Integrand size = 46, antiderivative size = 336 \[ \int \frac {(f+g x)^4 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{(d+e x)^{3/2}} \, dx=-\frac {128 (c d f-a e g)^3 \left (2 a e^2 g-c d (7 e f-5 d g)\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{15015 c^5 d^5 e (d+e x)^{5/2}}+\frac {128 g (c d f-a e g)^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{3003 c^4 d^4 e (d+e x)^{3/2}}+\frac {32 (c d f-a e g)^2 (f+g x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{429 c^3 d^3 (d+e x)^{5/2}}+\frac {16 (c d f-a e g) (f+g x)^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{143 c^2 d^2 (d+e x)^{5/2}}+\frac {2 (f+g x)^4 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{13 c d (d+e x)^{5/2}} \]
-128/15015*(-a*e*g+c*d*f)^3*(2*a*e^2*g-c*d*(-5*d*g+7*e*f))*(a*d*e+(a*e^2+c *d^2)*x+c*d*e*x^2)^(5/2)/c^5/d^5/e/(e*x+d)^(5/2)+128/3003*g*(-a*e*g+c*d*f) ^3*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/c^4/d^4/e/(e*x+d)^(3/2)+32/429* (-a*e*g+c*d*f)^2*(g*x+f)^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/c^3/d^3 /(e*x+d)^(5/2)+16/143*(-a*e*g+c*d*f)*(g*x+f)^3*(a*d*e+(a*e^2+c*d^2)*x+c*d* e*x^2)^(5/2)/c^2/d^2/(e*x+d)^(5/2)+2/13*(g*x+f)^4*(a*d*e+(a*e^2+c*d^2)*x+c *d*e*x^2)^(5/2)/c/d/(e*x+d)^(5/2)
Time = 0.20 (sec) , antiderivative size = 195, normalized size of antiderivative = 0.58 \[ \int \frac {(f+g x)^4 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{(d+e x)^{3/2}} \, dx=\frac {2 ((a e+c d x) (d+e x))^{5/2} \left (128 a^4 e^4 g^4-64 a^3 c d e^3 g^3 (13 f+5 g x)+16 a^2 c^2 d^2 e^2 g^2 \left (143 f^2+130 f g x+35 g^2 x^2\right )-8 a c^3 d^3 e g \left (429 f^3+715 f^2 g x+455 f g^2 x^2+105 g^3 x^3\right )+c^4 d^4 \left (3003 f^4+8580 f^3 g x+10010 f^2 g^2 x^2+5460 f g^3 x^3+1155 g^4 x^4\right )\right )}{15015 c^5 d^5 (d+e x)^{5/2}} \]
(2*((a*e + c*d*x)*(d + e*x))^(5/2)*(128*a^4*e^4*g^4 - 64*a^3*c*d*e^3*g^3*( 13*f + 5*g*x) + 16*a^2*c^2*d^2*e^2*g^2*(143*f^2 + 130*f*g*x + 35*g^2*x^2) - 8*a*c^3*d^3*e*g*(429*f^3 + 715*f^2*g*x + 455*f*g^2*x^2 + 105*g^3*x^3) + c^4*d^4*(3003*f^4 + 8580*f^3*g*x + 10010*f^2*g^2*x^2 + 5460*f*g^3*x^3 + 11 55*g^4*x^4)))/(15015*c^5*d^5*(d + e*x)^(5/2))
Time = 0.69 (sec) , antiderivative size = 351, normalized size of antiderivative = 1.04, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.109, Rules used = {1253, 1253, 1253, 1221, 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.
| \(\displaystyle \int \frac {(f+g x)^4 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{(d+e x)^{3/2}} \, dx\) |
| \(\Big \downarrow \) 1253 |
| \(\displaystyle \frac {8 (c d f-a e g) \int \frac {(f+g x)^3 \left (c d e x^2+\left (c d^2+a e^2\right ) x+a d e\right )^{3/2}}{(d+e x)^{3/2}}dx}{13 c d}+\frac {2 (f+g x)^4 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{13 c d (d+e x)^{5/2}}\) |
| \(\Big \downarrow \) 1253 |
| \(\displaystyle \frac {8 (c d f-a e g) \left (\frac {6 (c d f-a e g) \int \frac {(f+g x)^2 \left (c d e x^2+\left (c d^2+a e^2\right ) x+a d e\right )^{3/2}}{(d+e x)^{3/2}}dx}{11 c d}+\frac {2 (f+g x)^3 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{11 c d (d+e x)^{5/2}}\right )}{13 c d}+\frac {2 (f+g x)^4 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{13 c d (d+e x)^{5/2}}\) |
| \(\Big \downarrow \) 1253 |
| \(\displaystyle \frac {8 (c d f-a e g) \left (\frac {6 (c d f-a e g) \left (\frac {4 (c d f-a e g) \int \frac {(f+g x) \left (c d e x^2+\left (c d^2+a e^2\right ) x+a d e\right )^{3/2}}{(d+e x)^{3/2}}dx}{9 c d}+\frac {2 (f+g x)^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{9 c d (d+e x)^{5/2}}\right )}{11 c d}+\frac {2 (f+g x)^3 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{11 c d (d+e x)^{5/2}}\right )}{13 c d}+\frac {2 (f+g x)^4 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{13 c d (d+e x)^{5/2}}\) |
| \(\Big \downarrow \) 1221 |
| \(\displaystyle \frac {8 (c d f-a e g) \left (\frac {6 (c d f-a e g) \left (\frac {4 (c d f-a e g) \left (\frac {1}{7} \left (-\frac {2 a e g}{c d}-\frac {5 d g}{e}+7 f\right ) \int \frac {\left (c d e x^2+\left (c d^2+a e^2\right ) x+a d e\right )^{3/2}}{(d+e x)^{3/2}}dx+\frac {2 g \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{7 c d e (d+e x)^{3/2}}\right )}{9 c d}+\frac {2 (f+g x)^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{9 c d (d+e x)^{5/2}}\right )}{11 c d}+\frac {2 (f+g x)^3 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{11 c d (d+e x)^{5/2}}\right )}{13 c d}+\frac {2 (f+g x)^4 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{13 c d (d+e x)^{5/2}}\) |
| \(\Big \downarrow \) 1122 |
| \(\displaystyle \frac {2 (f+g x)^4 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{13 c d (d+e x)^{5/2}}+\frac {8 (c d f-a e g) \left (\frac {2 (f+g x)^3 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{11 c d (d+e x)^{5/2}}+\frac {6 (c d f-a e g) \left (\frac {2 (f+g x)^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{9 c d (d+e x)^{5/2}}+\frac {4 (c d f-a e g) \left (\frac {2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2} \left (-\frac {2 a e g}{c d}-\frac {5 d g}{e}+7 f\right )}{35 c d (d+e x)^{5/2}}+\frac {2 g \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{7 c d e (d+e x)^{3/2}}\right )}{9 c d}\right )}{11 c d}\right )}{13 c d}\) |
(2*(f + g*x)^4*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2))/(13*c*d*(d + e*x)^(5/2)) + (8*(c*d*f - a*e*g)*((2*(f + g*x)^3*(a*d*e + (c*d^2 + a*e^2) *x + c*d*e*x^2)^(5/2))/(11*c*d*(d + e*x)^(5/2)) + (6*(c*d*f - a*e*g)*((2*( f + g*x)^2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2))/(9*c*d*(d + e*x) ^(5/2)) + (4*(c*d*f - a*e*g)*((2*(7*f - (5*d*g)/e - (2*a*e*g)/(c*d))*(a*d* e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2))/(35*c*d*(d + e*x)^(5/2)) + (2*g* (a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2))/(7*c*d*e*(d + e*x)^(3/2)))) /(9*c*d)))/(11*c*d)))/(13*c*d)
3.7.89.3.1 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_)*((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]
Int[((d_) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-e)*(d + e*x)^(m - 1)*(f + g*x)^n* ((a + b*x + c*x^2)^(p + 1)/(c*(m - n - 1))), x] - Simp[n*((c*e*f + c*d*g - b*e*g)/(c*e*(m - n - 1))) Int[(d + e*x)^m*(f + g*x)^(n - 1)*(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] && EqQ[m + p, 0] && GtQ[n, 0] && NeQ[m - n - 1, 0] && (Intege rQ[2*p] || IntegerQ[n])
Time = 0.54 (sec) , antiderivative size = 275, normalized size of antiderivative = 0.82
| method | result | size |
| default | \(\frac {2 \sqrt {\left (c d x +a e \right ) \left (e x +d \right )}\, \left (c d x +a e \right )^{2} \left (1155 g^{4} x^{4} c^{4} d^{4}-840 a \,c^{3} d^{3} e \,g^{4} x^{3}+5460 c^{4} d^{4} f \,g^{3} x^{3}+560 a^{2} c^{2} d^{2} e^{2} g^{4} x^{2}-3640 a \,c^{3} d^{3} e f \,g^{3} x^{2}+10010 c^{4} d^{4} f^{2} g^{2} x^{2}-320 a^{3} c d \,e^{3} g^{4} x +2080 a^{2} c^{2} d^{2} e^{2} f \,g^{3} x -5720 a \,c^{3} d^{3} e \,f^{2} g^{2} x +8580 c^{4} d^{4} f^{3} g x +128 a^{4} e^{4} g^{4}-832 a^{3} c d \,e^{3} f \,g^{3}+2288 a^{2} c^{2} d^{2} e^{2} f^{2} g^{2}-3432 a \,c^{3} d^{3} e \,f^{3} g +3003 f^{4} c^{4} d^{4}\right )}{15015 \sqrt {e x +d}\, c^{5} d^{5}}\) | \(275\) |
| gosper | \(\frac {2 \left (c d x +a e \right ) \left (1155 g^{4} x^{4} c^{4} d^{4}-840 a \,c^{3} d^{3} e \,g^{4} x^{3}+5460 c^{4} d^{4} f \,g^{3} x^{3}+560 a^{2} c^{2} d^{2} e^{2} g^{4} x^{2}-3640 a \,c^{3} d^{3} e f \,g^{3} x^{2}+10010 c^{4} d^{4} f^{2} g^{2} x^{2}-320 a^{3} c d \,e^{3} g^{4} x +2080 a^{2} c^{2} d^{2} e^{2} f \,g^{3} x -5720 a \,c^{3} d^{3} e \,f^{2} g^{2} x +8580 c^{4} d^{4} f^{3} g x +128 a^{4} e^{4} g^{4}-832 a^{3} c d \,e^{3} f \,g^{3}+2288 a^{2} c^{2} d^{2} e^{2} f^{2} g^{2}-3432 a \,c^{3} d^{3} e \,f^{3} g +3003 f^{4} c^{4} d^{4}\right ) \left (c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e \right )^{\frac {3}{2}}}{15015 c^{5} d^{5} \left (e x +d \right )^{\frac {3}{2}}}\) | \(283\) |
2/15015*((c*d*x+a*e)*(e*x+d))^(1/2)/(e*x+d)^(1/2)*(c*d*x+a*e)^2*(1155*c^4* d^4*g^4*x^4-840*a*c^3*d^3*e*g^4*x^3+5460*c^4*d^4*f*g^3*x^3+560*a^2*c^2*d^2 *e^2*g^4*x^2-3640*a*c^3*d^3*e*f*g^3*x^2+10010*c^4*d^4*f^2*g^2*x^2-320*a^3* c*d*e^3*g^4*x+2080*a^2*c^2*d^2*e^2*f*g^3*x-5720*a*c^3*d^3*e*f^2*g^2*x+8580 *c^4*d^4*f^3*g*x+128*a^4*e^4*g^4-832*a^3*c*d*e^3*f*g^3+2288*a^2*c^2*d^2*e^ 2*f^2*g^2-3432*a*c^3*d^3*e*f^3*g+3003*c^4*d^4*f^4)/c^5/d^5
Time = 0.35 (sec) , antiderivative size = 472, normalized size of antiderivative = 1.40 \[ \int \frac {(f+g x)^4 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{(d+e x)^{3/2}} \, dx=\frac {2 \, {\left (1155 \, c^{6} d^{6} g^{4} x^{6} + 3003 \, a^{2} c^{4} d^{4} e^{2} f^{4} - 3432 \, a^{3} c^{3} d^{3} e^{3} f^{3} g + 2288 \, a^{4} c^{2} d^{2} e^{4} f^{2} g^{2} - 832 \, a^{5} c d e^{5} f g^{3} + 128 \, a^{6} e^{6} g^{4} + 210 \, {\left (26 \, c^{6} d^{6} f g^{3} + 7 \, a c^{5} d^{5} e g^{4}\right )} x^{5} + 35 \, {\left (286 \, c^{6} d^{6} f^{2} g^{2} + 208 \, a c^{5} d^{5} e f g^{3} + a^{2} c^{4} d^{4} e^{2} g^{4}\right )} x^{4} + 20 \, {\left (429 \, c^{6} d^{6} f^{3} g + 715 \, a c^{5} d^{5} e f^{2} g^{2} + 13 \, a^{2} c^{4} d^{4} e^{2} f g^{3} - 2 \, a^{3} c^{3} d^{3} e^{3} g^{4}\right )} x^{3} + 3 \, {\left (1001 \, c^{6} d^{6} f^{4} + 4576 \, a c^{5} d^{5} e f^{3} g + 286 \, a^{2} c^{4} d^{4} e^{2} f^{2} g^{2} - 104 \, a^{3} c^{3} d^{3} e^{3} f g^{3} + 16 \, a^{4} c^{2} d^{2} e^{4} g^{4}\right )} x^{2} + 2 \, {\left (3003 \, a c^{5} d^{5} e f^{4} + 858 \, a^{2} c^{4} d^{4} e^{2} f^{3} g - 572 \, a^{3} c^{3} d^{3} e^{3} f^{2} g^{2} + 208 \, a^{4} c^{2} d^{2} e^{4} f g^{3} - 32 \, a^{5} c d e^{5} g^{4}\right )} x\right )} \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} \sqrt {e x + d}}{15015 \, {\left (c^{5} d^{5} e x + c^{5} d^{6}\right )}} \]
integrate((g*x+f)^4*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)/(e*x+d)^(3/2), x, algorithm="fricas")
2/15015*(1155*c^6*d^6*g^4*x^6 + 3003*a^2*c^4*d^4*e^2*f^4 - 3432*a^3*c^3*d^ 3*e^3*f^3*g + 2288*a^4*c^2*d^2*e^4*f^2*g^2 - 832*a^5*c*d*e^5*f*g^3 + 128*a ^6*e^6*g^4 + 210*(26*c^6*d^6*f*g^3 + 7*a*c^5*d^5*e*g^4)*x^5 + 35*(286*c^6* d^6*f^2*g^2 + 208*a*c^5*d^5*e*f*g^3 + a^2*c^4*d^4*e^2*g^4)*x^4 + 20*(429*c ^6*d^6*f^3*g + 715*a*c^5*d^5*e*f^2*g^2 + 13*a^2*c^4*d^4*e^2*f*g^3 - 2*a^3* c^3*d^3*e^3*g^4)*x^3 + 3*(1001*c^6*d^6*f^4 + 4576*a*c^5*d^5*e*f^3*g + 286* a^2*c^4*d^4*e^2*f^2*g^2 - 104*a^3*c^3*d^3*e^3*f*g^3 + 16*a^4*c^2*d^2*e^4*g ^4)*x^2 + 2*(3003*a*c^5*d^5*e*f^4 + 858*a^2*c^4*d^4*e^2*f^3*g - 572*a^3*c^ 3*d^3*e^3*f^2*g^2 + 208*a^4*c^2*d^2*e^4*f*g^3 - 32*a^5*c*d*e^5*g^4)*x)*sqr t(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*sqrt(e*x + d)/(c^5*d^5*e*x + c^5* d^6)
\[ \int \frac {(f+g x)^4 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{(d+e x)^{3/2}} \, dx=\int \frac {\left (\left (d + e x\right ) \left (a e + c d x\right )\right )^{\frac {3}{2}} \left (f + g x\right )^{4}}{\left (d + e x\right )^{\frac {3}{2}}}\, dx \]
Time = 0.26 (sec) , antiderivative size = 413, normalized size of antiderivative = 1.23 \[ \int \frac {(f+g x)^4 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{(d+e x)^{3/2}} \, dx=\frac {2 \, {\left (c^{2} d^{2} x^{2} + 2 \, a c d e x + a^{2} e^{2}\right )} \sqrt {c d x + a e} f^{4}}{5 \, c d} + \frac {8 \, {\left (5 \, c^{3} d^{3} x^{3} + 8 \, a c^{2} d^{2} e x^{2} + a^{2} c d e^{2} x - 2 \, a^{3} e^{3}\right )} \sqrt {c d x + a e} f^{3} g}{35 \, c^{2} d^{2}} + \frac {4 \, {\left (35 \, c^{4} d^{4} x^{4} + 50 \, a c^{3} d^{3} e x^{3} + 3 \, a^{2} c^{2} d^{2} e^{2} x^{2} - 4 \, a^{3} c d e^{3} x + 8 \, a^{4} e^{4}\right )} \sqrt {c d x + a e} f^{2} g^{2}}{105 \, c^{3} d^{3}} + \frac {8 \, {\left (105 \, c^{5} d^{5} x^{5} + 140 \, a c^{4} d^{4} e x^{4} + 5 \, a^{2} c^{3} d^{3} e^{2} x^{3} - 6 \, a^{3} c^{2} d^{2} e^{3} x^{2} + 8 \, a^{4} c d e^{4} x - 16 \, a^{5} e^{5}\right )} \sqrt {c d x + a e} f g^{3}}{1155 \, c^{4} d^{4}} + \frac {2 \, {\left (1155 \, c^{6} d^{6} x^{6} + 1470 \, a c^{5} d^{5} e x^{5} + 35 \, a^{2} c^{4} d^{4} e^{2} x^{4} - 40 \, a^{3} c^{3} d^{3} e^{3} x^{3} + 48 \, a^{4} c^{2} d^{2} e^{4} x^{2} - 64 \, a^{5} c d e^{5} x + 128 \, a^{6} e^{6}\right )} \sqrt {c d x + a e} g^{4}}{15015 \, c^{5} d^{5}} \]
integrate((g*x+f)^4*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)/(e*x+d)^(3/2), x, algorithm="maxima")
2/5*(c^2*d^2*x^2 + 2*a*c*d*e*x + a^2*e^2)*sqrt(c*d*x + a*e)*f^4/(c*d) + 8/ 35*(5*c^3*d^3*x^3 + 8*a*c^2*d^2*e*x^2 + a^2*c*d*e^2*x - 2*a^3*e^3)*sqrt(c* d*x + a*e)*f^3*g/(c^2*d^2) + 4/105*(35*c^4*d^4*x^4 + 50*a*c^3*d^3*e*x^3 + 3*a^2*c^2*d^2*e^2*x^2 - 4*a^3*c*d*e^3*x + 8*a^4*e^4)*sqrt(c*d*x + a*e)*f^2 *g^2/(c^3*d^3) + 8/1155*(105*c^5*d^5*x^5 + 140*a*c^4*d^4*e*x^4 + 5*a^2*c^3 *d^3*e^2*x^3 - 6*a^3*c^2*d^2*e^3*x^2 + 8*a^4*c*d*e^4*x - 16*a^5*e^5)*sqrt( c*d*x + a*e)*f*g^3/(c^4*d^4) + 2/15015*(1155*c^6*d^6*x^6 + 1470*a*c^5*d^5* e*x^5 + 35*a^2*c^4*d^4*e^2*x^4 - 40*a^3*c^3*d^3*e^3*x^3 + 48*a^4*c^2*d^2*e ^4*x^2 - 64*a^5*c*d*e^5*x + 128*a^6*e^6)*sqrt(c*d*x + a*e)*g^4/(c^5*d^5)
Leaf count of result is larger than twice the leaf count of optimal. 2535 vs. \(2 (306) = 612\).
Time = 0.38 (sec) , antiderivative size = 2535, normalized size of antiderivative = 7.54 \[ \int \frac {(f+g x)^4 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{(d+e x)^{3/2}} \, dx=\text {Too large to display} \]
integrate((g*x+f)^4*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)/(e*x+d)^(3/2), x, algorithm="giac")
2/45045*(15015*a*f^4*((sqrt(-c*d^2*e + a*e^3)*c*d^2 - sqrt(-c*d^2*e + a*e^ 3)*a*e^2)/(c*d) + ((e*x + d)*c*d*e - c*d^2*e + a*e^3)^(3/2)/(c*d*e))*abs(e )/e + 1716*c*d*f^3*g*((15*sqrt(-c*d^2*e + a*e^3)*c^3*d^6 - 3*sqrt(-c*d^2*e + a*e^3)*a*c^2*d^4*e^2 - 4*sqrt(-c*d^2*e + a*e^3)*a^2*c*d^2*e^4 - 8*sqrt( -c*d^2*e + a*e^3)*a^3*e^6)/(c^3*d^3*e^2) + (35*((e*x + d)*c*d*e - c*d^2*e + a*e^3)^(3/2)*a^2*e^6 - 42*((e*x + d)*c*d*e - c*d^2*e + a*e^3)^(5/2)*a*e^ 3 + 15*((e*x + d)*c*d*e - c*d^2*e + a*e^3)^(7/2))/(c^3*d^3*e^5))*abs(e)/e^ 2 + 2574*a*f^2*g^2*((15*sqrt(-c*d^2*e + a*e^3)*c^3*d^6 - 3*sqrt(-c*d^2*e + a*e^3)*a*c^2*d^4*e^2 - 4*sqrt(-c*d^2*e + a*e^3)*a^2*c*d^2*e^4 - 8*sqrt(-c *d^2*e + a*e^3)*a^3*e^6)/(c^3*d^3*e^2) + (35*((e*x + d)*c*d*e - c*d^2*e + a*e^3)^(3/2)*a^2*e^6 - 42*((e*x + d)*c*d*e - c*d^2*e + a*e^3)^(5/2)*a*e^3 + 15*((e*x + d)*c*d*e - c*d^2*e + a*e^3)^(7/2))/(c^3*d^3*e^5))*abs(e)/e - 858*c*d*f^2*g^2*((35*sqrt(-c*d^2*e + a*e^3)*c^4*d^8 - 5*sqrt(-c*d^2*e + a* e^3)*a*c^3*d^6*e^2 - 6*sqrt(-c*d^2*e + a*e^3)*a^2*c^2*d^4*e^4 - 8*sqrt(-c* d^2*e + a*e^3)*a^3*c*d^2*e^6 - 16*sqrt(-c*d^2*e + a*e^3)*a^4*e^8)/(c^4*d^4 *e^3) + (105*((e*x + d)*c*d*e - c*d^2*e + a*e^3)^(3/2)*a^3*e^9 - 189*((e*x + d)*c*d*e - c*d^2*e + a*e^3)^(5/2)*a^2*e^6 + 135*((e*x + d)*c*d*e - c*d^ 2*e + a*e^3)^(7/2)*a*e^3 - 35*((e*x + d)*c*d*e - c*d^2*e + a*e^3)^(9/2))/( c^4*d^4*e^7))*abs(e)/e^2 - 572*a*f*g^3*((35*sqrt(-c*d^2*e + a*e^3)*c^4*d^8 - 5*sqrt(-c*d^2*e + a*e^3)*a*c^3*d^6*e^2 - 6*sqrt(-c*d^2*e + a*e^3)*a^...
Time = 12.56 (sec) , antiderivative size = 445, normalized size of antiderivative = 1.32 \[ \int \frac {(f+g x)^4 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{(d+e x)^{3/2}} \, dx=\frac {\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}\,\left (\frac {4\,g^3\,x^5\,\left (7\,a\,e\,g+26\,c\,d\,f\right )}{143}+\frac {256\,a^6\,e^6\,g^4-1664\,a^5\,c\,d\,e^5\,f\,g^3+4576\,a^4\,c^2\,d^2\,e^4\,f^2\,g^2-6864\,a^3\,c^3\,d^3\,e^3\,f^3\,g+6006\,a^2\,c^4\,d^4\,e^2\,f^4}{15015\,c^5\,d^5}+\frac {x^2\,\left (96\,a^4\,c^2\,d^2\,e^4\,g^4-624\,a^3\,c^3\,d^3\,e^3\,f\,g^3+1716\,a^2\,c^4\,d^4\,e^2\,f^2\,g^2+27456\,a\,c^5\,d^5\,e\,f^3\,g+6006\,c^6\,d^6\,f^4\right )}{15015\,c^5\,d^5}+\frac {x\,\left (-128\,a^5\,c\,d\,e^5\,g^4+832\,a^4\,c^2\,d^2\,e^4\,f\,g^3-2288\,a^3\,c^3\,d^3\,e^3\,f^2\,g^2+3432\,a^2\,c^4\,d^4\,e^2\,f^3\,g+12012\,a\,c^5\,d^5\,e\,f^4\right )}{15015\,c^5\,d^5}+\frac {2\,c\,d\,g^4\,x^6}{13}+\frac {8\,g\,x^3\,\left (-2\,a^3\,e^3\,g^3+13\,a^2\,c\,d\,e^2\,f\,g^2+715\,a\,c^2\,d^2\,e\,f^2\,g+429\,c^3\,d^3\,f^3\right )}{3003\,c^2\,d^2}+\frac {2\,g^2\,x^4\,\left (a^2\,e^2\,g^2+208\,a\,c\,d\,e\,f\,g+286\,c^2\,d^2\,f^2\right )}{429\,c\,d}\right )}{\sqrt {d+e\,x}} \]
((x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1/2)*((4*g^3*x^5*(7*a*e*g + 26*c *d*f))/143 + (256*a^6*e^6*g^4 + 6006*a^2*c^4*d^4*e^2*f^4 - 6864*a^3*c^3*d^ 3*e^3*f^3*g - 1664*a^5*c*d*e^5*f*g^3 + 4576*a^4*c^2*d^2*e^4*f^2*g^2)/(1501 5*c^5*d^5) + (x^2*(6006*c^6*d^6*f^4 + 96*a^4*c^2*d^2*e^4*g^4 - 624*a^3*c^3 *d^3*e^3*f*g^3 + 27456*a*c^5*d^5*e*f^3*g + 1716*a^2*c^4*d^4*e^2*f^2*g^2))/ (15015*c^5*d^5) + (x*(12012*a*c^5*d^5*e*f^4 - 128*a^5*c*d*e^5*g^4 + 3432*a ^2*c^4*d^4*e^2*f^3*g + 832*a^4*c^2*d^2*e^4*f*g^3 - 2288*a^3*c^3*d^3*e^3*f^ 2*g^2))/(15015*c^5*d^5) + (2*c*d*g^4*x^6)/13 + (8*g*x^3*(429*c^3*d^3*f^3 - 2*a^3*e^3*g^3 + 715*a*c^2*d^2*e*f^2*g + 13*a^2*c*d*e^2*f*g^2))/(3003*c^2* d^2) + (2*g^2*x^4*(a^2*e^2*g^2 + 286*c^2*d^2*f^2 + 208*a*c*d*e*f*g))/(429* c*d)))/(d + e*x)^(1/2)