\(\int \frac {(a d e+(c d^2+a e^2) x+c d e x^2)^{5/2}}{(d+e x)^{5/2} (f+g x)^{15/2}} \, dx\) [83]

Optimal result

Integrand size = 48, antiderivative size = 267 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^{5/2} (f+g x)^{15/2}} \, dx=\frac {2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{13 (c d f-a e g) (d+e x)^{7/2} (f+g x)^{13/2}}+\frac {12 c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{143 (c d f-a e g)^2 (d+e x)^{7/2} (f+g x)^{11/2}}+\frac {16 c^2 d^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{429 (c d f-a e g)^3 (d+e x)^{7/2} (f+g x)^{9/2}}+\frac {32 c^3 d^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{3003 (c d f-a e g)^4 (d+e x)^{7/2} (f+g x)^{7/2}} \] Output:

2/13*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(7/2)/(-a*e*g+c*d*f)/(e*x+d)^(7/2)/ 
(g*x+f)^(13/2)+12/143*c*d*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(7/2)/(-a*e*g+ 
c*d*f)^2/(e*x+d)^(7/2)/(g*x+f)^(11/2)+16/429*c^2*d^2*(a*d*e+(a*e^2+c*d^2)* 
x+c*d*e*x^2)^(7/2)/(-a*e*g+c*d*f)^3/(e*x+d)^(7/2)/(g*x+f)^(9/2)+32/3003*c^ 
3*d^3*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(7/2)/(-a*e*g+c*d*f)^4/(e*x+d)^(7/ 
2)/(g*x+f)^(7/2)
 

Mathematica [A] (verified)

Time = 0.43 (sec) , antiderivative size = 162, normalized size of antiderivative = 0.61 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^{5/2} (f+g x)^{15/2}} \, dx=\frac {2 (a e+c d x)^3 \sqrt {(a e+c d x) (d+e x)} \left (-231 a^3 e^3 g^3+63 a^2 c d e^2 g^2 (13 f+2 g x)-7 a c^2 d^2 e g \left (143 f^2+52 f g x+8 g^2 x^2\right )+c^3 d^3 \left (429 f^3+286 f^2 g x+104 f g^2 x^2+16 g^3 x^3\right )\right )}{3003 (c d f-a e g)^4 \sqrt {d+e x} (f+g x)^{13/2}} \] Input:

Integrate[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)/((d + e*x)^(5/2)*( 
f + g*x)^(15/2)),x]
 

Output:

(2*(a*e + c*d*x)^3*Sqrt[(a*e + c*d*x)*(d + e*x)]*(-231*a^3*e^3*g^3 + 63*a^ 
2*c*d*e^2*g^2*(13*f + 2*g*x) - 7*a*c^2*d^2*e*g*(143*f^2 + 52*f*g*x + 8*g^2 
*x^2) + c^3*d^3*(429*f^3 + 286*f^2*g*x + 104*f*g^2*x^2 + 16*g^3*x^3)))/(30 
03*(c*d*f - a*e*g)^4*Sqrt[d + e*x]*(f + g*x)^(13/2))
 

Rubi [A] (verified)

Time = 0.97 (sec) , antiderivative size = 293, normalized size of antiderivative = 1.10, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {1254, 1254, 1254, 1248}

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.

\(\displaystyle \int \frac {\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{(d+e x)^{5/2} (f+g x)^{15/2}} \, dx\)

\(\Big \downarrow \) 1254

\(\displaystyle \frac {6 c d \int \frac {\left (c d e x^2+\left (c d^2+a e^2\right ) x+a d e\right )^{5/2}}{(d+e x)^{5/2} (f+g x)^{13/2}}dx}{13 (c d f-a e g)}+\frac {2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{13 (d+e x)^{7/2} (f+g x)^{13/2} (c d f-a e g)}\)

\(\Big \downarrow \) 1254

\(\displaystyle \frac {6 c d \left (\frac {4 c d \int \frac {\left (c d e x^2+\left (c d^2+a e^2\right ) x+a d e\right )^{5/2}}{(d+e x)^{5/2} (f+g x)^{11/2}}dx}{11 (c d f-a e g)}+\frac {2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{11 (d+e x)^{7/2} (f+g x)^{11/2} (c d f-a e g)}\right )}{13 (c d f-a e g)}+\frac {2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{13 (d+e x)^{7/2} (f+g x)^{13/2} (c d f-a e g)}\)

\(\Big \downarrow \) 1254

\(\displaystyle \frac {6 c d \left (\frac {4 c d \left (\frac {2 c d \int \frac {\left (c d e x^2+\left (c d^2+a e^2\right ) x+a d e\right )^{5/2}}{(d+e x)^{5/2} (f+g x)^{9/2}}dx}{9 (c d f-a e g)}+\frac {2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{9 (d+e x)^{7/2} (f+g x)^{9/2} (c d f-a e g)}\right )}{11 (c d f-a e g)}+\frac {2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{11 (d+e x)^{7/2} (f+g x)^{11/2} (c d f-a e g)}\right )}{13 (c d f-a e g)}+\frac {2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{13 (d+e x)^{7/2} (f+g x)^{13/2} (c d f-a e g)}\)

\(\Big \downarrow \) 1248

\(\displaystyle \frac {2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{13 (d+e x)^{7/2} (f+g x)^{13/2} (c d f-a e g)}+\frac {6 c d \left (\frac {2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{11 (d+e x)^{7/2} (f+g x)^{11/2} (c d f-a e g)}+\frac {4 c d \left (\frac {4 c d \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{63 (d+e x)^{7/2} (f+g x)^{7/2} (c d f-a e g)^2}+\frac {2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{9 (d+e x)^{7/2} (f+g x)^{9/2} (c d f-a e g)}\right )}{11 (c d f-a e g)}\right )}{13 (c d f-a e g)}\)

Input:

Int[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)/((d + e*x)^(5/2)*(f + g* 
x)^(15/2)),x]
 

Output:

(2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(7/2))/(13*(c*d*f - a*e*g)*(d + 
 e*x)^(7/2)*(f + g*x)^(13/2)) + (6*c*d*((2*(a*d*e + (c*d^2 + a*e^2)*x + c* 
d*e*x^2)^(7/2))/(11*(c*d*f - a*e*g)*(d + e*x)^(7/2)*(f + g*x)^(11/2)) + (4 
*c*d*((2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(7/2))/(9*(c*d*f - a*e*g) 
*(d + e*x)^(7/2)*(f + g*x)^(9/2)) + (4*c*d*(a*d*e + (c*d^2 + a*e^2)*x + c* 
d*e*x^2)^(7/2))/(63*(c*d*f - a*e*g)^2*(d + e*x)^(7/2)*(f + g*x)^(7/2))))/( 
11*(c*d*f - a*e*g))))/(13*(c*d*f - a*e*g))
 

Defintions of rubi rules used

rule 1248

Int[((d_) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_.) + (b_.)*(x_) 
+ (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-e^2)*(d + e*x)^(m - 1)*(f + g*x)^ 
(n + 1)*((a + b*x + c*x^2)^(p + 1)/((n + 1)*(c*e*f + c*d*g - b*e*g))), x] / 
; FreeQ[{a, b, c, d, e, f, g, m, n, p}, x] && EqQ[c*d^2 - b*d*e + a*e^2, 0] 
 && EqQ[m + p, 0] && EqQ[m - n - 2, 0]
 

rule 1254

Int[((d_) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_.) + (b_.)*(x_) 
+ (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-e^2)*(d + e*x)^(m - 1)*(f + g*x)^ 
(n + 1)*((a + b*x + c*x^2)^(p + 1)/((n + 1)*(c*e*f + c*d*g - b*e*g))), x] - 
 Simp[c*e*((m - n - 2)/((n + 1)*(c*e*f + c*d*g - b*e*g)))   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] && LtQ[n, -1 
] && IntegerQ[2*p]
 

Maple [A] (verified)

Time = 2.52 (sec) , antiderivative size = 260, normalized size of antiderivative = 0.97

method	result	size
gosper	\(-\frac {2 \left (c d x +a e \right ) \left (-16 x^{3} g^{3} d^{3} c^{3}+56 a \,c^{2} d^{2} e \,g^{3} x^{2}-104 c^{3} d^{3} f \,g^{2} x^{2}-126 a^{2} c d \,e^{2} g^{3} x +364 a \,c^{2} d^{2} e f \,g^{2} x -286 c^{3} d^{3} f^{2} g x +231 a^{3} e^{3} g^{3}-819 a^{2} c d \,e^{2} f \,g^{2}+1001 a \,c^{2} d^{2} e \,f^{2} g -429 f^{3} d^{3} c^{3}\right ) \left (c d \,x^{2} e +a \,e^{2} x +c \,d^{2} x +a d e \right )^{\frac {5}{2}}}{3003 \left (g x +f \right )^{\frac {13}{2}} \left (a^{4} e^{4} g^{4}-4 a^{3} c d \,e^{3} f \,g^{3}+6 a^{2} c^{2} d^{2} e^{2} f^{2} g^{2}-4 a \,c^{3} d^{3} e \,f^{3} g +f^{4} d^{4} c^{4}\right ) \left (e x +d \right )^{\frac {5}{2}}}\)	\(260\)
orering	\(-\frac {2 \left (-16 x^{3} g^{3} d^{3} c^{3}+56 a \,c^{2} d^{2} e \,g^{3} x^{2}-104 c^{3} d^{3} f \,g^{2} x^{2}-126 a^{2} c d \,e^{2} g^{3} x +364 a \,c^{2} d^{2} e f \,g^{2} x -286 c^{3} d^{3} f^{2} g x +231 a^{3} e^{3} g^{3}-819 a^{2} c d \,e^{2} f \,g^{2}+1001 a \,c^{2} d^{2} e \,f^{2} g -429 f^{3} d^{3} c^{3}\right ) \left (c d x +a e \right ) {\left (a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d \,x^{2} e \right )}^{\frac {5}{2}}}{3003 \left (a^{4} e^{4} g^{4}-4 a^{3} c d \,e^{3} f \,g^{3}+6 a^{2} c^{2} d^{2} e^{2} f^{2} g^{2}-4 a \,c^{3} d^{3} e \,f^{3} g +f^{4} d^{4} c^{4}\right ) \left (g x +f \right )^{\frac {13}{2}} \left (e x +d \right )^{\frac {5}{2}}}\)	\(261\)
default	\(-\frac {2 \sqrt {\left (e x +d \right ) \left (c d x +a e \right )}\, \left (-16 c^{5} d^{5} g^{3} x^{5}+24 a \,c^{4} d^{4} e \,g^{3} x^{4}-104 c^{5} d^{5} f \,g^{2} x^{4}-30 a^{2} c^{3} d^{3} e^{2} g^{3} x^{3}+156 a \,c^{4} d^{4} e f \,g^{2} x^{3}-286 c^{5} d^{5} f^{2} g \,x^{3}+35 a^{3} c^{2} d^{2} e^{3} g^{3} x^{2}-195 a^{2} c^{3} d^{3} e^{2} f \,g^{2} x^{2}+429 a \,c^{4} d^{4} e \,f^{2} g \,x^{2}-429 c^{5} d^{5} f^{3} x^{2}+336 a^{4} c d \,e^{4} g^{3} x -1274 a^{3} c^{2} d^{2} e^{3} f \,g^{2} x +1716 a^{2} c^{3} d^{3} e^{2} f^{2} g x -858 a \,c^{4} d^{4} e \,f^{3} x +231 a^{5} e^{5} g^{3}-819 a^{4} c d \,e^{4} f \,g^{2}+1001 a^{3} c^{2} d^{2} e^{3} f^{2} g -429 a^{2} c^{3} d^{3} e^{2} f^{3}\right ) \left (c d x +a e \right )}{3003 \sqrt {e x +d}\, \left (g x +f \right )^{\frac {13}{2}} \left (a e g -d f c \right )^{4}}\)	\(349\)

Input:

int((a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^(5/2)/(e*x+d)^(5/2)/(g*x+f)^(15/2),x 
,method=_RETURNVERBOSE)
 

Output:

-2/3003*(c*d*x+a*e)*(-16*c^3*d^3*g^3*x^3+56*a*c^2*d^2*e*g^3*x^2-104*c^3*d^ 
3*f*g^2*x^2-126*a^2*c*d*e^2*g^3*x+364*a*c^2*d^2*e*f*g^2*x-286*c^3*d^3*f^2* 
g*x+231*a^3*e^3*g^3-819*a^2*c*d*e^2*f*g^2+1001*a*c^2*d^2*e*f^2*g-429*c^3*d 
^3*f^3)*(c*d*e*x^2+a*e^2*x+c*d^2*x+a*d*e)^(5/2)/(g*x+f)^(13/2)/(a^4*e^4*g^ 
4-4*a^3*c*d*e^3*f*g^3+6*a^2*c^2*d^2*e^2*f^2*g^2-4*a*c^3*d^3*e*f^3*g+c^4*d^ 
4*f^4)/(e*x+d)^(5/2)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1648 vs. \(2 (235) = 470\).

Time = 1.17 (sec) , antiderivative size = 1648, normalized size of antiderivative = 6.17 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^{5/2} (f+g x)^{15/2}} \, dx=\text {Too large to display} \] Input:

integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/(e*x+d)^(5/2)/(g*x+f)^(1 
5/2),x, algorithm="fricas")
 

Output:

2/3003*(16*c^6*d^6*g^3*x^6 + 429*a^3*c^3*d^3*e^3*f^3 - 1001*a^4*c^2*d^2*e^ 
4*f^2*g + 819*a^5*c*d*e^5*f*g^2 - 231*a^6*e^6*g^3 + 8*(13*c^6*d^6*f*g^2 - 
a*c^5*d^5*e*g^3)*x^5 + 2*(143*c^6*d^6*f^2*g - 26*a*c^5*d^5*e*f*g^2 + 3*a^2 
*c^4*d^4*e^2*g^3)*x^4 + (429*c^6*d^6*f^3 - 143*a*c^5*d^5*e*f^2*g + 39*a^2* 
c^4*d^4*e^2*f*g^2 - 5*a^3*c^3*d^3*e^3*g^3)*x^3 + (1287*a*c^5*d^5*e*f^3 - 2 
145*a^2*c^4*d^4*e^2*f^2*g + 1469*a^3*c^3*d^3*e^3*f*g^2 - 371*a^4*c^2*d^2*e 
^4*g^3)*x^2 + (1287*a^2*c^4*d^4*e^2*f^3 - 2717*a^3*c^3*d^3*e^3*f^2*g + 209 
3*a^4*c^2*d^2*e^4*f*g^2 - 567*a^5*c*d*e^5*g^3)*x)*sqrt(c*d*e*x^2 + a*d*e + 
 (c*d^2 + a*e^2)*x)*sqrt(e*x + d)*sqrt(g*x + f)/(c^4*d^5*f^11 - 4*a*c^3*d^ 
4*e*f^10*g + 6*a^2*c^2*d^3*e^2*f^9*g^2 - 4*a^3*c*d^2*e^3*f^8*g^3 + a^4*d*e 
^4*f^7*g^4 + (c^4*d^4*e*f^4*g^7 - 4*a*c^3*d^3*e^2*f^3*g^8 + 6*a^2*c^2*d^2* 
e^3*f^2*g^9 - 4*a^3*c*d*e^4*f*g^10 + a^4*e^5*g^11)*x^8 + (7*c^4*d^4*e*f^5* 
g^6 + a^4*d*e^4*g^11 + (c^4*d^5 - 28*a*c^3*d^3*e^2)*f^4*g^7 - 2*(2*a*c^3*d 
^4*e - 21*a^2*c^2*d^2*e^3)*f^3*g^8 + 2*(3*a^2*c^2*d^3*e^2 - 14*a^3*c*d*e^4 
)*f^2*g^9 - (4*a^3*c*d^2*e^3 - 7*a^4*e^5)*f*g^10)*x^7 + 7*(3*c^4*d^4*e*f^6 
*g^5 + a^4*d*e^4*f*g^10 + (c^4*d^5 - 12*a*c^3*d^3*e^2)*f^5*g^6 - 2*(2*a*c^ 
3*d^4*e - 9*a^2*c^2*d^2*e^3)*f^4*g^7 + 6*(a^2*c^2*d^3*e^2 - 2*a^3*c*d*e^4) 
*f^3*g^8 - (4*a^3*c*d^2*e^3 - 3*a^4*e^5)*f^2*g^9)*x^6 + 7*(5*c^4*d^4*e*f^7 
*g^4 + 3*a^4*d*e^4*f^2*g^9 + (3*c^4*d^5 - 20*a*c^3*d^3*e^2)*f^6*g^5 - 6*(2 
*a*c^3*d^4*e - 5*a^2*c^2*d^2*e^3)*f^5*g^6 + 2*(9*a^2*c^2*d^3*e^2 - 10*a...
 

Sympy [F(-1)]

Timed out. \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^{5/2} (f+g x)^{15/2}} \, dx=\text {Timed out} \] Input:

integrate((a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(5/2)/(e*x+d)**(5/2)/(g*x+ 
f)**(15/2),x)
                                                                                    
                                                                                    
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^{5/2} (f+g x)^{15/2}} \, dx=\int { \frac {{\left (c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x\right )}^{\frac {5}{2}}}{{\left (e x + d\right )}^{\frac {5}{2}} {\left (g x + f\right )}^{\frac {15}{2}}} \,d x } \] Input:

integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/(e*x+d)^(5/2)/(g*x+f)^(1 
5/2),x, algorithm="maxima")
 

Output:

integrate((c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^(5/2)/((e*x + d)^(5/2)*( 
g*x + f)^(15/2)), x)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 915 vs. \(2 (235) = 470\).

Time = 0.91 (sec) , antiderivative size = 915, normalized size of antiderivative = 3.43 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^{5/2} (f+g x)^{15/2}} \, dx =\text {Too large to display} \] Input:

integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/(e*x+d)^(5/2)/(g*x+f)^(1 
5/2),x, algorithm="giac")
 

Output:

2/3003*(c*d*x + a*e)^(7/2)*(2*(c*d*x + a*e)*(4*(c*d*x + a*e)*(2*(c^14*d^14 
*f^2*g^9*abs(c)*abs(d) - 2*a*c^13*d^13*e*f*g^10*abs(c)*abs(d) + a^2*c^12*d 
^12*e^2*g^11*abs(c)*abs(d))*(c*d*x + a*e)/(c^6*d^6*f^6*g^6 - 6*a*c^5*d^5*e 
*f^5*g^7 + 15*a^2*c^4*d^4*e^2*f^4*g^8 - 20*a^3*c^3*d^3*e^3*f^3*g^9 + 15*a^ 
4*c^2*d^2*e^4*f^2*g^10 - 6*a^5*c*d*e^5*f*g^11 + a^6*e^6*g^12) + 13*(c^15*d 
^15*f^3*g^8*abs(c)*abs(d) - 3*a*c^14*d^14*e*f^2*g^9*abs(c)*abs(d) + 3*a^2* 
c^13*d^13*e^2*f*g^10*abs(c)*abs(d) - a^3*c^12*d^12*e^3*g^11*abs(c)*abs(d)) 
/(c^6*d^6*f^6*g^6 - 6*a*c^5*d^5*e*f^5*g^7 + 15*a^2*c^4*d^4*e^2*f^4*g^8 - 2 
0*a^3*c^3*d^3*e^3*f^3*g^9 + 15*a^4*c^2*d^2*e^4*f^2*g^10 - 6*a^5*c*d*e^5*f* 
g^11 + a^6*e^6*g^12)) + 143*(c^16*d^16*f^4*g^7*abs(c)*abs(d) - 4*a*c^15*d^ 
15*e*f^3*g^8*abs(c)*abs(d) + 6*a^2*c^14*d^14*e^2*f^2*g^9*abs(c)*abs(d) - 4 
*a^3*c^13*d^13*e^3*f*g^10*abs(c)*abs(d) + a^4*c^12*d^12*e^4*g^11*abs(c)*ab 
s(d))/(c^6*d^6*f^6*g^6 - 6*a*c^5*d^5*e*f^5*g^7 + 15*a^2*c^4*d^4*e^2*f^4*g^ 
8 - 20*a^3*c^3*d^3*e^3*f^3*g^9 + 15*a^4*c^2*d^2*e^4*f^2*g^10 - 6*a^5*c*d*e 
^5*f*g^11 + a^6*e^6*g^12)) + 429*(c^17*d^17*f^5*g^6*abs(c)*abs(d) - 5*a*c^ 
16*d^16*e*f^4*g^7*abs(c)*abs(d) + 10*a^2*c^15*d^15*e^2*f^3*g^8*abs(c)*abs( 
d) - 10*a^3*c^14*d^14*e^3*f^2*g^9*abs(c)*abs(d) + 5*a^4*c^13*d^13*e^4*f*g^ 
10*abs(c)*abs(d) - a^5*c^12*d^12*e^5*g^11*abs(c)*abs(d))/(c^6*d^6*f^6*g^6 
- 6*a*c^5*d^5*e*f^5*g^7 + 15*a^2*c^4*d^4*e^2*f^4*g^8 - 20*a^3*c^3*d^3*e^3* 
f^3*g^9 + 15*a^4*c^2*d^2*e^4*f^2*g^10 - 6*a^5*c*d*e^5*f*g^11 + a^6*e^6*...
 

Mupad [B] (verification not implemented)

Time = 7.16 (sec) , antiderivative size = 627, normalized size of antiderivative = 2.35 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^{5/2} (f+g x)^{15/2}} \, dx=-\frac {\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}\,\left (\frac {462\,a^6\,e^6\,g^3-1638\,a^5\,c\,d\,e^5\,f\,g^2+2002\,a^4\,c^2\,d^2\,e^4\,f^2\,g-858\,a^3\,c^3\,d^3\,e^3\,f^3}{3003\,g^6\,{\left (a\,e\,g-c\,d\,f\right )}^4}-\frac {x^3\,\left (-10\,a^3\,c^3\,d^3\,e^3\,g^3+78\,a^2\,c^4\,d^4\,e^2\,f\,g^2-286\,a\,c^5\,d^5\,e\,f^2\,g+858\,c^6\,d^6\,f^3\right )}{3003\,g^6\,{\left (a\,e\,g-c\,d\,f\right )}^4}-\frac {32\,c^6\,d^6\,x^6}{3003\,g^3\,{\left (a\,e\,g-c\,d\,f\right )}^4}-\frac {4\,c^4\,d^4\,x^4\,\left (3\,a^2\,e^2\,g^2-26\,a\,c\,d\,e\,f\,g+143\,c^2\,d^2\,f^2\right )}{3003\,g^5\,{\left (a\,e\,g-c\,d\,f\right )}^4}+\frac {16\,c^5\,d^5\,x^5\,\left (a\,e\,g-13\,c\,d\,f\right )}{3003\,g^4\,{\left (a\,e\,g-c\,d\,f\right )}^4}+\frac {2\,a^2\,c\,d\,e^2\,x\,\left (567\,a^3\,e^3\,g^3-2093\,a^2\,c\,d\,e^2\,f\,g^2+2717\,a\,c^2\,d^2\,e\,f^2\,g-1287\,c^3\,d^3\,f^3\right )}{3003\,g^6\,{\left (a\,e\,g-c\,d\,f\right )}^4}+\frac {2\,a\,c^2\,d^2\,e\,x^2\,\left (371\,a^3\,e^3\,g^3-1469\,a^2\,c\,d\,e^2\,f\,g^2+2145\,a\,c^2\,d^2\,e\,f^2\,g-1287\,c^3\,d^3\,f^3\right )}{3003\,g^6\,{\left (a\,e\,g-c\,d\,f\right )}^4}\right )}{x^6\,\sqrt {f+g\,x}\,\sqrt {d+e\,x}+\frac {f^6\,\sqrt {f+g\,x}\,\sqrt {d+e\,x}}{g^6}+\frac {6\,f\,x^5\,\sqrt {f+g\,x}\,\sqrt {d+e\,x}}{g}+\frac {6\,f^5\,x\,\sqrt {f+g\,x}\,\sqrt {d+e\,x}}{g^5}+\frac {15\,f^2\,x^4\,\sqrt {f+g\,x}\,\sqrt {d+e\,x}}{g^2}+\frac {20\,f^3\,x^3\,\sqrt {f+g\,x}\,\sqrt {d+e\,x}}{g^3}+\frac {15\,f^4\,x^2\,\sqrt {f+g\,x}\,\sqrt {d+e\,x}}{g^4}} \] Input:

int((x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(5/2)/((f + g*x)^(15/2)*(d + e 
*x)^(5/2)),x)
 

Output:

-((x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1/2)*((462*a^6*e^6*g^3 - 858*a^ 
3*c^3*d^3*e^3*f^3 + 2002*a^4*c^2*d^2*e^4*f^2*g - 1638*a^5*c*d*e^5*f*g^2)/( 
3003*g^6*(a*e*g - c*d*f)^4) - (x^3*(858*c^6*d^6*f^3 - 10*a^3*c^3*d^3*e^3*g 
^3 + 78*a^2*c^4*d^4*e^2*f*g^2 - 286*a*c^5*d^5*e*f^2*g))/(3003*g^6*(a*e*g - 
 c*d*f)^4) - (32*c^6*d^6*x^6)/(3003*g^3*(a*e*g - c*d*f)^4) - (4*c^4*d^4*x^ 
4*(3*a^2*e^2*g^2 + 143*c^2*d^2*f^2 - 26*a*c*d*e*f*g))/(3003*g^5*(a*e*g - c 
*d*f)^4) + (16*c^5*d^5*x^5*(a*e*g - 13*c*d*f))/(3003*g^4*(a*e*g - c*d*f)^4 
) + (2*a^2*c*d*e^2*x*(567*a^3*e^3*g^3 - 1287*c^3*d^3*f^3 + 2717*a*c^2*d^2* 
e*f^2*g - 2093*a^2*c*d*e^2*f*g^2))/(3003*g^6*(a*e*g - c*d*f)^4) + (2*a*c^2 
*d^2*e*x^2*(371*a^3*e^3*g^3 - 1287*c^3*d^3*f^3 + 2145*a*c^2*d^2*e*f^2*g - 
1469*a^2*c*d*e^2*f*g^2))/(3003*g^6*(a*e*g - c*d*f)^4)))/(x^6*(f + g*x)^(1/ 
2)*(d + e*x)^(1/2) + (f^6*(f + g*x)^(1/2)*(d + e*x)^(1/2))/g^6 + (6*f*x^5* 
(f + g*x)^(1/2)*(d + e*x)^(1/2))/g + (6*f^5*x*(f + g*x)^(1/2)*(d + e*x)^(1 
/2))/g^5 + (15*f^2*x^4*(f + g*x)^(1/2)*(d + e*x)^(1/2))/g^2 + (20*f^3*x^3* 
(f + g*x)^(1/2)*(d + e*x)^(1/2))/g^3 + (15*f^4*x^2*(f + g*x)^(1/2)*(d + e* 
x)^(1/2))/g^4)
 

Reduce [B] (verification not implemented)

Time = 1.99 (sec) , antiderivative size = 1624, normalized size of antiderivative = 6.08 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^{5/2} (f+g x)^{15/2}} \, dx =\text {Too large to display} \] Input:

int((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/(e*x+d)^(5/2)/(g*x+f)^(15/2),x 
)
 

Output:

(2*( - 231*sqrt(f + g*x)*sqrt(a*e + c*d*x)*a**6*e**6*g**7 + 819*sqrt(f + g 
*x)*sqrt(a*e + c*d*x)*a**5*c*d*e**5*f*g**6 - 567*sqrt(f + g*x)*sqrt(a*e + 
c*d*x)*a**5*c*d*e**5*g**7*x - 1001*sqrt(f + g*x)*sqrt(a*e + c*d*x)*a**4*c* 
*2*d**2*e**4*f**2*g**5 + 2093*sqrt(f + g*x)*sqrt(a*e + c*d*x)*a**4*c**2*d* 
*2*e**4*f*g**6*x - 371*sqrt(f + g*x)*sqrt(a*e + c*d*x)*a**4*c**2*d**2*e**4 
*g**7*x**2 + 429*sqrt(f + g*x)*sqrt(a*e + c*d*x)*a**3*c**3*d**3*e**3*f**3* 
g**4 - 2717*sqrt(f + g*x)*sqrt(a*e + c*d*x)*a**3*c**3*d**3*e**3*f**2*g**5* 
x + 1469*sqrt(f + g*x)*sqrt(a*e + c*d*x)*a**3*c**3*d**3*e**3*f*g**6*x**2 - 
 5*sqrt(f + g*x)*sqrt(a*e + c*d*x)*a**3*c**3*d**3*e**3*g**7*x**3 + 1287*sq 
rt(f + g*x)*sqrt(a*e + c*d*x)*a**2*c**4*d**4*e**2*f**3*g**4*x - 2145*sqrt( 
f + g*x)*sqrt(a*e + c*d*x)*a**2*c**4*d**4*e**2*f**2*g**5*x**2 + 39*sqrt(f 
+ g*x)*sqrt(a*e + c*d*x)*a**2*c**4*d**4*e**2*f*g**6*x**3 + 6*sqrt(f + g*x) 
*sqrt(a*e + c*d*x)*a**2*c**4*d**4*e**2*g**7*x**4 + 1287*sqrt(f + g*x)*sqrt 
(a*e + c*d*x)*a*c**5*d**5*e*f**3*g**4*x**2 - 143*sqrt(f + g*x)*sqrt(a*e + 
c*d*x)*a*c**5*d**5*e*f**2*g**5*x**3 - 52*sqrt(f + g*x)*sqrt(a*e + c*d*x)*a 
*c**5*d**5*e*f*g**6*x**4 - 8*sqrt(f + g*x)*sqrt(a*e + c*d*x)*a*c**5*d**5*e 
*g**7*x**5 + 429*sqrt(f + g*x)*sqrt(a*e + c*d*x)*c**6*d**6*f**3*g**4*x**3 
+ 286*sqrt(f + g*x)*sqrt(a*e + c*d*x)*c**6*d**6*f**2*g**5*x**4 + 104*sqrt( 
f + g*x)*sqrt(a*e + c*d*x)*c**6*d**6*f*g**6*x**5 + 16*sqrt(f + g*x)*sqrt(a 
*e + c*d*x)*c**6*d**6*g**7*x**6 - 16*sqrt(g)*sqrt(d)*sqrt(c)*c**6*d**6*...