e x + d\right )}^{\frac {3}{2}} + 15015 \, {\left (7 \, B c^{3} d^{6} - 6 \, A c^{3} d^{5} e + 15 \, B a c^{2} d^{4} e^{2} - 12 \, A a c^{2} d^{3} e^{3} + 9 \, B a^{2} c d^{2} e^{4} - 6 \, A a^{2} c d e^{5} + B a^{3} e^{6}\right )} \sqrt {e x + d}}{e^{7}} + \frac {15015 \, {\left (B c^{3} d^{7} - A c^{3} d^{6} e + 3 \, B a c^{2} d^{5} e^{2} - 3 \, A a c^{2} d^{4} e^{3} + 3 \, B a^{2} c d^{3} e^{4} - 3 \, A a^{2} c d^{2} e^{5} + B a^{3} d e^{6} - A a^{3} e^{7}\right )}}{\sqrt {e x + d} e^{7}}\right )}}{15015 \, e} \] Input:
integrate((B*x+A)*(c*x^2+a)^3/(e*x+d)^(3/2),x, algorithm="maxima")
Output:
2/15015*((1155*(e*x + d)^(13/2)*B*c^3 - 1365*(7*B*c^3*d - A*c^3*e)*(e*x + d)^(11/2) + 5005*(7*B*c^3*d^2 - 2*A*c^3*d*e + B*a*c^2*e^2)*(e*x + d)^(9/2) - 2145*(35*B*c^3*d^3 - 15*A*c^3*d^2*e + 15*B*a*c^2*d*e^2 - 3*A*a*c^2*e^3) *(e*x + d)^(7/2) + 3003*(35*B*c^3*d^4 - 20*A*c^3*d^3*e + 30*B*a*c^2*d^2*e^ 2 - 12*A*a*c^2*d*e^3 + 3*B*a^2*c*e^4)*(e*x + d)^(5/2) - 15015*(7*B*c^3*d^5 - 5*A*c^3*d^4*e + 10*B*a*c^2*d^3*e^2 - 6*A*a*c^2*d^2*e^3 + 3*B*a^2*c*d*e^ 4 - A*a^2*c*e^5)*(e*x + d)^(3/2) + 15015*(7*B*c^3*d^6 - 6*A*c^3*d^5*e + 15 *B*a*c^2*d^4*e^2 - 12*A*a*c^2*d^3*e^3 + 9*B*a^2*c*d^2*e^4 - 6*A*a^2*c*d*e^ 5 + B*a^3*e^6)*sqrt(e*x + d))/e^7 + 15015*(B*c^3*d^7 - A*c^3*d^6*e + 3*B*a *c^2*d^5*e^2 - 3*A*a*c^2*d^4*e^3 + 3*B*a^2*c*d^3*e^4 - 3*A*a^2*c*d^2*e^5 + B*a^3*d*e^6 - A*a^3*e^7)/(sqrt(e*x + d)*e^7))/e
Time = 0.15 (sec) , antiderivative size = 621, normalized size of antiderivative = 1.81 \[ \int \frac {(A+B x) \left (a+c x^2\right )^3}{(d+e x)^{3/2}} \, dx =\text {Too large to display} \] Input:
integrate((B*x+A)*(c*x^2+a)^3/(e*x+d)^(3/2),x, algorithm="giac")
Output:
2*(B*c^3*d^7 - A*c^3*d^6*e + 3*B*a*c^2*d^5*e^2 - 3*A*a*c^2*d^4*e^3 + 3*B*a ^2*c*d^3*e^4 - 3*A*a^2*c*d^2*e^5 + B*a^3*d*e^6 - A*a^3*e^7)/(sqrt(e*x + d) *e^8) + 2/15015*(1155*(e*x + d)^(13/2)*B*c^3*e^96 - 9555*(e*x + d)^(11/2)* B*c^3*d*e^96 + 35035*(e*x + d)^(9/2)*B*c^3*d^2*e^96 - 75075*(e*x + d)^(7/2 )*B*c^3*d^3*e^96 + 105105*(e*x + d)^(5/2)*B*c^3*d^4*e^96 - 105105*(e*x + d )^(3/2)*B*c^3*d^5*e^96 + 105105*sqrt(e*x + d)*B*c^3*d^6*e^96 + 1365*(e*x + d)^(11/2)*A*c^3*e^97 - 10010*(e*x + d)^(9/2)*A*c^3*d*e^97 + 32175*(e*x + d)^(7/2)*A*c^3*d^2*e^97 - 60060*(e*x + d)^(5/2)*A*c^3*d^3*e^97 + 75075*(e* x + d)^(3/2)*A*c^3*d^4*e^97 - 90090*sqrt(e*x + d)*A*c^3*d^5*e^97 + 5005*(e *x + d)^(9/2)*B*a*c^2*e^98 - 32175*(e*x + d)^(7/2)*B*a*c^2*d*e^98 + 90090* (e*x + d)^(5/2)*B*a*c^2*d^2*e^98 - 150150*(e*x + d)^(3/2)*B*a*c^2*d^3*e^98 + 225225*sqrt(e*x + d)*B*a*c^2*d^4*e^98 + 6435*(e*x + d)^(7/2)*A*a*c^2*e^ 99 - 36036*(e*x + d)^(5/2)*A*a*c^2*d*e^99 + 90090*(e*x + d)^(3/2)*A*a*c^2* d^2*e^99 - 180180*sqrt(e*x + d)*A*a*c^2*d^3*e^99 + 9009*(e*x + d)^(5/2)*B* a^2*c*e^100 - 45045*(e*x + d)^(3/2)*B*a^2*c*d*e^100 + 135135*sqrt(e*x + d) *B*a^2*c*d^2*e^100 + 15015*(e*x + d)^(3/2)*A*a^2*c*e^101 - 90090*sqrt(e*x + d)*A*a^2*c*d*e^101 + 15015*sqrt(e*x + d)*B*a^3*e^102)/e^104
Time = 6.42 (sec) , antiderivative size = 394, normalized size of antiderivative = 1.15 \[ \int \frac {(A+B x) \left (a+c x^2\right )^3}{(d+e x)^{3/2}} \, dx=\frac {{\left (d+e\,x\right )}^{5/2}\,\left (6\,B\,a^2\,c\,e^4+60\,B\,a\,c^2\,d^2\,e^2-24\,A\,a\,c^2\,d\,e^3+70\,B\,c^3\,d^4-40\,A\,c^3\,d^3\,e\right )}{5\,e^8}-\frac {-2\,B\,a^3\,d\,e^6+2\,A\,a^3\,e^7-6\,B\,a^2\,c\,d^3\,e^4+6\,A\,a^2\,c\,d^2\,e^5-6\,B\,a\,c^2\,d^5\,e^2+6\,A\,a\,c^2\,d^4\,e^3-2\,B\,c^3\,d^7+2\,A\,c^3\,d^6\,e}{e^8\,\sqrt {d+e\,x}}+\frac {{\left (d+e\,x\right )}^{9/2}\,\left (42\,B\,c^3\,d^2-12\,A\,c^3\,d\,e+6\,B\,a\,c^2\,e^2\right )}{9\,e^8}+\frac {2\,{\left (c\,d^2+a\,e^2\right )}^2\,\sqrt {d+e\,x}\,\left (7\,B\,c\,d^2-6\,A\,c\,d\,e+B\,a\,e^2\right )}{e^8}+\frac {2\,B\,c^3\,{\left (d+e\,x\right )}^{13/2}}{13\,e^8}+\frac {2\,c^2\,{\left (d+e\,x\right )}^{7/2}\,\left (-35\,B\,c\,d^3+15\,A\,c\,d^2\,e-15\,B\,a\,d\,e^2+3\,A\,a\,e^3\right )}{7\,e^8}+\frac {2\,c^3\,\left (A\,e-7\,B\,d\right )\,{\left (d+e\,x\right )}^{11/2}}{11\,e^8}+\frac {2\,c\,\left (c\,d^2+a\,e^2\right )\,{\left (d+e\,x\right )}^{3/2}\,\left (-7\,B\,c\,d^3+5\,A\,c\,d^2\,e-3\,B\,a\,d\,e^2+A\,a\,e^3\right )}{e^8} \] Input:
int(((a + c*x^2)^3*(A + B*x))/(d + e*x)^(3/2),x)
Output:
((d + e*x)^(5/2)*(70*B*c^3*d^4 + 6*B*a^2*c*e^4 - 40*A*c^3*d^3*e + 60*B*a*c ^2*d^2*e^2 - 24*A*a*c^2*d*e^3))/(5*e^8) - (2*A*a^3*e^7 - 2*B*c^3*d^7 - 2*B *a^3*d*e^6 + 2*A*c^3*d^6*e + 6*A*a*c^2*d^4*e^3 + 6*A*a^2*c*d^2*e^5 - 6*B*a *c^2*d^5*e^2 - 6*B*a^2*c*d^3*e^4)/(e^8*(d + e*x)^(1/2)) + ((d + e*x)^(9/2) *(42*B*c^3*d^2 - 12*A*c^3*d*e + 6*B*a*c^2*e^2))/(9*e^8) + (2*(a*e^2 + c*d^ 2)^2*(d + e*x)^(1/2)*(B*a*e^2 + 7*B*c*d^2 - 6*A*c*d*e))/e^8 + (2*B*c^3*(d + e*x)^(13/2))/(13*e^8) + (2*c^2*(d + e*x)^(7/2)*(3*A*a*e^3 - 35*B*c*d^3 - 15*B*a*d*e^2 + 15*A*c*d^2*e))/(7*e^8) + (2*c^3*(A*e - 7*B*d)*(d + e*x)^(1 1/2))/(11*e^8) + (2*c*(a*e^2 + c*d^2)*(d + e*x)^(3/2)*(A*a*e^3 - 7*B*c*d^3 - 3*B*a*d*e^2 + 5*A*c*d^2*e))/e^8
Time = 0.27 (sec) , antiderivative size = 490, normalized size of antiderivative = 1.42 \[ \int \frac {(A+B x) \left (a+c x^2\right )^3}{(d+e x)^{3/2}} \, dx=\frac {\frac {4096}{429} b \,c^{3} d^{7}-8 a^{3} c d \,e^{6} x +\frac {6}{5} a^{2} b c \,e^{7} x^{3}+4 a^{3} b d \,e^{6}-16 a^{3} c \,d^{2} e^{5}-\frac {768}{35} a^{2} c^{2} d^{4} e^{3}-\frac {2048}{231} a \,c^{3} d^{6} e -2 a^{4} e^{7}+\frac {48}{5} a^{2} b c \,d^{2} e^{5} x -\frac {12}{5} a^{2} b c d \,e^{6} x^{2}+\frac {256}{21} a b \,c^{2} d^{4} e^{3} x -\frac {64}{21} a b \,c^{2} d^{3} e^{4} x^{2}+\frac {32}{21} a b \,c^{2} d^{2} e^{5} x^{3}-\frac {20}{21} a b \,c^{2} d \,e^{6} x^{4}+2 a^{3} b \,e^{7} x +2 a^{3} c \,e^{7} x^{2}+\frac {6}{7} a^{2} c^{2} e^{7} x^{4}+\frac {2}{11} a \,c^{3} e^{7} x^{6}+\frac {2}{13} b \,c^{3} e^{7} x^{7}-\frac {384}{35} a^{2} c^{2} d^{3} e^{4} x +\frac {96}{35} a^{2} c^{2} d^{2} e^{5} x^{2}-\frac {48}{35} a^{2} c^{2} d \,e^{6} x^{3}+\frac {2}{3} a b \,c^{2} e^{7} x^{5}-\frac {1024}{231} a \,c^{3} d^{5} e^{2} x +\frac {256}{231} a \,c^{3} d^{4} e^{3} x^{2}-\frac {128}{231} a \,c^{3} d^{3} e^{4} x^{3}+\frac {80}{231} a \,c^{3} d^{2} e^{5} x^{4}-\frac {8}{33} a \,c^{3} d \,e^{6} x^{5}+\frac {2048}{429} b \,c^{3} d^{6} e x -\frac {512}{429} b \,c^{3} d^{5} e^{2} x^{2}+\frac {256}{429} b \,c^{3} d^{4} e^{3} x^{3}-\frac {160}{429} b \,c^{3} d^{3} e^{4} x^{4}+\frac {112}{429} b \,c^{3} d^{2} e^{5} x^{5}-\frac {28}{143} b \,c^{3} d \,e^{6} x^{6}+\frac {96}{5} a^{2} b c \,d^{3} e^{4}+\frac {512}{21} a b \,c^{2} d^{5} e^{2}}{\sqrt {e x +d}\, e^{8}} \] Input:
int((B*x+A)*(c*x^2+a)^3/(e*x+d)^(3/2),x)
Output:
(2*( - 15015*a**4*e**7 + 30030*a**3*b*d*e**6 + 15015*a**3*b*e**7*x - 12012 0*a**3*c*d**2*e**5 - 60060*a**3*c*d*e**6*x + 15015*a**3*c*e**7*x**2 + 1441 44*a**2*b*c*d**3*e**4 + 72072*a**2*b*c*d**2*e**5*x - 18018*a**2*b*c*d*e**6 *x**2 + 9009*a**2*b*c*e**7*x**3 - 164736*a**2*c**2*d**4*e**3 - 82368*a**2* c**2*d**3*e**4*x + 20592*a**2*c**2*d**2*e**5*x**2 - 10296*a**2*c**2*d*e**6 *x**3 + 6435*a**2*c**2*e**7*x**4 + 183040*a*b*c**2*d**5*e**2 + 91520*a*b*c **2*d**4*e**3*x - 22880*a*b*c**2*d**3*e**4*x**2 + 11440*a*b*c**2*d**2*e**5 *x**3 - 7150*a*b*c**2*d*e**6*x**4 + 5005*a*b*c**2*e**7*x**5 - 66560*a*c**3 *d**6*e - 33280*a*c**3*d**5*e**2*x + 8320*a*c**3*d**4*e**3*x**2 - 4160*a*c **3*d**3*e**4*x**3 + 2600*a*c**3*d**2*e**5*x**4 - 1820*a*c**3*d*e**6*x**5 + 1365*a*c**3*e**7*x**6 + 71680*b*c**3*d**7 + 35840*b*c**3*d**6*e*x - 8960 *b*c**3*d**5*e**2*x**2 + 4480*b*c**3*d**4*e**3*x**3 - 2800*b*c**3*d**3*e** 4*x**4 + 1960*b*c**3*d**2*e**5*x**5 - 1470*b*c**3*d*e**6*x**6 + 1155*b*c** 3*e**7*x**7))/(15015*sqrt(d + e*x)*e**8)