Integrand size = 46, antiderivative size = 190 \[ \int \sqrt {d+e x} (f+g x) \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2} \, dx=-\frac {2 (2 c d-b e) (c e f+c d g-b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{3 c^3 e^2 (d+e x)^{3/2}}+\frac {2 (c e f+3 c d g-2 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{5 c^3 e^2 (d+e x)^{5/2}}-\frac {2 g \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{7 c^3 e^2 (d+e x)^{7/2}} \] Output:
-2/3*(-b*e+2*c*d)*(-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^3/e^2/(e*x+d)^(3/2)+2/5*(-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)^(5/2)/c^3/e^2/(e*x+d)^(5/2)-2/7*g*(d*(-b*e+c*d)-b*e^2*x-c*e^ 2*x^2)^(7/2)/c^3/e^2/(e*x+d)^(7/2)
Time = 0.13 (sec) , antiderivative size = 119, normalized size of antiderivative = 0.63 \[ \int \sqrt {d+e x} (f+g x) \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2} \, dx=\frac {2 (-c d+b e+c e x) \sqrt {(d+e x) (-b e+c (d-e x))} \left (8 b^2 e^2 g-2 b c e (7 e f+15 d g+6 e g x)+c^2 \left (22 d^2 g+3 e^2 x (7 f+5 g x)+d e (49 f+33 g x)\right )\right )}{105 c^3 e^2 \sqrt {d+e x}} \] Input:
Integrate[Sqrt[d + e*x]*(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))]*(8*b^2*e^ 2*g - 2*b*c*e*(7*e*f + 15*d*g + 6*e*g*x) + c^2*(22*d^2*g + 3*e^2*x*(7*f + 5*g*x) + d*e*(49*f + 33*g*x))))/(105*c^3*e^2*Sqrt[d + e*x])
Time = 0.67 (sec) , antiderivative size = 187, normalized size of antiderivative = 0.98, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.065, Rules used = {1221, 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.
| \(\displaystyle \int \sqrt {d+e x} (f+g x) \sqrt {-b d e-b e^2 x+c d^2-c e^2 x^2} \, dx\) |
| \(\Big \downarrow \) 1221 |
| \(\displaystyle \frac {(-4 b e g+c d g+7 c e f) \int \sqrt {d+e x} \sqrt {-c x^2 e^2-b x e^2+d (c d-b e)}dx}{7 c e}-\frac {2 g \sqrt {d+e x} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{7 c e^2}\) |
| \(\Big \downarrow \) 1128 |
| \(\displaystyle \frac {(-4 b e g+c d g+7 c e f) \left (\frac {2 (2 c d-b e) \int \frac {\sqrt {-c x^2 e^2-b x e^2+d (c d-b e)}}{\sqrt {d+e x}}dx}{5 c}-\frac {2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{5 c e \sqrt {d+e x}}\right )}{7 c e}-\frac {2 g \sqrt {d+e x} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{7 c e^2}\) |
| \(\Big \downarrow \) 1122 |
| \(\displaystyle \frac {\left (-\frac {4 (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{15 c^2 e (d+e x)^{3/2}}-\frac {2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{5 c e \sqrt {d+e x}}\right ) (-4 b e g+c d g+7 c e f)}{7 c e}-\frac {2 g \sqrt {d+e x} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{7 c e^2}\) |
Input:
Int[Sqrt[d + e*x]*(f + g*x)*Sqrt[c*d^2 - b*d*e - b*e^2*x - c*e^2*x^2],x]
Output:
(-2*g*Sqrt[d + e*x]*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(3/2))/(7*c*e^2) + ((7*c*e*f + c*d*g - 4*b*e*g)*((-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*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 = 2.48 (sec) , antiderivative size = 131, normalized size of antiderivative = 0.69
| method | result | size |
| default | \(\frac {2 \left (c e x +b e -c d \right ) \left (15 g \,x^{2} c^{2} e^{2}-12 b c \,e^{2} g x +33 c^{2} d e g x +21 c^{2} e^{2} f x +8 b^{2} e^{2} g -30 b c d e g -14 b c \,e^{2} f +22 c^{2} d^{2} g +49 c^{2} d e f \right ) \sqrt {-\left (e x +d \right ) \left (c e x +b e -c d \right )}}{105 c^{3} e^{2} \sqrt {e x +d}}\) | \(131\) |
| gosper | \(\frac {2 \left (c e x +b e -c d \right ) \left (15 g \,x^{2} c^{2} e^{2}-12 b c \,e^{2} g x +33 c^{2} d e g x +21 c^{2} e^{2} f x +8 b^{2} e^{2} g -30 b c d e g -14 b c \,e^{2} f +22 c^{2} d^{2} g +49 c^{2} d e f \right ) \sqrt {-x^{2} c \,e^{2}-x b \,e^{2}-b d e +c \,d^{2}}}{105 c^{3} e^{2} \sqrt {e x +d}}\) | \(139\) |
| orering | \(\frac {2 \left (c e x +b e -c d \right ) \left (15 g \,x^{2} c^{2} e^{2}-12 b c \,e^{2} g x +33 c^{2} d e g x +21 c^{2} e^{2} f x +8 b^{2} e^{2} g -30 b c d e g -14 b c \,e^{2} f +22 c^{2} d^{2} g +49 c^{2} d e f \right ) \sqrt {-x^{2} c \,e^{2}-x b \,e^{2}-b d e +c \,d^{2}}}{105 c^{3} e^{2} \sqrt {e x +d}}\) | \(139\) |
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)^(1/2),x,method= _RETURNVERBOSE)
Output:
2/105*(c*e*x+b*e-c*d)*(15*c^2*e^2*g*x^2-12*b*c*e^2*g*x+33*c^2*d*e*g*x+21*c ^2*e^2*f*x+8*b^2*e^2*g-30*b*c*d*e*g-14*b*c*e^2*f+22*c^2*d^2*g+49*c^2*d*e*f )*(-(e*x+d)*(c*e*x+b*e-c*d))^(1/2)/c^3/e^2/(e*x+d)^(1/2)
Time = 0.10 (sec) , antiderivative size = 233, normalized size of antiderivative = 1.23 \[ \int \sqrt {d+e x} (f+g x) \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2} \, dx=\frac {2 \, {\left (15 \, c^{3} e^{3} g x^{3} + 3 \, {\left (7 \, c^{3} e^{3} f + {\left (6 \, c^{3} d e^{2} + b c^{2} e^{3}\right )} g\right )} x^{2} - 7 \, {\left (7 \, c^{3} d^{2} e - 9 \, b c^{2} d e^{2} + 2 \, b^{2} c e^{3}\right )} f - 2 \, {\left (11 \, c^{3} d^{3} - 26 \, b c^{2} d^{2} e + 19 \, b^{2} c d e^{2} - 4 \, b^{3} e^{3}\right )} g + {\left (7 \, {\left (4 \, c^{3} d e^{2} + b c^{2} e^{3}\right )} f - {\left (11 \, c^{3} d^{2} e - 15 \, b c^{2} d e^{2} + 4 \, b^{2} c e^{3}\right )} g\right )} x\right )} \sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e} \sqrt {e x + d}}{105 \, {\left (c^{3} e^{3} x + c^{3} d e^{2}\right )}} \] 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)^(1/2),x, algorithm="fricas")
Output:
2/105*(15*c^3*e^3*g*x^3 + 3*(7*c^3*e^3*f + (6*c^3*d*e^2 + b*c^2*e^3)*g)*x^ 2 - 7*(7*c^3*d^2*e - 9*b*c^2*d*e^2 + 2*b^2*c*e^3)*f - 2*(11*c^3*d^3 - 26*b *c^2*d^2*e + 19*b^2*c*d*e^2 - 4*b^3*e^3)*g + (7*(4*c^3*d*e^2 + b*c^2*e^3)* f - (11*c^3*d^2*e - 15*b*c^2*d*e^2 + 4*b^2*c*e^3)*g)*x)*sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)*sqrt(e*x + d)/(c^3*e^3*x + c^3*d*e^2)
\[ \int \sqrt {d+e x} (f+g x) \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2} \, dx=\int \sqrt {- \left (d + e x\right ) \left (b e - c d + c e x\right )} \sqrt {d + e x} \left (f + g x\right )\, dx \] 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)**(1/ 2),x)
Output:
Integral(sqrt(-(d + e*x)*(b*e - c*d + c*e*x))*sqrt(d + e*x)*(f + g*x), x)
Time = 0.09 (sec) , antiderivative size = 236, normalized size of antiderivative = 1.24 \[ \int \sqrt {d+e x} (f+g x) \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2} \, dx=\frac {2 \, {\left (3 \, c^{2} e^{2} x^{2} - 7 \, c^{2} d^{2} + 9 \, b c d e - 2 \, b^{2} e^{2} + {\left (4 \, c^{2} d e + b c e^{2}\right )} x\right )} \sqrt {-c e x + c d - b e} {\left (e x + d\right )} f}{15 \, {\left (c^{2} e^{2} x + c^{2} d e\right )}} + \frac {2 \, {\left (15 \, c^{3} e^{3} x^{3} - 22 \, c^{3} d^{3} + 52 \, b c^{2} d^{2} e - 38 \, b^{2} c d e^{2} + 8 \, b^{3} e^{3} + 3 \, {\left (6 \, c^{3} d e^{2} + b c^{2} e^{3}\right )} x^{2} - {\left (11 \, c^{3} d^{2} e - 15 \, b c^{2} d e^{2} + 4 \, b^{2} c e^{3}\right )} x\right )} \sqrt {-c e x + c d - b e} {\left (e x + d\right )} g}{105 \, {\left (c^{3} e^{3} x + c^{3} d e^{2}\right )}} \] 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)^(1/2),x, algorithm="maxima")
Output:
2/15*(3*c^2*e^2*x^2 - 7*c^2*d^2 + 9*b*c*d*e - 2*b^2*e^2 + (4*c^2*d*e + b*c *e^2)*x)*sqrt(-c*e*x + c*d - b*e)*(e*x + d)*f/(c^2*e^2*x + c^2*d*e) + 2/10 5*(15*c^3*e^3*x^3 - 22*c^3*d^3 + 52*b*c^2*d^2*e - 38*b^2*c*d*e^2 + 8*b^3*e ^3 + 3*(6*c^3*d*e^2 + b*c^2*e^3)*x^2 - (11*c^3*d^2*e - 15*b*c^2*d*e^2 + 4* b^2*c*e^3)*x)*sqrt(-c*e*x + c*d - b*e)*(e*x + d)*g/(c^3*e^3*x + c^3*d*e^2)
Leaf count of result is larger than twice the leaf count of optimal. 804 vs. \(2 (172) = 344\).
Time = 0.33 (sec) , antiderivative size = 804, normalized size of antiderivative = 4.23 \[ \int \sqrt {d+e x} (f+g x) \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2} \, dx=-\frac {2 \, {\left (105 \, \sqrt {-c e x + c d - b e} c d^{2} e f - 105 \, \sqrt {-c e x + c d - b e} b d e^{2} f - \frac {35 \, {\left (3 \, \sqrt {-c e x + c d - b e} c d - 3 \, \sqrt {-c e x + c d - b e} b e - {\left (-c e x + c d - b e\right )}^{\frac {3}{2}}\right )} b e^{2} f}{c} + 35 \, {\left (3 \, \sqrt {-c e x + c d - b e} c d - 3 \, \sqrt {-c e x + c d - b e} b e - {\left (-c e x + c d - b e\right )}^{\frac {3}{2}}\right )} d^{2} g - \frac {35 \, {\left (3 \, \sqrt {-c e x + c d - b e} c d - 3 \, \sqrt {-c e x + c d - b e} b e - {\left (-c e x + c d - b e\right )}^{\frac {3}{2}}\right )} b d e g}{c} - \frac {7 \, {\left (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 \, {\left (-c e x + c d - b e\right )}^{\frac {3}{2}} c d + 10 \, {\left (-c e x + c d - b e\right )}^{\frac {3}{2}} b e + 3 \, {\left (c e x - c d + b e\right )}^{2} \sqrt {-c e x + c d - b e}\right )} e f}{c} - \frac {7 \, {\left (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 \, {\left (-c e x + c d - b e\right )}^{\frac {3}{2}} c d + 10 \, {\left (-c e x + c d - b e\right )}^{\frac {3}{2}} b e + 3 \, {\left (c e x - c d + b e\right )}^{2} \sqrt {-c e x + c d - b e}\right )} b e g}{c^{2}} - \frac {3 \, {\left (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 \, {\left (-c e x + c d - b e\right )}^{\frac {3}{2}} c^{2} d^{2} + 70 \, {\left (-c e x + c d - b e\right )}^{\frac {3}{2}} b c d e - 35 \, {\left (-c e x + c d - b e\right )}^{\frac {3}{2}} b^{2} e^{2} + 21 \, {\left (c e x - c d + b e\right )}^{2} \sqrt {-c e x + c d - b e} c d - 21 \, {\left (c e x - c d + b e\right )}^{2} \sqrt {-c e x + c d - b e} b e + 5 \, {\left (c e x - c d + b e\right )}^{3} \sqrt {-c e x + c d - b e}\right )} g}{c^{2}}\right )}}{105 \, c e^{2}} \] 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)^(1/2),x, algorithm="giac")
Output:
-2/105*(105*sqrt(-c*e*x + c*d - b*e)*c*d^2*e*f - 105*sqrt(-c*e*x + c*d - b *e)*b*d*e^2*f - 35*(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*e^2*f/c + 35*(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^2*g - 35*(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*e*g/c - 7*(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))*e*f/c - 7*( 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*e*g/c^2 - 3*(35*sqrt(-c*e*x + c*d - b*e)*c^3*d^3 - 105*sqr t(-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*(-c*e*x + c*d - b*e)^(3/2)*c ^2*d^2 + 70*(-c*e*x + c*d - b*e)^(3/2)*b*c*d*e - 35*(-c*e*x + c*d - b*e)^( 3/2)*b^2*e^2 + 21*(c*e*x - c*d + b*e)^2*sqrt(-c*e*x + c*d - b*e)*c*d - 21* (c*e*x - c*d + b*e)^2*sqrt(-c*e*x + c*d - b*e)*b*e + 5*(c*e*x - c*d + b*e) ^3*sqrt(-c*e*x + c*d - b*e))*g/c^2)/(c*e^2)
Time = 11.04 (sec) , antiderivative size = 219, normalized size of antiderivative = 1.15 \[ \int \sqrt {d+e x} (f+g x) \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2} \, dx=\frac {\left (\frac {2\,g\,x^3\,\sqrt {d+e\,x}}{7}+\frac {2\,x^2\,\sqrt {d+e\,x}\,\left (b\,e\,g+6\,c\,d\,g+7\,c\,e\,f\right )}{35\,c\,e}+\frac {2\,\left (b\,e-c\,d\right )\,\sqrt {d+e\,x}\,\left (8\,g\,b^2\,e^2-30\,g\,b\,c\,d\,e-14\,f\,b\,c\,e^2+22\,g\,c^2\,d^2+49\,f\,c^2\,d\,e\right )}{105\,c^3\,e^3}+\frac {x\,\sqrt {d+e\,x}\,\left (-8\,g\,b^2\,c\,e^3+30\,g\,b\,c^2\,d\,e^2+14\,f\,b\,c^2\,e^3-22\,g\,c^3\,d^2\,e+56\,f\,c^3\,d\,e^2\right )}{105\,c^3\,e^3}\right )\,\sqrt {c\,d^2-b\,d\,e-c\,e^2\,x^2-b\,e^2\,x}}{x+\frac {d}{e}} \] 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)^(1/2), x)
Output:
(((2*g*x^3*(d + e*x)^(1/2))/7 + (2*x^2*(d + e*x)^(1/2)*(b*e*g + 6*c*d*g + 7*c*e*f))/(35*c*e) + (2*(b*e - c*d)*(d + e*x)^(1/2)*(8*b^2*e^2*g + 22*c^2* d^2*g - 14*b*c*e^2*f + 49*c^2*d*e*f - 30*b*c*d*e*g))/(105*c^3*e^3) + (x*(d + e*x)^(1/2)*(14*b*c^2*e^3*f - 8*b^2*c*e^3*g + 56*c^3*d*e^2*f - 22*c^3*d^ 2*e*g + 30*b*c^2*d*e^2*g))/(105*c^3*e^3))*(c*d^2 - c*e^2*x^2 - b*d*e - b*e ^2*x)^(1/2))/(x + d/e)
Time = 0.21 (sec) , antiderivative size = 200, normalized size of antiderivative = 1.05 \[ \int \sqrt {d+e x} (f+g x) \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2} \, dx=\frac {2 \sqrt {-c e x -b e +c d}\, \left (15 c^{3} e^{3} g \,x^{3}+3 b \,c^{2} e^{3} g \,x^{2}+18 c^{3} d \,e^{2} g \,x^{2}+21 c^{3} e^{3} f \,x^{2}-4 b^{2} c \,e^{3} g x +15 b \,c^{2} d \,e^{2} g x +7 b \,c^{2} e^{3} f x -11 c^{3} d^{2} e g x +28 c^{3} d \,e^{2} f x +8 b^{3} e^{3} g -38 b^{2} c d \,e^{2} g -14 b^{2} c \,e^{3} f +52 b \,c^{2} d^{2} e g +63 b \,c^{2} d \,e^{2} f -22 c^{3} d^{3} g -49 c^{3} d^{2} e f \right )}{105 c^{3} e^{2}} \] 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)^(1/2),x)
Output:
(2*sqrt( - b*e + c*d - c*e*x)*(8*b**3*e**3*g - 38*b**2*c*d*e**2*g - 14*b** 2*c*e**3*f - 4*b**2*c*e**3*g*x + 52*b*c**2*d**2*e*g + 63*b*c**2*d*e**2*f + 15*b*c**2*d*e**2*g*x + 7*b*c**2*e**3*f*x + 3*b*c**2*e**3*g*x**2 - 22*c**3 *d**3*g - 49*c**3*d**2*e*f - 11*c**3*d**2*e*g*x + 28*c**3*d*e**2*f*x + 18* c**3*d*e**2*g*x**2 + 21*c**3*e**3*f*x**2 + 15*c**3*e**3*g*x**3))/(105*c**3 *e**2)