\(\int \frac {(a d e+(c d^2+a e^2) x+c d e x^2)^{5/2}}{(d+e x)^{5/2} (f+g x)^2} \, dx\) [28]

Optimal result

Integrand size = 46, antiderivative size = 235 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^{5/2} (f+g x)^2} \, dx=-\frac {5 c d (c d f-a e g) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{g^3 \sqrt {d+e x}}+\frac {5 c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{3 g^2 (d+e x)^{3/2}}-\frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{g (d+e x)^{5/2} (f+g x)}+\frac {5 c d (c d f-a e g)^{3/2} \arctan \left (\frac {\sqrt {g} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {c d f-a e g} \sqrt {d+e x}}\right )}{g^{7/2}} \] Output:

-5*c*d*(-a*e*g+c*d*f)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/g^3/(e*x+d)^ 
(1/2)+5/3*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)-(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)+5*c*d*(-a*e* 
g+c*d*f)^(3/2)*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)
 

Mathematica [A] (verified)

Time = 0.73 (sec) , antiderivative size = 183, normalized size of antiderivative = 0.78 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^{5/2} (f+g x)^2} \, dx=\frac {\sqrt {a e+c d x} \sqrt {d+e x} \left (\sqrt {g} \sqrt {a e+c d x} \left (-3 a^2 e^2 g^2+2 a c d e g (10 f+7 g x)+c^2 d^2 \left (-15 f^2-10 f g x+2 g^2 x^2\right )\right )+15 c d (c d f-a e g)^{3/2} (f+g x) \arctan \left (\frac {\sqrt {g} \sqrt {a e+c d x}}{\sqrt {c d f-a e g}}\right )\right )}{3 g^{7/2} \sqrt {(a e+c d x) (d+e x)} (f+g x)} \] 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)^2),x]
 

Output:

(Sqrt[a*e + c*d*x]*Sqrt[d + e*x]*(Sqrt[g]*Sqrt[a*e + c*d*x]*(-3*a^2*e^2*g^ 
2 + 2*a*c*d*e*g*(10*f + 7*g*x) + c^2*d^2*(-15*f^2 - 10*f*g*x + 2*g^2*x^2)) 
 + 15*c*d*(c*d*f - a*e*g)^(3/2)*(f + g*x)*ArcTan[(Sqrt[g]*Sqrt[a*e + c*d*x 
])/Sqrt[c*d*f - a*e*g]]))/(3*g^(7/2)*Sqrt[(a*e + c*d*x)*(d + e*x)]*(f + g* 
x))
 

Rubi [A] (verified)

Time = 1.03 (sec) , antiderivative size = 245, normalized size of antiderivative = 1.04, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.109, Rules used = {1249, 1250, 1250, 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.

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

\(\Big \downarrow \) 1249

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

\(\Big \downarrow \) 1250

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

\(\Big \downarrow \) 1250

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

\(\Big \downarrow \) 1255

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

\(\Big \downarrow \) 218

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

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)^2),x]
 

Output:

-((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))) + (5*c*d*((2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(3*g*(d + 
 e*x)^(3/2)) - ((c*d*f - a*e*g)*((2*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e 
*x^2])/(g*Sqrt[d + e*x]) - (2*Sqrt[c*d*f - a*e*g]*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])]) 
/g^(3/2)))/g))/(2*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 1250

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*(m - n - 1))), x] - Simp[m*((c*e*f + c*d*g - b*e*g)/( 
e^2*g*(m - n - 1)))   Int[(d + e*x)^(m + 1)*(f + g*x)^n*(a + b*x + c*x^2)^( 
p - 1), x], x] /; FreeQ[{a, b, c, d, e, f, g, n}, x] && EqQ[c*d^2 - b*d*e + 
 a*e^2, 0] && EqQ[m + p, 0] && GtQ[p, 0] && NeQ[m - n - 1, 0] &&  !IGtQ[n, 
0] &&  !(IntegerQ[n + p] && LtQ[n + p + 2, 0]) && RationalQ[n]
 

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. \(512\) vs. \(2(209)=418\).

Time = 2.76 (sec) , antiderivative size = 513, normalized size of antiderivative = 2.18

method	result	size
default	\(-\frac {\sqrt {\left (e x +d \right ) \left (c d x +a e \right )}\, \left (15 \,\operatorname {arctanh}\left (\frac {g \sqrt {c d x +a e}}{\sqrt {\left (a e g -d f c \right ) g}}\right ) a^{2} c d \,e^{2} g^{3} x -30 \,\operatorname {arctanh}\left (\frac {g \sqrt {c d x +a e}}{\sqrt {\left (a e g -d f c \right ) g}}\right ) a \,c^{2} d^{2} e f \,g^{2} x +15 \,\operatorname {arctanh}\left (\frac {g \sqrt {c d x +a e}}{\sqrt {\left (a e g -d f c \right ) g}}\right ) c^{3} d^{3} f^{2} g x +15 \,\operatorname {arctanh}\left (\frac {g \sqrt {c d x +a e}}{\sqrt {\left (a e g -d f c \right ) g}}\right ) a^{2} c d \,e^{2} f \,g^{2}-30 \,\operatorname {arctanh}\left (\frac {g \sqrt {c d x +a e}}{\sqrt {\left (a e g -d f c \right ) g}}\right ) a \,c^{2} d^{2} e \,f^{2} g +15 \,\operatorname {arctanh}\left (\frac {g \sqrt {c d x +a e}}{\sqrt {\left (a e g -d f c \right ) g}}\right ) c^{3} d^{3} f^{3}-2 c^{2} d^{2} g^{2} x^{2} \sqrt {c d x +a e}\, \sqrt {\left (a e g -d f c \right ) g}-14 a c d e \,g^{2} x \sqrt {c d x +a e}\, \sqrt {\left (a e g -d f c \right ) g}+10 c^{2} d^{2} f g x \sqrt {c d x +a e}\, \sqrt {\left (a e g -d f c \right ) g}+3 \sqrt {\left (a e g -d f c \right ) g}\, \sqrt {c d x +a e}\, a^{2} e^{2} g^{2}-20 \sqrt {\left (a e g -d f c \right ) g}\, \sqrt {c d x +a e}\, a c d e f g +15 \sqrt {\left (a e g -d f c \right ) g}\, \sqrt {c d x +a e}\, c^{2} d^{2} f^{2}\right )}{3 \sqrt {e x +d}\, \sqrt {c d x +a e}\, g^{3} \left (g x +f \right ) \sqrt {\left (a e g -d f c \right ) g}}\)	\(513\)

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)^2,x,meth 
od=_RETURNVERBOSE)
 

Output:

-1/3*((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))*a^2*c*d*e^2*g^3*x-30*arctanh(g*(c*d*x+a*e)^(1/2)/((a*e*g-c 
*d*f)*g)^(1/2))*a*c^2*d^2*e*f*g^2*x+15*arctanh(g*(c*d*x+a*e)^(1/2)/((a*e*g 
-c*d*f)*g)^(1/2))*c^3*d^3*f^2*g*x+15*arctanh(g*(c*d*x+a*e)^(1/2)/((a*e*g-c 
*d*f)*g)^(1/2))*a^2*c*d*e^2*f*g^2-30*arctanh(g*(c*d*x+a*e)^(1/2)/((a*e*g-c 
*d*f)*g)^(1/2))*a*c^2*d^2*e*f^2*g+15*arctanh(g*(c*d*x+a*e)^(1/2)/((a*e*g-c 
*d*f)*g)^(1/2))*c^3*d^3*f^3-2*c^2*d^2*g^2*x^2*(c*d*x+a*e)^(1/2)*((a*e*g-c* 
d*f)*g)^(1/2)-14*a*c*d*e*g^2*x*(c*d*x+a*e)^(1/2)*((a*e*g-c*d*f)*g)^(1/2)+1 
0*c^2*d^2*f*g*x*(c*d*x+a*e)^(1/2)*((a*e*g-c*d*f)*g)^(1/2)+3*((a*e*g-c*d*f) 
*g)^(1/2)*(c*d*x+a*e)^(1/2)*a^2*e^2*g^2-20*((a*e*g-c*d*f)*g)^(1/2)*(c*d*x+ 
a*e)^(1/2)*a*c*d*e*f*g+15*((a*e*g-c*d*f)*g)^(1/2)*(c*d*x+a*e)^(1/2)*c^2*d^ 
2*f^2)/(e*x+d)^(1/2)/(c*d*x+a*e)^(1/2)/g^3/(g*x+f)/((a*e*g-c*d*f)*g)^(1/2)
 

Fricas [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 704, normalized size of antiderivative = 3.00 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^{5/2} (f+g x)^2} \, dx=\left [-\frac {15 \, {\left (c^{2} d^{3} f^{2} - a c d^{2} e f g + {\left (c^{2} d^{2} e f g - a c d e^{2} g^{2}\right )} x^{2} + {\left (c^{2} d^{2} e f^{2} - a c d^{2} e g^{2} + {\left (c^{2} d^{3} - a c d e^{2}\right )} f g\right )} x\right )} \sqrt {-\frac {c d f - a e g}{g}} \log \left (-\frac {c d e g x^{2} - c d^{2} f + 2 \, a d e g - 2 \, \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} \sqrt {e x + d} g \sqrt {-\frac {c d f - a e g}{g}} - {\left (c d e f - {\left (c d^{2} + 2 \, a e^{2}\right )} g\right )} x}{e g x^{2} + d f + {\left (e f + d g\right )} x}\right ) - 2 \, {\left (2 \, c^{2} d^{2} g^{2} x^{2} - 15 \, c^{2} d^{2} f^{2} + 20 \, a c d e f g - 3 \, a^{2} e^{2} g^{2} - 2 \, {\left (5 \, c^{2} d^{2} f g - 7 \, a c d e g^{2}\right )} x\right )} \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} \sqrt {e x + d}}{6 \, {\left (e g^{4} x^{2} + d f g^{3} + {\left (e f g^{3} + d g^{4}\right )} x\right )}}, -\frac {15 \, {\left (c^{2} d^{3} f^{2} - a c d^{2} e f g + {\left (c^{2} d^{2} e f g - a c d e^{2} g^{2}\right )} x^{2} + {\left (c^{2} d^{2} e f^{2} - a c d^{2} e g^{2} + {\left (c^{2} d^{3} - a c d e^{2}\right )} f g\right )} x\right )} \sqrt {\frac {c d f - a e g}{g}} \arctan \left (-\frac {\sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} \sqrt {e x + d} g \sqrt {\frac {c d f - a e g}{g}}}{c d^{2} f - a d e g + {\left (c d e f - a e^{2} g\right )} x}\right ) - {\left (2 \, c^{2} d^{2} g^{2} x^{2} - 15 \, c^{2} d^{2} f^{2} + 20 \, a c d e f g - 3 \, a^{2} e^{2} g^{2} - 2 \, {\left (5 \, c^{2} d^{2} f g - 7 \, a c d e g^{2}\right )} x\right )} \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} \sqrt {e x + d}}{3 \, {\left (e g^{4} x^{2} + d f g^{3} + {\left (e f g^{3} + d g^{4}\right )} x\right )}}\right ] \] 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)^2, 
x, algorithm="fricas")
 

Output:

[-1/6*(15*(c^2*d^3*f^2 - a*c*d^2*e*f*g + (c^2*d^2*e*f*g - a*c*d*e^2*g^2)*x 
^2 + (c^2*d^2*e*f^2 - a*c*d^2*e*g^2 + (c^2*d^3 - a*c*d*e^2)*f*g)*x)*sqrt(- 
(c*d*f - a*e*g)/g)*log(-(c*d*e*g*x^2 - c*d^2*f + 2*a*d*e*g - 2*sqrt(c*d*e* 
x^2 + a*d*e + (c*d^2 + a*e^2)*x)*sqrt(e*x + d)*g*sqrt(-(c*d*f - a*e*g)/g) 
- (c*d*e*f - (c*d^2 + 2*a*e^2)*g)*x)/(e*g*x^2 + d*f + (e*f + d*g)*x)) - 2* 
(2*c^2*d^2*g^2*x^2 - 15*c^2*d^2*f^2 + 20*a*c*d*e*f*g - 3*a^2*e^2*g^2 - 2*( 
5*c^2*d^2*f*g - 7*a*c*d*e*g^2)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2) 
*x)*sqrt(e*x + d))/(e*g^4*x^2 + d*f*g^3 + (e*f*g^3 + d*g^4)*x), -1/3*(15*( 
c^2*d^3*f^2 - a*c*d^2*e*f*g + (c^2*d^2*e*f*g - a*c*d*e^2*g^2)*x^2 + (c^2*d 
^2*e*f^2 - a*c*d^2*e*g^2 + (c^2*d^3 - a*c*d*e^2)*f*g)*x)*sqrt((c*d*f - a*e 
*g)/g)*arctan(-sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*sqrt(e*x + d)*g 
*sqrt((c*d*f - a*e*g)/g)/(c*d^2*f - a*d*e*g + (c*d*e*f - a*e^2*g)*x)) - (2 
*c^2*d^2*g^2*x^2 - 15*c^2*d^2*f^2 + 20*a*c*d*e*f*g - 3*a^2*e^2*g^2 - 2*(5* 
c^2*d^2*f*g - 7*a*c*d*e*g^2)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x 
)*sqrt(e*x + d))/(e*g^4*x^2 + d*f*g^3 + (e*f*g^3 + d*g^4)*x)]
 

Sympy [F(-1)]

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

Output:

Timed out
 

Maxima [F]

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

Giac [A] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 392, normalized size of antiderivative = 1.67 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^{5/2} (f+g x)^2} \, dx=\frac {5 \, {\left (c^{3} d^{3} f^{2} {\left | e \right |} - 2 \, a c^{2} d^{2} e f g {\left | e \right |} + a^{2} c d e^{2} g^{2} {\left | e \right |}\right )} \arctan \left (\frac {\sqrt {{\left (e x + d\right )} c d e - c d^{2} e + a e^{3}} g}{\sqrt {c d f g - a e g^{2}} e}\right )}{\sqrt {c d f g - a e g^{2}} e g^{3}} - \frac {\sqrt {{\left (e x + d\right )} c d e - c d^{2} e + a e^{3}} c^{3} d^{3} f^{2} {\left | e \right |} - 2 \, \sqrt {{\left (e x + d\right )} c d e - c d^{2} e + a e^{3}} a c^{2} d^{2} e f g {\left | e \right |} + \sqrt {{\left (e x + d\right )} c d e - c d^{2} e + a e^{3}} a^{2} c d e^{2} g^{2} {\left | e \right |}}{{\left (c d e^{2} f - a e^{3} g + {\left ({\left (e x + d\right )} c d e - c d^{2} e + a e^{3}\right )} g\right )} g^{3}} - \frac {2 \, {\left (6 \, \sqrt {{\left (e x + d\right )} c d e - c d^{2} e + a e^{3}} c^{2} d^{2} e^{10} f g^{3} {\left | e \right |} - 6 \, \sqrt {{\left (e x + d\right )} c d e - c d^{2} e + a e^{3}} a c d e^{11} g^{4} {\left | e \right |} - {\left ({\left (e x + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {3}{2}} c d e^{8} g^{4} {\left | e \right |}\right )}}{3 \, e^{12} g^{6}} \] 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)^2, 
x, algorithm="giac")
 

Output:

5*(c^3*d^3*f^2*abs(e) - 2*a*c^2*d^2*e*f*g*abs(e) + a^2*c*d*e^2*g^2*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))/(sqrt(c*d*f*g - a*e*g^2)*e*g^3) - (sqrt((e*x + d)*c*d*e - c*d^2*e + a 
*e^3)*c^3*d^3*f^2*abs(e) - 2*sqrt((e*x + d)*c*d*e - c*d^2*e + a*e^3)*a*c^2 
*d^2*e*f*g*abs(e) + sqrt((e*x + d)*c*d*e - c*d^2*e + a*e^3)*a^2*c*d*e^2*g^ 
2*abs(e))/((c*d*e^2*f - a*e^3*g + ((e*x + d)*c*d*e - c*d^2*e + a*e^3)*g)*g 
^3) - 2/3*(6*sqrt((e*x + d)*c*d*e - c*d^2*e + a*e^3)*c^2*d^2*e^10*f*g^3*ab 
s(e) - 6*sqrt((e*x + d)*c*d*e - c*d^2*e + a*e^3)*a*c*d*e^11*g^4*abs(e) - ( 
(e*x + d)*c*d*e - c*d^2*e + a*e^3)^(3/2)*c*d*e^8*g^4*abs(e))/(e^12*g^6)
 

Mupad [F(-1)]

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

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

Reduce [B] (verification not implemented)

Time = 0.28 (sec) , antiderivative size = 345, normalized size of antiderivative = 1.47 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^{5/2} (f+g x)^2} \, dx=\frac {-15 \sqrt {g}\, \sqrt {-a e g +c d f}\, \mathit {atan} \left (\frac {\sqrt {c d x +a e}\, g}{\sqrt {g}\, \sqrt {-a e g +c d f}}\right ) a c d e f g -15 \sqrt {g}\, \sqrt {-a e g +c d f}\, \mathit {atan} \left (\frac {\sqrt {c d x +a e}\, g}{\sqrt {g}\, \sqrt {-a e g +c d f}}\right ) a c d e \,g^{2} x +15 \sqrt {g}\, \sqrt {-a e g +c d f}\, \mathit {atan} \left (\frac {\sqrt {c d x +a e}\, g}{\sqrt {g}\, \sqrt {-a e g +c d f}}\right ) c^{2} d^{2} f^{2}+15 \sqrt {g}\, \sqrt {-a e g +c d f}\, \mathit {atan} \left (\frac {\sqrt {c d x +a e}\, g}{\sqrt {g}\, \sqrt {-a e g +c d f}}\right ) c^{2} d^{2} f g x -3 \sqrt {c d x +a e}\, a^{2} e^{2} g^{3}+20 \sqrt {c d x +a e}\, a c d e f \,g^{2}+14 \sqrt {c d x +a e}\, a c d e \,g^{3} x -15 \sqrt {c d x +a e}\, c^{2} d^{2} f^{2} g -10 \sqrt {c d x +a e}\, c^{2} d^{2} f \,g^{2} x +2 \sqrt {c d x +a e}\, c^{2} d^{2} g^{3} x^{2}}{3 g^{4} \left (g x +f \right )} \] 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)^2,x)
 

Output:

( - 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)))*a*c*d*e*f*g - 15*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)))*a*c*d*e*g**2*x 
 + 15*sqrt(g)*sqrt( - a*e*g + c*d*f)*atan((sqrt(a*e + c*d*x)*g)/(sqrt(g)*s 
qrt( - a*e*g + c*d*f)))*c**2*d**2*f**2 + 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**2*d**2*f* 
g*x - 3*sqrt(a*e + c*d*x)*a**2*e**2*g**3 + 20*sqrt(a*e + c*d*x)*a*c*d*e*f* 
g**2 + 14*sqrt(a*e + c*d*x)*a*c*d*e*g**3*x - 15*sqrt(a*e + c*d*x)*c**2*d** 
2*f**2*g - 10*sqrt(a*e + c*d*x)*c**2*d**2*f*g**2*x + 2*sqrt(a*e + c*d*x)*c 
**2*d**2*g**3*x**2)/(3*g**4*(f + g*x))