Integrand size = 24, antiderivative size = 216 \[ \int \frac {(A+B x) \left (a+c x^2\right )^2}{\sqrt {d+e x}} \, dx=-\frac {2 (B d-A e) \left (c d^2+a e^2\right )^2 \sqrt {d+e x}}{e^6}+\frac {2 \left (c d^2+a e^2\right ) \left (5 B c d^2-4 A c d e+a B e^2\right ) (d+e x)^{3/2}}{3 e^6}-\frac {4 c \left (5 B c d^3-3 A c d^2 e+3 a B d e^2-a A e^3\right ) (d+e x)^{5/2}}{5 e^6}+\frac {4 c \left (5 B c d^2-2 A c d e+a B e^2\right ) (d+e x)^{7/2}}{7 e^6}-\frac {2 c^2 (5 B d-A e) (d+e x)^{9/2}}{9 e^6}+\frac {2 B c^2 (d+e x)^{11/2}}{11 e^6} \]
2/3*(a*e^2+c*d^2)*(-4*A*c*d*e+B*a*e^2+5*B*c*d^2)*(e*x+d)^(3/2)/e^6-4/5*c*( -A*a*e^3-3*A*c*d^2*e+3*B*a*d*e^2+5*B*c*d^3)*(e*x+d)^(5/2)/e^6+4/7*c*(-2*A* c*d*e+B*a*e^2+5*B*c*d^2)*(e*x+d)^(7/2)/e^6-2/9*c^2*(-A*e+5*B*d)*(e*x+d)^(9 /2)/e^6+2/11*B*c^2*(e*x+d)^(11/2)/e^6-2*(-A*e+B*d)*(a*e^2+c*d^2)^2*(e*x+d) ^(1/2)/e^6
Time = 0.13 (sec) , antiderivative size = 213, normalized size of antiderivative = 0.99 \[ \int \frac {(A+B x) \left (a+c x^2\right )^2}{\sqrt {d+e x}} \, dx=\frac {2 \sqrt {d+e x} \left (11 A e \left (315 a^2 e^4+42 a c e^2 \left (8 d^2-4 d e x+3 e^2 x^2\right )+c^2 \left (128 d^4-64 d^3 e x+48 d^2 e^2 x^2-40 d e^3 x^3+35 e^4 x^4\right )\right )+B \left (1155 a^2 e^4 (-2 d+e x)+198 a c e^2 \left (-16 d^3+8 d^2 e x-6 d e^2 x^2+5 e^3 x^3\right )-5 c^2 \left (256 d^5-128 d^4 e x+96 d^3 e^2 x^2-80 d^2 e^3 x^3+70 d e^4 x^4-63 e^5 x^5\right )\right )\right )}{3465 e^6} \]
(2*Sqrt[d + e*x]*(11*A*e*(315*a^2*e^4 + 42*a*c*e^2*(8*d^2 - 4*d*e*x + 3*e^ 2*x^2) + c^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)) + B*(1155*a^2*e^4*(-2*d + e*x) + 198*a*c*e^2*(-16*d^3 + 8*d^2*e*x - 6*d*e^2*x^2 + 5*e^3*x^3) - 5*c^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))))/(3465*e^6)
Time = 0.34 (sec) , antiderivative size = 216, 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.
| \(\displaystyle \int \frac {\left (a+c x^2\right )^2 (A+B x)}{\sqrt {d+e x}} \, dx\) |
| \(\Big \downarrow \) 652 |
| \(\displaystyle \int \left (-\frac {2 c (d+e x)^{5/2} \left (-a B e^2+2 A c d e-5 B c d^2\right )}{e^5}+\frac {\sqrt {d+e x} \left (a e^2+c d^2\right ) \left (a B e^2-4 A c d e+5 B c d^2\right )}{e^5}+\frac {\left (a e^2+c d^2\right )^2 (A e-B d)}{e^5 \sqrt {d+e x}}+\frac {2 c (d+e x)^{3/2} \left (a A e^3-3 a B d e^2+3 A c d^2 e-5 B c d^3\right )}{e^5}+\frac {c^2 (d+e x)^{7/2} (A e-5 B d)}{e^5}+\frac {B c^2 (d+e x)^{9/2}}{e^5}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {4 c (d+e x)^{7/2} \left (a B e^2-2 A c d e+5 B c d^2\right )}{7 e^6}+\frac {2 (d+e x)^{3/2} \left (a e^2+c d^2\right ) \left (a B e^2-4 A c d e+5 B c d^2\right )}{3 e^6}-\frac {2 \sqrt {d+e x} \left (a e^2+c d^2\right )^2 (B d-A e)}{e^6}-\frac {4 c (d+e x)^{5/2} \left (-a A e^3+3 a B d e^2-3 A c d^2 e+5 B c d^3\right )}{5 e^6}-\frac {2 c^2 (d+e x)^{9/2} (5 B d-A e)}{9 e^6}+\frac {2 B c^2 (d+e x)^{11/2}}{11 e^6}\) |
(-2*(B*d - A*e)*(c*d^2 + a*e^2)^2*Sqrt[d + e*x])/e^6 + (2*(c*d^2 + a*e^2)* (5*B*c*d^2 - 4*A*c*d*e + a*B*e^2)*(d + e*x)^(3/2))/(3*e^6) - (4*c*(5*B*c*d ^3 - 3*A*c*d^2*e + 3*a*B*d*e^2 - a*A*e^3)*(d + e*x)^(5/2))/(5*e^6) + (4*c* (5*B*c*d^2 - 2*A*c*d*e + a*B*e^2)*(d + e*x)^(7/2))/(7*e^6) - (2*c^2*(5*B*d - A*e)*(d + e*x)^(9/2))/(9*e^6) + (2*B*c^2*(d + e*x)^(11/2))/(11*e^6)
3.15.36.3.1 Defintions of rubi rules used
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 = 0.34 (sec) , antiderivative size = 170, normalized size of antiderivative = 0.79
| method | result | size |
| pseudoelliptic | \(\frac {2 \left (\left (\frac {x^{4} \left (\frac {9 B x}{11}+A \right ) c^{2}}{9}+\frac {2 \left (\frac {5 B x}{7}+A \right ) x^{2} a c}{5}+a^{2} \left (\frac {B x}{3}+A \right )\right ) e^{5}-\frac {8 d \left (\frac {5 x^{3} \left (\frac {35 B x}{44}+A \right ) c^{2}}{21}+a x \left (\frac {9 B x}{14}+A \right ) c +\frac {5 B \,a^{2}}{4}\right ) e^{4}}{15}+\frac {16 c \,d^{2} \left (\frac {x^{2} \left (\frac {25 B x}{33}+A \right ) c}{7}+a \left (\frac {3 B x}{7}+A \right )\right ) e^{3}}{15}-\frac {64 c \left (x \left (\frac {15 B x}{22}+A \right ) c +\frac {9 B a}{2}\right ) d^{3} e^{2}}{315}+\frac {128 c^{2} d^{4} \left (\frac {5 B x}{11}+A \right ) e}{315}-\frac {256 B \,c^{2} d^{5}}{693}\right ) \sqrt {e x +d}}{e^{6}}\) | \(170\) |
| derivativedivides | \(\frac {\frac {2 B \,c^{2} \left (e x +d \right )^{\frac {11}{2}}}{11}+\frac {2 \left (\left (A e -B d \right ) c^{2}-4 B \,c^{2} d \right ) \left (e x +d \right )^{\frac {9}{2}}}{9}+\frac {2 \left (-4 \left (A e -B d \right ) c^{2} d +B \left (2 \left (e^{2} a +c \,d^{2}\right ) c +4 c^{2} d^{2}\right )\right ) \left (e x +d \right )^{\frac {7}{2}}}{7}+\frac {2 \left (\left (A e -B d \right ) \left (2 \left (e^{2} a +c \,d^{2}\right ) c +4 c^{2} d^{2}\right )-4 B \left (e^{2} a +c \,d^{2}\right ) c d \right ) \left (e x +d \right )^{\frac {5}{2}}}{5}+\frac {2 \left (-4 \left (A e -B d \right ) \left (e^{2} a +c \,d^{2}\right ) c d +B \left (e^{2} a +c \,d^{2}\right )^{2}\right ) \left (e x +d \right )^{\frac {3}{2}}}{3}+2 \left (A e -B d \right ) \left (e^{2} a +c \,d^{2}\right )^{2} \sqrt {e x +d}}{e^{6}}\) | \(233\) |
| default | \(\frac {\frac {2 B \,c^{2} \left (e x +d \right )^{\frac {11}{2}}}{11}+\frac {2 \left (\left (A e -B d \right ) c^{2}-4 B \,c^{2} d \right ) \left (e x +d \right )^{\frac {9}{2}}}{9}+\frac {2 \left (-4 \left (A e -B d \right ) c^{2} d +B \left (2 \left (e^{2} a +c \,d^{2}\right ) c +4 c^{2} d^{2}\right )\right ) \left (e x +d \right )^{\frac {7}{2}}}{7}+\frac {2 \left (\left (A e -B d \right ) \left (2 \left (e^{2} a +c \,d^{2}\right ) c +4 c^{2} d^{2}\right )-4 B \left (e^{2} a +c \,d^{2}\right ) c d \right ) \left (e x +d \right )^{\frac {5}{2}}}{5}+\frac {2 \left (-4 \left (A e -B d \right ) \left (e^{2} a +c \,d^{2}\right ) c d +B \left (e^{2} a +c \,d^{2}\right )^{2}\right ) \left (e x +d \right )^{\frac {3}{2}}}{3}+2 \left (A e -B d \right ) \left (e^{2} a +c \,d^{2}\right )^{2} \sqrt {e x +d}}{e^{6}}\) | \(233\) |
| gosper | \(\frac {2 \sqrt {e x +d}\, \left (315 B \,x^{5} c^{2} e^{5}+385 A \,x^{4} c^{2} e^{5}-350 B \,x^{4} c^{2} d \,e^{4}-440 A \,x^{3} c^{2} d \,e^{4}+990 B \,x^{3} a c \,e^{5}+400 B \,x^{3} c^{2} d^{2} e^{3}+1386 A \,x^{2} a c \,e^{5}+528 A \,x^{2} c^{2} d^{2} e^{3}-1188 B \,x^{2} a c d \,e^{4}-480 B \,x^{2} c^{2} d^{3} e^{2}-1848 A x a c d \,e^{4}-704 A x \,c^{2} d^{3} e^{2}+1155 B x \,a^{2} e^{5}+1584 B x a c \,d^{2} e^{3}+640 B x \,c^{2} d^{4} e +3465 A \,a^{2} e^{5}+3696 A a c \,d^{2} e^{3}+1408 A \,c^{2} d^{4} e -2310 B \,a^{2} d \,e^{4}-3168 B a c \,d^{3} e^{2}-1280 B \,c^{2} d^{5}\right )}{3465 e^{6}}\) | \(259\) |
| trager | \(\frac {2 \sqrt {e x +d}\, \left (315 B \,x^{5} c^{2} e^{5}+385 A \,x^{4} c^{2} e^{5}-350 B \,x^{4} c^{2} d \,e^{4}-440 A \,x^{3} c^{2} d \,e^{4}+990 B \,x^{3} a c \,e^{5}+400 B \,x^{3} c^{2} d^{2} e^{3}+1386 A \,x^{2} a c \,e^{5}+528 A \,x^{2} c^{2} d^{2} e^{3}-1188 B \,x^{2} a c d \,e^{4}-480 B \,x^{2} c^{2} d^{3} e^{2}-1848 A x a c d \,e^{4}-704 A x \,c^{2} d^{3} e^{2}+1155 B x \,a^{2} e^{5}+1584 B x a c \,d^{2} e^{3}+640 B x \,c^{2} d^{4} e +3465 A \,a^{2} e^{5}+3696 A a c \,d^{2} e^{3}+1408 A \,c^{2} d^{4} e -2310 B \,a^{2} d \,e^{4}-3168 B a c \,d^{3} e^{2}-1280 B \,c^{2} d^{5}\right )}{3465 e^{6}}\) | \(259\) |
| risch | \(\frac {2 \sqrt {e x +d}\, \left (315 B \,x^{5} c^{2} e^{5}+385 A \,x^{4} c^{2} e^{5}-350 B \,x^{4} c^{2} d \,e^{4}-440 A \,x^{3} c^{2} d \,e^{4}+990 B \,x^{3} a c \,e^{5}+400 B \,x^{3} c^{2} d^{2} e^{3}+1386 A \,x^{2} a c \,e^{5}+528 A \,x^{2} c^{2} d^{2} e^{3}-1188 B \,x^{2} a c d \,e^{4}-480 B \,x^{2} c^{2} d^{3} e^{2}-1848 A x a c d \,e^{4}-704 A x \,c^{2} d^{3} e^{2}+1155 B x \,a^{2} e^{5}+1584 B x a c \,d^{2} e^{3}+640 B x \,c^{2} d^{4} e +3465 A \,a^{2} e^{5}+3696 A a c \,d^{2} e^{3}+1408 A \,c^{2} d^{4} e -2310 B \,a^{2} d \,e^{4}-3168 B a c \,d^{3} e^{2}-1280 B \,c^{2} d^{5}\right )}{3465 e^{6}}\) | \(259\) |
2*((1/9*x^4*(9/11*B*x+A)*c^2+2/5*(5/7*B*x+A)*x^2*a*c+a^2*(1/3*B*x+A))*e^5- 8/15*d*(5/21*x^3*(35/44*B*x+A)*c^2+a*x*(9/14*B*x+A)*c+5/4*B*a^2)*e^4+16/15 *c*d^2*(1/7*x^2*(25/33*B*x+A)*c+a*(3/7*B*x+A))*e^3-64/315*c*(x*(15/22*B*x+ A)*c+9/2*B*a)*d^3*e^2+128/315*c^2*d^4*(5/11*B*x+A)*e-256/693*B*c^2*d^5)*(e *x+d)^(1/2)/e^6
Time = 0.34 (sec) , antiderivative size = 247, normalized size of antiderivative = 1.14 \[ \int \frac {(A+B x) \left (a+c x^2\right )^2}{\sqrt {d+e x}} \, dx=\frac {2 \, {\left (315 \, B c^{2} e^{5} x^{5} - 1280 \, B c^{2} d^{5} + 1408 \, A c^{2} d^{4} e - 3168 \, B a c d^{3} e^{2} + 3696 \, A a c d^{2} e^{3} - 2310 \, B a^{2} d e^{4} + 3465 \, A a^{2} e^{5} - 35 \, {\left (10 \, B c^{2} d e^{4} - 11 \, A c^{2} e^{5}\right )} x^{4} + 10 \, {\left (40 \, B c^{2} d^{2} e^{3} - 44 \, A c^{2} d e^{4} + 99 \, B a c e^{5}\right )} x^{3} - 6 \, {\left (80 \, B c^{2} d^{3} e^{2} - 88 \, A c^{2} d^{2} e^{3} + 198 \, B a c d e^{4} - 231 \, A a c e^{5}\right )} x^{2} + {\left (640 \, B c^{2} d^{4} e - 704 \, A c^{2} d^{3} e^{2} + 1584 \, B a c d^{2} e^{3} - 1848 \, A a c d e^{4} + 1155 \, B a^{2} e^{5}\right )} x\right )} \sqrt {e x + d}}{3465 \, e^{6}} \]
2/3465*(315*B*c^2*e^5*x^5 - 1280*B*c^2*d^5 + 1408*A*c^2*d^4*e - 3168*B*a*c *d^3*e^2 + 3696*A*a*c*d^2*e^3 - 2310*B*a^2*d*e^4 + 3465*A*a^2*e^5 - 35*(10 *B*c^2*d*e^4 - 11*A*c^2*e^5)*x^4 + 10*(40*B*c^2*d^2*e^3 - 44*A*c^2*d*e^4 + 99*B*a*c*e^5)*x^3 - 6*(80*B*c^2*d^3*e^2 - 88*A*c^2*d^2*e^3 + 198*B*a*c*d* e^4 - 231*A*a*c*e^5)*x^2 + (640*B*c^2*d^4*e - 704*A*c^2*d^3*e^2 + 1584*B*a *c*d^2*e^3 - 1848*A*a*c*d*e^4 + 1155*B*a^2*e^5)*x)*sqrt(e*x + d)/e^6
Time = 0.89 (sec) , antiderivative size = 371, normalized size of antiderivative = 1.72 \[ \int \frac {(A+B x) \left (a+c x^2\right )^2}{\sqrt {d+e x}} \, dx=\begin {cases} \frac {2 \left (\frac {B c^{2} \left (d + e x\right )^{\frac {11}{2}}}{11 e^{5}} + \frac {\left (d + e x\right )^{\frac {9}{2}} \left (A c^{2} e - 5 B c^{2} d\right )}{9 e^{5}} + \frac {\left (d + e x\right )^{\frac {7}{2}} \left (- 4 A c^{2} d e + 2 B a c e^{2} + 10 B c^{2} d^{2}\right )}{7 e^{5}} + \frac {\left (d + e x\right )^{\frac {5}{2}} \cdot \left (2 A a c e^{3} + 6 A c^{2} d^{2} e - 6 B a c d e^{2} - 10 B c^{2} d^{3}\right )}{5 e^{5}} + \frac {\left (d + e x\right )^{\frac {3}{2}} \left (- 4 A a c d e^{3} - 4 A c^{2} d^{3} e + B a^{2} e^{4} + 6 B a c d^{2} e^{2} + 5 B c^{2} d^{4}\right )}{3 e^{5}} + \frac {\sqrt {d + e x} \left (A a^{2} e^{5} + 2 A a c d^{2} e^{3} + A c^{2} d^{4} e - B a^{2} d e^{4} - 2 B a c d^{3} e^{2} - B c^{2} d^{5}\right )}{e^{5}}\right )}{e} & \text {for}\: e \neq 0 \\\frac {A a^{2} x + \frac {2 A a c x^{3}}{3} + \frac {A c^{2} x^{5}}{5} + \frac {B a^{2} x^{2}}{2} + \frac {B a c x^{4}}{2} + \frac {B c^{2} x^{6}}{6}}{\sqrt {d}} & \text {otherwise} \end {cases} \]
Piecewise((2*(B*c**2*(d + e*x)**(11/2)/(11*e**5) + (d + e*x)**(9/2)*(A*c** 2*e - 5*B*c**2*d)/(9*e**5) + (d + e*x)**(7/2)*(-4*A*c**2*d*e + 2*B*a*c*e** 2 + 10*B*c**2*d**2)/(7*e**5) + (d + e*x)**(5/2)*(2*A*a*c*e**3 + 6*A*c**2*d **2*e - 6*B*a*c*d*e**2 - 10*B*c**2*d**3)/(5*e**5) + (d + e*x)**(3/2)*(-4*A *a*c*d*e**3 - 4*A*c**2*d**3*e + B*a**2*e**4 + 6*B*a*c*d**2*e**2 + 5*B*c**2 *d**4)/(3*e**5) + sqrt(d + e*x)*(A*a**2*e**5 + 2*A*a*c*d**2*e**3 + A*c**2* d**4*e - B*a**2*d*e**4 - 2*B*a*c*d**3*e**2 - B*c**2*d**5)/e**5)/e, Ne(e, 0 )), ((A*a**2*x + 2*A*a*c*x**3/3 + A*c**2*x**5/5 + B*a**2*x**2/2 + B*a*c*x* *4/2 + B*c**2*x**6/6)/sqrt(d), True))
Time = 0.19 (sec) , antiderivative size = 248, normalized size of antiderivative = 1.15 \[ \int \frac {(A+B x) \left (a+c x^2\right )^2}{\sqrt {d+e x}} \, dx=\frac {2 \, {\left (315 \, {\left (e x + d\right )}^{\frac {11}{2}} B c^{2} - 385 \, {\left (5 \, B c^{2} d - A c^{2} e\right )} {\left (e x + d\right )}^{\frac {9}{2}} + 990 \, {\left (5 \, B c^{2} d^{2} - 2 \, A c^{2} d e + B a c e^{2}\right )} {\left (e x + d\right )}^{\frac {7}{2}} - 1386 \, {\left (5 \, B c^{2} d^{3} - 3 \, A c^{2} d^{2} e + 3 \, B a c d e^{2} - A a c e^{3}\right )} {\left (e x + d\right )}^{\frac {5}{2}} + 1155 \, {\left (5 \, B c^{2} d^{4} - 4 \, A c^{2} d^{3} e + 6 \, B a c d^{2} e^{2} - 4 \, A a c d e^{3} + B a^{2} e^{4}\right )} {\left (e x + d\right )}^{\frac {3}{2}} - 3465 \, {\left (B c^{2} d^{5} - A c^{2} d^{4} e + 2 \, B a c d^{3} e^{2} - 2 \, A a c d^{2} e^{3} + B a^{2} d e^{4} - A a^{2} e^{5}\right )} \sqrt {e x + d}\right )}}{3465 \, e^{6}} \]
2/3465*(315*(e*x + d)^(11/2)*B*c^2 - 385*(5*B*c^2*d - A*c^2*e)*(e*x + d)^( 9/2) + 990*(5*B*c^2*d^2 - 2*A*c^2*d*e + B*a*c*e^2)*(e*x + d)^(7/2) - 1386* (5*B*c^2*d^3 - 3*A*c^2*d^2*e + 3*B*a*c*d*e^2 - A*a*c*e^3)*(e*x + d)^(5/2) + 1155*(5*B*c^2*d^4 - 4*A*c^2*d^3*e + 6*B*a*c*d^2*e^2 - 4*A*a*c*d*e^3 + B* a^2*e^4)*(e*x + d)^(3/2) - 3465*(B*c^2*d^5 - A*c^2*d^4*e + 2*B*a*c*d^3*e^2 - 2*A*a*c*d^2*e^3 + B*a^2*d*e^4 - A*a^2*e^5)*sqrt(e*x + d))/e^6
Time = 0.28 (sec) , antiderivative size = 280, normalized size of antiderivative = 1.30 \[ \int \frac {(A+B x) \left (a+c x^2\right )^2}{\sqrt {d+e x}} \, dx=\frac {2 \, {\left (3465 \, \sqrt {e x + d} A a^{2} + \frac {1155 \, {\left ({\left (e x + d\right )}^{\frac {3}{2}} - 3 \, \sqrt {e x + d} d\right )} B a^{2}}{e} + \frac {462 \, {\left (3 \, {\left (e x + d\right )}^{\frac {5}{2}} - 10 \, {\left (e x + d\right )}^{\frac {3}{2}} d + 15 \, \sqrt {e x + d} d^{2}\right )} A a c}{e^{2}} + \frac {198 \, {\left (5 \, {\left (e x + d\right )}^{\frac {7}{2}} - 21 \, {\left (e x + d\right )}^{\frac {5}{2}} d + 35 \, {\left (e x + d\right )}^{\frac {3}{2}} d^{2} - 35 \, \sqrt {e x + d} d^{3}\right )} B a c}{e^{3}} + \frac {11 \, {\left (35 \, {\left (e x + d\right )}^{\frac {9}{2}} - 180 \, {\left (e x + d\right )}^{\frac {7}{2}} d + 378 \, {\left (e x + d\right )}^{\frac {5}{2}} d^{2} - 420 \, {\left (e x + d\right )}^{\frac {3}{2}} d^{3} + 315 \, \sqrt {e x + d} d^{4}\right )} A c^{2}}{e^{4}} + \frac {5 \, {\left (63 \, {\left (e x + d\right )}^{\frac {11}{2}} - 385 \, {\left (e x + d\right )}^{\frac {9}{2}} d + 990 \, {\left (e x + d\right )}^{\frac {7}{2}} d^{2} - 1386 \, {\left (e x + d\right )}^{\frac {5}{2}} d^{3} + 1155 \, {\left (e x + d\right )}^{\frac {3}{2}} d^{4} - 693 \, \sqrt {e x + d} d^{5}\right )} B c^{2}}{e^{5}}\right )}}{3465 \, e} \]
2/3465*(3465*sqrt(e*x + d)*A*a^2 + 1155*((e*x + d)^(3/2) - 3*sqrt(e*x + d) *d)*B*a^2/e + 462*(3*(e*x + d)^(5/2) - 10*(e*x + d)^(3/2)*d + 15*sqrt(e*x + d)*d^2)*A*a*c/e^2 + 198*(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*c/e^3 + 11*(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*c^2/e^4 + 5*(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 + 115 5*(e*x + d)^(3/2)*d^4 - 693*sqrt(e*x + d)*d^5)*B*c^2/e^5)/e
Time = 10.53 (sec) , antiderivative size = 197, normalized size of antiderivative = 0.91 \[ \int \frac {(A+B x) \left (a+c x^2\right )^2}{\sqrt {d+e x}} \, dx=\frac {{\left (d+e\,x\right )}^{7/2}\,\left (20\,B\,c^2\,d^2-8\,A\,c^2\,d\,e+4\,B\,a\,c\,e^2\right )}{7\,e^6}+\frac {4\,c\,{\left (d+e\,x\right )}^{5/2}\,\left (-5\,B\,c\,d^3+3\,A\,c\,d^2\,e-3\,B\,a\,d\,e^2+A\,a\,e^3\right )}{5\,e^6}+\frac {2\,B\,c^2\,{\left (d+e\,x\right )}^{11/2}}{11\,e^6}+\frac {2\,\left (c\,d^2+a\,e^2\right )\,{\left (d+e\,x\right )}^{3/2}\,\left (5\,B\,c\,d^2-4\,A\,c\,d\,e+B\,a\,e^2\right )}{3\,e^6}+\frac {2\,c^2\,\left (A\,e-5\,B\,d\right )\,{\left (d+e\,x\right )}^{9/2}}{9\,e^6}+\frac {2\,{\left (c\,d^2+a\,e^2\right )}^2\,\left (A\,e-B\,d\right )\,\sqrt {d+e\,x}}{e^6} \]
((d + e*x)^(7/2)*(20*B*c^2*d^2 + 4*B*a*c*e^2 - 8*A*c^2*d*e))/(7*e^6) + (4* c*(d + e*x)^(5/2)*(A*a*e^3 - 5*B*c*d^3 - 3*B*a*d*e^2 + 3*A*c*d^2*e))/(5*e^ 6) + (2*B*c^2*(d + e*x)^(11/2))/(11*e^6) + (2*(a*e^2 + c*d^2)*(d + e*x)^(3 /2)*(B*a*e^2 + 5*B*c*d^2 - 4*A*c*d*e))/(3*e^6) + (2*c^2*(A*e - 5*B*d)*(d + e*x)^(9/2))/(9*e^6) + (2*(a*e^2 + c*d^2)^2*(A*e - B*d)*(d + e*x)^(1/2))/e ^6