\(\int \frac {(a d e+(c d^2+a e^2) x+c d e x^2)^{5/2}}{(d+e x)^{5/2} (f+g x)^5} \, dx\) [31]

Optimal result

Integrand size = 46, antiderivative size = 323 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^{5/2} (f+g x)^5} \, dx=-\frac {5 c^2 d^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{32 g^3 \sqrt {d+e x} (f+g x)^2}+\frac {5 c^3 d^3 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{64 g^3 (c d f-a e g) \sqrt {d+e x} (f+g x)}-\frac {5 c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{24 g^2 (d+e x)^{3/2} (f+g x)^3}-\frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{4 g (d+e x)^{5/2} (f+g x)^4}+\frac {5 c^4 d^4 \arctan \left (\frac {\sqrt {g} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {c d f-a e g} \sqrt {d+e x}}\right )}{64 g^{7/2} (c d f-a e g)^{3/2}} \] Output:

-5/32*c^2*d^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/g^3/(e*x+d)^(1/2)/(g 
*x+f)^2+5/64*c^3*d^3*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/g^3/(-a*e*g+c 
*d*f)/(e*x+d)^(1/2)/(g*x+f)-5/24*c*d*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/ 
2)/g^2/(e*x+d)^(3/2)/(g*x+f)^3-1/4*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2) 
/g/(e*x+d)^(5/2)/(g*x+f)^4+5/64*c^4*d^4*arctan(g^(1/2)*(a*d*e+(a*e^2+c*d^2 
)*x+c*d*e*x^2)^(1/2)/(-a*e*g+c*d*f)^(1/2)/(e*x+d)^(1/2))/g^(7/2)/(-a*e*g+c 
*d*f)^(3/2)
 

Mathematica [A] (verified)

Time = 2.02 (sec) , antiderivative size = 244, normalized size of antiderivative = 0.76 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^{5/2} (f+g x)^5} \, dx=\frac {c^4 d^4 ((a e+c d x) (d+e x))^{5/2} \left (\frac {\sqrt {g} \left (48 a^3 e^3 g^3-8 a^2 c d e^2 g^2 (f-17 g x)+2 a c^2 d^2 e g \left (-5 f^2-18 f g x+59 g^2 x^2\right )-c^3 d^3 \left (15 f^3+55 f^2 g x+73 f g^2 x^2-15 g^3 x^3\right )\right )}{c^4 d^4 (c d f-a e g) (a e+c d x)^2 (f+g x)^4}+\frac {15 \arctan \left (\frac {\sqrt {g} \sqrt {a e+c d x}}{\sqrt {c d f-a e g}}\right )}{(c d f-a e g)^{3/2} (a e+c d x)^{5/2}}\right )}{192 g^{7/2} (d+e x)^{5/2}} \] 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)^5),x]
 

Output:

(c^4*d^4*((a*e + c*d*x)*(d + e*x))^(5/2)*((Sqrt[g]*(48*a^3*e^3*g^3 - 8*a^2 
*c*d*e^2*g^2*(f - 17*g*x) + 2*a*c^2*d^2*e*g*(-5*f^2 - 18*f*g*x + 59*g^2*x^ 
2) - c^3*d^3*(15*f^3 + 55*f^2*g*x + 73*f*g^2*x^2 - 15*g^3*x^3)))/(c^4*d^4* 
(c*d*f - a*e*g)*(a*e + c*d*x)^2*(f + g*x)^4) + (15*ArcTan[(Sqrt[g]*Sqrt[a* 
e + c*d*x])/Sqrt[c*d*f - a*e*g]])/((c*d*f - a*e*g)^(3/2)*(a*e + c*d*x)^(5/ 
2))))/(192*g^(7/2)*(d + e*x)^(5/2))
 

Rubi [A] (verified)

Time = 1.15 (sec) , antiderivative size = 326, normalized size of antiderivative = 1.01, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {1249, 1249, 1249, 1254, 1255, 218}

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.

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

\(\Big \downarrow \) 1249

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

\(\Big \downarrow \) 1249

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

\(\Big \downarrow \) 1249

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

\(\Big \downarrow \) 1254

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

\(\Big \downarrow \) 1255

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

\(\Big \downarrow \) 218

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

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)^5),x]
 

Output:

-1/4*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)/(g*(d + e*x)^(5/2)*(f + 
 g*x)^4) + (5*c*d*(-1/3*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)/(g*( 
d + e*x)^(3/2)*(f + g*x)^3) + (c*d*(-1/2*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + 
c*d*e*x^2]/(g*Sqrt[d + e*x]*(f + g*x)^2) + (c*d*(Sqrt[a*d*e + (c*d^2 + a*e 
^2)*x + c*d*e*x^2]/((c*d*f - a*e*g)*Sqrt[d + e*x]*(f + g*x)) + (c*d*ArcTan 
[(Sqrt[g]*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(Sqrt[c*d*f - a*e*g 
]*Sqrt[d + e*x])])/(Sqrt[g]*(c*d*f - a*e*g)^(3/2))))/(4*g)))/(2*g)))/(8*g)
 

Defintions of rubi rules used

rule 218

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/R 
t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[a/b]
 

rule 1249

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

rule 1255

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(654\) vs. \(2(285)=570\).

Time = 2.91 (sec) , antiderivative size = 655, normalized size of antiderivative = 2.03

method	result	size
default	\(\frac {\sqrt {\left (e x +d \right ) \left (c d x +a e \right )}\, \left (15 \,\operatorname {arctanh}\left (\frac {g \sqrt {c d x +a e}}{\sqrt {\left (a e g -d f c \right ) g}}\right ) c^{4} d^{4} g^{4} x^{4}+60 \,\operatorname {arctanh}\left (\frac {g \sqrt {c d x +a e}}{\sqrt {\left (a e g -d f c \right ) g}}\right ) c^{4} d^{4} f \,g^{3} x^{3}+90 \,\operatorname {arctanh}\left (\frac {g \sqrt {c d x +a e}}{\sqrt {\left (a e g -d f c \right ) g}}\right ) c^{4} d^{4} f^{2} g^{2} x^{2}+60 \,\operatorname {arctanh}\left (\frac {g \sqrt {c d x +a e}}{\sqrt {\left (a e g -d f c \right ) g}}\right ) c^{4} d^{4} f^{3} g x -15 c^{3} d^{3} g^{3} x^{3} \sqrt {c d x +a e}\, \sqrt {\left (a e g -d f c \right ) g}+15 \,\operatorname {arctanh}\left (\frac {g \sqrt {c d x +a e}}{\sqrt {\left (a e g -d f c \right ) g}}\right ) c^{4} d^{4} f^{4}-118 a \,c^{2} d^{2} e \,g^{3} x^{2} \sqrt {c d x +a e}\, \sqrt {\left (a e g -d f c \right ) g}+73 c^{3} d^{3} f \,g^{2} x^{2} \sqrt {c d x +a e}\, \sqrt {\left (a e g -d f c \right ) g}-136 a^{2} c d \,e^{2} g^{3} x \sqrt {c d x +a e}\, \sqrt {\left (a e g -d f c \right ) g}+36 a \,c^{2} d^{2} e f \,g^{2} x \sqrt {c d x +a e}\, \sqrt {\left (a e g -d f c \right ) g}+55 c^{3} d^{3} f^{2} g x \sqrt {c d x +a e}\, \sqrt {\left (a e g -d f c \right ) g}-48 \sqrt {\left (a e g -d f c \right ) g}\, \sqrt {c d x +a e}\, a^{3} e^{3} g^{3}+8 \sqrt {\left (a e g -d f c \right ) g}\, \sqrt {c d x +a e}\, a^{2} c d \,e^{2} f \,g^{2}+10 \sqrt {\left (a e g -d f c \right ) g}\, \sqrt {c d x +a e}\, a \,c^{2} d^{2} e \,f^{2} g +15 \sqrt {\left (a e g -d f c \right ) g}\, \sqrt {c d x +a e}\, c^{3} d^{3} f^{3}\right )}{192 \sqrt {e x +d}\, \sqrt {c d x +a e}\, \left (a e g -d f c \right ) g^{3} \left (g x +f \right )^{4} \sqrt {\left (a e g -d f c \right ) g}}\)	\(655\)

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)^5,x,meth 
od=_RETURNVERBOSE)
 

Output:

1/192*((e*x+d)*(c*d*x+a*e))^(1/2)*(15*arctanh(g*(c*d*x+a*e)^(1/2)/((a*e*g- 
c*d*f)*g)^(1/2))*c^4*d^4*g^4*x^4+60*arctanh(g*(c*d*x+a*e)^(1/2)/((a*e*g-c* 
d*f)*g)^(1/2))*c^4*d^4*f*g^3*x^3+90*arctanh(g*(c*d*x+a*e)^(1/2)/((a*e*g-c* 
d*f)*g)^(1/2))*c^4*d^4*f^2*g^2*x^2+60*arctanh(g*(c*d*x+a*e)^(1/2)/((a*e*g- 
c*d*f)*g)^(1/2))*c^4*d^4*f^3*g*x-15*c^3*d^3*g^3*x^3*(c*d*x+a*e)^(1/2)*((a* 
e*g-c*d*f)*g)^(1/2)+15*arctanh(g*(c*d*x+a*e)^(1/2)/((a*e*g-c*d*f)*g)^(1/2) 
)*c^4*d^4*f^4-118*a*c^2*d^2*e*g^3*x^2*(c*d*x+a*e)^(1/2)*((a*e*g-c*d*f)*g)^ 
(1/2)+73*c^3*d^3*f*g^2*x^2*(c*d*x+a*e)^(1/2)*((a*e*g-c*d*f)*g)^(1/2)-136*a 
^2*c*d*e^2*g^3*x*(c*d*x+a*e)^(1/2)*((a*e*g-c*d*f)*g)^(1/2)+36*a*c^2*d^2*e* 
f*g^2*x*(c*d*x+a*e)^(1/2)*((a*e*g-c*d*f)*g)^(1/2)+55*c^3*d^3*f^2*g*x*(c*d* 
x+a*e)^(1/2)*((a*e*g-c*d*f)*g)^(1/2)-48*((a*e*g-c*d*f)*g)^(1/2)*(c*d*x+a*e 
)^(1/2)*a^3*e^3*g^3+8*((a*e*g-c*d*f)*g)^(1/2)*(c*d*x+a*e)^(1/2)*a^2*c*d*e^ 
2*f*g^2+10*((a*e*g-c*d*f)*g)^(1/2)*(c*d*x+a*e)^(1/2)*a*c^2*d^2*e*f^2*g+15* 
((a*e*g-c*d*f)*g)^(1/2)*(c*d*x+a*e)^(1/2)*c^3*d^3*f^3)/(e*x+d)^(1/2)/(c*d* 
x+a*e)^(1/2)/(a*e*g-c*d*f)/g^3/(g*x+f)^4/((a*e*g-c*d*f)*g)^(1/2)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 911 vs. \(2 (285) = 570\).

Time = 0.32 (sec) , antiderivative size = 1863, normalized size of antiderivative = 5.77 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^{5/2} (f+g x)^5} \, dx=\text {Too large to display} \] 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)^5, 
x, algorithm="fricas")
 

Output:

[1/384*(15*(c^4*d^4*e*g^4*x^5 + c^4*d^5*f^4 + (4*c^4*d^4*e*f*g^3 + c^4*d^5 
*g^4)*x^4 + 2*(3*c^4*d^4*e*f^2*g^2 + 2*c^4*d^5*f*g^3)*x^3 + 2*(2*c^4*d^4*e 
*f^3*g + 3*c^4*d^5*f^2*g^2)*x^2 + (c^4*d^4*e*f^4 + 4*c^4*d^5*f^3*g)*x)*sqr 
t(-c*d*f*g + a*e*g^2)*log(-(c*d*e*g*x^2 - c*d^2*f + 2*a*d*e*g - (c*d*e*f - 
 (c*d^2 + 2*a*e^2)*g)*x + 2*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*sq 
rt(-c*d*f*g + a*e*g^2)*sqrt(e*x + d))/(e*g*x^2 + d*f + (e*f + d*g)*x)) - 2 
*(15*c^4*d^4*f^4*g - 5*a*c^3*d^3*e*f^3*g^2 - 2*a^2*c^2*d^2*e^2*f^2*g^3 - 5 
6*a^3*c*d*e^3*f*g^4 + 48*a^4*e^4*g^5 - 15*(c^4*d^4*f*g^4 - a*c^3*d^3*e*g^5 
)*x^3 + (73*c^4*d^4*f^2*g^3 - 191*a*c^3*d^3*e*f*g^4 + 118*a^2*c^2*d^2*e^2* 
g^5)*x^2 + (55*c^4*d^4*f^3*g^2 - 19*a*c^3*d^3*e*f^2*g^3 - 172*a^2*c^2*d^2* 
e^2*f*g^4 + 136*a^3*c*d*e^3*g^5)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^ 
2)*x)*sqrt(e*x + d))/(c^2*d^3*f^6*g^4 - 2*a*c*d^2*e*f^5*g^5 + a^2*d*e^2*f^ 
4*g^6 + (c^2*d^2*e*f^2*g^8 - 2*a*c*d*e^2*f*g^9 + a^2*e^3*g^10)*x^5 + (4*c^ 
2*d^2*e*f^3*g^7 + a^2*d*e^2*g^10 + (c^2*d^3 - 8*a*c*d*e^2)*f^2*g^8 - 2*(a* 
c*d^2*e - 2*a^2*e^3)*f*g^9)*x^4 + 2*(3*c^2*d^2*e*f^4*g^6 + 2*a^2*d*e^2*f*g 
^9 + 2*(c^2*d^3 - 3*a*c*d*e^2)*f^3*g^7 - (4*a*c*d^2*e - 3*a^2*e^3)*f^2*g^8 
)*x^3 + 2*(2*c^2*d^2*e*f^5*g^5 + 3*a^2*d*e^2*f^2*g^8 + (3*c^2*d^3 - 4*a*c* 
d*e^2)*f^4*g^6 - 2*(3*a*c*d^2*e - a^2*e^3)*f^3*g^7)*x^2 + (c^2*d^2*e*f^6*g 
^4 + 4*a^2*d*e^2*f^3*g^7 + 2*(2*c^2*d^3 - a*c*d*e^2)*f^5*g^5 - (8*a*c*d^2* 
e - a^2*e^3)*f^4*g^6)*x), -1/192*(15*(c^4*d^4*e*g^4*x^5 + c^4*d^5*f^4 +...
 

Sympy [F(-1)]

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

Output:

Timed out
 

Maxima [F]

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

Giac [A] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 566, normalized size of antiderivative = 1.75 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^{5/2} (f+g x)^5} \, dx=\frac {5 \, c^{4} d^{4} {\left | e \right |} \arctan \left (\frac {\sqrt {{\left (e x + d\right )} c d e - c d^{2} e + a e^{3}} g}{\sqrt {c d f g - a e g^{2}} e}\right )}{64 \, {\left (c d f g^{3} - a e g^{4}\right )} \sqrt {c d f g - a e g^{2}} e} - \frac {15 \, \sqrt {{\left (e x + d\right )} c d e - c d^{2} e + a e^{3}} c^{7} d^{7} e^{6} f^{3} {\left | e \right |} - 45 \, \sqrt {{\left (e x + d\right )} c d e - c d^{2} e + a e^{3}} a c^{6} d^{6} e^{7} f^{2} g {\left | e \right |} + 45 \, \sqrt {{\left (e x + d\right )} c d e - c d^{2} e + a e^{3}} a^{2} c^{5} d^{5} e^{8} f g^{2} {\left | e \right |} - 15 \, \sqrt {{\left (e x + d\right )} c d e - c d^{2} e + a e^{3}} a^{3} c^{4} d^{4} e^{9} g^{3} {\left | e \right |} + 55 \, {\left ({\left (e x + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {3}{2}} c^{6} d^{6} e^{4} f^{2} g {\left | e \right |} - 110 \, {\left ({\left (e x + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {3}{2}} a c^{5} d^{5} e^{5} f g^{2} {\left | e \right |} + 55 \, {\left ({\left (e x + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {3}{2}} a^{2} c^{4} d^{4} e^{6} g^{3} {\left | e \right |} + 73 \, {\left ({\left (e x + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {5}{2}} c^{5} d^{5} e^{2} f g^{2} {\left | e \right |} - 73 \, {\left ({\left (e x + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {5}{2}} a c^{4} d^{4} e^{3} g^{3} {\left | e \right |} - 15 \, {\left ({\left (e x + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {7}{2}} c^{4} d^{4} g^{3} {\left | e \right |}}{192 \, {\left (c d f g^{3} - a e g^{4}\right )} {\left (c d e^{2} f - a e^{3} g + {\left ({\left (e x + d\right )} c d e - c d^{2} e + a e^{3}\right )} g\right )}^{4}} \] 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)^5, 
x, algorithm="giac")
 

Output:

5/64*c^4*d^4*abs(e)*arctan(sqrt((e*x + d)*c*d*e - c*d^2*e + a*e^3)*g/(sqrt 
(c*d*f*g - a*e*g^2)*e))/((c*d*f*g^3 - a*e*g^4)*sqrt(c*d*f*g - a*e*g^2)*e) 
- 1/192*(15*sqrt((e*x + d)*c*d*e - c*d^2*e + a*e^3)*c^7*d^7*e^6*f^3*abs(e) 
 - 45*sqrt((e*x + d)*c*d*e - c*d^2*e + a*e^3)*a*c^6*d^6*e^7*f^2*g*abs(e) + 
 45*sqrt((e*x + d)*c*d*e - c*d^2*e + a*e^3)*a^2*c^5*d^5*e^8*f*g^2*abs(e) - 
 15*sqrt((e*x + d)*c*d*e - c*d^2*e + a*e^3)*a^3*c^4*d^4*e^9*g^3*abs(e) + 5 
5*((e*x + d)*c*d*e - c*d^2*e + a*e^3)^(3/2)*c^6*d^6*e^4*f^2*g*abs(e) - 110 
*((e*x + d)*c*d*e - c*d^2*e + a*e^3)^(3/2)*a*c^5*d^5*e^5*f*g^2*abs(e) + 55 
*((e*x + d)*c*d*e - c*d^2*e + a*e^3)^(3/2)*a^2*c^4*d^4*e^6*g^3*abs(e) + 73 
*((e*x + d)*c*d*e - c*d^2*e + a*e^3)^(5/2)*c^5*d^5*e^2*f*g^2*abs(e) - 73*( 
(e*x + d)*c*d*e - c*d^2*e + a*e^3)^(5/2)*a*c^4*d^4*e^3*g^3*abs(e) - 15*((e 
*x + d)*c*d*e - c*d^2*e + a*e^3)^(7/2)*c^4*d^4*g^3*abs(e))/((c*d*f*g^3 - a 
*e*g^4)*(c*d*e^2*f - a*e^3*g + ((e*x + d)*c*d*e - c*d^2*e + a*e^3)*g)^4)
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.47 (sec) , antiderivative size = 850, normalized size of antiderivative = 2.63 \[ \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)^5} \, dx=\frac {15 \sqrt {g}\, \sqrt {-a e g +c d f}\, \mathit {atan} \left (\frac {\sqrt {c d x +a e}\, g}{\sqrt {g}\, \sqrt {-a e g +c d f}}\right ) c^{4} d^{4} f^{4}+60 \sqrt {g}\, \sqrt {-a e g +c d f}\, \mathit {atan} \left (\frac {\sqrt {c d x +a e}\, g}{\sqrt {g}\, \sqrt {-a e g +c d f}}\right ) c^{4} d^{4} f^{3} g x +90 \sqrt {g}\, \sqrt {-a e g +c d f}\, \mathit {atan} \left (\frac {\sqrt {c d x +a e}\, g}{\sqrt {g}\, \sqrt {-a e g +c d f}}\right ) c^{4} d^{4} f^{2} g^{2} x^{2}+60 \sqrt {g}\, \sqrt {-a e g +c d f}\, \mathit {atan} \left (\frac {\sqrt {c d x +a e}\, g}{\sqrt {g}\, \sqrt {-a e g +c d f}}\right ) c^{4} d^{4} f \,g^{3} x^{3}+15 \sqrt {g}\, \sqrt {-a e g +c d f}\, \mathit {atan} \left (\frac {\sqrt {c d x +a e}\, g}{\sqrt {g}\, \sqrt {-a e g +c d f}}\right ) c^{4} d^{4} g^{4} x^{4}-48 \sqrt {c d x +a e}\, a^{4} e^{4} g^{5}+56 \sqrt {c d x +a e}\, a^{3} c d \,e^{3} f \,g^{4}-136 \sqrt {c d x +a e}\, a^{3} c d \,e^{3} g^{5} x +2 \sqrt {c d x +a e}\, a^{2} c^{2} d^{2} e^{2} f^{2} g^{3}+172 \sqrt {c d x +a e}\, a^{2} c^{2} d^{2} e^{2} f \,g^{4} x -118 \sqrt {c d x +a e}\, a^{2} c^{2} d^{2} e^{2} g^{5} x^{2}+5 \sqrt {c d x +a e}\, a \,c^{3} d^{3} e \,f^{3} g^{2}+19 \sqrt {c d x +a e}\, a \,c^{3} d^{3} e \,f^{2} g^{3} x +191 \sqrt {c d x +a e}\, a \,c^{3} d^{3} e f \,g^{4} x^{2}-15 \sqrt {c d x +a e}\, a \,c^{3} d^{3} e \,g^{5} x^{3}-15 \sqrt {c d x +a e}\, c^{4} d^{4} f^{4} g -55 \sqrt {c d x +a e}\, c^{4} d^{4} f^{3} g^{2} x -73 \sqrt {c d x +a e}\, c^{4} d^{4} f^{2} g^{3} x^{2}+15 \sqrt {c d x +a e}\, c^{4} d^{4} f \,g^{4} x^{3}}{192 g^{4} \left (a^{2} e^{2} g^{6} x^{4}-2 a c d e f \,g^{5} x^{4}+c^{2} d^{2} f^{2} g^{4} x^{4}+4 a^{2} e^{2} f \,g^{5} x^{3}-8 a c d e \,f^{2} g^{4} x^{3}+4 c^{2} d^{2} f^{3} g^{3} x^{3}+6 a^{2} e^{2} f^{2} g^{4} x^{2}-12 a c d e \,f^{3} g^{3} x^{2}+6 c^{2} d^{2} f^{4} g^{2} x^{2}+4 a^{2} e^{2} f^{3} g^{3} x -8 a c d e \,f^{4} g^{2} x +4 c^{2} d^{2} f^{5} g x +a^{2} e^{2} f^{4} g^{2}-2 a c d e \,f^{5} g +c^{2} d^{2} f^{6}\right )} \] 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)^5,x)
 

Output:

(15*sqrt(g)*sqrt( - a*e*g + c*d*f)*atan((sqrt(a*e + c*d*x)*g)/(sqrt(g)*sqr 
t( - a*e*g + c*d*f)))*c**4*d**4*f**4 + 60*sqrt(g)*sqrt( - a*e*g + c*d*f)*a 
tan((sqrt(a*e + c*d*x)*g)/(sqrt(g)*sqrt( - a*e*g + c*d*f)))*c**4*d**4*f**3 
*g*x + 90*sqrt(g)*sqrt( - a*e*g + c*d*f)*atan((sqrt(a*e + c*d*x)*g)/(sqrt( 
g)*sqrt( - a*e*g + c*d*f)))*c**4*d**4*f**2*g**2*x**2 + 60*sqrt(g)*sqrt( - 
a*e*g + c*d*f)*atan((sqrt(a*e + c*d*x)*g)/(sqrt(g)*sqrt( - a*e*g + c*d*f)) 
)*c**4*d**4*f*g**3*x**3 + 15*sqrt(g)*sqrt( - a*e*g + c*d*f)*atan((sqrt(a*e 
 + c*d*x)*g)/(sqrt(g)*sqrt( - a*e*g + c*d*f)))*c**4*d**4*g**4*x**4 - 48*sq 
rt(a*e + c*d*x)*a**4*e**4*g**5 + 56*sqrt(a*e + c*d*x)*a**3*c*d*e**3*f*g**4 
 - 136*sqrt(a*e + c*d*x)*a**3*c*d*e**3*g**5*x + 2*sqrt(a*e + c*d*x)*a**2*c 
**2*d**2*e**2*f**2*g**3 + 172*sqrt(a*e + c*d*x)*a**2*c**2*d**2*e**2*f*g**4 
*x - 118*sqrt(a*e + c*d*x)*a**2*c**2*d**2*e**2*g**5*x**2 + 5*sqrt(a*e + c* 
d*x)*a*c**3*d**3*e*f**3*g**2 + 19*sqrt(a*e + c*d*x)*a*c**3*d**3*e*f**2*g** 
3*x + 191*sqrt(a*e + c*d*x)*a*c**3*d**3*e*f*g**4*x**2 - 15*sqrt(a*e + c*d* 
x)*a*c**3*d**3*e*g**5*x**3 - 15*sqrt(a*e + c*d*x)*c**4*d**4*f**4*g - 55*sq 
rt(a*e + c*d*x)*c**4*d**4*f**3*g**2*x - 73*sqrt(a*e + c*d*x)*c**4*d**4*f** 
2*g**3*x**2 + 15*sqrt(a*e + c*d*x)*c**4*d**4*f*g**4*x**3)/(192*g**4*(a**2* 
e**2*f**4*g**2 + 4*a**2*e**2*f**3*g**3*x + 6*a**2*e**2*f**2*g**4*x**2 + 4* 
a**2*e**2*f*g**5*x**3 + a**2*e**2*g**6*x**4 - 2*a*c*d*e*f**5*g - 8*a*c*d*e 
*f**4*g**2*x - 12*a*c*d*e*f**3*g**3*x**2 - 8*a*c*d*e*f**2*g**4*x**3 - 2...