Integrand size = 46, antiderivative size = 347 \[ \int (d+e x)^{5/2} (f+g x) \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2} \, dx=-\frac {32 (2 c d-b e)^3 (11 c e f+5 c d g-8 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{3465 c^5 e^2 (d+e x)^{3/2}}-\frac {16 (2 c d-b e)^2 (11 c e f+5 c d g-8 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{1155 c^4 e^2 \sqrt {d+e x}}-\frac {4 (2 c d-b e) (11 c e f+5 c d g-8 b e g) \sqrt {d+e x} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{231 c^3 e^2}-\frac {2 (11 c e f+5 c d g-8 b e g) (d+e x)^{3/2} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{99 c^2 e^2}-\frac {2 g (d+e x)^{5/2} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{11 c e^2} \]
-32/3465*(-b*e+2*c*d)^3*(-8*b*e*g+5*c*d*g+11*c*e*f)*(d*(-b*e+c*d)-b*e^2*x- c*e^2*x^2)^(3/2)/c^5/e^2/(e*x+d)^(3/2)-2/99*(-8*b*e*g+5*c*d*g+11*c*e*f)*(e *x+d)^(3/2)*(d*(-b*e+c*d)-b*e^2*x-c*e^2*x^2)^(3/2)/c^2/e^2-2/11*g*(e*x+d)^ (5/2)*(d*(-b*e+c*d)-b*e^2*x-c*e^2*x^2)^(3/2)/c/e^2-16/1155*(-b*e+2*c*d)^2* (-8*b*e*g+5*c*d*g+11*c*e*f)*(d*(-b*e+c*d)-b*e^2*x-c*e^2*x^2)^(3/2)/c^4/e^2 /(e*x+d)^(1/2)-4/231*(-b*e+2*c*d)*(-8*b*e*g+5*c*d*g+11*c*e*f)*(d*(-b*e+c*d )-b*e^2*x-c*e^2*x^2)^(3/2)*(e*x+d)^(1/2)/c^3/e^2
Time = 0.18 (sec) , antiderivative size = 262, normalized size of antiderivative = 0.76 \[ \int (d+e x)^{5/2} (f+g x) \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2} \, dx=\frac {2 (-c d+b e+c e x) \sqrt {(d+e x) (-b e+c (d-e x))} \left (128 b^4 e^4 g-16 b^3 c e^3 (11 e f+65 d g+12 e g x)+24 b^2 c^2 e^2 \left (131 d^2 g+e^2 x (11 f+10 g x)+d e (55 f+57 g x)\right )-2 b c^3 e \left (2071 d^3 g+5 e^3 x^2 (33 f+28 g x)+3 d e^2 x (286 f+245 g x)+3 d^2 e (583 f+558 g x)\right )+c^4 \left (1910 d^4 g+35 e^4 x^3 (11 f+9 g x)+5 d e^3 x^2 (363 f+287 g x)+3 d^2 e^2 x (1177 f+905 g x)+d^3 e (3509 f+2865 g x)\right )\right )}{3465 c^5 e^2 \sqrt {d+e x}} \]
(2*(-(c*d) + b*e + c*e*x)*Sqrt[(d + e*x)*(-(b*e) + c*(d - e*x))]*(128*b^4* e^4*g - 16*b^3*c*e^3*(11*e*f + 65*d*g + 12*e*g*x) + 24*b^2*c^2*e^2*(131*d^ 2*g + e^2*x*(11*f + 10*g*x) + d*e*(55*f + 57*g*x)) - 2*b*c^3*e*(2071*d^3*g + 5*e^3*x^2*(33*f + 28*g*x) + 3*d*e^2*x*(286*f + 245*g*x) + 3*d^2*e*(583* f + 558*g*x)) + c^4*(1910*d^4*g + 35*e^4*x^3*(11*f + 9*g*x) + 5*d*e^3*x^2* (363*f + 287*g*x) + 3*d^2*e^2*x*(1177*f + 905*g*x) + d^3*e*(3509*f + 2865* g*x))))/(3465*c^5*e^2*Sqrt[d + e*x])
Time = 0.58 (sec) , antiderivative size = 322, normalized size of antiderivative = 0.93, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.109, Rules used = {1221, 1128, 1128, 1128, 1122}
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.
| \(\displaystyle \int (d+e x)^{5/2} (f+g x) \sqrt {-b d e-b e^2 x+c d^2-c e^2 x^2} \, dx\) |
| \(\Big \downarrow \) 1221 |
| \(\displaystyle \frac {(-8 b e g+5 c d g+11 c e f) \int (d+e x)^{5/2} \sqrt {-c x^2 e^2-b x e^2+d (c d-b e)}dx}{11 c e}-\frac {2 g (d+e x)^{5/2} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{11 c e^2}\) |
| \(\Big \downarrow \) 1128 |
| \(\displaystyle \frac {(-8 b e g+5 c d g+11 c e f) \left (\frac {2 (2 c d-b e) \int (d+e x)^{3/2} \sqrt {-c x^2 e^2-b x e^2+d (c d-b e)}dx}{3 c}-\frac {2 (d+e x)^{3/2} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{9 c e}\right )}{11 c e}-\frac {2 g (d+e x)^{5/2} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{11 c e^2}\) |
| \(\Big \downarrow \) 1128 |
| \(\displaystyle \frac {(-8 b e g+5 c d g+11 c e f) \left (\frac {2 (2 c d-b e) \left (\frac {4 (2 c d-b e) \int \sqrt {d+e x} \sqrt {-c x^2 e^2-b x e^2+d (c d-b e)}dx}{7 c}-\frac {2 \sqrt {d+e x} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{7 c e}\right )}{3 c}-\frac {2 (d+e x)^{3/2} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{9 c e}\right )}{11 c e}-\frac {2 g (d+e x)^{5/2} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{11 c e^2}\) |
| \(\Big \downarrow \) 1128 |
| \(\displaystyle \frac {(-8 b e g+5 c d g+11 c e f) \left (\frac {2 (2 c d-b e) \left (\frac {4 (2 c d-b e) \left (\frac {2 (2 c d-b e) \int \frac {\sqrt {-c x^2 e^2-b x e^2+d (c d-b e)}}{\sqrt {d+e x}}dx}{5 c}-\frac {2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{5 c e \sqrt {d+e x}}\right )}{7 c}-\frac {2 \sqrt {d+e x} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{7 c e}\right )}{3 c}-\frac {2 (d+e x)^{3/2} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{9 c e}\right )}{11 c e}-\frac {2 g (d+e x)^{5/2} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{11 c e^2}\) |
| \(\Big \downarrow \) 1122 |
| \(\displaystyle \frac {\left (\frac {2 (2 c d-b e) \left (\frac {4 (2 c d-b e) \left (-\frac {4 (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{15 c^2 e (d+e x)^{3/2}}-\frac {2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{5 c e \sqrt {d+e x}}\right )}{7 c}-\frac {2 \sqrt {d+e x} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{7 c e}\right )}{3 c}-\frac {2 (d+e x)^{3/2} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{9 c e}\right ) (-8 b e g+5 c d g+11 c e f)}{11 c e}-\frac {2 g (d+e x)^{5/2} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{11 c e^2}\) |
(-2*g*(d + e*x)^(5/2)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(3/2))/(11*c*e ^2) + ((11*c*e*f + 5*c*d*g - 8*b*e*g)*((-2*(d + e*x)^(3/2)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(3/2))/(9*c*e) + (2*(2*c*d - b*e)*((-2*Sqrt[d + e*x ]*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(3/2))/(7*c*e) + (4*(2*c*d - b*e)* ((-4*(2*c*d - b*e)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(3/2))/(15*c^2*e* (d + e*x)^(3/2)) - (2*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(3/2))/(5*c*e* Sqrt[d + e*x])))/(7*c)))/(3*c)))/(11*c*e)
3.23.31.3.1 Defintions of rubi rules used
Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S ymbol] :> Simp[e*(d + e*x)^(m - 1)*((a + b*x + c*x^2)^(p + 1)/(c*(p + 1))), x] /; FreeQ[{a, b, c, d, e, m, p}, x] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && EqQ[m + p, 0]
Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S ymbol] :> Simp[e*(d + e*x)^(m - 1)*((a + b*x + c*x^2)^(p + 1)/(c*(m + 2*p + 1))), x] + Simp[Simplify[m + p]*((2*c*d - b*e)/(c*(m + 2*p + 1))) Int[(d + e*x)^(m - 1)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && IGtQ[Simplify[m + p], 0]
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c _.)*(x_)^2)^(p_), x_Symbol] :> Simp[g*(d + e*x)^m*((a + b*x + c*x^2)^(p + 1 )/(c*(m + 2*p + 2))), x] + Simp[(m*(g*(c*d - b*e) + c*e*f) + e*(p + 1)*(2*c *f - b*g))/(c*e*(m + 2*p + 2)) Int[(d + e*x)^m*(a + b*x + c*x^2)^p, x], x ] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && NeQ[m + 2*p + 2, 0]
Time = 0.35 (sec) , antiderivative size = 359, normalized size of antiderivative = 1.03
| method | result | size |
| default | \(\frac {2 \left (x c e +b e -c d \right ) \left (315 g \,e^{4} x^{4} c^{4}-280 b \,c^{3} e^{4} g \,x^{3}+1435 c^{4} d \,e^{3} g \,x^{3}+385 c^{4} e^{4} f \,x^{3}+240 b^{2} c^{2} e^{4} g \,x^{2}-1470 b \,c^{3} d \,e^{3} g \,x^{2}-330 b \,c^{3} e^{4} f \,x^{2}+2715 c^{4} d^{2} e^{2} g \,x^{2}+1815 c^{4} d \,e^{3} f \,x^{2}-192 b^{3} c \,e^{4} g x +1368 b^{2} c^{2} d \,e^{3} g x +264 b^{2} c^{2} e^{4} f x -3348 b \,c^{3} d^{2} e^{2} g x -1716 b \,c^{3} d \,e^{3} f x +2865 c^{4} d^{3} e g x +3531 c^{4} d^{2} e^{2} f x +128 b^{4} e^{4} g -1040 b^{3} c d \,e^{3} g -176 b^{3} c \,e^{4} f +3144 b^{2} c^{2} d^{2} e^{2} g +1320 b^{2} c^{2} d \,e^{3} f -4142 b \,c^{3} d^{3} e g -3498 b \,c^{3} d^{2} e^{2} f +1910 c^{4} d^{4} g +3509 d^{3} f \,c^{4} e \right ) \sqrt {-\left (e x +d \right ) \left (x c e +b e -c d \right )}}{3465 c^{5} e^{2} \sqrt {e x +d}}\) | \(359\) |
| gosper | \(\frac {2 \left (x c e +b e -c d \right ) \left (315 g \,e^{4} x^{4} c^{4}-280 b \,c^{3} e^{4} g \,x^{3}+1435 c^{4} d \,e^{3} g \,x^{3}+385 c^{4} e^{4} f \,x^{3}+240 b^{2} c^{2} e^{4} g \,x^{2}-1470 b \,c^{3} d \,e^{3} g \,x^{2}-330 b \,c^{3} e^{4} f \,x^{2}+2715 c^{4} d^{2} e^{2} g \,x^{2}+1815 c^{4} d \,e^{3} f \,x^{2}-192 b^{3} c \,e^{4} g x +1368 b^{2} c^{2} d \,e^{3} g x +264 b^{2} c^{2} e^{4} f x -3348 b \,c^{3} d^{2} e^{2} g x -1716 b \,c^{3} d \,e^{3} f x +2865 c^{4} d^{3} e g x +3531 c^{4} d^{2} e^{2} f x +128 b^{4} e^{4} g -1040 b^{3} c d \,e^{3} g -176 b^{3} c \,e^{4} f +3144 b^{2} c^{2} d^{2} e^{2} g +1320 b^{2} c^{2} d \,e^{3} f -4142 b \,c^{3} d^{3} e g -3498 b \,c^{3} d^{2} e^{2} f +1910 c^{4} d^{4} g +3509 d^{3} f \,c^{4} e \right ) \sqrt {-c \,e^{2} x^{2}-b \,e^{2} x -b d e +c \,d^{2}}}{3465 c^{5} e^{2} \sqrt {e x +d}}\) | \(367\) |
2/3465*(c*e*x+b*e-c*d)*(315*c^4*e^4*g*x^4-280*b*c^3*e^4*g*x^3+1435*c^4*d*e ^3*g*x^3+385*c^4*e^4*f*x^3+240*b^2*c^2*e^4*g*x^2-1470*b*c^3*d*e^3*g*x^2-33 0*b*c^3*e^4*f*x^2+2715*c^4*d^2*e^2*g*x^2+1815*c^4*d*e^3*f*x^2-192*b^3*c*e^ 4*g*x+1368*b^2*c^2*d*e^3*g*x+264*b^2*c^2*e^4*f*x-3348*b*c^3*d^2*e^2*g*x-17 16*b*c^3*d*e^3*f*x+2865*c^4*d^3*e*g*x+3531*c^4*d^2*e^2*f*x+128*b^4*e^4*g-1 040*b^3*c*d*e^3*g-176*b^3*c*e^4*f+3144*b^2*c^2*d^2*e^2*g+1320*b^2*c^2*d*e^ 3*f-4142*b*c^3*d^3*e*g-3498*b*c^3*d^2*e^2*f+1910*c^4*d^4*g+3509*c^4*d^3*e* f)*(-(e*x+d)*(c*e*x+b*e-c*d))^(1/2)/c^5/e^2/(e*x+d)^(1/2)
Time = 0.33 (sec) , antiderivative size = 499, normalized size of antiderivative = 1.44 \[ \int (d+e x)^{5/2} (f+g x) \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2} \, dx=\frac {2 \, {\left (315 \, c^{5} e^{5} g x^{5} + 35 \, {\left (11 \, c^{5} e^{5} f + {\left (32 \, c^{5} d e^{4} + b c^{4} e^{5}\right )} g\right )} x^{4} + 5 \, {\left (11 \, {\left (26 \, c^{5} d e^{4} + b c^{4} e^{5}\right )} f + {\left (256 \, c^{5} d^{2} e^{3} + 49 \, b c^{4} d e^{4} - 8 \, b^{2} c^{3} e^{5}\right )} g\right )} x^{3} + 3 \, {\left (11 \, {\left (52 \, c^{5} d^{2} e^{3} + 13 \, b c^{4} d e^{4} - 2 \, b^{2} c^{3} e^{5}\right )} f + {\left (50 \, c^{5} d^{3} e^{2} + 279 \, b c^{4} d^{2} e^{3} - 114 \, b^{2} c^{3} d e^{4} + 16 \, b^{3} c^{2} e^{5}\right )} g\right )} x^{2} - 11 \, {\left (319 \, c^{5} d^{4} e - 637 \, b c^{4} d^{3} e^{2} + 438 \, b^{2} c^{3} d^{2} e^{3} - 136 \, b^{3} c^{2} d e^{4} + 16 \, b^{4} c e^{5}\right )} f - 2 \, {\left (955 \, c^{5} d^{5} - 3026 \, b c^{4} d^{4} e + 3643 \, b^{2} c^{3} d^{3} e^{2} - 2092 \, b^{3} c^{2} d^{2} e^{3} + 584 \, b^{4} c d e^{4} - 64 \, b^{5} e^{5}\right )} g - {\left (11 \, {\left (2 \, c^{5} d^{3} e^{2} - 159 \, b c^{4} d^{2} e^{3} + 60 \, b^{2} c^{3} d e^{4} - 8 \, b^{3} c^{2} e^{5}\right )} f + {\left (955 \, c^{5} d^{4} e - 2071 \, b c^{4} d^{3} e^{2} + 1572 \, b^{2} c^{3} d^{2} e^{3} - 520 \, b^{3} c^{2} d e^{4} + 64 \, b^{4} c e^{5}\right )} g\right )} x\right )} \sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e} \sqrt {e x + d}}{3465 \, {\left (c^{5} e^{3} x + c^{5} d e^{2}\right )}} \]
2/3465*(315*c^5*e^5*g*x^5 + 35*(11*c^5*e^5*f + (32*c^5*d*e^4 + b*c^4*e^5)* g)*x^4 + 5*(11*(26*c^5*d*e^4 + b*c^4*e^5)*f + (256*c^5*d^2*e^3 + 49*b*c^4* d*e^4 - 8*b^2*c^3*e^5)*g)*x^3 + 3*(11*(52*c^5*d^2*e^3 + 13*b*c^4*d*e^4 - 2 *b^2*c^3*e^5)*f + (50*c^5*d^3*e^2 + 279*b*c^4*d^2*e^3 - 114*b^2*c^3*d*e^4 + 16*b^3*c^2*e^5)*g)*x^2 - 11*(319*c^5*d^4*e - 637*b*c^4*d^3*e^2 + 438*b^2 *c^3*d^2*e^3 - 136*b^3*c^2*d*e^4 + 16*b^4*c*e^5)*f - 2*(955*c^5*d^5 - 3026 *b*c^4*d^4*e + 3643*b^2*c^3*d^3*e^2 - 2092*b^3*c^2*d^2*e^3 + 584*b^4*c*d*e ^4 - 64*b^5*e^5)*g - (11*(2*c^5*d^3*e^2 - 159*b*c^4*d^2*e^3 + 60*b^2*c^3*d *e^4 - 8*b^3*c^2*e^5)*f + (955*c^5*d^4*e - 2071*b*c^4*d^3*e^2 + 1572*b^2*c ^3*d^2*e^3 - 520*b^3*c^2*d*e^4 + 64*b^4*c*e^5)*g)*x)*sqrt(-c*e^2*x^2 - b*e ^2*x + c*d^2 - b*d*e)*sqrt(e*x + d)/(c^5*e^3*x + c^5*d*e^2)
\[ \int (d+e x)^{5/2} (f+g x) \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2} \, dx=\int \sqrt {- \left (d + e x\right ) \left (b e - c d + c e x\right )} \left (d + e x\right )^{\frac {5}{2}} \left (f + g x\right )\, dx \]
Time = 0.23 (sec) , antiderivative size = 501, normalized size of antiderivative = 1.44 \[ \int (d+e x)^{5/2} (f+g x) \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2} \, dx=\frac {2 \, {\left (35 \, c^{4} e^{4} x^{4} - 319 \, c^{4} d^{4} + 637 \, b c^{3} d^{3} e - 438 \, b^{2} c^{2} d^{2} e^{2} + 136 \, b^{3} c d e^{3} - 16 \, b^{4} e^{4} + 5 \, {\left (26 \, c^{4} d e^{3} + b c^{3} e^{4}\right )} x^{3} + 3 \, {\left (52 \, c^{4} d^{2} e^{2} + 13 \, b c^{3} d e^{3} - 2 \, b^{2} c^{2} e^{4}\right )} x^{2} - {\left (2 \, c^{4} d^{3} e - 159 \, b c^{3} d^{2} e^{2} + 60 \, b^{2} c^{2} d e^{3} - 8 \, b^{3} c e^{4}\right )} x\right )} \sqrt {-c e x + c d - b e} {\left (e x + d\right )} f}{315 \, {\left (c^{4} e^{2} x + c^{4} d e\right )}} + \frac {2 \, {\left (315 \, c^{5} e^{5} x^{5} - 1910 \, c^{5} d^{5} + 6052 \, b c^{4} d^{4} e - 7286 \, b^{2} c^{3} d^{3} e^{2} + 4184 \, b^{3} c^{2} d^{2} e^{3} - 1168 \, b^{4} c d e^{4} + 128 \, b^{5} e^{5} + 35 \, {\left (32 \, c^{5} d e^{4} + b c^{4} e^{5}\right )} x^{4} + 5 \, {\left (256 \, c^{5} d^{2} e^{3} + 49 \, b c^{4} d e^{4} - 8 \, b^{2} c^{3} e^{5}\right )} x^{3} + 3 \, {\left (50 \, c^{5} d^{3} e^{2} + 279 \, b c^{4} d^{2} e^{3} - 114 \, b^{2} c^{3} d e^{4} + 16 \, b^{3} c^{2} e^{5}\right )} x^{2} - {\left (955 \, c^{5} d^{4} e - 2071 \, b c^{4} d^{3} e^{2} + 1572 \, b^{2} c^{3} d^{2} e^{3} - 520 \, b^{3} c^{2} d e^{4} + 64 \, b^{4} c e^{5}\right )} x\right )} \sqrt {-c e x + c d - b e} {\left (e x + d\right )} g}{3465 \, {\left (c^{5} e^{3} x + c^{5} d e^{2}\right )}} \]
2/315*(35*c^4*e^4*x^4 - 319*c^4*d^4 + 637*b*c^3*d^3*e - 438*b^2*c^2*d^2*e^ 2 + 136*b^3*c*d*e^3 - 16*b^4*e^4 + 5*(26*c^4*d*e^3 + b*c^3*e^4)*x^3 + 3*(5 2*c^4*d^2*e^2 + 13*b*c^3*d*e^3 - 2*b^2*c^2*e^4)*x^2 - (2*c^4*d^3*e - 159*b *c^3*d^2*e^2 + 60*b^2*c^2*d*e^3 - 8*b^3*c*e^4)*x)*sqrt(-c*e*x + c*d - b*e) *(e*x + d)*f/(c^4*e^2*x + c^4*d*e) + 2/3465*(315*c^5*e^5*x^5 - 1910*c^5*d^ 5 + 6052*b*c^4*d^4*e - 7286*b^2*c^3*d^3*e^2 + 4184*b^3*c^2*d^2*e^3 - 1168* b^4*c*d*e^4 + 128*b^5*e^5 + 35*(32*c^5*d*e^4 + b*c^4*e^5)*x^4 + 5*(256*c^5 *d^2*e^3 + 49*b*c^4*d*e^4 - 8*b^2*c^3*e^5)*x^3 + 3*(50*c^5*d^3*e^2 + 279*b *c^4*d^2*e^3 - 114*b^2*c^3*d*e^4 + 16*b^3*c^2*e^5)*x^2 - (955*c^5*d^4*e - 2071*b*c^4*d^3*e^2 + 1572*b^2*c^3*d^2*e^3 - 520*b^3*c^2*d*e^4 + 64*b^4*c*e ^5)*x)*sqrt(-c*e*x + c*d - b*e)*(e*x + d)*g/(c^5*e^3*x + c^5*d*e^2)
Leaf count of result is larger than twice the leaf count of optimal. 2695 vs. \(2 (317) = 634\).
Time = 0.37 (sec) , antiderivative size = 2695, normalized size of antiderivative = 7.77 \[ \int (d+e x)^{5/2} (f+g x) \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2} \, dx=\text {Too large to display} \]
-2/3465*(1155*d^3*f*((-(e*x + d)*c + 2*c*d - b*e)^(3/2)/c - (2*sqrt(2*c*d - b*e)*c*d - sqrt(2*c*d - b*e)*b*e)/c) - 99*d*e^2*f*((22*sqrt(2*c*d - b*e) *c^3*d^3 - 19*sqrt(2*c*d - b*e)*b*c^2*d^2*e + 20*sqrt(2*c*d - b*e)*b^2*c*d *e^2 - 8*sqrt(2*c*d - b*e)*b^3*e^3)/(c^3*e^2) - (35*(-(e*x + d)*c + 2*c*d - b*e)^(3/2)*c^2*d^2 - 70*(-(e*x + d)*c + 2*c*d - b*e)^(3/2)*b*c*d*e + 35* (-(e*x + d)*c + 2*c*d - b*e)^(3/2)*b^2*e^2 - 42*((e*x + d)*c - 2*c*d + b*e )^2*sqrt(-(e*x + d)*c + 2*c*d - b*e)*c*d + 42*((e*x + d)*c - 2*c*d + b*e)^ 2*sqrt(-(e*x + d)*c + 2*c*d - b*e)*b*e - 15*((e*x + d)*c - 2*c*d + b*e)^3* sqrt(-(e*x + d)*c + 2*c*d - b*e))/(c^3*e^2)) - 99*d^2*e*g*((22*sqrt(2*c*d - b*e)*c^3*d^3 - 19*sqrt(2*c*d - b*e)*b*c^2*d^2*e + 20*sqrt(2*c*d - b*e)*b ^2*c*d*e^2 - 8*sqrt(2*c*d - b*e)*b^3*e^3)/(c^3*e^2) - (35*(-(e*x + d)*c + 2*c*d - b*e)^(3/2)*c^2*d^2 - 70*(-(e*x + d)*c + 2*c*d - b*e)^(3/2)*b*c*d*e + 35*(-(e*x + d)*c + 2*c*d - b*e)^(3/2)*b^2*e^2 - 42*((e*x + d)*c - 2*c*d + b*e)^2*sqrt(-(e*x + d)*c + 2*c*d - b*e)*c*d + 42*((e*x + d)*c - 2*c*d + b*e)^2*sqrt(-(e*x + d)*c + 2*c*d - b*e)*b*e - 15*((e*x + d)*c - 2*c*d + b *e)^3*sqrt(-(e*x + d)*c + 2*c*d - b*e))/(c^3*e^2)) + 11*e^3*f*((26*sqrt(2* c*d - b*e)*c^4*d^4 + 47*sqrt(2*c*d - b*e)*b*c^3*d^3*e - 78*sqrt(2*c*d - b* e)*b^2*c^2*d^2*e^2 + 56*sqrt(2*c*d - b*e)*b^3*c*d*e^3 - 16*sqrt(2*c*d - b* e)*b^4*e^4)/(c^4*e^3) + (105*(-(e*x + d)*c + 2*c*d - b*e)^(3/2)*c^3*d^3 - 315*(-(e*x + d)*c + 2*c*d - b*e)^(3/2)*b*c^2*d^2*e + 315*(-(e*x + d)*c ...
Time = 12.00 (sec) , antiderivative size = 501, normalized size of antiderivative = 1.44 \[ \int (d+e x)^{5/2} (f+g x) \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2} \, dx=\frac {\sqrt {c\,d^2-b\,d\,e-c\,e^2\,x^2-b\,e^2\,x}\,\left (\frac {2\,x^3\,\sqrt {d+e\,x}\,\left (-8\,g\,b^2\,e^2+49\,g\,b\,c\,d\,e+11\,f\,b\,c\,e^2+256\,g\,c^2\,d^2+286\,f\,c^2\,d\,e\right )}{693\,c^2}+\frac {2\,e^2\,g\,x^5\,\sqrt {d+e\,x}}{11}+\frac {2\,\left (b\,e-c\,d\right )\,\sqrt {d+e\,x}\,\left (128\,g\,b^4\,e^4-1040\,g\,b^3\,c\,d\,e^3-176\,f\,b^3\,c\,e^4+3144\,g\,b^2\,c^2\,d^2\,e^2+1320\,f\,b^2\,c^2\,d\,e^3-4142\,g\,b\,c^3\,d^3\,e-3498\,f\,b\,c^3\,d^2\,e^2+1910\,g\,c^4\,d^4+3509\,f\,c^4\,d^3\,e\right )}{3465\,c^5\,e^3}-\frac {x\,\sqrt {d+e\,x}\,\left (128\,g\,b^4\,c\,e^5-1040\,g\,b^3\,c^2\,d\,e^4-176\,f\,b^3\,c^2\,e^5+3144\,g\,b^2\,c^3\,d^2\,e^3+1320\,f\,b^2\,c^3\,d\,e^4-4142\,g\,b\,c^4\,d^3\,e^2-3498\,f\,b\,c^4\,d^2\,e^3+1910\,g\,c^5\,d^4\,e+44\,f\,c^5\,d^3\,e^2\right )}{3465\,c^5\,e^3}+\frac {x^2\,\sqrt {d+e\,x}\,\left (96\,g\,b^3\,c^2\,e^5-684\,g\,b^2\,c^3\,d\,e^4-132\,f\,b^2\,c^3\,e^5+1674\,g\,b\,c^4\,d^2\,e^3+858\,f\,b\,c^4\,d\,e^4+300\,g\,c^5\,d^3\,e^2+3432\,f\,c^5\,d^2\,e^3\right )}{3465\,c^5\,e^3}+\frac {2\,e\,x^4\,\sqrt {d+e\,x}\,\left (b\,e\,g+32\,c\,d\,g+11\,c\,e\,f\right )}{99\,c}\right )}{x+\frac {d}{e}} \]
((c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(1/2)*((2*x^3*(d + e*x)^(1/2)*(256* c^2*d^2*g - 8*b^2*e^2*g + 11*b*c*e^2*f + 286*c^2*d*e*f + 49*b*c*d*e*g))/(6 93*c^2) + (2*e^2*g*x^5*(d + e*x)^(1/2))/11 + (2*(b*e - c*d)*(d + e*x)^(1/2 )*(128*b^4*e^4*g + 1910*c^4*d^4*g - 176*b^3*c*e^4*f + 3509*c^4*d^3*e*f - 4 142*b*c^3*d^3*e*g - 1040*b^3*c*d*e^3*g - 3498*b*c^3*d^2*e^2*f + 1320*b^2*c ^2*d*e^3*f + 3144*b^2*c^2*d^2*e^2*g))/(3465*c^5*e^3) - (x*(d + e*x)^(1/2)* (44*c^5*d^3*e^2*f - 176*b^3*c^2*e^5*f + 128*b^4*c*e^5*g + 1910*c^5*d^4*e*g - 3498*b*c^4*d^2*e^3*f + 1320*b^2*c^3*d*e^4*f - 4142*b*c^4*d^3*e^2*g - 10 40*b^3*c^2*d*e^4*g + 3144*b^2*c^3*d^2*e^3*g))/(3465*c^5*e^3) + (x^2*(d + e *x)^(1/2)*(96*b^3*c^2*e^5*g - 132*b^2*c^3*e^5*f + 3432*c^5*d^2*e^3*f + 300 *c^5*d^3*e^2*g + 858*b*c^4*d*e^4*f + 1674*b*c^4*d^2*e^3*g - 684*b^2*c^3*d* e^4*g))/(3465*c^5*e^3) + (2*e*x^4*(d + e*x)^(1/2)*(b*e*g + 32*c*d*g + 11*c *e*f))/(99*c)))/(x + d/e)