Integrand size = 46, antiderivative size = 190 \[ \int \frac {(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}}{(d+e x)^{3/2}} \, dx=-\frac {2 (2 c d-b e) (c e f+c d g-b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{7 c^3 e^2 (d+e x)^{7/2}}+\frac {2 (c e f+3 c d g-2 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{9/2}}{9 c^3 e^2 (d+e x)^{9/2}}-\frac {2 g \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{11/2}}{11 c^3 e^2 (d+e x)^{11/2}} \] Output:
-2/7*(-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)^(7 /2)/c^3/e^2/(e*x+d)^(7/2)+2/9*(-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)^(9/2)/c^3/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^3/e^2/(e*x+d)^(11/2)
Time = 0.20 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.64 \[ \int \frac {(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}}{(d+e x)^{3/2}} \, dx=\frac {2 (-c d+b e+c e x)^3 \sqrt {(d+e x) (-b e+c (d-e x))} \left (8 b^2 e^2 g-2 b c e (11 e f+19 d g+14 e g x)+c^2 \left (30 d^2 g+7 e^2 x (11 f+9 g x)+d e (121 f+105 g x)\right )\right )}{693 c^3 e^2 \sqrt {d+e x}} \] Input:
Integrate[((f + g*x)*(c*d^2 - b*d*e - b*e^2*x - c*e^2*x^2)^(5/2))/(d + e*x )^(3/2),x]
Output:
(2*(-(c*d) + b*e + c*e*x)^3*Sqrt[(d + e*x)*(-(b*e) + c*(d - e*x))]*(8*b^2* e^2*g - 2*b*c*e*(11*e*f + 19*d*g + 14*e*g*x) + c^2*(30*d^2*g + 7*e^2*x*(11 *f + 9*g*x) + d*e*(121*f + 105*g*x))))/(693*c^3*e^2*Sqrt[d + e*x])
Time = 0.72 (sec) , antiderivative size = 188, normalized size of antiderivative = 0.99, 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.
| \(\displaystyle \int \frac {(f+g x) \left (-b d e-b e^2 x+c d^2-c e^2 x^2\right )^{5/2}}{(d+e x)^{3/2}} \, dx\) |
| \(\Big \downarrow \) 1221 |
| \(\displaystyle \frac {(-4 b e g-3 c d g+11 c e f) \int \frac {\left (-c x^2 e^2-b x e^2+d (c d-b e)\right )^{5/2}}{(d+e x)^{3/2}}dx}{11 c e}-\frac {2 g \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{11 c e^2 (d+e x)^{3/2}}\) |
| \(\Big \downarrow \) 1128 |
| \(\displaystyle \frac {(-4 b e g-3 c d g+11 c e f) \left (\frac {2 (2 c d-b e) \int \frac {\left (-c x^2 e^2-b x e^2+d (c d-b e)\right )^{5/2}}{(d+e x)^{5/2}}dx}{9 c}-\frac {2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{9 c e (d+e x)^{5/2}}\right )}{11 c e}-\frac {2 g \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{11 c e^2 (d+e x)^{3/2}}\) |
| \(\Big \downarrow \) 1122 |
| \(\displaystyle \frac {\left (-\frac {4 (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{63 c^2 e (d+e x)^{7/2}}-\frac {2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{9 c e (d+e x)^{5/2}}\right ) (-4 b e g-3 c d g+11 c e f)}{11 c e}-\frac {2 g \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{11 c e^2 (d+e x)^{3/2}}\) |
Input:
Int[((f + g*x)*(c*d^2 - b*d*e - b*e^2*x - c*e^2*x^2)^(5/2))/(d + e*x)^(3/2 ),x]
Output:
(-2*g*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(7/2))/(11*c*e^2*(d + e*x)^(3/ 2)) + ((11*c*e*f - 3*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)^(7/2))/(63*c^2*e*(d + e*x)^(7/2)) - (2*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(7/2))/(9*c*e*(d + e*x)^(5/2))))/(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 = 2.01 (sec) , antiderivative size = 133, normalized size of antiderivative = 0.70
| method | result | size |
| default | \(\frac {2 \sqrt {-\left (e x +d \right ) \left (c e x +b e -c d \right )}\, \left (c e x +b e -c d \right )^{3} \left (63 g \,x^{2} c^{2} e^{2}-28 b c \,e^{2} g x +105 c^{2} d e g x +77 c^{2} e^{2} f x +8 b^{2} e^{2} g -38 b c d e g -22 b c \,e^{2} f +30 c^{2} d^{2} g +121 c^{2} d e f \right )}{693 \sqrt {e x +d}\, c^{3} e^{2}}\) | \(133\) |
| gosper | \(\frac {2 \left (c e x +b e -c d \right ) \left (63 g \,x^{2} c^{2} e^{2}-28 b c \,e^{2} g x +105 c^{2} d e g x +77 c^{2} e^{2} f x +8 b^{2} e^{2} g -38 b c d e g -22 b c \,e^{2} f +30 c^{2} d^{2} g +121 c^{2} d e f \right ) \left (-x^{2} c \,e^{2}-x b \,e^{2}-b d e +c \,d^{2}\right )^{\frac {5}{2}}}{693 c^{3} e^{2} \left (e x +d \right )^{\frac {5}{2}}}\) | \(139\) |
| orering | \(\frac {2 \left (c e x +b e -c d \right ) \left (63 g \,x^{2} c^{2} e^{2}-28 b c \,e^{2} g x +105 c^{2} d e g x +77 c^{2} e^{2} f x +8 b^{2} e^{2} g -38 b c d e g -22 b c \,e^{2} f +30 c^{2} d^{2} g +121 c^{2} d e f \right ) \left (-x^{2} c \,e^{2}-x b \,e^{2}-b d e +c \,d^{2}\right )^{\frac {5}{2}}}{693 c^{3} e^{2} \left (e x +d \right )^{\frac {5}{2}}}\) | \(139\) |
Input:
int((g*x+f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2)/(e*x+d)^(3/2),x,method= _RETURNVERBOSE)
Output:
2/693*(-(e*x+d)*(c*e*x+b*e-c*d))^(1/2)/(e*x+d)^(1/2)*(c*e*x+b*e-c*d)^3*(63 *c^2*e^2*g*x^2-28*b*c*e^2*g*x+105*c^2*d*e*g*x+77*c^2*e^2*f*x+8*b^2*e^2*g-3 8*b*c*d*e*g-22*b*c*e^2*f+30*c^2*d^2*g+121*c^2*d*e*f)/c^3/e^2
Leaf count of result is larger than twice the leaf count of optimal. 500 vs. \(2 (172) = 344\).
Time = 0.09 (sec) , antiderivative size = 500, normalized size of antiderivative = 2.63 \[ \int \frac {(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}}{(d+e x)^{3/2}} \, dx=\frac {2 \, {\left (63 \, c^{5} e^{5} g x^{5} + 7 \, {\left (11 \, c^{5} e^{5} f - {\left (12 \, c^{5} d e^{4} - 23 \, b c^{4} e^{5}\right )} g\right )} x^{4} - {\left (11 \, {\left (10 \, c^{5} d e^{4} - 19 \, b c^{4} e^{5}\right )} f + {\left (96 \, c^{5} d^{2} e^{3} + 17 \, b c^{4} d e^{4} - 113 \, b^{2} c^{3} e^{5}\right )} g\right )} x^{3} - 3 \, {\left (11 \, {\left (4 \, c^{5} d^{2} e^{3} + b c^{4} d e^{4} - 5 \, b^{2} c^{3} e^{5}\right )} f - {\left (54 \, c^{5} d^{3} e^{2} - 107 \, b c^{4} d^{2} e^{3} + 52 \, b^{2} c^{3} d e^{4} + b^{3} c^{2} e^{5}\right )} g\right )} x^{2} - 11 \, {\left (11 \, c^{5} d^{4} e - 35 \, b c^{4} d^{3} e^{2} + 39 \, b^{2} c^{3} d^{2} e^{3} - 17 \, b^{3} c^{2} d e^{4} + 2 \, b^{4} c e^{5}\right )} f - 2 \, {\left (15 \, c^{5} d^{5} - 64 \, b c^{4} d^{4} e + 106 \, b^{2} c^{3} d^{3} e^{2} - 84 \, b^{3} c^{2} d^{2} e^{3} + 31 \, b^{4} c d e^{4} - 4 \, b^{5} e^{5}\right )} g + {\left (11 \, {\left (26 \, c^{5} d^{3} e^{2} - 51 \, b c^{4} d^{2} e^{3} + 24 \, b^{2} c^{3} d e^{4} + b^{3} c^{2} e^{5}\right )} f - {\left (15 \, c^{5} d^{4} e - 49 \, b c^{4} d^{3} e^{2} + 57 \, b^{2} c^{3} d^{2} e^{3} - 27 \, b^{3} c^{2} d e^{4} + 4 \, b^{4} c e^{5}\right )} g\right )} x\right )} \sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e} \sqrt {e x + d}}{693 \, {\left (c^{3} e^{3} x + c^{3} d e^{2}\right )}} \] Input:
integrate((g*x+f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2)/(e*x+d)^(3/2),x, algorithm="fricas")
Output:
2/693*(63*c^5*e^5*g*x^5 + 7*(11*c^5*e^5*f - (12*c^5*d*e^4 - 23*b*c^4*e^5)* g)*x^4 - (11*(10*c^5*d*e^4 - 19*b*c^4*e^5)*f + (96*c^5*d^2*e^3 + 17*b*c^4* d*e^4 - 113*b^2*c^3*e^5)*g)*x^3 - 3*(11*(4*c^5*d^2*e^3 + b*c^4*d*e^4 - 5*b ^2*c^3*e^5)*f - (54*c^5*d^3*e^2 - 107*b*c^4*d^2*e^3 + 52*b^2*c^3*d*e^4 + b ^3*c^2*e^5)*g)*x^2 - 11*(11*c^5*d^4*e - 35*b*c^4*d^3*e^2 + 39*b^2*c^3*d^2* e^3 - 17*b^3*c^2*d*e^4 + 2*b^4*c*e^5)*f - 2*(15*c^5*d^5 - 64*b*c^4*d^4*e + 106*b^2*c^3*d^3*e^2 - 84*b^3*c^2*d^2*e^3 + 31*b^4*c*d*e^4 - 4*b^5*e^5)*g + (11*(26*c^5*d^3*e^2 - 51*b*c^4*d^2*e^3 + 24*b^2*c^3*d*e^4 + b^3*c^2*e^5) *f - (15*c^5*d^4*e - 49*b*c^4*d^3*e^2 + 57*b^2*c^3*d^2*e^3 - 27*b^3*c^2*d* e^4 + 4*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^3*e^3*x + c^3*d*e^2)
\[ \int \frac {(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}}{(d+e x)^{3/2}} \, dx=\int \frac {\left (- \left (d + e x\right ) \left (b e - c d + c e x\right )\right )^{\frac {5}{2}} \left (f + g x\right )}{\left (d + e x\right )^{\frac {3}{2}}}\, dx \] Input:
integrate((g*x+f)*(-c*e**2*x**2-b*e**2*x-b*d*e+c*d**2)**(5/2)/(e*x+d)**(3/ 2),x)
Output:
Integral((-(d + e*x)*(b*e - c*d + c*e*x))**(5/2)*(f + g*x)/(d + e*x)**(3/2 ), x)
Leaf count of result is larger than twice the leaf count of optimal. 465 vs. \(2 (172) = 344\).
Time = 0.08 (sec) , antiderivative size = 465, normalized size of antiderivative = 2.45 \[ \int \frac {(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}}{(d+e x)^{3/2}} \, dx=\frac {2 \, {\left (7 \, c^{4} e^{4} x^{4} - 11 \, c^{4} d^{4} + 35 \, b c^{3} d^{3} e - 39 \, b^{2} c^{2} d^{2} e^{2} + 17 \, b^{3} c d e^{3} - 2 \, b^{4} e^{4} - {\left (10 \, c^{4} d e^{3} - 19 \, b c^{3} e^{4}\right )} x^{3} - 3 \, {\left (4 \, c^{4} d^{2} e^{2} + b c^{3} d e^{3} - 5 \, b^{2} c^{2} e^{4}\right )} x^{2} + {\left (26 \, c^{4} d^{3} e - 51 \, b c^{3} d^{2} e^{2} + 24 \, b^{2} c^{2} d e^{3} + b^{3} c e^{4}\right )} x\right )} \sqrt {-c e x + c d - b e} f}{63 \, c^{2} e} + \frac {2 \, {\left (63 \, c^{5} e^{5} x^{5} - 30 \, c^{5} d^{5} + 128 \, b c^{4} d^{4} e - 212 \, b^{2} c^{3} d^{3} e^{2} + 168 \, b^{3} c^{2} d^{2} e^{3} - 62 \, b^{4} c d e^{4} + 8 \, b^{5} e^{5} - 7 \, {\left (12 \, c^{5} d e^{4} - 23 \, b c^{4} e^{5}\right )} x^{4} - {\left (96 \, c^{5} d^{2} e^{3} + 17 \, b c^{4} d e^{4} - 113 \, b^{2} c^{3} e^{5}\right )} x^{3} + 3 \, {\left (54 \, c^{5} d^{3} e^{2} - 107 \, b c^{4} d^{2} e^{3} + 52 \, b^{2} c^{3} d e^{4} + b^{3} c^{2} e^{5}\right )} x^{2} - {\left (15 \, c^{5} d^{4} e - 49 \, b c^{4} d^{3} e^{2} + 57 \, b^{2} c^{3} d^{2} e^{3} - 27 \, b^{3} c^{2} d e^{4} + 4 \, b^{4} c e^{5}\right )} x\right )} \sqrt {-c e x + c d - b e} g}{693 \, c^{3} e^{2}} \] Input:
integrate((g*x+f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2)/(e*x+d)^(3/2),x, algorithm="maxima")
Output:
2/63*(7*c^4*e^4*x^4 - 11*c^4*d^4 + 35*b*c^3*d^3*e - 39*b^2*c^2*d^2*e^2 + 1 7*b^3*c*d*e^3 - 2*b^4*e^4 - (10*c^4*d*e^3 - 19*b*c^3*e^4)*x^3 - 3*(4*c^4*d ^2*e^2 + b*c^3*d*e^3 - 5*b^2*c^2*e^4)*x^2 + (26*c^4*d^3*e - 51*b*c^3*d^2*e ^2 + 24*b^2*c^2*d*e^3 + b^3*c*e^4)*x)*sqrt(-c*e*x + c*d - b*e)*f/(c^2*e) + 2/693*(63*c^5*e^5*x^5 - 30*c^5*d^5 + 128*b*c^4*d^4*e - 212*b^2*c^3*d^3*e^ 2 + 168*b^3*c^2*d^2*e^3 - 62*b^4*c*d*e^4 + 8*b^5*e^5 - 7*(12*c^5*d*e^4 - 2 3*b*c^4*e^5)*x^4 - (96*c^5*d^2*e^3 + 17*b*c^4*d*e^4 - 113*b^2*c^3*e^5)*x^3 + 3*(54*c^5*d^3*e^2 - 107*b*c^4*d^2*e^3 + 52*b^2*c^3*d*e^4 + b^3*c^2*e^5) *x^2 - (15*c^5*d^4*e - 49*b*c^4*d^3*e^2 + 57*b^2*c^3*d^2*e^3 - 27*b^3*c^2* d*e^4 + 4*b^4*c*e^5)*x)*sqrt(-c*e*x + c*d - b*e)*g/(c^3*e^2)
Leaf count of result is larger than twice the leaf count of optimal. 3083 vs. \(2 (172) = 344\).
Time = 0.41 (sec) , antiderivative size = 3083, normalized size of antiderivative = 16.23 \[ \int \frac {(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}}{(d+e x)^{3/2}} \, dx=\text {Too large to display} \] Input:
integrate((g*x+f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2)/(e*x+d)^(3/2),x, algorithm="giac")
Output:
-2/3465*(1155*(-(e*x + d)*c + 2*c*d - b*e)^(3/2)*c*d^3*e*f - 2310*(-(e*x + d)*c + 2*c*d - b*e)^(3/2)*b*d^2*e^2*f + 1155*(-(e*x + d)*c + 2*c*d - b*e) ^(3/2)*b^2*d*e^3*f/c - 231*(5*(-(e*x + d)*c + 2*c*d - b*e)^(3/2)*c*d - 5*( -(e*x + d)*c + 2*c*d - b*e)^(3/2)*b*e - 3*((e*x + d)*c - 2*c*d + b*e)^2*sq rt(-(e*x + d)*c + 2*c*d - b*e))*d^2*e*f + 231*(5*(-(e*x + d)*c + 2*c*d - b *e)^(3/2)*c*d - 5*(-(e*x + d)*c + 2*c*d - b*e)^(3/2)*b*e - 3*((e*x + d)*c - 2*c*d + b*e)^2*sqrt(-(e*x + d)*c + 2*c*d - b*e))*b^2*e^3*f/c^2 + 231*(5* (-(e*x + d)*c + 2*c*d - b*e)^(3/2)*c*d - 5*(-(e*x + d)*c + 2*c*d - b*e)^(3 /2)*b*e - 3*((e*x + d)*c - 2*c*d + b*e)^2*sqrt(-(e*x + d)*c + 2*c*d - b*e) )*d^3*g - 462*(5*(-(e*x + d)*c + 2*c*d - b*e)^(3/2)*c*d - 5*(-(e*x + d)*c + 2*c*d - b*e)^(3/2)*b*e - 3*((e*x + d)*c - 2*c*d + b*e)^2*sqrt(-(e*x + d) *c + 2*c*d - b*e))*b*d^2*e*g/c + 231*(5*(-(e*x + d)*c + 2*c*d - b*e)^(3/2) *c*d - 5*(-(e*x + d)*c + 2*c*d - b*e)^(3/2)*b*e - 3*((e*x + d)*c - 2*c*d + b*e)^2*sqrt(-(e*x + d)*c + 2*c*d - b*e))*b^2*d*e^2*g/c^2 - 33*(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))*d*e*f/c + 66*(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...
Time = 11.36 (sec) , antiderivative size = 320, normalized size of antiderivative = 1.68 \[ \int \frac {(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}}{(d+e x)^{3/2}} \, dx=\frac {\sqrt {c\,d^2-b\,d\,e-c\,e^2\,x^2-b\,e^2\,x}\,\left (\frac {2\,c^2\,e^3\,g\,x^5}{11}+\frac {2\,c\,e^2\,x^4\,\left (23\,b\,e\,g-12\,c\,d\,g+11\,c\,e\,f\right )}{99}+\frac {2\,x^2\,\left (b\,e-c\,d\right )\,\left (g\,b^2\,e^2+53\,g\,b\,c\,d\,e+55\,f\,b\,c\,e^2-54\,g\,c^2\,d^2+44\,f\,c^2\,d\,e\right )}{231\,c}-\frac {x^3\,\left (-226\,g\,b^2\,c^3\,e^5+34\,g\,b\,c^4\,d\,e^4-418\,f\,b\,c^4\,e^5+192\,g\,c^5\,d^2\,e^3+220\,f\,c^5\,d\,e^4\right )}{693\,c^3\,e^2}+\frac {2\,{\left (b\,e-c\,d\right )}^3\,\left (8\,g\,b^2\,e^2-38\,g\,b\,c\,d\,e-22\,f\,b\,c\,e^2+30\,g\,c^2\,d^2+121\,f\,c^2\,d\,e\right )}{693\,c^3\,e^2}+\frac {2\,x\,{\left (b\,e-c\,d\right )}^2\,\left (-4\,g\,b^2\,e^2+19\,g\,b\,c\,d\,e+11\,f\,b\,c\,e^2-15\,g\,c^2\,d^2+286\,f\,c^2\,d\,e\right )}{693\,c^2\,e}\right )}{\sqrt {d+e\,x}} \] Input:
int(((f + g*x)*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(5/2))/(d + e*x)^(3/2 ),x)
Output:
((c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(1/2)*((2*c^2*e^3*g*x^5)/11 + (2*c* e^2*x^4*(23*b*e*g - 12*c*d*g + 11*c*e*f))/99 + (2*x^2*(b*e - c*d)*(b^2*e^2 *g - 54*c^2*d^2*g + 55*b*c*e^2*f + 44*c^2*d*e*f + 53*b*c*d*e*g))/(231*c) - (x^3*(192*c^5*d^2*e^3*g - 226*b^2*c^3*e^5*g - 418*b*c^4*e^5*f + 220*c^5*d *e^4*f + 34*b*c^4*d*e^4*g))/(693*c^3*e^2) + (2*(b*e - c*d)^3*(8*b^2*e^2*g + 30*c^2*d^2*g - 22*b*c*e^2*f + 121*c^2*d*e*f - 38*b*c*d*e*g))/(693*c^3*e^ 2) + (2*x*(b*e - c*d)^2*(11*b*c*e^2*f - 15*c^2*d^2*g - 4*b^2*e^2*g + 286*c ^2*d*e*f + 19*b*c*d*e*g))/(693*c^2*e)))/(d + e*x)^(1/2)
Time = 0.20 (sec) , antiderivative size = 500, normalized size of antiderivative = 2.63 \[ \int \frac {(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}}{(d+e x)^{3/2}} \, dx=\frac {2 \sqrt {-c e x -b e +c d}\, \left (63 c^{5} e^{5} g \,x^{5}+161 b \,c^{4} e^{5} g \,x^{4}-84 c^{5} d \,e^{4} g \,x^{4}+77 c^{5} e^{5} f \,x^{4}+113 b^{2} c^{3} e^{5} g \,x^{3}-17 b \,c^{4} d \,e^{4} g \,x^{3}+209 b \,c^{4} e^{5} f \,x^{3}-96 c^{5} d^{2} e^{3} g \,x^{3}-110 c^{5} d \,e^{4} f \,x^{3}+3 b^{3} c^{2} e^{5} g \,x^{2}+156 b^{2} c^{3} d \,e^{4} g \,x^{2}+165 b^{2} c^{3} e^{5} f \,x^{2}-321 b \,c^{4} d^{2} e^{3} g \,x^{2}-33 b \,c^{4} d \,e^{4} f \,x^{2}+162 c^{5} d^{3} e^{2} g \,x^{2}-132 c^{5} d^{2} e^{3} f \,x^{2}-4 b^{4} c \,e^{5} g x +27 b^{3} c^{2} d \,e^{4} g x +11 b^{3} c^{2} e^{5} f x -57 b^{2} c^{3} d^{2} e^{3} g x +264 b^{2} c^{3} d \,e^{4} f x +49 b \,c^{4} d^{3} e^{2} g x -561 b \,c^{4} d^{2} e^{3} f x -15 c^{5} d^{4} e g x +286 c^{5} d^{3} e^{2} f x +8 b^{5} e^{5} g -62 b^{4} c d \,e^{4} g -22 b^{4} c \,e^{5} f +168 b^{3} c^{2} d^{2} e^{3} g +187 b^{3} c^{2} d \,e^{4} f -212 b^{2} c^{3} d^{3} e^{2} g -429 b^{2} c^{3} d^{2} e^{3} f +128 b \,c^{4} d^{4} e g +385 b \,c^{4} d^{3} e^{2} f -30 c^{5} d^{5} g -121 c^{5} d^{4} e f \right )}{693 c^{3} e^{2}} \] Input:
int((g*x+f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2)/(e*x+d)^(3/2),x)
Output:
(2*sqrt( - b*e + c*d - c*e*x)*(8*b**5*e**5*g - 62*b**4*c*d*e**4*g - 22*b** 4*c*e**5*f - 4*b**4*c*e**5*g*x + 168*b**3*c**2*d**2*e**3*g + 187*b**3*c**2 *d*e**4*f + 27*b**3*c**2*d*e**4*g*x + 11*b**3*c**2*e**5*f*x + 3*b**3*c**2* e**5*g*x**2 - 212*b**2*c**3*d**3*e**2*g - 429*b**2*c**3*d**2*e**3*f - 57*b **2*c**3*d**2*e**3*g*x + 264*b**2*c**3*d*e**4*f*x + 156*b**2*c**3*d*e**4*g *x**2 + 165*b**2*c**3*e**5*f*x**2 + 113*b**2*c**3*e**5*g*x**3 + 128*b*c**4 *d**4*e*g + 385*b*c**4*d**3*e**2*f + 49*b*c**4*d**3*e**2*g*x - 561*b*c**4* d**2*e**3*f*x - 321*b*c**4*d**2*e**3*g*x**2 - 33*b*c**4*d*e**4*f*x**2 - 17 *b*c**4*d*e**4*g*x**3 + 209*b*c**4*e**5*f*x**3 + 161*b*c**4*e**5*g*x**4 - 30*c**5*d**5*g - 121*c**5*d**4*e*f - 15*c**5*d**4*e*g*x + 286*c**5*d**3*e* *2*f*x + 162*c**5*d**3*e**2*g*x**2 - 132*c**5*d**2*e**3*f*x**2 - 96*c**5*d **2*e**3*g*x**3 - 110*c**5*d*e**4*f*x**3 - 84*c**5*d*e**4*g*x**4 + 77*c**5 *e**5*f*x**4 + 63*c**5*e**5*g*x**5))/(693*c**3*e**2)