Integrand size = 24, antiderivative size = 348 \[ \int \frac {(A+B x) \left (a+c x^2\right )^3}{\sqrt {d+e x}} \, dx=-\frac {2 (B d-A e) \left (c d^2+a e^2\right )^3 \sqrt {d+e x}}{e^8}+\frac {2 \left (c d^2+a e^2\right )^2 \left (7 B c d^2-6 A c d e+a B e^2\right ) (d+e x)^{3/2}}{3 e^8}-\frac {6 c \left (c d^2+a e^2\right ) \left (7 B c d^3-5 A c d^2 e+3 a B d e^2-a A e^3\right ) (d+e x)^{5/2}}{5 e^8}-\frac {2 c \left (4 A c d e \left (5 c d^2+3 a e^2\right )-B \left (35 c^2 d^4+30 a c d^2 e^2+3 a^2 e^4\right )\right ) (d+e x)^{7/2}}{7 e^8}-\frac {2 c^2 \left (35 B c d^3-15 A c d^2 e+15 a B d e^2-3 a A e^3\right ) (d+e x)^{9/2}}{9 e^8}+\frac {6 c^2 \left (7 B c d^2-2 A c d e+a B e^2\right ) (d+e x)^{11/2}}{11 e^8}-\frac {2 c^3 (7 B d-A e) (d+e x)^{13/2}}{13 e^8}+\frac {2 B c^3 (d+e x)^{15/2}}{15 e^8} \] Output:
-2*(-A*e+B*d)*(a*e^2+c*d^2)^3*(e*x+d)^(1/2)/e^8+2/3*(a*e^2+c*d^2)^2*(-6*A* c*d*e+B*a*e^2+7*B*c*d^2)*(e*x+d)^(3/2)/e^8-6/5*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)^(5/2)/e^8-2/7*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)^(7/2)/e^8-2/ 9*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)^(9/2)/e^8+ 6/11*c^2*(-2*A*c*d*e+B*a*e^2+7*B*c*d^2)*(e*x+d)^(11/2)/e^8-2/13*c^3*(-A*e+ 7*B*d)*(e*x+d)^(13/2)/e^8+2/15*B*c^3*(e*x+d)^(15/2)/e^8
Time = 0.50 (sec) , antiderivative size = 373, normalized size of antiderivative = 1.07 \[ \int \frac {(A+B x) \left (a+c x^2\right )^3}{\sqrt {d+e x}} \, dx=\frac {2 \sqrt {d+e x} \left (3 A e \left (15015 a^3 e^6+3003 a^2 c e^4 \left (8 d^2-4 d e x+3 e^2 x^2\right )+143 a c^2 e^2 \left (128 d^4-64 d^3 e x+48 d^2 e^2 x^2-40 d e^3 x^3+35 e^4 x^4\right )+5 c^3 \left (1024 d^6-512 d^5 e x+384 d^4 e^2 x^2-320 d^3 e^3 x^3+280 d^2 e^4 x^4-252 d e^5 x^5+231 e^6 x^6\right )\right )+B \left (15015 a^3 e^6 (-2 d+e x)+3861 a^2 c e^4 \left (-16 d^3+8 d^2 e x-6 d e^2 x^2+5 e^3 x^3\right )+195 a c^2 e^2 \left (-256 d^5+128 d^4 e x-96 d^3 e^2 x^2+80 d^2 e^3 x^3-70 d e^4 x^4+63 e^5 x^5\right )-7 c^3 \left (2048 d^7-1024 d^6 e x+768 d^5 e^2 x^2-640 d^4 e^3 x^3+560 d^3 e^4 x^4-504 d^2 e^5 x^5+462 d e^6 x^6-429 e^7 x^7\right )\right )\right )}{45045 e^8} \] Input:
Integrate[((A + B*x)*(a + c*x^2)^3)/Sqrt[d + e*x],x]
Output:
(2*Sqrt[d + e*x]*(3*A*e*(15015*a^3*e^6 + 3003*a^2*c*e^4*(8*d^2 - 4*d*e*x + 3*e^2*x^2) + 143*a*c^2*e^2*(128*d^4 - 64*d^3*e*x + 48*d^2*e^2*x^2 - 40*d* e^3*x^3 + 35*e^4*x^4) + 5*c^3*(1024*d^6 - 512*d^5*e*x + 384*d^4*e^2*x^2 - 320*d^3*e^3*x^3 + 280*d^2*e^4*x^4 - 252*d*e^5*x^5 + 231*e^6*x^6)) + B*(150 15*a^3*e^6*(-2*d + e*x) + 3861*a^2*c*e^4*(-16*d^3 + 8*d^2*e*x - 6*d*e^2*x^ 2 + 5*e^3*x^3) + 195*a*c^2*e^2*(-256*d^5 + 128*d^4*e*x - 96*d^3*e^2*x^2 + 80*d^2*e^3*x^3 - 70*d*e^4*x^4 + 63*e^5*x^5) - 7*c^3*(2048*d^7 - 1024*d^6*e *x + 768*d^5*e^2*x^2 - 640*d^4*e^3*x^3 + 560*d^3*e^4*x^4 - 504*d^2*e^5*x^5 + 462*d*e^6*x^6 - 429*e^7*x^7))))/(45045*e^8)
Time = 0.53 (sec) , antiderivative size = 348, 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.
| \(\displaystyle \int \frac {\left (a+c x^2\right )^3 (A+B x)}{\sqrt {d+e x}} \, dx\) |
| \(\Big \downarrow \) 652 |
| \(\displaystyle \int \left (-\frac {c (d+e x)^{5/2} \left (-3 a^2 B e^4+12 a A c d e^3-30 a B c d^2 e^2+20 A c^2 d^3 e-35 B c^2 d^4\right )}{e^7}-\frac {3 c^2 (d+e x)^{9/2} \left (-a B e^2+2 A c d e-7 B c d^2\right )}{e^7}+\frac {c^2 (d+e x)^{7/2} \left (3 a A e^3-15 a B d e^2+15 A c d^2 e-35 B c d^3\right )}{e^7}+\frac {\sqrt {d+e x} \left (a e^2+c d^2\right )^2 \left (a B e^2-6 A c d e+7 B c d^2\right )}{e^7}+\frac {\left (a e^2+c d^2\right )^3 (A e-B d)}{e^7 \sqrt {d+e x}}+\frac {3 c (d+e x)^{3/2} \left (a e^2+c d^2\right ) \left (a A e^3-3 a B d e^2+5 A c d^2 e-7 B c d^3\right )}{e^7}+\frac {c^3 (d+e x)^{11/2} (A e-7 B d)}{e^7}+\frac {B c^3 (d+e x)^{13/2}}{e^7}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {2 c (d+e x)^{7/2} \left (4 A c d e \left (3 a e^2+5 c d^2\right )-B \left (3 a^2 e^4+30 a c d^2 e^2+35 c^2 d^4\right )\right )}{7 e^8}+\frac {6 c^2 (d+e x)^{11/2} \left (a B e^2-2 A c d e+7 B c d^2\right )}{11 e^8}-\frac {2 c^2 (d+e x)^{9/2} \left (-3 a A e^3+15 a B d e^2-15 A c d^2 e+35 B c d^3\right )}{9 e^8}+\frac {2 (d+e x)^{3/2} \left (a e^2+c d^2\right )^2 \left (a B e^2-6 A c d e+7 B c d^2\right )}{3 e^8}-\frac {2 \sqrt {d+e x} \left (a e^2+c d^2\right )^3 (B d-A e)}{e^8}-\frac {6 c (d+e x)^{5/2} \left (a e^2+c d^2\right ) \left (-a A e^3+3 a B d e^2-5 A c d^2 e+7 B c d^3\right )}{5 e^8}-\frac {2 c^3 (d+e x)^{13/2} (7 B d-A e)}{13 e^8}+\frac {2 B c^3 (d+e x)^{15/2}}{15 e^8}\) |
Input:
Int[((A + B*x)*(a + c*x^2)^3)/Sqrt[d + e*x],x]
Output:
(-2*(B*d - A*e)*(c*d^2 + a*e^2)^3*Sqrt[d + e*x])/e^8 + (2*(c*d^2 + a*e^2)^ 2*(7*B*c*d^2 - 6*A*c*d*e + a*B*e^2)*(d + e*x)^(3/2))/(3*e^8) - (6*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)^(5/2 ))/(5*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)^(7/2))/(7*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)^(9/2))/(9*e^8) + (6*c^2*(7* B*c*d^2 - 2*A*c*d*e + a*B*e^2)*(d + e*x)^(11/2))/(11*e^8) - (2*c^3*(7*B*d - A*e)*(d + e*x)^(13/2))/(13*e^8) + (2*B*c^3*(d + e*x)^(15/2))/(15*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 = 2.18 (sec) , antiderivative size = 301, normalized size of antiderivative = 0.86
| method | result | size |
| pseudoelliptic | \(\frac {2 \left (\left (\left (\frac {1}{15} B \,x^{7}+\frac {1}{13} A \,x^{6}\right ) c^{3}+\frac {\left (\frac {9 B x}{11}+A \right ) x^{4} a \,c^{2}}{3}+\frac {3 x^{2} a^{2} \left (\frac {5 B x}{7}+A \right ) c}{5}+a^{3} \left (\frac {B x}{3}+A \right )\right ) e^{7}-\frac {4 d \left (\left (\frac {7}{78} B \,x^{6}+\frac {15}{143} A \,x^{5}\right ) c^{3}+\frac {10 \left (\frac {35 B x}{44}+A \right ) x^{3} a \,c^{2}}{21}+a^{2} x \left (\frac {9 B x}{14}+A \right ) c +\frac {5 B \,a^{3}}{6}\right ) e^{6}}{5}+\frac {8 c \,d^{2} \left (\frac {25 \left (\frac {21 B x}{25}+A \right ) x^{4} c^{2}}{429}+\frac {2 x^{2} \left (\frac {25 B x}{33}+A \right ) a c}{7}+a^{2} \left (\frac {3 B x}{7}+A \right )\right ) e^{5}}{5}-\frac {64 c \,d^{3} \left (\frac {25 x^{3} \left (\frac {49 B x}{60}+A \right ) c^{2}}{143}+a x \left (\frac {15 B x}{22}+A \right ) c +\frac {9 a^{2} B}{4}\right ) e^{4}}{105}+\frac {128 c^{2} \left (\frac {15 \left (\frac {7 B x}{9}+A \right ) x^{2} c}{143}+a \left (\frac {5 B x}{11}+A \right )\right ) d^{4} e^{3}}{105}-\frac {512 c^{2} d^{5} \left (x \left (\frac {7 B x}{10}+A \right ) c +\frac {13 B a}{2}\right ) e^{2}}{3003}+\frac {1024 c^{3} d^{6} \left (\frac {7 B x}{15}+A \right ) e}{3003}-\frac {2048 B \,c^{3} d^{7}}{6435}\right ) \sqrt {e x +d}}{e^{8}}\) | \(301\) |
| gosper | \(\frac {2 \sqrt {e x +d}\, \left (3003 B \,x^{7} c^{3} e^{7}+3465 A \,x^{6} c^{3} e^{7}-3234 B \,x^{6} c^{3} d \,e^{6}-3780 A \,x^{5} c^{3} d \,e^{6}+12285 B \,x^{5} a \,c^{2} e^{7}+3528 B \,x^{5} c^{3} d^{2} e^{5}+15015 A \,x^{4} a \,c^{2} e^{7}+4200 A \,x^{4} c^{3} d^{2} e^{5}-13650 B \,x^{4} a \,c^{2} d \,e^{6}-3920 B \,x^{4} c^{3} d^{3} e^{4}-17160 A \,x^{3} a \,c^{2} d \,e^{6}-4800 A \,x^{3} c^{3} d^{3} e^{4}+19305 B \,x^{3} a^{2} c \,e^{7}+15600 B \,x^{3} a \,c^{2} d^{2} e^{5}+4480 B \,x^{3} c^{3} d^{4} e^{3}+27027 A \,x^{2} a^{2} c \,e^{7}+20592 A \,x^{2} a \,c^{2} d^{2} e^{5}+5760 A \,x^{2} c^{3} d^{4} e^{3}-23166 B \,x^{2} a^{2} c d \,e^{6}-18720 B \,x^{2} a \,c^{2} d^{3} e^{4}-5376 B \,x^{2} c^{3} d^{5} e^{2}-36036 A x \,a^{2} c d \,e^{6}-27456 A x a \,c^{2} d^{3} e^{4}-7680 A x \,c^{3} d^{5} e^{2}+15015 B x \,a^{3} e^{7}+30888 B x \,a^{2} c \,d^{2} e^{5}+24960 B x a \,c^{2} d^{4} e^{3}+7168 B x \,c^{3} d^{6} e +45045 A \,a^{3} e^{7}+72072 A \,a^{2} c \,d^{2} e^{5}+54912 A a \,c^{2} d^{4} e^{3}+15360 A \,c^{3} d^{6} e -30030 B \,a^{3} d \,e^{6}-61776 B \,a^{2} c \,d^{3} e^{4}-49920 B a \,c^{2} d^{5} e^{2}-14336 B \,c^{3} d^{7}\right )}{45045 e^{8}}\) | \(489\) |
| trager | \(\frac {2 \sqrt {e x +d}\, \left (3003 B \,x^{7} c^{3} e^{7}+3465 A \,x^{6} c^{3} e^{7}-3234 B \,x^{6} c^{3} d \,e^{6}-3780 A \,x^{5} c^{3} d \,e^{6}+12285 B \,x^{5} a \,c^{2} e^{7}+3528 B \,x^{5} c^{3} d^{2} e^{5}+15015 A \,x^{4} a \,c^{2} e^{7}+4200 A \,x^{4} c^{3} d^{2} e^{5}-13650 B \,x^{4} a \,c^{2} d \,e^{6}-3920 B \,x^{4} c^{3} d^{3} e^{4}-17160 A \,x^{3} a \,c^{2} d \,e^{6}-4800 A \,x^{3} c^{3} d^{3} e^{4}+19305 B \,x^{3} a^{2} c \,e^{7}+15600 B \,x^{3} a \,c^{2} d^{2} e^{5}+4480 B \,x^{3} c^{3} d^{4} e^{3}+27027 A \,x^{2} a^{2} c \,e^{7}+20592 A \,x^{2} a \,c^{2} d^{2} e^{5}+5760 A \,x^{2} c^{3} d^{4} e^{3}-23166 B \,x^{2} a^{2} c d \,e^{6}-18720 B \,x^{2} a \,c^{2} d^{3} e^{4}-5376 B \,x^{2} c^{3} d^{5} e^{2}-36036 A x \,a^{2} c d \,e^{6}-27456 A x a \,c^{2} d^{3} e^{4}-7680 A x \,c^{3} d^{5} e^{2}+15015 B x \,a^{3} e^{7}+30888 B x \,a^{2} c \,d^{2} e^{5}+24960 B x a \,c^{2} d^{4} e^{3}+7168 B x \,c^{3} d^{6} e +45045 A \,a^{3} e^{7}+72072 A \,a^{2} c \,d^{2} e^{5}+54912 A a \,c^{2} d^{4} e^{3}+15360 A \,c^{3} d^{6} e -30030 B \,a^{3} d \,e^{6}-61776 B \,a^{2} c \,d^{3} e^{4}-49920 B a \,c^{2} d^{5} e^{2}-14336 B \,c^{3} d^{7}\right )}{45045 e^{8}}\) | \(489\) |
| risch | \(\frac {2 \sqrt {e x +d}\, \left (3003 B \,x^{7} c^{3} e^{7}+3465 A \,x^{6} c^{3} e^{7}-3234 B \,x^{6} c^{3} d \,e^{6}-3780 A \,x^{5} c^{3} d \,e^{6}+12285 B \,x^{5} a \,c^{2} e^{7}+3528 B \,x^{5} c^{3} d^{2} e^{5}+15015 A \,x^{4} a \,c^{2} e^{7}+4200 A \,x^{4} c^{3} d^{2} e^{5}-13650 B \,x^{4} a \,c^{2} d \,e^{6}-3920 B \,x^{4} c^{3} d^{3} e^{4}-17160 A \,x^{3} a \,c^{2} d \,e^{6}-4800 A \,x^{3} c^{3} d^{3} e^{4}+19305 B \,x^{3} a^{2} c \,e^{7}+15600 B \,x^{3} a \,c^{2} d^{2} e^{5}+4480 B \,x^{3} c^{3} d^{4} e^{3}+27027 A \,x^{2} a^{2} c \,e^{7}+20592 A \,x^{2} a \,c^{2} d^{2} e^{5}+5760 A \,x^{2} c^{3} d^{4} e^{3}-23166 B \,x^{2} a^{2} c d \,e^{6}-18720 B \,x^{2} a \,c^{2} d^{3} e^{4}-5376 B \,x^{2} c^{3} d^{5} e^{2}-36036 A x \,a^{2} c d \,e^{6}-27456 A x a \,c^{2} d^{3} e^{4}-7680 A x \,c^{3} d^{5} e^{2}+15015 B x \,a^{3} e^{7}+30888 B x \,a^{2} c \,d^{2} e^{5}+24960 B x a \,c^{2} d^{4} e^{3}+7168 B x \,c^{3} d^{6} e +45045 A \,a^{3} e^{7}+72072 A \,a^{2} c \,d^{2} e^{5}+54912 A a \,c^{2} d^{4} e^{3}+15360 A \,c^{3} d^{6} e -30030 B \,a^{3} d \,e^{6}-61776 B \,a^{2} c \,d^{3} e^{4}-49920 B a \,c^{2} d^{5} e^{2}-14336 B \,c^{3} d^{7}\right )}{45045 e^{8}}\) | \(489\) |
| orering | \(\frac {2 \sqrt {e x +d}\, \left (3003 B \,x^{7} c^{3} e^{7}+3465 A \,x^{6} c^{3} e^{7}-3234 B \,x^{6} c^{3} d \,e^{6}-3780 A \,x^{5} c^{3} d \,e^{6}+12285 B \,x^{5} a \,c^{2} e^{7}+3528 B \,x^{5} c^{3} d^{2} e^{5}+15015 A \,x^{4} a \,c^{2} e^{7}+4200 A \,x^{4} c^{3} d^{2} e^{5}-13650 B \,x^{4} a \,c^{2} d \,e^{6}-3920 B \,x^{4} c^{3} d^{3} e^{4}-17160 A \,x^{3} a \,c^{2} d \,e^{6}-4800 A \,x^{3} c^{3} d^{3} e^{4}+19305 B \,x^{3} a^{2} c \,e^{7}+15600 B \,x^{3} a \,c^{2} d^{2} e^{5}+4480 B \,x^{3} c^{3} d^{4} e^{3}+27027 A \,x^{2} a^{2} c \,e^{7}+20592 A \,x^{2} a \,c^{2} d^{2} e^{5}+5760 A \,x^{2} c^{3} d^{4} e^{3}-23166 B \,x^{2} a^{2} c d \,e^{6}-18720 B \,x^{2} a \,c^{2} d^{3} e^{4}-5376 B \,x^{2} c^{3} d^{5} e^{2}-36036 A x \,a^{2} c d \,e^{6}-27456 A x a \,c^{2} d^{3} e^{4}-7680 A x \,c^{3} d^{5} e^{2}+15015 B x \,a^{3} e^{7}+30888 B x \,a^{2} c \,d^{2} e^{5}+24960 B x a \,c^{2} d^{4} e^{3}+7168 B x \,c^{3} d^{6} e +45045 A \,a^{3} e^{7}+72072 A \,a^{2} c \,d^{2} e^{5}+54912 A a \,c^{2} d^{4} e^{3}+15360 A \,c^{3} d^{6} e -30030 B \,a^{3} d \,e^{6}-61776 B \,a^{2} c \,d^{3} e^{4}-49920 B a \,c^{2} d^{5} e^{2}-14336 B \,c^{3} d^{7}\right )}{45045 e^{8}}\) | \(489\) |
| derivativedivides | \(\frac {\frac {2 B \,c^{3} \left (e x +d \right )^{\frac {15}{2}}}{15}+\frac {2 \left (\left (A e -B d \right ) c^{3}-6 B \,c^{3} d \right ) \left (e x +d \right )^{\frac {13}{2}}}{13}+\frac {2 \left (-6 \left (A e -B d \right ) c^{3} d +B \left (\left (a \,e^{2}+c \,d^{2}\right ) c^{2}+8 c^{3} d^{2}+c \left (2 \left (a \,e^{2}+c \,d^{2}\right ) c +4 c^{2} d^{2}\right )\right )\right ) \left (e x +d \right )^{\frac {11}{2}}}{11}+\frac {2 \left (\left (A e -B d \right ) \left (\left (a \,e^{2}+c \,d^{2}\right ) c^{2}+8 c^{3} d^{2}+c \left (2 \left (a \,e^{2}+c \,d^{2}\right ) c +4 c^{2} d^{2}\right )\right )+B \left (-8 \left (a \,e^{2}+c \,d^{2}\right ) c^{2} d -2 c d \left (2 \left (a \,e^{2}+c \,d^{2}\right ) c +4 c^{2} d^{2}\right )\right )\right ) \left (e x +d \right )^{\frac {9}{2}}}{9}+\frac {2 \left (\left (A e -B d \right ) \left (-8 \left (a \,e^{2}+c \,d^{2}\right ) c^{2} d -2 c d \left (2 \left (a \,e^{2}+c \,d^{2}\right ) c +4 c^{2} d^{2}\right )\right )+B \left (\left (a \,e^{2}+c \,d^{2}\right ) \left (2 \left (a \,e^{2}+c \,d^{2}\right ) c +4 c^{2} d^{2}\right )+8 c^{2} d^{2} \left (a \,e^{2}+c \,d^{2}\right )+c \left (a \,e^{2}+c \,d^{2}\right )^{2}\right )\right ) \left (e x +d \right )^{\frac {7}{2}}}{7}+\frac {2 \left (\left (A e -B d \right ) \left (\left (a \,e^{2}+c \,d^{2}\right ) \left (2 \left (a \,e^{2}+c \,d^{2}\right ) c +4 c^{2} d^{2}\right )+8 c^{2} d^{2} \left (a \,e^{2}+c \,d^{2}\right )+c \left (a \,e^{2}+c \,d^{2}\right )^{2}\right )-6 B \left (a \,e^{2}+c \,d^{2}\right )^{2} c d \right ) \left (e x +d \right )^{\frac {5}{2}}}{5}+\frac {2 \left (-6 \left (A e -B d \right ) \left (a \,e^{2}+c \,d^{2}\right )^{2} c d +B \left (a \,e^{2}+c \,d^{2}\right )^{3}\right ) \left (e x +d \right )^{\frac {3}{2}}}{3}+2 \left (A e -B d \right ) \left (a \,e^{2}+c \,d^{2}\right )^{3} \sqrt {e x +d}}{e^{8}}\) | \(561\) |
| default | \(\frac {\frac {2 B \,c^{3} \left (e x +d \right )^{\frac {15}{2}}}{15}+\frac {2 \left (\left (A e -B d \right ) c^{3}-6 B \,c^{3} d \right ) \left (e x +d \right )^{\frac {13}{2}}}{13}+\frac {2 \left (-6 \left (A e -B d \right ) c^{3} d +B \left (\left (a \,e^{2}+c \,d^{2}\right ) c^{2}+8 c^{3} d^{2}+c \left (2 \left (a \,e^{2}+c \,d^{2}\right ) c +4 c^{2} d^{2}\right )\right )\right ) \left (e x +d \right )^{\frac {11}{2}}}{11}+\frac {2 \left (\left (A e -B d \right ) \left (\left (a \,e^{2}+c \,d^{2}\right ) c^{2}+8 c^{3} d^{2}+c \left (2 \left (a \,e^{2}+c \,d^{2}\right ) c +4 c^{2} d^{2}\right )\right )+B \left (-8 \left (a \,e^{2}+c \,d^{2}\right ) c^{2} d -2 c d \left (2 \left (a \,e^{2}+c \,d^{2}\right ) c +4 c^{2} d^{2}\right )\right )\right ) \left (e x +d \right )^{\frac {9}{2}}}{9}+\frac {2 \left (\left (A e -B d \right ) \left (-8 \left (a \,e^{2}+c \,d^{2}\right ) c^{2} d -2 c d \left (2 \left (a \,e^{2}+c \,d^{2}\right ) c +4 c^{2} d^{2}\right )\right )+B \left (\left (a \,e^{2}+c \,d^{2}\right ) \left (2 \left (a \,e^{2}+c \,d^{2}\right ) c +4 c^{2} d^{2}\right )+8 c^{2} d^{2} \left (a \,e^{2}+c \,d^{2}\right )+c \left (a \,e^{2}+c \,d^{2}\right )^{2}\right )\right ) \left (e x +d \right )^{\frac {7}{2}}}{7}+\frac {2 \left (\left (A e -B d \right ) \left (\left (a \,e^{2}+c \,d^{2}\right ) \left (2 \left (a \,e^{2}+c \,d^{2}\right ) c +4 c^{2} d^{2}\right )+8 c^{2} d^{2} \left (a \,e^{2}+c \,d^{2}\right )+c \left (a \,e^{2}+c \,d^{2}\right )^{2}\right )-6 B \left (a \,e^{2}+c \,d^{2}\right )^{2} c d \right ) \left (e x +d \right )^{\frac {5}{2}}}{5}+\frac {2 \left (-6 \left (A e -B d \right ) \left (a \,e^{2}+c \,d^{2}\right )^{2} c d +B \left (a \,e^{2}+c \,d^{2}\right )^{3}\right ) \left (e x +d \right )^{\frac {3}{2}}}{3}+2 \left (A e -B d \right ) \left (a \,e^{2}+c \,d^{2}\right )^{3} \sqrt {e x +d}}{e^{8}}\) | \(561\) |
Input:
int((B*x+A)*(c*x^2+a)^3/(e*x+d)^(1/2),x,method=_RETURNVERBOSE)
Output:
2*(((1/15*B*x^7+1/13*A*x^6)*c^3+1/3*(9/11*B*x+A)*x^4*a*c^2+3/5*x^2*a^2*(5/ 7*B*x+A)*c+a^3*(1/3*B*x+A))*e^7-4/5*d*((7/78*B*x^6+15/143*A*x^5)*c^3+10/21 *(35/44*B*x+A)*x^3*a*c^2+a^2*x*(9/14*B*x+A)*c+5/6*B*a^3)*e^6+8/5*c*d^2*(25 /429*(21/25*B*x+A)*x^4*c^2+2/7*x^2*(25/33*B*x+A)*a*c+a^2*(3/7*B*x+A))*e^5- 64/105*c*d^3*(25/143*x^3*(49/60*B*x+A)*c^2+a*x*(15/22*B*x+A)*c+9/4*a^2*B)* e^4+128/105*c^2*(15/143*(7/9*B*x+A)*x^2*c+a*(5/11*B*x+A))*d^4*e^3-512/3003 *c^2*d^5*(x*(7/10*B*x+A)*c+13/2*B*a)*e^2+1024/3003*c^3*d^6*(7/15*B*x+A)*e- 2048/6435*B*c^3*d^7)*(e*x+d)^(1/2)/e^8
Time = 0.08 (sec) , antiderivative size = 454, normalized size of antiderivative = 1.30 \[ \int \frac {(A+B x) \left (a+c x^2\right )^3}{\sqrt {d+e x}} \, dx=\frac {2 \, {\left (3003 \, B c^{3} e^{7} x^{7} - 14336 \, B c^{3} d^{7} + 15360 \, A c^{3} d^{6} e - 49920 \, B a c^{2} d^{5} e^{2} + 54912 \, A a c^{2} d^{4} e^{3} - 61776 \, B a^{2} c d^{3} e^{4} + 72072 \, A a^{2} c d^{2} e^{5} - 30030 \, B a^{3} d e^{6} + 45045 \, A a^{3} e^{7} - 231 \, {\left (14 \, B c^{3} d e^{6} - 15 \, A c^{3} e^{7}\right )} x^{6} + 63 \, {\left (56 \, B c^{3} d^{2} e^{5} - 60 \, A c^{3} d e^{6} + 195 \, B a c^{2} e^{7}\right )} x^{5} - 35 \, {\left (112 \, B c^{3} d^{3} e^{4} - 120 \, A c^{3} d^{2} e^{5} + 390 \, B a c^{2} d e^{6} - 429 \, A a c^{2} e^{7}\right )} x^{4} + 5 \, {\left (896 \, B c^{3} d^{4} e^{3} - 960 \, A c^{3} d^{3} e^{4} + 3120 \, B a c^{2} d^{2} e^{5} - 3432 \, A a c^{2} d e^{6} + 3861 \, B a^{2} c e^{7}\right )} x^{3} - 3 \, {\left (1792 \, B c^{3} d^{5} e^{2} - 1920 \, A c^{3} d^{4} e^{3} + 6240 \, B a c^{2} d^{3} e^{4} - 6864 \, A a c^{2} d^{2} e^{5} + 7722 \, B a^{2} c d e^{6} - 9009 \, A a^{2} c e^{7}\right )} x^{2} + {\left (7168 \, B c^{3} d^{6} e - 7680 \, A c^{3} d^{5} e^{2} + 24960 \, B a c^{2} d^{4} e^{3} - 27456 \, A a c^{2} d^{3} e^{4} + 30888 \, B a^{2} c d^{2} e^{5} - 36036 \, A a^{2} c d e^{6} + 15015 \, B a^{3} e^{7}\right )} x\right )} \sqrt {e x + d}}{45045 \, e^{8}} \] Input:
integrate((B*x+A)*(c*x^2+a)^3/(e*x+d)^(1/2),x, algorithm="fricas")
Output:
2/45045*(3003*B*c^3*e^7*x^7 - 14336*B*c^3*d^7 + 15360*A*c^3*d^6*e - 49920* B*a*c^2*d^5*e^2 + 54912*A*a*c^2*d^4*e^3 - 61776*B*a^2*c*d^3*e^4 + 72072*A* a^2*c*d^2*e^5 - 30030*B*a^3*d*e^6 + 45045*A*a^3*e^7 - 231*(14*B*c^3*d*e^6 - 15*A*c^3*e^7)*x^6 + 63*(56*B*c^3*d^2*e^5 - 60*A*c^3*d*e^6 + 195*B*a*c^2* e^7)*x^5 - 35*(112*B*c^3*d^3*e^4 - 120*A*c^3*d^2*e^5 + 390*B*a*c^2*d*e^6 - 429*A*a*c^2*e^7)*x^4 + 5*(896*B*c^3*d^4*e^3 - 960*A*c^3*d^3*e^4 + 3120*B* a*c^2*d^2*e^5 - 3432*A*a*c^2*d*e^6 + 3861*B*a^2*c*e^7)*x^3 - 3*(1792*B*c^3 *d^5*e^2 - 1920*A*c^3*d^4*e^3 + 6240*B*a*c^2*d^3*e^4 - 6864*A*a*c^2*d^2*e^ 5 + 7722*B*a^2*c*d*e^6 - 9009*A*a^2*c*e^7)*x^2 + (7168*B*c^3*d^6*e - 7680* A*c^3*d^5*e^2 + 24960*B*a*c^2*d^4*e^3 - 27456*A*a*c^2*d^3*e^4 + 30888*B*a^ 2*c*d^2*e^5 - 36036*A*a^2*c*d*e^6 + 15015*B*a^3*e^7)*x)*sqrt(e*x + d)/e^8
Time = 1.20 (sec) , antiderivative size = 643, normalized size of antiderivative = 1.85 \[ \int \frac {(A+B x) \left (a+c x^2\right )^3}{\sqrt {d+e x}} \, dx=\begin {cases} \frac {2 \left (\frac {B c^{3} \left (d + e x\right )^{\frac {15}{2}}}{15 e^{7}} + \frac {\left (d + e x\right )^{\frac {13}{2}} \left (A c^{3} e - 7 B c^{3} d\right )}{13 e^{7}} + \frac {\left (d + e x\right )^{\frac {11}{2}} \left (- 6 A c^{3} d e + 3 B a c^{2} e^{2} + 21 B c^{3} d^{2}\right )}{11 e^{7}} + \frac {\left (d + e x\right )^{\frac {9}{2}} \cdot \left (3 A a c^{2} e^{3} + 15 A c^{3} d^{2} e - 15 B a c^{2} d e^{2} - 35 B c^{3} d^{3}\right )}{9 e^{7}} + \frac {\left (d + e x\right )^{\frac {7}{2}} \left (- 12 A a c^{2} d e^{3} - 20 A c^{3} d^{3} e + 3 B a^{2} c e^{4} + 30 B a c^{2} d^{2} e^{2} + 35 B c^{3} d^{4}\right )}{7 e^{7}} + \frac {\left (d + e x\right )^{\frac {5}{2}} \cdot \left (3 A a^{2} c e^{5} + 18 A a c^{2} d^{2} e^{3} + 15 A c^{3} d^{4} e - 9 B a^{2} c d e^{4} - 30 B a c^{2} d^{3} e^{2} - 21 B c^{3} d^{5}\right )}{5 e^{7}} + \frac {\left (d + e x\right )^{\frac {3}{2}} \left (- 6 A a^{2} c d e^{5} - 12 A a c^{2} d^{3} e^{3} - 6 A c^{3} d^{5} e + B a^{3} e^{6} + 9 B a^{2} c d^{2} e^{4} + 15 B a c^{2} d^{4} e^{2} + 7 B c^{3} d^{6}\right )}{3 e^{7}} + \frac {\sqrt {d + e x} \left (A a^{3} e^{7} + 3 A a^{2} c d^{2} e^{5} + 3 A a c^{2} d^{4} e^{3} + A c^{3} d^{6} e - B a^{3} d e^{6} - 3 B a^{2} c d^{3} e^{4} - 3 B a c^{2} d^{5} e^{2} - B c^{3} d^{7}\right )}{e^{7}}\right )}{e} & \text {for}\: e \neq 0 \\\frac {A a^{3} x + A a^{2} c x^{3} + \frac {3 A a c^{2} x^{5}}{5} + \frac {A c^{3} x^{7}}{7} + \frac {B a^{3} x^{2}}{2} + \frac {3 B a^{2} c x^{4}}{4} + \frac {B a c^{2} x^{6}}{2} + \frac {B c^{3} x^{8}}{8}}{\sqrt {d}} & \text {otherwise} \end {cases} \] Input:
integrate((B*x+A)*(c*x**2+a)**3/(e*x+d)**(1/2),x)
Output:
Piecewise((2*(B*c**3*(d + e*x)**(15/2)/(15*e**7) + (d + e*x)**(13/2)*(A*c* *3*e - 7*B*c**3*d)/(13*e**7) + (d + e*x)**(11/2)*(-6*A*c**3*d*e + 3*B*a*c* *2*e**2 + 21*B*c**3*d**2)/(11*e**7) + (d + e*x)**(9/2)*(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)/(9*e**7) + (d + e* x)**(7/2)*(-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)/(7*e**7) + (d + e*x)**(5/2)*(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)/(5*e**7) + (d + e*x)**(3/2)*(-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)/(3*e**7) + sqrt( d + e*x)*(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)/e**7)/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)/sqrt(d), True))
Time = 0.04 (sec) , antiderivative size = 453, normalized size of antiderivative = 1.30 \[ \int \frac {(A+B x) \left (a+c x^2\right )^3}{\sqrt {d+e x}} \, dx=\frac {2 \, {\left (3003 \, {\left (e x + d\right )}^{\frac {15}{2}} B c^{3} - 3465 \, {\left (7 \, B c^{3} d - A c^{3} e\right )} {\left (e x + d\right )}^{\frac {13}{2}} + 12285 \, {\left (7 \, B c^{3} d^{2} - 2 \, A c^{3} d e + B a c^{2} e^{2}\right )} {\left (e x + d\right )}^{\frac {11}{2}} - 5005 \, {\left (35 \, B c^{3} d^{3} - 15 \, A c^{3} d^{2} e + 15 \, B a c^{2} d e^{2} - 3 \, A a c^{2} e^{3}\right )} {\left (e x + d\right )}^{\frac {9}{2}} + 6435 \, {\left (35 \, B c^{3} d^{4} - 20 \, A c^{3} d^{3} e + 30 \, B a c^{2} d^{2} e^{2} - 12 \, A a c^{2} d e^{3} + 3 \, B a^{2} c e^{4}\right )} {\left (e x + d\right )}^{\frac {7}{2}} - 27027 \, {\left (7 \, B c^{3} d^{5} - 5 \, A c^{3} d^{4} e + 10 \, B a c^{2} d^{3} e^{2} - 6 \, A a c^{2} d^{2} e^{3} + 3 \, B a^{2} c d e^{4} - A a^{2} c e^{5}\right )} {\left (e x + d\right )}^{\frac {5}{2}} + 15015 \, {\left (7 \, B c^{3} d^{6} - 6 \, A c^{3} d^{5} e + 15 \, B a c^{2} d^{4} e^{2} - 12 \, A a c^{2} d^{3} e^{3} + 9 \, B a^{2} c d^{2} e^{4} - 6 \, A a^{2} c d e^{5} + B a^{3} e^{6}\right )} {\left (e x + d\right )}^{\frac {3}{2}} - 45045 \, {\left (B c^{3} d^{7} - A c^{3} d^{6} e + 3 \, B a c^{2} d^{5} e^{2} - 3 \, A a c^{2} d^{4} e^{3} + 3 \, B a^{2} c d^{3} e^{4} - 3 \, A a^{2} c d^{2} e^{5} + B a^{3} d e^{6} - A a^{3} e^{7}\right )} \sqrt {e x + d}\right )}}{45045 \, e^{8}} \] Input:
integrate((B*x+A)*(c*x^2+a)^3/(e*x+d)^(1/2),x, algorithm="maxima")
Output:
2/45045*(3003*(e*x + d)^(15/2)*B*c^3 - 3465*(7*B*c^3*d - A*c^3*e)*(e*x + d )^(13/2) + 12285*(7*B*c^3*d^2 - 2*A*c^3*d*e + B*a*c^2*e^2)*(e*x + d)^(11/2 ) - 5005*(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)^(9/2) + 6435*(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)^(7/2) - 27027*(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)^(5/2) + 15015*(7*B*c^3*d^6 - 6*A*c^3*d^5*e + 1 5*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)*(e*x + d)^(3/2) - 45045*(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
Time = 0.16 (sec) , antiderivative size = 476, normalized size of antiderivative = 1.37 \[ \int \frac {(A+B x) \left (a+c x^2\right )^3}{\sqrt {d+e x}} \, dx=\frac {2 \, {\left (45045 \, \sqrt {e x + d} A a^{3} + \frac {15015 \, {\left ({\left (e x + d\right )}^{\frac {3}{2}} - 3 \, \sqrt {e x + d} d\right )} B a^{3}}{e} + \frac {9009 \, {\left (3 \, {\left (e x + d\right )}^{\frac {5}{2}} - 10 \, {\left (e x + d\right )}^{\frac {3}{2}} d + 15 \, \sqrt {e x + d} d^{2}\right )} A a^{2} c}{e^{2}} + \frac {3861 \, {\left (5 \, {\left (e x + d\right )}^{\frac {7}{2}} - 21 \, {\left (e x + d\right )}^{\frac {5}{2}} d + 35 \, {\left (e x + d\right )}^{\frac {3}{2}} d^{2} - 35 \, \sqrt {e x + d} d^{3}\right )} B a^{2} c}{e^{3}} + \frac {429 \, {\left (35 \, {\left (e x + d\right )}^{\frac {9}{2}} - 180 \, {\left (e x + d\right )}^{\frac {7}{2}} d + 378 \, {\left (e x + d\right )}^{\frac {5}{2}} d^{2} - 420 \, {\left (e x + d\right )}^{\frac {3}{2}} d^{3} + 315 \, \sqrt {e x + d} d^{4}\right )} A a c^{2}}{e^{4}} + \frac {195 \, {\left (63 \, {\left (e x + d\right )}^{\frac {11}{2}} - 385 \, {\left (e x + d\right )}^{\frac {9}{2}} d + 990 \, {\left (e x + d\right )}^{\frac {7}{2}} d^{2} - 1386 \, {\left (e x + d\right )}^{\frac {5}{2}} d^{3} + 1155 \, {\left (e x + d\right )}^{\frac {3}{2}} d^{4} - 693 \, \sqrt {e x + d} d^{5}\right )} B a c^{2}}{e^{5}} + \frac {15 \, {\left (231 \, {\left (e x + d\right )}^{\frac {13}{2}} - 1638 \, {\left (e x + d\right )}^{\frac {11}{2}} d + 5005 \, {\left (e x + d\right )}^{\frac {9}{2}} d^{2} - 8580 \, {\left (e x + d\right )}^{\frac {7}{2}} d^{3} + 9009 \, {\left (e x + d\right )}^{\frac {5}{2}} d^{4} - 6006 \, {\left (e x + d\right )}^{\frac {3}{2}} d^{5} + 3003 \, \sqrt {e x + d} d^{6}\right )} A c^{3}}{e^{6}} + \frac {7 \, {\left (429 \, {\left (e x + d\right )}^{\frac {15}{2}} - 3465 \, {\left (e x + d\right )}^{\frac {13}{2}} d + 12285 \, {\left (e x + d\right )}^{\frac {11}{2}} d^{2} - 25025 \, {\left (e x + d\right )}^{\frac {9}{2}} d^{3} + 32175 \, {\left (e x + d\right )}^{\frac {7}{2}} d^{4} - 27027 \, {\left (e x + d\right )}^{\frac {5}{2}} d^{5} + 15015 \, {\left (e x + d\right )}^{\frac {3}{2}} d^{6} - 6435 \, \sqrt {e x + d} d^{7}\right )} B c^{3}}{e^{7}}\right )}}{45045 \, e} \] Input:
integrate((B*x+A)*(c*x^2+a)^3/(e*x+d)^(1/2),x, algorithm="giac")
Output:
2/45045*(45045*sqrt(e*x + d)*A*a^3 + 15015*((e*x + d)^(3/2) - 3*sqrt(e*x + d)*d)*B*a^3/e + 9009*(3*(e*x + d)^(5/2) - 10*(e*x + d)^(3/2)*d + 15*sqrt( e*x + d)*d^2)*A*a^2*c/e^2 + 3861*(5*(e*x + d)^(7/2) - 21*(e*x + d)^(5/2)*d + 35*(e*x + d)^(3/2)*d^2 - 35*sqrt(e*x + d)*d^3)*B*a^2*c/e^3 + 429*(35*(e *x + d)^(9/2) - 180*(e*x + d)^(7/2)*d + 378*(e*x + d)^(5/2)*d^2 - 420*(e*x + d)^(3/2)*d^3 + 315*sqrt(e*x + d)*d^4)*A*a*c^2/e^4 + 195*(63*(e*x + d)^( 11/2) - 385*(e*x + d)^(9/2)*d + 990*(e*x + d)^(7/2)*d^2 - 1386*(e*x + d)^( 5/2)*d^3 + 1155*(e*x + d)^(3/2)*d^4 - 693*sqrt(e*x + d)*d^5)*B*a*c^2/e^5 + 15*(231*(e*x + d)^(13/2) - 1638*(e*x + d)^(11/2)*d + 5005*(e*x + d)^(9/2) *d^2 - 8580*(e*x + d)^(7/2)*d^3 + 9009*(e*x + d)^(5/2)*d^4 - 6006*(e*x + d )^(3/2)*d^5 + 3003*sqrt(e*x + d)*d^6)*A*c^3/e^6 + 7*(429*(e*x + d)^(15/2) - 3465*(e*x + d)^(13/2)*d + 12285*(e*x + d)^(11/2)*d^2 - 25025*(e*x + d)^( 9/2)*d^3 + 32175*(e*x + d)^(7/2)*d^4 - 27027*(e*x + d)^(5/2)*d^5 + 15015*( e*x + d)^(3/2)*d^6 - 6435*sqrt(e*x + d)*d^7)*B*c^3/e^7)/e
Time = 0.08 (sec) , antiderivative size = 324, normalized size of antiderivative = 0.93 \[ \int \frac {(A+B x) \left (a+c x^2\right )^3}{\sqrt {d+e x}} \, dx=\frac {{\left (d+e\,x\right )}^{7/2}\,\left (6\,B\,a^2\,c\,e^4+60\,B\,a\,c^2\,d^2\,e^2-24\,A\,a\,c^2\,d\,e^3+70\,B\,c^3\,d^4-40\,A\,c^3\,d^3\,e\right )}{7\,e^8}+\frac {{\left (d+e\,x\right )}^{11/2}\,\left (42\,B\,c^3\,d^2-12\,A\,c^3\,d\,e+6\,B\,a\,c^2\,e^2\right )}{11\,e^8}+\frac {2\,{\left (c\,d^2+a\,e^2\right )}^2\,{\left (d+e\,x\right )}^{3/2}\,\left (7\,B\,c\,d^2-6\,A\,c\,d\,e+B\,a\,e^2\right )}{3\,e^8}+\frac {2\,B\,c^3\,{\left (d+e\,x\right )}^{15/2}}{15\,e^8}+\frac {2\,c^2\,{\left (d+e\,x\right )}^{9/2}\,\left (-35\,B\,c\,d^3+15\,A\,c\,d^2\,e-15\,B\,a\,d\,e^2+3\,A\,a\,e^3\right )}{9\,e^8}+\frac {2\,c^3\,\left (A\,e-7\,B\,d\right )\,{\left (d+e\,x\right )}^{13/2}}{13\,e^8}+\frac {2\,{\left (c\,d^2+a\,e^2\right )}^3\,\left (A\,e-B\,d\right )\,\sqrt {d+e\,x}}{e^8}+\frac {6\,c\,\left (c\,d^2+a\,e^2\right )\,{\left (d+e\,x\right )}^{5/2}\,\left (-7\,B\,c\,d^3+5\,A\,c\,d^2\,e-3\,B\,a\,d\,e^2+A\,a\,e^3\right )}{5\,e^8} \] Input:
int(((a + c*x^2)^3*(A + B*x))/(d + e*x)^(1/2),x)
Output:
((d + e*x)^(7/2)*(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))/(7*e^8) + ((d + e*x)^(11/2)*(42*B*c^3*d^2 - 12*A*c^3*d*e + 6*B*a*c^2*e^2))/(11*e^8) + (2*(a*e^2 + c*d^2)^2*(d + e*x) ^(3/2)*(B*a*e^2 + 7*B*c*d^2 - 6*A*c*d*e))/(3*e^8) + (2*B*c^3*(d + e*x)^(15 /2))/(15*e^8) + (2*c^2*(d + e*x)^(9/2)*(3*A*a*e^3 - 35*B*c*d^3 - 15*B*a*d* e^2 + 15*A*c*d^2*e))/(9*e^8) + (2*c^3*(A*e - 7*B*d)*(d + e*x)^(13/2))/(13* e^8) + (2*(a*e^2 + c*d^2)^3*(A*e - B*d)*(d + e*x)^(1/2))/e^8 + (6*c*(a*e^2 + c*d^2)*(d + e*x)^(5/2)*(A*a*e^3 - 7*B*c*d^3 - 3*B*a*d*e^2 + 5*A*c*d^2*e ))/(5*e^8)
Time = 0.22 (sec) , antiderivative size = 488, normalized size of antiderivative = 1.40 \[ \int \frac {(A+B x) \left (a+c x^2\right )^3}{\sqrt {d+e x}} \, dx=\frac {2 \sqrt {e x +d}\, \left (3003 b \,c^{3} e^{7} x^{7}+3465 a \,c^{3} e^{7} x^{6}-3234 b \,c^{3} d \,e^{6} x^{6}+12285 a b \,c^{2} e^{7} x^{5}-3780 a \,c^{3} d \,e^{6} x^{5}+3528 b \,c^{3} d^{2} e^{5} x^{5}+15015 a^{2} c^{2} e^{7} x^{4}-13650 a b \,c^{2} d \,e^{6} x^{4}+4200 a \,c^{3} d^{2} e^{5} x^{4}-3920 b \,c^{3} d^{3} e^{4} x^{4}+19305 a^{2} b c \,e^{7} x^{3}-17160 a^{2} c^{2} d \,e^{6} x^{3}+15600 a b \,c^{2} d^{2} e^{5} x^{3}-4800 a \,c^{3} d^{3} e^{4} x^{3}+4480 b \,c^{3} d^{4} e^{3} x^{3}+27027 a^{3} c \,e^{7} x^{2}-23166 a^{2} b c d \,e^{6} x^{2}+20592 a^{2} c^{2} d^{2} e^{5} x^{2}-18720 a b \,c^{2} d^{3} e^{4} x^{2}+5760 a \,c^{3} d^{4} e^{3} x^{2}-5376 b \,c^{3} d^{5} e^{2} x^{2}+15015 a^{3} b \,e^{7} x -36036 a^{3} c d \,e^{6} x +30888 a^{2} b c \,d^{2} e^{5} x -27456 a^{2} c^{2} d^{3} e^{4} x +24960 a b \,c^{2} d^{4} e^{3} x -7680 a \,c^{3} d^{5} e^{2} x +7168 b \,c^{3} d^{6} e x +45045 a^{4} e^{7}-30030 a^{3} b d \,e^{6}+72072 a^{3} c \,d^{2} e^{5}-61776 a^{2} b c \,d^{3} e^{4}+54912 a^{2} c^{2} d^{4} e^{3}-49920 a b \,c^{2} d^{5} e^{2}+15360 a \,c^{3} d^{6} e -14336 b \,c^{3} d^{7}\right )}{45045 e^{8}} \] Input:
int((B*x+A)*(c*x^2+a)^3/(e*x+d)^(1/2),x)
Output:
(2*sqrt(d + e*x)*(45045*a**4*e**7 - 30030*a**3*b*d*e**6 + 15015*a**3*b*e** 7*x + 72072*a**3*c*d**2*e**5 - 36036*a**3*c*d*e**6*x + 27027*a**3*c*e**7*x **2 - 61776*a**2*b*c*d**3*e**4 + 30888*a**2*b*c*d**2*e**5*x - 23166*a**2*b *c*d*e**6*x**2 + 19305*a**2*b*c*e**7*x**3 + 54912*a**2*c**2*d**4*e**3 - 27 456*a**2*c**2*d**3*e**4*x + 20592*a**2*c**2*d**2*e**5*x**2 - 17160*a**2*c* *2*d*e**6*x**3 + 15015*a**2*c**2*e**7*x**4 - 49920*a*b*c**2*d**5*e**2 + 24 960*a*b*c**2*d**4*e**3*x - 18720*a*b*c**2*d**3*e**4*x**2 + 15600*a*b*c**2* d**2*e**5*x**3 - 13650*a*b*c**2*d*e**6*x**4 + 12285*a*b*c**2*e**7*x**5 + 1 5360*a*c**3*d**6*e - 7680*a*c**3*d**5*e**2*x + 5760*a*c**3*d**4*e**3*x**2 - 4800*a*c**3*d**3*e**4*x**3 + 4200*a*c**3*d**2*e**5*x**4 - 3780*a*c**3*d* e**6*x**5 + 3465*a*c**3*e**7*x**6 - 14336*b*c**3*d**7 + 7168*b*c**3*d**6*e *x - 5376*b*c**3*d**5*e**2*x**2 + 4480*b*c**3*d**4*e**3*x**3 - 3920*b*c**3 *d**3*e**4*x**4 + 3528*b*c**3*d**2*e**5*x**5 - 3234*b*c**3*d*e**6*x**6 + 3 003*b*c**3*e**7*x**7))/(45045*e**8)