Integrand size = 46, antiderivative size = 267 \[ \int \sqrt {d+e x} (f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2} \, dx=-\frac {2 (2 c d-b e)^2 (c e f+c d g-b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{5 c^4 e^2 (d+e x)^{5/2}}+\frac {2 (2 c d-b e) (2 c e f+4 c d g-3 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{7 c^4 e^2 (d+e x)^{7/2}}-\frac {2 (c e f+5 c d g-3 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{9/2}}{9 c^4 e^2 (d+e x)^{9/2}}+\frac {2 g \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{11/2}}{11 c^4 e^2 (d+e x)^{11/2}} \] Output:
-2/5*(-b*e+2*c*d)^2*(-b*e*g+c*d*g+c*e*f)*(d*(-b*e+c*d)-b*e^2*x-c*e^2*x^2)^ (5/2)/c^4/e^2/(e*x+d)^(5/2)+2/7*(-b*e+2*c*d)*(-3*b*e*g+4*c*d*g+2*c*e*f)*(d *(-b*e+c*d)-b*e^2*x-c*e^2*x^2)^(7/2)/c^4/e^2/(e*x+d)^(7/2)-2/9*(-3*b*e*g+5 *c*d*g+c*e*f)*(d*(-b*e+c*d)-b*e^2*x-c*e^2*x^2)^(9/2)/c^4/e^2/(e*x+d)^(9/2) +2/11*g*(d*(-b*e+c*d)-b*e^2*x-c*e^2*x^2)^(11/2)/c^4/e^2/(e*x+d)^(11/2)
Time = 0.24 (sec) , antiderivative size = 183, normalized size of antiderivative = 0.69 \[ \int \sqrt {d+e x} (f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2} \, dx=-\frac {2 (-c d+b e+c e x)^2 \sqrt {(d+e x) (-b e+c (d-e x))} \left (-48 b^3 e^3 g+8 b^2 c e^2 (11 e f+40 d g+15 e g x)-2 b c^2 e \left (347 d^2 g+5 e^2 x (22 f+21 g x)+d e (286 f+340 g x)\right )+c^3 \left (422 d^3 g+35 e^3 x^2 (11 f+9 g x)+10 d e^2 x (121 f+98 g x)+d^2 e (1177 f+1055 g x)\right )\right )}{3465 c^4 e^2 \sqrt {d+e x}} \] Input:
Integrate[Sqrt[d + e*x]*(f + g*x)*(c*d^2 - b*d*e - b*e^2*x - c*e^2*x^2)^(3 /2),x]
Output:
(-2*(-(c*d) + b*e + c*e*x)^2*Sqrt[(d + e*x)*(-(b*e) + c*(d - e*x))]*(-48*b ^3*e^3*g + 8*b^2*c*e^2*(11*e*f + 40*d*g + 15*e*g*x) - 2*b*c^2*e*(347*d^2*g + 5*e^2*x*(22*f + 21*g*x) + d*e*(286*f + 340*g*x)) + c^3*(422*d^3*g + 35* e^3*x^2*(11*f + 9*g*x) + 10*d*e^2*x*(121*f + 98*g*x) + d^2*e*(1177*f + 105 5*g*x))))/(3465*c^4*e^2*Sqrt[d + e*x])
Time = 0.81 (sec) , antiderivative size = 254, normalized size of antiderivative = 0.95, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {1221, 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.
| \(\displaystyle \int \sqrt {d+e x} (f+g x) \left (-b d e-b e^2 x+c d^2-c e^2 x^2\right )^{3/2} \, dx\) |
| \(\Big \downarrow \) 1221 |
| \(\displaystyle \frac {(-6 b e g+c d g+11 c e f) \int \sqrt {d+e x} \left (-c x^2 e^2-b x e^2+d (c d-b e)\right )^{3/2}dx}{11 c e}-\frac {2 g \sqrt {d+e x} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{11 c e^2}\) |
| \(\Big \downarrow \) 1128 |
| \(\displaystyle \frac {(-6 b e g+c d g+11 c e f) \left (\frac {4 (2 c d-b e) \int \frac {\left (-c x^2 e^2-b x e^2+d (c d-b e)\right )^{3/2}}{\sqrt {d+e x}}dx}{9 c}-\frac {2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{9 c e \sqrt {d+e x}}\right )}{11 c e}-\frac {2 g \sqrt {d+e x} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{11 c e^2}\) |
| \(\Big \downarrow \) 1128 |
| \(\displaystyle \frac {(-6 b e g+c d g+11 c e f) \left (\frac {4 (2 c d-b e) \left (\frac {2 (2 c d-b e) \int \frac {\left (-c x^2 e^2-b x e^2+d (c d-b e)\right )^{3/2}}{(d+e x)^{3/2}}dx}{7 c}-\frac {2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{7 c e (d+e x)^{3/2}}\right )}{9 c}-\frac {2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{9 c e \sqrt {d+e x}}\right )}{11 c e}-\frac {2 g \sqrt {d+e x} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{11 c e^2}\) |
| \(\Big \downarrow \) 1122 |
| \(\displaystyle \frac {\left (\frac {4 (2 c d-b e) \left (-\frac {4 (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{35 c^2 e (d+e x)^{5/2}}-\frac {2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{7 c e (d+e x)^{3/2}}\right )}{9 c}-\frac {2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{9 c e \sqrt {d+e x}}\right ) (-6 b e g+c d g+11 c e f)}{11 c e}-\frac {2 g \sqrt {d+e x} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{11 c e^2}\) |
Input:
Int[Sqrt[d + e*x]*(f + g*x)*(c*d^2 - b*d*e - b*e^2*x - c*e^2*x^2)^(3/2),x]
Output:
(-2*g*Sqrt[d + e*x]*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(5/2))/(11*c*e^2 ) + ((11*c*e*f + c*d*g - 6*b*e*g)*((-2*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^ 2)^(5/2))/(9*c*e*Sqrt[d + e*x]) + (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)^(5/2))/(35*c^2*e*(d + e*x)^(5/2)) - (2*( d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(5/2))/(7*c*e*(d + e*x)^(3/2))))/(9*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 = 2.02 (sec) , antiderivative size = 229, normalized size of antiderivative = 0.86
| method | result | size |
| default | \(\frac {2 \sqrt {-\left (e x +d \right ) \left (c e x +b e -c d \right )}\, \left (c e x +b e -c d \right )^{2} \left (-315 e^{3} g \,x^{3} c^{3}+210 b \,c^{2} e^{3} g \,x^{2}-980 c^{3} d \,e^{2} g \,x^{2}-385 c^{3} e^{3} f \,x^{2}-120 b^{2} c \,e^{3} g x +680 b \,c^{2} d \,e^{2} g x +220 b \,c^{2} e^{3} f x -1055 c^{3} d^{2} e g x -1210 c^{3} d \,e^{2} f x +48 b^{3} e^{3} g -320 b^{2} c d \,e^{2} g -88 b^{2} c \,e^{3} f +694 b \,c^{2} d^{2} e g +572 b \,c^{2} d \,e^{2} f -422 c^{3} d^{3} g -1177 d^{2} f \,c^{3} e \right )}{3465 \sqrt {e x +d}\, c^{4} e^{2}}\) | \(229\) |
| gosper | \(-\frac {2 \left (c e x +b e -c d \right ) \left (-315 e^{3} g \,x^{3} c^{3}+210 b \,c^{2} e^{3} g \,x^{2}-980 c^{3} d \,e^{2} g \,x^{2}-385 c^{3} e^{3} f \,x^{2}-120 b^{2} c \,e^{3} g x +680 b \,c^{2} d \,e^{2} g x +220 b \,c^{2} e^{3} f x -1055 c^{3} d^{2} e g x -1210 c^{3} d \,e^{2} f x +48 b^{3} e^{3} g -320 b^{2} c d \,e^{2} g -88 b^{2} c \,e^{3} f +694 b \,c^{2} d^{2} e g +572 b \,c^{2} d \,e^{2} f -422 c^{3} d^{3} g -1177 d^{2} f \,c^{3} e \right ) \left (-x^{2} c \,e^{2}-x b \,e^{2}-b d e +c \,d^{2}\right )^{\frac {3}{2}}}{3465 c^{4} e^{2} \left (e x +d \right )^{\frac {3}{2}}}\) | \(235\) |
| orering | \(-\frac {2 \left (c e x +b e -c d \right ) \left (-315 e^{3} g \,x^{3} c^{3}+210 b \,c^{2} e^{3} g \,x^{2}-980 c^{3} d \,e^{2} g \,x^{2}-385 c^{3} e^{3} f \,x^{2}-120 b^{2} c \,e^{3} g x +680 b \,c^{2} d \,e^{2} g x +220 b \,c^{2} e^{3} f x -1055 c^{3} d^{2} e g x -1210 c^{3} d \,e^{2} f x +48 b^{3} e^{3} g -320 b^{2} c d \,e^{2} g -88 b^{2} c \,e^{3} f +694 b \,c^{2} d^{2} e g +572 b \,c^{2} d \,e^{2} f -422 c^{3} d^{3} g -1177 d^{2} f \,c^{3} e \right ) \left (-x^{2} c \,e^{2}-x b \,e^{2}-b d e +c \,d^{2}\right )^{\frac {3}{2}}}{3465 c^{4} e^{2} \left (e x +d \right )^{\frac {3}{2}}}\) | \(235\) |
Input:
int((e*x+d)^(1/2)*(g*x+f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2),x,method= _RETURNVERBOSE)
Output:
2/3465/(e*x+d)^(1/2)*(-(e*x+d)*(c*e*x+b*e-c*d))^(1/2)*(c*e*x+b*e-c*d)^2*(- 315*c^3*e^3*g*x^3+210*b*c^2*e^3*g*x^2-980*c^3*d*e^2*g*x^2-385*c^3*e^3*f*x^ 2-120*b^2*c*e^3*g*x+680*b*c^2*d*e^2*g*x+220*b*c^2*e^3*f*x-1055*c^3*d^2*e*g *x-1210*c^3*d*e^2*f*x+48*b^3*e^3*g-320*b^2*c*d*e^2*g-88*b^2*c*e^3*f+694*b* c^2*d^2*e*g+572*b*c^2*d*e^2*f-422*c^3*d^3*g-1177*c^3*d^2*e*f)/c^4/e^2
Leaf count of result is larger than twice the leaf count of optimal. 505 vs. \(2 (243) = 486\).
Time = 0.09 (sec) , antiderivative size = 505, normalized size of antiderivative = 1.89 \[ \int \sqrt {d+e x} (f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2} \, dx=-\frac {2 \, {\left (315 \, c^{5} e^{5} g x^{5} + 35 \, {\left (11 \, c^{5} e^{5} f + 2 \, {\left (5 \, c^{5} d e^{4} + 6 \, b c^{4} e^{5}\right )} g\right )} x^{4} + 5 \, {\left (22 \, {\left (4 \, c^{5} d e^{4} + 5 \, b c^{4} e^{5}\right )} f - {\left (118 \, c^{5} d^{2} e^{3} - 214 \, b c^{4} d e^{4} - 3 \, b^{2} c^{3} e^{5}\right )} g\right )} x^{3} - 3 \, {\left (11 \, {\left (26 \, c^{5} d^{2} e^{3} - 46 \, b c^{4} d e^{4} - b^{2} c^{3} e^{5}\right )} f + 2 \, {\left (118 \, c^{5} d^{3} e^{2} - 101 \, b c^{4} d^{2} e^{3} - 20 \, b^{2} c^{3} d e^{4} + 3 \, b^{3} c^{2} e^{5}\right )} g\right )} x^{2} + 11 \, {\left (107 \, c^{5} d^{4} e - 266 \, b c^{4} d^{3} e^{2} + 219 \, b^{2} c^{3} d^{2} e^{3} - 68 \, b^{3} c^{2} d e^{4} + 8 \, b^{4} c e^{5}\right )} f + 2 \, {\left (211 \, c^{5} d^{5} - 769 \, b c^{4} d^{4} e + 1065 \, b^{2} c^{3} d^{3} e^{2} - 691 \, b^{3} c^{2} d^{2} e^{3} + 208 \, b^{4} c d e^{4} - 24 \, b^{5} e^{5}\right )} g - {\left (22 \, {\left (52 \, c^{5} d^{3} e^{2} - 39 \, b c^{4} d^{2} e^{3} - 15 \, b^{2} c^{3} d e^{4} + 2 \, b^{3} c^{2} e^{5}\right )} f - {\left (211 \, c^{5} d^{4} e - 558 \, b c^{4} d^{3} e^{2} + 507 \, b^{2} c^{3} d^{2} e^{3} - 184 \, b^{3} c^{2} d e^{4} + 24 \, b^{4} c e^{5}\right )} g\right )} x\right )} \sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e} \sqrt {e x + d}}{3465 \, {\left (c^{4} e^{3} x + c^{4} d e^{2}\right )}} \] Input:
integrate((e*x+d)^(1/2)*(g*x+f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2),x, algorithm="fricas")
Output:
-2/3465*(315*c^5*e^5*g*x^5 + 35*(11*c^5*e^5*f + 2*(5*c^5*d*e^4 + 6*b*c^4*e ^5)*g)*x^4 + 5*(22*(4*c^5*d*e^4 + 5*b*c^4*e^5)*f - (118*c^5*d^2*e^3 - 214* b*c^4*d*e^4 - 3*b^2*c^3*e^5)*g)*x^3 - 3*(11*(26*c^5*d^2*e^3 - 46*b*c^4*d*e ^4 - b^2*c^3*e^5)*f + 2*(118*c^5*d^3*e^2 - 101*b*c^4*d^2*e^3 - 20*b^2*c^3* d*e^4 + 3*b^3*c^2*e^5)*g)*x^2 + 11*(107*c^5*d^4*e - 266*b*c^4*d^3*e^2 + 21 9*b^2*c^3*d^2*e^3 - 68*b^3*c^2*d*e^4 + 8*b^4*c*e^5)*f + 2*(211*c^5*d^5 - 7 69*b*c^4*d^4*e + 1065*b^2*c^3*d^3*e^2 - 691*b^3*c^2*d^2*e^3 + 208*b^4*c*d* e^4 - 24*b^5*e^5)*g - (22*(52*c^5*d^3*e^2 - 39*b*c^4*d^2*e^3 - 15*b^2*c^3* d*e^4 + 2*b^3*c^2*e^5)*f - (211*c^5*d^4*e - 558*b*c^4*d^3*e^2 + 507*b^2*c^ 3*d^2*e^3 - 184*b^3*c^2*d*e^4 + 24*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^4*e^3*x + c^4*d*e^2)
\[ \int \sqrt {d+e x} (f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2} \, dx=\int \left (- \left (d + e x\right ) \left (b e - c d + c e x\right )\right )^{\frac {3}{2}} \sqrt {d + e x} \left (f + g x\right )\, dx \] Input:
integrate((e*x+d)**(1/2)*(g*x+f)*(-c*e**2*x**2-b*e**2*x-b*d*e+c*d**2)**(3/ 2),x)
Output:
Integral((-(d + e*x)*(b*e - c*d + c*e*x))**(3/2)*sqrt(d + e*x)*(f + g*x), x)
Leaf count of result is larger than twice the leaf count of optimal. 502 vs. \(2 (243) = 486\).
Time = 0.10 (sec) , antiderivative size = 502, normalized size of antiderivative = 1.88 \[ \int \sqrt {d+e x} (f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2} \, dx=-\frac {2 \, {\left (35 \, c^{4} e^{4} x^{4} + 107 \, c^{4} d^{4} - 266 \, b c^{3} d^{3} e + 219 \, b^{2} c^{2} d^{2} e^{2} - 68 \, b^{3} c d e^{3} + 8 \, b^{4} e^{4} + 10 \, {\left (4 \, c^{4} d e^{3} + 5 \, b c^{3} e^{4}\right )} x^{3} - 3 \, {\left (26 \, c^{4} d^{2} e^{2} - 46 \, b c^{3} d e^{3} - b^{2} c^{2} e^{4}\right )} x^{2} - 2 \, {\left (52 \, c^{4} d^{3} e - 39 \, b c^{3} d^{2} e^{2} - 15 \, b^{2} c^{2} d e^{3} + 2 \, b^{3} c e^{4}\right )} x\right )} \sqrt {-c e x + c d - b e} {\left (e x + d\right )} f}{315 \, {\left (c^{3} e^{2} x + c^{3} d e\right )}} - \frac {2 \, {\left (315 \, c^{5} e^{5} x^{5} + 422 \, c^{5} d^{5} - 1538 \, b c^{4} d^{4} e + 2130 \, b^{2} c^{3} d^{3} e^{2} - 1382 \, b^{3} c^{2} d^{2} e^{3} + 416 \, b^{4} c d e^{4} - 48 \, b^{5} e^{5} + 70 \, {\left (5 \, c^{5} d e^{4} + 6 \, b c^{4} e^{5}\right )} x^{4} - 5 \, {\left (118 \, c^{5} d^{2} e^{3} - 214 \, b c^{4} d e^{4} - 3 \, b^{2} c^{3} e^{5}\right )} x^{3} - 6 \, {\left (118 \, c^{5} d^{3} e^{2} - 101 \, b c^{4} d^{2} e^{3} - 20 \, b^{2} c^{3} d e^{4} + 3 \, b^{3} c^{2} e^{5}\right )} x^{2} + {\left (211 \, c^{5} d^{4} e - 558 \, b c^{4} d^{3} e^{2} + 507 \, b^{2} c^{3} d^{2} e^{3} - 184 \, b^{3} c^{2} d e^{4} + 24 \, b^{4} c e^{5}\right )} x\right )} \sqrt {-c e x + c d - b e} {\left (e x + d\right )} g}{3465 \, {\left (c^{4} e^{3} x + c^{4} d e^{2}\right )}} \] Input:
integrate((e*x+d)^(1/2)*(g*x+f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2),x, algorithm="maxima")
Output:
-2/315*(35*c^4*e^4*x^4 + 107*c^4*d^4 - 266*b*c^3*d^3*e + 219*b^2*c^2*d^2*e ^2 - 68*b^3*c*d*e^3 + 8*b^4*e^4 + 10*(4*c^4*d*e^3 + 5*b*c^3*e^4)*x^3 - 3*( 26*c^4*d^2*e^2 - 46*b*c^3*d*e^3 - b^2*c^2*e^4)*x^2 - 2*(52*c^4*d^3*e - 39* b*c^3*d^2*e^2 - 15*b^2*c^2*d*e^3 + 2*b^3*c*e^4)*x)*sqrt(-c*e*x + c*d - b*e )*(e*x + d)*f/(c^3*e^2*x + c^3*d*e) - 2/3465*(315*c^5*e^5*x^5 + 422*c^5*d^ 5 - 1538*b*c^4*d^4*e + 2130*b^2*c^3*d^3*e^2 - 1382*b^3*c^2*d^2*e^3 + 416*b ^4*c*d*e^4 - 48*b^5*e^5 + 70*(5*c^5*d*e^4 + 6*b*c^4*e^5)*x^4 - 5*(118*c^5* d^2*e^3 - 214*b*c^4*d*e^4 - 3*b^2*c^3*e^5)*x^3 - 6*(118*c^5*d^3*e^2 - 101* b*c^4*d^2*e^3 - 20*b^2*c^3*d*e^4 + 3*b^3*c^2*e^5)*x^2 + (211*c^5*d^4*e - 5 58*b*c^4*d^3*e^2 + 507*b^2*c^3*d^2*e^3 - 184*b^3*c^2*d*e^4 + 24*b^4*c*e^5) *x)*sqrt(-c*e*x + c*d - b*e)*(e*x + d)*g/(c^4*e^3*x + c^4*d*e^2)
Leaf count of result is larger than twice the leaf count of optimal. 3758 vs. \(2 (243) = 486\).
Time = 0.37 (sec) , antiderivative size = 3758, normalized size of antiderivative = 14.07 \[ \int \sqrt {d+e x} (f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2} \, dx=\text {Too large to display} \] Input:
integrate((e*x+d)^(1/2)*(g*x+f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2),x, algorithm="giac")
Output:
-2/3465*(3465*sqrt(-c*e*x + c*d - b*e)*c^2*d^4*e*f - 6930*sqrt(-c*e*x + c* d - b*e)*b*c*d^3*e^2*f + 3465*sqrt(-c*e*x + c*d - b*e)*b^2*d^2*e^3*f - 231 0*(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 + 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))*b^2*d*e^3*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))*c*d^4*g - 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))*b*d^3*e *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^2*d^2*e^2*g/c - 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^2*e* f + 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*sqr t(-c*e*x + c*d - b*e))*b*d*e^2*f/c + 231*(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^2*e^3*f/c^2 -...
Time = 12.11 (sec) , antiderivative size = 441, normalized size of antiderivative = 1.65 \[ \int \sqrt {d+e x} (f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2} \, dx=-\frac {\sqrt {c\,d^2-b\,d\,e-c\,e^2\,x^2-b\,e^2\,x}\,\left (\frac {2\,e\,x^4\,\sqrt {d+e\,x}\,\left (12\,b\,e\,g+10\,c\,d\,g+11\,c\,e\,f\right )}{99}+\frac {2\,x^3\,\sqrt {d+e\,x}\,\left (3\,g\,b^2\,e^2+214\,g\,b\,c\,d\,e+110\,f\,b\,c\,e^2-118\,g\,c^2\,d^2+88\,f\,c^2\,d\,e\right )}{693\,c}+\frac {2\,c\,e^2\,g\,x^5\,\sqrt {d+e\,x}}{11}+\frac {x^2\,\sqrt {d+e\,x}\,\left (-36\,g\,b^3\,c^2\,e^5+240\,g\,b^2\,c^3\,d\,e^4+66\,f\,b^2\,c^3\,e^5+1212\,g\,b\,c^4\,d^2\,e^3+3036\,f\,b\,c^4\,d\,e^4-1416\,g\,c^5\,d^3\,e^2-1716\,f\,c^5\,d^2\,e^3\right )}{3465\,c^4\,e^3}+\frac {2\,{\left (b\,e-c\,d\right )}^2\,\sqrt {d+e\,x}\,\left (-48\,g\,b^3\,e^3+320\,g\,b^2\,c\,d\,e^2+88\,f\,b^2\,c\,e^3-694\,g\,b\,c^2\,d^2\,e-572\,f\,b\,c^2\,d\,e^2+422\,g\,c^3\,d^3+1177\,f\,c^3\,d^2\,e\right )}{3465\,c^4\,e^3}+\frac {2\,x\,\left (b\,e-c\,d\right )\,\sqrt {d+e\,x}\,\left (24\,g\,b^3\,e^3-160\,g\,b^2\,c\,d\,e^2-44\,f\,b^2\,c\,e^3+347\,g\,b\,c^2\,d^2\,e+286\,f\,b\,c^2\,d\,e^2-211\,g\,c^3\,d^3+1144\,f\,c^3\,d^2\,e\right )}{3465\,c^3\,e^2}\right )}{x+\frac {d}{e}} \] Input:
int((f + g*x)*(d + e*x)^(1/2)*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(3/2), x)
Output:
-((c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(1/2)*((2*e*x^4*(d + e*x)^(1/2)*(1 2*b*e*g + 10*c*d*g + 11*c*e*f))/99 + (2*x^3*(d + e*x)^(1/2)*(3*b^2*e^2*g - 118*c^2*d^2*g + 110*b*c*e^2*f + 88*c^2*d*e*f + 214*b*c*d*e*g))/(693*c) + (2*c*e^2*g*x^5*(d + e*x)^(1/2))/11 + (x^2*(d + e*x)^(1/2)*(66*b^2*c^3*e^5* f - 36*b^3*c^2*e^5*g - 1716*c^5*d^2*e^3*f - 1416*c^5*d^3*e^2*g + 3036*b*c^ 4*d*e^4*f + 1212*b*c^4*d^2*e^3*g + 240*b^2*c^3*d*e^4*g))/(3465*c^4*e^3) + (2*(b*e - c*d)^2*(d + e*x)^(1/2)*(422*c^3*d^3*g - 48*b^3*e^3*g + 88*b^2*c* e^3*f + 1177*c^3*d^2*e*f - 572*b*c^2*d*e^2*f - 694*b*c^2*d^2*e*g + 320*b^2 *c*d*e^2*g))/(3465*c^4*e^3) + (2*x*(b*e - c*d)*(d + e*x)^(1/2)*(24*b^3*e^3 *g - 211*c^3*d^3*g - 44*b^2*c*e^3*f + 1144*c^3*d^2*e*f + 286*b*c^2*d*e^2*f + 347*b*c^2*d^2*e*g - 160*b^2*c*d*e^2*g))/(3465*c^3*e^2)))/(x + d/e)
Time = 0.25 (sec) , antiderivative size = 500, normalized size of antiderivative = 1.87 \[ \int \sqrt {d+e x} (f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2} \, dx=\frac {2 \sqrt {-c e x -b e +c d}\, \left (-315 c^{5} e^{5} g \,x^{5}-420 b \,c^{4} e^{5} g \,x^{4}-350 c^{5} d \,e^{4} g \,x^{4}-385 c^{5} e^{5} f \,x^{4}-15 b^{2} c^{3} e^{5} g \,x^{3}-1070 b \,c^{4} d \,e^{4} g \,x^{3}-550 b \,c^{4} e^{5} f \,x^{3}+590 c^{5} d^{2} e^{3} g \,x^{3}-440 c^{5} d \,e^{4} f \,x^{3}+18 b^{3} c^{2} e^{5} g \,x^{2}-120 b^{2} c^{3} d \,e^{4} g \,x^{2}-33 b^{2} c^{3} e^{5} f \,x^{2}-606 b \,c^{4} d^{2} e^{3} g \,x^{2}-1518 b \,c^{4} d \,e^{4} f \,x^{2}+708 c^{5} d^{3} e^{2} g \,x^{2}+858 c^{5} d^{2} e^{3} f \,x^{2}-24 b^{4} c \,e^{5} g x +184 b^{3} c^{2} d \,e^{4} g x +44 b^{3} c^{2} e^{5} f x -507 b^{2} c^{3} d^{2} e^{3} g x -330 b^{2} c^{3} d \,e^{4} f x +558 b \,c^{4} d^{3} e^{2} g x -858 b \,c^{4} d^{2} e^{3} f x -211 c^{5} d^{4} e g x +1144 c^{5} d^{3} e^{2} f x +48 b^{5} e^{5} g -416 b^{4} c d \,e^{4} g -88 b^{4} c \,e^{5} f +1382 b^{3} c^{2} d^{2} e^{3} g +748 b^{3} c^{2} d \,e^{4} f -2130 b^{2} c^{3} d^{3} e^{2} g -2409 b^{2} c^{3} d^{2} e^{3} f +1538 b \,c^{4} d^{4} e g +2926 b \,c^{4} d^{3} e^{2} f -422 c^{5} d^{5} g -1177 c^{5} d^{4} e f \right )}{3465 c^{4} e^{2}} \] Input:
int((e*x+d)^(1/2)*(g*x+f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2),x)
Output:
(2*sqrt( - b*e + c*d - c*e*x)*(48*b**5*e**5*g - 416*b**4*c*d*e**4*g - 88*b **4*c*e**5*f - 24*b**4*c*e**5*g*x + 1382*b**3*c**2*d**2*e**3*g + 748*b**3* c**2*d*e**4*f + 184*b**3*c**2*d*e**4*g*x + 44*b**3*c**2*e**5*f*x + 18*b**3 *c**2*e**5*g*x**2 - 2130*b**2*c**3*d**3*e**2*g - 2409*b**2*c**3*d**2*e**3* f - 507*b**2*c**3*d**2*e**3*g*x - 330*b**2*c**3*d*e**4*f*x - 120*b**2*c**3 *d*e**4*g*x**2 - 33*b**2*c**3*e**5*f*x**2 - 15*b**2*c**3*e**5*g*x**3 + 153 8*b*c**4*d**4*e*g + 2926*b*c**4*d**3*e**2*f + 558*b*c**4*d**3*e**2*g*x - 8 58*b*c**4*d**2*e**3*f*x - 606*b*c**4*d**2*e**3*g*x**2 - 1518*b*c**4*d*e**4 *f*x**2 - 1070*b*c**4*d*e**4*g*x**3 - 550*b*c**4*e**5*f*x**3 - 420*b*c**4* e**5*g*x**4 - 422*c**5*d**5*g - 1177*c**5*d**4*e*f - 211*c**5*d**4*e*g*x + 1144*c**5*d**3*e**2*f*x + 708*c**5*d**3*e**2*g*x**2 + 858*c**5*d**2*e**3* f*x**2 + 590*c**5*d**2*e**3*g*x**3 - 440*c**5*d*e**4*f*x**3 - 350*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**4*e**2 )