Integrand size = 46, antiderivative size = 343 \[ \int (d+e x)^{5/2} (f+g x) \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2} \, dx=-\frac {2 (2 c d-b e)^3 (c e f+c d g-b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{3 c^5 e^2 (d+e x)^{3/2}}+\frac {2 (2 c d-b e)^2 (3 c e f+5 c d g-4 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{5 c^5 e^2 (d+e x)^{5/2}}-\frac {6 (2 c d-b e) (c e f+3 c d g-2 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{7 c^5 e^2 (d+e x)^{7/2}}+\frac {2 (c e f+7 c d g-4 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{9/2}}{9 c^5 e^2 (d+e x)^{9/2}}-\frac {2 g \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{11/2}}{11 c^5 e^2 (d+e x)^{11/2}} \] Output:
-2/3*(-b*e+2*c*d)^3*(-b*e*g+c*d*g+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)+2/5*(-b*e+2*c*d)^2*(-4*b*e*g+5*c*d*g+3*c*e*f)* (d*(-b*e+c*d)-b*e^2*x-c*e^2*x^2)^(5/2)/c^5/e^2/(e*x+d)^(5/2)-6/7*(-b*e+2*c *d)*(-2*b*e*g+3*c*d*g+c*e*f)*(d*(-b*e+c*d)-b*e^2*x-c*e^2*x^2)^(7/2)/c^5/e^ 2/(e*x+d)^(7/2)+2/9*(-4*b*e*g+7*c*d*g+c*e*f)*(d*(-b*e+c*d)-b*e^2*x-c*e^2*x ^2)^(9/2)/c^5/e^2/(e*x+d)^(9/2)-2/11*g*(d*(-b*e+c*d)-b*e^2*x-c*e^2*x^2)^(1 1/2)/c^5/e^2/(e*x+d)^(11/2)
Time = 0.30 (sec) , antiderivative size = 262, normalized size of antiderivative = 0.76 \[ \int (d+e x)^{5/2} (f+g x) \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2} \, dx=\frac {2 (-c d+b e+c e x) \sqrt {(d+e x) (-b e+c (d-e x))} \left (128 b^4 e^4 g-16 b^3 c e^3 (11 e f+65 d g+12 e g x)+24 b^2 c^2 e^2 \left (131 d^2 g+e^2 x (11 f+10 g x)+d e (55 f+57 g x)\right )-2 b c^3 e \left (2071 d^3 g+5 e^3 x^2 (33 f+28 g x)+3 d e^2 x (286 f+245 g x)+3 d^2 e (583 f+558 g x)\right )+c^4 \left (1910 d^4 g+35 e^4 x^3 (11 f+9 g x)+5 d e^3 x^2 (363 f+287 g x)+3 d^2 e^2 x (1177 f+905 g x)+d^3 e (3509 f+2865 g x)\right )\right )}{3465 c^5 e^2 \sqrt {d+e x}} \] Input:
Integrate[(d + e*x)^(5/2)*(f + g*x)*Sqrt[c*d^2 - b*d*e - b*e^2*x - c*e^2*x ^2],x]
Output:
(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 = 0.99 (sec) , antiderivative size = 322, normalized size of antiderivative = 0.94, 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 |
| \(\displaystyle \frac {\left (\frac {2 (2 c d-b e) \left (\frac {4 (2 c d-b e) \left (-\frac {4 (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{15 c^2 e (d+e x)^{3/2}}-\frac {2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{5 c e \sqrt {d+e x}}\right )}{7 c}-\frac {2 \sqrt {d+e x} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{7 c e}\right )}{3 c}-\frac {2 (d+e x)^{3/2} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{9 c e}\right ) (-8 b e g+5 c d g+11 c e f)}{11 c e}-\frac {2 g (d+e x)^{5/2} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{11 c e^2}\) |
Input:
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],x]
Output:
(-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)
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 = 2.89 (sec) , antiderivative size = 359, normalized size of antiderivative = 1.05
| method | result | size |
| default | \(\frac {2 \left (c e x +b e -c d \right ) \left (315 g \,e^{4} x^{4} c^{4}-280 b \,c^{3} e^{4} g \,x^{3}+1435 c^{4} d \,e^{3} g \,x^{3}+385 c^{4} e^{4} f \,x^{3}+240 b^{2} c^{2} e^{4} g \,x^{2}-1470 b \,c^{3} d \,e^{3} g \,x^{2}-330 b \,c^{3} e^{4} f \,x^{2}+2715 c^{4} d^{2} e^{2} g \,x^{2}+1815 c^{4} d \,e^{3} f \,x^{2}-192 b^{3} c \,e^{4} g x +1368 b^{2} c^{2} d \,e^{3} g x +264 b^{2} c^{2} e^{4} f x -3348 b \,c^{3} d^{2} e^{2} g x -1716 b \,c^{3} d \,e^{3} f x +2865 c^{4} d^{3} e g x +3531 c^{4} d^{2} e^{2} f x +128 b^{4} e^{4} g -1040 b^{3} c d \,e^{3} g -176 b^{3} c \,e^{4} f +3144 b^{2} c^{2} d^{2} e^{2} g +1320 b^{2} c^{2} d \,e^{3} f -4142 b \,c^{3} d^{3} e g -3498 b \,c^{3} d^{2} e^{2} f +1910 c^{4} d^{4} g +3509 d^{3} f \,c^{4} e \right ) \sqrt {-\left (e x +d \right ) \left (c e x +b e -c d \right )}}{3465 c^{5} e^{2} \sqrt {e x +d}}\) | \(359\) |
| gosper | \(\frac {2 \left (c e x +b e -c d \right ) \left (315 g \,e^{4} x^{4} c^{4}-280 b \,c^{3} e^{4} g \,x^{3}+1435 c^{4} d \,e^{3} g \,x^{3}+385 c^{4} e^{4} f \,x^{3}+240 b^{2} c^{2} e^{4} g \,x^{2}-1470 b \,c^{3} d \,e^{3} g \,x^{2}-330 b \,c^{3} e^{4} f \,x^{2}+2715 c^{4} d^{2} e^{2} g \,x^{2}+1815 c^{4} d \,e^{3} f \,x^{2}-192 b^{3} c \,e^{4} g x +1368 b^{2} c^{2} d \,e^{3} g x +264 b^{2} c^{2} e^{4} f x -3348 b \,c^{3} d^{2} e^{2} g x -1716 b \,c^{3} d \,e^{3} f x +2865 c^{4} d^{3} e g x +3531 c^{4} d^{2} e^{2} f x +128 b^{4} e^{4} g -1040 b^{3} c d \,e^{3} g -176 b^{3} c \,e^{4} f +3144 b^{2} c^{2} d^{2} e^{2} g +1320 b^{2} c^{2} d \,e^{3} f -4142 b \,c^{3} d^{3} e g -3498 b \,c^{3} d^{2} e^{2} f +1910 c^{4} d^{4} g +3509 d^{3} f \,c^{4} e \right ) \sqrt {-x^{2} c \,e^{2}-x b \,e^{2}-b d e +c \,d^{2}}}{3465 c^{5} e^{2} \sqrt {e x +d}}\) | \(367\) |
| orering | \(\frac {2 \left (c e x +b e -c d \right ) \left (315 g \,e^{4} x^{4} c^{4}-280 b \,c^{3} e^{4} g \,x^{3}+1435 c^{4} d \,e^{3} g \,x^{3}+385 c^{4} e^{4} f \,x^{3}+240 b^{2} c^{2} e^{4} g \,x^{2}-1470 b \,c^{3} d \,e^{3} g \,x^{2}-330 b \,c^{3} e^{4} f \,x^{2}+2715 c^{4} d^{2} e^{2} g \,x^{2}+1815 c^{4} d \,e^{3} f \,x^{2}-192 b^{3} c \,e^{4} g x +1368 b^{2} c^{2} d \,e^{3} g x +264 b^{2} c^{2} e^{4} f x -3348 b \,c^{3} d^{2} e^{2} g x -1716 b \,c^{3} d \,e^{3} f x +2865 c^{4} d^{3} e g x +3531 c^{4} d^{2} e^{2} f x +128 b^{4} e^{4} g -1040 b^{3} c d \,e^{3} g -176 b^{3} c \,e^{4} f +3144 b^{2} c^{2} d^{2} e^{2} g +1320 b^{2} c^{2} d \,e^{3} f -4142 b \,c^{3} d^{3} e g -3498 b \,c^{3} d^{2} e^{2} f +1910 c^{4} d^{4} g +3509 d^{3} f \,c^{4} e \right ) \sqrt {-x^{2} c \,e^{2}-x b \,e^{2}-b d e +c \,d^{2}}}{3465 c^{5} e^{2} \sqrt {e x +d}}\) | \(367\) |
Input:
int((e*x+d)^(5/2)*(g*x+f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2),x,method= _RETURNVERBOSE)
Output:
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 = 0.09 (sec) , antiderivative size = 499, normalized size of antiderivative = 1.45 \[ \int (d+e x)^{5/2} (f+g x) \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2} \, dx=\frac {2 \, {\left (315 \, c^{5} e^{5} g x^{5} + 35 \, {\left (11 \, c^{5} e^{5} f + {\left (32 \, c^{5} d e^{4} + b c^{4} e^{5}\right )} g\right )} x^{4} + 5 \, {\left (11 \, {\left (26 \, c^{5} d e^{4} + b c^{4} e^{5}\right )} f + {\left (256 \, c^{5} d^{2} e^{3} + 49 \, b c^{4} d e^{4} - 8 \, b^{2} c^{3} e^{5}\right )} g\right )} x^{3} + 3 \, {\left (11 \, {\left (52 \, c^{5} d^{2} e^{3} + 13 \, b c^{4} d e^{4} - 2 \, b^{2} c^{3} e^{5}\right )} f + {\left (50 \, c^{5} d^{3} e^{2} + 279 \, b c^{4} d^{2} e^{3} - 114 \, b^{2} c^{3} d e^{4} + 16 \, b^{3} c^{2} e^{5}\right )} g\right )} x^{2} - 11 \, {\left (319 \, c^{5} d^{4} e - 637 \, b c^{4} d^{3} e^{2} + 438 \, b^{2} c^{3} d^{2} e^{3} - 136 \, b^{3} c^{2} d e^{4} + 16 \, b^{4} c e^{5}\right )} f - 2 \, {\left (955 \, c^{5} d^{5} - 3026 \, b c^{4} d^{4} e + 3643 \, b^{2} c^{3} d^{3} e^{2} - 2092 \, b^{3} c^{2} d^{2} e^{3} + 584 \, b^{4} c d e^{4} - 64 \, b^{5} e^{5}\right )} g - {\left (11 \, {\left (2 \, c^{5} d^{3} e^{2} - 159 \, b c^{4} d^{2} e^{3} + 60 \, b^{2} c^{3} d e^{4} - 8 \, b^{3} c^{2} e^{5}\right )} f + {\left (955 \, c^{5} d^{4} e - 2071 \, b c^{4} d^{3} e^{2} + 1572 \, b^{2} c^{3} d^{2} e^{3} - 520 \, b^{3} c^{2} d e^{4} + 64 \, b^{4} c e^{5}\right )} g\right )} x\right )} \sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e} \sqrt {e x + d}}{3465 \, {\left (c^{5} e^{3} x + c^{5} d e^{2}\right )}} \] Input:
integrate((e*x+d)^(5/2)*(g*x+f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2),x, algorithm="fricas")
Output:
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)
\[ \int (d+e x)^{5/2} (f+g x) \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2} \, dx=\int \sqrt {- \left (d + e x\right ) \left (b e - c d + c e x\right )} \left (d + e x\right )^{\frac {5}{2}} \left (f + g x\right )\, dx \] Input:
integrate((e*x+d)**(5/2)*(g*x+f)*(-c*e**2*x**2-b*e**2*x-b*d*e+c*d**2)**(1/ 2),x)
Output:
Integral(sqrt(-(d + e*x)*(b*e - c*d + c*e*x))*(d + e*x)**(5/2)*(f + g*x), x)
Time = 0.12 (sec) , antiderivative size = 501, normalized size of antiderivative = 1.46 \[ \int (d+e x)^{5/2} (f+g x) \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2} \, dx=\frac {2 \, {\left (35 \, c^{4} e^{4} x^{4} - 319 \, c^{4} d^{4} + 637 \, b c^{3} d^{3} e - 438 \, b^{2} c^{2} d^{2} e^{2} + 136 \, b^{3} c d e^{3} - 16 \, b^{4} e^{4} + 5 \, {\left (26 \, c^{4} d e^{3} + b c^{3} e^{4}\right )} x^{3} + 3 \, {\left (52 \, c^{4} d^{2} e^{2} + 13 \, b c^{3} d e^{3} - 2 \, b^{2} c^{2} e^{4}\right )} x^{2} - {\left (2 \, c^{4} d^{3} e - 159 \, b c^{3} d^{2} e^{2} + 60 \, b^{2} c^{2} d e^{3} - 8 \, b^{3} c e^{4}\right )} x\right )} \sqrt {-c e x + c d - b e} {\left (e x + d\right )} f}{315 \, {\left (c^{4} e^{2} x + c^{4} d e\right )}} + \frac {2 \, {\left (315 \, c^{5} e^{5} x^{5} - 1910 \, c^{5} d^{5} + 6052 \, b c^{4} d^{4} e - 7286 \, b^{2} c^{3} d^{3} e^{2} + 4184 \, b^{3} c^{2} d^{2} e^{3} - 1168 \, b^{4} c d e^{4} + 128 \, b^{5} e^{5} + 35 \, {\left (32 \, c^{5} d e^{4} + b c^{4} e^{5}\right )} x^{4} + 5 \, {\left (256 \, c^{5} d^{2} e^{3} + 49 \, b c^{4} d e^{4} - 8 \, b^{2} c^{3} e^{5}\right )} x^{3} + 3 \, {\left (50 \, c^{5} d^{3} e^{2} + 279 \, b c^{4} d^{2} e^{3} - 114 \, b^{2} c^{3} d e^{4} + 16 \, b^{3} c^{2} e^{5}\right )} x^{2} - {\left (955 \, c^{5} d^{4} e - 2071 \, b c^{4} d^{3} e^{2} + 1572 \, b^{2} c^{3} d^{2} e^{3} - 520 \, b^{3} c^{2} d e^{4} + 64 \, b^{4} c e^{5}\right )} x\right )} \sqrt {-c e x + c d - b e} {\left (e x + d\right )} g}{3465 \, {\left (c^{5} e^{3} x + c^{5} d e^{2}\right )}} \] Input:
integrate((e*x+d)^(5/2)*(g*x+f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2),x, algorithm="maxima")
Output:
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. 3520 vs. \(2 (313) = 626\).
Time = 0.37 (sec) , antiderivative size = 3520, normalized size of antiderivative = 10.26 \[ \int (d+e x)^{5/2} (f+g x) \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2} \, dx=\text {Too large to display} \] Input:
integrate((e*x+d)^(5/2)*(g*x+f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2),x, algorithm="giac")
Output:
-2/3465*(3465*sqrt(-c*e*x + c*d - b*e)*c*d^4*e*f - 3465*sqrt(-c*e*x + c*d - b*e)*b*d^3*e^2*f + 2310*(3*sqrt(-c*e*x + c*d - b*e)*c*d - 3*sqrt(-c*e*x + c*d - b*e)*b*e - (-c*e*x + c*d - b*e)^(3/2))*d^3*e*f - 3465*(3*sqrt(-c*e *x + c*d - b*e)*c*d - 3*sqrt(-c*e*x + c*d - b*e)*b*e - (-c*e*x + c*d - b*e )^(3/2))*b*d^2*e^2*f/c + 1155*(3*sqrt(-c*e*x + c*d - b*e)*c*d - 3*sqrt(-c* e*x + c*d - b*e)*b*e - (-c*e*x + c*d - b*e)^(3/2))*d^4*g - 1155*(3*sqrt(-c *e*x + c*d - b*e)*c*d - 3*sqrt(-c*e*x + c*d - b*e)*b*e - (-c*e*x + c*d - b *e)^(3/2))*b*d^3*e*g/c - 693*(15*sqrt(-c*e*x + c*d - b*e)*c^2*d^2 - 30*sqr t(-c*e*x + c*d - b*e)*b*c*d*e + 15*sqrt(-c*e*x + c*d - b*e)*b^2*e^2 - 10*( -c*e*x + c*d - b*e)^(3/2)*c*d + 10*(-c*e*x + c*d - b*e)^(3/2)*b*e + 3*(c*e *x - c*d + b*e)^2*sqrt(-c*e*x + c*d - b*e))*b*d*e^2*f/c^2 + 462*(15*sqrt(- c*e*x + c*d - b*e)*c^2*d^2 - 30*sqrt(-c*e*x + c*d - b*e)*b*c*d*e + 15*sqrt (-c*e*x + c*d - b*e)*b^2*e^2 - 10*(-c*e*x + c*d - b*e)^(3/2)*c*d + 10*(-c* e*x + c*d - b*e)^(3/2)*b*e + 3*(c*e*x - c*d + b*e)^2*sqrt(-c*e*x + c*d - b *e))*d^3*g/c - 693*(15*sqrt(-c*e*x + c*d - b*e)*c^2*d^2 - 30*sqrt(-c*e*x + c*d - b*e)*b*c*d*e + 15*sqrt(-c*e*x + c*d - b*e)*b^2*e^2 - 10*(-c*e*x + c *d - b*e)^(3/2)*c*d + 10*(-c*e*x + c*d - b*e)^(3/2)*b*e + 3*(c*e*x - c*d + b*e)^2*sqrt(-c*e*x + c*d - b*e))*b*d^2*e*g/c^2 - 198*(35*sqrt(-c*e*x + c* d - b*e)*c^3*d^3 - 105*sqrt(-c*e*x + c*d - b*e)*b*c^2*d^2*e + 105*sqrt(-c* e*x + c*d - b*e)*b^2*c*d*e^2 - 35*sqrt(-c*e*x + c*d - b*e)*b^3*e^3 - 35...
Time = 11.57 (sec) , antiderivative size = 501, normalized size of antiderivative = 1.46 \[ \int (d+e x)^{5/2} (f+g x) \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2} \, dx=\frac {\sqrt {c\,d^2-b\,d\,e-c\,e^2\,x^2-b\,e^2\,x}\,\left (\frac {2\,x^3\,\sqrt {d+e\,x}\,\left (-8\,g\,b^2\,e^2+49\,g\,b\,c\,d\,e+11\,f\,b\,c\,e^2+256\,g\,c^2\,d^2+286\,f\,c^2\,d\,e\right )}{693\,c^2}+\frac {2\,e^2\,g\,x^5\,\sqrt {d+e\,x}}{11}+\frac {2\,\left (b\,e-c\,d\right )\,\sqrt {d+e\,x}\,\left (128\,g\,b^4\,e^4-1040\,g\,b^3\,c\,d\,e^3-176\,f\,b^3\,c\,e^4+3144\,g\,b^2\,c^2\,d^2\,e^2+1320\,f\,b^2\,c^2\,d\,e^3-4142\,g\,b\,c^3\,d^3\,e-3498\,f\,b\,c^3\,d^2\,e^2+1910\,g\,c^4\,d^4+3509\,f\,c^4\,d^3\,e\right )}{3465\,c^5\,e^3}-\frac {x\,\sqrt {d+e\,x}\,\left (128\,g\,b^4\,c\,e^5-1040\,g\,b^3\,c^2\,d\,e^4-176\,f\,b^3\,c^2\,e^5+3144\,g\,b^2\,c^3\,d^2\,e^3+1320\,f\,b^2\,c^3\,d\,e^4-4142\,g\,b\,c^4\,d^3\,e^2-3498\,f\,b\,c^4\,d^2\,e^3+1910\,g\,c^5\,d^4\,e+44\,f\,c^5\,d^3\,e^2\right )}{3465\,c^5\,e^3}+\frac {x^2\,\sqrt {d+e\,x}\,\left (96\,g\,b^3\,c^2\,e^5-684\,g\,b^2\,c^3\,d\,e^4-132\,f\,b^2\,c^3\,e^5+1674\,g\,b\,c^4\,d^2\,e^3+858\,f\,b\,c^4\,d\,e^4+300\,g\,c^5\,d^3\,e^2+3432\,f\,c^5\,d^2\,e^3\right )}{3465\,c^5\,e^3}+\frac {2\,e\,x^4\,\sqrt {d+e\,x}\,\left (b\,e\,g+32\,c\,d\,g+11\,c\,e\,f\right )}{99\,c}\right )}{x+\frac {d}{e}} \] Input:
int((f + g*x)*(d + e*x)^(5/2)*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(1/2), x)
Output:
((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)
Time = 0.25 (sec) , antiderivative size = 500, normalized size of antiderivative = 1.46 \[ \int (d+e x)^{5/2} (f+g x) \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2} \, dx=\frac {2 \sqrt {-c e x -b e +c d}\, \left (315 c^{5} e^{5} g \,x^{5}+35 b \,c^{4} e^{5} g \,x^{4}+1120 c^{5} d \,e^{4} g \,x^{4}+385 c^{5} e^{5} f \,x^{4}-40 b^{2} c^{3} e^{5} g \,x^{3}+245 b \,c^{4} d \,e^{4} g \,x^{3}+55 b \,c^{4} e^{5} f \,x^{3}+1280 c^{5} d^{2} e^{3} g \,x^{3}+1430 c^{5} d \,e^{4} f \,x^{3}+48 b^{3} c^{2} e^{5} g \,x^{2}-342 b^{2} c^{3} d \,e^{4} g \,x^{2}-66 b^{2} c^{3} e^{5} f \,x^{2}+837 b \,c^{4} d^{2} e^{3} g \,x^{2}+429 b \,c^{4} d \,e^{4} f \,x^{2}+150 c^{5} d^{3} e^{2} g \,x^{2}+1716 c^{5} d^{2} e^{3} f \,x^{2}-64 b^{4} c \,e^{5} g x +520 b^{3} c^{2} d \,e^{4} g x +88 b^{3} c^{2} e^{5} f x -1572 b^{2} c^{3} d^{2} e^{3} g x -660 b^{2} c^{3} d \,e^{4} f x +2071 b \,c^{4} d^{3} e^{2} g x +1749 b \,c^{4} d^{2} e^{3} f x -955 c^{5} d^{4} e g x -22 c^{5} d^{3} e^{2} f x +128 b^{5} e^{5} g -1168 b^{4} c d \,e^{4} g -176 b^{4} c \,e^{5} f +4184 b^{3} c^{2} d^{2} e^{3} g +1496 b^{3} c^{2} d \,e^{4} f -7286 b^{2} c^{3} d^{3} e^{2} g -4818 b^{2} c^{3} d^{2} e^{3} f +6052 b \,c^{4} d^{4} e g +7007 b \,c^{4} d^{3} e^{2} f -1910 c^{5} d^{5} g -3509 c^{5} d^{4} e f \right )}{3465 c^{5} e^{2}} \] Input:
int((e*x+d)^(5/2)*(g*x+f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2),x)
Output:
(2*sqrt( - b*e + c*d - c*e*x)*(128*b**5*e**5*g - 1168*b**4*c*d*e**4*g - 17 6*b**4*c*e**5*f - 64*b**4*c*e**5*g*x + 4184*b**3*c**2*d**2*e**3*g + 1496*b **3*c**2*d*e**4*f + 520*b**3*c**2*d*e**4*g*x + 88*b**3*c**2*e**5*f*x + 48* b**3*c**2*e**5*g*x**2 - 7286*b**2*c**3*d**3*e**2*g - 4818*b**2*c**3*d**2*e **3*f - 1572*b**2*c**3*d**2*e**3*g*x - 660*b**2*c**3*d*e**4*f*x - 342*b**2 *c**3*d*e**4*g*x**2 - 66*b**2*c**3*e**5*f*x**2 - 40*b**2*c**3*e**5*g*x**3 + 6052*b*c**4*d**4*e*g + 7007*b*c**4*d**3*e**2*f + 2071*b*c**4*d**3*e**2*g *x + 1749*b*c**4*d**2*e**3*f*x + 837*b*c**4*d**2*e**3*g*x**2 + 429*b*c**4* d*e**4*f*x**2 + 245*b*c**4*d*e**4*g*x**3 + 55*b*c**4*e**5*f*x**3 + 35*b*c* *4*e**5*g*x**4 - 1910*c**5*d**5*g - 3509*c**5*d**4*e*f - 955*c**5*d**4*e*g *x - 22*c**5*d**3*e**2*f*x + 150*c**5*d**3*e**2*g*x**2 + 1716*c**5*d**2*e* *3*f*x**2 + 1280*c**5*d**2*e**3*g*x**3 + 1430*c**5*d*e**4*f*x**3 + 1120*c* *5*d*e**4*g*x**4 + 385*c**5*e**5*f*x**4 + 315*c**5*e**5*g*x**5))/(3465*c** 5*e**2)