\(\int \frac {(f+g x)^4 \sqrt {a d e+(c d^2+a e^2) x+c d e x^2}}{\sqrt {d+e x}} \, dx\) [1]

Optimal result

Integrand size = 46, antiderivative size = 297 \[ \int \frac {(f+g x)^4 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {d+e x}} \, dx=\frac {2 (c d f-a e g)^4 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{3 c^5 d^5 (d+e x)^{3/2}}+\frac {8 g (c d f-a e g)^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{5 c^5 d^5 (d+e x)^{5/2}}+\frac {12 g^2 (c d f-a e g)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{7 c^5 d^5 (d+e x)^{7/2}}+\frac {8 g^3 (c d f-a e g) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{9/2}}{9 c^5 d^5 (d+e x)^{9/2}}+\frac {2 g^4 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{11/2}}{11 c^5 d^5 (d+e x)^{11/2}} \] Output:

2/3*(-a*e*g+c*d*f)^4*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)/c^5/d^5/(e*x+ 
d)^(3/2)+8/5*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^ 
5/d^5/(e*x+d)^(5/2)+12/7*g^2*(-a*e*g+c*d*f)^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e 
*x^2)^(7/2)/c^5/d^5/(e*x+d)^(7/2)+8/9*g^3*(-a*e*g+c*d*f)*(a*d*e+(a*e^2+c*d 
^2)*x+c*d*e*x^2)^(9/2)/c^5/d^5/(e*x+d)^(9/2)+2/11*g^4*(a*d*e+(a*e^2+c*d^2) 
*x+c*d*e*x^2)^(11/2)/c^5/d^5/(e*x+d)^(11/2)
 

Mathematica [A] (verified)

Time = 0.26 (sec) , antiderivative size = 195, normalized size of antiderivative = 0.66 \[ \int \frac {(f+g x)^4 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {d+e x}} \, dx=\frac {2 ((a e+c d x) (d+e x))^{3/2} \left (128 a^4 e^4 g^4-64 a^3 c d e^3 g^3 (11 f+3 g x)+48 a^2 c^2 d^2 e^2 g^2 \left (33 f^2+22 f g x+5 g^2 x^2\right )-8 a c^3 d^3 e g \left (231 f^3+297 f^2 g x+165 f g^2 x^2+35 g^3 x^3\right )+c^4 d^4 \left (1155 f^4+2772 f^3 g x+2970 f^2 g^2 x^2+1540 f g^3 x^3+315 g^4 x^4\right )\right )}{3465 c^5 d^5 (d+e x)^{3/2}} \] Input:

Integrate[((f + g*x)^4*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/Sqrt[d 
 + e*x],x]
 

Output:

(2*((a*e + c*d*x)*(d + e*x))^(3/2)*(128*a^4*e^4*g^4 - 64*a^3*c*d*e^3*g^3*( 
11*f + 3*g*x) + 48*a^2*c^2*d^2*e^2*g^2*(33*f^2 + 22*f*g*x + 5*g^2*x^2) - 8 
*a*c^3*d^3*e*g*(231*f^3 + 297*f^2*g*x + 165*f*g^2*x^2 + 35*g^3*x^3) + c^4* 
d^4*(1155*f^4 + 2772*f^3*g*x + 2970*f^2*g^2*x^2 + 1540*f*g^3*x^3 + 315*g^4 
*x^4)))/(3465*c^5*d^5*(d + e*x)^(3/2))
 

Rubi [A] (verified)

Time = 1.18 (sec) , antiderivative size = 351, normalized size of antiderivative = 1.18, 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.

\(\displaystyle \int \frac {(f+g x)^4 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{\sqrt {d+e x}} \, dx\)

\(\Big \downarrow \) 1253

\(\displaystyle \frac {8 (c d f-a e g) \int \frac {(f+g x)^3 \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}{\sqrt {d+e x}}dx}{11 c d}+\frac {2 (f+g x)^4 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{11 c d (d+e x)^{3/2}}\)

\(\Big \downarrow \) 1253

\(\displaystyle \frac {8 (c d f-a e g) \left (\frac {2 (c d f-a e g) \int \frac {(f+g x)^2 \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}{\sqrt {d+e x}}dx}{3 c d}+\frac {2 (f+g x)^3 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{9 c d (d+e x)^{3/2}}\right )}{11 c d}+\frac {2 (f+g x)^4 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{11 c d (d+e x)^{3/2}}\)

\(\Big \downarrow \) 1253

\(\displaystyle \frac {8 (c d f-a e g) \left (\frac {2 (c d f-a e g) \left (\frac {4 (c d f-a e g) \int \frac {(f+g x) \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}{\sqrt {d+e x}}dx}{7 c d}+\frac {2 (f+g x)^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{7 c d (d+e x)^{3/2}}\right )}{3 c d}+\frac {2 (f+g x)^3 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{9 c d (d+e x)^{3/2}}\right )}{11 c d}+\frac {2 (f+g x)^4 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{11 c d (d+e x)^{3/2}}\)

\(\Big \downarrow \) 1221

\(\displaystyle \frac {8 (c d f-a e g) \left (\frac {2 (c d f-a e g) \left (\frac {4 (c d f-a e g) \left (\frac {1}{5} \left (-\frac {2 a e g}{c d}-\frac {3 d g}{e}+5 f\right ) \int \frac {\sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}{\sqrt {d+e x}}dx+\frac {2 g \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{5 c d e \sqrt {d+e x}}\right )}{7 c d}+\frac {2 (f+g x)^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{7 c d (d+e x)^{3/2}}\right )}{3 c d}+\frac {2 (f+g x)^3 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{9 c d (d+e x)^{3/2}}\right )}{11 c d}+\frac {2 (f+g x)^4 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{11 c d (d+e x)^{3/2}}\)

\(\Big \downarrow \) 1122

\(\displaystyle \frac {2 (f+g x)^4 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{11 c d (d+e x)^{3/2}}+\frac {8 (c d f-a e g) \left (\frac {2 (f+g x)^3 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{9 c d (d+e x)^{3/2}}+\frac {2 (c d f-a e g) \left (\frac {2 (f+g x)^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{7 c d (d+e x)^{3/2}}+\frac {4 (c d f-a e g) \left (\frac {2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2} \left (-\frac {2 a e g}{c d}-\frac {3 d g}{e}+5 f\right )}{15 c d (d+e x)^{3/2}}+\frac {2 g \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{5 c d e \sqrt {d+e x}}\right )}{7 c d}\right )}{3 c d}\right )}{11 c d}\)

Input:

Int[((f + g*x)^4*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/Sqrt[d + e*x 
],x]
 

Output:

(2*(f + g*x)^4*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(11*c*d*(d + 
 e*x)^(3/2)) + (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)^(3/2))/(9*c*d*(d + e*x)^(3/2)) + (2*(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)^(3/2))/(7*c*d*(d + e*x)^ 
(3/2)) + (4*(c*d*f - a*e*g)*((2*(5*f - (3*d*g)/e - (2*a*e*g)/(c*d))*(a*d*e 
 + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(15*c*d*(d + e*x)^(3/2)) + (2*g*( 
a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(5*c*d*e*Sqrt[d + e*x])))/(7 
*c*d)))/(3*c*d)))/(11*c*d)
 

Defintions of rubi rules used

rule 1122

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]
 

rule 1221

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]
 

rule 1253

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])
 

Maple [A] (verified)

Time = 2.95 (sec) , antiderivative size = 273, normalized size of antiderivative = 0.92

method	result	size
default	\(\frac {2 \left (c d x +a e \right ) \left (315 g^{4} x^{4} d^{4} c^{4}-280 a \,c^{3} d^{3} e \,g^{4} x^{3}+1540 c^{4} d^{4} f \,g^{3} x^{3}+240 a^{2} c^{2} d^{2} e^{2} g^{4} x^{2}-1320 a \,c^{3} d^{3} e f \,g^{3} x^{2}+2970 c^{4} d^{4} f^{2} g^{2} x^{2}-192 a^{3} c d \,e^{3} g^{4} x +1056 a^{2} c^{2} d^{2} e^{2} f \,g^{3} x -2376 a \,c^{3} d^{3} e \,f^{2} g^{2} x +2772 c^{4} d^{4} f^{3} g x +128 a^{4} e^{4} g^{4}-704 a^{3} c d \,e^{3} f \,g^{3}+1584 a^{2} c^{2} d^{2} e^{2} f^{2} g^{2}-1848 a \,c^{3} d^{3} e \,f^{3} g +1155 f^{4} d^{4} c^{4}\right ) \sqrt {\left (e x +d \right ) \left (c d x +a e \right )}}{3465 d^{5} c^{5} \sqrt {e x +d}}\)	\(273\)
gosper	\(\frac {2 \left (c d x +a e \right ) \left (315 g^{4} x^{4} d^{4} c^{4}-280 a \,c^{3} d^{3} e \,g^{4} x^{3}+1540 c^{4} d^{4} f \,g^{3} x^{3}+240 a^{2} c^{2} d^{2} e^{2} g^{4} x^{2}-1320 a \,c^{3} d^{3} e f \,g^{3} x^{2}+2970 c^{4} d^{4} f^{2} g^{2} x^{2}-192 a^{3} c d \,e^{3} g^{4} x +1056 a^{2} c^{2} d^{2} e^{2} f \,g^{3} x -2376 a \,c^{3} d^{3} e \,f^{2} g^{2} x +2772 c^{4} d^{4} f^{3} g x +128 a^{4} e^{4} g^{4}-704 a^{3} c d \,e^{3} f \,g^{3}+1584 a^{2} c^{2} d^{2} e^{2} f^{2} g^{2}-1848 a \,c^{3} d^{3} e \,f^{3} g +1155 f^{4} d^{4} c^{4}\right ) \sqrt {c d \,x^{2} e +a \,e^{2} x +c \,d^{2} x +a d e}}{3465 d^{5} c^{5} \sqrt {e x +d}}\)	\(283\)
orering	\(\frac {2 \left (315 g^{4} x^{4} d^{4} c^{4}-280 a \,c^{3} d^{3} e \,g^{4} x^{3}+1540 c^{4} d^{4} f \,g^{3} x^{3}+240 a^{2} c^{2} d^{2} e^{2} g^{4} x^{2}-1320 a \,c^{3} d^{3} e f \,g^{3} x^{2}+2970 c^{4} d^{4} f^{2} g^{2} x^{2}-192 a^{3} c d \,e^{3} g^{4} x +1056 a^{2} c^{2} d^{2} e^{2} f \,g^{3} x -2376 a \,c^{3} d^{3} e \,f^{2} g^{2} x +2772 c^{4} d^{4} f^{3} g x +128 a^{4} e^{4} g^{4}-704 a^{3} c d \,e^{3} f \,g^{3}+1584 a^{2} c^{2} d^{2} e^{2} f^{2} g^{2}-1848 a \,c^{3} d^{3} e \,f^{3} g +1155 f^{4} d^{4} c^{4}\right ) \left (c d x +a e \right ) \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d \,x^{2} e}}{3465 d^{5} c^{5} \sqrt {e x +d}}\)	\(284\)

Input:

int((g*x+f)^4*(a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^(1/2)/(e*x+d)^(1/2),x,meth 
od=_RETURNVERBOSE)
 

Output:

2/3465*(c*d*x+a*e)*(315*c^4*d^4*g^4*x^4-280*a*c^3*d^3*e*g^4*x^3+1540*c^4*d 
^4*f*g^3*x^3+240*a^2*c^2*d^2*e^2*g^4*x^2-1320*a*c^3*d^3*e*f*g^3*x^2+2970*c 
^4*d^4*f^2*g^2*x^2-192*a^3*c*d*e^3*g^4*x+1056*a^2*c^2*d^2*e^2*f*g^3*x-2376 
*a*c^3*d^3*e*f^2*g^2*x+2772*c^4*d^4*f^3*g*x+128*a^4*e^4*g^4-704*a^3*c*d*e^ 
3*f*g^3+1584*a^2*c^2*d^2*e^2*f^2*g^2-1848*a*c^3*d^3*e*f^3*g+1155*c^4*d^4*f 
^4)*((e*x+d)*(c*d*x+a*e))^(1/2)/d^5/c^5/(e*x+d)^(1/2)
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 375, normalized size of antiderivative = 1.26 \[ \int \frac {(f+g x)^4 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {d+e x}} \, dx=\frac {2 \, {\left (315 \, c^{5} d^{5} g^{4} x^{5} + 1155 \, a c^{4} d^{4} e f^{4} - 1848 \, a^{2} c^{3} d^{3} e^{2} f^{3} g + 1584 \, a^{3} c^{2} d^{2} e^{3} f^{2} g^{2} - 704 \, a^{4} c d e^{4} f g^{3} + 128 \, a^{5} e^{5} g^{4} + 35 \, {\left (44 \, c^{5} d^{5} f g^{3} + a c^{4} d^{4} e g^{4}\right )} x^{4} + 10 \, {\left (297 \, c^{5} d^{5} f^{2} g^{2} + 22 \, a c^{4} d^{4} e f g^{3} - 4 \, a^{2} c^{3} d^{3} e^{2} g^{4}\right )} x^{3} + 6 \, {\left (462 \, c^{5} d^{5} f^{3} g + 99 \, a c^{4} d^{4} e f^{2} g^{2} - 44 \, a^{2} c^{3} d^{3} e^{2} f g^{3} + 8 \, a^{3} c^{2} d^{2} e^{3} g^{4}\right )} x^{2} + {\left (1155 \, c^{5} d^{5} f^{4} + 924 \, a c^{4} d^{4} e f^{3} g - 792 \, a^{2} c^{3} d^{3} e^{2} f^{2} g^{2} + 352 \, a^{3} c^{2} d^{2} e^{3} f g^{3} - 64 \, a^{4} c d e^{4} g^{4}\right )} x\right )} \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} \sqrt {e x + d}}{3465 \, {\left (c^{5} d^{5} e x + c^{5} d^{6}\right )}} \] Input:

integrate((g*x+f)^4*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/(e*x+d)^(1/2), 
x, algorithm="fricas")
 

Output:

2/3465*(315*c^5*d^5*g^4*x^5 + 1155*a*c^4*d^4*e*f^4 - 1848*a^2*c^3*d^3*e^2* 
f^3*g + 1584*a^3*c^2*d^2*e^3*f^2*g^2 - 704*a^4*c*d*e^4*f*g^3 + 128*a^5*e^5 
*g^4 + 35*(44*c^5*d^5*f*g^3 + a*c^4*d^4*e*g^4)*x^4 + 10*(297*c^5*d^5*f^2*g 
^2 + 22*a*c^4*d^4*e*f*g^3 - 4*a^2*c^3*d^3*e^2*g^4)*x^3 + 6*(462*c^5*d^5*f^ 
3*g + 99*a*c^4*d^4*e*f^2*g^2 - 44*a^2*c^3*d^3*e^2*f*g^3 + 8*a^3*c^2*d^2*e^ 
3*g^4)*x^2 + (1155*c^5*d^5*f^4 + 924*a*c^4*d^4*e*f^3*g - 792*a^2*c^3*d^3*e 
^2*f^2*g^2 + 352*a^3*c^2*d^2*e^3*f*g^3 - 64*a^4*c*d*e^4*g^4)*x)*sqrt(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)
 

Sympy [F]

\[ \int \frac {(f+g x)^4 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {d+e x}} \, dx=\int \frac {\sqrt {\left (d + e x\right ) \left (a e + c d x\right )} \left (f + g x\right )^{4}}{\sqrt {d + e x}}\, dx \] Input:

integrate((g*x+f)**4*(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(1/2)/(e*x+d)** 
(1/2),x)
 

Output:

Integral(sqrt((d + e*x)*(a*e + c*d*x))*(f + g*x)**4/sqrt(d + e*x), x)
 

Maxima [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 320, normalized size of antiderivative = 1.08 \[ \int \frac {(f+g x)^4 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {d+e x}} \, dx=\frac {2 \, {\left (c d x + a e\right )}^{\frac {3}{2}} f^{4}}{3 \, c d} + \frac {8 \, {\left (3 \, c^{2} d^{2} x^{2} + a c d e x - 2 \, a^{2} e^{2}\right )} \sqrt {c d x + a e} f^{3} g}{15 \, c^{2} d^{2}} + \frac {4 \, {\left (15 \, c^{3} d^{3} x^{3} + 3 \, a c^{2} d^{2} e x^{2} - 4 \, a^{2} c d e^{2} x + 8 \, a^{3} e^{3}\right )} \sqrt {c d x + a e} f^{2} g^{2}}{35 \, c^{3} d^{3}} + \frac {8 \, {\left (35 \, c^{4} d^{4} x^{4} + 5 \, a c^{3} d^{3} e x^{3} - 6 \, a^{2} c^{2} d^{2} e^{2} x^{2} + 8 \, a^{3} c d e^{3} x - 16 \, a^{4} e^{4}\right )} \sqrt {c d x + a e} f g^{3}}{315 \, c^{4} d^{4}} + \frac {2 \, {\left (315 \, c^{5} d^{5} x^{5} + 35 \, a c^{4} d^{4} e x^{4} - 40 \, a^{2} c^{3} d^{3} e^{2} x^{3} + 48 \, a^{3} c^{2} d^{2} e^{3} x^{2} - 64 \, a^{4} c d e^{4} x + 128 \, a^{5} e^{5}\right )} \sqrt {c d x + a e} g^{4}}{3465 \, c^{5} d^{5}} \] Input:

integrate((g*x+f)^4*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/(e*x+d)^(1/2), 
x, algorithm="maxima")
 

Output:

2/3*(c*d*x + a*e)^(3/2)*f^4/(c*d) + 8/15*(3*c^2*d^2*x^2 + a*c*d*e*x - 2*a^ 
2*e^2)*sqrt(c*d*x + a*e)*f^3*g/(c^2*d^2) + 4/35*(15*c^3*d^3*x^3 + 3*a*c^2* 
d^2*e*x^2 - 4*a^2*c*d*e^2*x + 8*a^3*e^3)*sqrt(c*d*x + a*e)*f^2*g^2/(c^3*d^ 
3) + 8/315*(35*c^4*d^4*x^4 + 5*a*c^3*d^3*e*x^3 - 6*a^2*c^2*d^2*e^2*x^2 + 8 
*a^3*c*d*e^3*x - 16*a^4*e^4)*sqrt(c*d*x + a*e)*f*g^3/(c^4*d^4) + 2/3465*(3 
15*c^5*d^5*x^5 + 35*a*c^4*d^4*e*x^4 - 40*a^2*c^3*d^3*e^2*x^3 + 48*a^3*c^2* 
d^2*e^3*x^2 - 64*a^4*c*d*e^4*x + 128*a^5*e^5)*sqrt(c*d*x + a*e)*g^4/(c^5*d 
^5)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 668 vs. \(2 (267) = 534\).

Time = 0.12 (sec) , antiderivative size = 668, normalized size of antiderivative = 2.25 \[ \int \frac {(f+g x)^4 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {d+e x}} \, dx =\text {Too large to display} \] Input:

integrate((g*x+f)^4*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/(e*x+d)^(1/2), 
x, algorithm="giac")
 

Output:

2/3465*(3465*sqrt(c*d*x + a*e)*a*e*f^4 - 1155*(3*sqrt(c*d*x + a*e)*a*e - ( 
c*d*x + a*e)^(3/2))*f^4 - 4620*(3*sqrt(c*d*x + a*e)*a*e - (c*d*x + a*e)^(3 
/2))*a*e*f^3*g/(c*d) + 924*(15*sqrt(c*d*x + a*e)*a^2*e^2 - 10*(c*d*x + a*e 
)^(3/2)*a*e + 3*(c*d*x + a*e)^(5/2))*f^3*g/(c*d) + 1386*(15*sqrt(c*d*x + a 
*e)*a^2*e^2 - 10*(c*d*x + a*e)^(3/2)*a*e + 3*(c*d*x + a*e)^(5/2))*a*e*f^2* 
g^2/(c^2*d^2) - 594*(35*sqrt(c*d*x + a*e)*a^3*e^3 - 35*(c*d*x + a*e)^(3/2) 
*a^2*e^2 + 21*(c*d*x + a*e)^(5/2)*a*e - 5*(c*d*x + a*e)^(7/2))*f^2*g^2/(c^ 
2*d^2) - 396*(35*sqrt(c*d*x + a*e)*a^3*e^3 - 35*(c*d*x + a*e)^(3/2)*a^2*e^ 
2 + 21*(c*d*x + a*e)^(5/2)*a*e - 5*(c*d*x + a*e)^(7/2))*a*e*f*g^3/(c^3*d^3 
) + 44*(315*sqrt(c*d*x + a*e)*a^4*e^4 - 420*(c*d*x + a*e)^(3/2)*a^3*e^3 + 
378*(c*d*x + a*e)^(5/2)*a^2*e^2 - 180*(c*d*x + a*e)^(7/2)*a*e + 35*(c*d*x 
+ a*e)^(9/2))*f*g^3/(c^3*d^3) + 11*(315*sqrt(c*d*x + a*e)*a^4*e^4 - 420*(c 
*d*x + a*e)^(3/2)*a^3*e^3 + 378*(c*d*x + a*e)^(5/2)*a^2*e^2 - 180*(c*d*x + 
 a*e)^(7/2)*a*e + 35*(c*d*x + a*e)^(9/2))*a*e*g^4/(c^4*d^4) - 5*(693*sqrt( 
c*d*x + a*e)*a^5*e^5 - 1155*(c*d*x + a*e)^(3/2)*a^4*e^4 + 1386*(c*d*x + a* 
e)^(5/2)*a^3*e^3 - 990*(c*d*x + a*e)^(7/2)*a^2*e^2 + 385*(c*d*x + a*e)^(9/ 
2)*a*e - 63*(c*d*x + a*e)^(11/2))*g^4/(c^4*d^4))/(c*d)
 

Mupad [B] (verification not implemented)

Time = 6.39 (sec) , antiderivative size = 347, normalized size of antiderivative = 1.17 \[ \int \frac {(f+g x)^4 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {d+e x}} \, dx=\frac {\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}\,\left (\frac {2\,g^4\,x^5}{11}+\frac {256\,a^5\,e^5\,g^4-1408\,a^4\,c\,d\,e^4\,f\,g^3+3168\,a^3\,c^2\,d^2\,e^3\,f^2\,g^2-3696\,a^2\,c^3\,d^3\,e^2\,f^3\,g+2310\,a\,c^4\,d^4\,e\,f^4}{3465\,c^5\,d^5}+\frac {x\,\left (-128\,a^4\,c\,d\,e^4\,g^4+704\,a^3\,c^2\,d^2\,e^3\,f\,g^3-1584\,a^2\,c^3\,d^3\,e^2\,f^2\,g^2+1848\,a\,c^4\,d^4\,e\,f^3\,g+2310\,c^5\,d^5\,f^4\right )}{3465\,c^5\,d^5}+\frac {4\,g\,x^2\,\left (8\,a^3\,e^3\,g^3-44\,a^2\,c\,d\,e^2\,f\,g^2+99\,a\,c^2\,d^2\,e\,f^2\,g+462\,c^3\,d^3\,f^3\right )}{1155\,c^3\,d^3}+\frac {4\,g^2\,x^3\,\left (-4\,a^2\,e^2\,g^2+22\,a\,c\,d\,e\,f\,g+297\,c^2\,d^2\,f^2\right )}{693\,c^2\,d^2}+\frac {2\,g^3\,x^4\,\left (a\,e\,g+44\,c\,d\,f\right )}{99\,c\,d}\right )}{\sqrt {d+e\,x}} \] Input:

int(((f + g*x)^4*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1/2))/(d + e*x)^ 
(1/2),x)
 

Output:

((x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1/2)*((2*g^4*x^5)/11 + (256*a^5* 
e^5*g^4 + 2310*a*c^4*d^4*e*f^4 - 3696*a^2*c^3*d^3*e^2*f^3*g - 1408*a^4*c*d 
*e^4*f*g^3 + 3168*a^3*c^2*d^2*e^3*f^2*g^2)/(3465*c^5*d^5) + (x*(2310*c^5*d 
^5*f^4 - 128*a^4*c*d*e^4*g^4 + 704*a^3*c^2*d^2*e^3*f*g^3 + 1848*a*c^4*d^4* 
e*f^3*g - 1584*a^2*c^3*d^3*e^2*f^2*g^2))/(3465*c^5*d^5) + (4*g*x^2*(8*a^3* 
e^3*g^3 + 462*c^3*d^3*f^3 + 99*a*c^2*d^2*e*f^2*g - 44*a^2*c*d*e^2*f*g^2))/ 
(1155*c^3*d^3) + (4*g^2*x^3*(297*c^2*d^2*f^2 - 4*a^2*e^2*g^2 + 22*a*c*d*e* 
f*g))/(693*c^2*d^2) + (2*g^3*x^4*(a*e*g + 44*c*d*f))/(99*c*d)))/(d + e*x)^ 
(1/2)
 

Reduce [B] (verification not implemented)

Time = 0.60 (sec) , antiderivative size = 349, normalized size of antiderivative = 1.18 \[ \int \frac {(f+g x)^4 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {d+e x}} \, dx=\frac {2 \sqrt {c d x +a e}\, \left (315 c^{5} d^{5} g^{4} x^{5}+35 a \,c^{4} d^{4} e \,g^{4} x^{4}+1540 c^{5} d^{5} f \,g^{3} x^{4}-40 a^{2} c^{3} d^{3} e^{2} g^{4} x^{3}+220 a \,c^{4} d^{4} e f \,g^{3} x^{3}+2970 c^{5} d^{5} f^{2} g^{2} x^{3}+48 a^{3} c^{2} d^{2} e^{3} g^{4} x^{2}-264 a^{2} c^{3} d^{3} e^{2} f \,g^{3} x^{2}+594 a \,c^{4} d^{4} e \,f^{2} g^{2} x^{2}+2772 c^{5} d^{5} f^{3} g \,x^{2}-64 a^{4} c d \,e^{4} g^{4} x +352 a^{3} c^{2} d^{2} e^{3} f \,g^{3} x -792 a^{2} c^{3} d^{3} e^{2} f^{2} g^{2} x +924 a \,c^{4} d^{4} e \,f^{3} g x +1155 c^{5} d^{5} f^{4} x +128 a^{5} e^{5} g^{4}-704 a^{4} c d \,e^{4} f \,g^{3}+1584 a^{3} c^{2} d^{2} e^{3} f^{2} g^{2}-1848 a^{2} c^{3} d^{3} e^{2} f^{3} g +1155 a \,c^{4} d^{4} e \,f^{4}\right )}{3465 c^{5} d^{5}} \] Input:

int((g*x+f)^4*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/(e*x+d)^(1/2),x)
 

Output:

(2*sqrt(a*e + c*d*x)*(128*a**5*e**5*g**4 - 704*a**4*c*d*e**4*f*g**3 - 64*a 
**4*c*d*e**4*g**4*x + 1584*a**3*c**2*d**2*e**3*f**2*g**2 + 352*a**3*c**2*d 
**2*e**3*f*g**3*x + 48*a**3*c**2*d**2*e**3*g**4*x**2 - 1848*a**2*c**3*d**3 
*e**2*f**3*g - 792*a**2*c**3*d**3*e**2*f**2*g**2*x - 264*a**2*c**3*d**3*e* 
*2*f*g**3*x**2 - 40*a**2*c**3*d**3*e**2*g**4*x**3 + 1155*a*c**4*d**4*e*f** 
4 + 924*a*c**4*d**4*e*f**3*g*x + 594*a*c**4*d**4*e*f**2*g**2*x**2 + 220*a* 
c**4*d**4*e*f*g**3*x**3 + 35*a*c**4*d**4*e*g**4*x**4 + 1155*c**5*d**5*f**4 
*x + 2772*c**5*d**5*f**3*g*x**2 + 2970*c**5*d**5*f**2*g**2*x**3 + 1540*c** 
5*d**5*f*g**3*x**4 + 315*c**5*d**5*g**4*x**5))/(3465*c**5*d**5)