\(\int \frac {(a d e+(c d^2+a e^2) x+c d e x^2)^{5/2}}{(d+e x)^{5/2} (f+g x)^{13/2}} \, dx\) [82]

Optimal result

Integrand size = 48, antiderivative size = 198 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^{5/2} (f+g x)^{13/2}} \, dx=\frac {2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{11 (c d f-a e g) (d+e x)^{7/2} (f+g x)^{11/2}}+\frac {8 c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{99 (c d f-a e g)^2 (d+e x)^{7/2} (f+g x)^{9/2}}+\frac {16 c^2 d^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{693 (c d f-a e g)^3 (d+e x)^{7/2} (f+g x)^{7/2}} \] Output:

2/11*(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)^(11/2)+8/99*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)^(9/2)+16/693*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)^(7/2)
 

Mathematica [A] (verified)

Time = 0.29 (sec) , antiderivative size = 115, normalized size of antiderivative = 0.58 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^{5/2} (f+g x)^{13/2}} \, dx=\frac {2 (a e+c d x)^3 \sqrt {(a e+c d x) (d+e x)} \left (63 a^2 e^2 g^2-14 a c d e g (11 f+2 g x)+c^2 d^2 \left (99 f^2+44 f g x+8 g^2 x^2\right )\right )}{693 (c d f-a e g)^3 \sqrt {d+e x} (f+g x)^{11/2}} \] 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)^(13/2)),x]
 

Output:

(2*(a*e + c*d*x)^3*Sqrt[(a*e + c*d*x)*(d + e*x)]*(63*a^2*e^2*g^2 - 14*a*c* 
d*e*g*(11*f + 2*g*x) + c^2*d^2*(99*f^2 + 44*f*g*x + 8*g^2*x^2)))/(693*(c*d 
*f - a*e*g)^3*Sqrt[d + e*x]*(f + g*x)^(11/2))
 

Rubi [A] (verified)

Time = 0.75 (sec) , antiderivative size = 211, normalized size of antiderivative = 1.07, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {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.

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

\(\Big \downarrow \) 1254

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

\(\Big \downarrow \) 1254

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

\(\Big \downarrow \) 1248

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

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)^(13/2)),x]
 

Output:

(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))
 

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 = 2.58 (sec) , antiderivative size = 169, normalized size of antiderivative = 0.85

method	result	size
gosper	\(-\frac {2 \left (c d x +a e \right ) \left (8 g^{2} x^{2} d^{2} c^{2}-28 a c d e \,g^{2} x +44 c^{2} d^{2} f g x +63 a^{2} e^{2} g^{2}-154 a c d e f g +99 f^{2} c^{2} d^{2}\right ) \left (c d \,x^{2} e +a \,e^{2} x +c \,d^{2} x +a d e \right )^{\frac {5}{2}}}{693 \left (g x +f \right )^{\frac {11}{2}} \left (a^{3} e^{3} g^{3}-3 a^{2} c d \,e^{2} f \,g^{2}+3 a \,c^{2} d^{2} e \,f^{2} g -f^{3} d^{3} c^{3}\right ) \left (e x +d \right )^{\frac {5}{2}}}\)	\(169\)
orering	\(-\frac {2 \left (8 g^{2} x^{2} d^{2} c^{2}-28 a c d e \,g^{2} x +44 c^{2} d^{2} f g x +63 a^{2} e^{2} g^{2}-154 a c d e f g +99 f^{2} c^{2} d^{2}\right ) \left (c d x +a e \right ) {\left (a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d \,x^{2} e \right )}^{\frac {5}{2}}}{693 \left (a^{3} e^{3} g^{3}-3 a^{2} c d \,e^{2} f \,g^{2}+3 a \,c^{2} d^{2} e \,f^{2} g -f^{3} d^{3} c^{3}\right ) \left (g x +f \right )^{\frac {11}{2}} \left (e x +d \right )^{\frac {5}{2}}}\)	\(170\)
default	\(-\frac {2 \sqrt {\left (e x +d \right ) \left (c d x +a e \right )}\, \left (8 c^{4} d^{4} g^{2} x^{4}-12 a \,c^{3} d^{3} e \,g^{2} x^{3}+44 c^{4} d^{4} f g \,x^{3}+15 a^{2} c^{2} d^{2} e^{2} g^{2} x^{2}-66 a \,c^{3} d^{3} e f g \,x^{2}+99 c^{4} d^{4} f^{2} x^{2}+98 a^{3} c d \,e^{3} g^{2} x -264 a^{2} c^{2} d^{2} e^{2} f g x +198 a \,c^{3} d^{3} e \,f^{2} x +63 a^{4} e^{4} g^{2}-154 a^{3} c d \,e^{3} f g +99 a^{2} c^{2} d^{2} e^{2} f^{2}\right ) \left (c d x +a e \right )}{693 \sqrt {e x +d}\, \left (g x +f \right )^{\frac {11}{2}} \left (a e g -d f c \right )^{3}}\)	\(231\)

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)^(13/2),x 
,method=_RETURNVERBOSE)
 

Output:

-2/693*(c*d*x+a*e)*(8*c^2*d^2*g^2*x^2-28*a*c*d*e*g^2*x+44*c^2*d^2*f*g*x+63 
*a^2*e^2*g^2-154*a*c*d*e*f*g+99*c^2*d^2*f^2)*(c*d*e*x^2+a*e^2*x+c*d^2*x+a* 
d*e)^(5/2)/(g*x+f)^(11/2)/(a^3*e^3*g^3-3*a^2*c*d*e^2*f*g^2+3*a*c^2*d^2*e*f 
^2*g-c^3*d^3*f^3)/(e*x+d)^(5/2)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1101 vs. \(2 (174) = 348\).

Time = 0.47 (sec) , antiderivative size = 1101, normalized size of antiderivative = 5.56 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^{5/2} (f+g x)^{13/2}} \, dx =\text {Too large to display} \] 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 
3/2),x, algorithm="fricas")
 

Output:

2/693*(8*c^5*d^5*g^2*x^5 + 99*a^3*c^2*d^2*e^3*f^2 - 154*a^4*c*d*e^4*f*g + 
63*a^5*e^5*g^2 + 4*(11*c^5*d^5*f*g - a*c^4*d^4*e*g^2)*x^4 + (99*c^5*d^5*f^ 
2 - 22*a*c^4*d^4*e*f*g + 3*a^2*c^3*d^3*e^2*g^2)*x^3 + (297*a*c^4*d^4*e*f^2 
 - 330*a^2*c^3*d^3*e^2*f*g + 113*a^3*c^2*d^2*e^3*g^2)*x^2 + (297*a^2*c^3*d 
^3*e^2*f^2 - 418*a^3*c^2*d^2*e^3*f*g + 161*a^4*c*d*e^4*g^2)*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^3*d^4*f^9 
- 3*a*c^2*d^3*e*f^8*g + 3*a^2*c*d^2*e^2*f^7*g^2 - a^3*d*e^3*f^6*g^3 + (c^3 
*d^3*e*f^3*g^6 - 3*a*c^2*d^2*e^2*f^2*g^7 + 3*a^2*c*d*e^3*f*g^8 - a^3*e^4*g 
^9)*x^7 + (6*c^3*d^3*e*f^4*g^5 - a^3*d*e^3*g^9 + (c^3*d^4 - 18*a*c^2*d^2*e 
^2)*f^3*g^6 - 3*(a*c^2*d^3*e - 6*a^2*c*d*e^3)*f^2*g^7 + 3*(a^2*c*d^2*e^2 - 
 2*a^3*e^4)*f*g^8)*x^6 + 3*(5*c^3*d^3*e*f^5*g^4 - 2*a^3*d*e^3*f*g^8 + (2*c 
^3*d^4 - 15*a*c^2*d^2*e^2)*f^4*g^5 - 3*(2*a*c^2*d^3*e - 5*a^2*c*d*e^3)*f^3 
*g^6 + (6*a^2*c*d^2*e^2 - 5*a^3*e^4)*f^2*g^7)*x^5 + 5*(4*c^3*d^3*e*f^6*g^3 
 - 3*a^3*d*e^3*f^2*g^7 + 3*(c^3*d^4 - 4*a*c^2*d^2*e^2)*f^5*g^4 - 3*(3*a*c^ 
2*d^3*e - 4*a^2*c*d*e^3)*f^4*g^5 + (9*a^2*c*d^2*e^2 - 4*a^3*e^4)*f^3*g^6)* 
x^4 + 5*(3*c^3*d^3*e*f^7*g^2 - 4*a^3*d*e^3*f^3*g^6 + (4*c^3*d^4 - 9*a*c^2* 
d^2*e^2)*f^6*g^3 - 3*(4*a*c^2*d^3*e - 3*a^2*c*d*e^3)*f^5*g^4 + 3*(4*a^2*c* 
d^2*e^2 - a^3*e^4)*f^4*g^5)*x^3 + 3*(2*c^3*d^3*e*f^8*g - 5*a^3*d*e^3*f^4*g 
^5 + (5*c^3*d^4 - 6*a*c^2*d^2*e^2)*f^7*g^2 - 3*(5*a*c^2*d^3*e - 2*a^2*c*d* 
e^3)*f^6*g^3 + (15*a^2*c*d^2*e^2 - 2*a^3*e^4)*f^5*g^4)*x^2 + (c^3*d^3*e...
 

Sympy [F(-1)]

Timed out. \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^{5/2} (f+g x)^{13/2}} \, dx=\text {Timed out} \] 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)**(13/2),x)
                                                                                    
                                                                                    
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^{5/2} (f+g x)^{13/2}} \, dx=\int { \frac {{\left (c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x\right )}^{\frac {5}{2}}}{{\left (e x + d\right )}^{\frac {5}{2}} {\left (g x + f\right )}^{\frac {13}{2}}} \,d x } \] 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 
3/2),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)^(13/2)), x)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 600 vs. \(2 (174) = 348\).

Time = 0.66 (sec) , antiderivative size = 600, normalized size of antiderivative = 3.03 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^{5/2} (f+g x)^{13/2}} \, dx=\frac {2 \, {\left (c d x + a e\right )}^{\frac {7}{2}} {\left (4 \, {\left (c d x + a e\right )} {\left (\frac {2 \, {\left (c^{12} d^{12} f^{2} g^{7} {\left | c \right |} {\left | d \right |} - 2 \, a c^{11} d^{11} e f g^{8} {\left | c \right |} {\left | d \right |} + a^{2} c^{10} d^{10} e^{2} g^{9} {\left | c \right |} {\left | d \right |}\right )} {\left (c d x + a e\right )}}{c^{5} d^{5} f^{5} g^{5} - 5 \, a c^{4} d^{4} e f^{4} g^{6} + 10 \, a^{2} c^{3} d^{3} e^{2} f^{3} g^{7} - 10 \, a^{3} c^{2} d^{2} e^{3} f^{2} g^{8} + 5 \, a^{4} c d e^{4} f g^{9} - a^{5} e^{5} g^{10}} + \frac {11 \, {\left (c^{13} d^{13} f^{3} g^{6} {\left | c \right |} {\left | d \right |} - 3 \, a c^{12} d^{12} e f^{2} g^{7} {\left | c \right |} {\left | d \right |} + 3 \, a^{2} c^{11} d^{11} e^{2} f g^{8} {\left | c \right |} {\left | d \right |} - a^{3} c^{10} d^{10} e^{3} g^{9} {\left | c \right |} {\left | d \right |}\right )}}{c^{5} d^{5} f^{5} g^{5} - 5 \, a c^{4} d^{4} e f^{4} g^{6} + 10 \, a^{2} c^{3} d^{3} e^{2} f^{3} g^{7} - 10 \, a^{3} c^{2} d^{2} e^{3} f^{2} g^{8} + 5 \, a^{4} c d e^{4} f g^{9} - a^{5} e^{5} g^{10}}\right )} + \frac {99 \, {\left (c^{14} d^{14} f^{4} g^{5} {\left | c \right |} {\left | d \right |} - 4 \, a c^{13} d^{13} e f^{3} g^{6} {\left | c \right |} {\left | d \right |} + 6 \, a^{2} c^{12} d^{12} e^{2} f^{2} g^{7} {\left | c \right |} {\left | d \right |} - 4 \, a^{3} c^{11} d^{11} e^{3} f g^{8} {\left | c \right |} {\left | d \right |} + a^{4} c^{10} d^{10} e^{4} g^{9} {\left | c \right |} {\left | d \right |}\right )}}{c^{5} d^{5} f^{5} g^{5} - 5 \, a c^{4} d^{4} e f^{4} g^{6} + 10 \, a^{2} c^{3} d^{3} e^{2} f^{3} g^{7} - 10 \, a^{3} c^{2} d^{2} e^{3} f^{2} g^{8} + 5 \, a^{4} c d e^{4} f g^{9} - a^{5} e^{5} g^{10}}\right )}}{693 \, {\left (c^{2} d^{2} f - a c d e g + {\left (c d x + a e\right )} c d g\right )}^{\frac {11}{2}}} \] 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 
3/2),x, algorithm="giac")
 

Output:

2/693*(c*d*x + a*e)^(7/2)*(4*(c*d*x + a*e)*(2*(c^12*d^12*f^2*g^7*abs(c)*ab 
s(d) - 2*a*c^11*d^11*e*f*g^8*abs(c)*abs(d) + a^2*c^10*d^10*e^2*g^9*abs(c)* 
abs(d))*(c*d*x + a*e)/(c^5*d^5*f^5*g^5 - 5*a*c^4*d^4*e*f^4*g^6 + 10*a^2*c^ 
3*d^3*e^2*f^3*g^7 - 10*a^3*c^2*d^2*e^3*f^2*g^8 + 5*a^4*c*d*e^4*f*g^9 - a^5 
*e^5*g^10) + 11*(c^13*d^13*f^3*g^6*abs(c)*abs(d) - 3*a*c^12*d^12*e*f^2*g^7 
*abs(c)*abs(d) + 3*a^2*c^11*d^11*e^2*f*g^8*abs(c)*abs(d) - a^3*c^10*d^10*e 
^3*g^9*abs(c)*abs(d))/(c^5*d^5*f^5*g^5 - 5*a*c^4*d^4*e*f^4*g^6 + 10*a^2*c^ 
3*d^3*e^2*f^3*g^7 - 10*a^3*c^2*d^2*e^3*f^2*g^8 + 5*a^4*c*d*e^4*f*g^9 - a^5 
*e^5*g^10)) + 99*(c^14*d^14*f^4*g^5*abs(c)*abs(d) - 4*a*c^13*d^13*e*f^3*g^ 
6*abs(c)*abs(d) + 6*a^2*c^12*d^12*e^2*f^2*g^7*abs(c)*abs(d) - 4*a^3*c^11*d 
^11*e^3*f*g^8*abs(c)*abs(d) + a^4*c^10*d^10*e^4*g^9*abs(c)*abs(d))/(c^5*d^ 
5*f^5*g^5 - 5*a*c^4*d^4*e*f^4*g^6 + 10*a^2*c^3*d^3*e^2*f^3*g^7 - 10*a^3*c^ 
2*d^2*e^3*f^2*g^8 + 5*a^4*c*d*e^4*f*g^9 - a^5*e^5*g^10))/(c^2*d^2*f - a*c* 
d*e*g + (c*d*x + a*e)*c*d*g)^(11/2)
 

Mupad [B] (verification not implemented)

Time = 7.05 (sec) , antiderivative size = 465, normalized size of antiderivative = 2.35 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^{5/2} (f+g x)^{13/2}} \, dx=-\frac {\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}\,\left (\frac {126\,a^5\,e^5\,g^2-308\,a^4\,c\,d\,e^4\,f\,g+198\,a^3\,c^2\,d^2\,e^3\,f^2}{693\,g^5\,{\left (a\,e\,g-c\,d\,f\right )}^3}+\frac {x^3\,\left (6\,a^2\,c^3\,d^3\,e^2\,g^2-44\,a\,c^4\,d^4\,e\,f\,g+198\,c^5\,d^5\,f^2\right )}{693\,g^5\,{\left (a\,e\,g-c\,d\,f\right )}^3}+\frac {16\,c^5\,d^5\,x^5}{693\,g^3\,{\left (a\,e\,g-c\,d\,f\right )}^3}-\frac {8\,c^4\,d^4\,x^4\,\left (a\,e\,g-11\,c\,d\,f\right )}{693\,g^4\,{\left (a\,e\,g-c\,d\,f\right )}^3}+\frac {2\,a^2\,c\,d\,e^2\,x\,\left (161\,a^2\,e^2\,g^2-418\,a\,c\,d\,e\,f\,g+297\,c^2\,d^2\,f^2\right )}{693\,g^5\,{\left (a\,e\,g-c\,d\,f\right )}^3}+\frac {2\,a\,c^2\,d^2\,e\,x^2\,\left (113\,a^2\,e^2\,g^2-330\,a\,c\,d\,e\,f\,g+297\,c^2\,d^2\,f^2\right )}{693\,g^5\,{\left (a\,e\,g-c\,d\,f\right )}^3}\right )}{x^5\,\sqrt {f+g\,x}\,\sqrt {d+e\,x}+\frac {f^5\,\sqrt {f+g\,x}\,\sqrt {d+e\,x}}{g^5}+\frac {5\,f\,x^4\,\sqrt {f+g\,x}\,\sqrt {d+e\,x}}{g}+\frac {5\,f^4\,x\,\sqrt {f+g\,x}\,\sqrt {d+e\,x}}{g^4}+\frac {10\,f^2\,x^3\,\sqrt {f+g\,x}\,\sqrt {d+e\,x}}{g^2}+\frac {10\,f^3\,x^2\,\sqrt {f+g\,x}\,\sqrt {d+e\,x}}{g^3}} \] Input:

int((x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(5/2)/((f + g*x)^(13/2)*(d + e 
*x)^(5/2)),x)
 

Output:

-((x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1/2)*((126*a^5*e^5*g^2 + 198*a^ 
3*c^2*d^2*e^3*f^2 - 308*a^4*c*d*e^4*f*g)/(693*g^5*(a*e*g - c*d*f)^3) + (x^ 
3*(198*c^5*d^5*f^2 + 6*a^2*c^3*d^3*e^2*g^2 - 44*a*c^4*d^4*e*f*g))/(693*g^5 
*(a*e*g - c*d*f)^3) + (16*c^5*d^5*x^5)/(693*g^3*(a*e*g - c*d*f)^3) - (8*c^ 
4*d^4*x^4*(a*e*g - 11*c*d*f))/(693*g^4*(a*e*g - c*d*f)^3) + (2*a^2*c*d*e^2 
*x*(161*a^2*e^2*g^2 + 297*c^2*d^2*f^2 - 418*a*c*d*e*f*g))/(693*g^5*(a*e*g 
- c*d*f)^3) + (2*a*c^2*d^2*e*x^2*(113*a^2*e^2*g^2 + 297*c^2*d^2*f^2 - 330* 
a*c*d*e*f*g))/(693*g^5*(a*e*g - c*d*f)^3)))/(x^5*(f + g*x)^(1/2)*(d + e*x) 
^(1/2) + (f^5*(f + g*x)^(1/2)*(d + e*x)^(1/2))/g^5 + (5*f*x^4*(f + g*x)^(1 
/2)*(d + e*x)^(1/2))/g + (5*f^4*x*(f + g*x)^(1/2)*(d + e*x)^(1/2))/g^4 + ( 
10*f^2*x^3*(f + g*x)^(1/2)*(d + e*x)^(1/2))/g^2 + (10*f^3*x^2*(f + g*x)^(1 
/2)*(d + e*x)^(1/2))/g^3)
 

Reduce [B] (verification not implemented)

Time = 1.01 (sec) , antiderivative size = 1106, normalized size of antiderivative = 5.59 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^{5/2} (f+g x)^{13/2}} \, dx =\text {Too large to display} \] 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)^(13/2),x 
)
 

Output:

(2*( - 63*sqrt(f + g*x)*sqrt(a*e + c*d*x)*a**5*e**5*g**6 + 154*sqrt(f + g* 
x)*sqrt(a*e + c*d*x)*a**4*c*d*e**4*f*g**5 - 161*sqrt(f + g*x)*sqrt(a*e + c 
*d*x)*a**4*c*d*e**4*g**6*x - 99*sqrt(f + g*x)*sqrt(a*e + c*d*x)*a**3*c**2* 
d**2*e**3*f**2*g**4 + 418*sqrt(f + g*x)*sqrt(a*e + c*d*x)*a**3*c**2*d**2*e 
**3*f*g**5*x - 113*sqrt(f + g*x)*sqrt(a*e + c*d*x)*a**3*c**2*d**2*e**3*g** 
6*x**2 - 297*sqrt(f + g*x)*sqrt(a*e + c*d*x)*a**2*c**3*d**3*e**2*f**2*g**4 
*x + 330*sqrt(f + g*x)*sqrt(a*e + c*d*x)*a**2*c**3*d**3*e**2*f*g**5*x**2 - 
 3*sqrt(f + g*x)*sqrt(a*e + c*d*x)*a**2*c**3*d**3*e**2*g**6*x**3 - 297*sqr 
t(f + g*x)*sqrt(a*e + c*d*x)*a*c**4*d**4*e*f**2*g**4*x**2 + 22*sqrt(f + g* 
x)*sqrt(a*e + c*d*x)*a*c**4*d**4*e*f*g**5*x**3 + 4*sqrt(f + g*x)*sqrt(a*e 
+ c*d*x)*a*c**4*d**4*e*g**6*x**4 - 99*sqrt(f + g*x)*sqrt(a*e + c*d*x)*c**5 
*d**5*f**2*g**4*x**3 - 44*sqrt(f + g*x)*sqrt(a*e + c*d*x)*c**5*d**5*f*g**5 
*x**4 - 8*sqrt(f + g*x)*sqrt(a*e + c*d*x)*c**5*d**5*g**6*x**5 + 8*sqrt(g)* 
sqrt(d)*sqrt(c)*c**5*d**5*f**6 + 48*sqrt(g)*sqrt(d)*sqrt(c)*c**5*d**5*f**5 
*g*x + 120*sqrt(g)*sqrt(d)*sqrt(c)*c**5*d**5*f**4*g**2*x**2 + 160*sqrt(g)* 
sqrt(d)*sqrt(c)*c**5*d**5*f**3*g**3*x**3 + 120*sqrt(g)*sqrt(d)*sqrt(c)*c** 
5*d**5*f**2*g**4*x**4 + 48*sqrt(g)*sqrt(d)*sqrt(c)*c**5*d**5*f*g**5*x**5 + 
 8*sqrt(g)*sqrt(d)*sqrt(c)*c**5*d**5*g**6*x**6))/(693*g**4*(a**3*e**3*f**6 
*g**3 + 6*a**3*e**3*f**5*g**4*x + 15*a**3*e**3*f**4*g**5*x**2 + 20*a**3*e* 
*3*f**3*g**6*x**3 + 15*a**3*e**3*f**2*g**7*x**4 + 6*a**3*e**3*f*g**8*x*...